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Microcanonical quantum cosmology: gauge fixing and functional determinants on a circle

Gauge fixing procedure in the path integral for the microcanonical statistical sum in quantum cosmology is considered in a special class of gauge conditions free from residual gauge ambiguities. Relevant functional determinants, path integrals and functional delta-functions on a circle are considered.


Introduction
The goal of this short note is to revisit the one-loop approximation for the microcanonical partition function in quantum cosmology [1,2]. Application of the density matrix in spatially closed cosmology shows that its statistical sum is mainly determined by the minisuperspace sector of the theory described by the scale factor a and lapse function N of the Euclidean FRW metric ds 2 = N 2 (τ )dτ 2 + a 2 (τ )d 2 Ω (3) . (1) Its time-parametrization invariant action has the form Γ [ a, N ] = dτ NL(a,ȧ/N) Here L(a,ȧ/N) is a local Lagrangian and F (η) is a free energy of quantum matter fields as a function of the effective inverse temperature η -the period of the Euclidean time τ measured in units of the conformal time parameter. Integration over τ runs over this period and is denoted by . Remarkable property of this expression is that for any Lagrangian L(a,ȧ/N) the quadratic part of Γ [ a, N ] on the classical background has a generic form [2] Γ 2 [n, ϕ] = ε 1 2 dτ φ 2 +g g ϕ 2 − 2gφn + 2ġϕn + g 2 n 2 + g ≡ aȧ |D|, D ≡ ∂ 2 L(a,ȧ) where the perturbations of the scale factor and lapse are parameterized by new variables 1 ϕ and n, and N = 1 gauge is used for background. The coefficients of this form are built in terms of the function g = g(τ ) and the parameter C -functionals of the gravitational background characterized by a periodic scale factor a(τ ). This background forms the cosmological instanton with the topology of S 1 × S 3 . The S 1 part is associated with the Euclidean time τ which parameterizes the instanton metric (1) in the gauge N = 1, and a(τ ) is a periodically oscillating radius of the sphere S 3 . The action (3) is gauge invariant under the local coordinate transformations Therefore, the path integral for the statistical sum of the model requires Faddeev-Popov gauge fixing procedure which was performed in [2] in the "relativistic" gauge χ(n, ϕ) ≡ṅ = 0. Since this gauge leaves unfixed residual gauge transformations associated with conformal Killing vectors of the background (1), additional gauge fixing procedure was necessary, which effectively has led to linearly dependent generators treated by the Batalin-Vilkovisky technique of reducible gauge theories [6]. This makes further analysis of the theory (in particular, the convexity issue for the action of the theory at the saddle points of the path integral) very cumbersome. So here we replace the gauge fixing procedure of [2] in the gauge (??) by using the alternative gauge free from residual gauge transformations.
Our main result will be a calculation of the gauge-fixed Gaussian path integral for the one-loop prefactor P of the statistical sum which is semiclassically dominated by the tree-level contribution e Γ 0 at the saddle point at which the on-shell value of the action (2) is given by Γ 0 . This involves treatment of a special delta-function type gauge, which is nonlocal in the Euclidean time, and finding closed algorithms for functional determinants of nonlocal operators on spaces of periodic functions. So this calculation turns out to be technically illuminating and deserves presentation.

