Finite temperature nonlocal effective action

[5mm] for quantum fields in curved space

Alberta-Thy-11-98

Yu. V. Gusev and A. I. Zelnikov

Theoretical Physics Institute, University of Alberta

Edmonton, Alberta, Canada T6G 2J1

E-mails: ygusev,

Canadian Institute for Theoretical Astrophysics

P.N. Lebedev Physics Institute

Leninskii prospect 53, Moscow 117 924 Russia

Abstract

Massless and massive scalar fields
and massless spinor fields are considered at arbitrary
temperatures in four dimensional ultrastatic curved spacetime.
Scalar models under consideration can be
either conformal or nonconformal
and include selfinteraction.
The one-loop nonlocal effective action at finite temperature
and free energy for these quantum fields
are found up to the second order in background field strengths
using the covariant perturbation theory.
The resulting expressions are free of infrared divergences.
Spectral representations for nonlocal terms
of high temperature expansions are obtained.

PACS: 04.62.+v, 11.10.Lm, 11.10.Wx

## 1 Introduction

Finite temperature field theory has been developed in a series of seminal papers [1, 2, 3]. Nowadays it is an actively growing branch of theoretical physics [4]. Thermodynamical properties of thermal quantum fields in the presence of background fields are very important for a large number of applications in high energy physics, astrophysics, and cosmology. However most of these studies are devoted to the situation when background fields are constant (homogeneous) [5, 6]. This particular form of the effective action, the effective potential [7, 8], when large background fields are taken into account nonperturbatively, is useful for study of phase transitions in the early Universe or quark-gluon plasma. For a long time, the opposite situation, when background fields are small but rapidly fluctuating, lacked investigation even in zero temperature field theory. Traditional tools of quantum field theory, like the short proper time Schwinger-DeWitt expansion [9, 10, 11], are intrinsically local, hence, they miss nonlocal contributions. As a consequence of this deficiency artificial infrared divergences appear in the perturbative effective action for massless fields, and perturbation theory breaks down. Finite temperature effects also contribute to infrared divergences [4], and methods of diagram summations have been developed to improve the perturbation series [12, 13].

To deal with massless field theories properly, such as gauge field theories or quantum gravity, Vilkovisky suggested a new powerful method [14] which is known as the covariant perturbation theory [15, 16, 17, 18]. In these papers it was shown that infrared divergences are artificial and brought into existence by a mode of calculation rather than by a field theory. They disappear after summation of terms with infinite number of derivatives acting on background fields, which results in nonlocal terms entering the effective action [15]. Such a summation can only be performed in a given order in background field strengths.

Thermodynamics of an ensemble of quantum fields in equilibrium in static curved spacetimes is well defined, and most properties of the system can be derived from its free energy [19, 20, 21]. In this paper we consider ensembles of scalar and spinor fields in the presence of external ultrastatic gravitational field. The scalar models may have an arbitrary interaction potential and an arbitrary coupling to gravity. We employ the method of covariant perturbation theory to find the finite temperature effective action and the corresponding free energy of these quantum fields on highly inhomogeneous gravitational backgrounds. An example of the situation when finite temperature effects on curved background are important, and, thus, nonlocal effective action is needed, is the Hawking radiation by black holes [22, 23].

The paper is organized as follows. In the next section we describe how to obtain nonlocal free energy at finite temperature with help of the covariant perturbation theory. In section 3 we derive the free energy of interacting massless scalar fields and study its high temperature behavior. Massive scalar fields at high temperatures are treated in section 4. In section 5 we derive free energy for massless spinor fields at finite and high temperatures. Conclusion and discussion of possible applications and extensions of obtained results can be found in section 6. We place necessary complicated computations into appendices A and B.

## 2 One-loop effective action and free energy of quantum fields in ultrastatic spacetimes

Let us consider fields described by the classical action and the corresponding canonical Hamiltonian in a generic curved static spacetime. Statistical free energy of the ensemble is defined as the trace of logarithm of eigenvalues of the normal-ordered Hamiltonian. In canonical quantization scheme, ultraviolet divergencies are traditionally subtracted from by the normal ordering prescription [21, 24]. On the other hand, in the imaginary time formalism of Matsubara [1] the problem of finding free energy of the system in equilibrium reduces to the computation of the path integral of the Euclidean quantum field theory and the corresponding effective action

(1) |

The temperature enters the calculation via the condition of (anti)periodicity in the Euclidean time imposed on quantum fields with (fermi) bose statistics

(2) |

(the Botzmann’s constant everywhere).

