One-Loop Effective Actions in Curved Spacetime

A one-loop determinant is not yet a complete statement about what is being calculated. A Euclidean determinant efficiently exposes ultraviolet counterterms; its Feynman continuation gives an in-out effective action for transition amplitudes. Causal equations for expectation values instead require a doubled, closed-time-path construction with an initial density operator. The three objects share local ultraviolet coefficients but differ in boundary conditions and physical meaning.

Required background. Gravitational Actions and Boundary Terms provides the variational role of bulk and boundary terms; Curvature supplies the invariant curvature basis. Helpful background. Flat-Spacetime Quantum Field Theory provides functional determinants and Feynman boundary conditions; Partial Differential Equations and Green Functions supplies elliptic and causal inverses; Causality and Global Structure establishes causal support and initial-value geometry; Quantum Fields in Curved Spacetime provides states and renormalized expectation values.

Quadratic fluctuations and determinant weights

Expand a Euclidean action around a background $\bar\Phi$:

$$S_E[\bar\Phi+\eta] =S_E[\bar\Phi] +\frac12(\eta,D\eta)+O(\eta^3).$$

For one real bosonic fluctuation with a positive elliptic operator,

$$\Gamma_E^{(1)}[\bar\Phi] =\frac12\operatorname{Tr}\ln\frac{D}{\mu^2}.$$

A complex Grassmann field contributes $-\operatorname{Tr}\ln D$, while a Dirac field is usually reduced to a Laplace-type square with the phase or spectral asymmetry treated separately. Gauge fields require a gauge-fixed vector operator and Faddeev–Popov ghosts; zero modes are removed from the determinant and integrated over or divided by their residual group volume. Negative modes require a contour prescription and can signal an instability rather than an ordinary real effective action.

Using the proper-time identity gives

$$\Gamma_E^{(1)} =-\frac12\int_0^\infty\frac{dt}{t} \operatorname{Tr}e^{-tD},$$

with regularization at $t=0$. Zeta regularization instead defines

$$\ln\det\frac{D}{\mu^2} =-\zeta_D'(0)-\zeta_D(0)\ln\mu^2.$$

The equivalence of these spectral descriptions, including the scale dependence and the qualifications about positive operators, is developed in Vassilevich 2003, §2.2, Eqs. (2.23)–(2.35), PDF.

Local divergences and finite nonlocal terms

If $D=D_0+m^2$, the short-proper-time expansion yields local ultraviolet poles built from $m^4a_0$, $m^2a_2$, and $a_4$ in four dimensions. The corresponding counterterms renormalize the cosmological, Newton, curvature-squared, matter, and boundary couplings allowed by the symmetries and boundary problem. On a manifold with boundary, the heat trace also contains boundary coefficients; cancelling only the bulk poles leaves the theory unrenormalized at the boundary.

The full determinant contains more. At quadratic order in curvature it can include structures such as

$$\int d^4x\sqrt{g_E}\, C_{\mu\nu\rho\sigma} F_C\!\left(-\frac{\nabla_E^2}{m^2}\right) C^{\mu\nu\rho\sigma},$$

where the form factor $F_C$ is nonpolynomial. A finite list of local heat-kernel coefficients fixes corresponding finite orders of its large-mass asymptotic expansion, not the complete function. The covariant perturbation and modified large-mass approaches are distinguished in Vassilevich 2003, §§8.1–8.2, PDF.

A useful scale check follows without evaluating the determinant:

$$\frac{\partial\Gamma_E^{(1)}}{\partial m^2} =\frac12\operatorname{Tr}D^{-1} =\frac12\int_0^\infty dt\, e^{-tm^2}K(t;D_0).$$

This identity holds with a common regulator before subtraction. For a renormalized action, differentiating mass-dependent local counterterms also contributes, so it does not remove renormalization-scheme dependence. It does expose the same heat-kernel coefficients. In flat space, a constant scalar background with positive fluctuation mass $M^2(\bar\Phi)$ gives, in the $\overline{\mathrm{MS}}$ convention for one real mode,

$$V_1=\frac{M^4}{64\pi^2} \left(\ln\frac{M^2}{\mu^2}-\frac32\right),$$

recovering the Coleman–Weinberg form. The additive polynomial and the constant inside the parentheses are scheme dependent; the logarithmic scale response is the more robust check.

