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Path-Integral Measure of Linearized Gravity in Curved Spacetime

The path-integral measure of linearized gravity around a saddle-point background with the cosmological term is considered in order to study the conformal rotation prescription proposed by Gibbons, Hawking and Perry. It is also argued that the most generally used measure, i.e., the covariant path-integral measure, does not give us a one-loop partition function which the only physical variables contribute and that its path integral fails to keep the cancellation of contributions between the Faddeev-Popov ghosts and the unphysical variables of the linearized gravitational field, although it has a coordinate invariant measure. In de~Sitter spacetime, it is shown that the uncancellation factor can be understood as a nontrivial (anomalous) Jacobian factor under the transformation of the path-integral measure from covariant one to canonical one.


Introduction
The truth that Euclidean path integrals are directly combined with classical actions has made them most popular as attractive and powerful tools for investigation of field theories in curved spacetime and quantum gravity [1][2][3][4][5]. They enable us to have suggestive discussions with respect to gauge symmetries of systems if the classical actions have them, and also manifest coordinate invariance which matches the spirit of general relativity. Under such circumstances, the issue of derivation of the Euclidean gravitational path-integral formula from the canonical quantization may be very important to obtain theoretical meanings of the conformal rotation prescription proposed by Gibbons, Hawking and Perry [6]. Furthermore, if it is done, objects given by the path integral of gravity may become more meaningful, for example, with respect to whether the path integral has some information on the ground state similarly to the case of any field theory with a positive-definite hamiltonian in the flat spacetime [7].
However, it is very difficult in the Einstein gravity to derive the path integral or to give the theoretical meanings to the conformal rotation. Because the Einstein theory is a reparametrization one, in which the physical time is a dynamical variable. Thus it is necessary to identify the physical time which plays a privileged role in the canonical formalism, when quantizing it according to that formalism [8][9][10].
With the identification it might be possible to construct a path-integral formula from the canonical approach. Unfortunately, the choice of the physical time is a difficult problem and even if it is done it is not so trivial how the choice is reflected in the path-integral formula, particularly in the covariant expression. Hence the choice of the physical time may be an important issue in quantum gravity.
Also in linearized gravity, we have to take the contour of integration for a variable associated with the conformal factor to be parallel to the imaginary axis in order to make the Euclidean path integral converge, since the action of linearized gravity is unbounded from below with respect to the variable. Here the time is not a dynamical variable, so the theoretical foundation of the conformal rotation in linearized gravity may be different from that in the case of gravity, but can be discussed from the standpoint as a gauge theory: indeed, as studied by many authors in the flat spacetime [11][12][13], the variable needing the conformal rotation is an unphysical one like A 0 in the Abelian gauge theory, and it is found that one has to rotate it to the imaginary axis, in addition to Wick rotating, in order to obtain the well-defined covariant formulation of the Euclidean path integral.
Furthermore, in linearized gravity around a nontrivial background such as de Sitter spacetime, it was shown by Griffin and Kosower [14] that the Euclidean action of the physical part is positive definite, which means that the conformal rotation is not necessary when taking a special gauge fixing condition so that redundant variables all may vanish like the Coulomb gauge in the Abelian gauge theory, although on this background the conformal rotation in covariant gauges is more complicated as compared with that on the flat spacetime background. They also pointed out in this de Sitter spacetime that the covariant path integral does not give the one-loop correction which is contributed by the only physical variables, in other words, that the covariant path integral is different from the canonical path integral. The same thing also happens in the Abelian gauge theory in curved spacetime [15][16][17]. This issue may be noticeable also from the point of view of reexamination of the Faddeev-Popov conjecture [18,19] that path integrals in various gauge choices are all same, because now the difference of the path-integral measures may emerge as differences of path integrals between various gauges [17].
The purpose of this paper is to study the relation between the covariant path integral and the canonical approach to linearized gravity in curved spacetime and the uncancellation of the one-loop contributions given from the unphysical variables and the Faddeev-Popov ghosts. It is shown that the uncancellation factor can be also understood as a nontrivial (anomalous) Jacobian factor under the transformation of the path-integral measure from covariant one to canonical one, similarly to the case of the Abelian gauge theory. This paper is organized as follows: in section 2 it is pointed out that the covariant path integral of linearized gravity around a saddle-point background with the cosmological term does not give us the one-loop partition function which the only physical modes contribute and also that its path integral fails to keep the cancellation of contributions from the Faddeev-Popov ghosts and the unphysical modes of the linearized gravitational field. In section 3 we study how the modes needing the conformal rotation prescription depend on the gauge fixing condition with gauge parameters α and η. In section 4 it is shown that the uncancellation factor can be understood as a nontrivial (anomalous) Jacobian factor under the transformation of the path-integral measure from canonical one to covariant one. Section 5 is devoted to conclusion, and eigenfunctions on S D are summarized in Appendix A.

