3 Path integral quantization of small perturbations: propagator and vacuum energy

To compute the vacuum energy we proceed further in the covariantly quantized fashion we started in the previous section. In a covariant $\alpha$-gauge the path integral for the partition function can be rewritten, using (21) and going to euclidean space, as

$$Z = \left [ \det \left( \Box + m _{\mathrm{A}} ^{2} \right) \right] ^... ...2 \alpha} \int f ^{2} d ^{4} x} \times \det \left[ \partial ^{2} \right] \times$$

$$\times \, \exp \left\{ - \int \frac{d^4k}{(2\pi)^4} A _{\mu} (k) ... ...^{\nu} \right)} {k ^{2} - m _{\mathrm{A}} ^{2}} \right] A _{\nu} (k) \right\} .$$ (22)

Here we have already performed the functional integration over $\phi$, we remember that we are approximating the higher order lagrangian up to the second order in the fluctuations and understand the path-integrals in the euclidean sector. The functional integration over $f$ can now be done and we get

$$Z = \frac{ \det \left[ \partial ^{2} \right] \int [ {\mathca... ...left( \Box + m _{\mathrm{A}} ^{2} \right) \right] ^{1/2} }\,,$$ (23)

where

$${{\mathcal{D}} ^{-1}} ^{\mu \nu} (k ; \alpha ) = \left( k ^{... ...^{-1}} ^{\mu \nu} (k) + \frac{1}{\alpha} k ^{\mu} k ^{\nu} \,.$$

In order to solve (23) we have to find the eigenvalues, $\lambda _{k}$, of ${{\mathcal{D}} ^{-1}} ^{\mu \nu} (k ; \alpha)$: $Z$ equals then the product of these eigenvalues. The requested eigenvalues are those related to the physical polarizations found in section 2.2, i.e. $\tilde{k} _{\mu}$, $\tilde{k} ^{\perp} _{\mu}$, $E _{\mu}$, with eigenvalues $k ^{2} - \kappa \bar{k} ^{2} / ( k ^{2} - m _{\mathrm{A}} ^{2} )$, $k ^{2}$ and $k ^{2}$ respectively. As already discussed at the end of the previous section, the eigenvector $k ^{\mu}$ has now a non-zero eigenvalue $1/\alpha$.

The vacuum energy, or free energy, is then

$$W = \ln Z = \frac{1}{2} \sum _{j} \ln \lambda _{j} = \frac{V... ...} \sum \int \frac{d^4k}{(2 \pi) ^{4}} \ln ( \lambda _{k} ) \,,$$

so that

$$W = \frac{V T}{2} \int \frac{d^4k}{(2 \pi) ^{4}} \ln \left( k ^{2} - ... ...{A}} ^{2}} \right) + V T \int \frac{d^4k}{(2\pi)^4} \ln \left( k ^{2} \right) +$$

$$+\, \frac{V T}{2} \int \frac{d ^{4} k}{(2 \pi) ^{4}} \ln \left( k ^{2} - m _{\mathrm{A}} ^{2} \right) .$$ (24)

The propagator, i.e. the inverse of ${{\mathcal{D}} ^{-1}} ^{\mu \nu} (k ; \alpha)$, can also be found exactly, as shown in appendix B, and results to be

$${\mathcal{D}} ^{\mu \nu} ( k ; \alpha ) = \frac{1}{k ^{2}} \left( g ^{\mu \nu} - \frac{k ^{\mu} k ^{\nu}}{k... ... \left( k ^{2} - m _{\mathrm{A}} ^{2} \right) - \kappa \bar{k} ^{2} \right) } +$$

$$- \, \frac{\kappa \bar{k} ^{\mu} \bar{k} ^{\nu}} { k ^{2} \left( ... ...a \bar{k} ^{2} \right) } + \alpha \frac{k ^{\mu} k ^{\nu}}{( k ^{2} ) ^{2}} \,.$$ (25)