Response of a canonical ensemble of quantum oscillators to a random metric

We calculate the susceptibility of a canonical ensemble of quantum oscillators to the singular random metric. If the covariance of the metric is $\vert {\bf x}-{\bf x}^{\prime}\vert^{-4\alpha}$ $0<\alpha<\frac{1}{2}$ then the expansion of the partition function in powers of the temperature involves non-integer indices.


Introduction
It is known that the dynamics in an irregular domain can be chaotic and conversely the chaotic motion can have a fractal attractor [1] [2]. Diffusion in fractal domains exhibits their fractal dimension which in general is not a natural number [3] [4][5] [6]. Thermodynamics of a canonical ensemble of particles in an irregular domain depends on the spectral dimension of the domain [7]. In such a case thermodynamic properties (e.g. critical indices) may be functions of a non-integer dimension. For some time fractal geometry has been associated with quantum gravity [5][8] [9].Quantum gravity can be expressed as a random geometry. It is believed that quantum gravity leads to a fractal geometry by a "foamy" behaviour of the metric at short distances [10]. Such irregular shapes of random figures have been at the basis of the fractal geometry [6].It has been suggested that irregular metric at the Planck scale can modify the short distance behaviour of quantum fields at short distances [5][10] [11].
In this paper we study quantum oscillators in a random singular metric ( or random position dependent mass). We define the susceptibility to the metric which has an expansion in powers of the inverse temperature β if the metric is a regular random field. We show that if the random metric is singular then the susceptibility has an expansion in non-integer powers of β. It is known that an ensemble of oscillators can serve as an approximation to field theory.
The quantum statistical mechanics of oscillators will resemble the quantum field theory at finite temperature.
We consider a Hamiltonian H γ perturbed around the free theory(harmonic oscillators) The statistical expectation value of an observable A at the temperature where the partition function For a small γ we have the expansion (till the first order in γ) In terms of the partition function we can define other thermodynamic functions as ,e.g.,the internal energy U The susceptibility to H 1 can be defined as We can use the expansion in eq.(5) to see that for a small β For H 0 of the harmonic oscillator with the frequency ω ( the ground state energy subtracted) we have and for a small β (high temperature).
If the oscillators are the modes of the electromagnetic field in a cavity then ω = |k|c where c is the velocity of light and k is the wave vector in the cavity. In such a case in eqs.(7)-(8) we have a sum over modes. The modified thermodynamics [7] comes from the modified distribution of modes in cavities with fractal geometry.
We consider Hamiltonians H such that its similarity transformation gives a generator of a diffusion process (exp(−βĤ) is a Markov semigroup). According to our assumption where ξ are the coordinates of the oscillators, ξ β (ξ) is a Markov process starting from ξ and the expectation value E[...] is over the paths of the process. In our models we consider ξ = (x, X) ∈ R n+d and assume that H 1 = V (P ) (a function of momentum). Then, according to eq.(6) So that

