Approach to Equal-Time Commutators in Quantum Field Theory

Abstract

Equal-time current commutators [$J\left(x\right)\mathrm{}$, $J^{\prime }\left(x^{\prime }\right)$] should be calculated as suitable equal-time limits of ordinary current commutators. Since this calculation is usually ambiguous or impractical, we propose instead to define them as limits of [$J(x; \xi )$, $J^{\prime }(x^{\prime }; {\xi }^{\prime })$] for $\xi , {\xi }^{\prime }\to 0$, where $J(x; \xi )$ is a suitable nonlocal expression in the fields which converges to $J\left(x\right)\mathrm{}$ for $\xi \to 0$. This alternative should be more reliable than the usual ones, such as taking equal-time limits inside of spectral representations or taking limits of time-ordered products from positive and negative time differences. The former procedure is invalid when the spectral function is nonintegrable, and the latter when equal-time $\delta$ functions are present. An analysis of two-point functions is presented which illustrates the above effects. In this connection, it is shown that the commutator $〈0\left|\right[j_k, j_4\left]\right|0\mathrm{}〉$ in electrodynamics has a ${\delta }_k\Delta \delta (\mathrm{x}-{\mathrm{x}}^{\prime })$ term in addition to the usual ${\delta }_k\delta (\mathrm{x}-{\mathrm{x}}^{\prime })$ term. Our definition is shown to give correct results in a number of soluble models. It is then used to calculate commutators for electrodynamics in all orders of perturbation theory. The main new result is that, contrary to previous assertions, the commutator [$j_k\mathrm{}\left(x\right), j_4\left(x^{\prime }\right)$] is a $q$-number—essentially $e^4:{\mathrm{A}}^2:{\partial }_k\delta (\mathrm{x}-{\mathrm{x}}^{\prime })$ in the Gupta-Bleuler gauge. This result, together with a similar one for [$j_k\mathrm{}\left(x\right), {\dot{A}}_l\left(x^{\prime }\right)$], is shown to be consistent with gauge invariance and to be suitable for use in equal-time commutators which arise in reduction formulas. Finally, reduction formulas are used to explicitly establish the correctness of our results in fourth order.