Spatio Temporal Chaos and Vacuum Fluctuations of Quantized Fields

P{N=3)W

=\fl(1

+ 9(l2-^))^2+0(92)

(1.66)

P{N=2)() = \ f l [ i + gH-2 + l^ -202+O(ff3/2)(167)

3

w

2s

(see [Hilgers et al. (1999b); Hilgers et al. (2001)] for details). Here J = 3 for N = 4 and J — 4 for N > 5. Note that the first-order correction to the Gaussian behaviour is of order y/g for N = 2 and of order g for iV > 3. For

* Perturbative

approach

25

N > 4, up to second order in ^g, only trivial trees contribute, and as a result of this the leading order perturbative expression for p( N - 4 ) (0) is the same as generated by independent discrete random variables \ n . In other words, only N = 2 and N = 3 yield nontrivial behaviour in leading order of chaotic quantization.

This page is intentionally left blank

Chapter 2

Chaotic strings

The principal idea of this book is to assume that the noise fields necessary for the quantization of standard model fields have dynamical origin. We will now spatially couple the chaotic noise. This leads to spatially extended chaotic noise objects, which, in order to have a name in the following, we will call 'chaotic strings'. Another suitable name might be 'discrete chaotic 1-dimensional 4>N+1 -theories', or simply 'turbulent quantum states'. From a mathematical point of view, chaotic strings are 1-dimensional coupled map lattices of diffusively coupled Tchebyscheff maps. We will show that such a dynamics can directly and intrinsically arise from a self-interacting continuum field theory in an appropriate limit of very large couplings.

2.1

Motivation for chaotic strings

Chaotic strings (or diffusively coupled Tchebyscheff maps on 1-dimensional lattices) are motivated by the assumption that there is a weakly coupled small-scale dynamics underlying the noise fields of the Parisi-Wu approach of stochastic quantization. Since in string theory ordinary particles are believed to have string-like structure, it is natural to assume that also the noise used for second quantization in the Parisi-Wu approach may have such a string-like structure on a very small scale (Fig. 2.1). On large scales, this more complicated noise just looks like Gaussian white noise. Among the many models that can be chosen to generate a coupled chaotic dynamics on a small scale certain criteria should be applied to select a particular system. First of all, for vanishing spatial coupling of the chaotic 'noise' one wants to have strongest possible random behavior 27

Chaotic

28

Standard model field

strings

Noise

i

Noise string

Fig. 2.1 A point-like standard model field is replaced by a string. Similarly, a point-like noise field used for stochastic quantization of the standard model field is replaced by a chaotic noise string. Each ordinary string may be 'shadowed' by a corresponding noise string.

with least possible higher-order correlations, in order to be closest to the Gaussian limit case (which corresponds to ordinary path integrals on a large scale). As shown in section 1.6 and 1.7, this selects as a local dynamics Tchebyscheff maps TN(x) of iV-th order (N > 2). We saw that Tchebyscheff maps satisfy a Central Limit Theorem which guarantees the convergence to the Wiener process (and hence to ordinary path integrals) if sums of iterates are looked at from large scales. Moreover, Tschebyscheff maps have least higher-order correlations among all systems conjugated to a Bernoulli shift, and are in that sense closest to Gaussian white noise, though being completely deterministic. Now let us discuss possible ways of spatially coupling the chaotic noise. Although in principle all types of coupling forms can be considered, physically it is most reasonable that the coupling should result from a Laplacian coupling rather than some other coupling, since this is the most relevant coupling form in quantum field and string theories. This leads to coupled

Motivation for chaotic

strings

29

map lattices of the diffusive coupling form. The resulting coupled map lattices can then be studied on lattices of arbitrary dimension, but motivated by the fact that ordinary strings are 1-dimensional objects we will mainly consider the 1-dimensional case. It will turn out in later chapters that our physical interpretation in terms of fluctuating virtual momenta will indeed very much favor 1-dimensional lattices. Further arguments for 1dimensional lattices will be presented in section 2.3. One obtains a 'chaotic string' denned by <^+1 = (1 - a)TN(K) + s^T^V-1)

+ T^« + 1 )).

(2.1)

$Jj is a discrete chaotic noise field variable taking continuous values on the interval [—1,1]. The initial values $Q are randomly distributed in this interval, i is a 1-dimensional spatial lattice coordinate and n a discrete time coordinate (in our case identified with the fictitious time of the Parisi-Wu approach). TV denotes the iV-th order Tchebyscheff polynomial. In the following we will mainly study T2(3>) = 2$ 2 - 1 and T3($) — 4 $ 3 - 3$. We consider both the positive and negative Tchebyscheff polynomial T ^ ( $ ) = ±Tjv(), but have suppressed the index ± in the above equation. The variable o is a coupling constant taking values in the interval [0,1]. Since o determines the strength of the Laplacian coupling, a - 1 can be regarded as a kind of metric in the 1-dimensional string space indexed by i. s is a sign variable taking on the values ± 1 . The choice s = + 1 is called 'diffusive coupling', but for symmetry reasons it also makes sense sense to study the choice s = —1, which we call 'anti-diffusive coupling'. The integer b distinguishes between the forward and backward coupling form, 6 = 1 corresponds to forward coupling (T^($) := T J V ( $ ) ) , b = 0 to backward coupling (T^($) := <£). We consider periodic boundary conditions and large lattices of size imax • The chaotic string dynamics (2.1) is deterministic chaotic and spatially extended. In later chapters we will use it as a model for rapidly evolving virtual momenta on a small scale. We are somewhat reminded of velocity fluctuations in fully developed turbulent hydrodynamic flows, which are also deterministic chaotic and spatially extended, and induced by strong nonlinearities. Physically we may indeed think of the chaotic string dynamics as describing a 'turbulent quantum state'. One can easily check in eq. (2.1) that for odd N the choice of s is irrelevant, whereas for even N the sign of s is relevant and a different

Chaotic

30

strings

dynamics arises (details in section 2.5). Hence, restricting ourselves to TV = 2 and N — 3, in total 6 different chaotic string theories arise, characterized by (N,b,s) = (2,1,+1), (2,0,+1), ( 2 , 1 , - 1 ) , (2, 0 , - 1 ) and (N,b) — (3,1), (3,0). For easier notation, in the following we will denote these string theories as 2A,2B, 2A~,2B~, 3A,3B, respectively. If the coupling a is sufficiently small, the chaotic variables $^ can be used to generate the noise of the Parisi-Wu approach of stochastic quantization on a very small scale. It can actually be shown, using the graph theoretical method described in section 1.7, that if chaotic Tchebyscheff noise on a time scale r is coupled to a slowly varying dynamics on a time scale m~2, in first and second order perturbation theory in Vm2T the cases N > 4 do not yield anything new compared to independent random variables (see section 1.8). Hence the above six chaotic string theories obtained for N = 2 and N = 3 are the most relevant ones to consider, yielding non-trivial behaviour in leading order of chaotic quantization.

2.2

Anti-integrable limit of a continuum JV+1-theory

So far our derivation of the chaotic string dynamics (2.1) was purely mathematical, with emphasis on the distinctive properties of Tchebyscheff maps. We will now provide a more physically motivated argument. We will show that the chaotic noise string dynamics formally originates from 1dimensional continuum (f>N+l-theories in the limit of infinite self-interaction strength. For example, the N — 3 string dynamics can be thought of as originating from a continuum A theory. Thus we may generate the dynamics of the chaotic noise strings by ordinary self-interacting scalar field theories, which, however, have much larger coupling constants and thus exhibit dynamics on a much faster time scale. In this sense, chaotic strings can also be regarded as degenerated Higgs-like fields with infinite self-interaction parameters, which are constraint to a 1-dimensional space. Let us start with the simplest example of an interacting field theory one can think of, a continuum 0 4 -theory in 1 dimension. The stochastically quantized field equation is J ^ ( M ) = (-^

- mA 3(x,t) + L(x,t).

(2.2)

Let us for the moment assume that m2 is a parameter taking on negative

Anti-integrable


limit of a continuum

31

values, whereas A is positive. This describes a double well potential. Let us investigate the behavior of eq. (2.2) in the limit of very large couplings —m2 and A. This limit clearly represents a departure from the ordinary perturbative treatment of self-interacting fields, where A is usually regarded as a small parameter. If m and A diverge, we have formally (j> — oo, and thus we expect a field that is strongly fluctuating in fictitious time. Let us consider a discretized version of eq. (1.6), we will perform the continuum limit later. Introducing the space-time lattice constant I and the fictitious time lattice constant r, we obtain ^n+l

^n

__ ^n

^^n

~ ^n

^,2

-m^-ASJ,

P

+ noise.

(2.3)

Here $J, denotes the discretized version of the field <j>, i is a discrete spacetime lattice coordinate, and n is a discrete fictitious time variable. Eq. (2.3) can be written as K+i = (1 - 2 ^ - m2T)¥n - A r < + ^(¥+l

+K~1)+T-

noise

(2.4)

Now let us proceed to the continuum limit. That means, both the fictitious time lattice constant r and the space-time lattice constant I go to zero. Suppose that r —> 0, / —> 0, —m2 —> oo, A —> oo such that T

|

I2 m2T AT

=

finite

(2.5)

m2rm

finite

(2.6)

Xren

finite.

(2.7)

We may call this an anti-integrable limit, similar to [Aubry et al. (1990)]. The constant a has similarities with a diffusion constant connecting fictitious time and space-time. uiren and Xren are 'renormalized' finite parameter values, which may be regarded as 'physical parameters' of the field theory under consideration. Eq. (2.4) determines a coupled map lattice, which, depending on the parameters, may exhibit either regular or chaotic behavior. The coupled map lattice has the diffusive coupling form < ^ + 1 = (1 - a)T{Vn) + ^ «

+ 1

+ Sjr 1 )-

(2-8)

The local map T is given by

r ( $ ) = (i _ J & a ) $ _ ^i£!L$3 V

1 — aJ

1— a

(29)

Chaotic

32

strings

The noise term r • noise actually vanishes in the type of continuum limit r —> 0, / —>• 0 that is considered here. Formally, we could still keep a very small noise term in eq. (2.8), but if the local map T in eq. (2.9) has chaotic properties then the time evolution of the system is completely dominated by this deterministic chaotic part, whereas the small noise level yields only a tiny correction. A particularly aesthetic choice of the parameters is -m2ren

=

Xren

2(1-a)

(2.10)

= 4(1-a)

(2.11)

In this case we obtain a local map conjugated to a Bernoulli shift, namely the (negative) third-order Tchebyscheff polynomial - T 3 ( $ ) = 3$ — 4 $ 3 . We then have a field theory with strongest possible chaotic properties. The remarkable fact is that an ordinary continuum 4 -theory degenerates to the discrete-time and discrete-space dynamics (2.8) under the assumption that m 2 and A diverge. The result is the chaotic 2>B string (N = 3, b = 0, 8 = 1).

Equally well we can also obtain the positive Tchebyscheff polynomial T 3 ($) for the choice m2ren

=

4(1-a)

(2.12)

Xren

=

-4(1-a)

(2.13)

Up to an alternating sign, +T3 basically generates the same dynamics. We could also start from a 3-theory, getting in a similar way an N = 2 string. Note that although a continuum 0 3 theory may be ill-defined, leading to unstable behaviour, the chaotic N = 2 string is a well-defined theory and has the same right of existence as the chaotic N = 3 string. Of course, more generally we can also start from <j>N+1 theories, getting Tchebyscheff maps of iV-th order for special values of the renormalized parameters. There are some ambiguities when deriving the coupled map dynamics in the anti-integrable limit. This is similar to the Ito-Stratonovich ambiguities that are well known for stochastic differential equations [van Kampen (1981)]. Instead of doing the nearest-neighbor coupling with §l~l and $^ + 1 , we could equally well choose the updated variables T{§1~1) and T($^ + 1 ). This yields a chaotic string with b = 1 rather than 6 = 0. For ordinary continuum theories the difference does not matter, since the field evolves in a smooth, infinitesimal way. For the coupled map lattices obtained in the

Possible

generalizations

33

anti-integrable limit, the difference between forward and backward updating is important and leads to something different. We can always study either positive or negative Tchebyscheff maps, both generating essentially the same dynamics up to a sign. But it is also possible to consider a dynamics where T and —T alternate in the spatial direction. Replacing T by — T at odd lattice sites yields the anti-diffusive coupling form s — —1, which is yet another degree of freedom for chaotic strings.

2.3

Possible generalizations

The attentive reader may have noticed that in principle we can also perform the anti-integrable limit for more general cases. First of all, we may start from a continuum field theory in d dimensions with a d-dimensional Laplacian. In the anti-integrable limit, this yields diffusively coupled maps on a d-dimensional lattice, of the form $j1+1 = ( l - a ) T « ) + ^ ^ < ,

(2.14)

i

where i denotes the nearest neighbors of i. i is now is a vector-valued lattice coordinate. Again, a $ 4 -theory leads to local cubic maps T of the form T($) = (1 - ^ A $ - ^ - $ 3 . \ 1 — aJ 1— a

(2.15)

Hence the study of these types of d-dimensional coupled maps can be regarded as being a necessary amendment for the complete understanding of 4-theories. Remember that eq. (2.14) corresponds to backward-coupling of nearest neighbors i, but we may also study the forward-coupling form, where £ • &n is replaced by YA T{&n)Rather than cubic maps we can of course also study other local 1dimensional maps T, for example quadratic ones, or maps that are of a more complicated non-polynomial form. All these dynamics can be thought of as arising from an anti-integrable limit of a self-interacting field theory which has a renormalized potential given by

Vre„(*) = (l-a)(-J

T($)d$ + 1$ 2 j + C,

(2.16)

34

Chaotic

strings

where C is an arbitrary constant. Note that Vren is finite but the original potential V of the continuum field theory diverges as V = -Vren for r —> 0. We may also look at vector-valued field variables $. One straightforward example is the following one: The scalar chaotic string field variable $^ can be written as coswNnuo in the uncoupled case, so it is natural to regard this as one component of a 2-dimensional unit vector e, the other component being given by sin7riVnuo. In a sense this unit vector e rotates with exponential acceleration and is viewed at discrete time points n. If we formally define another time variable t := irNn, then this vector rotates with constant frequency UQ in the time coordinate t, similar to a spin in a constant magnetic field. Many other generalizations to vector-valued field variables are possible (see chapter 4 for further examples). In principle, all kinds of local maps T can be studied which determine the local dynamics as $„+i = T($ r l ). Sometimes also here a potential Vren may exist, whose partial derivatives generate the dynamics of all components of T. Chaotic behaviour can also occur in various other quantum field theoretical models [Knill (1996); Matinyan et al. (1997); Heinz et al. (1997)]. One clearly needs mathematical and physical arguments to select from the infinity of such possible models. Tchebyscheff maps are clearly distinguished as having the strongest chaotic properties combined with highest symmetry standards. Still we can study these maps on lattices of arbitrary dimension d. When considering possible values for d, a simple physical argument immediately leads us to 1-dimensional lattices. Namely, in quantum field theories the potential V is an energy density, so for a d-dimensional space it has dimension energy/volume = energyd+1, using units where h = c = 1. The fictitious time t of the Parisi-Wu approach has dimension energy-2. The anti-integrable limit requires that the strength of V diverges and that r goes to 0 such that Vren — TV stays finite. Apparently for dimensional reasons this can only be achieved if d = 1, i.e. we end up with 1-dimensional lattices in the most natural way. This means, we are back to chaotic strings: 1-dimensional turbulent quantum states seem to be the most natural ones to consider. Only in more advanced theories it might be appropriate to include higher-dimensional noise objects as well.

Yet another way to derive the chaotic

2.4

string

35

Yet another way to derive the chaotic string

Let us now once again 'derive' the chaotic string dynamics in a completely different way, this time being lead by analogies with ordinary string theories. In these theories point-like particles are replaced by little string-like structures. More precisely, the position XM of a particle is thought to be a string XM(£,cr). Here fi labels the various space-time coordinates, and t and a are internal time and position coordinates which parametrize the string. In other words, a particle has not any more just one position X^(t) but several positions which are parametrized by the additional coordinate er. For consistency reasons, superstring theories require a 10-dimensional space-time {\i = 1,2,..., 10). Much more details on strings can e.g. be found in [Green et al. (1987)]. In string theories one studies wave equations for X^. The simplest wave equation is that of a bosonic string, which in suitable coordinates takes on the simple form d2

d2

W2^ = 8^X">

^

subject to suitable boundary conditions. For fermionic strings the relevant equations contain first rather than second derivatives. In suitable coordinates one has an equation of the form

^

=±An.

(2.18)

For superstrings there are first and second order derivatives. Now let us proceed to other, new types of strings. Let us consider the momentum of a particle, or better, the momentum uncertainty, which exists due to the uncertainty relation. Rather than considering a point momentum P^{t) we may consider a momentum string P^(t,a). In other words, one particle has several momenta (or momentum uncertainties), parametrized by t and a. In our approach, t is the fictitious time of the Parisi-Wu approach, which has dimension GeV~2, rather than GeV~l as ordinary coordinates have. For dimensionality reasons it is thus reasonable to combine a first derivative in t with a second derivative in a, and to write down an equation of the form d -

r

d2 = —P>.

(2.19)

36

Chaotic

strings

We may now discretize with a fictitious time lattice constant r and a spatial lattice constant / and write P"(nT,tO = **nPm«*, where pmax

(2.20)

is an arbitrary momentum scale. Eq. (2.19) leads to $* ,, - $ i = - f $ i + 1 - 2$* + $ i _ 1 )

(2.21)

where a/2 := T/12 is a kind of inverse metric in the string space, which determines the strength of the Laplacian coupling. Eq. (2.21) can be written as

$n+1 = (1 - om^j.) + I m ^ 1 ) + T^" 1 )),

(2-22)

where Xi ($) = $ is the first-order Tchebyscheff map. Clearly, this dynamics is not at all fluctuating. It is just a discretized heat equation. So it cannot be a model for strongly fluctuating virtual momenta $^ that are allowed due to the uncertainty relation. However, a straightforward idea is that most generally we should also consider higher values of N, i.e. higher-order Tchebyscheff maps, where we obtain just the chaotic string string dynamics eq. (2.1). Note that in eq. (2.22) it makes no difference whether the neighbored variables are given by Ti (^J^ 1 ) or $ ^ t l , since 7\ is just the identity. But for N > 2 it becomes important to distinguish between forward and backward coupling. While in this book we shall mainly concentrate on the case N > 2, the TV = 1 strings will also be briefly discussed in section 11.3 The analogies between chaotic strings and superstrings should not be over-emphasized. Clearly, chaotic strings are very different from superstrings. They are strongly self-interacting, chaotic, and discrete in space. None of these properties is shared by superstrings. Still one might think if ultimately it might be possible to establish a connection, also in view of contacts between string theory and stochastic quantization that were pointed out recently (see, e.g., [Polyakov (2001)]). Perhaps the chaotic strings play a similar role for the superstrings as the Higgs fields do for ordinary pointlike particles. Ultimately the chaotic strings would be responsible for physically observed masses and coupling constants. Each ordinary string could be 'shadowed' by a chaotic string, yielding the noise for 2nd quantization. In recent years it has become clear that superstrings are only the edge points of a more advanced theory, which is yet not fully formulated and

Yet another way to derive the chaotic

string

37

Fig. 2.2 A (speculative) 1-1 correspondence between superstrings and chaotic strings. The left hand side shows the 5 known superstring theories plus the so-called 11dimensional theory, which compactified on a circle is dual to the IIA string and compactified on an interval dual to the HET ES x £ 8 string. The right hand side shows the 6 chaotic string theories.

usually called M-theory. One could speculate that in the ultimate theory there is a symmetry between the 6 components that make up M-theory in moduli space and the 6 chaotic string theories, used for second quantization (Fig 2.2). At the present stage, however, any diagram of the kind of Fig 2.2 is pure speculation. What, however, is clear is that an ordinary string winding around a compactified space has a discrete momentum spectrum, and the string field variable XM(£,cr) is a kind of position variable taking on continuous values. On the contrary, a chaotic string has a discrete position spectrum i and the field variable $^ is a kind of momentum variable taking on continuous values. Hence in that sense the role of position and momentum is exchanged. Should higher-dimensional objects contained in superstring theories such as D-branes turn out to yield the correct decription of nature, then it is probably also necessary to study higher-dimensional generalizations of chaotic strings ('chaotic D-branes').

Chaotic

38

2.5

strings

Symmetry properties

Let us now discuss some elementary properties of chaotic strings. We say that the chaotic string dynamics <

+ 1

= (1 - a)TN(Vn)

+ ^ ( T j ^ r 1 ) + TbN(K+1))

(2-23)

is invariant under a certain symmetry transformation if for all i and n the corresponding field variables $^ differ just by a possible sign (and nothing else). For easier notation, in the following we denote the positive Tchebyscheff polynomial by XJV and the negative one by T_JV, i.e. T.N($) := - T J V ( $ ) . Looking at the string dynamics (2.23) for general N, we notice the following. If b = 1 then (2.23) is invariant under the transformation iV —> —N. If b — 0 then it is invariant under the simultaneous transformation N —» — N and s —> —s. These are all the symmetries the even-TV theories have. Hence there are, for example, 4 different N = 2 theories (b = 0,1; s = 1,-1), and negative N need not be considered, since they do not yield anything new. For odd TV, there is an additional symmetry due to the fact that TJV(-$)

=

(N odd).

-TJV($)

(2.24)

This effectively means that the theory is also invariant under s —>- —s (keeping N constant). To see this, consider an odd-TV theory with a given s and a given set of random initial values $Q . Then consider a different set of initial values where $Q is replaced by — $Q at odd lattice sites, i.e.

Using eq. (2.23) and (2.24) one immediately sees that after one iteration step one has

By induction we can proceed to arbitrary n $

f $«

« = { _\in

i i,

1 eeven ^ odd •

(2-27)

Symmetry

properties

39

4.5 r-

4

i

3.5

3

2.5 •

2 -

1.5 -'

1 -

0.5 0 I

-1

1

l

l

1

1

1

1

i

1

1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

* Fig. 2.3 Invariant 1-point density of the coupled map lattice (2.1) with N = 2,s = 1 and a = 0.00755.

Hence we get just the same dynamics is we replace s by — s and replace all initial values at odd lattice sites by — $Q- Since generally the initial values are arbitrary (and their memory is lost exponentially fast for mixing systems) this means that there are only two different odd-Af theories, corresponding to 6 = 0 and 6 = 1 . In particular, there are only two different N = 3 theories. The fact that odd-N theories are more symmetric is also reflected by the invariant density. For example, one notices that generally the singlesite invariant density of the variables $J, is symmetric for coupled maps T 3 , whereas it is not symmetric for coupled maps T2 (Figs. 2.3-2.6). This can be easily understood by performing the transformation $—)•—$ in the evolution equation of the field variable. Since TN is an odd function for odd N, the evolution equation is invariant under the replacement $—>—$, hence the invariant density is symmetric (ergodicity presumed). On the other hand, for the even functions TJV obtained for even A7" this symmetry property of the coupled map dynamics does not exist, hence for a ^ 0 the density is generically asymmetric.

40

Chaotic

4.5 I

1

1

1

1

strings

1

1

1

r

3.5 -

3 -

2.5 -

g Q.

2 -

1.5 1 -

0.5 -

0

*•

'

'

'

'

'

'

'

'

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

— •

0.8

'

1

* Fig. 2.4

Same as Fig. 2.3, but a = 0.120093.

In principle, the invariant densities of coupled map lattices can be understood by finding fixed points of the Perron-Frobenius operator of these very high-dimensional dynamical systems [Baladi (2000); Bricmont et al. (1996); Jarvenpaa (2001); Keller et al. (1992b); Lemaitre et al. (1997); Chate et al. (1997); Bunimovich et al. (1988)]. In practice, this is a very hard task to do. But numerically the densities are easily obtained by making histograms of the iterates. If we couple the Tchebyscheff noise to standard model fields <j>, then one observes similar differences of the symmetry properties for the probability density p((p) (see Fig. 1.1-1.8 in chapter 1) — this is also true for uncoupled Tchebyscheff noise. T3 generates symmetric p(
Stability

3 j

1

1

1

1

properties

41

r

2.5

2

G

1.5 •

1 A

0.5 -

0

^ ^ ^

I

1

I

1

I

l

I

l

I

l

I

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

* Fig. 2.5 Invariant 1-point density of the coupled map lattice (2.1) with N = 3, s = 1 and a = 0.0073038.

only one, not several attractors. But for large couplings a (or for small lattice sizes) chaotic strings can loose this property. For example, for the 3A string numerically a loss of ergodicity is observed for a close to 1. As a consequence, for large a suddenly the sign of s becomes relevant, and the 3 A and 3A~ theories exhibit inequivalent behaviour since suddenly the initial values matter. We will come back to this in chapter 4. Similarly, for the N = 1 strings there is no chaotic behaviour at all, hence the initial values remain relevant, and s = ± 1 (as well as N = ±1) have to be considered, in spite of the fact that N is odd. Hence there are four different N = 1 theories, due to lack of ergodicity (more details in section 11.3).

2.6

Stability properties

When strongly chaotic maps such as Tchebyscheff maps are weakly coupled one expects the chaotic behaviour to persist in a slightly perturbed way. Numerically, this is indeed observed if the coupling is smaller than

Chaotic

42

strings

\ -1

j

i

i

i

1

i

i

i

i

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

* Fig. 2.6

Same as Fig. 2.5, but a - 0.0953.

something of the order a sa 0.1. The situation, however, can significantly change for larger couplings a. Here, despite the strong chaotic properties of uncoupled Tchebyscheff maps, the coupled system may even have stable periodic orbits. The stability properties of a chaotic string, or of a turbulent quantum state, are determined by the Jacobian J. Consider a string with L lattice sites and periodic boundary conditions. A small perturbation d$ln of a given string field configuration $^ evolves according to the linearized equation

6Vn+1 = (1 - a ) T ' « ) < 5 ^ + * | ( r 6 ' ^ - 1 ) * ^ - 1 + T 6 ' « + 1 ) < 5 < + 1 ) . (2.28) In matrix form this can be written as

<5#„+i =

Jn5$

(2.29)

Stability

properties

43

where Jn is the Jacobian at step n. It is given by Jn f(l-a)T'(K) b fT '($l) 2

fTb'(f>2n) (l-a)T'(O fT»'(<S>2n)

fT6'($£)

\

6

^T '(0 (l-a)T'(^)

V fTb'($l)

fTb'($4n)

fT 6 '^- 1 )

(l-a)T'(^)/

Let 77fc (n), k = 0 , . . . , L -1 denote the eigenvalues of the matrix j(") • j("), where T denotes the transposed matrix and J("):= J„_1J„_2---J0.

(2.30)

The Liapunov exponents of the coupled map lattice are given by Xk=

lim — log77fc(n).

(2.31)

n—• oo ZTl

There is chaotic behaviour if at least one Liapunov exponent is positive. This is the generic case for chaotic strings. However, parameter regions with stable periodic orbits may exist as well, in particular if a is large. The 2A~ and 2B~ strings are examples where there is a stable synchronized fixed point for certain regions of the parameter a. First, let us define what we mean by a synchronized state. This is a state where one has $J, = 3>^ = . . . $^ =: $„ for all i, i.e. the field variable has the same value for all lattice sites i at a given time n. The temporal dynamics for such a state is just that of a 1-dimensional map given by $ n + 1 = (1 - a)T($n)

+ saTb(^n).

(2.32)

While these states always exist for special initial conditions they are usually unstable. To check for stability one has to look at the Jacobian. For a synchronized state this is a cyclic matrix, i.e. a matrix with matrix elements of the form Mij = m;_j (see also [Amritkar et al. (1991); Dettmann (2001)]). Generally, for such a matrix the eigenvectors v^ are given by v, k

=

(l,eik,ei2k,...,e~ik)

= ^

(J = 0 , 1 , . . . , L - 1 ) .

(2.33) (2.34)

44

Chaotic strings

The corresponding eigenvalues r/k are £-1

1=0

For a coupled map lattice in a synchronized state this implies that the eigenvalues of J^ are given by n-l

Vk(n) = J J ( ( l - a ) r ' ( * t ) + s a T 6 ' ( * t ) c o s f c ) t=o

(2.36)

A synchronized periodic orbit of length n is stable if |%(n)| < 1 for all k. Generic synchronized states of our coupled map lattices are unstable, but there also exist a few stable states of higher period. In spite of the fact that these states are stable the basin of attraction of these states is extremely small so that they generically don't show in a numerical simulation. The question of real interest for coupled map lattices is how large the basin of attraction of a given stable periodic orbit is. Numerically one sees that often it is extremely small and generic orbits are not attracted. Note that generally for coupled map lattices many different attractors (chaotic or periodic) may co-exist, so proving the existence of a stable periodic orbit does not automatically mean that this orbit is approached by a large amount of trajectories from generic initial values. Moreover, the transient time until a periodic orbit is reached can sometimes be extremely long (see, e.g., [Livi et al. (1990)]), in particular if the lattice size is large.

2.7

Fixed points

Of particular interest are the synchronized fixed points $* of the coupled map lattices. These are stationary states given by ** = (1 - o)T(**) + s|(T"($*) + T 6 ($*)),

(2.37)

where T is either the positive or negative Tchebyscheff map of N-th order. Usually these states are unstable. For s = 1 (strings of NA and NB type) the above equation reduces to $* = T ( $ * ) ,

(2.38)

Fixed points

45

i.e. the fixed points of the coupled system are given by those of the uncoupled maps, whereas for the strings with anti-diffusive coupling (s = - 1 ) one obtains fixed points with a non-trivial a-dependence. These are determined by the equation $* = (1 - a)T($*)

(NA~),

(2.39)

respectively (l + a)** = ( l - a ) T ( $ * )

(NB~).

(2.40)

Clearly fixed points of a system with s = — 1 can also be regarded as periodic orbits of spatial period 2 and temporal period 1 for a system with s = 1, with signs alternating in the spatial direction. Solving the above equations for $* and considering all possible sign combinations T = ±T)v and s = ± 1 one ends up with $*

=

±1,±-

(2,4,25)

$*

=

±1


=

0

$

= ± — - — ( l ± v ^ - 32a + 32a 2 )

(2.41)

{3A,3B)

(2.42)

{3A,3B,SA-,3B')

(2 A3) (2A~)

** - 4 \ / 3 ± r k <3A"> $*

=

± — * — ( l + a ± ^ 9 - 14a + 9a 2 ) 4 — 4a V /

*' = 4f^a

(3^-)-

(2.44)

(2 45)

-

(2B~)

(2.46)

(2-47)

All these curves are plotted in Fig. 2.7. Since unstable fixed points certainly also exist outside the interval $* £ [-1,1] and a £ [0,1], in Fig. 2.7 we have plotted a larger range of possible values, namely $* £ [—6,6] and a £ [-2,4]. Note the singular behaviour of <J?*(a) near a = | and a = 1. Almost all of the above fixed points are unstable, but there are two notable exceptions. One can easily check that the fixed point **

=

W 9 - 3 2 a + 32a2

(24g)

46

'©

Chaotic

strings

0

Fig. 2.7 Union of all fixed points $* of the JV — 2 and N = 3 strings as a function of a. Almost all of the fixed points are unstable.

of the 2A

string is stable for a e

5 9 14'14

(2.49)

Similarly, the fixed point $* = of the 2B

1 + a - -y/9 - 14a + 9a 2 4-4a

(2.50)

string is stable for a6|-,l

(2.51)

Going to higher periods than period 1, the other strings can have stable periodic orbits in certain parameter regions as well [Dettmann (2001)]. But as already mentioned, these usually have extremely small basins of attraction, and they generically don't show in a numerical simulation.

* Spatio-temporal

2.8

patterns

47

* Spatio-temporal patterns

The complexity inherent in chaotic string theories is immense. This is best illustrated by some color pictures showing examples of spatio-temporal patterns. Fig. 2.8 shows the color code that we use in the following. Starting with the uncoupled case, in Fig. 2.9 we see the pattern generated 20 uncoupled maps T2(<&) = 2$ 2 — 1 for random initial conditions. The horizontal direction is space i, the vertical direction is time n. The color of each pixel in the 20x20 lattice represents the state of the field variable $^ at position i at time n according to the color code of Fig. 2.8. Fig. 2.10 shows the same pattern as Fig. 2.9, but this time on a 100x100 lattice. Note that we use the same colors in Fig. 2.9 and 2.10, but the human eye is not able to resolve these in Fig. 2.10 and basically just sees some yellow and blue. On an even larger scale, a colorless picture arises. This can be regarded as a simple illustration of the fact that on large scales the deterministic chaotic Tchebyscheff noise looks like Gaussian white noise, as rigorously proved in section 1.2. A different color pattern arises for uncoupled third-order Tchebyscheff maps T 3 ($) = 3 $ 3 - 4$ (Fig 2.11). This once again illustrates the fact that the higher-order correlations of T3 are different from those of T^ (see section 1.7). For non-zero coupling, a large variety of different patterns can be generated. Figs. 2.12-2.17 just show some interesting examples. We always used random initial conditions and iterated 10000 times before plotting. All pictures were produced using the same color code. Hence the overall color obtained for a certain parameter in comparison to that for another parameter is a true property of the dynamics. For the various parameters we observe spatio-temporal chaos, turbulentlike states, intermittency, periodic and quasi-periodic behaviour, waves, breathers, kinks, defects, and so on. Clearly, for large a the variables $^ are not a good approximation of Gaussian white noise any more. In fact, here often ergodicity is lost and the outcome of the numerical experiment depends on the initial values chosen. As a consequence of this, one has different qualitative behaviour for e.g. the 3 A and 3A" string for a —• 1. Apparently much more complicated behaviour can arise if strongly rather than weakly coupled variables $^ are formally used for second quantization. Physically, this strongly coupled situation is expected to arise during the Planck epoch (see section 11.6). But for most applications described in

Chaotic

48

strings

- 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0.0 + 0.2 + 0.4 + 0.6 + 0.8+1.0 Fig. 2.8

Color code used for all spatio-temporal patterns shown in this book.

Fig. 2.9 Spatio-temporal pattern as generated by uncoupled maps T2(*) = 2 # 2 — 1 on a space-time lattice of size 20x20.

1

Fig. 2.10

Fig. 2.11

Spatio-temporal

patterns

49

Same as Fig. 2.9 but on a 100x100 lattice.

Spatio-temporal pattern as generated by uncoupled maps T s ( * ) = 4 * 3 - 3 * .

50

Fig. 2.12 0.375.

Chaotic

strings

Spatio-temporal pattern as generated by the 3/1 string dynamics with a =

Fig. 2.13

Same as Fig. 2.12 but o = 1.

Spatio-temporal

Fig. 2.14

Fig. 2.15

patterns

51

Spatio-temporal pattern as generated by the 3B string dynamics for o = 0.663.

Spatio-temporal pattern as generated by the 2/1 string dynamics for 0 = 1 .

52

Chaotic strings

"-"•"•'. "•"• v _ v •'•'•'•"•"•"•'• ;;::;:. •' v . " "•"•'•"• v . • • •

.

.v. .V. .V. .V.

• • • • • • • :

Fig. 2.16 Spatio-temporal pattern as generated by the IB

string dynamics for a = 0.3.

Fig. 2.17 Spatio-temporal pattern as generated by the 2/1 string dynamics for a — 0.9.

* Spatio-temporal

patterns

53

this book the coupling a will indeed be very small and the relevant pictures of vacuum fluctuations then look very similar to Fig. 2.10 and 2.11. To give a rough idea on what is going on for the chaotic strings as a function of the coupling parameter a, Figs. 2.18-2.23 show a subset of 100 spatial variables $^ as obtained after 10000 iterations on a lattice of size L = 10000. The generic behaviour is chaotic, but there are also some parameter regions where stable periodic orbits exist or coexist. Generally, for large couplings a the coupled map lattices can have many different attractors. Periodic attractors can coexist with chaotic ones. Figs. 2.20 and 2.21 clearly show the stable synchronized fixed points of the 2A~ and 2JB~ strings, whose existence we proved in the previous section. These synchronized fixed points bifurcate at a = -^ and a = ~ (2A~), respectively a — | (2B~). Generally, a large variety of complex phenomena is possible. The feather-like regions, seen e.g. in Fig. 2.19 for a € [0.6,0.7], correspond to frozen spatial patterns with very slow relaxation. We refer to the literature (e.g. [Amritkar et al. (1991); Amritkar (1996); Bagnoli et al. (1999); Binder et al. (1992); Bohr et al. (1989); Carretero-Gonzalez et al. (1997); Chate et al. (1988a); Chate et al. (1988b); Ding et al. (1997); Fernandez (1995); Fernandez (1996); Gade et al. (1993); Gorbon et al. (1997); He et al. (1994); He et al. (1997); Just (1995); Kaneko (1985); Kapral (1985); Lai et al. (1994); Lai et al. (1999); Lambert et al. (1994); Parekhetal. (1993); Pikovsky (1991); Piatt et al. (1997); Quet al. (1994a); Qu et al. (1994b); Raghavachari et al. (1995); Rudzick et al. (1997); Schreiber (1990); Volevich (1991); Wang et al. (1996); Xie et al. (1997); Bohretal. (2001); Kapral et al. (1994); Kuznetsov et al. (2001); Schmuser et al. (2001)]) for much more detailed results describing the complexity inherent in coupled maps.

Chaotic

54

strings

0.8 0.6 0.4 0.2

©

«4

""*'* - »

o

^ * # >/ .

-0.2 -0.4 -0.6

"^^^few@^

-0.8

Fig. 2.18 Plot of 100 variables <J>^ of the 2A string as a function of a. The lattice size is L - 10000.

0.1

0.2

Fig. 2.19

0.3

0.4

0.5

0.6

0.7

0.8

Same as Fig. 2.18, but for the 2B string.

0.9

1

* Spatio-temporal

patterns

55

•rf

m

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 2.20

Same as Fig. 2.18, but for the 1A~ string.

Fig. 2.21

Same as Fig. 2.18, but for the 2 B ^ string.

0.9

Chaotic

56

0.1

0.2

Fig. 2.22

0.1

0.2

Fig. 2.23

0.3

0.4

strings

0.5

0.6

0.7

0.8

0.9

Same as Fig. 2.18, but for the 3A string.

0.3

0.4

0.5

0.6

0.7

0.8

Same as Fig. 2.18, but for the 3B string.

0.9

Chapter 3

Vacuum energy of chaotic strings

We introduce formal potentials that generate the dynamics of chaotic strings. Expectations of these potentials yield two important types of vacuum energies associated with the chaotic dynamics, the self energy and the interaction energy of chaotic strings. In later chapters we will see that local minima of these quantities fix and stabilize the standard model parameters.

3.1

Self energy of the N = 3 string

For all that we will be doing in the rest of this book it will be useful to introduce formal potentials that generate the dynamics of the chaotic string field variable <&\. It should be clear that the dynamics is dissipative, but in spite of that it makes sense to introduce formal potentials. The dynamics generated by the 3rd order Tchebyscheff map $n+i=T3($n) = 4$«-3*n

(3.1)

can actually be written as *„+l - *n = ^ l - 4*n-

(3-2)

The change $„+i - <£„ of the field is formally caused by the 'force'

FJ 3 ) ($ n )=4$3_ 4 $ n . 57

(3.3)

Vacuum energy of chaotic

58

strings

We may regard this force to be formally generated by a potential V\_ defined by F } 3 ) ( $ ) = - ^ y | 3 ) ( $ ) . Apparently Vl3) ($) = 2 $ 2 - $ 4 + G+.

(3.4)

Here C+ is an arbitrary additive constant. Similarly, the dynamics * n + 1 = -T3($„) = - 4 $ 3 + 3 $ n

(3.5)

$ n + 1 - $ „ = - 4 $ 3 + 2 $ „ = Fl3)($„).

(3.6)

can be written as

The force F_ ($) is generated by the potential y( 3 )($) = - $ 2

+

$4

+

C_,

(3.7)

where C_ is another arbitrary additive constant. Remember that ultimately we want to use the chaotic dynamics to generate the noise fields in the Parisi-Wu approach of stochastic quantization. Hence expectations with respect to the chaotic dynamics correspond to quantum mechanical expectations. Let us now calculate the expectation of the vacuum energy associated with the chaotic noise fields themselves. First, let us consider the uncoupled case o = 0. We have from eq. (1.50) ($ 2 ) = \ and ($ 4 ) = §, thus

(V[3)($)) = -($ 2 ) + <$4)+C_ = - J + C_

(3.8)

o

(4 3) ($))

= 2($ 2 )-($ 4 )+C + = ^ + C+.

(3.9)

We notice that there is non-vanishing vacuum energy associated with both potentials. Of course, it could be made to vanish by choosing C-

=

+\

(3.10)

C+

=

-5-

(3.11)

However, it seems unaesthetic that C+ ^ C_. If there is any reason to choose a certain value of the additive constant, why should it be different for the two maps T3 and — T3, which basically generate the same dynamics, up to a sign?

Self energy of the N = 3 string

59

Nevertheless, we notice that there is one distinguished choice of C+ = C- that gives full symmetry to the problem, namely (3.12) In this case we get

(vi3)m

= +^(&) -<$4> = +

3 8 3 8

(3.13) (3.14)

This means, each potential has just the negative expectation value of the other one. If we assume that both the negative and the positive Tchebyscheff dynamics are of physical relevance, the two vacuum energies can compensate: (^3)($)) + (y[3)($))=0.

(3.15)

This reminds us of similarly good effects that supersymmetric partners have in ordinary quantum field theories, and hence, as a working hypothesis, we may associate the above ± symmetry with a kind of supersymmetry in the chaotic noise space. Notice that the sum of the two potentials V_ and V+ ' is actually just a simple harmonic oscillator potential with expectation 0: V($) := Vl 3) ($) + y [ 3 ) ($) = $ 2 + C- + C+ = $ 2 - i

(3.16)

If a ^ 0, the expectations ($ 2 ) and ($ 4 ) will slightly deviate from their uncoupled values \ and | , since the invariant density changes due to the coupling. However, we can still keep the symmetry condition (3.15) by defining quite generally for arbitrary a

vl3)($) ==

-$2 + $4-i($2)

(3.17)

Vf>(*) ==

2$2-$4-^($2)

(3.18)

vi3)(*)+KJi3)($) = $ 2 --<* 2 >

(3.19)

V(*)

~-=

For arbitrary a, the expectations of the vacuum energies associated with (3)

VI

(3)

and V+ can compensate each other. The three potentials V_',V+ and V are plotted in Fig. 3.1. For the

Vacuum energy of chaotic

60

0.8

strings

--,

y

/ v <3>

0.6

0.4

-""

\

\

\

/v

-

\

-

_..'.">•—'<'.'.

-0.4

""""" -0.6

1

1

-0.5

Fig. 3.1

"v.<3)

0.5

The potentials V^ 3 ) ($), v i 3 ) ( $ ) and V ( $ ) for the N = 3 string.

plot we have chosen the case o = 0, where the additive constant is given by

3.2

Self energy of the N = 2 string

Now let us study the 2nd-order Tchebyscheff map T 2 ($) = 2 $ 2 - 1. Let us find the corresponding potentials and their expectations, just analogous to our previous discussion for T3. The dynamics $n+1 $„

+ 1

-$

n

2$ 2 - 1 = + T 2 ( $ n )

= =

2 $

2

- $ „ - l

(3.20) (3.21)

is formally generated by the potential

F[2)($) = - ^

3

+ i $ 2 + $ + C+,

(3.22)

Self energy of the N — 2 string

61

where again C+ is an arbitrary constant. It generates the force F

+ } (*) = ~J^V+

a*'

( $ ) = 2 $ 2 - $ - 1.

(3.23)

Similarly, the dynamics generated by the negative Tchebyscheff map -T2 = 1 - 2$ 2

$n+1 ^ $

n + 1

- $

=

-T2($n)

2

= -2$ -$„ + l

n

(3.24) (3.25)

is generated by the potential V[2)($) = \<j>3 + i $ 2 - $ + C_.

(3.26)

It generates the force F™ {$) = - — y_($) = - 2 $ 2 - $ + l.

(3.27)

The most symmetric choice of the additive constants C+ = C_ is the same as for the 3rd-order Tchebyscheff polynomial, namely C+=C-=-\(&).

(3.28)

We then obtain potentials whose sum is again a simple harmonic oscillator potential with average zero: V_|2)($)

= -?$3 + i$2 + $-i($2)

Vl 2 ) (*)

=

|$ O

V($)

:=

3

+ i$2-$-i($2) Z

(3.29) (3.30)

Z

Vi 2 ) ($) + y | 2 ) ( $ ) = $ 2 - ($ 2 )

(3.31)

For a = 0 we have (<j>2) = | . The corresponding potentials are plotted in Fig. 3.2. Though many properties of the chaotic N = 2 string are similar to those of the chaotic N = 3 string, there are some significant differences. This has to do with the fact that the potential of the N = 2 string consists of odd powers. The odd moments of the $-field change sign if T2 is replaced by —T2. This means that the expectations of the potentials generating the positive and negative T2-dynamics satisfy (W 2 ) ($)) + r 2 - (Vd 2) (*))-T 2 = 0,

(3.32)

Vacuum energy of chaotic

62

Fig. 3.2

strings

The potentials V J 2 ) ( S ) , VJ 2 ) ($) and V(*) for the N = 2 string.

whereas for the potentials generating the positive and negative T3-dynamics we have ( V f ($)> +T 3 + ( F i 3 ) ( $ ) ) _ T 3 = 0 .

(3.33)

In a sense, these different relations remind us of commutator and anticommutator relations.

