QED vacuum loops and vacuum energy

Abstract

A QED-based “bootstrap” mechanism is suggested as a possible source of vacuum energy. In place of the conventional assumption that the vacuum expectation value of the current operator j _μ vanishes in the absence of a classical, external field, one notes the possibility that, on very small scales, the vacuum fluctuations can generate an equation for an effective, C-number ${A}_{\mu}^{\mathrm{ vac}}$ giving rise to a finite and computable vacuum energy.

Appendix

Defining the linkage operator as

$$\exp\mathcal{D} = \exp \biggl[{i\over2}\int dx\,dy {\delta\over \delta j(x)}A(x,y){\delta\over\delta j(y)} \biggr] $$

we have the following relation [9, 10]:

$$\begin{aligned} & {\exp\mathcal{D}\exp \biggl[-{i\over2}\int du\,dv \, j(u)B(u,v)j(v) \biggr] }\\ & {\quad {}= \exp \biggl[-{i\over2}\int du \,dv\, dx\,j(u)B(u,x) ( 1 - AB )^{-1} }\\ & {\qquad {}\times (x,v)j(v)-{1\over2} \mathop{\mathrm{Tr}}\log( 1 - AB ) \biggr] } \end{aligned}$$

This formula, used in Eqs. (2.5) and (2.6) with the approximation (3.1) leads to (3.4).