Vacuum Polarization and Curved-Space Casimir Effects

Vacuum polarization is a local response of a specified state to geometry, topology, gauge background, or boundary data. Casimir observables are controlled differences or forces, not an absolute assignment of zero-point energy. The comparison must use the same local subtraction and finite bulk and surface couplings on both sides.

Required background. Renormalized Stress Tensor: Axioms and Curvature Ambiguities supplies the finite curvature freedom; Curved-Space Renormalization Schemes: Domains and Translation supplies scheme matching; Green Functions and Causal Propagators supplies the two-point kernels.

Helpful background. Local Covariance, Isometries, and Background Embeddings explains comparisons between backgrounds; Spectra, Resolvents, and Functional Calculus supplies the spectral representation.

Relative local observables

For two Hadamard states or boundary configurations whose local field operator agrees near $x$, define

$$\Delta W(x,x') = W_1(x,x')-W_0(x,x').$$

Their common Hadamard singularity cancels, so in a region where $\Delta W$ is smooth,

$$\Delta\langle\phi^2(x)\rangle =\lim_{x'\to x}\Delta W(x,x'),$$

and

$$\Delta\langle T_{\mu\nu}(x)\rangle =\lim_{x'\to x} \mathcal D_{\mu\nu}(x,x')\,\Delta W(x,x').$$

On this page the stress convention is

$$\delta\Gamma_{\mathrm m} = \frac12\int\mathrm d^4x\,\sqrt{-g}\, \langle T_{\mu\nu}\rangle\,\delta g^{\mu\nu}.$$

Because the same local finite curvature tensor is added to both terms, it cancels in a same-background state difference. It need not cancel when the geometries, boundary embeddings, or finite surface couplings differ. This is why a relative observable is stronger than an unqualified “renormalized vacuum energy.”

Boundary-induced singularities also deserve a separate qualification. At fixed positive proper distance from a smooth boundary, the relative two-point function is well defined under standard local boundary conditions. As the boundary is approached, additional surface singularities appear and require the boundary treatment developed later in this chapter. Deutsch and Candelas exhibit this near-surface stress behavior explicitly Deutsch and Candelas 1979, §§II–IV, pp. 3065–3074.

First application: an ultrastatic spectral difference

Let

$$\mathrm ds^2=\mathrm dt^2-h_{ij}(x)\,\mathrm dx^i\mathrm dx^j$$

and let $\mathcal L_B=-\Delta_h+m^2+\xi R$ be positive and self-adjoint with boundary condition $B$. Write

$$\mathcal L_Bu_{n,B}=\omega_{n,B}^2u_{n,B}, \qquad \int_\Sigma\mathrm d^3x\,\sqrt h\, u_{n,B}^*u_{m,B}=\delta_{nm}.$$

The ground-state equal-time kernel is formally

$$W_B(t,x;t,x') = \sum_n\frac{u_{n,B}(x)u_{n,B}^*(x')}{2\omega_{n,B}}.$$

Choose two configurations $B_1,B_0$ with identical local geometry and matched high-frequency boundary data in the comparison region. After subtracting a common analytic high-mode expansion, compute

$$\Delta\langle\phi^2(x)\rangle = \lim_{N\to\infty} \left[ \sum_{n\le N}\frac{|u_{n,B_1}(x)|^2}{2\omega_{n,B_1}} -\sum_{n\le N}\frac{|u_{n,B_0}(x)|^2}{2\omega_{n,B_0}} -\Delta S_N(x) \right] +\Delta S_\infty(x).$$

Here $\Delta S_N$ is the integrated local spectral asymptotic through the required order and $\Delta S_\infty$ restores its analytically summed remainder. The two spectra need not admit a meaningful term-by-term pairing; the subtraction is organized by a common spectral parameter or heat kernel.

For a one-parameter family such as a boundary separation $L$, the relative vacuum energy is formally

$$\Delta E(L,L_0) = \frac12\sum_n \left[\omega_n(L)-\omega_n(L_0)\right]_{\mathrm{matched}},$$

and the force is

$$\mathcal F(L)=-\frac{\mathrm d}{\mathrm dL}\Delta E(L,L_0).$$

This derivative is insensitive to $L$-independent bulk constants, but it can retain dependence on allowed $L$-dependent surface or geometric counterterms. Heat-kernel coefficients identify precisely which local bulk and boundary pieces require matching Vassilevich 2003, §§2.2 and 5.2, pp. 288–294, 326–331.

Useful checks are the flat limit at fixed proper separation, the large-mass suppression of nonlocal polarization, symmetry under the isometries preserved by $B$, stress conservation in the interior, and agreement between the stress discontinuity and $-\mathrm dE/\mathrm dL$ after surface terms are included.

Adversarial absolute-energy test

Quote one finite value of $\langle T_{00}\rangle$ and change the finite coefficients of $m^4g_{\mu\nu}$ or the conserved curvature tensors. The number changes while the state, field equation, and local singular subtraction remain the same. Calling that number an absolute observable overstates the calculation.

Now form a same-background state difference or a boundary force with identical finite couplings. The common bulk ambiguity cancels; only configuration-dependent local surface terms and the nonlocal response remain. The strongest claim is therefore the explicitly matched difference or force, with its domain away from unresolved boundary singularities—not an absolute local vacuum energy.

Polarization and failure maps

The structure map shows the common local subtraction shared by both configurations and the later formation of a difference.

Scheme-independent content comes from a matched state, topology, or boundary comparison, while surface terms remain explicit; the map is schematic and not to scale.

The failure map asks whether a finite number survived a change of local curvature or surface prescription.

Relative stress, force, and topology dependence have different cancellation conditions and must be reported separately; the map is schematic and not to scale.

Use Domain and failure conditions. Record the background-identification map, state, boundary operator, proper distances, spectral measure, common subtraction, bulk and surface finite couplings, convergence, and the observable actually compared.

Mode-Sum and Numerical Renormalization develops the high-mode algorithm. Boundaries, Surface Counterterms, and Boundary Stress owns the surface limit. Chapter 13 owns cosmological-constant naturalness rather than this local comparison.

References

• David Deutsch and Paul Candelas, “Boundary Effects in Quantum Field Theory,” Physical Review D 20 (1979), 3063–3080, DOI.
• Dmitri V. Vassilevich, “Heat Kernel Expansion: User’s Manual,” Physics Reports 388 (2003), 279–360, DOI, arXiv:hep-th/0306138.