Gauge fixing
Since the system is a gauge one the correct expression for one-loop contribution according to Faddeev-Popov technique is where χ n, ϕ is some gauge condition completely fixing the gauge invariance, and Q -corresponding Faddeev-Popov determinant. The integration is performed over minisuperspace fields n(τ ), ϕ(τ ) defined on a circle 2 S 1 of length T . One of possible gauges (a family of gauges) completely fixing the gauge ambiguity is χ(n, ϕ) = βṅ − γ k ϕ (8) 1 New variables are related to perturbations δa and δN as: n = 1 a δN N − δa a , ϕ = |D| δa. 2 Functional space of (real-valued) functions on a circle S 1 of length T is equivalent to that of (real-valued) functions on R periodic with period T . so somewhere we call such functions "periodic".
where k(τ ) is some function on S 1 and β, γ are some constants. 3 In what follows for the sake of convenience we will omit denoting the measure dτ in all further integrals dτ where it can not lead to ambiguity.
Absence of residual gauge invariance in this gauge follows from a simple fact that for periodic n(τ ) (i.e. n(τ ) on S 1 ) the equation χ(n, ϕ) = 0 implies that both terms of (8) separately vanish, and thus forbid transformations (5) with a periodic f (τ ). The Faddeev-Popov operator Q for such gauge χ, acting on the gauge transformation parameter f , is defined by the equation It does not have periodic zero modes and, therefore, non-degenerate. Its determinant on the space of square integrable functions on a circle L 2 (S 1 ) This can be shown by direct spectrum analysis (see Appendix A). Calculation of the rest part of the prefactor simplifies due to the decoupling of the of functional delta-function, which is a corollary of the relation (9). Nontrivial element here is the proportionality coefficient in the right hand side and the definition of the rule how the functional delta-function δ [βṅ] acts on a test functional f [n] in the path integral Here the integration over square-integrable functions on a circle ("periodic" functions) is denoted by L 2 (S 1 ) and numerical integration over the real axis is labeled by R. Below we prove this property.
Integrating these delta-functions in (12) we reduce the functional integral over n(τ ) to a numerical integral over its homogeneous mode and restrict ϕ(τ ) to the subspace of functions transversal to k(τ ) which we denote L 2 ⊥k (S 1 ) 3 The result easily generalizes for β, γ → β(τ ), γ(τ ) with β(τ ) = 0 and γ(τ ) β(τ ) = 0. In particular this generalization allows to make Faddeev-Popov operator symmetric. But on the other hand it causes complications in calculations. In the article we preserve these constants as the trace of this possible generalization and for the reason of additional dimensionality check. where is the operator introduced in [4], and integration over repeated time arguments τ and τ ′ is assumed in the exponential of this expression (in accordance with the Einstein rule for the condensed index notation). Factors in the denominators in front of the integral originate from nontrivial arguments of delta-functions. Calculational details for (15), in particular decoupling of delta-functions and their definitions, zeta-renormalization and the origin of prefactors can be found in Appendix B. The Gaussian integration in (15) gives where G k (τ, τ ′ ) -Green's function [4] for operator F (16). Since the latter is degenerate there is an ambiguity in choosing the Green's function, so G k (τ, τ ′ ) is the one, which is orthogonal to k(τ ): The determinant in the first factor of (17), as shown in Appendix C, can be expressed as where in the last equality we substituted Det ⊥g F = △ g 2 . The latter was obtained in [4] and expresses the determinant of operator with a zero mode on S 1 in terms of its the monodromy coefficient △. The power of ε was obtained via the zeta-function renormalization.
Using the explicit representation of G k (τ, τ ′ ) obtained in [4] one can show that so the latter factor in (17) equals Assembling Eqs.(10,17,18,20) one finally gets which gives the preexponential factor P (7) to the amplitude of the Universe (6). Needless to say the result actually is independent of the gauge chosen, As a check of consistency note that in the absence of nonlocal "matter" contribution (C = 0) the contribution of the homogeneous sector is trivial.

Conclusion
Remarkably, in the model of the CFT driven cosmology with the result (21) fully confirms the answer for the statistical sum prefactor initially obtained in [2] for a particular class of instantons. It reads where △ ≡ −I/ε is the monodromy factor of the operator (16) and a numerical constant coefficient is independent of the parameters of the background instanton and, therefore, completely irrelevant.
Calculations of [2], where the functional determinants were obtained by the variational method, did not control their overall coefficients which depend on one such parameter -the period of the Euclidean time T (which is a nontrivial nonlocal functional of the instanton metric). Nontrivial factors of T contained in the quantities (11), (17) and (20), however, completely cancel out in the final answer (21) and, thus, do not affect the validity of conclusions of [2].
In contrast to the technique of [2] the present gauge fixing procedure avoids residual gauge symmetries and related complications of the quantization formalism for theories with reducible generators [6]. This opens prospects for the analysis of convexity properties of the Euclidean action of the theory and the selection criterion for physically relevant saddle points of the cosmological statistical sum.

Acknowledgements
The work of Dmitry Nesterov was partially supported by RFBR grant 12-02-31838 mol a and Andrei Barvinsky is grateful to RFBR grant 14-01-00489 a.