The canonical free energy and the thermal renormalized Euclidean effective action are closely related to each other and differ [19, 24] only by terms that are independent of temperature

(3) |

where are mean fields. The effective action is usually regularized using covariant methods, e.g., zeta-function [25, 26], dimensional [10], etc., while the canonical free energy is regularized via normal ordering of operators. The difference is related to different ways of taking into account vacuum energy contributions in covariant and canonical regularization schemes. The covariant approach is more appropriate to our problems since it is consistent with calculations of the stress tensor and vacuum polarization effects in external fields. In any case, it is easy to compute which is temperature independent and local [24]. Henceforth, we restrict our consideration to calculation of the one-loop Euclidean effective action and the corresponding covariant Euclidean free energy

(4) |

We calculate the free energy of quantum fields on static background fields which include mean field and static gravitational field. The Tolman temperature of such a field system in equilibrium is not constant throughout the static space. It is more convenient to perform calculations of temperature effects in the Euclidean ultrastatic (optical) spacetimes,

(5) |

where local temperature is constant throughout the space. Ultrastatic and static spacetimes are related to each other by a conformal transformation of the metric. Conformal properties of the effective action have been studied in detail [27], and applied to free energy calculations by Dowker and Schofield [28, 29]. Using scaling properties of the finite temperature zeta functions it was shown that the difference of free energies in two conformally related spaces does not depend on temperature. Then, all temperature dependent terms can be found from the free energy in an ultrastatic space, where solution of the problem simplifies significantly. This difference can be found by integrating the conformal anomaly, but the method of Ref. [28] works for generic nonconformal operators as well.

The brief outline of our research program is to calculate free energy in an ultrastatic space, and then using a relation between free energies in static and ultrastatic spaces to express the final result in terms of quantities defined in a physical (static) spacetime. In this paper we implement the first and most complicated step of obtaining and on the ultrastatic metric (5).

Let us consider quantum -component scalar field , , which satisfies the equation,

(6) |

Our notations correspond to those of Refs. [10, 16]: the Laplacian is constructed of covariant derivatives which are characterized by the commutator curvature

(7) |

This quantity is, of course, zero for scalar fields, but we will need it in section 5 where spinor fields are considered. The potential may depend on the metric and mean field , which is a part of classical background. Thus, this class of models includes self-interacting fields. The vanishing potential corresponds to the case of free conformal scalar fields, and - to the minimally coupled free scalar fields. The overhat symbol indicates the matrix structures, , and term in (6) is explicitly singled out for convenience. The three field strengths will be also referred to as curvatures. This massless field theory will be generalized to the case of massive fields in section 4.

The one-loop Euclidean effective actions is defined in terms of the functional trace of the heat kernel,

(8) |

where the heat kernel is the periodic in Euclidean time solution of the problem

(9) |

(10) |

The functional trace is understood as

(11) |

with the standing for the matrix trace, e.g., , .

The thermal (periodic in the Euclidean time) heat kernel can be expressed as an infinite sum of zero temperature (vacuum) heat kernels [5, 30]

(12) |

This image sum is equivalent to summation over Matsubara frequencies in a momentum space representation in thermal field theory. The image sum in the context of Casimir energy calculations was introduced in Ref. [31].

Temperature effects are inherently connected to the imaginary time. It is convenient to factorize the heat kernel into temporal and spatial parts,

(13) |

which is possible to do in ultrastatic spacetimes. Then, the trace of the heat kernel takes a form [19],

(14) |

when expressed in terms of the Jacobi theta function [32], which is defined in a usual way,

(15) |

The free energy of quantum fields in static spacetime is defined via the finite temperature Euclidean effective action and can be written in the form,

(16) |

The vacuum mode in the infinite sum (14)-(15) corresponds to the zero temperature effective action which suffers ultraviolet divergencies [9, 10, 26]. Fortunately, this is the only divergent term of the sum [19], so it is convenient to treat it separately. We subtract the zero temperature () free energy from and renormalize it with the use of the zeta function regularization [25, 26, 33],

(17) |

where is a mass-like regularization parameter and is the gamma function. will be combined with terms at the end of our derivations. Therefore, we compute

(18) |

The heat kernel is defined as a solution of Eq. (9) with the three dimensional operator,

(19) |

In this case the three dimensional Laplacian , and potential and the curvature are defined on a three dimensional hypersurface of the ultrastatic spacetime.