Euclidean continuation and the in-out action

Under a controlled Wick rotation, $e^{iS_L}$ becomes $e^{-S_E}$ and the continued effective actions obey $i\Gamma_L=-\Gamma_E$. Curvature signs, normal vectors, epsilon tensors, and background gauge components must be translated with the declared continuation; one cannot simply replace $g_E$ by $g_L$ in an intermediate coefficient table. Vassilevich explains why the Euclidean local coefficients remain useful for Lorentzian counterterms while a Lorentzian heat kernel lacks the same damping argument in Vassilevich 2003, §2.3, PDF.

The ordinary Lorentzian Legendre transform uses vacuum-to-vacuum boundary conditions. Its two-point function is Feynman, and its stationary equation describes an in-out matrix element. If particle production occurs, the in-vacuum and out-vacuum need not coincide; the resulting action can be complex. Its imaginary part carries transition or instability information, but varying it does not generally yield a real, causal expectation-value equation for a prepared initial state.

Closed-time-path action for causal expectation values

For an initial density operator $\rho_0$, define

$$Z[J_+,J_-] =\operatorname{Tr}\!\left( U_{J_+}\rho_0 U_{J_-}^{\dagger} \right).$$

The forward field $\Phi_+$ and backward field $\Phi_-$ are independent before the physical limit, and the closed-time-path effective action has the schematic one-loop form

$$\Gamma_{\mathrm{CTP}} =S[\Phi_+]-S[\Phi_-] +\frac{i}{2}\operatorname{Tr}_{\mathrm{CTP}} \ln G^{-1}_{ab}.$$

After variation, set $\Phi_+=\Phi_-=\Phi$. The contour components combine so that the equation depends only on earlier initial data. This in-in equation generates expectation values in $\rho_0$, not an in-to-out transition amplitude. The initial state and its correlations are indispensable inputs; “retarded” is not a regulator choice.

Normalization gives $Z[J,J]=\operatorname{Tr}\rho_0=1$: forward and backward evolution cancel for equal sources. Therefore setting the two branches equal before varying would erase precisely the response being sought. The separate-branch variation and the prepared initial state are explained in Schwinger–Keldysh and In-In Dynamics.

A revealing structural argument is that the second derivative of an ordinary single-field action is symmetric under exchanging its two variations. A purely retarded kernel is not symmetric. Doubling the fields allows a causal equation in the physical limit while retaining the contour identities of a variational construction.

Comparing the three effective actions

Object	Boundary or state data	Primary output	Causality status
Euclidean determinant	Elliptic boundary conditions and Euclidean saddle	Spectral determinant, ultraviolet poles, equilibrium quantities	No real-time causal claim until continuation
In-out action	In and out vacua, Feynman prescription	Transition amplitudes and vacuum persistence	Variation is generally not a causal expectation equation
In-in action	Initial density operator and closed time path	Real-time expectation values and response	Physical-limit equation is causal when the setup is consistent

The local ultraviolet counterterms agree after consistent continuation because they are controlled by the same short-distance singularities. The finite nonlocal kernels need not agree: their imaginary parts, homogeneous solutions, and state dependence encode the different boundary conditions. Parker and Toms organize one-loop effective actions, curved-space renormalization, and the continuation conventions in Parker and Toms 2009, Chapters 3 and 6.

Diagnostic: why “make the logarithm retarded” fails

Suppose a Euclidean calculation produces the quadratic nonlocal term $\frac12\int \phi(x)F(x,y)\phi(y)$. Explain why replacing $F$ by a retarded kernel after the calculation does not define an ordinary Lorentzian action, and identify the construction that gives a causal equation.

Solution

Only the symmetric part of $F(x,y)$ contributes to a single-field quadratic action, because interchanging the two integration variables leaves $\phi(x)\phi(y)$ unchanged. A genuinely retarded kernel is not symmetric: it vanishes when $y$ is not in the causal past of $x$, whereas its transpose is advanced. Therefore a post hoc retarded replacement cannot be the Hessian of the claimed ordinary action. The closed-time-path construction supplies forward and backward fields and four contour kernels. Varying before taking the equal-field physical limit combines those kernels into a retarded response, with the initial density operator fixing the remaining state data. This is why causal expectation equations come from the in-in, not merely the in-out, effective action.

References

• Parker, Leonard E., and David J. Toms. Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge University Press, 2009. DOI.
• Vassilevich, Dmitri V. “Heat Kernel Expansion: User’s Manual.” Physics Reports 388 (2003): 279–360. DOI. Open PDF, arXiv:hep-th/0306138.