A Determinant of the Fadeev-Popov operator
Here we prove (11) that for nondegenerate operator Q (10) on L 2 (S 1 ) For constant β, γ one can prove that by analyzing the spectrum of the operator. For simplicity consider it is positively definite, which is for example guaranteed by βγ kg > 0.
Right eigenmodes 4 can be constructed from that of operator − d 2 dτ 2 : The structure of Q implies that right eigenfunctions should be of the form Thus the product of eigenvalues equals to the product of nonzero eigenvalues of operator −β d 2 dτ 2 and homogeneous mode eigenvalue γ kg. Denoting operator − d 2 dτ 2 on the space of functions ζ ⊟ (0) and Det ⊟ can be expressed via Riemann zeta function ζ R (s) as ζ ⊟ (0) = 2ζ R (0) = −1 and which confirms (24).

B Path integral with delta-functions
In this section we discuss more precisely the calculation of the path integral (12)
where we used decomposition of functions on S 1 into homogeneous and inhomogeneous subspaces ρ(τ ) = ρ 0 e 0 (τ ) +ρ(τ ) and e 0 (τ ) -homogeneous basis function (25) As was noted, the decoupling of delta-functions could be predicted by the fact that gauge-fixing condition βṅ + γ k ϕ = 0 implies separate vanishing ofṅ and k ϕ. The main purpose of the theorem is to fix extra factors under given normalization of path integrals and delta-functions.
Going further in calculation of (7) with the help of two following theorems one can perform integration with correspondent delta-functions in n and ϕ sectors.
Proof. Using spectral decompositionṅ(τ ) = m∈Z c mėm = m∈Z c m 2π T (−m)e −m (τ ) one comes to Product factor m∈Z =0 T 2πmβ is nothing but the inverse of (absolute value of) the operator's β∂ τ determinant on L 2 ⊥const (S 1 ) ≡ {h(τ ) : h(τ ) = 0} -the space of functions on S 1 orthogonal to constant functions. This determinant can be expressed in terms of Riemann zeta-function [7]: Also the product factor in calculations above could be expressed as zeta-renormalized inverse of square root of determinant of operator −β 2 d 2 dτ 2 (29) on the same space which leads to the same result.
Next theorem shows the result of integrating one-dimensional delta-function: Dϕ δ γ kϕ f [ϕ] = 1 Proof. Here it is convenient to use method of dealing with delta-functions in path integrals analogous to that used in Appendix in [2].
√ 2π where S[ϕ(τ ), ρ 0 ] = −iρ 0 γ k ϕ − i ϕ J + ǫ 1 2 ϕ 2 . To make integrals well defined we have introduced the regularization of the initial integrals with regularization parameter ǫ (which is auxiliary and should not be mixed with ε parameter in the main part of the article).
applying f 1 i δ δJ e i ϕ ⊥k J = f [ϕ ⊥k ] e i ϕ ⊥k J and setting J(τ ) to zero one finally comes to the statement of the theorem (35) .

C Changing the gauge in Gaussian integration
Here we justify the relation (18) using the analogy with the finite-dimensional vector case. Let A be degenerate finite dimensional symmetric operator on linear space V with one zero eigenvector |g >.
Determinant of A is zero, but one is often interested in 'regularized' determinant det * A which is defined as the product of its nonzero eigenvalues.
The latter object has the integral representation (to simplify the formulae we will assume A does not have negative eigenvalues, and consider the simplest norm in V ): where < ., . > is the Euclidean inner product in V = R n and x ′ runs over V ⊥g , which is (n−1)dimensional subspace of initial R n which is orthogonal to |g >. In the exponent |x ′ > is assumed to be immersed in V as |x ′ , 0 >. In |x >= |x ′ , z > the one-dimensional parameter z is the coordinate along |g >. Such integral representation can be interpreted as the determinant of operator over particular subspace of initial vector space. One can write the 'covariant' integral over all V introducing the delta-function where |e g >= |g> |g| . Note that δ(< x|e g >) is nothing but δ(z). Let k be some vector so that < k, g > = 0 and define the following object -the determinant of A over subspace, orthogonal to |e k >= |k> |k| : The algebraic relation between such objects is det ⊥k A = < k, g > 2 |k| 2 |g| 2 det ⊥g A .
In the article we use the generalization of the property above for the infinite dimensional vector space L 2 (S 1 ) where integral definitions of determinants are with correspondent f (τ ), and its orthogonal complement L 2 ⊥f (S 1 ) = {ϕ(τ ) ∈ L 2 (S 1 ) | ϕf = 0}. The reasoning for the path integral case is analogous.