Many methods have been developed for calculation of the trace of the heat kernel [9, 34]. Most of them (see reviews [11, 10]) reduce to various representations of its small expansion,

(20) |

However, as soon as the inverse temperature is finite, the behavior of the heat kernel at large values of proper time becomes very important [19]. Therefore, expansion (20) is not suitable for our task of finding the free energy at finite temperature. Besides, Schwinger-DeWitt coefficients are local functions of background fields, henceforth, nonlocal free energy cannot be derived using (20). To solve the problem of obtaining nonlocal free energy at finite temperature we have to resort to the covariant perturbation theory [15, 16, 17, 18]. There is no need to repeat derivations of the covariant perturbation theory here because an expression for is already known in arbitrary -dimensions [16, 18]. In this paper we will take it up to terms quadratic in curvatures,

(21) | |||||

Analytic functions (form factors) have the dimensionless argument . (The appearance of nonlocal form factors in the momentum space representation of the effective action originates in the classical paper of Schwinger [35]). The form factors act on tensor invariants constructed of the set of field strengths characterizing background. The collective notation will be used for these curvatures. First two terms of the sum (21) are purely local and coincide with first two coefficients of the short proper time expansion (20). Formally, the expansion (21) is valid only in an asymptotically flat Euclidean spacetime with the topology . All background curvatures are supposed to vanish at spacetime infinity [16]. Since we use a perturbation theory, all calculations are carried out with accuracy , i.e., up to terms of -th and higher power in the curvatures . The very structure of this curvature expansion restricts its validity to background fields satisfying the relation,

(22) |

Physically it means that gravitational fields are small in magnitude but quickly oscillate.

All form factors in (21) can be expressed in terms of one basic form factor

(23) |

Their explicit form reads [16]

(24) | |||||

(25) | |||||

(26) | |||||

(27) | |||||

(28) |

Even though, in the following consideration general covariance is broken because of the presence of temperature, we will refer to this curvature expansion as to the covariant perturbation theory. In spatial dimensions the covariance remains explicit.

A few words about validity of this approximation are in order. Since we consider quantum fields at some fixed temperature, one can say that the field system in question represents a canonical ensemble. To define a canonical ensemble rigorously we have to assume that the fields are in some cavity of a finite volume, as it is usually assumed in the presence of a black hole [21]. This assumption should be reconciled, however, with our method of computation described above, which in the present form works only for asymptotically flat spacetimes and requires vanishing background fields at spacetime infinity. It is important to note that background field strengths, sources of vacuum polarization, have a compact support on a manifold, thus, providing an effective volume cut-off. In regard to gravitational field this property is due to the presence of the Ricci tensor rather than the Riemann tensor [16].

## 3 Free energy of massless scalar fields

Let us now compute free energy (18) of massless scalar fields at finite temperature. This case was briefly reported in our letter [36]. After introducing a new variable , first two terms of the trace of the heat kernel (21) take the form of the integral,

(29) |

where is the Riemann zeta function, is the gamma function. When is taking values 2 and 1, expression (29) gives for the zeroth and first curvature orders coefficients and correspondingly. These local contributions to the free energy are well known [19, 37] and coincide with the first two terms of high temperature expansion. Since all information about temperature is separated from tensor invariants, we can write down an anticipated form of free energy up to second order in the field strengths,

(30) | |||||

Then, the problem with the second curvature order is reduced now to calculation of the thermal form factors,

(31) |

where are given by (24)–(27). We show how to compute (31) when is the basic form factor (23). After substituting (23) into (31) and writing down the theta function (15) explicitly we get,

(32) |

Integration over produces the modified Bessel function of the second kind

(33) |

where . Change of variables, , allows us to express (33) in terms of the exponential integrals,