Statistical mechanics of oscillators
Let us consider in R n+d the coordinates ξ A and a diffusion operator of the form (sum over repeated indices) By means of we obtain the Hamiltonian H of eq.(9) We divide the coordinates ξ = (x, X) into two classes x and X where x ∈ R n and X ∈ R d . In order to simplify the model we assume that only the X coordinates are coupled to the random metric (or a random mass). So, (g AB ) = (1, g µν ) and in g µν (x) the dependence on X is negligible (the coordinates x j of x have Latin indices j = 1, ..., n and the coordinates of X the Greek indices µ = n + 1, .., n + d). We can imagine a random mass distribution (producing the metric) which depends only on some coordinates. To make the model simple we assume ω j = ν and ω µ = ω. If ω A are the modes of a massless field in a cavity which is a rectangular box of sides L A then ω A = c 2πnA LA where n A are integers. We could arrange the model so that ω j are small and the non-linear terms in eq. (14) with ω j = ν are negligible. Then,Ĥ is of the form where the similarity transformation (9) gives exp(−tĤ) can be expressed (according to eq.(10)) by the solution of the stochastic equations [12] where we expressed the metric g by vierbeins (tetrads) e g µν = e µ a e ν a .
is the Brownian motion on R n+d ,i.e., the Gaussian process with mean zero and the covariance E[b j t b l s ] = min(t, s)δ jl (and similarly for B t ). In order to take the expectation value over the metric we need an explicit solution of eq. (20). For x t we require only some estimates on the behaviour in t . However, for a simplicity of the arguments we neglect the non-linear term in eq.(19) (we assume that e µ a = δ µ a + κǫ µ a , where κ is a small parameter, then ∇g ≃ κ). After a negligence of the non-linear term the solution of eq. (19) with the initial condition x is The solution of eq.(20) reads The kernel K of exp(−βĤ) can be expressed by means of the Fourier transform 3 Random metric We assume that e µ a are Gaussian variables with the mean δ µ a e µ a = δ µ a + κǫ µ a where We calculate the Gaussian integral in eqs. (13) and (24)(with δ µν ac = δ µν δ ac chosen for simplicity) In eq.(27) we used the formula for Gaussian expectation value of ǫ with This expression can be written in a different way using the Ito calculus [12][14] where we have (here P 2 = P µ P µ ) Next, we consider a singular covariance G. For this purpose at the beginning we treat ǫ µ a as a regularized random field. Then, we remove the regularization. In order to make Hψ a well-defined random field we need the normal ordering of H with : g µν :=: e µ a (x)e ν a (x) := e µ a (x)e ν a (x) − κ 2 < ǫ µ a (x)ǫ ν a (x) > . The normal ordering in the exponential of eq.(27) removes the second term dκ 2 G(0)P 2 t on the rhs of eq.(30) whereas the first term there (i.e., β 0 F s dF s ) becomes a time-ordered integral (renormalization of such expressions appear also in QED [15] [16]). Owing to the normal ordering in eq.(27) and because of the subtraction (32) we can define the action of exp(−tH) upon a test function ψ (decaying fast in the momentum space) so that exp(−Ht)ψ is a well-defined random field. After the averaging over the translation invariant random field e µ a (x) and the renormalization (32) we can write < exp(−β : H : )ψ > in terms of Fourier transforms as where H (0) is the Hamiltonian of uncoupled harmonic oscillators and K (0) is the well-known Mehler kernel of the harmonic oscillator. So, at κ = 0 we obtain the formula (7). We are interested in the κ-term as a perturbation resulting from an interaction with a random metric. In eq.(33) we apply the identity for B s and b s (in the sense that both sides have the same probability law) After rescaling It is easy to see that for a small t in eq.(22) (set λ = s ′ in eq.(36)) where q t ≃ a + c √ t + ... with |a| > 0 for a small t. This scaling behaviour is all what we need to assume about solutions of eq.(19) with a random metric g of eq.(26). In eq.(33) we rescale the Brownian motion B s ′ and x s ′ as in (36) (with λ = s) and subsequently B s and x s with λ = β. After such a change of variables the integral of the κ-dependent part in the formula (33) reads √ βq s and x s ′ = x + √ βq s ′ where q s = 0 and q s = 0 at β = 0. We consider the covariance (the upper bound on α > 0 will be discussed at the end of this section) Then, from eq.(38) for a small β where s ′ ≤ s ∈ [0, 1] and g ≃ A + Cβ (with A > 0) for a small β. Hence,eq.(33) is of the form where |f µν (β)P µ P ν | is bounded from below by a constant.
We can now estimate the behaviour of the partition function in the metric field (26) ( note that : g µν : − <: g µν :> has the covariance |x − x ′ | −4α if ǫ µ a has the covariance (39)). We have The Hamiltonian (16) is of the form H = H 0 + κH 1 + κ 2 H 2 .From eq.(27) we can see that the contribution to the partition function Z is of order κ 2 . We consider H γ = H + γV (P ) then in the approximation (6) The anomalous (fractional) dependence β 1−α in eq.(41) of the partition function is a characteristic of the coupling to a singular metric field. We can calculate the κ 2 -derivative of the susceptibility (6) to the metric for small β (high temperature) and (45) For a small κ and small β the partition function has the expansion in non-integer powers of β with certain constants χ 1 and χ 2 .
We still have to estimate the integrals in eqs.(43)-(46). The integral (43) is expressed by kernels in eq. (13). The expectation value in eq.(13) after the renormalization (32) involves the time-ordered stochastic integral in eq.(33) which fails to be positive definite. Hence, if the expectation value in eq.(43) is to be finite V (P ) must decrease faster than exp(−RP 2 ) for any R (the derivatives in eqs.(44)-(46) impose milder requirements on the decay of V (P ) for a large P ). There is still the problem of the convergence of the stochastic integrals in the definition of f µν P µ P ν . This stochastic integral is of the form The stochastic integrals in eq. (47) can be estimated by ordinary integrals using the formula [13](better estimates on the multiple stochastic integrals (33) and (47) can be obtained using the results of ref. [17]) with certain constants C k . The rhs of eq.(48) involves the Ornstein-Uhlenbeck process (22) [18] whose transition function is expressed by the Mehler formula. In calculations in eq.(48) for small β we can approximate the Ornstein-Uhlenbeck process x s by the Brownian motion with the transition function p(s, x) = (2πs) − n 2 exp(− |x| 2 2s ). Then, the integral in eq.(47) can be estimated by (use eq.(48) twice with k = 1 ) if 2α < 1. The expansion (46) must be terminated at the k-th order if 2kα > 1 because the rhs of the estimate (48) is infinite.

The outlook
Some approximate calculations [8][9] [11] indicate that the singularity of the quantum gravitational field at small distances can be different than the canonical one which in n dimensions is of the form |x − x ′ | −n+2 ( where n = 4 corresponds to α = 1 2 in eq.(39)). The fractal dimensionality of the physical space-time has been discussed in [19][20] on the basis of the Cosmic Microwave Background measurements. Some limits on the deviation of the observational space-time dimension from the physical four dimensions have been obtained. The effect of quantum gravity could be observed either at small distances or at high energies (which in cosmology are connected with high temperatures). The non-integer indices in the expansion of the partition function in eqs.(44)-(46) could indicate the relevance of quantum gravity for some extremal processes in astrophysics ( which possibly could be tested on the quantum level in gravitational wave interferometers [21]). The model of a random mass distribution which according to eqs. (14) and (23) is equivalent to a random diffusivity is of interest in condensed matter physics [22] [23]. An anomalous behaviour of the partition function (46) or other thermodynamic functions of complex systems (e.g. molecules or crystals) could be an indication of the random mass or random metric present in these systems.