3.3

Self energy for general TV

Let us now quite generally consider Tchebyscheff polynomials Tjv($) of arbitrary order N > 2. The first few polynomials are given by T 2 ($)

=

2$2-l

T 3 ($)

=

4$3-3$

T 4 ($)

=

8$4-8$2 + l

T 5 ($)

=

16$ 5 - 20$ 3 + 5$

Self energy for general N

T 6 ($)

=

32$ 6 - 48$ 4 + 18$ 2 - 1

and the corresponding potentials V± integration. We have Vl2)($) =

v£\$)

± (~$3

63

(3.34)

can be obtained by straightforward

^$ 2 + C±

+ *\+

= ± (-$* + 1&\ + I$ 2 + c±

(3.35)

(3.36)

- -4- f 8*55 , 8^,3J _x,\: V4*)(S>\ ^*) = ± l - - $ + ^ $ - $ J + -1*2 $ 2 + C±

(3.37)

K<5>($)

=

± f _ | $ « + 5 $ 4 - ^ $ 2 ] + i $ 2 + C±

(3.38)

yW($)

=

± (-^*7 + ^*

5

- 6* 3 + $") + | * 2 + C ± . (3.39)

The potentials for N = 2,3,4,5,6 are plotted in Fig. 3.1-3.5. Note that only for N = 2 and AT = 3 the 'classical' minimum of the potential V_($) is smaller than or equal to that of V($). This once again distinguishes the N = 2,3 strings as compared to higher-iV strings. For $ € [—1,1] we may generally write TN($)

= cos(iVarccos$).

(3.40)

To obtain an expression for the potential V± ($) that is valid for arbitrary N, we have to evaluate the indefinite integral 1 = / cos(A r arccos$)d$. Substituting $ = cos x we obtain I

=

— I cos Nx sin xdx

(3.41)

Vacuum energy of chaotic

64

strings

i

i

0.4 V

/'

0.2

I

v 0

-0.2

-

-0.4

-

--_.\-'""

" \

'

+

"•••/L....--<jW

'., \ / . 1

' -

i

i

i

0.5

Fig. 3.3

The potentials V\M); ( $ ) , V^Wi; ( $ ) and V($) for the N = 4 string cos(7V + l)x

cos(7V -

2(iV + l)

l)x

2{N-1) '

(3.42)

From this one obtains the general formula

vlN\$)

= ±\ (j^T^W

- j^TN+^y^+const.

(3.43)

Full symmetry under the transformation TJV —>• —Tp? is achieved if the additive constant is chosen as C+ = C_ = - | ( $ 2 ) , which evaluates to | for a — 0. For even N, we then have r(N),

r(N),

+rA, = < ^ ( * ) ) - T w ,

(3-44)

and this is equal to 0 provided a = 0. For odd TV we have (^(*))+T, - - ( ^ ( * ) ) - T „

(3.45)

and this is not equal to 0 even for a = 0. If we couple the maps spatially, the invariant density at each lattice site will become asymmetric for even N, whereas it stays symmetric for odd N (see Figs. 2.3-2.6). However,

Interaction

H.

energy of chaotic

strings

65

0

Fig. 3.4

,/(j 5 ;)( $ ) and V(*) for the JV = 5 string. The potentials VAV(
what remains valid for arbitrary coupling and arbitrary N is the general symmetry condition NnAN),

AN)

{-i)N
(*)hTs = 3.4

(3.46)

Interaction energy of chaotic strings

Next, we also want to regard the diffusive coupling as being produced by a suitable potential. Consider, for example, a string with forward diffusive coupling. The time evolution can be regarded as consisting of two fictitious time steps. First the variables $^ evolve to <S>^ = ±Tjv($^) due to the self-interacting potentials V± '. In the next time step, there is diffusive coupling $*71+1 ^n+1

^r,

^n

r

o ^

n

n

' ^n

I

(3.47)

Vacuum energy of chaotic

66

0.6

1

....

strings

1

!

-\ /

S.

'> '"'•• \

>'''..

0.2

\

0

-0.2

-

-0.4

-

-0.6

/ /'

\---\'

\ v ^ " ~

/

v

+

-

-

--''''

1 ..

1

The potentials v[6)(), V^^)

.

">--

/

0.5

-0.5

Fig. 3.5

/

''

v

and V($) for the N = 6 string.

To simplify the notation, let us abbreviate <3? := $^ _ 1 and \P := $^ . Apparently, the interaction of two nearest neighbors yields the 'force' § ( $ — $ ) =: aFL onto * and the force § ( # - $ ) =: aFR = - a F L onto $. Both forces can be thought of as being produced by the interaction potential

W_ (*,*) = i($-tf) 2 +c_

(3.48)

We have (3.49) (3.50) The actual strength of the interaction is determined by the value of the coupling constant a. To select the most aesthetic value of the additive constant c_, let us proceed in just the same way as we did for the self-interacting potentials

* Double

strings

67

V±. Can we find another potential W+($, \P) such that the potential W ( * , ¥ ) := W _ ( $ , * ) + W + ( $ , * )

(3.51)

is just a harmonic oscillator potential with average 0, similar as the sum of V_ ' and V^ ' was? Yes, we can. Consider 1 W + ( $ , * ) = -^($ + # ) 2 + c + .

(3.52)

:=

W_($,tf) + W + ( $ , ¥ )

(3.53)

=

- ( $ - * ) 2 + -($ + *)2+c_+c+

=

i ( $ 2 + *2) + c_+c+.

We have W($,¥)

(3.54)

Choosing c+ = C- = - | ( $ 2 ) = - | ( \ ? 2 ) , we indeed end up with a harmonic oscillator potential with average 0: iy($,$) = I($2

+

$2_2($2))

(3.55)

For the expectations of the interaction potentials W+ and W_ we obtain (W±($,¥)>

=

i(($2)±2($*) + (*2))+c±

(3.56)

=

±^($$>,

(3.57)

i.e. they are just given by the correlation function of nearest neighbours $,\I>. Notice that we again have the same additive constant —1($ 2 ) as for the self-interacting potentials VJ. '. Moreover, note that the effect of replacing W_ -»• W+ is just the same as s-> —s, or replacing T -> —T at odd lattice sites. 3.5

* Double strings

So far we did not talk about the actual size of the vacuum energy represented by the potentials (V± ) and (W±($,i£)). Our considerations were independent of that value. But if there is a lot of vacuum energy concentrated on the chaotic string (as we would expect if we consider a string model at the Planck scale) then a natural question is why the gravitational

68

Vacuum energy of chaotic

strings

Fig. 3.6 Two chaotic strings corresponding to positive and negative Tchebyscheff maps are forming a double string. At the interaction point the double string structure breaks up and the two single strings are immediately covered by a new pair of strings.

attraction of the vacuum energy does not make the string contract into one point. One possible answer is that two strings with positive and negative vacuum energy (represented by V+ and V_) usually lie together so that the vacuum energies cancel. They form a double string. Only in exceptional cases the joint structure breaks up, and then the vacuum energy of each single string becomes relevant. This is similar to a DNA string in biology, which also consists of two strings whose chemical potentials usually cancel each other, except when the double helix is replicated [Kornberg et al. (1992)]. A possible picture for chaotic strings is shown in Fig. 3.6. If one tries to split the double string, the large vacuum energy may immediately create another pair of chaotic strings which form double strings with the original strings. This is similar to a hadronization process (forming mesons out of free quarks). Since Vr)W) + V^N) = $ 2 - ($ 2 ) for all N, the effective behaviour of a double string is similar to that of an N = 1 string, i.e.

* Rotating

strings

69

a string where the dynamics is generated by V($) = $ 2 - ($ 2 ). The expectation of the vacuum energy of the double string is just (V($)) = 0. This may help to avoid problems with the cosmological constant [Weinberg (1989)].

3.6

* Rotating strings

Another possibility of cancelling gravitational attraction of vacuum energy on a single string is to let the single chaotic string rotate. Let us work out this picture in somewhat more detail. The approach of section 2.4 actually suggests that the chaotic string describes fluctuating momenta of a single particle, say an electron. The electron is assumed not to be point-like but to have string structure. The typical radius R of this string is much smaller than the Compton wave length A ~ l / m e of the electron. It may be of the order of the Planck length or similar. In that sense, while (in first quantization) the electron is a wave packet smeared over a region of size A, we are now looking at a kind of 'core' of the electron of much smaller size R (Fig. 3.7). The chaotic string winds around the core and actually keeps it together: The centrifugal force due to the spin of the electron is compensated by the gravitational attraction of the vacuum energy on the string. In this kind of model, the spin of the electron is represented by the angular momentum of the vacuum energy that is concentrated on the rotating string. Let the total vacuum energy associated with the chaotic string be M = 2-KR • V

(3.58)

Here V is the energy density on the string, which has dimension energy/ length = energy2. If the coupling a is small, then the interaction energy can be neglected and M (in suitable units) is essentially given by |(V±($))| times the number of lattice sites. If the string does not rotate, gravitational attraction of the vacuum energy on the string will just let the string contract into a point and make the electron point-like. To avoid this, we may assume that the string rotates with frequency ui. Let us estimate the equilibrium state of this simple model system. Since we are only interested in the order of magnitude, we will just use purely classical formulas and ignore numerical factors of 0(1). The balance of

70

Vacuum energy of chaotic

strings

Fig. 3.7 Chaotic string of radius R describing a string-like electron. The radius R is much smaller than the Compton wave length A.

centrifugal and gravitational forces requires MLJ2R

=

M2 G—z R n

(3.59) r

mpiR?

(3.60)

Here G = m^ 2 is the gravitational constant (in natural units h = c = 1). We identify the angular momentum of this gravitational system with the spin S of the particle under consideration (S = 1/2 for an electron): MR2LO

= Sh

(3.61)

In natural units we get —

MR,-

(3-62)

* Rotating

strings

71

Equating eq.(3.60) and (3.62) we end up with * = ^ P

(»•«»)

W

(3 64)

and =

< ^

'

Suppose M is of the order of the Planck mass mpi, then we get a radius R of order m~p'l and a frequency u of the order of mpi. All this appears reasonable. The infinite self energy of the electron in QED is just replaced by a finite value M. We may also consider heavier particles than just a single electron, or systems of many particles. Suppose M is large enough that a black hole can be formed. In this case it seems reasonable to identify the string radius R with the event horizon (Schwarzschild radius) TQ of the black hole. For a rotating Kerr black hole with mass M, spin S, and charge 0 the Schwarzschild radius is given by [Kenyon (1990); Wald (1984)] (3.65) The chaotic string just lives at the edge of the event horizon of the black hole and ties it together. Combining eq. (3.63) and eq. (3.65) one obtains S2mPl M3

_ M ~~ mpi

1 M2

Y

S2 mpl " M2'

(3.66)

which yields M

=

u

=

R

=

vSmpi rnpi

Vs mpi'

(3.67) (3.68) (3.69)

This means, heavy black holes of this type are only allowed if the spin S is large. For the velocity we obtain v=uR

= l,

(3.70)

72

Vacuum energy of chaotic

strings

i.e. independently of S the string rotates with velocity of light (remember that we use natural units where c = 1). A particle of mass M and charge e rotates in a constant magnetic field B with Larmor frequency

w = IS-

(3-71)

To generate a frequency as given by eq. (3.68) one needs a moderately strong B-field, namely eB = 2MUJ = 2m2Pl

(3.72)

Numerically, one obtains B — 5879 Tesla. Such a 5-field and a string with charge would be an alternative to guarantee the stability of our rotating system, formally replacing gravitational interaction by electromagnetic interaction. The black hole considered may have incorporated a large number of particles. If all these particles have their spin in the same direction, then clearly S is proportional to the number Np of particles forming the black hole: S ~ Np

(3.73)

The reason for all spins pointing in the same direction could be the strong magnetic field B, which makes one spin direction much more energetically favorable than the other. However, without the assumption of a magnetic field the spins of the various particles will show in both directions, hence the sum of the spins is Gaussian distributed, and to achieve a total spin S of order y/Np one needs Np particles. In other words, we then have Np.

(3.74)

For the event horizon this implies N tJL

r0=R~

1/4

-.

(3.75)

mpi

For the energy density V on the string we obtain in either case

V = 1L = ?& = ?*.

(3.76)

* Rotating strings

Note that this is independent of the spin S (or the since both M and R are proportional to y/S. Though we do not need rotating chaotic strings still represent an interesting theoretical concept, and nection with black holes. Our concept allows for the vacuum energy density on the string.

73

particle number Np), in the following, they we see a possible conestimate (3.76) of the

This page is intentionally left blank

Chapter 4

Phase transitions and spontaneous symmetry breaking

We will investigate phase transition-like phenomena and spontaneous symmetry breaking for chaotic strings and their higher-dimensional extensions, illustrating the complexity inherent in turbulent quantum states. We will consider lattices of various dimension d, and also consider some examples of vector-valued field variables ^in. For large enough values of the coupling a it is observed that often the symmetry is broken due to the existence of several attractors with different averages. Generally, coupled map lattices can exhibit a very diverse spectrum of phase transitions (see, e.g., [Jiang (1997); Cuche et al. (1997); Blank (1997); Marcq et al. (1997); Boldrighini et al. (1995); Keller et al. (1992a); Houlrik et al. (1990); Grassberger et al. (1991); Miller et al. (1993); Gielis et al. (2000); Beck (1998)]).

4.1

Some general remarks on phase transitions

The phase transition behaviour of continuum 4-theories is a well-established area of research, and several precision results on critical exponents are known (see, e.g., [Guida et al. (1998)] and references therein). As we have seen in previous sections, in the anti-integrable limit ^-theories lead to coupled cubic maps, and hence it is natural to investigate phase transition phenomena for these systems as well. In fact, for coupled maps much less is known then for continuum field theories, and the spectrum of possibilities is extremely rich. Most results that are described in the literature are just numerical rather than analytic. Even the answer to the question of what phenomena should deserve the name 'phase transition' is 75

76

Phase transitions

Fig. 4.1

and spontaneous symmetry

breaking

Attractor of the map 3>n+i = WAr($n), N — 2, as a function of b.

not completely clear for coupled maps. In the thermodynamic formalism of dynamical systems (see, e.g., [Beck et al. (1993)] for an introduction), bifurcations occurring for just a single map (as in Fig. 4.1, 4.2) can already be described in the language of phase transitions. There are scaling laws for e.g. the Liapunov exponent near the accumulation point of bifurcations, which allow for the definition of 'critical exponents' etc. On the other hand, for a spatially extended system of coupled maps, one may want to reserve the name 'phase transition' for more significant changes in the behaviour of the system than just bifurcations. For example, one could define a phase transition by a diverging spatial correlation length in the thermodynamic limit. Relevant phase transition phenomena should always survive the limit of lattice size going to infinity. Mentioning the thermodynamic limit we are immediately facing a fundamental difficulty. There are two limits that have to be performed for a coupled map lattice: Lattice size going to infinity and number of iterations going to infinity. For any computer experiment, necessarily both numbers are finite. What requires particular attention is the fact that the two limits

Some general remarks on phase

77

transitions

1 0.8 0.6 0.4 0.2 ©

0

-0.2

-0.4 -0.6

-0.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.9

b

Fig. 4.2

Same as Fig. 4.1 but AT = 3.

usually do not commute. Many results depend on the order in which the two limits are performed. The class of dynamics we are interested in is quite generally coupled maps on d-dimensional lattices as given by

n + i = ( i - « m ^ ) + ^E/(*n)-

(4.1)

i = (ii,...,id) labels the lattice sites and T and / are some functions. i denotes the 2d nearest neighbors of i on a lattice of dimension d. For example, for the 2A string one has d = 1, T($) = / ( $ ) = T 2 ($) = 2 $ 2 - 1, and i takes on the values i — 1 and i + 1. Tchebyscheff maps Tjy can be regarded as the endpoint of a bifurcation scenario exhibited by maps of the form T($) = bT]v($), where b varies from 0 to 1. This scenario is plotted in Fig. 4.1 for N — 2 and in Fig. 4.2 for N — 3. The typical features of these scenarios are only slightly perturbed if a small spatial coupling a is introduced (Fig 4.3). Clearly, once a spatial coupling a is introduced then bifurcations can occur by either varying b or

78

Phase transitions

0

0.1

and spontaneous

symmetry

breaking

i

1

1

1

1

1

1

r

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

b

Fig. 4.3 Iterates ^\ of weakly coupled logistic maps * 5 , + 1 = (1 — a)bT2($ll) + § ( * n * + $ „ ) a s function of 6. The spatial coupling is a = 0.01. The picture is very similar to that of a logistic map with noise.

varying a. Whether these abrupt changes in behaviour deserve the name 'phase transition' is merely a matter of personal taste, depending on which aspect one is most interested in. Let us now come to a phenomenon that we may call 'spontaneous symmetry breaking', in certain analogy to the way this word is used in elementary particle physics [Kane (1987); Weinberg (1976)]. The simplest example is already provided by an uncoupled odd map, such as T($) = bTs($) = 6(4<j>3 _ 3$) xhe equation of motion $n+l = T ( * n )

(4.2)

is invariant under the transformation $ n -> — $ n for all n, since T ( - $ „ ) = - T ( $ n ) = -
(4.3)

In this sense the dynamics of an odd map has a global Z2 symmetry, which is not present for an even map. However, a single attractor of an odd

Vacuum expectation on 1-dimensional

lattices

79

map need not to have this symmetry. In fact, what in Fig. 4.2 looks like a bifurcation from a stable period 1 orbit to a stable period 2 orbit at b = bc — | is in reality the creation of two different stable period 1 orbits. This phenomenon definitely differs from the quadratic map, where only one attractor exists. Which one of the two attractors of the cubic map is reached depends on the initial values. Given a certain initial condition, only one of the fixed points is reached, and this particular choice breaks the Z2 symmetry of the problem: The fixed point is either positive or negative, depending on the initial values. We may generally look at the average value ($) of the iterates for a given initial condition. At b = bc this quantity switches from zero to a non-zero value.

4.2

Vacuum expectation on 1-dimensional lattices

Generally, one may now look at symmetry breaking phenomena for spatially coupled fields $ln. One of the simplest observables (or order parameters, in the language of phase transitions) is again the average of the field $^. Since averages with respect to the chaotic dynamics correspond to quantum mechanical expectations in the chaotic quantization approach (see section 1.2), this average can also be regarded as the vacuum expectation of the degenerated self interacting field for which we performed the anti-integrable limit in section 2.2. Numerically, the vacuum expectation v — ($) of the field is obtained by averaging the variable $^ over all n and i for random initial conditions $Q. Since there are usually many different attractors, this quantity does depend on the initial conditions, as well as on the lattice size. To illustrate this, Fig. 4.4 shows the quantity v as a function of the coupling a on a small 1-dimensional lattice of size imax — 10. The local map is ±T^. The plots are obtained by averaging the variable 3>^ over all lattice sites i and over a large number of iteration steps after some initial iteration time to avoid transients. The initial values $Q are chosen to be randomly distributed according to the invariant density p($) = l/(ny/l — $ 2 ) of the Tchebyscheff map. A very complicated set of structures arises for v as a function of a. The details of the patterns depend on the model studied, i.e. whether we either study —T3 or +T3 with backward or forward coupling. They also depend on the size of the lattice. For example, a lattice of size 20 yields different pictures (Fig 4.5). v(a) switches from 0 to a non-zero value at various critical coupling parameters a. Whether the positive or

Phase transitions

80

0.8

'

0.6

and spontaneous symmetry

'

breaking

,a>

0.4 0.2 i

0

1

>*.

-0.2

vf;k

-0.4

—^ ':

-0.6

'

-0.8

0

-

— >

,-'

—*— i

i

i

i

i

'

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.3 0.2 0.1

0 -0.1 -0.2 -0.3

_J

0

1

I

I

I

I

I

I

L_

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a

Fig. 4.4 Vacuum expectation of the field Qxn on a lattice of small S1ZG %max = 10 with periodic boundary conditions. The local map is —T3 in a),b) and +X3 and in c),d). The coupling is forward coupling in a),c) and backward coupling in b),d).

negative branch is chosen depends on the initial conditions. One expects these complicated phenomena to persist on lattices of larger size. However, the basin of attraction of periodic orbits with non-vanishing v will become smaller, and often a chaotic attractor with v = 0 will dominate the behaviour although some stable orbits with v ^ 0 are still formally present. How can we understand the structures in Fig. 4.4, 4.5? Once again note that our coupled dynamical system has a global Z2 symmetry (as well as an extra dihedral symmetry due to the periodic boundary conditions). The dynamics (4.1) is invariant under the transformation $ -> — $ since the local map is an odd mapping, i.e. T ( - $ ) = - T ( $ ) . For this reason a first guess would be that the average of $ is zero. In spite of that, a nonvanishing average ($) arises in certain parameter regions due to the fact that ergodicity is lost and many different attractors with different averages co-exist. These can either be stable periodic or quasiperiodic orbits or chaotic attractors. In Fig. 4.4, 4.5 we observe several bifurcation points ac where the vacuum expectation of the field suddenly starts to become

Vacuum expectation on 1-dimensional

Fig. 4.5

Same as Fig. 4.4, but for a lattice of

lattices

S1Z6

81

Irnax

= 20.

non-zero, obeying a scaling law of type v(a) = C • \a — ac\z for a in the vicinity of ac. For example, in Fig. 4.4d bifurcations are observed at ac = | , |,0.5276,0.5748 (among others). A very simple example is a = | . Here the non-vanishing average for o slightly larger than | is due to the creation of a stable periodic orbit of temporal period 1 and spatial period 2. We denote it by AB. The orbit elements A and B satisfy A + B ^ 0. There is also a stable periodic orbit consisting of orbit elements of opposite sign, and with equal probability the system reaches either of these two attractors. The ball-like structure for a slightly less than 0.5748 is due to a periodic orbit of temporal period 2 and spatial period 10, of the form ABCDEEDCBA FGHIJJIHGF.

(4.4)

Here columns represent time and rows represent space. Again the sum of all orbit elements is non-zero and again a mirror-orbit with opposite sign exists. Other structures do not correspond to periodic orbits, but to several chaotic attractors with different averages. In certain parameter regions, for

82

Phase transitions

and spontaneous symmetry

breaking

a lattice of size imax, we observe a splitting into imax -1 almost equidistant values v (see e.g. Fig. 4.4d, 4.5d near a — 0.17). The above bifurcations can formally be regarded as so-called external phase transitions in the thermodynamic formalism of dynamical systems, a mathematical theory that uses tools from statistical mechanics for the analysis of nonlinear dynamical systems (see, e.g. [Beck et al. (1993)]). However, they do not represent phase transitions if seen in analogy with Ising spin systems [Miller et al. (1993); Marcq et al. (1996); Marcq et al. (1997); Schmuser et al. (2000)], where one wants to have a diverging spatial correlation length in the thermodynamic limit. As already mentioned, for coupled maps the thermodynamic limit imax —> oo,n —> oo is a delicate business. For example, the above stable spatio-temporal periodic orbit with broken symmetry, as given by (4.4), is also observed on larger lattices whose sizes are multiples of 10. However, since the transient time until the orbit is reached from random initial conditions grows exponentially with lattice size imaxi we have an infinite transient time for imax —^ oo> which means that the stable periodic orbit is not seen if we first perform the limit imax —> oo, then the limit n —> oo. However, if we perform the limits in opposite order, the orbit is always there and asymptotically reachable, since the transient time is finite on a finite lattice. In practice, this symmetry breaking orbit is already unobservable on lattices of medium size, due to an extremely large transient time.

4.3

* Real scalar field on d-dimensional lattices

Let us now look at higher-dimensional lattices. Figs. 4.6-4.13 show typical states of the 2-dimensional system (4.1) with T = T3 and T — T2 and / ( $ ) = $ on a 100x100 lattice after 10000 iterations. We have chosen the same color code as in section 2.8. Note that the horizontal and vertical direction now correspond to the two spatial directions, hence the pictures represent snapshots at a fixed time and reveal no information on the time evolution. What we see reminds us of 2-dimensional spin systems exhibiting phase transitions. Clearly the correlation length of structures increases rapidly with increasing a. To extract some information on the time evolution, Figs. 4.14-4.17 show some states of the same coupled map lattices for a much smaller number of iterations, starting from random initial conditions. Interesting enough,

Real scalar field on d-dimensional

lattices

83

Fig. 4.6 Spatial pattern as generated by the 2-dimensional coupled map lattice (4.1) with T = T3 and o = 0.55 after a large number of iterations.

Fig. 4.7

Same as Fig. 4.6 but a = 0.60.

Phase transitions

and spontaneous

symmetry

breaking

mm&

Fig. 4.8 Same as Fig. 4.6 but a = 0.65. '•"•'•KwXvX'. v w c .

111!

mm xXvXv::

8

^^^^^B ^ '.•.•.•.•.•.'.•.•.".•.

wmmmmr I-:::::-:-

> • •

•-•TOOOOOO

Mi Fig. 4.9

Same as Fig. 4.6 but a = 0.70.

Real scalar field on d-dimensional

lattices

85

Fig. 4.10 Spatial pattern as generated by the 2-dimensional coupled map lattice (4.1) with T = T2 and a = 0.40 after a large number of iterations.

Fig. 4.11

Same as Fig. 4.10 but a = 0.45.

Phase transitions

Fig. 4.12

and spontaneous

symmetry

breaking

Same as Fig. 4.10 but a - 0 . 5 0 .

%™mr

mi *3» W -

v.;.1

w Fig. 4.13

Same as Fig. 4.10 but a = 0.55.

Real scalar field on d-dimensional

lattices

87

Fig. 4.14 Spatial pattern of t h e 2-dimensional coupled m a p lattice (4.1) with T = T$ and a = 0.70 after n — 6 iterations.

Fig. 4.15

Same as Fig. 4.14 but n = 20.

Phase transitions

and spontaneous symmetry

breaking

Fig. 4.16 Spatial pattern of the 2-dimensional coupled map lattice (4.1) with T = Tt and a = 0.55 after n = 10 iterations.

Fig. 4.17

Same as Fig. 4.16 but n = 20.

Real scalar field on d-dimensional

lattices

89

-0.2 -

-0.6 -

Fig. 4.18 Vacuum expectation (<E>) of a scalar field $ on a 4-dimensional lattice with backward coupling. The local dynamics is given by T = —T3.

after a few iterations some string-like structures arise (Fig. 4.15 and 4.17), which thin out in the long-term run. These structures somewhat remind us of other topological defects in field theories such as cosmic strings [Vilenkin et al. (1994)]. From the string point of view emphasized in section 2.4, the step from d — 1 to d = 2 is like a generalization of a chaotic string to a 'chaotic membrane'. We might actually also be interested in larger values of d. The case d — 4 is interesting since there are 4 non-compactified space-time dimensions in our physical world. This case corresponds to the anti-integrable limit of an Euclidean real scalar field 0 in 4 space-time dimensions. Fig. 4.18 shows the vacuum expectation v = ($) obtained with the dynamics (4.1) for T($) = - T 3 ( $ ) , / ( $ ) = $ on a 4-dimensional lattice (lattice size 10 x 10 x 10 x 10, periodic boundary conditions). For small couplings a the system behaves similar to slightly perturbed uncoupled Tchebyscheff maps and there is only one chaotic attractor with (<£) = 0. For large couplings a « 0.8 there are at least two different attractors and the system approaches one of two symmetry breaking stable periodic orbits of the form

90

Phase transitions

and spontaneous symmetry

breaking

, with A + B ^ 0, hence ($) ^ 0. This periodic pattern extends to all 4 space directions and the (fictitious) time direction. Inbetween, spontaneous symmetry breaking sets in, possibly already at a sa 0.16, but at latest at the coupling a « 0.48 (see Fig. 4.18). It is natural to conjecture that this 4-dimensional lattice system exhibits phase transitions with diverging correlation lengths in a suitably defined thermodynamic limit of infinite lattice size and infinite iteration time. But clearly the thermodynamic limit of these types of coupled map lattices is a difficult subject, and frankly, not fully understood. Some careful numerical evidence for phase transitions has been provided for 2-dimensional lattices of similar type [Marcq et al. (1996); Marcq et al. (1997)], and some ideas to obtain the thermodynamic limit from small lattices were presented in [Carretero-Gonzalez et al. (1999)].

4.4

* Complex scalar field with U(l)

symmetry

For typical Lagrangians studied in particle physics, one often replaces the real scalar field 0 by a complex field 0 consisting of two real components $ and * : 0 = - ^ ( $ + i$) v2

(4.5)

For a complex self-interacting field with 0 4 self-interaction the Lagrangian reads [Kane (1987); Ramond (1981)] L = (dM0r(3"0)-m2<^-A(0>)2.

(4.6)

The action S[(f>] is the space-time integral of the Lagrangian. The above Lagrangian is invariant under a global U(l) gauge transformation 0 -> 0' = e i a 0 .

(4.7)

In terms of the real components L reads L = \{d^)2

+

l

-{d^)2

- l-m2{& + tf2) - ^(d>2 + * 2 ) 2 .

(4.8)

If we quantize by means of the Langevin equation, then discretize and perform the anti-integrable limit - m 2 , A ->• oo as described in section 2.2,

* Complex scalar field with U(l)

symmetry

91

the result is a coupled map lattice of the form

-Are„($i2+<)) +

- ^ < (4.9)

*«+i

=

K(l-a-m2ren i

(4.10)

The local map is now 2-dimensional. Again we may study this system for the choice of parameters

m2ren

=

- 2 ( 1 -a)

(4.11)

Xren

=

4(1-a),

(4.12)

where the local map has strongest possible chaotic properties. As can be seen by squaring and adding eq. (4.9) and (4.10), the 'radius' rn := ^ $ 2 _|_ v[/2 t n e n obeyg t h e i-dim Tchebyscheff dynamics r n +i = ± r „ ( 3 4r^) for a = 0, the sign ± being chosen such that the radius is positive. Fig. 4.19 shows a snapshot of the field (^^Vt^) on a 2-dimensional lattice after a large number of initial iterations for various values of the coupling constant a. For the initial values, we have chosen at each lattice site $o = ros'm/3, $ 0 = r0cos/3, where ro is randomly chosen according to the invariant density p(ro) = 1/(7TA/1 - r^), and /? is chosen randomly in [0,2n] with uniform probability density. Note that whereas for small a the field vectors remain randomly scattered and hardly change their direction (though evolving chaotically in the local field direction), for larger a vortex-like structures arise, quite similar to a turbulent quantum state. Indeed, this coupled map system stands in some relationship to a discrete version of the complex Ginzburg-Landau equation, where these types of phenomena are well known [Bohr et al. (1998); Collet (1994); Torcini et al. (1997a)]. The vacuum expectation v of the complex field of Fig. 4.19 is plotted in Fig. 4.20. The figure provides evidence for spontaneous symmetry breaking near ac « 0.2.

Phase transitions

92

and spontaneous symmetry

breaking

mmmm\\\\i§immmmm.

iimiMmmmmmm

llillilffllffll||l HiiiftF™

^\Wm\mmm wMmMmmwmmm

n\

msmmmm%mmmmm\\\\s\

Fig. 4.19 Complex field on a 2-dimensional field of size 50 x 50. The two real components determine the direction of the little lines at each lattice point. Five consecutive time step are plotted for a = 0 (a), 0.2 (b), 0.3 (c) and 0.6 (d).

4.5

* Chaotic Higgs field with SU(2) symmetry

Finally, we may start from a continuum field theory that describes the Higgs sector of the standard model. The Higgs field actually consists of two complex fields (f>+ and cf>°, with V + iX

V2 '

(4.13)

where $,*,
L = (^)t(^)-/iV^-A(0V) 2 .

(4.14)

It has a global 377(2) symmetry. In the standard model, spontaneous symmetry breaking of the Higgs field gives mass to the W and Z boson. When doing the anti-integrable limit as before, we end up with a coupled map

* Chaotic Higgs field with SU(2)

symmetry

93

-0.2 -

-0.4 -

Fig. 4.20 Vacuum expectation of the complex field 4> on a 2-dimensional lattice. The quantity v = \Z(<£) 2 + (^)2 sgn($) is plotted as a function of a.

lattice of 4-dimensional local maps
a

~

m2

ren ~ *r

? a

»i ) + ^ E ^ -

<4-15)

on a 4-dimensional lattice. =($,$,
ft,

—=tan/?2,

—

Vn

Xn

tan /33

(4.16)

are conserved quantities, whereas for a > 0 they start evolving in time. On a 4-dimensional lattice, the dynamics (4.15) generates pictures similar to Fig. 4.19 when being projected onto a 2-dimensional plane. The vacuum expectation on a finite lattice is shown in Fig. 4.21 (lattice size 10 x 10 x 10 x 10). For large couplings a, the symmetry is spontaneously broken and the

Phase transitions

94

i

and spontaneous

1

symmetry

breaking

i

i .;.-^l

*•?

^,-^f?P'ft-^

. . ~'. .. .

,#::-:

"

S?''

*":*fl?ly.>*'Cy*^'.*.'3vT''3?:-*"v'.'*»''.: .-•'..".-'.' ~/:

^

--

';" / '

"N

-

\ \ .

*"*;;'t;iijijis.*is"'*" ,yj3M^*5;iss,iKss:5srsrr'7: •

,

....J,*

'*$'* •^_i/

' 0

0.2

0.4

1 0.6

1 0.8

1

Fig. 4.21 Vacuum expectation of a Higgs field as obtained in the anti-integrable limit on a 4-dimensional lattice. The quantity v = •\/(3 > ) 2 + ( * ) 2 + {
concrete value of the expectation of the vector-valued field depends on the initial conditions. Again there is a critical point ac « 0.15 where symmetry breaking sets in. A precise evaluation of the critical parameter ac would require larger lattice sites and much more computing time. It should be clear that a large variety of models can be studied. We only looked at some interesting examples. More generally, we may systematically vary the dimension of the lattice, the parameters m2ren and A of the potential, the number of real components of the ^-vector, the power of in the potential, and many other things. The complexity inherent in these types of field theories is immense. A lot of work is still needed until we will have precise numerical results for these discrete nonlinear theories on large lattices.

Chapter 5

Stochastic interpretation of the uncertainty relation

We will now return to 1-dimensional lattices and present a slightly modified physical interpretation of the chaotic string dynamics. Rather than associating n with the fictitious time of the Parisi-Wu approach, n will now be thought of as being a physical time variable. Moreover, the index i will label phase space cells rather than space-time cells. In this sense our interpretation is more related to a thermodynamic description of the vacuum. Our model will be a statistical model of vacuum fluctuations of virtual momenta in real time. The mathematics of our nonlinear dynamical system is the same as before, just the physical interpretation is slightly different. We will see that the new physical interpretation requires 1-dimensional lattices. Moreover, within this new interpretation it is reasonable to identify the coupling constant a with suitable standard model coupling constants.

5.1

Fluctuations of momenta and positions

Let us construct a probabilistic model of vacuum fluctuations. Consider an arbitrary spatial direction in 3-dimensional empty space, described by a unit vector u. Assuming that superstring or M-theory is the correct theory to describe nature, then u may also point into a direction spanned up by the extra dimensions. We know that the vacuum is full of virtual particleantiparticle pairs, for example e+e~~ pairs. Consider virtual momenta of such particles, which exist for short time intervals due to the uncertainty relation. From quantum mechanics (indeed, already from 1st quantization) we know that the phase space is effectively divided into cells of size of 0{%). 95

96

Stochastic interpretation

of the uncertainty

relation

The uncertainty principle implies ApAx = 0(h).

(5.1)

Here Ap is some momentum uncertainty in u direction, and Aa; is a position uncertainty in u direction. In the following, we will regard Ap and Aa; as random variables. A priori we do not know anything about the dynamics of these random variables. But three basic facts are clear: (i) Ap and Aa; are not independent. Rather, they are strongly correlated: A large Ap implies a small Aa;, and vice versa. (ii) There are lots of virtual particles in the vacuum. For each of those particles, and for each of the directions of space, there is a momentum uncertainty Ap* and a position uncertainty Aa;1, making up a phase space cell i. (iii) If the particle of a virtual particle-antiparticle pair has momentum Ap 1 , then the corresponding antiparticle has momentum — Ap 1 in the rest frame. We label the phase space cells in u-direction by the discrete lattice coordinate i, taking values in Z. With each phase space cell i we associate a rapidly fluctuating scalar field $ln. It represents the momentum uncertainty Ap = pln in u direction in cell i at time t — nr in units of some maximum momentum pmax '• Pn=PmaX&n

<€[-l,l]

(5.2)

r is some appropriate time unit. Whether t is physical time or fictitious time is irrelevant at the moment— we just regard it as some suitable time variable. We may call pln the 'spontaneous momentum' associated with cell i at time n. The particle and antiparticle associated with cell i have momentum pln and — pln. Let us introduce a position uncertainty variable xln by writing

p^xi = T-'h

(5.3)

Here T is a constant of order 1. Indeed, eq. (5.3) states that the position uncertainty random variable xln is essentially the same as the inverse momentum uncertainty random variable l/pln, up to a constant times h. By this we certainly realize property (i). The variable xln is related to the

Newton's law and self interaction

97

rapidly fluctuating field &n by

x

ymax

^n

By definition, the sign of x*n is equal to the sign of the field &n. The phase space cells, labeled by the index i, can be represented as intervals of constant length h/T. Each phase space cell is 2-dimensional (1 momentum, 1 position coordinate). Although the volume of the cells is constant, the shapes of the phase space cells fluctuate rapidly, since (in suitable units) their side lengths are given by $^ and 1/$^. Indeed, pln and xln fluctuate both in time n and in the lattice direction i, and are uniquely determined by the random variable $ln. It is clear that the $ln must have strong stochastic or chaotic properties in order to serve as a good model for vacuum fluctuations. On the other hand, since no physical measurements are able to determine the precise momentum and position within a phase space cell (due to the uncertainty relation), the dynamics of the $ln is a priori unknown. We cannot measure a concrete time sequence of vacuum fluctuations. Nevertheless, we are able to measure expectations of vacuum fluctuations.

5.2

Newton's law and self interaction

Let us now introduce a dynamics for the field $^. It will ultimately be the Tchebyscheff dynamics of the previous chapters, just the physical interpretation is slightly different. It should be clear that the dynamics itself is not observable, due to the uncertainty relation, but that expectations with respect to the dynamics should be measurable in experiments. We attribute to each phase space cell i a self-interacting potential that generates the dynamics. For example, we may choose a ^ 4 -theory, where the potential is of the form y ( $ ) = (^M 2 * 2 +

T**4)

rnc2 + C.

(5.5)

In our new physical interpretation $ is dimensionless, m is of the order of the mass of the virtual particles under consideration (say electrons), fi2 and A are dimensionless parameters, and C is an additive constant. The 'force'

98

Stochastic interpretation of the uncertainty relation

in u-direction due to this self-interacting potential is given by F($) =

"c7^y(#)

=(_V$

~ A$3) V

(56)

(the factor 1/(CT) is needed for dimensional reasons, regarding t as physical time). We assume that the change of momentum is given by Newton's law Q-tP = Pmax$ = F{$).

(5.7)

Note that Newton's law is also valid in a relativistic setting, provided t denotes proper time (see, e.g., [Kenyon (1990)]). The smallest time unit of our model of vacuum fluctuations is r . This means that it does not make sense to consider Newton's law for infinitesimally small time differences At, since these would yield infinite energies AE, from AEAt — 0(h). Thus we write eq. (5.7) in the finite-difference form n+1

Pmax*

~*n

= (-»2Zn-\$l)

—. (5.8) r r It is remarkable that the unknown time lattice constant r drops out. For arbitrary r we get a dynamics that is given by the cubic mapping

* n+1 = (i - £ ) *B - ^l

(5.9)

where v := Pmax. The dynamics of the $-field is time-scale invariant, it does not depend on the arbitrarily chosen time lattice constant r . We can obtain a cubic mapping of type (5.9) in the following two very different situations. Either v — 0(1), i.e. pmax = 0(mc), and the potential parameters /x2 and A are of 0(1) as well. In this case we consider a lowenergy model of vacuum fluctuations. We can, however, also let pmax ->• oo, thus considering a high-energy theory. In this case v = Vzm- —^ oo. However, we can get the same finite cubic mapping if at the same time the parameters //2 and A of the self-interacting potential diverge such that fi/u and X/u remain finite. This is just the anti-integrable limit mentioned before. Again there are distinguished parameter values leading to Tchebyscheff maps and thus to strongest possible chaotic behaviour. The negative 3rdorder Tchebyscheff map is obtained from the potential Vi 3 ) (*) = i / ( - $ 2 + $l)mc2

+ C-,

(5.10)

Coulomb forces and Laplacian

coupling

99

the corresponding force is i
7{3)($) = i/(2$2 - f4)mc2 + (7+ 3)

3

FJ ($) = i/(-4$ + 4$ ) —.

(5.11)

(5.12) (5.13)

In fact, we can get any Tchebyscheff map ±Tjv if we consider one of the potentials of section 3.3, multiplied with the energy scale vm
5.3

Coulomb forces and Laplacian coupling

We may associate the strongly fluctuating variables pln with the momenta of charged virtual particles. For example, we may think of electrons and positrons, or any other types of fermions. Actually, we should think of a collective system of such charged particles, similar to a Dirac lake. The following consideration will make it plausible why there should be 1-dimensional Laplacian (= diffusive) coupling between the fluctuating momenta. We will see that within our many-particle interpretation the coupling constant o is related to the strength of an 1/r-potential. Suppose that (for example) $^ represents the momentum of an electron and §ln+x the momentum of a neighbored positron. The Coulomb potential between two opposite charges at distance r = \r\ is Vei{r) = -hca-.

(5.14)

100

Stochastic interpretation

of the uncertainty

relation

a « 1/137 is the fine structure constant. The force (= momentum exchange per time unit At) is

«=> Apei

=

-hca — .

(5.16)

Again, due to the uncertainty relation AEAt — 0(h) it does not make sense to choose an infinitesimally small time unit At. It is more reasonable to choose cAt = r

(5.17)

since photons move with the velocity of light. We then end up with the fact that the Coulomb potential gives rise to the momentum transfer APel

= -ha-

(5.18) r

during the time unit At = r/c. In our picture of vacuum fluctuations, distances Ax and hence also inverse distances 1/r are strongly fluctuating due to the uncertainty relation. In section 5.1 we attributed the strongly fluctuating inverse distance variable | jrpln | to each phase space cell i. The maximum value of the inverse distance, corresponding to the smallest possible distance at a given energy scale pmax, is given by j-pma,x • What should we now take for the inverse interaction distance between two neighbored particles i and i + 1? Obviously, the relevant quantity is the momentum difference between them. According to the uncertainty relation, local momentum differences always correspond to local inverse distances between cells. Hence we define the inverse interaction distance - . .,, between the electron in cell % and the positron in cell i + 1 as the following strongly fluctuating random variable:

i

rPn

n,i+i

2h

l*n + 1 " * n l

(5-19)

The factor | is needed to let the inverse interaction distance not exceed the largest possible value ^pmax of the inverse distance. It follows that the absolute value of the momentum transfer between cell i and cell i + 1 is \APiti+1\

= ha-

=PmaxT^irt1 n,i+i I

- K\

(5-20)

Coulomb forces and Laplacian

101

coupling

The momentum transfer can be positive or negative with equal probability, depending on whether we have equal or opposite charges in the neighbored cells. A possible choice for the signs is to take Ap M + i = f w r | «

+ 1

- K)

(5-21)

This, indeed, causes inhomogeneities of the $-field to be smoothed out: If $*+i > $ ^ <j>« increases. If $ l = $ , + 1 (homogeneity), there is no change at all. If <S>i+1 < $% $* decreases. Similarly, the momentum transfer from the left neighbor is ApM_i = PmaxT^teiT1 ~ K)

(5-22)

Thus A p i + M = PmaXT^(¥n

- &+l) = -Aft.i+1.

(5.23)

The momentum exchange is antisymmetric under the exchange of the two particles, as it should be. We now have a direct physical interpretation associated with the signs of the $-field: If $^ + 1 > $J,, we have opposite charges in cell i and i + 1, causing attraction. Otherwise, the charges have equal signs, causing repulsion. The above approach corresponds to diffusive coupling as generated by the potential W_($,\IJ) (see section 3.4). However, we could also look at a momentum transfer given by —p m a i r^($J 1 _ 1 + $JJ, as generated by W+($, \£). In this case one assumes that the average momentum | ( $ ^ _ 1 + $n)pmax determines the inverse interaction distance l/n-i^. Note that this approach still generates diffusive coupling if at the same time one electron is re-interpreted as a positron, which formally, according to Feynman, has a momentum of opposite sign. So both coupling forms are physically relevant. In total, the momentum balance equation for cell i is Pn+l ^¥n+1

= Pn + Api,i-l + Api.i+1 1

(5.24) 1

= *jl + r|(±$jr -2$*n±$j+ ),

(5.25)

where the ± sign corresponds to diffusive or anti-diffusive coupling, respectively. Remarkably, the momentum cutoff pmax drops out, and we end up with an evolution equation where only dimensionless quantities $, a and T enter. Notice that eq. (5.25) is a discretized diffusion equation with diffusion constant Fa. Also notice that T is just the constant of 0(1) in the

102

Stochastic interpretation

of the uncertainty

relation

uncertainty relation. T _ 1 is the size of the phase space cells in units of h. If we define phase space cells to have size h, this implies T = 1. Then the only relevant constant remaining is the coupling strength a of the 1/r-potential. Finally, we have to combine the chaotic self-interaction and with the diffusive interaction. Since, due to the uncertainty principle, our time variable is effectively discrete, both interactions must alternate. First, in each cell i there is a 'spontaneous' creation of momentum due to the self-interacting potential V± . Vn+1

= ±TN(*in).