(34) |

Now we can use for the right hand side of (34) its standard form in terms of elementary functions [32] and obtain

(35) |

The sum over can be evaluated [32],

(36) |

and the resulting expression reads,

(37) |

As can be seen from (24)–(27), there are two other types of basic thermal form factors (with one and with two subtractions). Their derivations can be found in appendix A. Applying results (37), (105), and (114) to the table of form factors we obtain for all thermal form factors the following expression,

(38) |

and are simple combinations of elementary functions

(39) | |||||

(40) | |||||

(41) | |||||

(42) |

The final result for renormalized free energy at finite temperature is presented by a sum of Eqs. (30)–(41) and renormalized free energy at zero temperature . After the zeta regularization (17), the latter one takes the form,

(43) | |||||

where zero temperature form factors , , are

(44) | |||

(45) | |||

(46) | |||

(47) |

This expression differs from the one obtained using dimensional regularization only by unessential additive constants [16].

Formulae (30)–(47), we have obtained, are valid at arbitrary finite temperature. Now we would like to study asymptotic behavior of the free energy (43) in high temperature regime, the most interesting and the best studied limit. In the framework of perturbation theory, the problem boils down to finding asymptotic of thermal form factors (38). We have to be careful while dealing with mutually compensating singularities. After relatively straightforward calculations the outcome for (37) is,

(48) | |||||

where is Euler’s constant and is the Riemann zeta function.

Now, expressions for the vacuum free energy (43) and the high temperature expansion of (30) match, and can be combined into a single formula. The resulting expansion of the renormalized one loop free energy takes a form,

(49) | |||||

All local terms of this result perfectly reproduce those of Refs. [19, 28]. The combination of quadratic in curvatures terms at the logarithm is just the trace of the second Schwinger-DeWitt coefficient , taken with Riemann curvature expressed via Ricci one [9, 10, 16]. Higher powers of in Eq. (49) are also quadratic in curvatures parts of and Schwinger-DeWitt coefficients [34, 38].

We obtained the explicit form of all nonlocal terms of the second order in curvatures. They are proportional to , and were known to exist [28]. The general structure of nonlocal terms is , and, therefore, techniques based on local (small ) expansions could not generate them. Terms of higher orders in curvatures [18, 38] will also give nonlocal contribution linear in temperature.

The meaning of nonlocal structures can be understood from spectral representations in terms of massive Green functions [39, 17, 18]. For this particular form we have the following spectral formula,

(50) |

A remarkable property of the expression (49) is that it contains the only kind of nonlocality, (50). All logarithmic nonlocalities , that are present in and , have mutually canceled, leaving logarithm temperature dependance in the form of , This local combination is well known in both flat [5, 40] and curved [19] space thermal field theory. The disappearance is still being analyzed in a different physical language and in a different setting [41].

Of course, we are not completely satisfied with the integral representation for the free energy at finite temperature (30). Although, it admits a closed form, we would prefer to see expressed entirely in terms of analytical and special functions. Indeed, it is possible to obtain such a form after applying the Poisson resummation [32],

(51) |

Then, the following identity holds,

(52) |

We compute now the basic thermal form factor, (31) with (23), using this identity and separating term out of the sum,

(53) |

The mode of the sum gives precisely the leading infinite temperature contribution, while the rest can be calculated by employing the following sum,

(54) |

Adding up the regularized zero temperature form factor,

(55) |

we obtain an expression which is valid at any temperature,

(56) |

Besides an obvious advantage of Eq. (56), namely, that it is the formula in terms of usual elementary and special functions, the leading infinite temperature contributions are present here explicitly. Taking and limits, one can readily find zero temperature (55) and high temperature (48) asymptotics of this basic thermal form factor. In fact, one can see that the logarithm of the gamma functions’ ratio in the main result (56) is a sum of all positive powers of in the high temperature limit (48). Hence, it gives a partial (in the given curvature order) summation formula for the series [19]. Eventually, one has to transform (56) into a spectral form, the procedure we can complete again only at high temperatures, (50). This is the reason why we refrain from deriving the total free energy in this new representation.