BRST Path Integral of Linearized Gravity
In this section, we formally calculate the one-loop correction by the use of the BRST path-integral method with the covariant measure, and a probability is then pointed out that the contribution of the Faddeev-Popov ghosts might not generally cancel out with one from the unphysical degrees of freedom of linearized gravitational field.
Our starting Euclidean path integral of linearized gravity in a D-dimensional curved spacetime is where I is the linearized gravity action in Euclidean spacetime [4]; which is obtained by expanding the Einstein-Hilbert action with the cosmological term around a classical background g µν which satisfies withḡ µν ≡ g µν + κh µν , and I GF is a gauge fixing term with gauge parameters α and η; and then the Faddeev-Popov determinate, ∆ FP , becomes where v is any vector field. Now, decompose h µν as where h and φ are scalar fields, ξ d µ is a divergenceless vector field, and h trd µν is a traceless and divergenceless tensor field: Then, the actions I and I GF are separated into four independent parts of their variables: and where C µρνσ is the Weyl tensor and Obviously, I ψ may be negative definite with respect to higher frequency modes than the mass 2DΛ owing to opposite sign of the kinetic term to ordinary scalar fields. Thus, in addition to Wick rotating, one has to always rotate the contour of integration over these modes of ψ to the imaginary axis so as to make integrations converge. But, it is not surprising because ψ is an unphysical degree of freedom of linearized gravitational field, i.e., we can make the variable vanish as a consistency condition for a suitable gauge fixing like the Coulomb gauge whenever that is wanted, although it is a gauge invariant variable. Hence, if we start with the operator formalism, the theoretical meaning of this rotation could be understood similarly to the case in the flat spacetime [11][12][13]. On the other hand, only h trd µν has the physical degrees of freedom, which means that the path integral with respect to the physical mode is well-defined within the positivity of I trd , which may be expected in any curved spacetime as well as the flat and de Sitter spacetimes.
Next, in order to carry on calculation of (2.1) by the use of the covariant measure [1], let us define the measure using the Polyakov measure for convenience sake as where Here h trd µν includes the physical and unphysical modes, hence, only if the unphysical contributions in Z trd cancel out with the Faddeev-Popov ghost contribution, i.e.,

16)
Z gives just the physical contributions. But it does not generally hold in curved spacetimes. Indeed, in de Sitter spacetime it happens, as suggested by Griffin and Kosower [14]. Such an uncancellation arises also in the Abelian gauge theory and it is shown that the uncancellation can be regarded as the difference of measure between the covariant and canonical path integrals [16,17]. We will study in section 4 the relation between the uncancellation factor of unphysical contributions in the covariant one-loop correction and an anomalous Jacobian factor arising under the change of variables in the Polyakov measure from covariant one to canonical one.

Conformal Rotation
In order to find out the modes needing the conformal rotation, let us study the following eigenvalue equation; where λ is an eigenvalue and may be regarded as a constant. Then we might find that the modes with a negative eigenvalue, whose contours of integration need to be rotated to the imaginary axis in order to make integrations over them converge, may change on values of positive α and real η, but that the difference of number between the positive and negative modes is, of course, always unchanged, which is obvious because of a simple reason that the negative and positive parts of I ψ in (2.9) does not change and the gauge fixing term in (2.5) ((2.10)) is always positive independently of the gauge fixing parameters.
Eq. (3.1) is separated into h trd µν , ξ d µ and scalar (ψ and φ) parts by inserting (2.7): 2) says that if all eigenvalues of the operator acting on traceless and divergenceless tensor fields, i.e., are positive, we have no trouble when integrating out h trd µν , that might be plausible as mentioned in section 3.
From (3.3), eigenvalues of a tensor part defined by ξ d µ are the same as eigenvalues of the operator acting on divergence vector fields, i.e., Here all eigenvalues of this operator are expected to be positive because of the positivity of the gauge fixing action I GF , so the Gaussian integrations with respect to this part also converge.
To discuss about other remaining parts, let us take, for simplicity, the trace part and the traceless part of h µν then are decoupled: Eq (3.4) is reduced to two eigenvalue equations (3.8) Thus eigenvalues of these tensor parts are the same as these of the operators acting on scalar fields, such that The η dependence of these operators is very significant, since the negative modes needing the conformal rotation depend on its value and particularly for η ≥ 1 the variable which must be rotated to the imaginary axis is not the trace h but φ.
It may be also easily found from Eq.

Anomalous Jacobian Factor
In this section, we take the Euclidean de Sitter background and discuss a relation between the uncanceled factor in the one-loop contributions given by the covariant path integral of linearized gravity and the nontrivial Jacobian factor under the transformation from the covariant measure to the canonical measure, choosing coordinates so that the metric has the form .
On this manifold, we have C µνρσ = 0, so the uncancellation factor we have to study becomes, from (2.14) together with formulae in Appendix A (or ref. [14]), as where v T is any divergenceless vector field with no ℓ = 1 modes of the covariant Laplacian on the unit S D−1 .