(5.26)

Then, momenta of neighboured particles smooth out due to Coulomb interaction. Setting T = 1 we have K+2 = K+i + § (±*n+\ - 2 < + i ± K++\)-

(5-27)

Combining eqs. (5.26) and (5.27) we obtain the coupled map lattice

< + 2 = (1 - a)TN(Vn) ± ICZMSJT1) + TN(K+1),

(5-28)

where Tyy can be either the positive or negative Tchebyscheff map. Indeed, this is just the (forward) chaotic string dynamics introduced before (regarding the two time steps as one). It now has a physical interpretation in terms of a dynamics of fluctuating virtual momenta of many particles in a suitable time coordinate. Most importantly, the new interpretation implies that the coupling constant a = a can now be identified with a standard model coupling constant. At the same time, it can also be regarded as an inverse string metric, using the interpretation of section 2.2. Certainly we can also obtain backward coupling rather than forward coupling. This just means that the Coulomb interaction does not take place simultaneously at all lattice sites, but alternates with the self interaction in i direction. All these chaotic string theories with their various degrees of freedom represent possible models of vacuum fluctuations. Our current formulation emphasizes phase space rather than space and deals with a large number of virtual particles. It has thus similarities with a thermodynamic description of the vacuum (for more details of this aspect, see chapter 6). In contrast to conventional statistical mechanics, where the momentum and position of a particle can be chosen independently, the momentum uncertainty and position uncertainty are not independent from each other but connected through the uncertainty relation. Notice that our interpretation

Duality of

interpretations

103

quantity

Interpretation 11

Interpretation 12

n i

fictitious time (dim GeV~2) space-time coordinate string field self interaction GeV2 — energy/length kinetic term of action inverse metric noise of Parisi-Wu approach 2nd quantization

physical time (dim GeV~L) phase space coordinate momentum uncertainty Newton's law GeV = energy per particle Coulomb interaction standard model coupling uncertainty relation 1st quantization

*»

vm

dim V($) W(*',*,+1) a physical embedding quantization scheme Table 5.1

Two different physical interpretations of the coupled map dynamics.

requires 1-dimensional lattices, since the uncertainty relation is valid for each space direction in an independent way. Moreover, the Coulomb potential just changes the momentum component in one direction, the radial u-direction between the particles. 5.4

Duality of interpretations

The attentive reader may have noticed that we ended up with two different physical interpretations of the chaotic string dynamics, which at first sight look contradictory. The first one (called II) is that of a self-interacting scalar field with very large mass (of the order of the Planck mass or larger) and a very strong self-interacting potential, which rapidly evolves in the fictitious time of the Parisi-Wu approach and generates the noise necessary for stochastic quantization (see section 2.2). The second one (12) is that of a many-particle system where the particles interact by a Coulomb potential at relatively low energies, with fluctuations evolving in a physical time variable n (see previous section). The two different interpretations are compared in Table 5.1. Should we restrict ourselves to just one interpretation, either II or 12? Probably not. As we can learn from the history of physics, it is sometimes useful to have seemingly contradictory interpretations. When quantum mechanics was developed 100 years ago, the famous wave-particle duality arose. The question 'Is an electron a wave or a particle?' turned out to be ill-posed. The correct view, which today we teach to undergraduates in a first course on quantum mechanics, is something like 'An electron is

104

Stochastic interpretation

of the uncertainty

relation

a wave, which, however, in certain circumstances has properties that are better described by the particle picture'. The Schrodinger equation is a well defined mathematical evolution equation, but it does not care which physical interpretation is given to the mathematical symbols by human beings. Similarly, the chaotic string dynamics is a well-defined mathematical evolution equation, but it does not care about which physical interpretation of the mathematical symbols we find to be the most appealing. Our answer to the question of what this chaotic dynamics physically describes could be 'A chaotic string is a strongly fluctuating scalar field at or beyond the Planck scale, which, however, in certain circumstances has properties that are better described by that of a many-particle system interacting at relatively low energies'.

5.5

Feynman webs

Let us now further work out interpretation 12 and proceed to a more detailed physical interpretation of the chaotic string dynamics. Remember that in this interpretation we regard $^ to be a fluctuating momentum component associated with a hypothetical particle i at time n that effectively lives in a 1-dimensional space. Neighbored particles i and i — \ exchange momenta due to the diffusive coupling. A more detailed physical interpretation would be that at each time step n a fermion-antifermion pair / i , / 2 is being created in cell i by the field energy of the self-interacting potential. In units of some arbitrary energy scale Pmax, the fermion has momentum <J>^, the antifermion momentum —$*n. They interact with particles in neighbored cells by exchange of a (hypothetical) gauge boson Bi, then they annihilate into another boson B\ and the next chaotic vacuum fluctuation (the next creation of a particleantiparticle pair) takes place. This can be symbolically described by the Feynman graph in Fig. 5.1. Actually, the graph continues ad infinitum in time and space and could thus be called a 'Feynman web', since it describes an extended spatio-temporal interaction state of the string, to which we have given a standard model-like interpretation. The important point is that in this interpretation a is a (hypothetical) standard model coupling constant, since it describes the strength of momentum exchange of neighbored particles. At the same time, a can also be regarded as an inverse

Feynman webs

Fig. 5.1

105

Feynman web interpretation of the coupled map dynamics.

metric in the 1-dimensional string space, since it determines the strength of the Laplacian coupling. It is well known that standard model interaction strengths actually depend on the relevant energy scale E. We have the running electroweak and strong coupling constants. What should we now take for the energy (or temperature) E of the chaotic string? A priori this is unknown. However, in chapter 7 and 8 we will present extensive numerical evidence that minima of the vacuum energy of the chaotic strings are observed for certain distinguished string couplings cjj, and these string couplings are numerically observed to coincide with running standard model couplings, the energy (or temperature) being given by E = -N(mBl + mh +mh).

(5.29)

Here N is the index of the Tchebyscheff map of the chaotic string/theory considered, and m ^ , m ^ , m,f2 denote the masses of the particles involved in the Feynman web interpretation. The surprising observation is that rather than yielding just some unknown exotic physics, the chaotic string

106

Stochastic

interpretation

of the uncertainty

relation

spectrum appears to reproduce the masses and coupling constants of the known quarks, leptons and gauge bosons of the standard model (plus possibly more). Formula (5.29) formally reminds us of the energy levels EN = y ?iw of a quantum mechanical harmonic oscillator, with low-energy levels (TV = 2,3) given by the masses of the standard model particles. In the Feynman web interpretation of Fig. 5.1, the formula is plausible. We expect the process of Fig. 5.1 to be possible as soon as the energy per cell i is of the order m s j + m/ t + m/ 2 . The boson B2 is virtual and does not contribute to the energy scale. The factor N can be understood as a multiplicity factor counting the number of degrees of freedom. Given some value $^ of the momentum in cell i, there are N different pre-images T / ^ 1 ($^) how this value of the momentum can be achieved. All these different channels contribute to the energy scale.

5.6

Physical interpretation of discrete string symmetries

We saw that there are several degrees of freedom in the choice of the dynamics of the chaotic string. Often, there are just two choices. For example, we can choose between either V+ or V_, between either W+ or W-, between forward or backward coupling, between odd N and even N, and so on. It is natural to try to find a physical interpretation for these degrees of freedom, using interpretation 12. The following list yields a useful guiding principle of what various discrete symmetry transformations mean, using interpretation 12. Of course, in interpretation II the same symmetry transformation may have a completely different physical meaning. Our statements will be confirmed by a large number of examples studied in later chapters of this book. The transformation

(or s -» -s) can be interpreted as replacing particle-antiparticle interaction by particle-particle interaction. This is related to the CP (charge parity) transformation. In fact, according to Feynman a particle of momentum $Jj corresponds to an anti-particle of momentum - $ j , . Replacing

Physical interpretation

of discrete string

symmetries

107

W- by W+ just means changing the sign of one of the momenta of the interacting particles.

y_-> v+ could be interpreted as changing the direction of time (either physical

or fictitious). This is related to the T transformation in quantum field theories. If t is fictitious time, this transformation can also be related to a supersymmetry transformation [Gozzi (1983)].

even N ->• odd N Even ^-theories could yield basic information on bosons, odd ./V-theories on fermions. In fact, the relations (V_) = (V+) for the T 2 ,T4,T 6 ..., and (V_) = -(V+) for the T$,T<s,Ti..., which we derived in section 3.3, somewhat remind us of commutator and anti-commutator relations, though this analogy is just formal. Remember that any fermion has an anti-particle with equal properties, whereas a boson doesn't. This coincides with the observation that under the transformation W- —> W+ we basically get the same dynamics for a string with odd N, up to a sign, whereas for even N a completely different theory arises.

forward coupling -» backward coupling (or A -» B). These two possibilities correspond to whether either the chaotic self interaction (denoted as 'c') due to V± takes place in all cells i at the same time, then being followed by diffusive coupling (denoted as 'd'), or whether there is an alternating pattern between those states. Forward coupling can be represented by a pattern of the form cccc dddd cccc dddd, backward coupling by cdcd d cdc cdcd d c d c.

108

Stochastic interpretation

Table 5.2 strings.

of the uncertainty

relation

transformation

physical interpretation

s —> —s V ->-V even N -¥ odd N A -^B

CP (at odd i) T boson —• fermion pure state —> mixed state

Physical interpretation of discrete symmetry transformations for chaotic

Here rows correspond to the phase space cells (or particles) i and columns to the time variable n. Actually, the physical time unit consists of two time steps cd in the above patterns (see eqs. (5.26),(5.27)). Our consideration makes it plausible that the forward coupling form usually describes just one particle state (d), whereas the backward form can describe mixtures of two particle states (c and d), co-existing at the same time n. It is also possible to associate the forward/backward degree of freedom with spins. If there are two particles (backward coupling), their spins can add up to a spinless state. If there is just one particle (forward coupling), the spin is generically non-zero. Table 5.2 summarizes the main results. Naturally, a chaotic string is CPT invariant, since the transformation s —»• — s is reversed by replacing V —• —V at odd lattice sites. For possible breaking of the CP symmetry, see section 9.4.

5.7

Fluctuations of the metric and a 1 + 1 dimensional model of quantum gravity

The reader may have noticed that our consideration in section 5.3 does not necessarily require a Coulomb potential

V(r) = -ahc-

(5.30)

Fluctuations

of the metric and a 1+1 dimensional

model of quantum gravity

109

but any 1/r potential can do the job. So we could also consider a gravitational potential V(r) = G^ r

= he (-=-V \mpij

•1 r

(5.31)

as being responsible for the interaction of nearest neighbors. Here G = hc/mpl is the gravitational constant and m denotes the mass of the particles at each lattice site. Basically the same consideration as in section 5.3 does apply to this case, and one ends up with the same class of coupled map lattices. Still we have to decide what the physically relevant value of the spatial coupling constant a of the coupled map lattice is in this case. As a first guess, comparison of eq. (5.30) and (5.31) would suggest that the fine structure constant a should be replaced by the dimensionless gravitational coupling (m/mpi)2. However, for electric interaction two different processes are described by the same coupling constant a. In case of an attractive force, there can either be an electron at position i and a positron at position i +1, or a positron at position i and an electron at position i + 1 . The simultaneous action of both processes had led us to identify a = a. For gravity, however, there is just one such process, namely a mass m at position i and yet another mass m at position i + \. This suggests that the physically relevant coupling occurring in the coupled map dynamics has a factor 1/2 as compared to the electric case, i.e. a = aG = \(^-)\

2

(5.32)

\mpij

We will see in later chapters that this is indeed the correct formula. For electric interaction there can be both attracting and repelling forces, depending on whether we have opposite or equal charges at neighbored lattice sites. Gravitational interaction, however, is always attracting. So how can the momentum transfer in our coupled map dynamics be positive and negative if the interaction potential is assumed to be the gravitational potential? One solution to this problem is to assume that the physical time direction is yet not specified in our model, it fluctuates as well. Clearly an attracting force with positive time evolution looks like a repelling force in the opposite time direction. Also, if we allow for vacuum fluctuations that can create black holes as well, then the Schwarzschild metric switches sign

110

Stochastic interpretation

of the uncertainty

relation

at the event horizon of the black hole. Both signs are possible, depending on whether we are inside or outside the Schwarzschild radius. The time-component g00 of the Schwarzschild metric is given by 200 = 1 - ^

= 1 - ^ ^ )

(5.33)

(see any textbook on general relativity, e.g. [Kenyon (1990)]). Recalling that in our model of vacuum fluctuations the inverse distance 1/r fluctuates and is given by 1

=

ri,i+i

P ^ * | ^

±

^

(5.34)

( }

2n

the spatio-temporal chaotic dynamics can be interpreted as representing quantum fluctuations of a metric. One obtains a fluctuating 500 given by 2Gm 900 =

?v~77

Tnpmax

=1

n

t

~~^~V^

-i+ii

^ '

/co^ (5 35)

'

''lpi^

t- '1,1+1

The spatial component
(5-36)

It also fluctuates rapidly. In this sense, we see that our chaotic string models can also be regarded as l+l-dimensional models of quantum gravity where suitable components of a metric tensor fluctuate in a deterministic chaotic way. Assuming that the Schwarzschild metric is the correct one to use, the fields $Jj are directly related to the fluctuating metric by eq. (5.35) and (5.36). The unstable fixed points $*(a) displayed in Fig. 2.7 in chapter 2 can be related to some highly unstable stationary states of this metric. The singularities of $*(a) near a = | and a = 1 could possibly be interpreted in terms of singularities that occur at the event horizons of black holes. It clearly makes sense to define expectations of this strongly fluctuating unstable metric. One obtains for the temporal component W ^ - ^ r d ^ i ^ 171

and for the spatial one

PI

C

1

! )

(5.37)

Fluctuations of the metric and a 1+1 dimensional model of quantum gravity 111 If Pmax is chosen as vac then the expectation of the deviation from the Minkowski metric is given by (3oo)-l = - 2 a G - ( | ^ ± ^ + 1 D

(5-39)

Alternatively, one can define an average metric goo based on the standard deviation of the fields $ln: _

= 1

_^J

( ( $

i

± $

W

) 2 )

=

771 p i C

! _ !^pV2(cI>2)±2<W+i). 771 p i C

(5.40) This yields for o « 0 and pmax — "ic 771

55b" = 1

o- = 1 - 2a G = 1 - 2a.

m

(5.41)

pi

In this sense, the coupling constants a of the coupled map lattices can be regarded as the average of a strongly fluctuating metric or inverse metric, up to some suitable multiplicative and additive constants.

This page is intentionally left blank

Chapter 6

Generalized statistical mechanics approach

We will work out the thermodynamic aspects of our turbulent models of vacuum fluctuations in more detail. This approach provides a more general physical embedding for chaotic strings and points out connections with statistical mechanics. Clearly the chaotic string dynamics, as any chaotic dynamics, produces information. It is irreversible and distinguishes a particular direction of time. The self energy can be regarded as thermodynamic potential, similar to an entropy function of the vacuum. Extrema of the entropy function correspond to states of maximum information that we have on the state of the vacuum. The invariant density of the TchebyschefF maps can be regarded as a canonical distribution in a generalized version of statistical mechanics, so-called nonextensive statistical mechanics.

6.1

Heat bath of the vacuum

Let us consider the vacuum as a kind of heat bath of virtual particles. We want to develop a model of vacuum fluctuations using concepts from statistical mechanics and information theory. First, consider ordinary statistical mechanics. Given a system of Np classical particles with Hamiltonian

113

Generalized statistical mechanics

114

approach

the probability density p to observe a certain microstate {q\,..., •••,PNP)

••=

(q,P)

gjvp, Pi, • • •

is

p(q,p) = ~)e-pHiq'p).

(6.2)

We assume that Boltzmann statistics is applicable. /J = 1/kT is the inverse temperature and Z((S) = /" dgi • • • dqNpdp-[ • • • dpNve^H^^

(6.3)

is the partition function. The internal energy U is denned as the expectation of if: c?
• • • dpNpp{q,p)H(q,p)

= - — log Z(/?)

(6.4) If we want to develop a thermodynamic description of the vacuum we need a statistical theory of vacuum fluctuations Aqi and Api allowed by the uncertainty relation. The situation, however, is different from ordinary statistical mechanics because the momentum and position variables Aqi,Api cannot be chosen independent from each other, as for ordinary statistical mechanics. If we choose a certain Api then Aqi = h/Api is already fixed. Moreover, since virtual momenta violate energy conservation (they are just defined as doing that), we cannot expect to have an ordinary Hamiltonian H(q,p) as in classical mechanics. If anything, the dynamics should be dissipative. We also have to decide what 'temperature' means for the vacuum. It appears most reasonable to identify kT ~ E with the energy scale E at which we look at the vacuum. Then qmin = O(ff0 is the smallest spatial scale resolution we can achieve at this temperature, and pmax = —— is the Qmin

maximum momentum. As worked out in chapter 5, the relevant information on the state of a phase space cell of size h is assumed to be given by a field variable $ln, which is the momentum uncertainty Api = p\ in units of pmax at time n in cell i. The corresponding position uncertainty is x%n = h/pln. Since the vacuum is isotropic, the direction in which we measure the momentum is irrelevant. If there are d spatial directions, the d components ApXl,..., ApXd of the momentum uncertainty into the d space directions are expected to be independent from each other. In empty space, we do not

Heat bath of the

vacuum

115

expect any interactions between ApXl and ApX2 for two different directions. Rather, 1-dimensional models are expected to do a good job. We can either construct models where $^ is a pure random field or where there is an underlying chaotic dynamics. The second type of models, which leads to chaotic strings in a natural way, reproduces observed standard model parameters with very high precision (see chapter 7 and 8) and thus appears to be the physically relevant one. We do not have a true Hamiltonian for the chaotic string dynamics since the dynamics is dissipative. But we can write down a kind of analogue of a Hamiltonian given by

H = Y/V±($i)+aW±{¥,$i+1),

(6.5)

i

with V acting first, then followed by W. V and W are the potentials introduced in chapter 3. Due to the uncertainty relation, the time variable is effectively discrete with a lattice constant of order At = h/E. In order to define an internal energy of vacuum fluctuations similar to eq. (6.4) we thus have to decide whether we relate it to V at one time step, to W at the next time step, or to the sum of both averaged over both time steps. These degrees of freedom are absent in classical statistical mechanics, where the time evolution is continuous. All three types of vacuum energies will turn out to be important (see chapter 7,8,9). The equilibrium distributions, replacing the canonical probability distributions of ordinary statistical mechanics, are the invariant densities p($ 1 , $ 2 , . . . ) of the coupled map dynamics. In contrast to ordinary statistical mechanics, there is no simple analytic expression for them, except for the uncoupled case a = 0, where we have

i 2 ($ P ,$ ,...)=n / 2-

(6-6)

These types of densities can be dealt with in the formalism of nonextensive statistical mechanics (see section 6.4). Generally, the invariant densities depend on the coupling a in a non-trivial way. All averages are formed with these densities. For ergodic systems the ensemble averages can be replaced by time averages. Note that a dynamics generated by a Tchebyscheff map does not have a unique inverse, hence an arrow of time arises in a natural way. This arrow of time of the heat bath of the vacuum may help to justify the arrow

116

Generalized statistical mechanics

approach

of time in ordinary statistical mechanics. Whereas classical mechanics is invariant under time reversal, the dynamics of the vacuum fluctuations considered here is not. Hence in our approach the arrow of time already enters at a microscopic quantum mechanical level (see also [Mackey (1992); Veneziano (1999)]). Depending on the physical embedding, this can either be physical time or the fictitious time of the Parisi-Wu approach.

6.2

* States of maximum information

The quantity (V($))(a) = V(a) measures the self energy of the vacuum per phase space cell (or per virtual particle). It can be regarded as a kind of thermodynamic potential of the vacuum. Numerically it is obtained by iterating the coupled map lattice (2.1) for random initial conditions and averaging V+ ($ln) over all n and i (disregarding the first few transients). Numerical results for V{a) will be presented in detail in chapter 8. It turns out that this function typically varies smoothly with a but has lots of local minima and maxima. What is the physical interpretation of such an extremum? First of all we notice that if a is interpreted as a standard model coupling, as suggested by our consideration in chapter 5, then a change of energy E implies a change of a, since a — a{E). We have the so-called running standard model couplings. The reason for this energy dependence are vacuum polarization effects (see any text book on particle physics or quantum field theory). In a perturbative approach, the form of the energy dependence of the standard model couplings is well known, and is given by the so-called beta functions. Looking at this the other way round, a change of a implies a change of temperature kT ~ E, since a = a(E). At an extremum of V{a) we have |;V(a)=0.

(6.7)

d _ d dE d^-dE^

(6 8)

Since "

and ~

= ^(b0a2

+ha3+

•••),

(6.9)

* States of maximum

117

information

where the bi are the coefficients of the beta function of the running standard model coupling, we obtain dV _ E dV da ~~ b0a2 + b1a3 + ---'dE' Similarly, if a := ^(E/mpi)2 E/m2P[ and

(

is a gravitational coupling, we obtain da/dE

da

E 8E

y

'

' =

'

At an extremum one has ^ = 0. This means that although we change the temperature E ~ kT, the self energy of the vacuum per virtual particle does not change. This reminds us of a phase transition in thermodynamics. Note that for this consideration to hold, a can be either a standard model or a gravitational coupling , as long as it is energy dependent. Let us give a more detailed interpretation of what is happening at a local minimum. Suppose, for example, that a = as is a strong coupling. If we increase the temperature then a decreases. Let us approach a local minimum of V(a) from the right by increasing the temperature. Since V(a) decreases, the self energy of the vacuum per virtual particle decreases. We could also re-interpret this fact in the sense that the total self-energy of the vacuum stays constant but that the number of virtual particles that make up the vacuum increases. This effect is reversed at the critical point where V(a) has a minimum. Here the self-energy per virtual particle V(a) suddenly starts increasing again. If we again assume that the total self-energy is constant this means that the vacuum starts loosing virtual particles. This can be interpreted as being due to the fact that some virtual fermion antifermion pairs fa, fa and the corresponding boson B\ become real particles at that temperature. They leave the vacuum, hence the energy per virtual particle in the vacuum will start increasing. We expect the threshold points where the particles become real to occur at energies E = yfcT, where kT = m,Bx + rafx + m/ 2 and N, in analogy to classical thermodynamics, can be interpreted as the number of 'degrees of freedom'. We will see in the next chapters that indeed N coincides with the index of the chaotic string theory considered. We may also interpret the self energy V(a) as a kind of entropy function of the vacuum. Clearly the Tchebyscheff maps, as any chaotic maps, produce information when being iterated. Or, looking at this the other

118

Generalized statistical mechanics

approach

way round, information on the precise initial value is lost in each iteration step due to the sensitivity on initial conditions. The potential V($) generates the chaotic dynamics and hence could be formally regarded as a kind of information potential. Its expectation measures the missing information (^entropy) we have on the particle contents of the phase space cells. At a minimum of V(a) we have minimum missing information. In other words, we have maximum information on the particle contents of the cells. Hence we can associate the dynamics with a particular Feynman web at this point, and a should then coincide with the corresponding standard model coupling.

6.3

* States of minimum correlation

Detailed numerical results for the interaction energy W(a) of chaotic strings will be presented in chapter 7. One observes that W(a) typically varies smoothly with a, and that it vanishes at certain distinguished couplings Oi ^ 0. States of the vacuum with vanishing interaction energy W(a) = ± | ( $ * $ I + 1 ) are clearly distinguished — they describe, in a sense, a ground state of the vacuum where the absolute value of the interaction energy has a minimum. A zero of the interaction energy also means that the correlation between the momenta of neighbored particles vanishes, meaning that we can clearly distinguish the particles in the various phase space cells, so that again a Feynman web with a definite particle contents makes sense. If a standard model coupling is chosen to coincide with a zero of the interaction energy, then this clearly represents a distinguished state of the heat bath of the vacuum, with a vanishing spatial 2-point function just as for uncoupled independent random variables, well suitable for stochastic quantization. We also have to consider the stability of a certain state of the vacuum with zero interaction energy. Imagine that the coupling constant a of the chaotic string changes slightly, then this change induces a change of the interaction energy of the vacuum. This change of the interaction energy is expected to influence the temperature E ~ kT, and this will once again induce another change of the running standard model coupling a(E). If the coupling constant is driven back to its original value, then the state of the vacuum is stable. We could also interpret the correlation function as describing the polarization of the vacuum. Suppose, for example, that $^ represents a momen-

Nonextensive

statistical

mechanics

119

turn component of an electron. Then, according to Feynman, — $J, could be interpreted as the momentum component of a positron. A negative correlation function ( $ ^ $ 5 J + 1 ) means that if there is an electron in cell i, then with slightly larger probability there is a positron in cell i + 1, since the expectation of the product ^ln^1^'1 is negative. A zero of the correlation function thus means the onset of vacuum polarization. Again we expect the threshold points where vacuum polarization sets in to occur at the energy E = ^kT, with kT = mBx + m/ x + m/ 2 . To summarize, our physical arguments based on statistical mechanics emphasize the importance of stable zeros of W(a) and of local minima of V(a). We used interpretation 12 for our physical argumentation (see section 5.4 for a definition of interpretation II and 12). Further arguments based on interpretation II will be given in sections 7.1 and 8.1.

6.4

Nonextensive statistical mechanics

We now show that the statistical mechanics aspects of chaotic strings can be embedded into the formalism of so-called nonextensive statistical mechanics. This is a generalization of ordinary statistical mechanics due to C. Tsallis [Tsallis (1988); Tsallis (1999); Abe et al. (2001)]. The generalized formalism is particularly useful to describe systems with long-range interactions, multifractal behaviour and fluctuations of temperature or energy dissipation. In particular, it can be successfully applied to nonequilibrium systems with a stationary state [Wilk et al. (2000); Beck (2001b); Cohen (2002)]. Let us sketch the main idea of the new formalism. It really goes back to the foundations of statistical mechanics. Whereas ordinary statistical mechanics is usually derived by extremizing the Shannon entropy S = — ^ j Pi In pi (subject to constraints), in nonextensive statistical mechanics the more general Tsallis entropies

s

« = 7=T (* - X>?)

(6 12)

-

are extremized. The pi are probabilities associated with the microstates of a physical system, and q ^ 1 is the so-called entropic index. The ordinary

120

Generalized statistical mechanics

approach

Shannon entropy is obtained in the limit q —>• 1: lim Sq = S

(6.13)

g->l

Hence ordinary statistical mechanics is contained as a special case in this more general formalism. The Tsallis entropies are closely related to (but different from) the Renyi information measures [Renyi (1970)] and have similarly nice properties as the Shannon entropy has. They are positive, concave, take on their extremum for the uniform distribution, and preserve the Legendre transform structure of thermodynamics. However, they are non-additive for independent subsystems (hence the name 'nonextensive' statistical mechanics). In a sense, they can be regarded as the 'second-best' information measures just after the Shannon entropy [Abe (2000)]. If a physical system, for some reason, cannot extremize the Shannon entropy then it is natural to assume that it will then choose the second-best information measures. In the mean time various physical applications have been reported for the formalism with q ^ 1. Examples are hydrodynamic flows exhibiting fully developed turbulence [Beck (2000a); Beck et al. (2001); Arimitsu et al. (2000); Ramos et al. (2001)], heavy ion collisions [Alberico et al. (2000)] and e+e~~ annihilation experiments [Bediaga et al. (2000); Beck (2000b)]. Possible theoretical implications for early universe physics were discussed in [Tsallis et al. (1995); Plastino et al. (1995); Pennini et al. (1995); Tirnakli et al. (1999)]. If one performs just the same extremization principle that one usually uses to 'derive' statistical mechanics, but replaces S by the more general Sq, one ends up with more general versions of canonical distributions. These are given by Pi=^(l + (q-l)peiy^,

(6.14)

Z, = X)(l + (
(6.15)

where

i

is the partition function, e; are the effective energy levels of the microstates and /3 — 1/kT is a suitable inverse temperature variable. Slightly different versions of the generalized formalism are known [Tsallis et al. (1998)], but they all essentially lead to eq. (6.14) after suitable re-definitions. In the limit

Nonextensive

statistical

mechanics

121

q -> 1, ordinary statistical mechanics is recovered, and the pi just reduce to the ordinary Boltzmann factors pt ~ e _/3£i . For q ^ 1, on the other hand, one ends up with generalized Boltzmann factors and generalized versions of the canonical ensemble. All this is a kind of g-deformed version of statistical mechanics. There are indeed some formal analogies with ^-deformations for quantum groups [Johal (1999); Tsallis (1994); Majid (1995)]. Given a set of probabilities pi one can proceed to another set of probabilities Pi defined by Pi = ^ c H -

(6.16)

These distributions Pi are called escort distributions [Beck et al. (1993)] and turn out to be useful in the generalized formalism. Of course, for ordinary statistical mechanics with q = 1 one always has Pi = pi. Now let us come to chaotic strings and turbulent quantum states. If the coupling constant a of the string vanishes, then the invariant 1-point density

of the Tchebyscheff maps can be regarded as a generalized canonical distribution in the above formalism, with q = 3. The effective energy e($) of a local momentum uncertainty $ is formally given by the (non-relativistic) kinetic energy expression c($) = \&

(6.18)

If there is a small spatial coupling a, there are small corrections to this effective energy (Fig 6.1). In the generalized thermodynamic framework we may either work with the original distribution p(*) = -^-(l + ( g - l ) 0 e ( $ ) ) - 5 ± r

(6.19)

or with the escort distribution P($) = - i ( l + ( g - l ) 0 e ( * ) ) - i ^ . 4q

(6.20)

Generalized statistical mechanics

122

i

l

approach

•

l

i

-

0.4

if

-

0.3

\ 0.2

-

-

%'•

0.1

^-0 -0.8

-0.6

-0.4

-0.2

0.2

i

i

'

0.4

0.6

0.8

Fig. 6.1 Effective energies e($) as obtained from the single-site density of the 2B string for a = 0,0.00007625,0.000305,0.00122, respectively (from bottom to top).

In the former case q — 3 and /3 = — 1, i.e. we have to work with negative temperatures, whereas in the latter case q = — 1 and /? = 1, i.e. a positive temperature is possible. Alternatively, for q = 3 we can also work with complex momenta i<£ instead of negative temperatures. These, for a = 0, generate negative energies e = —1$ 2 . Both approaches, q — 3 and q — — 1, are possible and yield the same invariant probability density. For small coupling a ^ 0 the invariant density and hence the effective energy levels e($) slightly deform, but still a thermodynamic description with q = 3, respectively q = — 1 makes sense. The effective energies e($) as shown in Fig 6.1 are obtained by writing the observed single-site invariant density of the coupled map lattice as p($) = -K~1(1 — 2e($))~ 1 / 2 , which can be solved for e($) to give V

'

2

27r2p($)2

(6.21)

As can be seen in Fig 6.2, increasing the coupling a by a factor 4 results in the fact that the distance from the uncoupled energy function e($) = i $ 2

123

Nonexstensive statistical mechanics

0.08

'a=0.00007625' 'a=0.000305' 'a=0.00122'

0.06

0.04

t^Sr/*ert^w#'**

/is

*.

/ \ /

I

^^A**^ \

/ V

/iV \

-0.02 -0,4

-0.2

0.2

0.4

0.6

Fig. 6.2 Difference e($) - | $ 2 for the 2B string. The spatial coupling is given by a, = 0.00007625,0.000305,0.00122, respectively (change by a factor 4).

increases by a factor 2. But besides that the shape of the deviation stays the same if a mod 4 is kept constant (see also section 8.6). Increasing a only slightly, the various local maxima of the deviation move with almost constant velocity to the left, similar to the movements of solitons (Fig 6.3). One can now use the known tools from nonexteasive statistical mechanics to work out a thermodynamic description of chaotic strings. In fact, most formulas used in ordinary thermodynamics have simple generalizations for 9 ^ 1 (see [Tsallis (1999); Abe et al. (2001)] for reviews). These formulas can be applied here, and can be used to define a pressure, specific heat etc. for the vacuum. Whether this is just a formal approach or whether there is physical meaning behind such definitions must still be clarified. We will not go into details here, but just refer to the g-generalized formulas in [Tsallis (1999); Abe et al. (2001)].

Generalized statistical mechanics

124

approach

0.08

'a=0.000244' 'a=0.000305' 'a=0.000381'

0.06

0.04 in o

© 0.02

-0.02 -0.2

0.6

0.8

Fig. 6.3 Traveling local maxima of the effective energy deviation e(<E>) — |<E> . The coupling is a = 0.000244,0.000305,0.000381, respectively (change by a factor 1.25).

6.5

Energy dependence of the entropic index q

In the physical applications for nonextensive statistical mechanics that have been reported so far, the entropic index q is often observed not to be constant but to depend on the spatial scale r. For example, in fully developed hydrodynamic turbulence, q is about 3/2 at the smallest scale (the Kolmogorov length scale) and decreases to g w 1 at the largest scales [Beck (2001a); Beck et al. (2001)]. In e+e' annihilation experiments [DELPHI coll. (1997); TASSO coll. (1984)], experimentally measured differential cross sections are very well fitted by formulas derived from nonextensive statistical mechanics (see Fig. 6.4), with a q given by 1(E) =

11 - e - g / g ° Q

+

e-E/B0

•

(6.22)

In this case, e in eq. (6.19) is given by the relativistic energy-momentum relation, and / 3 _ 1 is given by the Hagedorn temperature [Hagedorn (1965)], a kind of 'boiling' temperature of nuclear matter. In eq. (6.22) E is the

Energy dependence of the entropic index q

0.1

1 PT

125

10

[GeV\

Fig. 6.4 Differential cross sections of transverse momenta as measured by the TASSO and DELPHI collaboration in e+e~ annihilation experiments at various center-of-mass energies, and comparison with predictions from nonextensive statistical mechanics with q given by eq. (6.22). Details in [Beck (2000b)].

center of mass energy of the beam, and EQ is a constant (of the order of half of the Z° mass). The spatial scale r that is seen by the experiments is given by r ~ h/E, so q decreases as a function of the scale, since q(E) in eq. (6.22) is a monotonously increasing function of E. Formula (6.22) can be derived using some plausible theoretical arguments based on the existence of moments [Beck (2000b)]. For a limited range of energies E between 14 and 161 GeV the formula is experimentally confirmed (see Fig. 6.4). Moreover, taking for E a much smaller energy, namely the recombination temperature, it also reproduces the correct order of magnitude of fluctuations in the cosmic microwave background [Beck (2002)]. Now we can of course evaluate this formula for all kinds of energies E, though we don't have experimental confirmation for this general

Generalized statistical mechanics

126

approach

validity. For E -t - c o , we obtain from eq. (6.22) the asymptotic entropic index q — — 1. Note that this value is obtained in a way that is completely independent from our previous consideration based on the invariant density of chaotic strings. Since we may regard the vacuum as a kind of Dirac lake with negative energies, formula (6.22) seems to indicate that a high-energy statistical mechanics theory of the vacuum should be one where the entropic index is q = — 1. This is indeed satisfied for our model based on chaotic strings. Next, we can also perform the limit E —> oo. We obtain the asymptotic value q = 11/9. Can we also give a physical interpretation to this value of q? Yes, we can. But for this we still have to learn a little bit more on the nonextensive formalism. The necessary prerequisites will be provided in the next section.

6.6

Fluctuations of temperature

Generally, statistics with q > 1 can arise from ordinary statistical mechanics (with ordinary Boltzmann factors e~@€i) if one assumes that the inverse temperature /3 is fluctuating [Wilk et al. (2000); Beck (2001a); Beck (2001b)]. To see this, consider the integral representation of the gamma function /•OO

T(z) = / Jo

e-H'^dt.

(6.23)

Substituting

i=

"le' +

ra>

(624)

and (6.25) one may write /•OO

(H-(
e-^fWdp,

(6.26)

Fluctuations

of

temperature

127

where

/w

= F(X){™} A ' s *" lexp {-ra}

(6 27)

-

is a probability density, the so-called \ 2 distribution (or gamma distribution). The physical interpretation of eq. (6.26) is that due to fluctuations of p with probability density /(/?) the Boltzmann factor e _ / 3 e i of ordinary statistical mechanics is effectively replaced by the generalized Boltzmann factor (l + (q — l)pVi)~5" 1 of nonextensive statistical mechanics. The generalized canonical distribution with average inverse temperature Po arises by integrating over all possible fluctuating inverse temperatures p\ The x2 distribution is well known to occur in many very common circumstances. For example, if one has n independent Gaussian random variables Xj, j = 1 , . . . ,n, with average 0, then, if j3 is given by the sum n

it has the probability density (6.27) with q-l

= -. n

(6.29)

The average of 0 is given by rOO

(P) = n(X2) = / Pf(PW Jo and the relative variance of P is given by

^L=A-1

= Po

(6.30)

(6 3l)

For n —> oo (or q —> 1), P becomes sharply peaked, all fluctuations of the n Gaussian random variables Xj average out, and we end up with ordinary statistical mechanics with q = 1. In that limit P is not fluctuating at all. For finite n, however, there are fluctuations. The physical idea behind these fluctuations is as follows. Partition the d-dimensional physical space into cubes of equal volume V ~ r d , where r is the scale we are interested in. For a nonequilibrium system with a stationary state each cube can have a different inverse temperature at different times. We assume that the probability to find a certain P in a given cube is given by f(P)dp.

128

Generalized statistical mechanics

approach

Moreover, we assume that each cube is in local equilibrium once a certain /? has been chosen, i.e. j3 changes on a long time scale. A test particle moves through these different cubes and in the long-term run it is described by nonextensive statistical mechanics with q = 1 + 2/n, since we have to average over the various fluctuating /?. This is just the physical meaning of eq. (6.26). It is now reasonable to assume that at the smallest possible scales r (described by the largest possible energy E) the number n of independent contributions X, to the fluctuating (3 is given by the number d of spatial dimensions of the space, since the momenta whose energies define the local temperature in the cubes can fluctuate in each space direction independently. By definition, there is no structure or degree of freedom within the smallest possible cubes other than dimensionality. Hence we obtain 2 q- 1= -

(at smallest scales).

(6.32)

q = 11/9, as obtained from formula (6.22), thus implies d — 9 at the smallest scales. 9 spatial dimensions mean that space-time is 10-dimensional. Remarkably, we get the number of space-time dimensions that are necessary for superstring theory to be formulated in a consistent way [Green et al. (1987)]. The fitting of the e+e~ annihilation data actually already shows a crossover from q « 1 to q « 11/9 at the electroweak energy scale of about 100 GeV [Bediaga et al. (2000); Beck (2000b)]. Should the saturation of q near the value 11/9 be confirmed by future experiments with larger center of mass energies E > 161 GeV, this could be interpreted as possible evidence that the compactified dimensions of superstring theory start already becoming visible at the electroweak scale. This means, their diameter would be of the order of r ~ (lOOGeV) -1 , much larger than the Planck length. This is compatible with other theoretical arguments, which also suggest relatively large sizes of the extra dimensions [Antoniadis et al. (1999a); Antoniadis et al. (1999b)]. We can also apply our dimensional consideration to the value q = 3 describing the chaotic string with negative energies (or complex momenta). q = 3 implies d = 1, using eq. (6.32). Thus we end up with the following possible picture. What the supersting is in a 9+1-dimensional space-time for E -¥ oo, could be the chaotic string in a 1 + 1-dimensional space-time for E -»• - c o . Superstrings describe positions of real particles, chaotic strings

Klein-Gordon field with

fluctuating

momenta

129

describe momenta of virtual particles.

6.7

Klein-Gordon field with fluctuating momenta

We may apply the considerations of the previous section to a massless KleinGordon field in d space-time dimensions. Stochastic quantization yields for the momentum amplitude 4>(k,t) of the field the stochastic differential equation — j>=-k2j> + L(k,t)

(6.33)

(see section 1.4). Here A; is a d-dimensional energy-momentum vector. One has k2 = kl + kl + ... + h?d,

(6.34)

where the kj are the components of k. lit is the fictitious time of the ParisiWu approach, then d is the number of space-time dimensions. On the other hand, if t is physical time (e.g. in some model with thermal noise), then d is the number of spatial dimensions. Both types of models make sense. If L(k,t) is Gaussian white noise of some fixed variance, then the stationary probability density of the field <j> is a Gaussian distribution with average 0 and variance depending on k2, p(0)~exp{-^202}-

(6.35)

Now assume that the momentum components kj are not fixed any more but fluctuate as well. Suppose they are given by independent Gaussian random variables with average 0. Then the formalism of the previous section applies. We may formally define /?:=fc 2 = £ > , 2 .

(6.36)

j=i 2

P is now a x -distributed random variable of degree n = d. In order to obtain the stationary probability density of in this fluctuating model, we have to integrate the Gaussian distributions (6.35) over all possible /?, with j3. weighted by the x 2 distribution. By this mechanism the stationary probability density of <> / now becomes a ^-generalized canonical distribution.

130

Generalized statistical mechanics

approach

The relevant Boltzmann factors are given by (6.26), with q — 1 = 2/n = 2/d and €i = \(j>2 • For this we have to assume that j3 fluctuates on a relatively long time scale, so that the local equilibrium distributions (6.35) are reached before /? changes again. For 1-dimensional systems (d — 1), where there is just one fluctuating Gaussian momentum component k\, we obtain q — 3 and the generalized canonical distribution is formally given by p{$) ~

l

2

(6.37)

VI + M

It can be made to coincide with the invariant density of the Tchebyscheff maps via analytic continuation <j> —> i<j>, and choosing j3o = 1.

Chapter 7

Interaction energy of chaotic strings

We will now consider suitable evolution equations for possible standard model couplings. These equations make a priori arbitrary standard model couplings evolve to the stable zeros of the interaction energy of the chaotic strings. The equations can be regarded as analogues of Einstein equations in the 1-dimensional string space. Numerical evidence will be given that the smallest stable zeros of the interaction energy of the chaotic strings coincide with running electric, weak, and strong coupling constants of the standard model, the energy scales being given by the masses of the lightest fermions and bosons. Inverting the argument, high-precision predictions on standard model parameters can be given.

7.1

Analogue of the Einstein field equations

We are now in a position to describe the central idea of this book. Consider an extremely early stage of the universe where there is no ordinary matter and space-time at all. All that is there may be just a shift of information in fictitious time, encoded by Tchebyscheff maps. This is a O-dimensional universe. In the next step, some space-time dimensions are created — we need at least one, which represents the spatial coordinate of the chaotic string. This pre-universe just consists of vacuum fluctuations, and the standard model parameters (coupling strengths of the three interactions, fermion and boson masses, mass mixing angles) are not yet fixed. At this early stage a priori arbitrary standard model couplings a are possible. Since there is no ordinary matter yet, just vacuum fluctuations described by chaotic strings, these pre-standard model couplings can only 131

132

Interaction

energy of chaotic

strings

be realized as couplings a in the string space. In the beginning of the universe we expect gravity and the three other interactions to be unified. So we cannot distinguish whether the string coupling is an inverse metric or a standard model coupling. This is just another way of saying that the two physical interpretations II and 12 described in section 5.4 should be valid simultaneously. If we regard a - 1 — a - 1 as a kind of metric in the 1-dimensional string space, then an Einstein equation should be valid. A stochastically quantized Einstein equation for this 1-dimensional (trivial) metric in the 1dimensional string space would have the form (-)

=

-a = T00 + noise.

(7.1)

Here Too is the energy density of the vacuum as created by the interaction energy W of the chaotic string, and t is the fictitious time of the Parisi-Wu approach, used for second quantization. Choosing Too = —T • W, where T is a positive constant and W(a) is the interaction energy of the relevant chaotic string, one ends up with d = a2YW(a) + noise.

(7.2)

The fictitious time t of the Parisi-Wu approach has dimension energy-2, hence if W(a) is defined to be dimensionless then the constant T should have dimension energy2, and we could for example choose F ~ G _ 1 or T ~ a^2G~1, where G is the gravitational constant. Alternatively, we could regard eq. (7.2) as an equation describing the renormalization flow under energy scale transformations. Strong local fluctuations of temperature are possible (see section 6.6), which allow for probing of all kinds of energy scales. Apparently all zeros of the function W(a) describe a stationary state for possible standard model couplings a, but only those zeros where W(a) has locally negative slope describe a stable stationary state. Arbitrary initial couplings o will evolve to these stable states (Fig. 7.1) and will stay there forever, thus fixing low-energy standard model parameters (in interpretation II) as inverse metrics in a high-energy scenario at or above the Planck scale (interpretation 12). More precisely, we should talk about the quantum mechanical expectation of a with respect to the noise. Of course the stability properties of the zeros depend on the choice of the sign of the constant T. This choice breaks the initial symmetry of the

The 3A string—electric

interaction

strengths of electrons and d-quarks

133

Fig. 7.1 Renormalization flow as generated by eq. (7.2). Arbitrary couplings a approach one of the stable zeros a\ or a,2 of the interaction energy W(a).

problem. Let us assume that for our world T > 0 is relevant and label the stable zeros of W(a) (i.e. those with negative slope) by a] ', i — 1,2,... Here N is the index of the chaotic string theory considered, and b — 1,0 (or A,B) distinguishes between forward and backward coupling. We will now describe our numerical results in detail.