## 4 Free energy of massive scalar fields

The use of the curvature expansion is crucial for derivation of the massless field free energy, because it allows one to avoid artificial infrared divergences. Two other advantages of perturbation theory, namely, that free energy can be found at arbitrary finite temperature and important nonlocal contributions can be obtained, work for a thermodynamic system of massive fields as well. Besides, this is the most studied field model, so let us investigate an ensemble of multi-component scalar massive fields satisfying equation,

(57) |

Because the mass term can be factorized out of the heat kernel, one can still use massless heat kernel (21) to derive the free energy,

(58) |

As usual, we subtract mode from the image sum, (18). Let us first treat local terms. The result in terms of the modified Bessel functions reads,

(59) | |||||

So far this expression is valid at any nonzero temperature. However, we are able to proceed and obtain explicit formulae only in high temperature limit. Simple expansions of the Bessel functions at with the subsequent -sum evaluation produces known local contributions [28, 29]. The total result for free energy of massive fields at high temperature looks like:

(60) | |||||

The computational procedure for second order terms is performed after Poisson resummation (51). Applying Eq. (52) to the basic form factor of nonlocal free energy for massive fields ,

(61) |

(vacuum contribution subtracted in (61) is dealt with at the end of the present section), and using the integral

(62) |

we get

(63) | |||||

This equation is valid at arbitrary finite temperature, therefore, free energy of massive fields is nonlocal at any temperature. The first term of Eq. (63) came from mode of the sum, and it is nothing but the leading term of high temperature expansion, . The difference of two divergent terms in the square brackets is finite, however, we are unable to give the result in a closed form. Thus, we restrict consideration to leading terms of high temperature expansion and understand the basic thermal form factor as,

(64) |

The main nonlocality is contained in the leading term (64). Subleading terms combined with vacuum contributions are not important at high temperatures. The full table of form factors , , in terms of reads

(65) | |||||

(66) | |||||

(67) | |||||

(68) |

For practical purposes of physical applications we need to know spectral form representations for (65)–(68). A spectral form for the basic form factor (64) is obvious,

(69) |

Its massless limit immediately gives (50). Spectral forms for form factors with subtractions are obtained similarly (see appendix B). Then, all form factors (65)–(68) admit the form,

(70) |

where mass spectral weights are given in the table,

(71) | |||||

(72) | |||||

(73) | |||||

(74) |

Now we need to complete our derivation with the regularized free energy at zero temperature . Nonlocal effective action for massive fields in an arbitrary spacetime dimension has been calculated first by Avramidi [42]. His approach is a direct summation of derivatives in a massive field theory, but we can make use of the massless heat kernel (21) obtained with the covariant perturbation theory and arrive at the same result. We compute zeta function regularized effective action according to the equation,

(75) |

Then, we get the following result for zero temperature free energy (the specific form of the effective action in Ref. [42] in four dimensions),

(76) | |||||

where form factors are given in terms of dimensionless argument by the following expressions,

(77) | |||||

(78) | |||||

(79) | |||||

(80) |

This effective action may look more similar to Eq. (43) if the inverse hyperbolic tangents in functions are expressed in terms of logarithms. We have to remark here that form factor is different from the others. Similarly to that of massless fields it does not depend on the regularization parameter . However, Eq. (79) is nonlocal in contrast to local (46). Of course, in the zero mass limit Eqs. (76)–(80) turn to Eqs. (43)–(47).

## 5 Free energy of massless spinor fields

In this section we consider the massless Dirac spinors in a Euclidean ultrastatic spacetime at finite temperature. The massless covariant Dirac equation is taken as

(82) |

where the standard notation is used (see [21] for general definitions). The method of calculation of the effective action for spin- fields, , is similar to the one for spin- fields. The main difference is that fermions are antiperiodic in the Euclidean time and, therefore, they satisfy boundary conditions (2) with the minus sign. The local form of the one-loop effective action was studied first in Ref. [9]. It is defined in terms of the heat kernel (or zeta function) of operator (82), however, following DeWitt’s idea we consider the squared operator , thus,

(83) |

with the heat kernel corresponding to the squared Dirac operator. It can be shown [9] that the heat kernel (Green function) of the operator is equivalent to the spinor heat kernel which is a solution of the equation,

(84) |