Covariant and Canonical Path-Integral Measures
In this subsection, we construct the Polyakov measure for tensor fields on S D by using two decompositions, i.e., the covariant and canonical decompositions.
The covariant decomposition in Eq. (2.7) may be rewritten as where components h tr µν and h trφ µν are defined by which can be separated, according to whether ξ d µ is a longitudinal vector ξ L µ or a transverse vector ξ T µ , into h trξ L µν and h trξ T µν : where each part, in the coordinate system (4.1), is defined by and (4.14) Then Eq. (2.13) becomes Therefore, the Gaussian integral, i.e., +∞ −∞ dxe −λx 2 = π/λ, says that the Polyakov measure (2.12) is represented in terms ofh,φ,ξ,h tdL DD ,ξ T µ ,h trdT µD and h tT µν as Then Eq. (2.13) is split into each part of them: Therefore, the Gaussian integral says that the Polyakov measure (2.12) in the canonical decomposition becomes τ ∆τ (CD + 2)a D−1 (τ ) 8πhD Eqs. (4.17) and (4.22) mean that the Jacobian factor under the transformation of measure from the covariant path integral to the canonical one would be naively 1, since both measures are defined with the Gaussian integral whose integration value is 1. However, as discussed in the following subsections, the Jacobian factor is unfortunately not 1 and takes an anomalous value. Such a situation happens also in the case of the gauge theory in Euclidean Robertson-Walker spacetimes with K = +1 [16,17]. The Jacobian factor is discussed below, separating each one of two decompositions into three parts, i.e., a traceless and transverse tensor part, a tensor part defined by transverse vector fieldsξ T i ,h trdT iD , h T iD and w T i , and that defined by scalar fieldsh,φ,ξ,h trdL DD , h, h tr DD , Φ and ψ.
As for the physical variable, i.e., the traceless and transverse tensor h trT µν , it is easily found that we have no anomalous thing under the transformation between the covariant and canonical measures: where it is noted that the time τ product in the l.h.s. is changed, in the r.h.s.,

Anomalous Jacobian Factor in Transverse Vector Part
The relations among the transverse vector fieldsξ T i ,h trdT iD , h T iD and w T i used in the covariant and canonical decompositions, i.e., (4.3) and (4.18), are where the relation between the former two equations and the later two equations is that of the inverse transformation. Now let us discuss an anomalous Jacobian factor with respect to these variables, separating them into ℓ = 1 modes and ℓ = 1 modes of .
where the indices ( 1) means that the variables are of ℓ = 1 modes and these equations have the inverse relationship to each other. From (4.29), a nontrivial Jacobian factor is obtained: or if Eq. (4.30) is used to change the variables, another Jacobian factor is where v T( 1) is any transverse vector field with ℓ = 1 modes of on the unit S D−1 .
With respect to ℓ = 1 part, if Eqs.
where v T is any transverse vector field without ℓ = 1 modes. Therefore, Eqs. (4.33) and (4.34) give τ ∆τ a D−3 (τ ) 2πh From Eqs. (4.35) and (4.36), therefore, the factor might be naively thought to take a trivial value 1 and both measures for tensor fields defined in terms of transverse vector fields without ℓ = 1 modes of on the unit S D−1 would be concluded to be equal to each other. However, the factor is obviously the same with the uncancellation factor (4.2) excluding the ℓ = 1 mode part, and in order to actually calculate the factor (4.37) one need to regularize it because the arguments of the determinates are infinite matrices. Unfortunately, some regularization fail to make it 1 [14].

Anomalous Jacobian Factor of Scalar Part
The relations among scalar fieldsh,φ,ξ,h trdL DD , h, h tr DD , Φ and ψ are When turning our discussion to the change of variables ofh trdL DD ,ξ,φ and h tr DD , Φ, ψ, it is important to note that h tr DD andφ have both modes of ℓ = 0 and 1, and the variables Φ,ξ have only ℓ = 1 modes among them, on the other hand, ψ andh trdL DD have no one of these modes. In this paper, we study the Jacobian factor with respect to only the part having no modes which are ℓ = 0 and 1 of on the unit S D−1 in order to check the relation between it and the uncancellation factor and these equations give . Then, an anomalous Jacobian factor is derived: which is equal to because the Jacobian factor can be separated into respectively independent parts of modes of . The last equation (4.49) is nothing but the longitudinal part of the uncancellation factor (4.2) excluding the ℓ = 1 modes.

Conclusion
In this paper, the conformal rotation in the Euclidean path integral of linearized gravity around a saddle-point background with the cosmological term was discussed and that it in a nontrivial curved spacetime will be more complicated than the flat spacetime because the sign of the mass term in the action I ψ (see Eq. Finally, we note that the following equations identically hold: (A.37)