7.2

The 3A string—electric interaction strengths of electrons and d-quarks

Figs. 7.2 and 7.3 show the interaction energy W(a) = i ( $ ^ $ ^ + 1 ) of the chaotic ZA string. Fig. 7.3 is a magnification of the low-coupling region a £ [0,0.018]. We observe the following stable zeros in the low-coupling region:

a^A)

=

0.0008164(8)

A)

=

0.0073038(17)

a^

Interaction

134

energy of chaotic

strings

1

0.1

^^~^y 0.05

1

_^w 3 _ A, _-_______

,

a

(3A)

4

/ -0.05

-0.1

-0.15

\1 -0.2

Fig. 7.2

1

i

0.2

0.6

1

Interaction energy of the 3A string in the region a £ [0,1],

The statistical error is estimated by repeating the iteration of the coupled map lattice (2.1) for different random initial conditions $Q £ [—1,1] m a small vicinity of the zero. We used coupled map lattices of size 10000 with periodic boundary conditions, typical iteration times over which the product $^$Jj +1 was averaged were nmax = 107. For our numerical experiments, we have always chosen spatially independent initial values $Q that were distributed according to the invariant density of the uncoupled Tchebyscheff map. Remarkably, the zero a\ ' appears to approximately coincide with the fine structure constant aei « 1/137. Arbitrary couplings a (in the basin of attraction) will evolve to this stable fixed point under the flow generated by eq. (7.2). To construct a suitable Feynman web interpretation, let us choose in Fig. 5.1 (in chapter 5) Bi to be any massless boson and B2 — 7 (photon), / i = e~,/2 = e + (electrons and positrons). The relevant energy scale underlying this Feynman web is given by E — (3/2)(m 7 +2m e ) = 3m e , according to eq. (5.29). Hence our standard model interpretation 12 of the

The 3A string—electric

I

u.uuuo

I

interaction

strengths of electrons and d-quarks

I

•

I

I

I

1

135

1

0.0006 + + + +

0.0004 + + + 0.0002

-

+ +

++

++

4+

+

n

±:

___i

+ 0 •±^- + 3. r -

a2(3A>\ +

+ -0.0002

+

+

\ ++

-0.0004

_n nr\na 0

Fig. 7.3

0.002

0.004

+

1

1

1

1

1

1

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Same as Fig. 7.2, but for the low-coupling region a £ [0,0.018],

string state described by a\

' suggests the numerical identity a)?

(7.3)

= a e ( (3m e ).

For a precise numerical comparison with today's experimentally measured value let us estimate the running electromagnetic coupling at this energy scale. It is sufficient to use the lst-order QED formula ael (E) = ael (0) 11 + ^ [

^

T

11

i

U f JO

dx x(l - x) log (l + ^ x ( l - x) \

m

i

(7.4)' (see, e.g., [Bjorken et al. (1965); Kane (1987)]). The sum is over all charged elementary particles, rrii denotes their (free) masses, and / , are charge factors given by 1 for e,/i, T-leptons, | for w,c, i-quarks and | for d,s,bquarks. Using this formula, we get a e ;(3m e ) = 0.007303, to be compared with d£A) = 0.0073038(17). There is excellent agreement. Next, we notice that the zero a[ ' has approximately the value | a e ; . This could mean that the chaotic 3A string also has a mode that provides

136

Interaction

energy of chaotic

strings

evidence for electrically interacting d-quarks. Our interpretation is a^A) = adel{imd) = -ael(3md),

(7.5)

where a.del — \aei denotes the electromagnetic interaction strength of dquarks. Arbitrary couplings (in the corresponding basin of attraction) will evolve to this stable fixed point under the flow (7.2). In the Feynman web interpretation of Fig 5.1, we can choose Bi massless, B2 — 7, /1 = d, f2 = d. Formula (7.4), as an estimate, yields for md = 9 MeV the value aet{3md) = 0.007349, which coincides very well with 9a[3A) = 0.007348(7). The value aei(E) = 9a[ actually translates to the energy scale Ed = (26.0±6.4) MeV. This yields md = \Ed = (8.7±2.1) MeV, which coincides with estimates of the MS current quark mass of the d quark at the proton mass renormalization scale [Groom et al. (2000)]. There are two further stable zeros in the large-coupling region, a 3 = 0.07318 and aA = 0.9141. These zeros cannot be identified with standard model couplings in a straightforward way. Presumably they describe physics beyond the standard model. If 03 is interpreted as a running strong coupling as(E), then formula (7.15) in section 7.5 yields the corresponding energy E as E RJ 7850 GeV.

7.3

The 3B string —weak interaction strengths of neutrinos and ti-quarks

The 3B string is a system with backward coupling. For backward coupling the action of the potentials V and W not only alternates in time n but also in discrete space i. In section 5.6 we provided arguments that this coupling form generally describes mixed states of two particles. The interaction energy W(a) of the 3B string is plotted in Fig. 7.4 and 7.5. Again Fig. 7.5 shows a magnification in the low-coupling region a € [0,0.018]. In the low-coupling region we observe the following stable zeros of W(a): af B ) 4

3B)

=

0.0018012(4)

=

0.017550(1)

We now show that it is possible to find an interpretation of a[ indicating the existence of u-quarks and neutrinos.

' and a>, '

The 3B string —weak interaction

strengths of neutrinos

and u-quarks

137

0.15

0.05

-0.05

-0.15

Fig. 7.4

Interaction energy of the 3B string in the region a 6 [0,1].

Let us start with a^ • For left-handed neutrinos v^ , the weak coupling due to the exchange of Z°-bosons is given by

a

weak

—

a

el

1 4 sin 8\y cos2 9w

(7.6)

Here 6w is the weak mixing angle. In the following we will treat sin 6w as an effective constant, and regard aei as the running electromagnetic coupling. Other renormalization schemes are also possible, but yield only minor numerical differences. Experimentally, the effective weak mixing angle is measured as sin2 9w = 0.2318(2) [Groom et al. (2000)]. Assuming that in addition to the left-handed neutrino interacting weakly there is an electron interacting electrically, the two interaction processes can add up independently if the electron is right-handed, since right-handed electrons cannot interact with left-handed neutrinos. Hence a possible standard

Interaction

138

energy of chaotic

strings

0.0006

0.0004

0.0002

a <3B>

t..„±+j-.?.t.

±

—-+_.

-0.0002

-0.0004

-0.0006 ++++

0

Fig. 7.5

0.002

0.004

0.006

-J 0.008

l 0.01

l 0.012

l_ 0.014

0.016

0.018

Same as Fig. 7.4, but for the low-coupling region a 6 [0,018].

model interpretation of the zero a2

' would be

1 4 sin 9w cos2 0w (7.7) In the Feynman web interpretation of Fig. 5.1, B2 = Z°, / i = VL, $2 = v~L i n addition to the process already described by a2 • B\ can again be any massless boson. Putting in the experimentally measured value of sin2 9W = 0.2318, we obtain for the right-hand side of eq. (7.7) the value 0.01755, which coincides perfectly with the observed string coupling a2 = 0.017550. A priori arbitrary standard model couplings a will evolve to this stable zero under the flow of eq. (7.2) and this will fix the electroweak mixing angle. Next, let us interpret a\ ' . I n analogy to the joint appearance of v and e, we should also expect to find a weakly interacting u-quark, together with a d-quark interacting electrically. Clearly, the u-quark could also interact electrically, but for symmetry reasons we expect the pair (u, d) to interact in a similar way as (y, e). A right-handed u-quark interacts weakly with ,(3B)

aei{Zme) + a£f eafc (3m„J = a2(3/1) + aei{2,mVc)

High-precision prediction of the electroweak parameters

139

the coupling _ 4

s i n \

Adding up the electrical interaction strength of a d-quark, a n a t u r a l interpretation, quite similar to t h a t of the zero a2 , is a[3B)

= adel(3md)

+ auw1ak(3mu)

= afA)

+ la y

e J

(3m

t t

)^^

COS

(7.9)

0\y

T h e Feynman web interpretation of this string state is B\ massless, Bi = Z°, fi = UR, f2 — UR in addition to the process underlying a[ '. Numerically, taking sin 2 8w = 0.2318 and evaluating the running aei using mu = 5 MeV, we obtain for the right-hand side of eq. (7.9) 0.001800, which should be compared with a2 = 0.001801. Again we have perfect agreement within the first 4 digits. It is remarkable t h a t the same universal effective value sin 2 6w = 0.2318 can be used consistently for b o t h leptons (couplings a2 , a2 ) and quarks (couplings a\ , a{ ). If, on the other hand, we assume t h a t sin 2 8w is fixed by a2 ' to be 0.2318, the running electric coupling can be used to extract the energy scale E from a\ . We obtain E = 3 m u = (21.3 ± 9) MeV, meaning mu = (7.1 ± 3.0) MeV. Note t h a t the backward coupling form of the N = 3 strings describes a spinless state (formed by CR and VL, respectively di and UR\ R and L add u p to 0), whereas the forward coupling form just describes one particle species with non-zero spin (e or d). A similar statement will t u r n out to hold for the N — 2 theories, replacing fermions by bosons. Again another stable zero, a3 = 0.3496, is observed in the largecoupling region, which might again describe physics beyond t h e s t a n d a r d model.

7.4

H i g h - p r e c i s i o n p r e d i c t i o n of t h e e l e c t r o w e a k p a r a m e ters

It appears t h a t the smallest stable zeros of the two N = 3 strings coincide with the electroweak coupling strengths at the mass scales of the lightest fermions d,u,e,ve. We can thus actually invert the formulas and give a rather precise prediction of the electroweak parameters. Once again, the general idea underlying this is t h a t at some very early stage of the universe

Interaction

140

energy of chaotic

strings

the electroweak couplings were not fixed but were following the evolution equation (7.2), being finally fixed as stable inverse metrics in the string space. The chaotic string dynamics may still persist today and stabilize the observed parameter values. The observed low-energy limit of the fine structure constant follows from a2 and formula (7.4) as (3A)

ael(0)

= 43A)(1=

rl

/

Q

2

\

^-Y,fi]

dxx(l-x)\og(l+^x(l-x)\\

0.0072979(17) = 1/137.03(3).

(7.10)

3

Here we neglected terms of order a , which are much smaller than the statistical error. Within the error bounds, our result agrees very well with the experimentally measured value a(0) = 0.00729735 = 1/137.036 (the experimentally measured number is known to quite precisely coincide with jiy • cos jfy = 0.00729735 [Gilson (1996)]). Next, the effective weak mixing angle can be predicted from eq. (7.7). Since mVc m 0 one obtains

^-iV-f-^r)

(7J1)

This yields sin2 6W = 0.23177(7).

(7.12)

The value perfectly coincides with today's experimental measurements, as listed in [Groom et al. (2000)]. In fact, this pre-universe prediction is much more precise and agrees much better with experimental values than predictions by supersymmetric GUT models. Within the error bars, we actually do get the same effective Weinberg angle for quarks and leptons .,,

(3B)

\

2a

(3B)

using either a\ ; or a2 '. Any high-precision result on sin2 8w can also be used to estimate gauge and Higgs boson masses from radiative corrections. For example we may use the results of a calculation by Degrassi et al. [Degrassi et al. (1997)]. ley n \

Our Feynman web interpretation of the zero a2 suggests to regard our sin2 @w as the effective mixing parameter sj 2 = sin2 Q*?. for Z°-lepton coupling. Table 1 in [Degrassi et al. (1997)] translates the value 0.23177(7) to mw — 80.35(1) GeV and yields a Higgs mass of the order of twice the W mass.

The 2A string —strong interaction

strength at the W-mass scale

141

0.35

0.25

0.15

-0.05 -

Fig. 7.6

Interaction energy of the 2A string.

Note that there is excellent agreement of the weak mixing angle obtained from the chaotic 3B string with the experimental measurements at LEP. At LEP the effective weak mixing angle si2 is measured as 0.23185(23), from this the Higgs mass is estimated as 166195° GeV and the W-mass as 80.35(6) GeV [Groom et al. (2000)]. Our predicted value (7.12) also very well agrees with the measurements listed as indirect standard model data in the 2001 web update of [Groom et al. (2000)].

7.5

The 2A string —strong interaction strength at the Wmass scale

If electroweak coupling strengths are fixed by suitable zeros of the interaction energy of chaotic strings, then something similar should also be the case for the strong coupling strength as. Let us now look at strings with N — 2. Fig. 7.6 shows the interaction energy W(a) of the 2A string. Only

Interaction

142

energy of chaotic

strings

one stable zero is observed: a f A ) = 0.120093(3)

(7.13)

Arbitrary couplings a will be attracted by this stable fixed point under the flow generated by eq. (7.2). We notice that a\ numerically seems to coincide with the strong coupling constant as at the W- or Z mass scale, which is experimentally measured as as(mz) — 0.1185(20) [Groom et al. (2000)]. For symmetry reasons, it seems plausible that if the N = 3 strings fix the electroweak couplings at the smallest fermionic mass scales, then N = 2 strings could fix the strong couplings at the smallest bosonic mass scales. The lightest massive gauge boson is indeed the W±. Hence our Feynman web interpretation associated with a\ ' is B\ = W * , B i — g (gluon), fi = u, fi = d (or / i — d, J2 = u), and since N — 2, formula (5.29) implies a\

= as(mw

+ mu + md) RS as(mw)-

(7.14)

Since the M^-mass is experimentally known with high precision, and since the free quark masses mu and md can be neglected, eq. (7.14) yields quite a precise prediction for the strong coupling as at the W mass scale. Provided our Feynman web interpretation is correct, this is a great theoretical progress, since as is experimentally only known with rather low precision. We can evolve our precision prediction to arbitrary energy scales, using the well known perturbative results from quantum chromodynamics. Let us just shortly review what we need to know from QCD on the running strong coupling. In a third-order calculation it is given by -2

as(E)

Mnf! +

1+

Abl ^(lnf)2L

+ O

2&i In In fj/,21TI

In In

£2

E2 A^

x) +

b0b2

(7.15) .(In A2 J

Here A is the Q C D scale parameter, which takes on different values in the various energy regions with ny active quark flavors, and the coefficients bi

The 2A string —strong interaction

strength at the W-mass scale

143

are given by

•> = -M™-^^)-

<718>

rif denotes the relevant number of quark flavors. The integer nj changes by 1 at thresholds given by the quark pole masses Mq. At the thresholds Mq, as(E) should be continuous. This determines the scale parameters A( n/ ) in the various flavor regions, given one of the parameters A(n-" for some nf (or, equivalently, given as(E*) at some fixed energy E*). The relation between pole quark masses Mq and free quark masses mq is

Mq=mq\l

l^+Kq(^^-)2+0^a^M^3S

+

(7.19) with tf, = 16.11 - 1.04 £

( l - ^ )

(7-20)

[Gray et al. (1990)]. By free quark mass we actually mean the so-called MS running quark mass fhq(E) at renormalization scale E = Mq. For the light quarks u, d, s we define the free masses to be the running MS masses at a renormalization scale given by the proton energy scale E s» 1 GeV. The reader not familiar with these terms may e.g. consult the reviews in [Groom et al. (2000)]. Now let us come back to the chaotic 2A string, which fixes the numerical value of as at one special energy scale, the W-mass scale. In 3rd-order perturbation theory eq. (7.14) is equivalent to an effective QCD scale parameter A^

= 0.20608(14)GeV,

(7.21)

neglecting higher-order terms. Here we use for the W-m&ss the value mw = 80.35(6) GeV [Groom et al. (2000)]. At E = mz = 91.188(2)

Interaction

144

energy of chaotic

strings

GeV eq. (7.15) yields as(mz)

= 0.117804(12).

(7.22)

This prediction of as from the zero of the chaotic 2A string is clearly consistent with the experimentally measured value 0.1185(20) and in fact much more precise than current experiments can verify. To evolve as(E) to energies E > Mt or E < Mf, one has to take into account the quark threshold effects. In chapter 8 we will obtain from the self energy of the chaotic strings the free bottom and top masses as m& = 4.23(1) GeV and mt — 164.5(2) GeV. The corresponding pole masses are Mb = 4.87(2) GeV and Mt = 174.3(3) GeV, using formula (7.19). Matching the running as(E) in a continuous way at Mt and Mb [Marciano (1984)], we obtain from the above value A<5) - 0.20608(14) GeV the values A<6) = 0.08705(6) GeV and A^ = 0.28913(17) GeV. Slight numerical differences can arise from the way the quark thresholds are treated. 7.6

The 2B string —the lightest scalar glueball

The interaction energy of the 2B string is shown in Fig. 7.7. W(a) has only one non-trivial zero o p f l ) = 0.3145(1).

(7.23)

It has negative slope, so it should describe an observable stable state. One possibility is to interpret this as a strong coupling at the lightest glueball mass scale. The lightest scalar glueball has spin Jpc = 0 + + and is denoted by 990++ m the following. If we choose a Feynman web consisting of Bi = gg0++, B2 =g, f\= u, f2 = u, meaning a\

= as(mggo++ + 2mu) « as(mggo++),

(7.24)

then indeed the 2B string describes two bosons (two gluons forming a glueball), similar to the 3B string, which described two fermions. In both cases a spin 0 state is formed. In lattice gauge calculations including dynamical fermions the smallest scalar glueball mass is estimated as mggo++ = (1.74 ± 0.07) GeV [Boglione et al. (1997)] and at this energy the running strong coupling constant is experimentally measured to be as — 0.32 ±0.05 [Peach et al. (1994)]. This clearly is consistent with the observed value of

a[ 2 B ) .

The 2A

and 2B

strings — towards a Higgs mass

prediction

Fig. 7.7

Interaction energy of the 2B string.

145

-0.2

We can estimate the lightest glueball mass predicted by the 2B string using once again the 3rd-order QCD formula (7.15) with the previously determined A^4). The value as(mggo++ + 2mu) = 0.3145(1) then translates to mggo++ + 2mu = 1.812(2) GeV, hence our prediction is mggo++ « 1.80 GeV. 7.7

The 2A and 2B prediction

strings — towards a Higgs mass

Finally let us look at the two remaining chaotic string theories, namely those with N = 2 and antidiffusive coupling. The interaction energies W(a) of the 2A ~ and 2B ~ strings are shown in Fig. 7.8 and 7.9. Let us now try to find a suitable Feynman web interpretation for the observed smallest stable zeros afA~] = 0.1758(1) and a(?B~) = 0.095370(1) of these strings. Again let us be guided by symmetry considerations. We saw that the smallest stable zero of the 2A string described a boson with non-zero spin

146

Interaction

energy of chaotic

1

:

-

!

/

rJ d

strings

1

i


-

\

r^

\

/

\ •

/

\ a 3~<2A">

a 2 <2A"> a

\

/

/

" 0

0.2

•

i

0.4

0.6

0.8

1

a

Fig. 7.8

Interaction energy of the 2A ~~ string.

(the lightest massive gauge boson W^) and the smallest stable zero of the 2B string a boson without spin (the lightest scalar glueball). Thus it seems reasonable to assume that the smallest stable zero of the 2A ~~ string describes yet another bosonic particle with non-zero spin, possibly the lightest glueball gg2++ with spin Jpc = 2 + + , and the smallest zero of the 2B ~ string yet another bosonic particle with spin 0, possibly the lightest Higgs boson. Let us start with a\ '. The Feynman web of this string state may consist of B\ — gg2++,B2 = g,fi = q, ]i — q, where q,q are suitable quarks. From the strong coupling interpretation a f A~] = a s ( E i )

(7.25)

the energy Ei = mgg2++ + 2mq can again be determined from the QCD formula (7.15), using our previously determined A^5^. One obtains Ei = 10.45(3) GeV. In lattice gauge calculations the mass of the lightest 2 + + glueball is estimated as 2.23(31) GeV [Groom et al. (2000)]. We thus get the correct order of magnitude of the 2 + + glueball mass if we assume that

The 2A

and 2B

strings — towards a Higgs mass

prediction

147

0.25

0.15

0.05

0.7

Fig. 7.9

Interaction energy of the 2B

0.8

0.9

string.

the quarks in the Feynman web interpretation are bottom quarks. In this case a glueball mass mgg2++ = (10.45(3) - 2 • 4.23(1)) GeV =1.99(4) GeV is predicted, using the value mb = 4.23(1) GeV, which we will derive later. Next, let us consider a\ . This zero is even more interesting, since it may provide evidence for the Higgs particle. Our Feynman web interpretation is Bi = H,B2 = g,fi — q, h — <7> where H is the lightest Higgs boson and q, q are suitable quarks. The strong coupling interpretation (2B~

=

as(E2)

(7.26)

yields E2 = 483.4(3) GeV —ma + 2mq. However, experimental and theoretical arguments [Groom et al. (2000)] imply that the Higgs mass should be in the region 100...200 GeV. Hence we only obtain a consistent value for the Higgs mass if we assume that the quarks involved are t quarks. This is similar to the zero a\ ', where the quarks involved were also heavy quarks, namely 6-quarks. Generally, strings with anti-diffusive coupling seem to have a tendency to describe heavy rather than light particles. In section 8.5 we will obtain quite a precise prediction of the free top

148

Interaction

energy of chaotic

strings

mass from the self energy of the chaotic 2B string, namely rat — 164.5(2) GeV, corresponding to a top pole mass of 174.4(3) GeV. With this value the zero a\ ' yields a Higgs mass prediction of mH = E2 - 2rm = 154.4(5)GeV.

(7.27)

This is a very precise prediction, the statistical error is very small. But of course the main source of uncertainty is a theoretical uncertainty, namely ; whether our Feynman web interpretation of the zero a\ is correct. For example, assuming that a supersymmetric extension of the standard model is correct, then the zero could also describe another scalar particle. A Higgs mass prediction of similar order of magnitude as in eq. (7.27) has also been obtained in [Roepstorff (1999)].

7.8

Gravitational interaction

Similar as for the N — 3 strings, the next larger zeros a2 and a2 should be considered as well. What is interesting here is the fact that near a = | (2A~) and near a = 1 (2B~) all chaotic fluctuations cease to exist, and the fields $^ approach a stable synchronized fixed point $ = 0 at a2 — \, respectively a2 = 1. This was analytically derived in section 2.6 and 2.7. From a stochastic quantization point of view this means that there are no vacuum fluctuations any more, and a purely classical theory arises if these strings with these parameter values are used for 2nd quantization. The zero cr2 = \ could be interpreted as a dimensionless gravitational coupling strength, defined by oto — ^(E/mpi)2, at the Planck mass scale E = mpi. Later, in section 10.6, we will consider a boson, denoted by W, with just that mass. The backward coupling form with a2 ' = 1 could then describe two such gravitational bosons. Note that the zeros a2 and a\ ' are only attracting for the flow (7.2) if a is smaller than the zero. If o is larger than the zero, then the renormalization flow drifts away from the zero. Total stability is only achieved if t (or T) switches sign at these zeros. This change of behaviour might be associated with a black hole singularity (see also section 11.5). If our interpretation is correct, then the smallest zeros of the interaction energies W(a) of the six chaotic string theories fix the strengths of all four

Gravitational

interaction

149

interactions. Electroweak interactions are fixed at the smallest fermionic mass scales (N = 3), strong interactions at the smallest bosonic mass scales (N = 2), gravitational interactions at the smallest gravitational (black hole) mass scales (N = 2, s = —1).

This page is intentionally left blank

Chapter 8

Self energy of chaotic strings

We will now study local minima of the self energy V(a) of the chaotic strings. Arbitrary initial standard model couplings o can also be regarded as homogeneous self-interacting scalar fields with an effective potential given by V(a). The local minima of V(a) describe possible stationary stable equilibrium states for these types of standard model couplings. We will provide extensive numerical evidence that the self energy has minima for couplings a that numerically coincide with various electroweak, strong, Yukawa and gravitational couplings of the known standard model particles. From this the fermion masses can be predicted quite precisely.

8.1

Self interacting scalar field equations

We saw in chapter 5 that the potentials V($) and W(^,^!) are of equal importance for chaotic strings, each of them generating one of the two steps of the discrete time evolution. Hence not only expectations of W but also expectations of V are expected to be relevant for fixing possible standard model parameters. Let us now look at the self energy V of the chaotic strings. Local minima of this quantity will turn out to fix masses (N = 2) and charges (N = 3). Consider again an extremely early stage of the universe where masses and charges are not yet fixed. There is yet no equilibrium metric and no ordinary space time. All that is there is a dynamics of vacuum fluctuations described by chaotic strings. The pre-standard model couplings a can take on any value and are realized as couplings in the chaotic string space. Consider an o priori free standard model coupling a that depends on a 151

152

Self energy of chaotic

strings

free mass parameter. For example, a may later become a Yukawa coupling after an equilibrium state has been reached. Or, it may become a strong or electroweak coupling. We may regard a as a homogeneous self-interacting scalar field variable (similar to a dilaton field in superstring theory) with a complicated effective potential given by the expectation of the self energy V(a) of the chaotic string. Alternatively, we may regard a as a kind of 1-dimensional homogeneous Higgs field containing information on fermion masses. The stochastically quantized field equation reads dV a = —Y— h noise, da

(8.1)

and if T > 0 then local minima of the function V(a) describe possible stable stationary states of standard model couplings (more precisely, of their quantum mechanical expectations with respect to the noise). Again the evolution takes place in a high-energy scenario (the Planck scale or above, in interpretation II), but nevertheless this evolution has the ability to fix low-energy parameters due to the simultaneous validity of interpretation 12. Any minimum of V is thus a candidate for observed standard model couplings. We will denote the local minima of V(a) as a'^ '. Indeed a large number of local minima is observed that precisely coincide with known standard model interaction strengths that we observe today. In particular, Yukawa couplings of heavy fermion flavors and gravitational couplings of light fermion flavors are observed, which allow for a very precise prediction of the fermion masses. In the following sections we will describe our numerical results in detail. First we will deal with minima describing strong and electroweak interaction states, then we will analyze minima describing Yukawa and gravitational interaction states.

8.2

The 3A string —weak and strong interactions of heavy fermion flavors

Let us start with the 3A string. Fig. 8.1 and 8.2 show the numerically determined function V(a). It is the temporal and spatial average of the observable §($),) 2 - ($J,) 4 under long-term iteration of the coupled map lattice. Fig 8.2 is a magnification of the low-coupling region a £ [0,0.022]. First let us first look at rather large values of the coupling a. The local

The 3A string —weak and strong interactions

1

r

0.4

0.5

of heavy fermion

-i

flavors

153

r

0.24 0.2

Fig. 8.1

0.3

0.6

0.7

0.8

0.9

Self energy of the 3A string in the region a € [0,1].

minima labeled as a'{3A) A)

af

0.0953(1)

(8.2)

0.1677(5)

(8.3)

0.2327(5)

(8.4)

seem to numerically coincide with strong couplings at the heavy quark mass energy scales. Arbitrary couplings (in the basin of attraction) evolve to these stable minima under the evolution equation (8.1). A suitable Feynman web interpretation of these stationary states is given by choosing in Fig. 5.1 B\ massless, Bi = g (gluon), fi = q, f2 = q, with q = t,b,c, respectively. Formula (5.29) with TV = 3 yields the relevant energy scale as E = |(2m g + mg) = 3m ? , i.e. a'S3A> =

as(3mq),

i = 6,7,8

(8.5)

The experimentally measured values of the free quark masses* are m c *See section 7.5 for the definition of 'free' quark masses.

Self energy of chaotic

154

strings

0.376 ,(3A)

0.374

0.372

0.37

a

,(3A)

\A a .,(3A)

0.368

0.366 J**1*1

(iA) 85--'

0.364

0.362

0.36 0.358

++

a •<3AU+++

+

0.356 0.005

Fig. 8.2

0.01

0.015

0.02

Same as Fig. 8.1, but in the region a e [0,0.022].

1.26(3) GeV, mb = 4.22(4) GeV, mt = 164(5) GeV. These numerical values are obtained if one performs the average over the various experimental results quoted in [Groom et al. (2000)]. With the scale parameters At™') determined earlier in section 7.5, the QCD formula (7.15) yields the strong couplings at these energy scales as as(3mt)

=

0.0952(3)

(8.6)

a s (3m 6 )

=

0.1684(4)

(8.7)

a s (3m c )

=

0.2323(20).

(8.8)

Apparently there is good coincidence with the observed local minima of the self energy of the 3A string. For smaller a, further minima are observed that seem to describe weak interaction states of right-handed fermions. In the standard model the weak coupling constant is given by (T3~Q sin2 6wy sin2 9w cos2 8w

&weak — O-el

.9)

The 3A string —weak and strong interactions

of heavy fermion

flavors

155

(see, e.g., [Kane (1987)]). Here Q is the electric charge of the particle (Q — - 1 for electrons, Q — 2/3 for u-like quarks, Q — - 1 / 3 for d-like quarks), and T3 is the third component of the isospin (T3 = 0 for righthanded particles, T3 = - | for eL and dL,T3 = +^ for vL and uL)- Consider right-handed fermions fR. With sin2 9W - 0.2318 (as obtained in section 7.4) and the running electric coupling aei(E) taken at energy scale E — 3my we obtain from formula (7.4) and (8.9) ^ a l ( ^ )
= =

°-000246 0.001013

(8-10) (8.11)


=

0.00220.

(8.12)

On the other hand, we observe that V(a) has minima at a[{3A) a'2{3A)

= =

0.000246(2) 0.00102(1)

(8.13) (8.14)

a'3(3j4)

=

0.00220(1)

(8.15)

(see Fig. 8.2. a'{; ' and o3( ' are actually small local minima on top of the hill). There is good agreement with the weak coupling constants of / R = UR,CR,eii, respectively. A suitable Feynman web interpretation would be Bi massless, B2 = Z°,fi = JR,/2 = }R- Basically, the minima yield statements on the charges of the particles involved. Generally, it is reasonable to assume that the above interaction states of d, c, and e do not describe pure mass eigenstates but that small components of other flavors can be mixed in as well. For example, c could also have a small t component, thus slightly increasing the relevant energy scale. If a'/ ' ,0.2 ,ag describe right-handed fermions of down, up and electron type then it is natural to assume that the next minimum a4v = 0.00965(1) describes a right-handed neutrino VR. While right-handed neutrinos are excluded in the simplest version of the standard model, there are natural extensions of it where they can exist — though with quite a large mass. 04 could (for example) describe a very heavy right-handed Majorana neutrino interacting gravitationally with a<3 = \{mVRlmpi)2 = a4 . From this one obtains the mass m„ R = 1.696(1) • 10 18 GeV. Symmetry with a'2 , which describes a c quark, suggests that this right-handed neutrino is essentially of /u-type. We will come back to left- and right-handed neutrinos in section 8.7.

156

0

Self energy of chaotic

strings

i

1

1

1

1

1

i

1

r

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a

Fig. 8.3

Self energy of the 3B string.

The remaining minimum ag i(3A)

i(3A)

i(3A)

,

,

' = 0.02145(3) inbetween weak minima . -

t(3A)

l(3A)

i(3A)

, ,

., ,

Oj , a2v , a3v ' and strong minima a6v , a7K , a8v ' could possibly decribe a unified coupling at the grand unified (GUT) scale. In fact, evolving the strong coupling as(3mt) = a'} to higher energies using the usual (non-supersymmetric) formulas (see section 10.3), one obtains {3A) for mQ « 1.73 • 1016 GeV, which is the GUT scale. At Q s (3m Q ) = a'5 E = m,Q the strong coupling as(E) and the weak coupling ^ ( - E ) are observed to merge into the common value as = a.2 — 0.02192. We will deal with possible grand unification scenarios in more detail in chapter 10. 8.3

The 3B string — further mixed states of heavy fermion flavors

V{a) for the 3B string is plotted in Fig. 8.3. This dynamics is of the backward coupling form, and our general arguments suggested that the backward coupling form describes mixed states of two particle species. We

The SB string — further mixed states of heavy fermion

flavors

157

observe (among others) the following minima of the vacuum energy. 4 3 B ) = 0.1027(1) ag

(3B)

(8.16)

=0.2916(5).

(8.17)

Arbitrary couplings (in the basin of attraction) are attracted by these stable minima under the evolution equation (8.1). If we choose the same numerical values of the quark masses and the parameters A( n/ ) as before, we obtain from the QCD-formula (7.15) ^{mt+mbyj

=

0.1027(4)

(8.18)

^(mc+msy\

=

0.2914(35).

(8.19)

For the s-quark mass we used the value ms = (160 ± 9) MeV [Narrison (1989); Narrison (1994)]. Again there is good coincidence between the above standard model couplings and the observed string couplings. The zeros Og ' and a.g ' seem to illustrate that up- and down quark flavors can mix flavor-independently for strong interaction. At lower coupling strengths, the minimum d£ = 0.00220 of the 3A string slightly changes to % ' = 0.00223 for the 3B string. This numerically coincides with the weak coupling Oi^eak(^(me + « v ) ) = 0.00223 of a state where e and ft mix with equal weights. If this interpretation is correct, then the minimum a 3 ' can be used to estimate the /j, mass as being of the order of magnitude 100 MeV. Similarly, the minimum a4( = 0.00965(1) describing the heavy right-handed Majorana neutrino changes by a small amount to 04 = 0.00972(1), which could be due to the fact that the heavy /x-type neutrino v^ gets mixed with a r-component. But since the gravitational coupling is proportional to the mass squared, and since a^ is almost the same as a 4 , this means that we must have mv» as m^ . Also the minima a'/ and d^ change by a very small positive amounts to a^ ,0-2 f° r the 3B string. It is difficult to numerically estimate this change, it is of the order of the precision by which we can determine the minima. It could again mean that small amounts of heavier flavors are mixing with d, c. For example, d might get a small s component and c a small t component. In total, one may conjecture that the four minima a'/ \d^ , a'i ,d^ contain the information on the four

158

Self energy of chaotic

0.05

i

i, T

i

strings

i

.(2A> 1

-

-I \\ aa 2 '<2A>

- \

-0.15 -

\

"

-0.2 2A 3 'l >

a a

-0.25

1

Fig. 8.4

I

i

I

Self energy of the 2A string.

mixing angles of the Kobayashi-Maskawa matrix. Generally, B-strings as compared to A-strings seem to contain information on all kinds of mixing angles. At larger couplings (a > 0.3), the N — 3 strings exhibit lots of further local minima. In chapter 9 we will see that these can be associated with baryonic and mesonic states. 8.4

The 2A string — further bosons

Let us now turn to N = 2 strings. The self-energy of the 2A string theory is shown in Fig. 8.4. A very strongly pronounced minimum is A2A)

0.1848(1)

.20)

It coincides with the running strong coupling as at an energy given by twice the free 6-quark mass. The Feynman web interpretation of this chaotic string state could be Bi = B2 = g, f\ = b, f2 = b. Since N - 2 the relevant energy scale is E — (N/2)(mg + 2mb) = 2mb. Numerically, we get from

The 2A string — further

bosons

159

eq. (7.15) for mb = 4.23 GeV a s (2m 6 ) = 0.1848,

(8.21)

which coincides precisely with the minimum a^ '. In fact, this minimum is very sharp, moreover, it is in the region where the scale parameter A^5) is of relevance, which we know with high precision from section 7.5. Hence we may go the other way round and use a'^ to predict mj with high 5 precision. The value A^ ^ = 0.20608(14) GeV obtained earlier translates 42A) = 0.1848(1) = as(2mb) to mb = 4.23(l)GeV.

(8.22)

This is in good agreement with estimates of the free 6-mass using current algebra techniques [Narrison (1994)]. Another minimum of interest is a$2A) = 0.03369(2)

(8.23)

This seems to coincide with the weak coupling constant a 2 K, 1/30 at the Z° mass scale. A possible Feynman web interpretation is B\ = Z°, Z?2 = W°, / i = b, fi — b, which implies a'2{2A) = a2(mz

+ 2mb).

(8.24)

We can estimate the value of the weak coupling Q 2 at this energy scale using the well-known formulas for the running electroweak couplings (see section 10.1). At the Z° mass scale mz = 91.188 GeV the QED formula (7.4) yields aei(mz)

=0.0078096.

(8.25)

The QED formula used does not explicitly take into account small corrections by hadronic states. However, later (in chapter 11) we will actually associate the chaotic string dynamics with a dynamics of vacuum fluctuations that takes place long before hadronic states are formed. In that sense, neglecting the small hadronic corrections in the QED formula makes sense for our purposes. The running electroweak mixing angle defined as s2 (E) : = aei (E) / a 2 (E) is known to very closely coincide with the effective mixing angle sin2 O^ff at the energy E = mz- One has [Groom et al. (2000)] s 2 (mz) = sin2 &$ - 0.00029

(8.26)

160

Self energy of chaotic

strings

Hence our result of sin 6lee$ = 0.23177(7) derived in section 7.4 yields s2(mz)

= 0.23148(7),

(8.27)

which implies a2(mz)

=

Qe;(

^ Z ) = 0.03374(1).

(8.28)

If further transferring the running a2 to the slightly larger energy scale mz + 2mt„ one obtains the numerical value a2 {mz + 2mb)= 0.03369(1)

(8.29) 2A

which precisely coincides with the observed string coupling a2'( ^ with a precision of 4 digits. Finally, there is also a rather weakly pronounced minimum ax = 0.00755(3) which seems to coincide with a e /(2m r ). In the Feynman web interpretation B\—B2 — 7, A = r, f2 = f. This minimum could be used to estimate that the order of magnitude of the r mass is about 2 GeV. Summarizing, we see that whereas the interaction energy of the 2A string provided evidence for the charged gauge bosons W ± , the self energy of the 2A string provides evidence for the uncharged gauge bosons g, Z° and 7. 8.5

The 2B string — Yukawa interaction of the top quark

The self energy of the 2B string is shown in Fig. 8.5. The most pronounced minimum a'^ ' = 0.2235(2) can be interpreted in terms of a strongly interacting quark state where b and d are equally mixed. However, most interesting for us is another minimum, namely 4 ( 2 B) = 0.03440(2).

(8.30)

This can be interpreted in terms of Yukawa interaction of the top quark. Our Feynman web interpretation is B\ — B2 = H (Higgs boson), f\ =

*, h = i. The Yukawa coupling of any fermion / is proportional to the square of its mass. It is given by

Q =

2

- Kfe) -

(831)

-

The 2B string — Yukawa interaction

0.05 I

1

1

1-

0.1

0.2

0.3

of the top quark

161

L .(28)

Fig. 8.5

0.4

0.5

0.6

_J 0.7

1_ 0.8

0.9

1

Self energy of the 2B string.

Still we have to decide on the energy scale E for the running a2{E). Our usual rules for the Feynman web imply that the energy is given by E = run + 2mt. Our interpretation of the minimum o2l is thus a2(2B) =atYu{mH

+ 2mt) = -Aa2{mH + 2mt)(^\

.

(8.32)

Accepting this interpretation the formula allows for a very precise prediction of the top mass mj. From our earlier consideration we know that a2(mz + 2mb) - 0.03369(1). Transferring the running a2 to the higher energy scale E — mn + 2rnt we obtain for m # « 154 GeV a2{E) =0.03284(1)

(8.33)

(the usual (non-supersymmetric) formula for a2(E) is used, see section 10.1). This result is insensitive to the precise value of the Higgs mass.

Self energy of chaotic

162

strings

Solving for mt we get a very precise prediction of the free top mass, namely /

^L

i(2B)

= 2\ \ ~

= 2.047(1)

(8.34)

Using the value mw = 80.35(6) GeV we thus get mt = 164.5(2)GeV.

(8.35)

The corresponding pole mass Mt, of relevance for experiments, can be quite precisely determined by a formula in [Degrassi et al. (1997)], which is based on A<5) and avoids A^6). We get Mt = 174A(3)GeV

(8.36)

The error only takes into account the precision by which we can determine a'2 , in addition there is the theoretical uncertainty whether our Feynman web interpretation of the minimum is correct. Nevertheless, our prediction is consistent with the experimentally measured value Mt = (174.3 ± 5.1) GeV [Groom et al. (2000)] and is in fact a much more precise prediction than the current experiments can confirm. Even if the Higgs mass prediction that we obtained in section 7.7 were wrong by a factor 2, the top mass prediction (8.35) or (8.36) would only change by less than 1 GeV. The very precise prediction of the top mass allows for a precise evaluation of the W mass as well, using standard model calculations. Fig. 10.2 in the section 'Electroweak model and constraints on new physics' of the 2001 web update of [Groom et al. (2000)] implies that a top mass of Mt = 174.4(3) GeV corresponds to a W mass of mw = 80.362(20) GeV, provided the Higgs mass is in the region 110...200 GeV. If we use our predicted value of TUH = 154.4(5) GeV as well then the error bars of this W mass prediction still significantly decreases. All mass values obtained from the chaotic strings fit very well into the allowed range given by the indirect experimental standard model data.

8.6

Yukawa and gravitational interactions of all quarks and leptons

Clearly not only the top-quark, but also the other heavy fermions are able to exhibit Yukawa interaction. The corresponding couplings are much smaller,

Yukawa and gravitational

interactions

of all quarks and leptons

163

log 9 a

Fig. 8.6 | log 9 \V(a) — | | | versus log g a for the 3A/B string in the scaling region.

since they all go with the mass squared of the particles involved. Nevertheless, they are indeed observed as suitable minima of the self energy of the 2A/2B string. Generally, for all chaotic strings scaling behaviour sets in if the coupling a approaches 0 (Fig 8.6, 8.7), and in this limit there is no difference between the forward and backward coupling form. One numerically observes for a —> 0 V{a)-V{0) = f(N\\na)-aK

(8.37)

where f^N\\na) is a periodic function of In a with period In A''2. Hence in a double logarithmic plot of \V(a) — V{Q)\ versus a one obtains a straight line that is modulated by oscillating behaviour. From the periodicity it follows that if there is some local minimum at o, then there is also a minimum at di/N2n for arbitrary n. Thus all local minima in the scaling region are only determined modulo A^2. In Fig. 8.6 the quantity \logg\V(a) - V(0)|| is plotted for the N = 3 string, in Fig. 8.7 the quantity \log4\V(a) - V(0)\\ for the N = 2 string (with diffusive coupling). For small enough a the self energy is invariant

Self energy of chaotic

164

4.5

1

1

strings

1

1

1

1

3.5 -

7

\

-

-

2.5

a2

-6

Fig. 8.7

-5.5

-5

-4.5 log 4 a

-4

-3.5

-3

:2B>

-2.5

| log 4 |V(a)|| versus log 4 a for the 2 A / B string in the scaling region.

under the transformation A —¥ B. One has V(0) = § for N = 3 and V(0) = 0 for N = 2. Whereas the 3A/B string has only 2 minima per period (essentially describing the charges of quarks and electrons, see section 8.2), the 2A/B string exhibits a much richer structure. Let us denote the local minima of the 2A/B string in the scaling region as &;. Within one period of length ln4 , one observes 11 different minima (Fig. 8.8). These we have numerically determined in

the region [0.000143,0.000572] as 6i

=

0.000199(1)

&2

=

0.000263(2)

b3

=

0.000291(1)

b4

=

0.000306(1)

h

=

0.000345(1)

be =

0.000368(3)

=

0.000399(1)

b7

Yukawa and gravitational

1

interactions

1

of all quarks and leptons

1

1

1

165

1

1

\

"\ \

X \

-

\

<~<"

b7 0.00015

0.0002

0.00025

0.0003

0.00035

0.0004

0.00045

0.0005

0.00055

a Fig. 8.8

One period of the self energy of the 2A/B string in the scaling region.

b8

=

0.000469(1)

b9

=

0.000482(1)

&io

=

0.000525(2)

6n

=

0.000558(1)

(only hi mod 4 is relevant). The remarkable fact is that these local minima of the self energy can be associated with Yukawa and gravitational couplings of all quark and lepton flavors modulo 4. Heavy particles turn out to minimize Yukawa couplings, light particles gravitational couplings. Leptons are found in the left part of Fig. 8.8, quarks in the right part (modulo 4). Note that we do observe the Yukawa couplings of the ordinary standard model and not those of a supersymmetric extension. Let us start with the heavy fermions t,b,r,c. The Yukawa coupling of the t quark was already described by the minimum a 2 = 0.03440 =: 65 outside the scaling region (see previous section). This minimum can be formally regarded as evolving out of fo5 in the scaling region. The minima ^2,&6;&io turn out to coincide with Yukawa couplings modulo 4 of r,b,c,

Self energy of chaotic

166

f t b T C

a2{mH + 2m,f) 0.03284 0.03340 0.03342 0.03342

i (5) 6 2 10

h • 104 344.0(2) 3.68(3) 2.63(2) 5.25(2)

n 0 2 3 4

strings

mf [GeV] 164.5(2) 4.22(2) 1.782(7) 1.259(4)

Table 8.1 Heavy fermion masses as obtained from the minima of the self energy of the 2A/B string.

respectively. We observe h = a-Yu - -a2{mHJr2mf)

4

(—-J

\mwJ

• 4n,

(8.38)

where / = r, b, c, respectively. Solving for rrif we get a prediction of the heavy fermion masses modulo 2. TUf

h a2(rriH + 2m/)

mw2' - n + l

(8.39)

The results obtained from the observed minima of the vacuum energy are listed in Tab. 8.1. Note the small error in the mass predictions, which is due to the fact that Yukawa couplings are very sensitive to the fermion masses. On the other hand, the results are almost independent of the Higgs mass, respectively the precise value of a 2 - Of course, the integer n, which fixes the order of magnitude of the predicted fermion masses, is not directly known from the scaling region of the N = 2 string. The scaling region can only fix masses modulo 2. However, the information on the integer n is already given by other minima of other strings for larger couplings. For example, the order of magnitude of the b mass follows from the minimum a'} (section 8.2) '(2A)

,(3A)

A3A)

or a^ (section 8.4), that of the c mass from the minimum ag or a 2 (section 8.2), and that of the r mass from the minimum a'/ (section 8.4). Hence n is known from other string states. Moreover, the integer i (i.e. which minimum is relevant for which particle) can essentially be deduced from symmetry considerations (see next section). Now let us proceed to the lighter fermions, and to the interpretation of the remaining minima 6,. Remarkably, for the light fermions where the mass is known with rather high precision (i.e. all light fermions except the

Yukawa and gravitational

/ V e d u s

i 1 4 7 8 9

bi • 1 0 4

1.99(1) 3.06(1) 3.99(1) 4.69(1) 4.82(1)

n 61 69 65 66 61

interactions

of all quarks and leptons

167

mf [MeV] 105.6(3) 0.5117(8) 9.35(1) 5.07(1) 164.4(2)

Table 8.2 Light fermion masses as obtained from the minima of the self energy of the 2A/B string.

neutrinos) we observe that the self energy has local minima for couplings that coincide with gravitational couplings modulo 4. We observe for i = 1,4,7,8,9 h — aG

rrif

•4n,

(8.40)

mpi

where f — fi,e, d, u, s, respectively. Solving for m / , we obtain the prediction mf — V 2bi mpi 2

(8.41)

of the light fermion masses. The result is shown in the Tab. 8.2. Again the integer n determining the order of magnitude of the mass is not known from the scaling region but already fixed by other string states. For example, the zeros a2 ,a\ ,a{ ' of the interaction energy W(a) yield the order of magnitude of the masses of e, d, u, respectively (see section 7.2 and 7.3) and hence the integer n. The order of magnitude of the fi mass follows from a^ (section 8.3), and that of the s mass from ag (section 8.3). Note that it is not some isolated minimum but the set of all minima of all chaotic strings that determine the fermion masses in an optimum way, i.e. that yield a standard model that has minimum vacuum energy in total. Comparing with experimentally observed fermion masses, we first notice the excellent agreement (3-4 digits) of our prediction for the electron mass me — 0.5117(8) MeV and muon mass mM = 105.6(3) MeV with the experimentally measured values me = 0.51099890(2) MeV and mM = 105.658357(5) MeV. The joint probability that just these two mass values obtained from the chaotic string agree that well with the experimental values by a pure random coincidence is of the order 10~ 6 . The joint

168

b0 »2

Self energy of chaotic strings

b{1] M

6< 2)

b2

b3

bi

h

h

y?,

T

V\

e

(t)

b

b7 d

bs u

b9 s

ho c

Table 8.3 Attribution of the various minima to the 12 fermions.

probability rapidly decreases further by taking into account all the other correctly reproduced masses. Hence a random coincidence can be excluded. Averaging the various experimental results on light quark masses in the particle data listing [Groom et al. (2000)], one obtains the experimental values mu = 4.7(9) MeV, md = 9.1(6) MeV, ms = 167(7) MeV at the proton mass renormalization scale. Our prediction of the light quark masses is consistent with these experimental values and yields in fact much more precise numbers. This is possible since our formula is based on gravitational rather than strong interaction. Generally, it is a very fortunate effect that scaling behaviour of the vacuum energy sets in for small a. For example, we would never be able to get any information on the gravitational coupling of an electron aG(me) = ^(me/mpi)2 = 8.76 • 1 0 - 4 5 in a direct simulation with a = ao- However, we can easily iterate the coupled map lattice with the much larger coupling a = a c ( m e ) • 4 69 = 0.000305 and then conclude onto aa{me) modulo 4 via the scaling argument. It is straightforward to conjecture that the remaining minima in Fig. 8.8 can be associated with massive neutrino states. This will be worked out in detail in the next section.

8.7

N e u t r i n o m a s s prediction

Let us try to find an interpretation of all minima in Fig. 8.8 that possesses the highest symmetry standards. It is in fact possible to provide a fully symmetric attribution of the 11 minima to the 12 known fermions if one assumes that the minimum b\ is degenerated, i.e. that it describes two different fermionic particles with the same mass modulo 2. This scheme of largest symmetry is shown in Tab. 8.3. Since all minima are only defined modulo 4, we have identified bo := bulA. The t quark is put into parenthesis since mt is so large that the corresponding Yukawa coupling falls out of the scaling region. Within the above scheme all up and down members of the same family are described

Neutrino mass prediction

169

by neighbored minima, and the up member always has a larger self energy than the down member (see Fig. 8.8). All leptons are grouped together (in family index order 2,3,1) and all quarks as well (in family index order 3,1,2). This scheme has the largest possible symmetry that is consistent with the observed masses. The fact that for the N = 2 string one minimum is degenerated is completely analogous and symmetric to the N = 3 string, where also one minimum was degenerated, consistent with the fact that electrons and d-quarks have the same charge modulo 3 (see section 8.2 and Fig. 8.6). The above scheme associates the three neutrino states ^1,^2, ^3 with the (2) minima b3,bo,b\ , respectively, and hence these minima fix the neutrino masses modulo 2 as mVl

= V^m

P /

2-

n i

n

(8.42)

m„2

=

^/Zb~QmPl2- *

(8.43)

m Vz

=

\[2^)mPl2-n>,

(8.44)

or equivalently

mVl

= 0.952(l)eV mod 2

(8.45)

m„2

=

1.318(l)eV mod 2

(8.46)

m„3

=

1.574(4)eV mod 2.

(8.47)

Since the relevant string coupling constant is the gravitational coupling aG = | ( m ^ / m p ; ) 2 one expects that these mass values represent the masses of the mass eigenstates v\,V2, ^3 (rather than those of the weak eigenstates To obtain concrete numerical values we still have to decide on the integers rij. Let us here again be guided by symmetry considerations. Define the open intervals I\ and 1% as 71

=

(2 4 ,2 5 )

/ 2 = (2 7 ,2 8 )

(8.48)

We observe ^ = 130.7 6 h mc ^ = 248.3 6 h mu — = 25.67 e h

(8-49) (8.50) (8-51)

170

Self energy of chaotic

strings

= 17.58 G A md mT = 16.83 G Ii

(8.53)

= 206.7 G/ 2 ,

(8.54)

me

i.e. all up-type quark mass ratios are in I2, all down-type quark mass ratios in Ix, 3rd/2nd family charged lepton mass ratios are in I\, 2nd/lst family charged lepton mass ratios in I2- The most reasonable assumption giving full symmetry to the problem is to assume that neutrinos either follow the charged lepton pattern as well, i.e. ^

Gh

(8.55)

^ G h, mVx

(8.56)

or they follow it in the opposite way, i.e. ^

Gh

(8-57)

^^G/i.

(8.58)

m„2

So far the most stringent experimental evidence for neutrino masses comes from atmospheric neutrinos, providing evidence for Am2a ~ 3 • 10~ 3 eV 2

(8.59)

and maximal mixing [Groom et al. (2000)]. Here Am 2 is the difference of squares of masses of atmospheric neutrino species. If there is a hierarchy of neutrino masses m„ 3 > > m„2 > > mVl then the above experimental result can be interpreted as m„3 « ^/Amf ~ 0.055eV. If this experimental estimate is correct with a precision of a factor 2, then eq. (8.44) together with (8.59) implies n 3 = 92, and eqs. (8.43), (8.42) together with (8.55), (8.56) imply 712 = 96, ni = 104. This means that the chaotic string spectrum yields the very precise neutrino mass predictions mVl m„2 m„3

= = =

1.452(3) • 10" 5 eV 3

2.574(3) • 10~ el/ 2

4.92(1) • 10^ eV.

(8.60) (8.61) (8.62)

Neutrino

mass

prediction

171

If instead neutrinos prefer the choice (8.57), (8.58), then only mU2 changes. In this case we get n-i = 99 and the value mV2 = 3.218(4) • 10~4eV.

(8.63)

Under the assumption that the degeneracy of the minimum b\ is an exact property, we can even further increase the precision of our prediction for m„ 3 , using the very precise experimentally known value mM = 105.658357(5) MeV of the muon mass. Exact degeneracy would imply mV3 = mM • 2" 3 1 = 4.9201006(2) • 10~2eV.

(8.64)

On theoretical grounds, we may also try to avoid the experimental input (8.59) and use a purely theoretical argument to fix the relevant neutrino mass scale. As described in section 8.2, the minima a'} and a'^ ' can be theoretically interpreted as indicating the existence of very heavy righthanded neutrinos VR of mass m„M w mv-r m 1.69 • 10 18 GeV. The seesaw mechanism [Gell-Mann et al. (1979); Falcone et al. (2001)] provides an estimate of light (left-handed) neutrino masses from heavy right-handed ones via the equation 771

mVL « —*-, mUR

(8.65)

where mq is of the order of magnitude of a typical quark mass. Choosing mq = mt, eq. (8.65) yields mVL « 1.59 • 10~ 5 eV. This is very close to the value of m„j in eq. (8.60), thus suggesting that VL in the above equation is an electron neutrino and that there is almost no mixing of v\ with v?, (or 1/3) to form ve, i.e. ve « v\. In this way one again ends up with n\ = 104 and eqs. (8.60)-(8.63) are obtained by purely theoretical arguments. Moreover, assuming that not only m„M K, mvr but also mv^ w m , ' (for the weak eigenstates t/% and i/£) this could be seen as evidence that there is maximal mixing between i/2 and u3 to form i/M and vT of nearly equal mass, consistent with the present experimental results [Groom et al. (2000)]. The predicted value for m„2 in eq. (8.61) is consistent with present experimental measurements for solar neutrinos, i.e. the small mixing-angle (SMA) solution of the solar neutrino problem. The experiments are consistent with Am2s ?s 5.4 • 10" 6 eV2 [Groom et al. (2000)], which (in case there is a mass hierarchy) could be interpreted as m„2 « ^Am2. RS 2.3 • 1 0 - 3 eV, in very good agreement with eq. (8.61). On the other hand, the other choice,

Self energy of chaotic

172

strings

-0.0045 -0.005 -0.0055 -0.006

\

0.0065 -0.007

\

X

-0.0075 -0.008 -

b,"

^

-0.0085 -0.009 0.00015

Fig. 8.9

0.0002

0.00025

0.0003

0.00035

0.0004

One period of the self energy of the 2A / B

0.00045

0.0005

0.00055

string in the scaling region.

eq. (8.63), is also consistent with the experimental data. In the so-called low mass solution (LOW) of the solar neutrino problem the experimental data allow for a squared mass difference of approximately 10~ 7 , and this agrees with eq. (8.63) squared.

8.8

The 2A

and 2B

strings — bosonic mass ratios

Finally, let us look at the scaling region of the 2A~/B~ string (Fig 8.9). Only two minima modulo 4 are observed. In the region [0.000143,0.000572] these are numerically determined as

b-

=

0.000335(1)

frf

=

0.000361(2).

The 2A

and 2B

strings — bosonic mass ratios

173

Remember that the fermion mass ratios were determined by the minima bi of the 2A/B string, according to the relation

k = (HtL.)

mod4

(866)

The quadratic dependence comes from the quadratic energy dependence of gravitational (or Yukawa) couplings. It now seems plausible that also bosonic mass ratios could be fixed in a similar way. The most plausible remaining candidate is the 2A~ /B~ string. The self energy of this string also exhibits periodic scaling behaviour with period In 4 for a —> 0, but there are only two rather than eleven minima per period. In analogy to the fermionic case we may assume that the bosonic mass ratios are given by

^=f^lVmod4. b2 \mBJ

(8.67)

Since the Z° mass is already indirectly fixed by the Weinberg angle, the free parameters are essentially the Higgs and the W mass. Suppose one of the minima describes the I^-boson, of mass 80.35(6) GeV, and the other one the Higgs boson. Then, depending on which minimum is identified with which particle (as well as the unknown power of 4), the above equation allows for the following masses of the Higgs boson: 77.4, 83.4, 154.8, 166.8, 309.6, 333.6 GeV. But experimental and theoretical bounds [Groom et al. (2000)] imply 95 GeV < mH < 190 GeV, hence only the values 154.8(7) or 166.8(7) GeV survive. It is remarkable that (within the error bars) the value 154.8(7) GeV coincides with the value 154.4(4) GeV that was predicted independently in section 7.7. This value implies that the minimum b± is attributed to W and b% to H. Numerically, one actually observes the relation =

i0/(3A) (mpiV

mod4=

(rn^V

m

(g6g)

which suggests that what is really fixed by the 2A~ /B~~ string is the ratio mVHlmBi between fermion and boson masses. Note that the coupling b~ is proportional to an inverse gravitational coupling. This reminds us of the concept of duality in superstring theories, where couplings are replaced by

174

Self energy of chaotic

strings

inverse couplings. We will come back to these types of dualities in section 11.5.

Chapter 9

Total vacuum energy of chaotic strings

Hadronic matter at low energies consists of bounded quark states rather than free quarks. We will show that chaotic strings contain information on these confined states as well. The relevant thermodynamic potential turns out to be the total vacuum energy of the string, i.e. the sum of self energy and interaction energy. Local minima of this observable reproduce the mass spectrum of light mesons (diffusive coupling) and light baryons (anti-diffusive coupling).

9.1

Hadronization of free quarks

Let us consider chaotic strings for rather large values of the coupling a. If the coupling describes strong interaction processes (within the scheme of interpretation 12, see section 5.4), this means we are looking at low energies. At low energies confinement sets in, and hence we may expect that the chaotic string also provides evidence for bounded hadronic states at low temperatures. This is indeed the case, as we shall see in the following. It should be clear that the low energy in interpretation 12 still correspond to a very large energy (of the order of the Planck mass) in the dual interpretation II (see section 5.4 for these two interpretations). We expect that the onset of confinement is accompanied by a huge change of the interaction energy of the vacuum. Indeed, the interaction energy W(a) of the 3A string is observed to rapidly increases at a « 0.3 from large negative to large positive values (see Fig. 7.2, chapter 7). In interpretation 12, a = as « 0.3 corresponds to an energy scale of the order of 2 GeV. This huge gain in interaction energy could be physically interpreted 175

176

Total vacuum energy of chaotic

strings

Fig. 9.1 Feynman web describing the formation of mesons at low energies. The dotted line describes a confined state.

as being the reason that confined states become energetically favorable. Also the self energy V(a) has a rather unusual peak for a slightly below 0.3 (Fig. 8.1), which could possibly be connected with the confinement phase transition. Generally, we saw in chapter 7 that the interaction energy of TV = 3 strings contained information on fermions, that of the N = 2 strings on bosons. Since baryons and mesons consist of fermions, we thus expect the TV = 3 strings to be relevant, rather than the N — 2 strings (which, however, can e.g. describe bounded states made up of bosons, such as glueballs). A possible way how the Feynman webs of section 5.5 could account for bounded quark states is sketched in Figs. 9.1-9.4. These figures describe possible hadronization processes for free quarks. A process relevant for mesons is shown in Fig. 9.1. In each cell of the Feynman web the free quarks q,q fly apart, and due to the large interaction energy at large distances another q, q pair is created. The result are two mesons at each lattice site i. One bounded quark within the meson at position i, formally standing for half of the meson energy, interacts with another bounded quark within

Hadronization

Fig. 9.2

of free quarks

177

Feynman web describing the formation of baryons.

the other meson at position i + 1. According to our general rules for the N = 3 string, the relevant energy scale should thus be given by E =

3 1 2^ms + 2 ' 2 m M )

=

3 2mM'

^'^

Here g stands for 'gluon' and M for 'meson'. Note that for these hadronized states we have just formally replaced the free quark mass by half of the meson mass. Similarly, Fig. 9.2 shows a typical process how baryons can be created. The free quarks q, q fly apart, and their interaction energy creates two further q, q pairs. In total, we then have three quarks and 3 anti-quarks at each lattice site. These can form a baryon and an anti-baryon. Since only one quark within each baryon interacts with another quark in the baryon at the neighbored position, the relevant energy scale is E=~{mg + 2-^mB)=mB.

(9.2)

Here B stands for 'baryon'. Again we have formally replaced the free quark

178

Total vacuum energy of chaotic

Fig. 9.3

strings

Alternative production process for baryons.

mass by one third of the baryon mass. There are also other possible production schemes for baryons and mesons. Examples are sketched in Fig. 9.3 and 9.4. In Fig. 9.3 there is just one inner fermion loop (just as for the mesonic case of Fig. 9.1), but still baryons may form by combination of 2 quarks at lattice site i and 1 quark at lattice site i + 1. It is clear that in this case the relevant energy scale should be smaller by a factor | compared to the standard baryonic case, Fig. 9.2. On the other hand, for mesons we may also have the process of Fig. 9.4, where there are two inner fermion loops (just as for the baryonic case of Fig. 9.2). At each lattice site 3 mesons can form, hence in this case the energy should be larger by a factor | as compared to the standard mesonic case, Fig. 9.1. Summarizing, for low energies we expect minima of the vacuum energy of the chaotic N = 3 string theories at couplings o = as(E) where the energy E is either \mM or \mM for mesons M and either mB or § m B for baryons B. Still we have to clarify what type of vacuum energy is relevant for confined states. Should it be the self energy or the interaction energy? Or a

Mesonic

Fig. 9.4

states

179

Alternative production process for mesons.

combination of both? Since hadrons obtain their mass both from the interaction energy of the quarks as well as from the constituent quark masses, it looks reasonable to look at the sum of interaction energy and self energy, and to regard this as the relevant thermodynamic potential for bounded quark states as soon as confinement has set in at low energies.

9.2

Mesonic states

The energy of the 1/r potential underlying the fluctuating approach of section 3.4 and 5.3 is aW. Let us thus define the total vacuum energy as H_(o)

:=

=

V(a)-aW(a) 2

= 4

^($ ) - <$ > - i a ( $ i $ i + 1 ) .

(9.3)

We regard this as the relevant observable to describe mesonic states. The physical idea behind this is that vacuum energy due to mesons consists of two different contributions. One is the contribution from the constituent

Total vacuum energy of chaotic

180

strings

0.45

0.25

0.15 -

Fig. 9.5

Total vacuum energy H-(a)

of the 3A string.

quarks (essentially described by V) and the other one is due to the interaction of the quarks (essentially described by aW). As mentioned before, we expect local minima of H_{a) at energy levels given by EM — \TUMWith smaller probability one can also expect the energy levels E = \THMThe quantities H-(a) for the 3A, respectively the 3B string, are plotted in Fig. 9.5 and 9.6. We observe minima at a = 0.341,0.351,0.356,0.375,0.412,0.416,0.615,0.676

(9.4)

Formula (7.15) with A^3) = 0.3333 GeV yields the corresponding energies as EM = 1-54,1.47,1.44,1.33,1.17,1.16,0.81,0.76 GeV

(9.5)

Two thirds of these energies give 2 -EM

= 1-03,0.98,0.96,0.89,0.78,0.77,0.54,0.51 GeV.

(9.6)

Mesonic

states

181

0.46

0.44

0.42

0.38 -

0.36

0.34

0.32

Fig. 9.6

Total vacuum energy H-(a)

of the 3B string.

This indeed coincides, within the error bars of about ±0.01 GeV, with the masses of the light mesons $,ao,r}',K*,u),p,ri,K. Their masses TUM, as measured in experiments, are [Groom et al. (2000)] mM = 1.02,0.98,0.96,0.89,0.78,0.77,0.55,0.50 GeV.

(9.7)

The two chaotic N = 3 strings thus correctly reproduce the mass spectrum of light mesons. It is quite amazing that even for these large values of as, where QCD perturbation theory starts breaking down, we still obtain sensible results. We also notice that for E smaller than about 0.9 GeV neglected higher-order terms in as(E) start becoming relevant, leading to slightly larger relative errors for the 77- and if-mass. Finally, the pion mass is too small to give us a sensible as(E„) from QCD perturbation theory. Nevertheless, the vacuum energy still has minima that presumably represent ir^ and 7T° at a = 0.771 and a = 0.781. There is also another, rather broad minimum left in the large-coupling region that has not yet been interpreted, a = 0.594. It might correspond to the only remaining light particle seen in experiments, the somewhat controversal entry /o(400 — 1200)

182

Total vacuum energy of chaotic

strings

[Groom et al. (2000)]. Generally, we observe that the 3A string has a tendency to describe charged mesons, whereas the 3B string mainly describes chargeless mesons. The value of the QCD parameter A^3' — 0.3333 GeV used above can (in principle) be deduced from the parameter A^5) = 0.20608(14) GeV determined very precisely in section 7.5. In practice, however, threshold effects and unknown higher-order contributions in as make it hard to determine A^3) precisely. We should also keep in mind that we are actually interested in an effective A^3' such that formula (7.15) with this A^3) and higher-oder terms neglected yields the best fit to the true running strong coupling as(E) in all orders of perturbation theory. To determine this optimum A^3) we can just take one hadron mass value, for example the 7/-mass mvi = 0.9578 GeV, as input and determine A^3) from the corresponding minimum a = 0.3565(3) of the vacuum energy. The result is A<3> = 0.3333(3) GeV. All other masses then follow as ratios to the ry'-mass. Proceeding two higher energies (i.e. smaller couplings a), further mesonic states can be identified with local minima. As can be seen in Fig. 9.5 and 9.6, there is particular evidence for signals from the heavy bottom mesons T, Bc and B (minima at E = ^TUM) and the charmed D mesons (minima at E — | T O M ) - TO extract the masses from the running as(E) and to identify the corresponding mesons, we use the QCD scale parameters A^™') of section 7.5. The <J> and rf mesons are observed to produce minima at E — |mjvf as well, in addition to those already observed at E = |TOMHence both production processes sketched in Fig. 9.1 and 9.4 seem to be relevant.

9.3

Baryonic states

After having obtained energy levels corresponding to mesons, we should also

expect to be able to obtain evidence for baryons. What could be a suitable modification of our present approach to describe baryons? Apparently, the quark-antiquark coupling of quarks in mesons is correctly described by the function H- — V — aW. In baryons quarks couple to quarks rather than to antiquarks. Hence it is straightforward to choose the opposite sign of the interaction energy relative to the self energy. For baryons, we thus consider H+(a)

:=

V(a)+aW{a)

Baryonic states

183

0.38

0.36

0.34

0.32 -

0.28

0.26

Fig. 9.7 Total vacuum energy H+ of the 3A string.

\{&) - ($4) + ia(^$ i + 1 )

(9.8)

to be the relevant thermodynamic potential. As mentioned before, we expect this quantity to have local minima for energy levels TUB or | m j 3 . T h e quantity H+(a) is plotted for the two types of strings in Fig. 9.7 and 9.8. Remarkably, for the 3A dynamics we observe a minimum at a = 0.508. T h e corresponding energy, obtained from formula (7.15) with A^3) = 0.3333 GeV, is 0.939 GeV and coincides with the nucleon mass. T h u s our theory provides evidence for the existence of the proton (or neutron). Further minima are observed, which can be identified with further baryons. In fact, for the two strings we observe the following minima in the large-coupling region: a = 0.356,0.367,0.378,0.451,0.468,0.508,0.609,0.628,0.679

(9.9)

(see Fig. 9.7, 9.8). T h e corresponding energies are EB = 1.44,1.38,1.32,1.06,1.02,0.939,0.813,0.796,0.758 GeV.

(9.10)

Total vacuum energy of chaotic

184

strings

0.45

0.35

l+

0.3

0.25 -

0.15

0.1

0.2

Fig. 9.8

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Total vacuum energy H+ of the 3B string.

Four out of these nine energy levels directly coincide with known light baryon masses: m N(144 o) = 1-44 GeV,m E . = 1.38 GeV,m H = 1.32 G e V , m p „ = 0.939 GeV (9.11) The remaining energy levels 1.06,1.02,0.813,0.796,0.758 GeV, multiplied with | , give -EB = 1.59,1.53,1.22,1.19,1.14 GeV.

(9.12)

These values coincide with the masses of the remaining light baryons A(1600), A(1520),A,£,A (the resonances A(1600) and A(1520) actually stand for several resonances of similar mass). In experiments, their masses are measured as [Groom et al. (2000)] mA(i6oo) = 1-6(1) GeV ,m A ( 1520 ) = 1-52 GeV,m A = 1-23 GeV, m E = 1.19 GeV ,m A = 1.12 GeV.

(9.13)

Thus the N = 3 strings reproduce the correct mass spectrum of light

Baryonic

states

185

baryons. No other light baryons are predicted, since no other minima of the vacuum energy are observed in the large-coupling region, and this again coincides with experimental observation. Actually, the A-baryon is described by a minimum corresponding to a rather large coupling a = 0.679, and the slight error of about 0.02 GeV for its mass is mainly produced by the unknown higher-order terms in the formula for as(E). To eliminate this error, we may take advantage of the fact that the minimum a = 0.679 representing the A-baryon almost coincides with the minimum a = 0.676 representing the if-meson. As a consequence, in very good approximation we should have 2 ^A = ^ A

3 = EK

= 2mK>

(9-14)

which means m\ = f m ^ . Taking as input the experimentally measured average value mx = \{mK± + mKo) = 0.4957 GeV, we get \mn = 1.115 GeV, and this indeed coincides quite precisely with the experimentally measured value of the A-mass m\ = 1.116 GeV [Groom et al. (2000)]. Similarly, the minimum a = 0.609 representing the A-baryon is very close to the minimum a — 0.615 representing the 77-meson, hence the corresponding energies should almost coincide. 2 3 EA = T W A = Ev - -mr,

(9.15)

yields m^. — \mT). Taking as input the experimentally measured 77-mass, we get jmv = 1.232 GeV, which agrees quite precisely with the experimentally measured mass of the A-baryon m& = 1.232(2) GeV. Summarizing, the N = 3 strings correctly reproduce the mass spectrum of light mesons and baryons. Again we may look at smaller couplings as well, thus proceeding to higher energies. As one can see in Fig. 9.7, there is also evidence for the charmed baryon Ac (observed energy level E = m s ) and the bottom baryon A;, (observed energy level E = § m s ) . At even larger energies, corresponding to a < 0.233, further minima are observed, which presumably describe further b-baryonic states, which have not yet been observed in experiments. There is also a minimum at o = 0.3030(2), which might provide evidence for a baryon-like particle of mass E = 1.96 GeV, respectively \E = 2.94 GeV.

186

9.4

Total vacuum energy of chaotic

strings

* CP violation

We mentioned in previous chapters that for large couplings a the N = 3 string can loose ergodicity, and expectations of observables can become dependent on the initial values. That is to say, for large a there is not necessarily invariance of expectations under the transformation s —> —s, which is equivalent to replacing T —> —T at odd lattice sites, and which means replacing quarks by anti-quarks at odd lattice sites in interpretation 12. Different results for the expectations of observables can occur if either a is large or if the lattice size is small (see, e.g., Fig. 4.4 and 4.5 in section 4.2). Indeed, for mesons and baryons relatively large couplings a are relevant. This means, the a-dependence of the functions V(a) and W(a) can slightly differ if s is replaced by — s and/or T by —T. In other words, due to loss of ergodicity the CP symmetry (or T symmetry) can be broken (see section 5.6). That can result in more baryons than anti-baryons being produced, if we think of these particles as being created from the vacuum energy of the chaotic strings (see also section 6.2). Generally, we know that CP violation is realized in nature, mainly in weak interactions, and that it helps to explain why our universe is baryon dominated. We see that chaotic strings also allow for this mechanism, at least in principle.

9.5

Planck scale interpretation

Of course meson and baryon masses are no free parameters of the standard model. In principle they can be calculated, using for example lattice QCD. So what is the physical interpretation of our results obtained in section 9.2 and 9.3? Is our theory over-determined? No, it is not. It just resembles some properties of low-energy QCD, using interpretation 12, but equally well we could think of it as describing events at or above the Planck scale, using interpretation II. Remember that in interpretation II we are considering a fluctuating scalar field at an extremely early stage of the universe where gravity and standard model interactions may still be united, and where ordinary space-time may yet not exist. In this interpretation the string dynamics is a kind of 1+1 dimensional quantum gravity dynamics (see also section 5.7). But since gravity and strong interactions are not distinguishable from each other at this early

Dark

matter

187

stage, it is clear that the quantum gravity dynamics must resemble some of the properties of QCD, provided the energy scale is re-defined in a suitable way. Thus the fluctuating <£-field may resemble some properties of strongly interacting quarks and anti-quarks, using interpretation 12. These are just the meson and baryon masses we see. Note that the running a interpreted as as can provide information on mass ratios only, rather than masses themselves. If, for example, we formally re-defined the QCD parameter A^3) to be 1019 GeV rather than 0.3 GeV, then only the absolute unit of energy would change in our approach — but all mass ratios would stay the same. Thus we may well look at a dynamics of vacuum fluctuations that takes place during the Planck era, but that already sketches in advance the mass spectrum of hadronic matter, which will be formed at a much later stage. There also seems to be some kind of universality. Clearly our chaotic dynamics is different from the field equations of QCD. In spite of that we observe a mass spectrum of allowed particle states that essentially coincides with that of QCD. Thus it seems that for certain aspects details of the field or string theory considered are not that important. Different field theories can approach the same fixed point under energy scale transformations and predict the same mass spectrum of particles.

9.6

Dark matter

Figs. 9.9-9.11 show the total vacuum energies H±(a) of all six chaotic strings of type 3A, 3B, 2A, IB, 2A~, 1B~. In all cases, for small a the functions H±{a) rapidly approach the self energy V(a) and the contribution of the interaction energy aW can be neglected. Hence, for small a our considerations of chapter 8 become valid with H± = V. For larger a and N = 3, the local minima of H± describe hadronic states. For N — 2, they may describe additional bosonic bounded states, e.g. glueballs or gravitational states. We have discussed many local minima of V(a) and H±(a) for the various strings, but clearly not all of them. One fact is clear. There are more local minima than known interaction states from the standard model. In fact, local minima of V(a) accumulate near a — 0 for all six chaotic string theories, due to the periodic scaling behaviour, which fixes minima only modulo N2. It is unlikely that for all observed minima of all six chaotic strings simple standard model interpretations can be found. Apparently,

188

Total vacuum energy of chaotic

strings

Fig. 9.9 Total vacuum energies H+ and if _ for the 3A, 3B dynamics. The figure, which looks a bit like a 'broken violin', is the union of the data of Fig. 9.5-9.8.

chaotic strings provide evidence for more massive particle states than we just know from standard model physics. It is natural to associate some of those additional states with dark matter. We can always associate an observed minimum a[ with a particle of mass rrii that interacts by gravitational interaction with a g = \{mi/mpi)2 = a^. In other words, any observed local minimum a\ could be interpreted in terms of a particle of mass mi — ^J2a'impi. In this sense the many minima observed could simply mean that there are lots of particles in the universe that purely interact gravitationally but do not take part in standard model interactions. Many of these particles may be stable and can then account for dark matter. They are the relics of the gravitational states that the chaotic strings describe in the early universe. Chaotic strings seem to predict a very broad (though discrete) allowed spectrum of dark matter particles, with masses covering all orders of magnitude. Only some of these gravitational states may be lucky enough to become 'bright' matter at a later stage.

Dark matter

Fig. 9.10

Fig. 9.11

189

Total vacuum energies H+ and H_ for the 2A, 2B dynamics.

Total vacuum energies H+ and H- for the 2A

,2B

dynamics.

This page is intentionally left blank

Chapter 10

Grand unification

We will use the precision results on standard model coupling constants obtained in previous chapters to extrapolate to much higher energies and to construct grand unification scenarios. We will consider both supersymmetric and non-supersymmetric scenarios. The observed minima of the self energy of the chaotic strings seem to support non-supersymmetric scenarios.

10.1

Supersymmetric versus non-supersymmetric theories

Due to our precision results on electroweak and strong coupling constants at energies of the order of the W-mass, it makes sense to extrapolate the running standard model couplings to much higher energies and to look for possible unification scenarios. Generally, the energy dependence of the three standard model couplings is given by the second-order formula di(E') = / 3 o l n | ;

E

+

- ^ + ^ln^^(E')+/3l//30 di(E) fo V V«i(£)+&/A)

(10.1)

The &i are the three standard model couplings, but for supersymmetric scenari scenarios it is convenient to introduce a factor | in front of ot.\ and to define dj

=

-ai

(10.2)

d2

=

a2

(10.3)

d3

= a3 = as. 191

(10-4)

Grand

192

unification

The variables /?o and fi\ stand for

* = ^( i "+ fea'(£, + fea'<£))

(la5)

ft = =£,

(10-6)

where the bi and b^ are suitable coefficients. The indices i,j, k take on values in {1,2,3} in such a way that they are all different. The firstorder coefficients bi and second-order coefficients b^ depend on whether one considers the standard model as it stands or a supersymmetric extension. For the ordinary standard model one has [Amaldi et al. (1991)] 10

+ NFam ( 1 1 + NHiggs |

\

|

(10.7)

and

K

=

+

0 -±3£

NHiggs

A

0

M o

\+NFa

.

(10.8)

Usually one works with just one Higgs particle, i.e. Nuiggs — 1, and the number of families is Npam — 3 for sufficiently high energies. Eq. (10.1) can be easily solved by iteration. Given the couplings a,\, a.2,63 at some reference energy scale E it provides the couplings at another energy scale E'. For supersymmetric extensions of the standard model, the coefficients bi and b^ are different [Amaldi et al. (1991); Langacker et al. (1993)]. In the simplest case one has

+ NFam

2

+ NHiggs

i

(10.9)

Super symmetric

versus non-supersymmetric

theories

193

and + NFa

+ Nm9gs\ U

»

•

38 15 2 5 11 15

6 5

88 15

14 3

68 3

8

(10-10)

The behaviour of the running couplings is very different in this case. For example, in the usual standard model a2(E) decreases with energy E, in the supersymmetric extension it increases with E. Which type of coefficients should we now choose to proceed to larger energies? So far we had little evidence for supersymmetry from the chaotic string spectrum. The main points against low-energy supersymmetry could be summarized as follows. (1) We observed the correct Yukawa couplings of the heavy fermions t, b, c, T as given by the ordinary standard model (section 8.6). They minimize the self energy of the 2A/B string. With supersymmetry, Yukawa couplings would (generically) change to different numerical values. But we observe the non-supersymmetric values. (2) Supersymmetric extensions of the standard model predict a lot of fermionic and bosonic particles with masses in the region 80-1000 GeV. If these particles interact strongly, they should lead to a lot of minima of the self energy of the N = 2 and TV = 3 strings in the region a = 0.080 - 0.120. However, for the TV = 3 strings we only observe one local minimum in this region, a = 0.0953, which was identified with the i-quark. For the TV = 2 strings there is also just one minimum in this region, a = 0.1186, but it roughly coincides with as(mw), and can thus probably be associated with a W or Z-decay. Hence no straightforward signatures of supersymmetric particles are seen. (3) Our prediction of the Higgs mass was mH = 154.4 GeV. This value was obtained in two independent ways in section 7.7 and 8.8, and it was also supported by the precision prediction of the electroweak mixing angle in section 7.4. For supersymmetric extensions of the standard model, however, there is an upper bound on the mass of the lightest Higgs particle, namely rnn < 125 GeV [Groom et al.

194

Grand

unification

(2000)]. This bound contradicts our value of the Higgs mass, as obtained from the chaotic string spectrum. In the ordinary standard model, the upper bound on the Higgs mass is much higher, and there is no contradiction in this case. All this looks as if chaotic strings do not support a supersymmetric standard model, but just the ordinary one. On the other hand, the assumption of supersymmetry has some theoretically attractive features for the standard model, mainly the fact that certain divergences in Feynman graphs cancel each other (though this advantage is dearly purchased by twice as many free parameters describing the masses of all supersymmetric partners). Also, there is the hope that with the supersymmetric beta functions starting to become relevant in the energy region Esup RJ 100 — 1000 GeV, the three running couplings 01,02,03 might meet in one point, allowing for the formulation of a simple grand unified theory. In the following, we will evolve our precision results on 01,(22,0:3 from the electroweak scale E — mw to much larger energies. First we will use supersymetric beta functions, and then non-supersymmetric ones.

10.2

A supersymmetric scenario

As a convenient starting point for a unification scenario, we may choose the energy scale Mt = 174.3(3) GeV. First we evolve a3(E) from a3(mw + mu + md) = 0.120093(3) to Mt using 3rd-order QCD with nn = 5 and A<5) = 0.20608(14) GeV. These were the precision parameters we obtained in section 7.5. Electroweak and supersymmetric corrections of the QCDbeta function can be neglected in this region of energy if compared to the higher-order QCD terms. Formula (7.15) yields a3(Mt)

= 0.10734(3)

(10.11)

We also evolve a2(mz) = aei(mz)/s2 = 0.03374(1) (see section 8.4) to Mt using eq. (10.1) with Npam — 3 (the result depends only very weakly on N Higgs)- Using the coefficients (10.9), (10.10) one gets a2(Mt) = 0.03389(1).

(10.12)

A supersymmetric

Similarly, we evolve a i ( m z ) = aei(mz)/{l

scenario

195

- s2) — 0.01016(1) to Mt:

ai{Mt) = 0.01028(1)

(10.13)

From Mt onwards, we evolve all three couplings using the second-order formula eq. (10.1) with Npam — 3 and with the supersymmetric coefficients (10.9), (10.10). It turns out that with this procedure the three couplings 01,02,0:3 do not meet at all in one point. Our precision is large enough to clearly say this. But the reason may simply be that we switched on the supersymmetric beta functions too early. Let us thus first evolve with non-supersymmetric beta functions and then switch to the supersymmetric ones at some energy scale Esup. The result is that the couplings still don't meet in one point if Esup < 1000 GeV, as one would expect from a supersymmetric scenario with typical supersymmetric particle masses in the region 100-1000 GeV. However, the three couplings can be made to meet in one point if Esup « 7.5 TeV. This yields the unified supersymmetric coupling di = o 2 = 03 = 0.03779(1)

(10.14)

at E = E^STY

= 7.9(2) • 1015 GeV.

(10.15)

The energy scale Esup = 7.5 TeV is larger than expected. Since generic supersymmetric particles should be lighter than 1 TeV, the value 7.5 TeV is an additional artificial parameter. There isn't really anything deep in the fact that the three couplings meet in one point, since one has introduced an additional parameter to achieve this. If this (minimal) supersymmetric grand unification scenario is correct, one should expect lots of particles with a mass of the order of the grand unification scale -E^fX which interact with the unified coupling. Hence these particles should be described by a minimum of the vacuum energy of suitable chaotic strings at or near a — 0.03779. However, none of the 6 chaotic string theories possesses a minimum for this numerical value of the coupling. The only (weak) candidate is a minimum of the self energy of the 2B string at a = 0.03805(2), which is different from 0.03779(1) within the precision of 4 digits that we can achieve. Also, this minimum is rather weakly pronounced and thus not really convincing. But we would expect

Grand unification

196

lots of particles to exist that interact with this grand unified coupling, yielding a pronounced minimum of the vacuum energy at a = 0.03779. All this seems to indicate that from the chaotic string spectrum there is neither evidence for a supersymmetric extension of the standard model nor for a supersymmetric grand unification scenario. For this reason, an alternative scenario will be studied in the next section. 10.3

A non-supersymmetric scenario

In the following we will construct a grand unification scheme that is based on the beta functions of the ordinary standard model rather than a supersymmetric extension. This model will lead to unification points of couplings where in fact local minima of the vacuum energy of the chaotic strings are observed, thus yielding a consistent picture. The only drawback is that the three couplings do not unify simultaneously at the same energy but in pairs one after the other. First, a2 will unify with a 3 at a GUT scale of order 1016 GeV, then a.\ will unify with az at the Planck scale. The scenario is sketched in Fig. 10.1. As a convenient starting point we again choose the energy scale Mt = 174.3(3) GeV. We have a3(Mt) = 0.10734(3).

(10.16)

ai(m^) and cx^ijnz) are evolved using the non-supersymmetric coefficients (10.7), (10.8) (we switch from Nniggs = 0 to Nniggs = 1 at E = m # = 154.4 GeV). One gets ai(Aft)

=

0.01023(1)

(10.17)

a2{Mt)

=

0.03337(1).

(10.18)

From Mt onwards, we evolve all three couplings with 2nd-order standard model beta-functions, i.e. eq. (10.1) with Npam = 3 and the non-supersymmetric coefficients bi and 6^. The couplings a2 and a$ meet at E = mGUT

= 1.73(2) • 1018GeV,

(10.19)

where we have a.2{mGUT) = a3(mGUT) and a i ( m G i 7 T ) = 0.01616(1).

= 0.02192(1)

(10.20)

A non-supersymmetric

scenario

197

140

120

100

-

40

20

i

1

1

log 10

Fig. 10.1

10 E [GeV]

1 C

GUT

C Ep, P I 20

Grand unification scenario based on non-supersymmetric beta functions.

At this energy scale, there is the possibility that ct2 and a-$ unify and follow a new common beta-function. This change could be accompanied by the creation of new (exotic) quark-like particles Q of mass TROUT- This should then create a minimum of the vacuum energy of the 3A string at the energy scale E = SmouT = 5.19(6) • 1016 GeV, similarly as the heavy quarks t,b,c produced minima at E = 3mq. We could for example regard Q as a kind of 4th-generation quark interacting strongly by exchange of gluons (a — as) or weakly by exchange of W® (a = a?) , both interactions being equivalent from the unification scale onwards. ot2 would then cease to exist as an independent interaction for energies larger than mauT- Looking at the 3B string, we should also observe mixed states of Q with lighter quarks such as the t-quark. Since TTIQ » rat the corresponding energy is E = \{mQ + mt) w f m Q = 2.60(3) • 1016 GeV. Remarkably, we observe minima Og and % of the self energy of the N = 3 strings that precisely fit into this picture, provided we choose for the new energy dependence from mauT onwards a strong coupling with NFam = 4. Doing 2nd-order standard model evolution of (10.20) using

Grand

198

(10.1) with NFam = 4,NHiggs

unification

— l,a2 = 0 we get

a3(3mGUT) a3 (JmauTj

=

0.02144(1)

(10.21)

=

0.02175(1),

(10.22)

to be compared with the observed minima of the vacuum energy 4(3A)

=

0.02145(2)

(10.23)

(3B)

=

0.02175(2).

(10.24)

4

Thus our picture of non-supersymmetric unification of a2 and a3 is supported by the observed minima of the N — 3 strings. They provide evidence for the existence of particles Q of mass TROUT- This precise coincidence with observed minima of the vacuum energy is in apparent contrast to the supersymmetric scenario of the previous section, where no such minima were observed. It makes the non-supersymmetric scenario much more likely than the supersymmetric one.

10.4

Final unification at the Planck scale

If we further evolve a.\ and a3 = a2 with the help of eq. (10.1) using NFam = 4, Nmggs - 1, OL2 = 0, they meet at E = 1.73(2) -10 19 GeV,

(10.25)

where we have ai=a3

= 0.01929(1).

(10.26)

Remarkably, we have E/V2 = 1.223(14) • 1019 GeV, whereas the Planck mass has the value mpi = 1.221 • 1019 GeV. Unification of a.\ and a3 seems to happen at the energy scale V2mpi —: Epi. Note that at this energy scale the gravitational coupling ao satisfies -o(EPl) = l ( ^ ) 2 2 y mpi J

= l.

(10.27)

In sections 7.8 and 8.6 we saw that the N = 2 strings are often coupled with a = aa- Hence at E = \/2mpi these strings have reached the largest coupling a = 1 where the dynamics is still well defined.

Final unification

at the Planck scale

199

No minimum of the vacuum energy is observed for the unified coupling Qi = C13 = 0.01929(1). However, we do observe the local minimum a'} = 0.00965(1) of the self energy of the 3A string, which seems to coincide with | o i = i a 3 = 0.009645(5). What could be an explanation for the factor | ? A fundamental conjecture would be that at E = Epi we once again have a phase transition with spontaneous symmetry breaking, similar to the electroweak phase transition at the electroweak scale. Remember that at the electroweak phase transition point the two couplings ot\ and 0:2 cease to exist, yielding instead electric interaction with strength

J_ _ J_ J_ Q-el

^ a

e l

Q.\

a2

QlQ2

=

.

(10.28)

Ot-i + Ct2

The electroweak mixing angle at the electroweak scale is given by sin2 6W = — — — « 0.232 OLI +

Now, at E = y/2mpi we seem to have unification strength a.\ and strong interaction of strength a$. regard, due to the earlier unification of 03 and a2 strong coupling 0:3 as a kind of weak coupling a2. angle at the Planck scale is thus given by sm2dw

(10.29)

a2

= ^ —

= l,

0:1+0:3

of weak interaction of However, we can also at the GUT scale, the Formally, the mixing

(10.30)

2

because a\ = a% at Epi = y/2mpi. By analogy with the electroweak scenario, it now makes sense to conjecture that at the Planck scale the two interactions of strength a.\ and 03 cease to exist, giving room to a new electric-like interaction of strength ael =

• = — (10.31) a.\ + 03 2 This yields the factor | we were looking for. The observed minimum of the vacuum energy, a'} ' — 0.00965(1), could describe new very heavy particles L of mass \/2mpi which interact with this electric-like unified interaction. As before, the relevant energy scale should be three times the given mass, since we are looking at an A^ = 3 string: a}3A)

=ael(3V2mPl).

(10.32)

200

Grand

unification

So far we approached the minimum a} ' from the right, starting at low energies with the minimum a 6 describing strong interactions of t-quarks and then continuing via the minimum a'^ ' describing strong interactions of Q-particles. But we can approach a^ also from the left. Remember that the minimum a'^ described a right-handed electron e# interacting weakly with strength

aweak(E)

= ael(E)S-^^, cos^ 0w

(10.33)

with E — 3m e . Now, if the energy E is increased, the electroweak mixing angle finally approaches the value sin2 dyv = cos2 8\y = \ at E = Epi. At E = Z\/2mpi the coupling (10.33) thus approaches aei(3V2mpi), if we assume that sin2 9w = § = const from Epi onwards. This is consistent with eq. (10.32). Our consideration suggests that the electric-like interaction of strength aei at the Planck scale can indeed be identified with ordinary electric interaction at this energy scale. Approaching from the left, the very heavy particle of mass y/2mpi described by a} ' can be regarded as a kind of 4th-family charged lepton L interacting electrically. Approaching from the right, it can be regarded as a very heavy quark that was originally interacting strongly, the strong interaction being spontaneously broken to unified electric interaction. In this (somewhat vague) sense, in our model quarks Q and leptons L are unified at the Planck scale. Finally, the unified gauge coupling aei = a'} — 0.00965(1) also has the same numerical value as the gravitational coupling of the very heavy right-handed Majorana neutrino introduced in section 8.2. It seems that at this point just all four interactions reduce to the same strength.

10.5

Simplification for sin 2 9w — \

Let us quite generally investigate the properties of a theory with sin2 6w = | , as realized in our model at the Planck scale. A surprising scenario arises. The weak coupling

OtWeak

= OCel

(r3-Qsin2^)2 .

2

„

7TZ

sin Vw cos^ tiw

(10.34)

Simplification

for sin 2 9\y = ^

201

becomes aweak = 4a e , (T3 - ±Q\

,

(10.35)

where Q is the charge of the particle, and T3 denotes isospin. Let us first consider right-handed particles, i.e. T3 = 0. For these particles we have aweak = aei(-Q)2

= aeiQ2,

(10.36)

i.e. weak interaction of right-handed fermions is like electric interaction. However, the charge has opposite sign! Now let us look at left-handed particles. For left-handed neutrinos VL, we have T3 = | , Q — 0, hence aweak = aei, hence the weak interaction of the neutrino is like electric interaction of an electron (with formally positive charge). For a left-handed electron ei, we have T$ = — | , Q = —1, which yields the vanishing coupling aweak = 0. This is just like the (non-existing) electric interaction of neutrinos. The left-handed u-quark (T3 = \,Q = | ) interacts with aWeak = 4a e / f ^ ~ g ) = -jT>

(10.37)

which is like electric interaction of a
(

1

1 \

A

~ 2 + fi ) = qael> (10.38) which is like electric interaction of an u-quark, again with opposite charge. Even the strength of the /3-decay, described the exchange of a W-boson, reduces to that of electric interaction. We have = - . o /. — ael(10.39) 2 sin 0w Summarizing, for a theory with sin2 8w = \ things become much simpler. Right-handed weak interaction is like electric interaction, and lefthanded weak interaction is like electric interaction under the replacement aweak

vL -4 e

(10.40)

eL^n>

(10.41)

uL -> d

(10.42)

Grand

202

unification

dL -> u.

(10.43)

All electric charges Q are formally replaced by —Q. There is no dependence on the spin of the particles, only 3 charge states with Q = | , — ^ , 1 are relevant.

10.6

Bosons at the Planck scale

In the previous sections we described a scenario where there is symmetry breaking towards an electric-like interaction at the Planck scale. If this is the correct picture, then we expect that there are also Planck scale analogues W, Z, H of the electroweak bosons W, Z, H. Their masses should be of the order mpi. The Higgs boson H gives mass to W and Z. Remember the Feynman web interpretation of the zero a[ = as(mw + mu + mrf) given in section 7.5. A W-boson decays into light quarks, which then interact strongly by exchange of gluons g. Also, remember that for the N = 3 string proceeding from the first to the second stable zero meant proceeding from light quarks to light leptons. In a certain analogy we could thus interpret the second zero a2 = | of the interaction energy of the 2A" string in terms of a Feynman web with B\ — W± ,fi=v (or v), + J2 = e (or e ), B 2 = G. That is to say, the W^ decays into electrons and neutrinos which then interact by exchange of gravitons G. According to our general rules for the N = 2 strings, this yields the relevant energy scale of the Feynman web as E = m^, + m „ +me « m^.

(10.44)

But since

^'=H-K^)2

(i 45)

°-

this yields the W mass as mw = mp[.

(10.46)

To estimate the Z mass, remember that in the electroweak case one has ^ f

= l-sin26V.

(10.47)

* Some thoughts on

supersymmetry

203

At the Planck scale, we have 2 m\^ = l-sm 9w

1 = -,

(10.48)

which implies V2mw

= V2mPl.

(10.49)

The Higgs particle H could possibly also have mass y/2mpi, associating it with the second zero o/2 — 1 of the IB" string and a Feynman web with B1 =H,h= v,h = v,B2= G. If the very heavy fermions L of mass \f2mpi introduced in section 10.4 exchange a virtual H, the corresponding Yukawa coupling is r

1

/ Till \

1

&

Yu = 4 Q 2(m^) I —— I = - a 2 ( m ^ ) = a e /(m^)

(10.50)

Hence this Yukawa-type of interaction of L reduces to the electric-like unified interaction as well. In this sense almost every interaction strength seems to reduce to the unified coupling d e /.

10.7

* Some thoughts on supersymmetry

We have seen that from the chaotic string spectrum there was no straightforward evidence for supersymmetric particles in the region 100-1000 GeV, and there was also no straightforward evidence for a supersymmetric grand unification scenario. But of course the assumption of supersymmetry has generally many attractive theoretical features. Do our results mean that there is no supersymmetry at all in our world? Although this is a possibility, the situation is not that simple. We should just be more precise with what we mean by supersymmetry and, in fact, by supersymmetry breaking. Supersymmetry says that for every fermionic (bosonic) particle there is a bosonic (fermionic) partner with exactly the same mass. How supersymmetry is broken is not yet clear at the moment, many different models coexist. One idea (different from the standard view) would be that the supersymmetric partners just live in a part of the world that is not accessible to us, for example, in a suitably compactified space of superstring theory, or within black holes, or in a .D-brane. In that case, the

204

Grand

unification

supersymmetric partners could just have the same mass as the original partners, but supersymmetry breaking is connected with a symmetry breaking phenomenon in space-time, which divides the accessible space-time volume into different separated regions. This could also be seen in connection with the 'mirror symmetry' in superstring theories [Polchinski (1998)]. We have seen in chapter 7 that the interaction energy of the N = 3 strings fixes coupling constants at the lightest fermionic mass scales, that of the N = 2 strings at the lightest bosonic mass scales. Thus, changing N, there is also a kind of symmetry between fermions and bosons for our dynamical systems. To further investigate this idea, let us consider natural units h — c — 1. Then wave functions have the dimension of an energy, measured in GeV. Pure dimensional analysis of the free Klein-Gordon and Dirac operator yields (<92-m2)$ (d-im)V

~

GeV3~$3

(10.51)

~

2

(10.52)

2

Ge^ ~* .

Thus one might link an N = 3 string, corresponding to a dynamics with terms of order $ 3 , to bosons and an N — 2 string, with terms of order $ 2 , to fermions. Also, the three possible pre-images (or the three unstable fixed points) of the map T3 remind us of three possible spin states, the two preimages (or the two unstable fixed points) of the map T2 of two possible spin states. Since the number of spin states is given by 2s + 1, this suggests to associate T3 with bosons (integer spin) and T2 with fermions (half-integer spin). But, remarkably, we saw in chapter 7 that it was just the other way round: The interaction energy of the N = 3 strings was fixing fermion masses, that of the N = 2 strings boson masses. All this could be interpreted in terms of a supersymmetry that is slightly different from the ordinary point of view. In this picture the supersymmetric partners can just have the same mass as the original partners, but the bosonic or fermionic character depends very much on whether we either choose interpretation II or interpretation 12. In fact, an unbroken supersymmetry with respect to masses may just be the reason that we were able to obtain the correct standard model masses out of the chaotic strings. What is broken, however, may be the symmetry of the space-time where the particles and their supersymmetric partners live. Perhaps we just live

Some thoughts on

supersymmetry

205

in a part of space-time where there are no supersymmetric partners. The splitting of space-time into two separated parts could also take place with respect to the fictitious time coordinate, regarding this time coordinate as a suitable dimension as well. Ordinary particles might correspond to positive fictitious time, supersymmetric partners (or mirror partners) to negative fictitious time. Again these two worlds can't interact with each other. Note that chaotic strings usually exhibit symmetry under the replacement TN -> — Tjv- In other words, the potential V+($) is replaced by V_($), which can be interpreted as a time reversal of fictitious time in interpretation II. But it is known that in stochastic quantization such a reversal of fictitious time can be associated with a supersymmetry transformation [Gozzi (1983)]. Now, when introducing the evolution equations (7.2) and (8.1) of the couplings one has to decide on the sign of the constant r , or, equivalently, the direction into which fictitious time t evolves. This choice effectively breaks the symmetry. Supersymmetric partners of ordinary particles might be formally described by maxima rather than minima of the effective potentials of the chaotic strings — but these string states are unstable states in fictitious time, once a certain time direction has been chosen. The instability could mean that supersymmetric partners, though formally there to cancel divergences in the Feynman diagrams as well as unwanted vacuum energy, are unobservable in our world. This world may correspond to positive (fictitious) time evolution, the supersymmetric partner world (or, alternatively, the mirror world) to negative (fictitious) time evolution. In principle there are at least two different ways how these ideas could be implemented. First of all we notice that maxima and minima of the function V(a) alternate, and so do stable and unstable zeros of W(a). Apparently a completely different world with different masses and different coupling strengths arises if we assume that the sign of T takes on the opposite value. This is a very radical form of symmetry breaking. On the other hand, a less radical approach would be to assume that all partner particles precisely have the same mass and interaction strengths as the original particles, so they are just described by the same zeros and extrema of the vacuum energy as the original particles, but they are by definition the unstable states obtained under fictitious time reversal. In the supersymmetric context it is interesting to note that the stochastic differential equations considered in the stochastic quantization approach can indeed provide a so-called Nicolai map [Nicolai (1980); Cecotti et al.

206

Grand

unification

(1983); Damgaard et al. (1984)]. A Nicolai map provides a very elegant 'change of coordinates' in supersymmetric field theories. If for such a field theory the fermionic fields are integrated out from the path integral, then a nontrivial determinant arises. The Nicolai map is a nonlinear transformation of the bosonic fields of the action whose Jacobian determinant exactly cancels the fermionic determinant, and, at the same time, transforms the bosonic part of the action into a free Gaussian bosonic action. In other word, there is a coordinate change from the original interacting bosonic fields of the supersymmetric field theory to free Gaussian fields r}. These fields 7] can be interpreted as the noise fields of the Parisi-Wu approach. In general, due to the central limit theorems mentioned before, on a large scale the chaotic noise fields can thus be regarded as Nicolai variables of the original fields of a supersymmetric field theory.

Chapter 11

11-dimensional space-time and quantum gravity

We will consider possible ways of embedding the chaotic dynamics into the 10 dimensional space-time of superstring theory, respectively the 11dimensional space-time of M-theory. One may consider a model where the 6 relevant chaotic strings obtained for N = 2 and 3 wind around the 6 compactified dimensions of superstring theory. The N = 1 theories are then related to 4-dimensional Minkowski space. For energies above the Planck scale the chaotic string dynamics can be interpreted in terms of black holes that emit Hawking radiation.

11.1

Chaotic dynamics in compactified dimensions

Any supersymmetric field theory can only be consistently formulated in d < 11 space-time dimensions. M-theory, the hypothetical theory of all interactions, requires 11-dimensional space-time [Banks et al. (1997); Gauntlett (1998)]. Usually one assumes that the 11 dimensions consist of the 4 ordinary space-time dimensions, 6 further dimensions that are compactified (i.e. curled up on small circles), and one additional dimension that may or may not be compactified, depending on the parameters of one of the 5 superstring theories contained in M-theory [Horava et al. (1996); Polchinski (1998)]. For chaotic strings it is tempting to relate the 11-th (non-compactified) dimension to the fictitious time of the Parisi-Wu approach, which played such an important role in our approach. This leaves us with 6 compactified 207

208

11-dimensional

space-time and quantum

gravity

dimensions *. The radii by which these additional dimensions are curled up are usually believed to be of the order of a few Planck lengths, but also much larger radii of the order of the inverse electroweak energy scale are under discussion [Antoniadis et al. (1999a); Antoniadis et al. (1999b)]. The topology of the compactified space as well as the mechanism of compactiflcation is unknown so far. Clearly the simplest topology would be the direct product of 6 single independent circles, but many other topologies are also possible. The existence of 4 generations of quarks and leptons (for which we had some theoretical evidence in chapter 10) and some further assumptions suggest e.g. the topology S2 x S2 x 52 of the compactified space [Green et al. (1987)]. To preserve what is generally believed to be the right amount of supersymmetry, one needs to compactify on special kinds of 6-manifolds, so-called Calabi-Yau manifolds [Polchinski (1998)]. In superstring theory the free parameters of the standard model are believed to be related to the so-called moduli (integration constants) in the compactified space, but no concrete predictions are given. We have 6 different relevant chaotic string theories, which we labeled as 2 A, 2B,2A~ ,2B~ ,3A, SB. These contain the information on the standard model couplings and the mass parameters and hence must be 'known' at each ordinary space-time point x^ of our world. Hence, if supersymmetric string theory is the correct theory to describe nature, then a straightforward idea is that each of the six chaotic strings winds around one of the six compactified dimensions. The periodic boundary condition of the chaotic string dynamics corresponds to the fact that each dimension is curled up (Fig. 11.1). All chaotic strings can formally be regarded as 1-dimensional
(11.1)

where ds = cdr' is a small difference in proper time r', the time a clock *For non-supersymmetric string theories there are 22 compactified dimensions.

Chaotic dynamics

f

t t +

\

in compactified

dimensions

209

X t -

+ + + + +

•

-' ^ i : : * n

X1

4

+

t t+

++

VJ

Fig. 11.1

t t

+ ++

y?

Chaotic string winding around a compactified dimension.

measures that travels between space time points separated by dx** [Kenyon (1990)]. Looking at the way we derived the string field equations in section 2.2 from a quantized self-interacting field theory it seems reasonable tc regard a = 2T/12 as a kind of metric gjj in the compactified space, if we identify ds1 = 2r and dx2 = I2. Here r and / are the small lattice constants we introduced in chapter 2 (remember that r had dimension length2). We can also exchange the role of ds and dx, obtaining the result that a = gj- is an inverse metric. Finally, the quantum gravity interpretation of section 5.7 suggested that 1 — const • a was the expectation of a strongly fluctuating metric. We will see that our considerations in the next section do not depend on the precise form of the relationship between a and gjj in a significant way. All we need is that the metric gjj in the compactified space is a strictly monotonous function / of the coupling a. Chaotic strings winding around the compactified space can be very well used for chaotic quantization. Consider an ordinary particle, for example an electron with wave function 9(x,t) in ordinary 4-dimensional space-time. In order to do second quantization, we can generate the necessary noise

210

11-dimensional

space-time and quantum

gravity

field L(x, t) by any of the many available chaotic noise field variables $ „ in any of the six compactified dimensions (labeled by j) at position x. For example, we could choose chaotic noise generated by the four N = 2 theories to 2nd-quantize the four components of the Dirac spinor \P, and similarly noise from the two N = 3 theories could be used to 2nd-quantize wave functions of bosons. The difference compared to chapter 1 is that now the chaotic noise fields <S>^ are just coupled within the compactified dimension, not in ordinary space-time. So no problem with correlations in ordinary space-time arises, and we rigorously get white noise under rescaling due to the central limit theorems mentioned in section 1.2, provided the coupling a is small enough. We can actually keep ordinary 4-dimensional space-time to be continuous. Just in the compactified dimensions the space coordinate of the chaotic string is effectively discrete. An unsolved problem in superstring theory is to explain why the compactified dimensions are actually compactified. One type of models one could think of is to assume that the chaotic strings winding around the compactified space are rotating. The balance of centrifugal forces and gravitational attraction of the vacuum energy on the chaotic string would then allow for a stationary state of the compactified dimension. After an equilibrium state has been reached the compactified dimension neither expands nor contracts but just has a finite diameter R (see section 3.6 for more details).

11.2

Quantized Einstein field equations

In ordinary 4-dimensional space-time, the stress-energy tensor T/iU(x) of gravitationally interacting matter determines the metric g^i/(x), according to the classical Einstein field equation GW - A
(11.2)

The metric tensor gM„ determines the way local distances ds are measured in curved space-time, see eq. (11.1). The stress-energy tensor TM„ essentially describes the energy density and the flow of 4-momentum components of real matter in the various space directions. G^„ is the Einstein tensor, describing the curvature of space-time. It is a nonlinear (cubic) function of the components gM„ of the metric tensor and their second derivatives in

Quantized Einstein field equations

211

the various space-time directions. G is the gravitational constant and A is the cosmological constant. If we stochastically quantize the Einstein field equation, a possible Langevin equation is the following one: 9i*v(x>t) - -^-g^v(x,t)

+Gllv(x,t)

^-r^„(a;,t) + noise

(11.3)

The metric g^u is now a random variable, which evolves in fictitious time towards a certain equilibrium metric, determined by the classical matter distribution. There are various possibilities to stochastically quantize gravity, so eq. (11.3) is just one possibility, see [Rumpf (1986)] for a survey. Now let us consider 10-dimensional space-time with 6 compactified dimensions plus the fictitious time needed for quantization. We have to extend the metric tensor and the Einstein equations to higher dimensions (see also [Polchinski (1998)]). The simplest tensor one can think of is a 10 x 10 matrix which is the ordinary metric tensor for fi, v = 0,1,2,3 (ordinary space-time) and which is diagonal in the additional compactified dimensions. We may assume that the diagonal elements gjj,j = 5 , . . . ,8 are simple functions fj (a) of the coupling constants a of the four types of N = 2 strings and the elements gjj,j = 8,9 simple functions fj(a) of the coupling constants of the two types of TV = 3 strings: 344 = 955 =

fM2A~]) / 5 (a< 2 B "))

(11.4) (11.5)

966 =

/6(a(2^)

(11.6)

377 988 g99

= = =

{2B)

)

(11.7)

{3A)

)

(11.8)

3B

(11.9)

fr(a h(a

/ 9 (a< >).

The non-diagonal elements are assumed to be zero. This is certainly reasonable for empty space, though most generally one could also think about non-diagonal theories. As we have seen before, typical choices of functions fj that physically make sense are e.g. f(a) — a, f(a) = a"1 or f(a) = 1 — const -a. In a similar way we can also associate the metric of ordinary 4-dimensional space-time with the couplings of the four types of N = 1 strings (see next section). If this picture is correct, then the 4 ordinary space-time dimensions are related to the 'ground state' of our (formal) quantum mechanical string oscillator (N = 1), and the compacti-

212

11-dimensional

space-time and quantum

gravity

fled dimensions to the first (N = 2) and second (N — 3) excited state (see also section 5.5). In the quantum gravity picture described in section 5.7, the metrics of the excited states are strongly fluctuating, and the above numbers gjj then stand for expectations of the strongly fluctuating metric. Now consider any of the compactified dimensions, labeled by j . One way to proceed further is to assume that the compactified space is just empty since the strings only describe virtual particles. We may thus assume that the average curvature Gjj and average energy density Tjj are just 0, and in each dimension j the cosmological constant A is given by Aj=TW^(a),

(11.10)

where F ~ G~l is a constant of dimensions energy2 and W^\a) is the interaction energy of the chaotic string considered (j takes on the values 2A~,2B~, 2A,2B, 3A,3B). We define the sign of T as sign T = -sign / j ( a ) / / j ( a ) . The function W^(a) = ±(in&+1) is defined to be dimensionless. Since gjj = fj{a) one obtains gjj = fUa) • a, and hence the Einstein field equation (11.3) with fi, v replaced by i, j leads to a=-^\rW^\a)+noise, Jj (a)

(11.11)

which makes arbitrary couplings a evolve to the stable zeros of the interaction energy. More precisely, the quantum mechanical expectations of a with respect to the noise evolve to the stable zeros. Alternatively, we could set Aj = 0 and Gjj = 0 but assume that the energy density Tjj (defined to have dimension energy4) is proportional to the interaction energy W^ (a) of the chaotic string. 8

-^Tjj=TWM(a)

(11.12)

Again T is a constant of dimension energy2, and clearly the real meaning of the quantities on both sides of eq. (11.12) is that of an energy per length, as associated with the energy density on the 1-dimensional string. This approach leads to the same evolution equation (11.11) provided we choose

r = fj(a)r.

(n.13)

Most generally, we could also allow for curvature in the compactified space

N = 1 strings and Minkowski

space

213

and define 8TTG

Tjj-Gjj

= TWW(a),

(11.14)

again obtaining eq. (11.11). In this way stable zeros of wW(a) fix all relevant equilibrium metrics in the compactifled space. As shown in detail in chapter 7, the corresponding distinguished couplings a appear to coincide with observed standard model couplings.

11.3

N = 1 strings and Minkowski space

Let us now consider strings with N — 1, i.e. coupled map lattices of lstorder Tchebyscheff maps T±($) = ± $ . It is interesting to see that the (non-chaotic) jV = 1 strings indeed resemble some of the properties of (empty) Minkowski space. For N = 1 the string field equation ¥n+l = (1 - a)TN(^J

+ 8^(TbN(*iTl)

+ TbN(K+1))

(11-15)

degenerates to a discrete diffusion equation, up to a possible sign. Four different dynamics are obtained:

N

iV = l , S = l

#j l + l = ( 1 - « ) * n + f ( * n - 1 + * J l + 1 )

(H-16)

N = 1,S = -1

*j l + l = ( l - o ) * n - f ( * n " 1 + * n + 1 )

(11-17)

N =-1,8

= 1

K+1

_i

$ji+i

=

_ljS

=

= - ( 1 - a)**„ - l ^ j , " 1 + <^+1) (11-18) =

_ ( i _ a)&n + £($^-1

+

$j+i) ( n . 1 9 )

Again we assume periodic boundary conditions. The notation A^ = ± 1 means that the local map is either the positive lst-order Tchebyscheff map Ti($) = + $ or the negative one T_i($) = - T i ( $ ) = - $ . Note that for the chaotic strings with N > 2 the asymptotic behaviour is invariant under N —• — TV, since these strings are ergodic if a is not too large. For N = 1, however, this is not the case, and a different dynamics arises if A^ is replaced by — N. Also note that for the N = 1 strings the backward coupling form (6 = 0) does not yield anything new, since Xi(<J>) = $ is the identity. We denote the four strings as 1 + , 1~, — 1 + , —1~, where the upper index is the value of s. We notice that the number of different N = 1 strings

11 -dimensional

214

space-time and quantum

gravity

as given by eq. (11.16) - (11.19) coincides with the number of (ordinary) space-time dimensions. Iterating the dynamics, for all strings it is observed that for arbitrary random initial values $Q € [—1,1] the fields <&ln smooth out in time — up to a sign pattern that is invariant under the dynamics. For arbitrary a £ (0,1) one observes after sufficiently long iteration the following patterns: + + ++

(iV = l , s = l) = l+

+ + ++ + + ++ + + ++ + - + -

(7V = l , s = - l ) = l -

+ -++ -++ -++ + ++

(JV = - l , s = l) = - l +

+ + ++

+ - + -

(W = - l , a = - l ) = - l -

- + -+ + -+- + -+ These patterns can be easily understood, since the dynamics of N = 1 strings is linear. Iteration of the coupled dynamics is just matrix multiplication of a cyclic matrix (see section 2.6). The first one of the above patterns (N = l , s = 1) can also be just minus signs — this depends on the initial values. Clearly the (N = l , s = 1) pattern is different from the other 3 patterns. It is the only one that consists of one and the same sign variable. The other patterns have both signs equally often. Our physical interpretation is as follows, n may again be the fictitious time of the Parisi-Wu approach. In the beginning of the universe, the spatial directions of the four N = 1 strings span up the 4 dimensions of ordinary

Potentials for the N = 1 strings and

inflation

215

space-time. The 1 + string corresponds to the physical time dimension. Due to spontaneous symmetry breaking (depending on the initial values) all fields $J, obtain the same sign for n ~> oo. This means time only goes in one direction, there is an arrow of time. The l - , — 1 + , — 1 ~ strings correspond to the 3 spatial directions of 4dimensional space-time. Here for n ->• oo the fields $^ alternate in sign (though in a different way for each dimension). This means one can go forward and backward in space. Note that by a rotation of +45° the 1~ pattern is transformed into the —1~ pattern, and by a rotation of —45° the — 1 + pattern into the —1~ pattern. This means all three space dimensions are equivalent. It is possible to 'rotate' them into each other, but it is not possible to 'rotate' them into the time pattern. All this suggests that in empty 4-dimensional space-time we can relate the string couplings o ^ 1 ) to the Minkowski metric, and define ffoo = 9ii 522 933

= = =

/o(a ( 1 + ) ) /i(a

(1_)

(11.20)

)

(11-21)

( 1+)

/2(a "

( 1_)

/3(a -

)

(11-22)

).

(11.23)

For example, a very simple choice would be fo(a) = —a and fj(a) = a,j = 1,2,3. If the 4-dimensional space-time is not empty but contains gravitationally interacting matter then the above simple relations will generally change. Non-diagonal terms can occur as well, and the relevant equilibrium metric is determined by the Einstein equations.

11.4

Potentials for the N = 1 strings and inflation

Similarly as for the JV = 2 and N = 3 strings, we may also define the potentials V± and W± for the N = 1 strings. For Ti($) = + $ we have $ n + 1 - $ „ = 0 = F£\Qn)

= -—K*1)^).

(11.24)

Hence

vi1\*) = c,

(11.25)

11-dimensional

216

space-time and quantum

gravity

where C is an arbitrary constant. For T_i($) = —$ we obtain $ n + 1 - $„ = - 2 $ „ = F l

1

^ ) = -g^VW^n),

(11.26)

which implies yl 1 )($) = $ 2 + ( 7 .

(11.27)

Again, for symmetry reasons we may choose C = —1($ 2 ), which formally yields (*)> = - ( y i 1 } ( $ ) ) = - ^ ( $ 2 ) .

(11.28)

The interaction potential between nearest neighbors $ l —: $ and $* +1 := W that generates diffusive or anti-diffusive coupling is again given by W±($, * ) = - ( $ ± * ) 2 + C = - ( $ 2 + * 2 ± 2 $ $ ) + C.

(11.29)

(see section 3.4). But there is a significant difference compared to the N > 2 case. The N = 1 dynamics is neither chaotic nor ergodic, and hence we first have to define what we mean by the expectation (• • •) in eq. (11.28). It is most reasonable to define that (• • •) again means averaging over all n and i, omitting the first few transients. For n —> oo, however, the field $^ becomes homogeneous and approaches 0. This is true for all four types of N — 1 strings and all couplings a > 0. This means we have asymptotically C = -I<$2)

= 0,

(11.30)

in marked contrast to the N > 2 case. Hence only one relevant self interaction potential remains, V_ ($) = $ 2 , whereas the other one, , is zero. In fact, when iterating the N = 1 strings one sees that there are two different relevant time scales, a fast one (which we may call tn) on which the field becomes homogeneous (up to the sign pattern) and a much slower one on which the field reaches the asymptotic value 0. Using eq. (11.29) as well as C — 0, one can now easily verify that after the time tu has elapsed one has 7i1)($) = $ 2 ^ 0

(11.31)

Black holes, Hawking radiation, and duality

217

whereas W±($,V)

= 0.

(11.32)

The latter equation is valid for all four strings since $\I/ = s $ 2 due to homogeneity and the sign pattern. Remember that stable zeros of the expectation (W+($, \&)) were fixing equilibrium metrics in the compactified space for N > 2. Now, for N = 1 in = 0,1,2,3) we see that W^\a) = \{W±($,*)) = 0 for all a. This means that a priori all couplings a are allowed as an equilibrium metrics, a marginal situation. Hence for N = 1 the metric is not fixed by zeros of the vacuum energy W^\a) but fixed by the next-order contribution, the distribution of real matter, which fixes the metric according to Einstein's theory. Generally, a homogeneous field $ with V($) > 0 can generate inflation in the (so-called) chaotic inflation scenario [Linde (1982); Linde (2001)]. This is a period of very fast expansion of the universe [Guth (1981); Brandenberger et al. (1990); Kolb et al. (1990)]. Starting from random initial conditions, and evolving according to the N = ± 1 string dynamics, the fields $^ become homogeneous after the time tn has elapsed. Thus the strings with N = —1 have a chance to drive inflation during a transient period, namely when the fields §ln have reached sufficiently large homogeneity in the spatial direction and when V_ ($) is still sufficiently large. Should all or part of this consideration turn out to be the correct physical picture, then one might be able to embed the fields $J, into inflationary scenarios. The chaotic fields could then simply be regarded as higher-./V excitations of inflaton fields [Beck (1995c)].

11.5

Black holes, Hawking radiation, and duality

Let us now come back to strings with N > 2 exhibiting chaotic behaviour. We want to understand what happens if (in interpretation 12) the temperature E ~ kT surrounding the chaotic string becomes significantly larger than the Planck mass rripi. Remember that the energy E — ^/2mPl was the end point of the scenario we described in chapter 10. If the energy E exceeds an energy of the order mpi, black holes of mass M ~ E > 0(mpi) can be created within the string. The remarkable property of black holes is that they emit Hawking radiation and behave like

218

11-dimensional

space-time and quantum

gravity

a black body at Hawking temperature TH- In natural units (% = c = 1) the Hawking temperature of a (non-rotating) black hole is given by

kT

» = h(k'

(1L33)

where M denotes the mass of the black hole. Hence the larger the mass M of a black hole, the smaller is the temperature and thus the energy density of the radiation it emits. In principle all kinds of particles can be emitted by a black hole, but in practice the masses of the emitted particles are of order O(kTn) or less. Hence only very small black holes can emit massive particles such as electrons, quarks or heavy bosons. It is only when the mass M of the black hole becomes as small as the Planck mass mpi that particles with a mass of the order of the Planck mass can be emitted. Naturally, for this reason there cannot be black holes with mass M < 0(mpi). Eq. (11.33) is the correct formula for black holes in 3 space dimensions. However, remember that in interpretation 12 we are considering 1-dimensional physics. In 1 dimension we might expect a black hole to radiate with kTH = C ^

(11.34)

where the constant C is not necessarily l/(87r). One readily observes that there is fixed point in eq. (11.34) where fcT/j = M. Since in natural units the gravitational constant is given by G = m ^ the fixed point is kTH = M = y/Cmpt.

(11.35)

A mini-black hole of this mass can immediately evaporate by emitting another black hole of this mass. Now suppose we further try to increase the energy E = kT in order to resolve distances smaller than the Planck length within our chaotic model of vacuum fluctuations. We will not succeed. The high energy will create black holes of mass M ~ E, but the event horizon (Schwarzschild radius) of these black holes is given by ro = 2GM. It increases with increasing M ~ E. All black holes created within the string will just start to interact via Hawking radiation, similar to the picture sketched in Fig. 11.2. They emit some ordinary particles (e.g. electrons) which then interact by exchange of some gauge bosons (e.g. photons). The relevant temperature that describes our system is not E = kT anymore, but the Hawking temperature kTn- At the

Black holes, Hawking radiation, and duality

219

Fig. 11.2 Feynman web interpretation of the coupled map dynamics for energies above the Planck scale. Back holes (denoted by B) emit low-energy particles (e.g. electrons) that interact (for example) electrically.

fixed point VCmpi the role of kT and kTjj will be exchanged. Formally, if we let E = kT ~ M -> oo, we will end up with a low-temperature theory kTH -> 0. The process of decreasing Hawking temperature can be significantly enhanced by the fact that two black holes of mass M within the string may combine into one bigger black hole of mass 2M. This process may repeat again and again in a cascade-like process, which leads to an exponentially increasing black hole mass and an exponentially decreasing Hawking temperature. For rapidly decreasing Hawking temperature the event horizon rn = 2GM rapidly increases. It coincides with the Compton wave length of the low-energy particles emitted. Our ideas can be seen in connection with the 'duality' principle (see, e.g., [Witten (1997)]). In superstring theory, a dual theory is obtained by replacing large couplings a by small couplings 1/a (S-duality), or by considering instead of a compactified dimension of radius R a compactified dimension of radius 1/R (T-duality). If for our chaotic theories we go to en-

220

11-dimensional

space-time and quantum

gravity

ergies E larger than y/2mpi then the gravitational coupling a = ^(E/mpi)2 formally becomes larger than 1. A chaotic string with a > 1 is ill-defined and leads to diverging behaviour. However, this apparently is not the right theory to study. Rather, we should replace ordinary temperature by Hawking temperature, which essentially means replacing E by 1/E. Also, the Compton wave length A ~ 1/E should be replaced by the event horizon rQ ~ E ~ 1/A, and so on. In this sense well-defined 1-dimensional chaotic string theories with metrics determined by low-energy standard model couplings can be obtained for very large energies E » mpi. They might be regarded as 'dual theories' of a high-energy theory, exhibiting effectively low-energy behaviour. 11.6

The limit E -)• oo

For E < \/2mpi we saw that the correct string coupling constant due to gravitational coupling is given by aG = ^(E/mpi)2. For E —> V^mpi, a = aa approaches 1 and the chaotic string dynamics does not exist for larger a. To define an effective temperature for larger energies, we proceed to the Hawking temperature

the temperature of a black hole of mass M. We want to have a fixed point E — UTH at E — M = \f2mpi. This implies C — 2. Hence the idea is that for E > V2mpi we should use the effective energy E = kTH = 2-^ E

(11.37)

rather than E itself. Generally, the gravitational coupling corresponding to the energy E is

<*=; £ - » m ' E_

mpi

38

<"- >

Hence a~G = — , (11.39) aG i.e. we have duality between the theories described by aG and aa-

The limit E —> oo

221

For E -> oo the (dual) gravitational coupling approaches 0, and if the N = 2 string keeps on being coupled gravitationally with Q.G then it just reduces to a free Bernoulli shift with 2 symbols in the limit E ->• oo. The N = 3 string was coupled with the unified electric coupling aei = 0.00965 at E = EPl = \f2mpi (see section 10.5). How does aei(E) further evolve with increasing E > y/2mpi? As shown at the end of section 10.5, we can also interpret aei as a Yukawa coupling, in which case aei is proportional to (E/mpi)2—but for E > V^mpi our recipe was to replace E/\/2mpi by \f2mpLJE. Hence we expect aei(E) ~ {mpi/E)2 for large E. We expect that the coupling decreases quadratically with E just like the dual gravitational coupling does. Hence, for E -> oo the N = 3 string also reduces to a free Bernoulli shift, this time with 3 symbols. Instead of making aei quadratically energy dependent we could also regard the unified gauge coupling constant to be (almost) constant from the Planck scale onwards and just let the charge grow in proportion to E. In units of e, we have Q = 1 at the minimum described by a 4 . From Epi onwards the dual charge decreases proportional to 1/E, the original charge Q increases proportional to E. In this picture, there is equivalence between electric and gravitational interaction strengths, or between mass and charge. We just have to measure the mass in units of \/2mpi and the charge Q in units of e. In section 8.6 we saw that for a —¥ 0 minima of the self energy of the 2A/B string occur at couplings given by a = | ( m / / m p i ) 2 • 4", where / is a light fermion. Generally, all minima of the vacuum energy were only determined modulo N2. How can this invariance under multiplication with a factor N 2 be physically understood? A simple picture would be that the strings fold due to the strong gravitational attraction. This is illustrated in Fig 11.3 for the N = 2 strings and in Fig. 11.4 for the N = 3 strings. It is apparent that such a folding process can occur arbitrary often, say n times. In each step the energy density of the string per unit length increases by a factor 2, respectively 3. Since we do not know how often the string has folded itself, all masses are only fixed up to a factor 2 n . Similarly, all charges are only fixed up to a factor 3". The dual gravitational coupling changes by a factor N~n in n steps of this cascade process. All this is precisely described by the log-periodic oscillations of the local minima of the vacuum energy in the scaling region. Another way of physically interpreting this is that for N = 2 two black

11-dimensional

222

f

+ ++ + + + + + + + + + + + + + + + + + + + + + + + ++

space-time and quantum

gravity

\+

X^+y /

++ + + + + + + + + + + + + + + + + +

+ + + + + ++

++?**%

->

++++ + + + + + + + + + + + + + + + + + + + + + +

* -H-H• + + + + * * -H-H-

\\+++%«#-#

Fig. 11.3 N = 2 string folding to a new string with half of the original diameter and double the energy density.

holes B in Fig. 11.2 can combine into one new black hole with twice that mass. For the N = 3 string three black holes can combine into one new allowed state with three times that mass (or charge). All this somewhat reminds us of the way mesons and baryons are formed out of 2, respectively 3 quarks. Just now we are well above the Planck scale, and black holes rather than quarks combine into bounded states. The TV = 2 strings presumably describe 'mesonic' black holes (with integer spin), the N = 3 strings 'baryonic' black holes (with half-integer spin). In contrast to hadronic matter at low energies, an infinite cascade of the combining process is possible.

11.7

Brief history of the universe — as seen from chaotic strings

The main idea of this book is that in the beginning the universe is neither matter nor radiation dominated, but information dominated. All that is there is a shift of information in fictitious time n. This initial shift of

Brief history of the universe — as seen from chaotic strings

/ ^ \

+ -K + + + + + + + + + + + + + + + + + + +

++ + + + + + + + + + + + + + + + + + + +

+ + + + + + + +

+ + + + + + + +

-f-

X

223

j f l W+ i ,

>

++# +-H+-H+-H+++ +++ -H-+ -H-+

+1^ +++ +++ +++ +++ +++ +++

-H-+

%#

-f

Fig. 11.4 N = 3 string folding to a new string with a third of the original diameter and three times the energy density.

information is a Bernoulli shift of JV symbols and it is mathematically described by iterations of Tchebyscheff maps T^'$n+1 =

TN($n)

(11.40)

This dynamics can also be regarded as a deterministic chaotic dynamics of vacuum fluctuations at a stage of the universe where space-time does not yet exist. A priori N can take on any value. This is just describing different ways of extracting information out of the initial value $n, using N different symbols (see section 1.6). From a quantum mechanical point of view, N has similarities with the quantum number of a harmonic oscillator, with N = 1 corresponding to the ground state and iV > 2 to higher excited states. As shown in section 1.6, we may write $„ = cosu0i, where i := irNn grows exponentially, similar to a de Sitter state. So far this is a 0-dimensional universe (or a l-dimensional one, if the fictitious time is counted as a dimension). The next step is that some space-time dimensions are created. These are created with the help of the

224

11-dimensional

space-time and quantum

gravity

lowest states N = 1, 2,3 of the above 'information shift oscillator'. The four N — 1 strings, the four N = 2 strings and the two N = 3 strings initially span up 10 space-time directions of this pre-universe. These may be the 10 space-time dimensions of superstring theory. Together with the fictitious time coordinate, they may represent the 11 dimensions of M-theory. A space-time dimension of type N is created by replacing a map Xjv by N identical copies and introducing a tiny little coupling between the maps. This process can repeat again and again in a cascade-like process. The dimension unfolds via the inverse of the process sketched in Figs. 11.3 and 11.4. The four N — 1 dimensions hardly develop at this early stage of evolution, they stay almost point-like. The initial coupling a of each chaotic string is extremely small. It is given by the dual gravitational coupling

«= 2 (^r

am)

where E is of the oder Mu, the entire mass of the universe. The current astrophysical observations suggest that the density of the universe is given by the critical density P c « 5 ^ ,

(11.42)

which corresponds to a flat universe. The age to of the universe is about t0 « l5Gyr = 4.7 • 10 17 s.

(11.43)

From this we can estimate the total mass of the universe as being of the order Mv as \Tt{ctofpc

« 6 • 1079 GeV.

(11.44)

This implies that initially the coupling of the chaotic strings is given by something of the order a = 2(mpi/Mu)2 ~ 8 • 10~ 122 . The initial average 'temperature' of this pre-universe is of the order kT « 1079 GeV, but the physics is much better described by the initial Hawking temperature, which is of the order kTu ~ 1 0 - 4 2 GeV, i.e. very small on average. Strong fluctuations of temperature are possible (see section 6.6), so in principle all kinds of couplings can occur locally. They evolve to local minima of the vacuum energy of the chaotic strings, and

Brief history of the universe — as seen from chaotic strings

225

particles with standard model properties can be created out of the vacuum energy. The local minima of the vacuum energy V(a) of the chaotic strings describe possible masses of particles in the universe at a given temperature. At this stage of evolution the relevant physics is mainly consisting of black holes that have very large mass and emit low-energy standard model particles by Hawking radiation. Black holes of N = 2 type decay into two black holes of half that mass, and in a cascade-like process we obtain black holes of mass 2~nM. Similarly, black holes of N = 3 type decay into 3 black holes with a third of that mass (or charge), and in a cascade-like process we obtain black holes with charges 3~ n M. Naturally, masses and charges of low-energy particles emmitted are only fixed modulo 2, respectively modulo 3 at this early stage. The average temperature T decreases rapidly, the average Hawking temperature T# increases rapidly. Although we think of this pre-universe as evolving out of an initial singularity, the question whether it is actually regarded as expanding or contracting is mainly a question of definition, due to T-duality and due to the fact that the physical time coordinate is not yet fully developed. Two different types of potentials are relevant, the self energy V(a) and the interaction energy W(a) of the chaotic strings. Masses (modulo 2) and charges (modulo 3) of possible particle states are fixed by local minima the self energy, and stationary states of coupling constants (= interaction strengths) are determined by stable zeros of the interaction energy. These two degrees of freedom are natural since the fictitious time evolution is discrete and consists of two steps, one generated by V and the other one generated by W. After a cascade process of about n 2 ~ log2 2fv « 201 unfolding steps for N = 2 strings, respectively n 3 ~ log3 ¥u

~ 127 steps for

*\J ^ lib P I

19

N - 3 strings, we end up with kT w 10 GeV « kTH, the temperature at the Planck scale. Typical particles now have a mass of the order mPi, but again there can be large inhomogeneities and large local fluctuations in temperature. On average, at this stage the black-hole character of the particles making up the universe ceases to exist. About Nv ~ 2 201 « 3 • 1060 particles have been created out of the initial vacuum energy in the cascade process, and in case eq. (3.75) in section 3.6 is valid then the typical size of the compactified dimensions has reached a value of the order R ~ Np/4/mPi ~ (104 GeV)-1. This a grees with typical radii discussed in

226

11-dimensional

space-time and quantum

gravity

[Antoniadis (2001)]. Note that for the special coupling a = 1 corresponding to E = \[2mpi the 2B~,2B and 3B string dynamics is actually equivalent to the N = 1 string dynamics. At this special point one has 2B~ = 1" = - 1 +

(11.45)

3B = 2B=l+ = -l~.

(11.46)

and

Hence at E = Epi = y/2mpi a kind of crossover from the N = 3 and N = 2 dimensions to the N = 1 dimensions is possible. The N > 2 IBstrings may generate 'random' initial conditions for the N = 1 strings and particles may directly flow from the 'excited' space-time described by N = 2 and N = 3 into ordinary 4-dimensional space-time, the ground state with N — 1. In fact, it may be just this point where it starts making sense to talk about ordinary 4-dimensional space-time. The N = 1 string dynamics will smooth out the inhomogeneous initial seeds from the B-strings. After some iterations sufficient smoothness is obtained. This means, ordinary inflation can set in shortly after the Planck epoch and expand the N — 1 universe (see section 11.4). It leads to a flat universe. Once the N = 1 dimensions have reached a significant size, particles created out of the vacuum energy of the chaotic strings (flowing into ordinary 4-dimensional space-time) 'decide' whether they continue to interact just gravitationally (making up dark matter) or whether they are distinguished particles that can also interact with a gauge coupling. This choice may depend on whether their origin is vacuum energy of the N — 2 strings (essentially producing dark matter) or of the N = 3 strings (essentially producing 'bright' matter). At E = \f2mpi the gravitational coupling is aa = 1 and the unified gauge coupling is given by aei = 0.00965. Since less mass is required to make matter with a gravitational coupling of the same size as aei, one might conjecture that the mass ratio between bright and dark matter created at this stage is given by ^bright Wl'dark

_

1

/aa

=0.049,

(11.47)

'&

the factor 1/2 originating from the fact that there are twice as many N = 2 theories as N = 3 theories. The above ratio coincides with current observational estimates.

Brief history of the universe — as seen from chaotic strings

227

At E = \^2mpi gravitational and electric forces split, formally by replacing y^mpi by e. The unified gauge coupling aei immediately splits into weak and strong forces, of strength a.\ — 013 = 2aei. The coupling a,\ decreases with decreasing temperature, whereas 0:3 increases. Also Yukawa couplings split from gravitational forces. A bit later, at EQVT — 1-7 • 1016 GeV the coupling ai splits from a 3 . We then have the electroweak phase transition at the electroweak scale, and later the quarks in ordinary 4dimensional space-time can form baryonic and mesonic states, as already described by the total vacuum energy of the N = 3 strings at an earlier stage. To describe the present stage of the universe, the chaotic strings are presumably still there, and if they keep on being coupled gravitationally then we expect that today the coupling constant a would be of the order E \2 a = aG = -

rnpij

=1.85-1(T64,

(11.48)

where E = kT0 = 2.35 • 10" 4 eV

(11.49)

is the present temperature of the cosmic microwave background. The strings can still generate the noise for stochastic quantization of ordinary particles in ordinary 4-dimensional space-time, and in a sense their existence could be the deeper reason why one has to do second quantization, why a classical description is not sufficient. Since the coupling is so small, the chaotic strings are quite a perfect source of noise, they look like uncorrected Gaussian white noise on a large scale. They presumably still span up the compactified space.

This page is intentionally left blank

Chapter 12

Summary

We summarize the most important results of this book in a self-consistent way, i.e. no knowledge of the previous chapters is required. 12.1

Motivation and main results

A fundamental problem of elementary particle physics is the fact that there are about 25 free fundamental constants which are not understood on a theoretical basis. These constants are essentially the values of the three coupling constants, the quark and lepton masses, the W and Higgs boson mass, and various mass mixing angles. An explanation of the observed numerical values is ultimately expected to come from a larger theory that embeds the standard model. Prime candidates for this are superstring and M-theory [Green et al. (1987); Kaku (1988); Polchinski (1998); Witten (1997); Banks et al. (1997)]. However, so far the predictive power of these and other theories is not large enough to allow for precise numerical predictions. In this book we have developed a dynamical theory of vacuum fluctuations that may provide possible answers to this problem. We have found that there is a simple class of 1+1-dimensional strongly self-interacting discrete field theories (which, in order to have a name, we have called 'chaotic strings') that have a remarkable property. The expectation of the vacuum energy of these strings is minimized for string couplings that numerically coincide with running standard model or gravitational couplings a(E), the energy E being given by the masses of the known quarks, leptons, and gauge bosons. Chaotic strings can thus be used to provide theoretical arguments 229

230

Summary

why certain standard model parameters are realized in nature, others are not. It is natural to assume that the a priori free parameters evolve to the local minima of the effective potentials generated by the chaotic strings. Out of the many possible vacua, chaotic strings seem to select the physically relevant states. The dynamics of the chaotic strings is discrete in both space and time and exhibits strongest possible chaotic behaviour [Beck (1995c)]. It can be regarded as a dynamics of vacuum fluctuations that can be used to 2nd-quantize other fields, for example ordinary standard model fields, or ordinary strings, by dynamically generating the noise of the Parisi-Wu approach of stochastic quantization [Parisi et al. (1981); Damgaard et al. (1987)] on a very small scale. Mathematically, chaotic strings are 1dimensional coupled map lattices [Kaneko (1984)] of diffusively coupled Tchebyscheff maps TM of order N. The dynamics describes a kind of 'turbulent quantum state'. It turns out that there are six different relevant chaotic string theories —similar to the six components that make up Mtheory in the moduli space of superstring theory [Gauntlett (1998)]. We have labeled these six chaotic string theories as 2A, 2B, 2A~~, 2B~, 3A, SB. Here the first number denotes the index N of the Tchebyscheff polynomial and the letter A,B distinguishes between a forward and backward coupling form. The index _ denotes anti-diffusive coupling (alternating signs of Tchebyscheff polynomials in spatial direction). In principle one can study these string theories for arbitrary N, but for stochastic quantization only the cases N = 2 and N = 3 yield non-trivial behaviour in a first and second order perturbative approach [Hilgers et al. (1999b); Hilgers et al. (2001)]. Chaotic strings can be used to generate effective potentials for possible standard model couplings, regarding the a priori free couplings as suitable scalar fields. The chaotic dynamics can be embedded into ordinary physics in various ways, ranging from a generalization of stochastic quantization (chapter 1) to an extension of statistical mechanics (chapter 6) and to a quantum gravity setting (chapter 11). Assuming that the a priori free standard model couplings evolve to the minima of the effective potentials generated by the chaotic strings, one can obtain a large number of very precise predictions. The smallest stable zeros of the expectation of the interaction energy of the chaotic 3A and W strings are numerically observed to coincide with the running electroweak couplings at the smallest fermionic mass scales. Inverting the argument, the chaotic 3A string can be

Motivation

and main results

231

used to theoretically predict that the low-energy limit of the fine structure constant has the numerical value a ei (0) = 0.0072979(17) = 1/137.03(3), to be compared with the experimental value 1/137.036. The SB string predicts that the effective electroweak mixing angle is numerically given by sf — sin26^FJ = 0.23177(7), in perfect agreement with the experimental measurements, which yield the value s 2 = 0.23185(23) [Groom et al. (2000)]. The smallest stable zeros of the interaction energy of the N — 2 strings are observed to coincide with strong couplings at the smallest bosonic mass scales. In particular, the smallest stable zero of the interaction energy of the 2A string yields a very precise prediction of the strong coupling at the W mass scale, which, if evolved to the Z° scale, corresponds to the prediction as(mZo) = 0.117804(12). The current experimentally measured value is as(mZo) = 0.1185(20) [Groom et al. (2000)]. Besides the coupling strengths of the three interactions, also the fermion mass spectrum can be obtained with high precision from chaotic strings. Here the expectation of the self energy of the chaotic strings is the relevant observable. One observes a large number of string couplings that locally minimize the self energy and at the same time numerically coincide with various running electroweak, strong, Yukawa and gravitational couplings, evaluated at the mass scales of the higher fermion families. The highest precision predictions for fermion masses comes from the self energy of the 2 A and 2B strings, which is observed to exhibit minima for string couplings that coincide with gravitational and Yukawa couplings of all known fermions. These minima of the vacuum energy yield the free masses of the six quarks as mu = 5.07(1) MeV, md = 9.35(1) MeV, ms = 164.4(2) MeV, m c = 1.259(4) GeV, mb = 4.22(2) GeV and mt - 164.5(2) GeV. Note that a free top mass prediction of 164.5(2) GeV corresponds to a top pole mass prediction of 174.4(3) GeV, in very good agreement with the experimentally measured value Mt = 174.3 ± 5.1 GeV. The masses of the charged leptons come out as me = 0.5117(8) MeV, mM = 105.6(3) MeV and mT = 1.782(7) GeV. All these theoretically obtained values of fermion masses are in perfect agreement with experimental measurements. To the best of our knowledge, there is no other theoretical model that has achieved theoretical predictions of similar precision. Chaotic strings also provide evidence for massive neutrinos, and yield concrete predictions for the masses of the neutrino mass eigenstates ^i, ^2, ^3. These are mVl = 1.452(3) • 10~ 5 eV, mV2 = 2.574(3) • 10~ 3 eV, m„ 3 = 4.92(1) • 1 0 - 2 eV (our symmetry considerations would also allow the mass m„2 to be smaller by a factor

232

Summary

1/8). Not only fermion masses, but also boson masses can be obtained from chaotic strings. The 2A string correctly reproduces the masses of the W and Z boson, and a suitable interpretation of the 2B~ string dynamics provides evidence for the existence of a scalar particle of mass mn = 154.4(5) GeV, which could be identified with the Higgs particle. The latter mass prediction is slightly larger than supersymmetric expectations but well within the experimental bounds based on the ordinary standard model. We also obtain estimates of the lightest glueball masses, which are consistent with estimates from lattice QCD.

12.2

The chaotic string dynamics

From a nonlinear dynamics point of view, chaotic strings are easily introduced. They are 1-dimensional coupled map lattices of diffusively coupled Tchebyscheff maps. In fact, for somebody with a background in dynamical systems the dynamics is a straightforward standard example of a spatially extended dynamical system exhibiting chaotic behaviour. On the other hand, for somebody working in high energy physics the equation may first look somewhat unfamiliar, but the results summarized in the previous section certainly indicate that it is worth learning new things. Consider a 1-dimensional lattice with lattice sites labeled by an integer i. At each lattice site i there is a variable $ln that takes on values in the interval [—1,1]. n is a discrete time variable. Given some initial value $Q the time evolution is given by deterministic recurrence relation K+I=TN(K)-

(12-1)

Here T/v($) is the iV-th order Tchebyscheff polynomial. One has T2($) = 2$ 2 - 1 and T 3 ($) = 4 $ 3 - 3$, generally TN($) = cos(iVarccos$). The Tchebyscheff maps Tjv with N > 2 are known to exhibit strongly chaotic behaviour. There is sensitive dependence on initial conditions: Small perturbations in the initial values will lead to completely different trajectories in the long-term run. The maps are conjugated to a Bernoulli shift with an alphabet of N symbols. This means, in suitable coordinates the iteration process is just like shifting symbols in a symbol sequence (see section 1.6 or any textbook on dynamical systems for more details). As shown in [Beck (1991a)] Tchebyscheff maps have least higher-order correlations among all

The chaotic string

dynamics

233

systems conjugated to a Bernoulli shift, and are in that sense closest to Gaussian white noise, though being completely deterministic. A graph theoretical method for this type of 'deterministic noise' has been developed in [Beck (1991a); Hilgers et al. (2001)]. We now couple the Tchebyscheff dynamics with a small coupling a in the spatial direction labeled by i, obtaining the chaotic string dynamics: K+1 = (1 - a)TNm

+ sffifa-1)

+ TbN(K+1))

(12-2)

We consider both the positive and negative Tchebyscheff polynomial T±N($) = ±Tjv($), but have suppressed the index ± in the above equation. The variable a is a coupling constant taking values in the interval [0,1]. s is a sign variable taking on the values ± 1 . The choice s = +1 is called 'diffusive coupling', but for symmetry reasons it also makes sense sense to study the choice s = —1, which we call 'anti-diffusive coupling'. The integer b distinguishes between the forward and backward coupling form, b — 1 corresponds to forward coupling (T^($) := T/v()), b = 0 to backward coupling (T^($) := $). We consider periodic boundary conditions and large lattices OI Size ''rnax •

The dynamics (12.2) is deterministic chaotic, spatially extended, and strongly nonlinear. The field variable $ln is physically interpreted in terms of rapidly fluctuating virtual momenta in units of some arbitrary maximum momentum scale. There are some analogies with velocity fluctuations in fully developed turbulent flows, which are also deterministic chaotic, spatially extended, and induced by strong nonlinearities. For this reason it makes sense to think of eq. (12.2) as describing a turbulent state of vacuum fluctuations, or in short a 'turbulent quantum state'. These states appear to have physical relevance, since they reproduce the observed values of the standard model parameters. We may also write the coupled dynamics as S\+1

= TN{Vn) + \ (sTU&n1)

- 2 T W « ) + sTbN(¥n+1))

.

(12.3)

This way of writing illustrates that the effect of the coupling is similar to the action a Laplacian operator. Since a determines the strength of the Laplacian coupling, and since in quantum field theories this role is usually attributed to a metric, a - 1 can be regarded as a kind of metric in the 1-dimensional string space indexed by i.

234

Summary

It is easy to see that for odd N the statistical properties of the coupled map lattice are independent of the choice of s (since odd Tchebyscheff maps satisfy TJV(—$) = — T J V ( $ ) ) , whereas for even N the sign of s is relevant and a different dynamics arises if s is replaced by - s . Hence, restricting ourselves to N = 2 and N = 3, in total 6 different chaotic string theories arise, characterized by (N,b,s) = (2,1,+1), (2,0,+1), ( 2 , 1 , - 1 ) , ( 2 , 0 , - 1 ) and (N, b) = (3,1), (3,0). For easier notation, we have labeled these string theories as 2A,2B,2A~ ,2B~~,3A,3B, respectively. If the coupling a is sufficiently small, the chaotic variables $^ can be used to generate the noise of the Parisi-Wu approach of stochastic quantization [Parisi et al. (1981); Damgaard et al. (1988)] on a very small scale. That is to say, we assume that on a very small (quantum gravity) scale the noise used for quantization purposes is not structureless but evolves in a deterministic chaotic way. Just on a large scale it looks like Gaussian white noise, and there is convergence to ordinary path integrals using this 'deterministic noise', as can be rigorously proved for a = 0 [Beck et al. (1987); Beck (1995a); Beck (1995b)]. In this interpretation the discrete time variable n corresponds to the fictitious time of the Parisi-Wu approach, an artificial time coordinate that is just introduced for quantization purposes (see chapter 1 for details). The chaotic string dynamics (12.2) formally originates from a 1-dimensional continuum <j>N+1 -theory in the limit of infinite self-interaction strength (see section 2.2 for details). In this sense, chaotic strings can also be regarded as degenerated Higgs-like fields with infinite self-interaction parameters, which are constraint to a 1-dimensional space. Another way to physically motivate chaotic strings is to emphasize certain analogies with ordinary strings (section 2.4), to connect them with fluctuating momenta that are allowed due to the uncertainty relation (sections 5.1-5.3), and to relate them to a 1+1 dimensional model of quantum gravity (section 5.7). One can also embed them into the compactified space of string theories (section 11.1).

12.3

Vacuum energy of chaotic strings

Though the chaotic string dynamics is dissipative, one can formally introduce potentials that generate the discrete time evolution. For a = 0 we

Vacuum energy of chaotic

235

strings

may write * n + i - * „ = ±rJV(*n) = - ^ - V ± ( $ n ) .

<9$n

(12.4)

For N — 2 the (formal) potential is given by

V®($) = ±(-l&

+ $\+^*2

+ C,

(12#5)

yW($) = ± (-$* + 1$A + i$ 2 + C.

(12.6)

for JV = 3 by

Here C is an arbitrary constant. The uncoupled case a = 0 is completely understood. The dynamics is ergodic and mixing. Any expectation of an observable A($) can be calculated as (A) = j _

d p()A(),

(12.7)

where

PW = -7T=^

<12-8)

is the natural invariant density describing the probability distribution of iterates of Tchebyscheff maps (see, e.g., [Beck et al. (1993)] for an introduction). In more general versions of statistical mechanics [Tsallis (1988)], this probability density can be regarded as a generalized canonical distribution (see section 6.4 for more details). If a spatial coupling a is introduced, things become much more complicated, and the invariant 1-point density deviates from the simple form (12.8). A spatial coupling is formally generated by the interaction potential aW±($,*), with

W± ($,¥) = J($±*) 2 + C.

(12.9)

Here $ and \& are neighbored noise field variables on the lattice. One has -7^rW±($i,$i+1)-1—-W±(§\<S>i-'L)

= ±-$i+1-i±-$i-1.

(12.10)

This generates diffusive (+), respectively anti-diffusive (—) coupling. Antidiffusive coupling can equivalently be obtained by keeping W- but replacing

236

Summary

TN —> —TN at odd lattice sites. The coupled map dynamics (12.2) is obtained by letting the action of V and W alternate in discrete time n, then regarding the two time steps as one. The expectations of the potentials V and W yield two types of vacuum energies V±(a) := (viN){¥)) (the self energy) and W±(a) := (W±($\¥+1)) (the interaction energy). Here (• • •) denotes the expectation with respect to the coupled chaotic dynamics. Numerically, any such expectation can be determined by averaging over all i and n for random initial conditions $o e [—1,1], omitting the first few transients. Note that in the stochastic quantization approach the chaotic noise is used for 2nd quantization of standard model fields (or ordinary strings) via the Parisi-Wu approach [Beck (1995c)]. Hence generally expectations with respect to the chaotic dynamics correspond to expectations with respect to 2nd quantization. The expectation of the vacuum energy of the string, given by the above functions W±(a) and V±(a), depends on the coupling a in a non-trivial way. Moreover, it also depends on the integers N, b, s that define the chaotic string theory. Since negative and positive Tchebyscheff maps essentially generate the same dynamics, up to a sign, any physically relevant observable should be invariant under the transformation TN —>• —TN- The vacuum energies V±(a) and W±(a) of the various strings exhibit full symmetry under the transformation TN -> — TN (respectively s —> —s) if the additive constant C is chosen to be C = -\($2)-

(12.11)

For that choice of C, the expectations of V+ and V_ as well as those of Wand W+ are the same, up to a sign. Choosing (by convention) the + sign one obtains from eq. (12.5), (12.6) and (12.9) for the expectations of the potentials VW(a)

=

- ^ ( $ 3 ) + ($)

(12.12)

VW(a)

=

-<$4) + f<*2>>

(12-13)

and W{a) = i ( W + 1 ) .

(12.14)

Fixing standard model

parameters

237

The above ± symmetry can actually be used to cancel unwanted vacuum energy and to avoid problems with the cosmological constant. If one assumes that strings with both TN and — TN are physically relevant, the two contributions V(a) and -V(a) (respectively W(a) and ~W(a)) may simply add up to zero. This reminds us of similarly good effects that supersymmetric partners have in ordinary quantum field and string theories. In fact, as a working hypothesis we may regard the above Zi symmetry as representing a kind of supersymmetry in the chaotic noise space. Similarly as for ordinary strings it also makes sense to consider certain conditions of constraints for the chaotic string. For ordinary strings (for example bosonic strings in covariant gauge [Green et al. (1987)]) one has the condition of constraint that the energy momentum tensor should vanish. The first diagonal component of the energy momentum tensor is an energy density. For chaotic strings, the evolution in space i is governed by the potential W±($, * ) and the corresponding expectation of the energy density is ±W(a). We should thus impose the condition of constraint that W(a) should vanish for physically observable states. Moreover, the evolution in fictitious time n is governed by the self-interacting potential V W ( $ ) . This potential generates a shift of information, since the Tchebyscheff maps TN are conjugated to a Bernoulli shift of iV symbols. Hence V(a) can be regarded as the expectation of a kind of information potential or entropy function, which, motivated by thermodynamics, should be extremized for physically observable states. Note that the action of V and W alternates in n and % direction. Both types of vacuum energies describe different relevant observables of the chaotic string and are of equal importance.

12.4

Fixing standard model parameters

In order to construct a link to standard model phenomenology it is useful to introduce a simple physical interpretation of the chaotic string dynamics in terms of fluctuating virtual momenta. Suppose we regard $ln to be a fluctuating virtual momentum component that can be associated with a hypothetical particle i at time n that lives in the constraint 1-dimensional string space, n can be either interpreted as fictitious time or as physical time, it doesn't really matter for our purposes. Neighbored particles i and i — 1 can exchange momenta due to the Laplacian coupling of the coupled map lattice. To make this model more concrete we may assume that at

238

Summary

Fig. 12.1 Interpretation of the coupled map dynamics in terms of fluctuating momenta exchanged by fermions / i , / 2 and bosons B\,B2-

each time step n a fermion-antifermion pair / i , J2 is spontaneously created in cell i. In units of some arbitrary energy scale pmax, the particle has momentum $^, the antiparticle momentum — <&^. They interact with particles in neighbored cells by exchange of a (hypothetical) gauge boson B2, then they annihilate into another boson B\ until the next vacuum fluctuation takes place. This can be symbolically described by the Feynman graph in Fig. 12.1. We called this graph a 'Feynman web', since it describes an extended spatio-temporal interaction state of the string, to which we have given a standard model-like interpretation. Note that in this interpretation a is a (hypothetical) standard model coupling constant, since it describes the strength of momentum exchange of neighbored particles. At the same time, a can also be regarded as an inverse metric in the 1-dimensional string space, since it determines the strength of the Laplacian coupling. What is now observed numerically for the various chaotic strings is that the interaction energy W(a) has zeros and the self energy V(a) has local minima for string couplings a that numerically coincide with running

Fixing standard model

parameters

239

standard model couplings a(E), the energy being given by E= -N-(mBl+mfl+mh).

(12.15)

Here N is the index of the chaotic string theory considered, and m ^ , mf1, m,f2 denote the masses of the standard model particles involved in the Feynman web interpretation. The surprising observation is that rather than yielding just some unknown exotic physics, the chaotic string spectrum appears to reproduce the masses and coupling constants of the known quarks, leptons and gauge bosons of the standard model with very high precision. Gravitational and Yukawa couplings are observed as well. We thus have the possibility to fix and predict the free standard model parameters by simply assuming that the entire set of parameters is chosen in such a way that it minimizes the vacuum energy of the chaotic strings. Although eq. (12.15) looks like a low-energy formula, the chaotic strings may well describe a scenario that takes place at very large energies well above the Planck scale. The reason is that alternatively the Feynman web dynamics can also be interpreted in terms of black holes that emit lowenergy particles by Hawking radiation, similar to the process sketched in Fig. 12.2. In this case a very large energy is given by something of the order of the mass M of the black holes, but what is physically relevant is the Hawking temperature kTn ~ Q ^ by which the black holes radiate. Here G is the gravitational constant. Thus very large energies M can be associated with very small Hawking temperatures kTn and allow for fixing of low-energy parameters in a pre-Planck epoch (see section 11.5 for more details). At a very early stage of the universe, where standard model parameters are not yet fixed and ordinary space-time may not yet exist as well, pre-standard model couplings may be realized as coupling constants a in the chaotic string space. The parameters are then fixed by an evolution equation (a renormalization flow) of the form o = const • W(a) + noise,

(12.16)

dV a = —const • — f- noise, da

(12.17)

respectively

where we assume that the constant const is positive. It can also depend

240

Summary

Fig. 12.2 Alternative interpretation of the coupled map dynamics. Black holes (denoted by B) emit low-energy particles / i , fa that interact with gauge bosons B2 at low temperatures.

on a, this doesn't matter as long as it does not switch sign. The equations make the expectations of a priori arbitrary standard model couplings a evolve to the stable zeros of W(a), respectively to the local the minima of V(a). Eq. (12.16) is a kind of Einstein equation and eq. (12.17) a kind of scalar self-interacting field equation for a (see sections 7.1, 8.1, 11.2 for more details).

12.5

Numerical findings

Our numerical results for zeros of W(a) and local minima of V(a) and the corresponding physical interpretations are described in detail in chapter 7-10. Let us here just summarize the most important points. The smallest non-trivial zeros of the interaction energy W(a) with negative slope at the zero, describing stable stationary states of couplings under the evolution equation (12.16), are listed in Tab. 12.1. The N — 3 ze-

Numerical

string 3A 3A 3B 3B 2A 2B 2A~ 2B~

stable zero a[iA) = 0.0008164(8) 4 3 J 4 ) = 0.0073038(17)

afB) = 0.0018012(4) 4 3 B ) = 0.017550(1) a^A) = 0.120093(3) 4 2 B ) = 0.3145(1) af- 4 "' = 0.1758(1) 4 2 B _ ) = 0.095370(1)

findings

241

running SM coupling adel(3md) =0.0008166 aeel(3me) = 0.007303
Table 12.1 The smallest stable zeros of the interaction energies of the 3A,3B,2A,2B,2A~ ,2B~ strings and comparison with experimentally measured standard model couplings (where known).

ros are observed to numerically coincide with the coupling constants of electroweak interactions, evaluated at four different energy scales. The four relevant energy scales are given by the masses of the lightest fermions d,e,u,ve. To calculate the concrete value of the energy scale, one uses eq. (12.15). For example, the zero a2 is associated with a Feynman web of the form / i = e~, fy = e + , B\ massless, B2 = 7- This yields E — | ( 2 m e + m ^ J = 3m e . Tab. 12.1 shows that there is coincidence of all four observed stable zeros with the corresponding standard model couplings with a precision of 4 digits (more details in section 7.2-7.4). Symmetry considerations with the N = 3 strings suggest that the smallest stable zeros of the N = 2 strings fix strong couplings at the smallest bosonic mass scales. These are given by the W boson, the Higgs boson, and the lightest glueballs of spin 0 and 2. Since only the W boson mass is precisely known, we can only compare the zero a\ with experimental data. It indeed coincides with as(mw)- The other zeros yield predictions for the Higgs and glueball masses (details in section 7.6 and 7.7). The observed zeros of the N = 2 and N — 3 strings allow for highprecision predictions of the fine structure constant, the Weinberg angle and the strong coupling constant. The fine structure constant and the effective Weinberg angle for Z°-lepton coupling are correctly obtained with a precision of 4-5 digits. The strong coupling constant at the W mass scale is predicted with about 5 valid digits, a much higher precision than can be confirmed by experiments at the present stage. as(E) can then be

242

Summary

easily evaluated at other energy scales as well, using the well-known QCD formulas. Generally, we observe interesting symmetries between the various zeros and the corresponding standard model interactions. These kind of symmetries often help to fill 'gaps' in the table, i.e. to find the right Feynman web interpretation for a given zero. For the N = 3 strings, proceeding from the smallest stable zeros a[ ' ' t o the next larger stable zeros a2 means replacing quarks by leptons, whereas for the iV = 2 string it means replacing ordinary gauge and Higgs bosons (W^ and H) by their Planckscale analogues {W± and H, see section 10.6). For the N = 3 strings, the forward coupling form (3A) describes just one species of fermions (either d or e). These fermions certainly have spin. On the other hand, the backward coupling form (3B) describes simultaneous states of two fermions ( ( u ^ d i ) and (vL,eR)) of opposite handedness, which coexist independently. Hence proceeding from forward to backward coupling means proceeding from a state with spin to a state with total spin 0, since R + L — 0. For the N = 2 strings proceeding from forward to backward coupling also means going from a state with spin (the W-boson, the glueball with spin 2) to a state without spin (the Higgs boson, the glueball with spin 0). Now let us look at the other relevant type of vacuum energy, the self energy V(a). The functions V(a) have plenty of local minima for the various strings. For small a, oscillating scaling behaviour sets in and all local minima of V(a) are only fixed modulo N2 (details in section 8.6). Also, in this limit there is no visible difference between the self energies of the forward and backward coupling form. The self energy of the 2A/B string turns out to have local minima for Yukawa couplings of heavy fermions (t, b, T, c) and gravitational couplings of light fermions (fi, s, d, u, e). This is shown in Tab. 12.2. The Yukawa coupling of the t quark falls out of the scaling region and is separately listed in Tab. 12.3. Clearly, since the minima of the vacuum energy can be determined with high precision, and since gravitational and Yukawa couplings depend quadratically on the mass, the minima allow for high-precision predictions of masses modulo 2, in particular of quark masses. Note that most of the values listed in column 1 of Tab. 12.2 have a much higher precision than the experimental values in column 3. There are further minima (not listed in Tab. 12.2) that can be used to make neutrino mass predictions (see section 8.7). For the attribution of the various minima to the various particles one uses simple discrete symmetry considerations.

Numerical

local minimum 61 = 0.000199(1) mod 4 62 = 0.000263(2) mod 4 h = 0.000306(1) mod 4 b6 = 0.000368(3) mod 4 67 = 0.000399(1) mod 4 b8 = 0.000469(1) mod 4 b9 = 0.000482(1) mod 4 610 = 0.000525(2) mod 4

fermion M T

e 6 d u s c

findings

243

SM gravitational/Yukawa coupling aG = 0.0001989 mod 4 aTYu = 0.0002615 mod 4 o& = 0.0003052 mod 4 < u = 0.000369(6) mod 4 a £ = 0.00038(5) mod 4 aG = 0.00040(17) mod 4 a'G = 0.00050(4) mod 4 acYu = 0.000526(25) mod 4

Table 12.2 Local minima of the self energy of the 2A/B string in the scaling region and comparison with gravitational and Yukawa couplings of standard model particles. For the experimental quark mass values that lead to the numbers and error bars in column 3 we have chosen mt = 164(5) GeV, mb = 4.22(4) GeV, m c = 1.26(3) GeV, ms = 167(7) MeV, md = 9.1(6) MeV, mu = 4.7(9) MeV.

For all states in the scaling region, the relevant power of 4 of the coupling (equivalent to a power of 2 for the mass) is a priori undetermined. It can be theoretically related to a folding number of the string in a quantum gravity epoch (section 11.6). In this epoch all masses are only fixed modulo 2. For standard model states, however, the relevant power of 2 can be deduced by postulating compatibility with other string states outside the scaling region (see section 8.6-8.7 for details). Outside the scaling region, lots of other minima are observed that can directly be identified with standard model interaction strengths (Tab. 12.3). For the N = 3 strings we observe minima describing strongly interacting heavy quarks q, i.e. f\ = q, f2 — q, B\ massless, B2 = g (gluon), where q = t, b, c, respectively. Pure flavor states are described by forward coupling and mixed flavor states (t, b) and (c, s) by backward coupling. Further minima can be identified with weak interaction states of right-handed fermions, fixing the three different charges of charged quarks and leptons. Small differences between the forward and backward coupling form may possibly provide information on mass mixing angles, i.e. the entries of the KobayashiMaskawa matrix. Between small (weak) and large (strong) couplings we have identified the remaining two minima as describing unified couplings at the GUT and Planck scale (see chapter 10). Summarizing, the two types of vacuum energy and the six types of chaotic strings considered seem to be sufficient to fix the most relevant pa-

244

string 3A 3A 3A 3A 3A 3A 3B 3B 2A 2B

Summary

local minimum a'}3A} = 0.000246(2) afA) = 0.00102(1) afA) = 0.00220(1) a$3A) = 0.0953(1) a'7{3A) = 0.1677(5) a£3A) = 0.2327(5) afB) = 0.1027(1) B) af = 0.2916(5) A) a'¥ = 0.03369(2) afB) = 0.03440(2)

running SM coupling ataki*™*)

= 0-000246

0 ( 3 m c ) = 0.000101
Table 12.3 Various observed local minima of the self energy of the 3A,3B, 2A, 2B strings outside the scaling region and comparison with running standard model couplings. The running strong coupling as (E) can be very precisely evaluated using the zero a[ ' listed in Tab. 12.1.

rameters of the standard model such as masses, charges and coupling constants. The standard model appears to have evolved to a state of minimum vacuum energy with respect to the chaotic strings. Tab. 12.4 summarizes all predictions on standard model parameters (together with the relevant error bars) that we have extracted from the zeros and minima of the vacuum energy. For comparison, also the experimentally measured values are listed. Some predictions made by the chaotic strings are much more precise than current experiments can confirm (e.g. as(mw), the free quark masses, neutrino masses, the Higgs mass), whereas other predictions are less precise than the experimental data (e.g. aei(0) and the charged lepton masses). All known masses are correctly reproduced with a precision of 3-4 digits. The unknown masses (Higgs and neutrino masses) are predicted with a similar precision, based on simple symmetry arguments on what type of minimum is relevant (see sections 7.7 and 8.7 for details). One can also study the total vacuum energy of the strings as given by H-(a) = V(a) - aW{a) and H+(a) = V(a) + aW(a). It turns out that for the N — 3 strings local minima of H-(a) essentially reproduce the spectrum of light mesons (Tab. 12.5) and local minima of H+(a) essentially that of light baryons (Tab. 12.6).

Numerical

parameter <*ej(0)

s2 as(mz) me m^ mT mVx m„2 m„3

mu md ms mc mb mt mw mH

findings

chaotic string prediction 0.0072979(17) 0.23177(7) 0.117804(12) 0.5117(8) MeV 105.6(3) MeV 1.782(7) GeV 1.452(3) • 10~ 5 eV 2.574(3) • 10" 3 eV 4.92(1) • 10" 2 eV 5.07(1) MeV 9.35(1) MeV 164.4(2) MeV 1.259(4) GeV 4.22(2) GeV 164.5(2) GeV 80.36(2) GeV 154.4(5) GeV

245

measured 0.00729735253(3)=1/137. 036 0.2318(2) 0.1185(20) 0.51099890(2) MeV 105.658357(5) MeV 1.7770(3) GeV ? ?

~ 5 • 10" 2 eV ? ~5MeV ~9MeV ~ 170 MeV 1.26(3) GeV 4.22(4) GeV 164(5) GeV 80.37(5) GeV ?

Table 12.4 Most important standard model parameters as obtained from the chaotic string spectrum and as measured in experiments. All quark masses denote free quark masses, mt = 164.5(2) GeV corresponds to a top pole mass of Mt = 174.4(3) GeV. Symmetry considerations also allow mV2 to be smaller by a factor g (see section 8.7).

As explained in section 9.1, the relevant energy scales for mesons M are given by |TTIM and |m^f, those for baryons B by | m s and ms- Apparently, for confined states the total vacuum energy H± (a) is the relevant quantity to look at, rather than the single vacuum energies V(a) and W(a). Another interesting observation is that from the chaotic string spectrum there is no straightforward evidence for supersymmetric particles with masses in the region 100-1000 GeV. There are simply no minima observed in the relevant energy region! Also, the observed minima of the vacuum energy of the strings seem to support grand unification scenarios based on the non-supersymmetric beta functions, rather than the supersymmetric ones. More on supersymmetry and possible grand unification scenarios can be found in chapter 10. In total, we found more than 30 zeros and minima that could be identified with known standard model and gravitational interaction strengths

246

string 3A 3A 3A 3A 3A 3A 3A 3B 3B 3B 3B 3B 3B 3B

Summary

local minimum of H_ (a) 0.163 0.186 0.282 0.375 0.416 0.615 0.676 0.161 0.182 0.224 0.290 0.341 0.356 0.412

strong coupling a s ( | m T ( l s ) ) =0.164 a s ( | m B ) = 0.188 a s ( f m $ ) = 0.282 as{\mK.) = 0.374 a , ( | m p ) = 0.418 as(^mn) = 0.60 ocs{\mK) =0.70 a s ( | m T ( 2 s ) ) = 0.161 cts{\mBc) = 0.179 a,{\mD)= 0.224 a , ( f n v ) = 0.291 a s ( | m $ ) = 0.344

a s (§77v) =0.356 ots{\mu)

= 0.413

Table 12.5 Local minima of the total vacuum energy H-{a) of the N = 3 strings and comparison with strong couplings at energy levels given by mesonic states.

string 3A 3A 3A 3A 3A 3A 3B 3B 3B

local minimum of H+(a) 0.233 0.282 0.508 0.609 0.628 0.70 0.356 0.367 0.378

strong coupling Q s ( | m A J = 0.233 a s ( m A c ) = 0.282 a s(fnPt„) = 0.508 a s ( | m A ) =0.60 as(|mE)=0.63 a s (|rriA) = 0.68 a s(mN(U40)) = 0.356 a2{mx*) = 0.366 as{mE) =0.379

Table 12.6 Local minima of the total vacuum energy H+(a) of the N = 3 strings and comparison with strong couplings at energy levels given by baryonic states.

in a straightforward way. Moreover, more than 20 minima could be identified with hadronic states. Could all this be a random coincidence? It can't. Let us estimate the probability to obtain all this by a pure random coincidence. In Tab. 12.1-12.3 we observe some 20 string couplings that

Physical embedding

247

coincide with experimentally measured standard model coupling constants with a precision of 3-4 digits. The joint probability for this being the result of a pure random coincidence is of the order (1CT3)20 = 10~ 60 . Here we still haven't taken into account the hadronic minima of Tab. 12.5 and 12.6. Even if we allow for different attributions of the minima to various possible standard model interaction states, the joint probability of obtaining randomly a joint coincidence of this order of magnitude is still extremely small. In other words, a random coincidence can be excluded. For this reason the chaotic string dynamics needs to be integrated into future theories of particle physics in one way or another, there is no way around it. The question of real interest is whether chaotic strings are already the exact and complete theory, or whether they are just the beginning (and perhaps the perturbative approximation) of a more advanced theory.

12.6

Physical embedding

We saw that the chaotic strings resemble the mass spectrum of elementary particles in a correct way. That means, although a complete picture certainly requires the field equations of QED, QCD, weak interactions, Einstein's gravity, and possibly superstring theory, the information on masses and coupling strengths at a certain energy scale is correctly encoded by the strings. The chaotic strings seem to contain the most important information on what is going on at a certain temperature. In fact they yield information on parameters that is not directly given by the other theories, thus providing a nice and necessary amendment. How can we understand the general role of these strings? Let us become a bit philosophical and make a simple comparison. Suppose you want to build a house. Before you can start, an architect has to draw a plan of the house. The plan contains the most important information about the house, but is certainly not the house itself. Moreover, the plan is 2-dimensional, whereas the house is 3-dimensional. Also, the plan is drawn before the house is being built. In our case, the plan are the chaotic strings and the house is the universe. The strings are 1+1 dimensional, the universe is 4 or 11-dimensional. The chaotic strings presumably fix lowenergy standard model parameters of the universe already at an extremely early stage, long before standard model field equations and a 4-dimensional space-time become relevant. The different dimensionalities somewhat re-

248

Summary

mind us of the holographic principle [Susskind (1995); Maldacena (1998); Polyakov (2001)], which relates degrees of freedom of quantum field theories in different dimensions. Keeping on being philosophical, we may even see some analogies with biological systems. The DNA string encodes the most important information about a living being, in fact already long before this living being is being born. This information is encoded using symbol sequences made up of 4 different symbols, which are the chemical substances called Adenine, Cytosine, Guanine, and Thymine. So for biological systems a kind of N = 4 string is realized. The DNA string is certainly not the living being itself, but it contains the most important information about it. In that sense, we may regard the chaotic string as a kind of 'DNA' string of the universe. It encodes the most important information on the universe, and this may be relevant already long before ordinary space-time is being created. What chemistry is for biology, is information theory for physics. Still one may ask how to concretely embed the chaotic string dynamics into ordinary physics and possible theories of quantum gravity. Several approaches are possible, all of which are somewhat related. The first and simplest approach is to regard the chaotic dynamics as a new and a priori standard model-independent dynamics of virtual momenta. We have shown in chapter 1 that it effectively reduces to spatiotemporal Gaussian white noise if seen on a large scale. This noise can then be coupled to the classical standard model field equations and, in fact, can be used to generate the noise fields of the stochastic quantization method. So the strings are embedded as a tool for quantization. The new thing is that the dynamics used for quantization is a deterministic chaotic one, rather than a purely random one. The second approach is to relate the chaotic dynamics to an effective thermodynamic description of vacuum fluctuations allowed by the uncertainty relation. Clearly ordinary statistical mechanics is not valid for a description of vacuum fluctuations, but the chaotic strings and their expectation values with respect to the invariant measures of the coupled dynamics may yield the correct tools for a statistical mechanics of vacuum fluctuations. In fact, free (a = 0) chaotic strings generate invariant densities that can be regarded as generalized canonical distributions in the formalism of non-extensive statistical mechanics, with an entropic index q given by either q = - 1 or q — 3. So the strings are embedded as a generalization of

Conclusion

249

statistical mechanics (more details in chapter 6). The third approach is to relate the chaotic strings to the structure of space-time itself. There are 6 relevant chaotic strings, which we labeled as 2A,2B,2A~,2B~ ,3A,3B. Also, there are 6 compactified dimensions necessary for the formulation of superstring theory or M-theory. If we let the 6 chaotic strings wind around (or even span up) the 6 compactified dimensions, then they do not 'disturb' our usual understanding of 4-dimensional space-time physics. Rather, they yield a very relevant amendment. Each coupling constant a can then be regarded as a kind of metric in the compactified space, and the analogue of the Einstein field equations makes the observed standard model parameters evolve to the minima of the effective potentials of the chaotic strings. In this picture the chaotic strings are related to a higher-dimensional extension of ordinary 4-dimensional spacetime, in fact to kind of 'excited states' of ordinary 4-dimensional space-time. The 'ground state' (4-dimensional Minkowski space) is represented by the (non-chaotic) N — 1 strings, whereas the chaotic strings with N > 2 span up higher states (higher dimensions), similar to the energy levels of a quantum mechanical harmonic oscillator with quantum number N (more details on this kind of approach in chapter 11). Which of these different physical embeddings is the most relevant one is not clear at the moment. Perhaps there is some truth in a combination of all three of them.

12.7

Conclusion

Instead of considering the standard model alone and putting in about 25 free parameters by hand, in this book we have postulated the existence of chaotic strings. The chaotic string dynamics can be physically interpreted as a 1-dimensional strongly fluctuating dynamics of vacuum fluctuations. It generates effective potentials which distinguish the observed standard model couplings from arbitrary ones. The dynamics may have already determined the standard model parameters at a very early stage of the universe (in a pre-Planck scenario) and it may still evolve today and stabilize the observed values of the parameters. There are 6 relevant chaotic string theories (Figs. 12.3, 12.4). Whereas for standard model fields, as well as for superstrings after compactification, continuous gauge symmetries such as t/(l), SU(2) or SU(3) are relevant,

Summary

250

strong (at lightest bosonic mass scales)

2B

2A

3A

electroweak (at lightest fermionic mass scales)

3B

2B" gravitational (at lightest black hole mass scales) 2A"

Fig. 12.3 The way in which the interaction energies W(a) theories fix the coupling strengths of the four interactions.

of the six chaotic string

for the chaotic strings a discrete Z2 symmetry is relevant. In fact, the total theory may be regarded as a SU(3) x 577(2) x £7(1) x Z2 theory. Whereas standard model fields or ordinary strings usually evolve in a regular way, the chaotic strings obtained for N > 1 evolve in a deterministic chaotic (turbulent) way. They arise out of strongly self-interacting 1-dimensional field theories and correspond to a Bernoulli shift of information for vanishing spatial coupling a. The constraint conditions on the vacuum energy (or the analogues of the Einstein and scalar field equations) fix certain equilibrium metrics in string space, which determine the strength of the Laplacian coupling. We have provided extensive numerical evidence that these equilibrium metrics reproduce the free standard model parameters with very high precision (see Tab. 12.4). Essentially coupling constants are fixed by the interaction energy W(a), and masses, mass mixing angles and charges by the self energy V(a). This is summarized in Fig. 12.3 and 12.4. The simplest physical interpretation is to regard the chaotic string dynamics as a dynamics of vacuum fluctuations, which is present everywhere but which is unobservable due to the uncertainty relation. Only expec-

Conclusion

fermion masses (via Yukawa and gravitational interaction)

251

2B

2A

3A

charges and mixing angles (via electroweak and strong interaction)

3B

2B' boson masses (via dual gravitational interaction) 2A"

Fig. 12.4 The way in which the self energies V(a) of the six chaotic string theories fix the charges, mass mixing angles and masses of the standard model particles.

tations of the dynamics can be measured, in terms of the fundamental constants of nature. The strings can be related to a generalized statistical mechanics description of vacuum fluctuations and may possibly wind around the compactified space of superstring theory. Generically, chaotic strings exhibit symmetry under the replacement V ->• —V (or TN -» —TN), which can be formally associated with a kind of supersymmetry transformation. However, when introducing the evolution equations (12.16) and (12.17) of the couplings one has to decide on the sign of the constant const. This choice effectively breaks the symmetry. Generally, with such a choice of sign the expectation of the vacuum energy of the chaotic strings singles out the physically relevant vacua, in the sense of stability. Supersymmetric partners of ordinary particles, if they exist at all, can be formally described by maxima rather than minima of the effective potentials. But they are unstable with respect to the fictitious time evolution, at least in our world. The instability might indicate that supersymmetric partners, though formally there to cancel divergences in the Feynman diagrams as well as unwanted vacuum energy, may turn out

252

Summary

to be unobservable in our world. In any case, chaotic dynamics of the type studied in this book seems to significantly enrich our understanding of standard model parameters and of quantum fluctuations in general. Assuming that on a very small scale quantum fluctuations are a deterministic chaotic process rather than a pure random process the most important free parameters of the standard model can be understood with high precision. Embedding the chaotic strings into the compactified space of a 10- or 11-dimensional theory we seem to be looking at an extremely early stage of the universe where neither matter nor radiation but information is the relevant concept.

Bibliography

Abe S, Axioms and uniqueness theorem for Tsallis entropy, Phys. Lett. 271A, 74 (2000) Abe S, Okamoto Y (eds.), Nonextensive Statistical Mechanics and its Applications, Springer, Berlin (2001) Aitchison IJR, Hey AJG, Gauge theories in particle physics, Adam Hilger, Bristol (1989) Alberico WN, Lavagno A, Quarati P, Non-extensive statistics, fluctuations and correlations in high-energy nuclear collisions, Eur. Phys. J. C12, 499 (2000) Albeverio S, Some applications of infinite dimensional analysis in mathematical physics, Helv. Phys. Acta 70, 479 (1997) Amaldi U, de Boer W, Fiirstenau H, Comparison of grand unified theories with electroweak and strong coupling constants measured at LEP, Phys. Lett. 260B, 447 (1991) Amritkar RE, Gade PM, Gangal AD, Nandkumaran VM, Stability of periodic orbits of coupled-map lattices, Phys. Rev. 44A, R3407 (1991) Amritkar RE, Gade PM, Wavelength doubling bifurcations in coupled map lattices, Phys. Rev. Lett. 70, 3408 (1993) Amritkar RE, Structures in coupled map lattices, Physica 224A, 382 (1996) Antoniadis I, Bachas C, Branes and the gauge hierarchy, Phys. Lett. 450B, 83 (1999) Antoniadis I, Benakliand K, Quiros M, Direct collider signatures of large extra dimensions, Phys. Lett. 460B, 176 (1999) Antoniadis I, String and D-brane physics at low energy, hep-th/0102202 Arad I, Biferale L, Celani A, Procaccia I, Vergassola M, Statistical conservation laws in turbulent transport, Phys. Rev. Lett. 87, 164502 (2001) Arimitsu T, Arimitsu N, Tsallis statistics and fully developed turbulence, J. Phys. 33A, L235 (2000) Aubry S, Abramovici G, Chaotic trajectories in the standard map - the concept of anti-integrability, Physica 43D, 199 (1990) 253

254

Bibliography

Baesens C, MacKay RS, Cantori for multiharmonic maps, Physica 69D, 59 (1993) Bagnoli F, Baroni L, Palmerini P, Synchronization and directed percolation in coupled map lattices, Phys. Rev. 59E, 409 (1999) Baladi V, Esposti MD, Isola S, Jarvenpaa E, Kupiainen A, The spectrum of weakly coupled map lattices, J. Math. Pur. Appl. 77, 539 (1998) Baladi V, Positive Transfer Operators and Decay of Correlations, World Scientific, Singapore (2000) Baladi V, Rugh HH, Floquet spectrum of weakly coupled map lattices, Commun. Math. Phys. 220, 561 (2001) Banks T, Fischler W, Shenker SH, Susskind L, M-theory as a matrix model: a conjecture, Phys. Rev. 55D, 5112 (1997) Baulieu L, Grassi PA, Zwanziger D, Gauge and topological symmetries in the bulk quantization of gauge theories, Nucl. Phys. 597B, 583 (2001) Batrouni GG, Katz GR, Kronfeld AS, Lepage GP, Svetitsky B, Wilson KG, Langevin simulations of lattice field theories, Phys. Rev. 32D, 2736 (1985) Beck C, Roepstorff G, From dynamical systems to the Langevin equation, Physica 145A, 1 (1987) Beck C, Ergodic properties of a kicked damped particle, Commun. Math. Phys. 130, 51 (1990) Beck C, Brownian motion from deterministic dynamics, Physica 169A, 324 (1990) Beck C, Higher order correlation functions of chaotic maps: A graph theoretical approach, Nonlinearity 4, 1131 (1991) Beck C, Thermodynamic formalism for quantum mechanical systems, Physica 50D, 1 (1991) Beck C, Graudenz D, Symbolic dynamics of successive quantum mechanical measurements, Phys. Rev. 46A, 6265 (1992) Beck C, Schlogl F, Thermodynamics of Chaotic Systems, Cambridge University Press, Cambridge (1993) Beck C, Chaotic cascade model for turbulent velocity distributions, Phys. Rev. 49E, 3641 (1994) Beck C, From Ruelle's transfer operator to the Schrodinger operator, Physica 85D, 459 (1995) Beck C, From the Perron-Frobenius equation to the Fokker-Planck equation, J. Stat. Phys. 79, 875 (1995) Beck C, Chaotic quantization of field theories, Nonlinearity 8, 423 (1995) Beck C, Tel T, Path integrals in the symbol space of chaotic mappings, J. Phys. 28A, 1889 (1995) Beck C, Dynamical systems of Langevin type, Physica 233A, 419 (1996) Beck C, Coupled map lattices simulating hydrodynamical turbulence, Physica 103D, 528 (1997) Beck C, Spontaneous symmetry breaking in a coupled map lattice simulation of quantized Higgs fields, Phys. Lett. 248A, 386 (1998) Beck C, Physical meaning for Mandelbrot and Julia sets, Physica 125D, 171 (1999)

Bibliography

255

Beck C, Application of generalized thermostatistics to fully developed turbulence, Physica 277A, 115 (2000) Beck C, Non-extensive statistical mechanics and particle spectra in elementary interactions, Physica 286A, 164 (2000) Beck C, Lewis G and Swinney HL, Measuring non-extensitivity parameters in a turbulent Taylor-Couette flow, Phys. Rev. 63E, 035303(R) (2001) Beck C, On the small scale statistics of Lagrangian turbulence, Phys. Lett. 287A, 240 (2001) Beck C, Dynamical foundations of nonextensive statistical mechanics, Phys. Rev. Lett. 87, 180601 (2001) Beck C, Scaling exponents in fully developed turbulence from nonextensive statistical mechanics, Physica 295A, 195 (2001) Beck C, Chaotic strings and standard model parameters, hep-th/0105152 Beck C, Nonextensive methods in turbulence and particle physics, condmat/0110071, to appear in Physica A (2002) Bediaga I, Curado EMP, de Miranda JM, A nonextensive thermodynamic equilibrium approach in e+e" —>• hadrons, Physica 286A, 156 (2000) Bergstroem L, Iguri S, Rubinstein H, Constraints on the variation of the fine structure constant from big bang nucleosynthesis, Phys. Rev. 60D, 045005 (1999) Bethke S, Hadronic physics in e + e~ annihilation, in [Peach et al. (1994)] Bevers M, Flather CH, Numerically exploring habitat fragmentation effects on populations using cell-based coupled map lattices, Theor. Pop. Biol. 55, 61 (1999) Billingsley P, Convergence of Probability Measures, Wiley, New York (1968) Binder PM, Privman V, Collective behavior in one-dimensional locally coupled map lattices, J. Phys. 25A, L775 (1992) Biro TS, Matinyan SG, Miiller B, Chaotic quantization of classical gauge fields, Found. Phys. Lett. 14, 471 (2001) Bjorken JD, Drell SD, Relativistic Quantum Mechanics, Mc Graw-Hill, New York (1965) Blank M, Generalized phase transitions in finite coupled map lattices, Physica 103D, 34 (1997) Boglione M, Pennington MR, Unquenching the scalar glueball, Phys. Rev. Lett. 79, 1998 (1997) Bohr T, Christensen OB, Size dependence, coherence, and scaling in turbulent coupled- map lattices, Phys. Rev. Lett. 63, 2161 (1989) Bohr T, Rand DA, A mechanism for localized turbulence, Physica 52D, 532 (1991) Bohr T, Jensen MH, Paladin G, Vulpiani A, Dynamical Systems Approach to Turbulence, Cambridge University Press, Cambridge (1998) Bohr T, van Hecke M, Mikkelsen R, Ipsen M, Breakdown of universality in transitions to spatiotemporal chaos, Phys. Rev. 86, 5482 (2001) Boldrighini C, Bunimovich LA, Cosimi G, Frigio S, Pellegrinotti A, Ising-type transitions in coupled map lattices, J. Stat. Phys. 80, 1185 (1995)

256

Bibliography

Born M, Atomic Physics, Blackie and Son, London (1953) Bottin S, Chate H, Statistical analysis of the transition to turbulence in plane Couette flow, Eur. Phys. J. 6B, 143 (1998) Brandenberger RH, Inflationary Universe Models and Cosmic Strings, in Physics of the Early Universe, SUSSP proceedings, eds. Peacock JA, Heavens AF, Davies AT, IOP Publishing, Bristol (1990) Breit JD, Gupta S, Zaks A, Stochastic quantization and regularization, Nucl. Phys. B233, 61 (1984) Bricmont J, Kupiainen A, High temperature expansions and dynamical systems, Commun. Math. Phys. 178, 703 (1996) Bunimovich A, Sinai YaG, Spacetime chaos in coupled map lattices, Nonlinearity 1, 491 (1988) Bunimovich LA, Lambert A, Lima R, The emergence of coherent structures in coupled map lattices, J. Stat. Phys. 61, 253 (1990) Bunimovich LA, Sinai YaG, Statistical mechanics of coupled map lattices, in [Kaneko (1993)] Bunimovich LA, Coupled map lattices - one step forward and 2 steps back, Physica 86D, 248 (1995) Bunimovich LA, Venkatagiri S, Onset of chaos in coupled map lattices via the peak-crossing bifurcation, Nonlinearity 9, 1281 (1996) Bunimovich LA, Coupled map lattices: Some topological and ergodic properties, Physica 103D, 1 (1997) Calogero F, Cosmic orign of quantization, Phys. Lett. 228A, 335 (1997) Carretero-Gonzalez R, Low dimensional traveling interfaces in coupled map lattices, Int. J. Bif. Chaos 7, 2745 (1997) Carretero-Gonzalez R, Arrowsmith DK, Vivaldi F, Mode-locking in coupled map lattices, Physica 103D, 381 (1997) Carretero-Gonzalez R, Arrowsmith DK, Vivaldi F, One-dimensional dynamics for traveling fronts in coupled map lattices, Phys. Rev. 61E, 1329 (2000) Carretero-Gonzalez R, Orstavik S, Huke J, Broomhead DS, Stark J, Thermodynamic limit from small lattices of coupled maps, Phys. Rev. Lett. 83, 3633 (1999) Cecotti S, Girardello L, Stochastic and parastochastic aspects of supersymmetric functional measures: A new non-perturbative approach to supersymmetry, Ann. Phys. (NY) 145, 81 (1983) Chate H, Manneville P, Spatio-temporal intermittency in coupled map lattices, Physica 32D, 409 (1988) Chate H, Manneville P, Continuous and discontinuous transition to spatiotemporal intermittency in two-dimensional coupled map lattices, Europhys. Lett. 6, 591 (1988) Chate H, Manneville P, Coupled map lattices as cellular automata, J. Stat. Phys. 56, 357 (1989) Chate H, Losson J, Non-trivial collective behavior in coupled map lattices: A transfer operator perspective, Physica 103D, 51 (1997)

Bibliography

257

Chernov NI, Limit theorems and Markov approximations for chaotic dynamical systems, Prob. Theory Relat. Fields, 101, 321 (1995) Chetyrkin KG, Kniehl BA, Steinhauser M, Strong coupling constant with flavor thresholds at four loops in the modified minimal-subtraction scheme, Phys. Rev. Lett. 79, 2184 (1997) Chew LY, Ting C, Microscopic chaos and Gaussian diffusion processes, to appear in Physica A (2002) Cohen EGD, Dynamical ensembles in statistical mechanics, Physica 240A, 43 (1997) Cohen EGD, Statistics and dynamics, to appear in Physica A (2002) Collet P, Thermodynamic limit of the Ginzburg-Landau equations, Nonlinearity 7, 1175 (1994) Cuche Y, Livi R, Politi A, Phase transitions in 2D linearly stable coupled map lattices, Physica 103D, 369 (1997) Damgaard PH, Tsokos K, Stochastic quantization, supersymmetry and the Nicolai map, Lett. Math. Phys. 8, 535 (1984) Damgaard PH, Hiiffel H, Stochastic quantization, Phys. Rep. 152, 227 (1987) Damgaard PH, Hiiffel H, eds., Stochastic Quantization, World Scientific, Singapore (1988) Degrassi G, Gambino P, Sirlin A, Precise calculation of Mw, sin 2 §w(Mz), and sin 2 0leef}, Phys. Lett. 394B, 188 (1997) DELPHI collaboration, Z. Phys. 73C, 229 (1997) Dens EJ, Van Impe JF, On the importance of taking space into account when modeling microbial competition in structured food products, Math. Comput. Simulat. 53, 443 (2000) Dettmann C, Stable synchronised states of coupled Tchebyscheff maps, nlin.CD/0110007 de San Roman FS, Boccaletti S, Maza D, Mancini H, Weak synchronization of chaotic coupled map lattices Phys. Rev. Lett. 81, 3639 (1998) Ding MZ, Yang WM, Stability of synchronous chaos and on-off intermittency in coupled map lattices, Phys. Rev. 56E, 4009 (1997) Djouadi A, Spira M, Zerwas PM, QCD corrections to hadronic Higgs decays, Zeitschr. Phys. 70C, 427 (1996) Dolgopyat D, Entropy of coupled map lattices, J. Stat. Phys. 86, 377 (1997) Dorfman JR, An Introduction to Chaos in Nonequilibrium Statistical Mechanics, Cambridge University Press, Cambridge (1999) Diirr D, Spohn H, Brownian motion and microscopic chaos, Nature 394, 831 (1998) Eckmann JP, Ruelle D, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57, 617 (1985) Eddington AS, Fundamental Theory, Cambridge University Press, Cambridge (1948) Egolf DA, Socolar JES, Failure of linear control in noisy coupled map lattices, Phys. Rev. 57E, 5271 (1998)

258

Bibliography

Ennyu D, Kawabe H, Nakazawa N, Algebraic structure in 0 < c < 1 open-closed string field theories, Phys. Lett. 454B, 43 (1999) Falcone F, Tramontane F, Leptogenesis and neutrino parameters, Phys. Rev. 63D, 073007 (2001) Fernandez B, Kink dynamics in one-dimensional coupled map lattices, Chaos 5, 602 (1995) Fernandez B, Existence and stability of steady fronts in bistable coupled map lattices, J. Stat. Phys. 82, 931 (1996) Frisch U, Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge (1995) Gade PM, Amritkar RE, Spatially periodic-orbits in coupled-map lattices, Phys. Rev. 47E, 143 (1993) Gade PM, Rai R, Singh H, Stochastic resonance in maps and coupled map lattices, Phys. Rev. 56E, 2518 (1997) Gade PM, Feedback control in coupled map lattices, Phys. Rev. 57E, 7309 (1998) Gaspard P, Chaos, Scattering and Statistical Mechanics, Cambridge University Press, Cambridge (1998) Gasser J, Leutwyler H, Quark masses, Physics Reports 87, 77 (1982) Gauntlett JP, M-theory: strings, duality and branes, Contemp. Phys. 39, 317 (1998) Gell-Mann M, Ramond P, Slansky R, in: Supergravity, eds. van Nieuwenhuizen P and Freedman D, North Holland, Amsterdam (1979) Ghosh A, Pollifrone G, Veneziano G, Pre-big bang cosmology and quantum fluctuations, Nucl. Phys. 88B (Proc. Suppl.), 341 (2000) Gielis G, MacKay RS, Coupled map lattices with phase transition, Nonlinearity 13, 867 (2000) Gilson JG, Calculating the fine structure constant, Physics Essays 9, 342 (1996) Gilson JG, Alpha and electro-weak coupling, Proceedings PIRT Conference, IC London (2000) Gorbon A, Erzan A, Multiscaling of the graph length in coupled map lattices, J. Phys. 30A, 785 (1997) Goto K, Yamaguchi A, Tsuda I, Examining the robustness of coupled map lattices against discretization, Int. J. Bif. Chaos 7, 1679 (1997) Gozzi E, Functional-integral approach to Parisi-Wu stochastic quantization: Scalar theory, Phys. Rev. 28D, 1922 (1983) Grassberger P, Schreiber T, Phase-transitions in coupled map lattices, Physica 50D, 177 (1991) Gray N, Broadhurst DJ, Grafe W, Schilcher K, Three-loop relation of quark MS and pole masses, Zeitschr. Phys. 48C, 673 (1990) Green MB, Schwarz JH, Witten E, Superstring Theory, vol.1 and II, Cambridge University Press, Cambridge (1987) Groom DE et al. (Particle Data Group), Eur. Phys. J. 15C, 1 (2000), web update: http://pdg.lbl.gov Gross DJ, Migdal AA, Nonperturbative 2-dimensional quantum gravity, Phys.

Bibliography

259

Rev. Lett. 64, 127 (1990) Gubser SS, Gukov S, Klebanov IR, Rangamani M, Witten E, The Hagedorn transition in noncommutative open string theory, J. Math. Phys. 42, 2749 (2001) Guida R, Zinn-Justin J, Critical exponents of the iV-vector model, J. Phys. 31A, 8103 (1998) Guth A, Inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. 23D, 347 (1981) Hagedorn R, Suppl. Nuovo Cim. 3, 147 (1965) Hasselmann K, The metron model: Elements of a unified deterministic theory of fields and particles, Physics Essays 9, 311 (1996) He GW, Li JC, The mechanism of pattern selection in coupled map lattices, Phys. Lett. 185A, 55 (1994) He GW, Lambert A, Lima R, Wavelike patterns in one-dimensional coupled map lattices, Physica 103D, 404 (1997) Heinz U, Hu CR, Leupold S, Matinyan SG, Miiller B, Thermalization and Lyapunov exponents in Yang-Mills-Higgs theory, Phys. Rev. 55D, 2464 (1997) Hilgers A, Beck C, Turbulent behavior of stock exchange indices and foreign currency exchange rates, Int. J. Bif. Chaos 7, 1855 (1997) Hilgers A, Beck C, Coupled map lattices simulating fully developed turbulent flows, in: Applied Nonlinear Dynamics and Stochastic Systems near the Millenium, eds. Kadtke JB, Bulsara A, American Institute of Physics p.11 (1997) Hilgers A, Beck C, Hierarchical coupled map lattices as cascade models for hydrodynamical turbulence, Europhys. Lett. 45, 552 (1999) Hilgers A, Beck C, Approach to Gaussian stochastic behavior for systems driven by deterministic chaotic forces, Phys. Rev. 60E, 5385 (1999) Hilgers A, Beck C, Higher-order correlations of Tchebyscheff maps, Physica 156D, 1 (2001) Houlrik JM, Webman I, Jensen MH, Mean-field theory and critical-behaviour of coupled map lattices, Phys. Rev. 41A, 4210 (1990) Houlrik JM, Jensen MH, Critical correlations in coupled map lattices, Phys. Lett. 163A, 275 (1992) Horava P, Witten E, Heterotic and type I string dynamics from eleven dimensions, Nucl. Phys. 460B, 506 (1996) Huner M, Erzan' A, The fixed scale transformation approach to spatiotemporal intermittency in coupled map lattices, Physica 212A, 314 (1994) Isola S, Politi A, Ruffo S, Torcini A, Lyapunov spectra of coupled map lattices, Phys. Lett. 143A, 365 (1990) Itzykson C, Zuber JP, Quantum Field Theory, McGraw Hill, New York (1980) Jarvenpaa E, Jarvenpaa M, On the definition of SRB-measures for coupled map lattices, Commun. Math. Phys. 220, 1 (2001) Jiang Y, Phase transitions in two-variable coupled map lattices, Phys. Rev. 56E, 2672 (1997)

260

Bibliography

Jiang Y, Collective behavior of coupled map lattices with asymmetrical coupling, Phys. Lett. 240A, 60 (1998) Jiang MH, Pesin YB, Equilibrium measures for coupled map lattices: Existence, uniqueness and finite-dimensional approximations, Commun. Math. Phys. 193, 675 (1998) Jiang Y, Parmananda P, Synchronization of spatiotemporal chaos in asymmetrically coupled map lattices, Phys. Rev. 57E, 4135 (1998) Johal RS, Modified exponential function: the connection between nonextensivity and q-deformation, Phys. Lett. 258A, 15 (1999) John JK, Amritkar RE, Coherent structures in coupled map lattices, Phys. Rev. 51E, 5103 (1995) Just W, Bifurcations in globally coupled map lattices, J. Stat. Phys. 79, 429 (1995) Just W, Analytical approach for piecewise linear coupled map lattices, J. Stat. Phys. 90, 727 (1998) Kane G, Modern Elementary Particle Physics, Addison-Wesley, Redwood City (1987) Kaku M, Introduction to Superstrings, Springer, New York (1988) Kaneko K, Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice - towards a prelude of a field-theory of chaos, Progr. Theor. Phys. 72, 480 (1984) Kaneko K, Spatiotemporal intermittency in coupled map lattices, Progr. Theor. Phys. 74, 1033 (1985) Kaneko K, Turbulence in coupled map lattices, Physica 18D, 475 (1986) Kaneko K, Lyapunov analysis and information-flow in coupled map lattices, Physica 23D, 436 (1986) Kaneko K, Pattern dynamics in spatiotemporal chaos - pattern selection, diffusion of defect and pattern competition intermittency, Physica 34D, 1 (1989) Kaneko K, Spatiotemporal chaos in one-dimensional and two-dimensional coupled map lattices, Physica 37D, 60 (1989) Kaneko K (ed.), Theory and Applications of Coupled Map Lattices, John Wiley and Sons, New York (1993) Kantz H, Schreiber T, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge (1997) Kantz H, Olbrich E, The transition from deterministic chaos to a stochastic process, Physica 253A, 105 (1998) Kaplan JL, Yorke J A, in: Functional Differential Equations and the Approximation of Fixed Points, Lecture Notes in Mathematics 730, 228, Springer, New York (1979) Kapral R, Pattern-formation in two-dimensional arrays of coupled discrete-time oscillators, Phys. Rev. 31A, 3868 (1985) Kapral R, Chemical waves and coupled map lattices, in [Kaneko (1993)] Kapral R, Livi R, Oppo GL, Politi A, Dynamics of complex interfaces, Phys. Rev.

Bibliography

261

49E, 2009 (1994) Keller G, Kunzle M, Nowicki T, Some phase-transitions in coupled map lattices, Physica 59D, 39 (1992) Keller G, Kunzle M, Transfer operator for coupled map lattices, Erg. Th. Dyn. Syst. 12, 297 (1992) Kenyon IR, General Relativity, Oxford University Press, Oxford (1990) Knill O, Nonlinear dynamics from the Wilson Lagrangian, J. Phys. 29A, L595 (1996) Kolb E, Turner M, The Early Universe, Addison-Wesley, New York (1990) Kornberg A, Baker TA, DNA Replication, WH Freeman and Company, New York (1992) Kuznetsov SP, Mosekilde E, Coupled map lattices with complex order parameters, Physica 292A, 182 (2001) Lai YC, Winslow RL, Fractal basin boundaries in coupled map lattices, Phys. Rev. 50E, 3470 (1994) Lai YC, Lerner D, Williams K, Grebogi C, Unstable dimension variability in coupled chaotic systems, Phys. Rev. 60E, 5445 (1999) Lambert A, Lima R, Stability of wavelengths and spatiotemporal intermittency in coupled map lattices, Physica 71D, 390 (1994) Landsberg PT, Seeking Ultimates, IOP, Bristol (2000) Langacker P, Polonsky N, Uncertainties in coupling constant unification, Phys. Rev. 47D, 4028 (1993) Lemaitre A, Chate H, Manneville P, Conditional mean field for chaotic coupled map lattices, Europhys. Lett. 39, 377 (1997) Lemaitre A, Chate H, Nonperturbative renormalization group for chaotic coupled map lattices, Phys. Rev. Lett. 80, 5528 (1998) Lemaitre A, Chate H, Phase ordering and onset of collective behavior in chaotic coupled map lattices, Phys. Rev. Lett. 82, 1140 (1999) Lepri S, Just W, Mean-field theory of critical coupled map lattices, J. Phys. 31A, 6175 (1998) Lidsey JE, Embedding of superstring backgrounds in Einstein gravity, Phys. Lett. 417B, 33 (1998) Linde A, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. 108B, 389 (1982) Linde A, Inflationary cosmology and creation of matter in the universe, Class. Quant. Grav. 18, 3275 (2001) Liu WJ, Yang JZ, Hu G, Numerical results for the defects in two-dimensional one-way coupled map lattices, Commun. Theor. Phys. 30, 405 (1998) Livi R, Martinezmekler G, Ruffo S, Periodic-orbits and long transients in coupled map lattices, Physica 45D, 452 (1990) Losson J, Mackey MC, Statistical cycling in coupled map lattices, Phys. Rev. 50E, 843 (1994) Losson J, Mackey MC, Coupled map lattices as models of deterministic and

262

Bibliography

stochastic differential-delay equations, Phys. Rev. 52E, 115 (1995) Losson J, Vannitsem S, Nicolis G, Aperiodic mean-field evolutions in coupled map lattices, Phys. Rev. 57E, 4921 (1998) MacKay RS, Renormalization in Area-Preserving Maps, World Scientific, Singapore (1993) Mackey MC, Time's Arrow: The Origins of Thermodynamic Behavior, Springer, New York (1992) Mackey M, Milton J, Asymptotic stability of densities in coupled map lattices, Physica 80D, 1 (1995) Majid S, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge (1995) Maldacena J, D-branes and near extremal black holes at low energies, Phys. Rev. 55D, 7645 (1997) Maldacena J, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998) Marciano WJ, Flavor thresholds and A in the modified minimal-subtraction scheme, Phys. Rev. 29D, 580 (1984) Marcq P, Chate H, Manneville P, Universal critical behavior in two-dimensional coupled map lattices, Phys. Rev. Lett. 77, 4003 (1996) Marcq P, Chate H, Manneville P, Universality in Ising-like phase transitions of lattices of coupled chaotic maps, Phys. Rev. 55E, 2606 (1997) Martinezmekler G, Cocho G, Gelover A, Bulajich R, Modeling genetic evolution with coupled map lattices, Rev. Mex. Fis. 38, 127 (1992) Matinyan SG, Miiller B, Quantum fluctuations and dynamical chaos, Phys. Rev. Lett. 78, 2515 (1997) Mayer-Kress G, Kaneko K, Spatiotemporal chaos and noise, J. Stat. Phys. 54, 1489 (1989) Melchiorri A, Vernizzi F, Durrer R, Veneziano G, Cosmic background anisotropies and extra dimensions in string cosmology, Phys. Rev. Lett. 83, 4464 (1999) Mezic I, FKG inequalities in cellular automata and coupled map lattices, Physica 103D, 491 (1997) Miller J, Huse D, Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled map lattice, Phys. Rev. 48E, 2528 (1993) Mondragon RJ, Arrowsmith DK, On control of coupled map lattices: Using local dynamics to predict controllability, Int. J. Bif. Chaos 7, 383 (1997) Namiki M, Ohba I, Okano K, Yamanaka Y, Stochastic quantization of non-abelian gauge field, Progr. Theor. Phys. 69, 1580 (1983) Namiki M, Kanenaga M, Evolution of fluctuations toward quantum-mechanical uncertainty in new quantization schemes, Phys. Lett. 249A, 13 (1998) Narison S, Light and heavy quark masses, test of PCAC and flavour breakings of condensates in QCD, Phys. Lett. 216B, 191 (1989) Narison S, A fresh look into heavy quark-mass values, Phys. Lett. 341B, 73 (1994) Nicolai H, On a new characterization of scalar supersymmetric theories, Phys. Lett. 89B, 341 (1980)

Bibliography

263

Nottale L, Scale relativity and gauge invariance, Chaos Sol. Pract. 12, 1577 (2001) Orstavik S, Stark J, Reconstruction and cross-prediction in coupled map lattices using spatio-temporal embedding techniques, Phys. Lett. 247A, 145 (1998) Ohishi Y, Ohashi H, Akiyama M, Controlling spatiotemporal chaos by pinning method in 2d- dimensional coupled map lattices, Jap. J. Appl. Phys. 34, L1420 (1995) Parekh N, Puri S, Velocity selection in coupled map lattices, Phys. Rev. 47E, 1415 (1993) Parekh N, Parthasarathy S, Sinha S, Global and local control of spatiotemporal chaos in coupled map lattices, Phys. Rev. Lett. 81, 1401 (1998) Parisi G, Sourlas N, Random magnetic fields, supersymmetry, and negative dimensions, Phys. Rev. Lett. 43, 744 (1979) Parisi G, Wu YS, Perturbation theory without gauge fixing, Sci. Sin. 24, 483 (1981) Peach KJ, Vick LLJ, eds., High Energy Phenomenology, Proc. 42nd Scottish Universities Summer School in Physics, St. Andrews, IOP Publishing, London (1994) Pennini F, Plastino A, Plastino AR, Tsallis entropy and quantal distribution functions, Phys. Lett. 208A, 309 (1995) Periwal V, Model of Polyakov duality: String field theory Hamiltonians from Yang-Mills theories, Phys. Rev. 62D, 047701 (2000) Pikovsky AS, Chaotic wave-front propagation in coupled map lattices, Phys. Lett. 156A, 223 (1991) Pikovsky AS, Grassberger P, Symmetry-breaking bifurcations for coupled chaotic attractors, J. Phys. 24A, 4587 (1991) Plastino AR, Plastino A, Vucetich H, A quantitative test of Gibbs statistical mechanics, Phys. Lett. 207A, 42 (1995) Piatt N, Hammel SM, Pattern formation in driven coupled map lattices, Physica 239A, 296 (1997) Polchinski J, String Theory, vol. I and II, Cambridge University Press, Cambridge (1998) Polchinski J, Quantum gravity at the Planck length, Int. J. Mod. Phys. 14A, 2633 (1999) Polyakov D, AdS/CFT correspondence, critical strings and stochastic quantization, Class. Quant. Grav. 18, 1979 (2001) Pope S, Turbulent Flows, Cambridge University Press, Cambridge (2000) Qu ZL, Xu CY, Ma BK, Hu G, Periodic windows and chaotic transient in coupled map lattices, Act. Phys. Sin. 3, 353 (1994) Qu ZL, Hu G, Spatiotemporally periodic states, periodic windows, and intermittency in coupled map lattices, Phys. Rev. 49E, 1099 (1994) Qu ZHL, Hu G, Ma BK, Tian G, Spatiotemporally periodic patterns in symmetrically coupled map lattices, Phys. Rev. 50E, 163 (1994) Raghavachaxi S, Glazier JA, Spatially coherent states in fractally coupled map lattices, Phys. Rev. Lett. 74, 3297 (1995)

264

Bibliography

Ramond P, Field Theory: A Modern Primer, Addison-Wesley, Redwood City (1989) Ramos FM, Rosa RR, Neto CR, Bolzan MJA, Sa LDA, Velho HFC, Non-extensive statistics and three-dimensional fully developed turbulence, Physica 295A, 250 (2001) Renyi A, Probability Theory, North-Holland, Amsterdam (1970) RoepstorfF G, Path Integral Approach to Quantum Physics, Springer, New York (1994) RoepstorfF G, Superconnections and the Higgs field, J. Math. Phys. 40, 2698 (1999) Roy M, Amritkar RE, Observation of stochastic coherence in coupled map lattices, Phys. Rev. 55E, 2422 (1997) Rudzick O, Pikovsky A, Scheffczyk C, Kurths J, Dynamics of chaos-order interface in coupled map lattices, Physica 103D, 330 (1997) Ruelle D, Large volume limit distribution of characteristic exponents in turbulence, Commun. Math. Phys. 87, 287 (1982) Ruelle D, Sinai YaG, From dynamical systems to statistical mechanics and back, Physica 140A, 1 (1986) Ruelle D, Chance and Chaos, Princeton University Press, Princeton (1991) Rumpf H, Stochastic quantization of Einstein gravity, Phys. Rev. 33D, 942 (1986) Ryang S, Saito T, Shigemoto K, Canonical stochastic quantization, Progr. Theor. Phys. 73, 1295 (1985) Salem R, Zygmund A, Proc. Nat. Acad. USA 33, 333 (1947) Schlogl F, Thermodynamic uncertainty relation, J. Chem. Solids 49, 679 (1988) Schlogl F, Probability and Heat, Vieweg, Braunschweig (1989) Schmuser F, Just W, Kantz H, Relation between coupled map lattices and kinetic Ising models, Phys. Rev. 61E, 3675 (2000) Schmuser F, Just W, Non-equilibrium behaviour in unidirectionally coupled map lattices, J. Stat. Phys. 105, 525 (2001) Schreiber T, Spatiotemporal structure in coupled map lattices - 2-point correlations versus mutual information, J. Phys. 23A, L393 (1990) Schulman LS, Techniques and Applications of Path Integration, Wiley, New York (1981) Sinai YaG (ed.), Mathematical Problems of Statistical Mechanics, World Scientific, Singapore (1991) Shen WX, Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices, Siam J. Appl. Math. 56, 1379 (1996) Shimizu T, Relaxation and bifurcation in Brownian motion driven by a chaotic force, Physica 164A, 123 (1990) Spira M, Djouadi A, Graudenz D, Zerwas PM, Higgs boson production at the LHC, Nucl. Phys. 453B, 17 (1995) Spohn H, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Mod. Phys. 52, 569 (1980) Steinhauser M, Higgs boson decay intoo gluons up to 0(a3sGFml), Phys. Rev.

Bibliography

265

59D, 054005 (1999) Susskind L, The world as a hologram, J. Math Phys. 36, 6377 (1995) TASSO collaboration, Z. Phys. 22C, 307 (1984) t'Hooft G, Isler K, Kalitzin S, Quantum-field theoretic behavior of a deterministic cellular automaton, Nucl. Phys. 386B, 495 (1992) t'Hooft G, In Search for the Ultimate Building Blocks, Cambridge University Press, Cambridge (1997) t'Hooft G, Quantum mechanical behaviour in a deterministic model, Found. Phys. Lett. 10, 105 (1997) Tirnakli U, Torres DF, Quantal distribution functions in non-extensive statistics and an early universe test revisited, Physica 268A, 225 (1999) Torcini A, Frauenkron H, Grassberger P, Studies of phase turbulence in the one-dimensional complex Ginzburg-Landau equation, Phys. Rev. 55E, 5073 (1997) Torcini A, Livi R, Politi A, Ruffo S, Universal scaling law for the largest Lyapunov exponent in coupled map lattices - Comment, Phys. Rev. 78, 1391 (1997) Tsallis C, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479 (1988) Tsallis C, Nonextensive physics: a possible connection between generalized statistical mechanics and quantum groups, Phys. Lett. 195A, 329 (1994) Tsallis C, Sa Barreto FCS, Loh ED, Generalization of the Planck radiation law and application to the cosmic microwave background-radiation, Phys. Rev. 52E, 1447 (1995) Tsallis C, Mendes RS, Plastino AR, The role of constraints within generalized nonextensive statistical mechanics, Physica 261A, 534 (1998) Tsallis C, Nonextensive statistics: Theoretical, experimental and computational evidences and connections, Braz. J. Phys. 29, 1 (1999) Ulam SM, von Neumann J, Bull. Am. Soc. 53, 1120 (1947) van Kampen NG, Stochastic Processes in Physics and Chemisty, North-Holland, Amsterdam (1981) Veneziaono G, Inhomogeneous pre-big bang string cosmology, Phys. Lett. 406B, 297 (1997) Veneziano G, Pre-bangian orign of our entropy and time arrow, Phys. Lett. 454B, 22 (1999) Vilenkin A, Shellard EPS, Cosmic Strings and Other Topological Defects, Cambridge University Press (1994) Volevich VL, Kinetics of coupled map lattices, Nonlinearity 4, 37 (1991) Wald RM, General Relativity, University of Chicago Press, Chicago (1984) Wang JL, Chen GZ, Qin TF, Ni WS, Wang XM, Synchronizing spatiotemporal chaos in coupled map lattices via active-passive decomposition, Phys. Rev. 58E, 3017 (1998) Wang W, Cerdeira HA, Dynamical behavior of the multiplicative diffusion coupled map lattices, Chaos 6, 200 (1996) Weinberg S, Implications of dynamical symmetry breaking, Phys. Rev. 13D, 974

266

Bibliography

(1976) Weinberg S, The cosmological constant problem, Rev. Mod. Phys. 61, 1 (1989) Wilk G, Wlodarczyk Z, Interpretation of the nonextensivity parameter q in some applications of Tsallis statistics and Levy distributions, Phys. Rev. Lett. 84, 2770 (2000) Witten E, Duality, spacetime and quantum mechanics, Physics Today 50, 28 (1997) Witten E, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Int. J. Mod. Phys. 16A, 2747 (2001) Xie FG, Hu G, Clustering dynamics in globally coupled map lattices, Phys. Rev. 56E, 1567 (1997) Yang WM, Ding EJ, Ding MZ, Universal scaling law for the largest Lyapunov exponent in coupled map lattices, Phys. Rev. Lett. 76, 1808 (1996) Zhilin Q and Gang H, Spatiotemporally periodic states, periodic windows, and intermittency in coupled map lattices, Phys. Rev. 49E, 1099 (1994) Zygmund A, Trigonometric Series, Cambridge University Press, Cambridge (1959)

Index

(^-mixing, 4,6 0 4 -theory, xv,2,30,58,97 JV+1 -theory, 27,30,234 (/) ^-distribution, 127,129

bosonic string, 35,237 boson masses, 140,159,172,245,251 bosons at Planck scale, 202 b-quark mass, 154,159,166,245 breathers, 47 Brownian motion, 4,7

action, 1,3,90 anti-diffusive coupling, 29,33,145,172,175,189,235 anti-integrable limit, 31,34,75,90,94 arrow of time, 115 atmospheric neutrinos, 170 attractor, 44,53 of logistic map, 76 of coupled map, 44,53,78 of cubic map, 77

Calabi-Yau manifold, 208 cascade in pre-Planck era, 219,221,225 canonical distribution, 115,120 chaotic D-brane, 37 chaotic Higgs field, 92 chaotic inflation, 217 chaotic map, 4,8,13,232 chaotic membrane, 89 chaotic noise, 3,27 chaotic quantization, xiv,l,9,27,36,234,248 chaotic string, vii,27,35,104,131,151,175,229,233 charges, 155,225,251 Central Limit Theorem, 4,16 centrifugal force, 70 chemical waves, xiv CP transformation, 106 CP violation, 186 CPT invariance, 108 c-quark mass, 154,166,245 compactified dimensions,

backward coupling, 29,107,163 baryon masses, 184,246 basin of attraction, 44,46 Bernoulli shift, 15,21,221,232,237 beta function, 117,191,194,197,245 bifurcations, 76,82 of logistic map, 76 of cubic map, 77,79 of coupled maps, xiii,53,55,78,80 binary shift map, 5 black holes, 71,203,217,222,225,239,250 Boltzmann factor, 121,127,129 Boltzmann statistics, 114 267

268

xvii,128,207,209,212,225,249 size of, 128,208,225 compactified space, xvii,203,208,210,234 complex Ginzburg-Landau equation, 91 complex structures, 47,50,55,83 complex scalar field, 90 Compton wave length, 69,220 confinement, 175 continued fraction map, 5 control, xiv Coulomb force, 99,102 Coulomb potential, 99,108 coupled map lattice, xiii,29,91,102 correlation function, 17,67 correlation length, 76,82 cosmic microwave background, 125,227 cosmic strings, 89 cosmological constant, 69,212,237 critical point, 81,91,94 cubic map, 31,75,77 curvature, 212 cyclic matrix, 43,214 dark matter, 187,226 D-brane, 37,203 degrees of freedom, 106,117,178 defects, 47,50,89 de Sitter state, 217,223 differential cross section, 125 diffusive coupling, xiii,29,99,101,175,235 dihedral symmetry, 80 dilaton, 152 Dirac lake, 99,126 Dirac operator, 204 discrete symmetry, 38,40,64,78,106,168 DNA, 68,248 double string, 67 double tree, 18,24 d-quark, 136,155,167,245

Index dual gravitational coupling, 221,224 duality, 103,217,220 for chaotic string, 217,220,225 in superstring theories, 173,219 of interpretations, 103 wave-particle, 103 e + e " annihilation, 120,124,128 effective energy levels, 122 effective weak mixing angle, 140,159,231,241,245 electric interaction, 134,136,160 at Planck scale, 199,203 electron mass, 135,155,167,245 electroweak interaction, 136,139,149,155,157,159 electroweak phase transition, 199,227 electroweak scale, 128,227 Einstein field equations, 131,210,212,240 entropic index, 119,124,248 entropy function, 117,119,237 ergodicity, 40,235 escort distribution, 121 event horizon, 71,218 expectation, 17,97 of field, 79,89,93 of observable, 17,115,235 of vacuum energy, 57,64,67,69,131,151,175,216,236 extra dimensions, xvii,128,207,224,227 fermion masses, 152,165,245,251 fermionic string, 35 Feynman webs, 104,176,178,238,240 fictitious time, xiv,2,29,205,207,234 financial markets, xiv fine structure constant, xvi, 100,134,140,231,241,245 first quantization, 95,103 fixed point, 44 of coupled map lattice, 44,46 of map, 44

Index of Perron-Frobenius operator, 40 of renormalization flow, xvi,133,239 fluctuations, 95 of cosmic microwave background, 125 of metric, 108,110 of temperature, 126,224 of virtual momenta, 95,104,234,237 folding of chaotic strings, 221 forward coupling, 29,107,163 Fourier transformed chaotic noise, 9 free field, 6,8,11,206 free Maxwell field, 11 free quark masses, 143,153,245 free standard model parameters, vii,xv,131,229,245,250 fully developed turbulence, xv,29,99,120 gamma distribution, 127 gauge field, 11,249 gauge transformation, 90 Gaussian random variable, 4,127,129 Gaussian white noise, 2,4,6,16,47,227,248 generalized canonical distribution, 120,127,130,248 generalized entropies, 119 generalized statistical mechanics, 113,119,248,251 Ginzburg-Landau equation, 91 global coupling, xiii glueball, 144,146,176,241 graph theoretical method, 17,23,233 grand unification, xvi,194,197,227 gravitational coupling, 24,109,148,167,188,227,242 gravitational interaction, 109,148,162,187,210,243 Greens function, 6,10 GUT scale, 194,196,243 hadron masses, 175,180,184,246 hadronization, 68,175

269

Hagedorn temperature, 124 Hamiltonian, 113,115 harmonic oscillator, 67,223 Hawking radiation, 217,219,225,239 Hawking temperature, 218,220,224,239 heat bath of the vacuum, 113 heavy ion collisions, 120 heavy quark masses, 154,162,166,245 higher-order correlations, 17,22,47 Higss field, 92,160,202 Higgs mass, 140,148,173,193,232,245 holographic principle, 248 inflation, 215,217,226 information shift, 16,222 interaction energy, 65,131,216,240,250 invariant density, 5,17,39,121,130,235 invariant measure, 5,39 Ising system, 82 isospin, 155 Ito-Stratonovich ambiguity, 32 Jacobian, 42 Kerr black hole, 71 Klein-Gordon field, 2,6,23,129 Klein-Gordon operator, 204 kinks, 47 Kobayashi-Maskawa matrix, 158,243 lacunary trigonometric series, 23 Lagrangian, 90,92 Langevin equation, 2,129,211 Larmor frequency, 34,72 lattice gauge theory, 144,186 Liapunov exponents, xiii,43,76 Laplacian coupling, 28,99,233,238 left-handed fermions, 137,139,171 lepton masses, 167,170,231,245,251 light quark masses, 136,139,167,245 logistic map, 76 magnetic field, 34,72

270

map, 4,8,14,76,223,232 mass, 97,225,245,251 of bosons, 241,245,251 of dark matter particles 188,226 of fermions, 241,245,251 of universe, 224 Maxwell field, 11 meson masses, 181, 246 metric, 108,208,233,238,250 metric tensor, 210 Minkowski metric, 111,215 Minkowski space, 213,249 mirror symmetry, 204 mixed state, 108,156,197,243 mixing, 4,39,235 moduli, 37,208 momentum space, 9 M-theory, xvi,37,207,224,229,249 multiplicity factor, 106,178 muon mass, 157,167,245

Index Planck mass, 8,24,109,148,167,202,220,225 Planck scale, 128,148,198,202,221,225 phase space cells, 95,97,114 phase transitions, 75,84,89,93,199 pole quark masses, 143 pre-big bang scenario, 131,222,239,249 pre-standard model parameters, 131,151,225,239 pre-universe, 131,151,222,239,249 propagator, 6,10 proper time, 98,208 proton mass, 183 pure state, 108

order parameter, 79 Ornstein-Uhlenbeck process, 13,129

quark flavor thresholds, 143 quark masses, 231,245 free, 143,153,245 light, 136,139,245 heavy, 154,162,166,245 pole, 143 running MS, 143 QCD, 142,154,186,194 QCD scale parameter, 142,144,187 QED, 3,135,159 quantization, 1,9,103,227 chaotic, xiv,l,9,27,36,234,248 first, 95,103 second, 1,37,103,227 stochastic, xiv,l,27,36,211,234,248 quantum field theory, 3,34 quantum gravity,xvii,108,207,234 quantum groups, 121 quantum mechanics, 103 quantum mechanical string oscillator, 106,211,223,249

Parisi-Wu approach, xiv,2,27,234 path integrals, 5,206 periodic orbits, 46,76,81 Perron-Probenius operator, 40 perturbation theory, 23,30,116,135,142,191,247

radiative corrections, 140 random coincidence, 167,247 real scalar field, 2,30,79,82 renormalization flow, xvi,133,152,212,239 renormalized parameters, 31

natural invariant measure, 5,17,40 neutrino masses, 168,170,231,242,244 neutrinos, 136,168 atmospheric, 170 left-handed, 137 right-handed, 155,157,200 solar, 171 neutron mass, 183 Newton's law, 97 Nicolai map, 205 noise string, 28,37 nonextensive statistical mechanics, xv,115,119,248

271

Index

renormalized potential, 33 Renyi information measures, 120 resonances, 184 right-handed fermions, 139,155,157,201 rotating strings, 69 running coupling constants, 105,116,135,142,191,197 scaling region, 163,243 Schrodinger equation, 104 Schwarzschild metric, 109 Schwarzschild radius, 71, 218 S-duality, 219 second quantization, 1,37,103,227 self energy, 57,151 of chaotic string, 57,60,62,151,216,242,251 of electron, 71 Shannon entropy, 120 shift of information, 16,222 spatio-temporal patterns, 47,87,214 spin, 34,69,242 spin system, 82 spontaneous symmetry breaking, 78,91,93,199,215 s-quark mass 157,167,245 stability, 41,251 of fixed point, 42,45 of periodic orbit, 42,81 of vacuum, 118,133,251 stable fixed points, 44 of coupled map lattices, 44,46,55 of maps, 76 of renormalization flow, 133,240 standard model, vii,3,194 standard model couplings, vii,131,151,231,238,241,243 standard model parameters, vii,132,229,237,239,245,249 states, 108,230 mixed, 156,197,243 of maximum information, 116,118 of minimum correlation, 118

pure, 243 synchronized, xiv,43 turbulent quantum, xv,27,29,230,233 stochastic quantization, xiv,1,27,36,211,234,248 stress-energy tensor, 210,212 string, 35 bosonic, 35,237 chaotic, vii,27,35,104,131,151,175,229,233 cosmic, 89 fermionic, 35 noise, 28,37 super, xvi,35,37,128,207,229 string cosmology, xvii,222 string field theory, xvii,37 strong interaction, 141,144,152,157,175 strong coupling constant 144,194,241,245 SC/(2)-symmetry, 92,249 5C/(3)-symmetry, 249 superstrings, xvi,35,37,128,207,229 sup ersy mmetry, 148,191,193,203,237,245,251 supersymmetry breaking, 203,205,251 symmetry, 37,39,59,64,168,242,244 CPT, 108 dihedral, 80 discrete, 38,40,64,78,106,168 gauge, 90,92,249 mirror, 204 super, 148,191,193,203,237,245,251 Z2, 38,78,249 symmetry breaking, 75,78,89,91,199,251 synchronized states, xiv,43 tau mass, 160,166,245 Tchebyscheff map, 13,29,32,34,62,91,98,121,223,232 Tchebyscheff polynomial, 14,32,62,232

272

thermodynamic formalism, 76,82 thermodynamic limit, 76,90 t-quark mass 154,161,166,245 transverse momentum spectrum, 125 traveling waves, viv,51 transients, 82,87 Tsallis entropies, 119 T-duality, 219 T-transformation, 107 turbulent quantum state, xv,27,29,230,233 [/(l)-symmetry, 90,249 uncertainty relation, 96,100,114,234,248,250 unified gauge coupling, 195,199,203,226 unstable fixed points, 45 of coupled map lattice, 45 of renormalization flow, 133,205,251 u-quark mass 139,167,245 vacuum energy, xvi,57,72,234,239,244 interaction, 65,131,236 self, 57,60,62,151,236 total, 175,179,182,188 vacuum expectation, 79,89,93 vacuum fluctuations, vii,xiii,95,108,110,230 vacuum polarization, 116,118,135,142 virtual momenta, 95,97,104,114,237 W-boson mass 140,142,162,245 weak coupling constant, 154,160,194,196,200 weak interaction, 136,139,154,157,159 at GUT scale, 195,197 at Planck scale, 199,201 left-handed, 137 right-handed, 139,155,157 weak mixing angle, 137,140,159,173,200,231,241,245 Weinberg mixing angle,

Index 137,140,159,173,200,231,241,245 Wiener process, 3,5,13 Yukawa couplings, 160,162,166,193,203,243 Yukawa interaction, 160,162,203 Z-boson mass, 92,125,143,159,173,202 Z 2 -symmetry, 38,78,80,237,250

Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields ['•';• ' : y - . ^ 4 : ^ ; : : : |

I .

-

• i •

-

.

~ • • •

[•;.cv.'./.':••'.

•'•

•

• • •

••:::•.:*::••

.—

•

•••;.

A spatio-temporal pattern as generated by the chaotic 3A~ string for a=I nop). and a-0.9 (bottom right).

This book describes new applications for spatio-temporal chaotic dynamical systems in elementary particle physics and q u a n t u m field theories. The stochastic quantization approach of Paris! and Wu is extended to more general deterministic chaotic processes as generated by coupled map lattices. In particular, so-called chaotic strings are introduced as a suitable smallscale dynamics of vacuum fluctuations. This more general approach to second quantization reduces to the ordinary stochastic quantization scheme on large scales, but it also opens u p interesting new perspectives: chaoticsi rings appear to minimize their vacuum energy for the observed numerical values of the free standard model parameters.

ISBN 981-02-4798-2

www. worldscientific. com 4853 he