Electric-magnetic duality in the QED effective action

Abstract

Recently we have obtained a non-perturbative but convergent series expression of the one loop effective action of QED, and discussed the renormalization of the effective action. In this paper we establish the electric-magnetic duality in the quantum effective action.PACS numbers: 11.10.Jj, 11.15.Tk The effective theory of QED plays a crucial role in our understanding of the non-linear effects in electrodynamics. Recently we have obtained a convergent series expression of the effective action of QED in one loop approximation,and established the renormalization group invariance of the effective action [1]. A remarkable feature of the effective action is the electric-magnetic duality,a fundamental symmetry of the quantum effective action of QED. The purpose of this paper is to provide a concrete proof of the duality in the effective action of QED.The effective action of QED has been studied by Euler and Heisenberg and by Schwinger long time ago [2,3],and by many others later [4,5].To derive the effective action one may start from the QED Lagrangianwhere m is the electron mass.With a proper gauge fixing one can show that the electron one loop correction of the effective action is given bySo for an arbitrary constant background one has [2,3] ∆L = − ab 8π 2 ∞+iǫ 0+iǫ dt t coth(at) cot(bt) exp(−m 2 t),

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“…This is exactly what we obtain from the non-perturbative effective action in the massless limit [11,12]. This confirms that in massless QED, one can calculate the imaginary part of the effective action either perturbatively or non-perturbatively, with identical results.…”

“…From this we finally have 12) and ∆L = lim (7.13) Obviously this has no infra-red divergence. This tells that, only when one mistakenly includes the unphysical eigenstates in the calculation of the functional determinant, one has to worry about which infra-red regularization is the correct one.…”

“…This amounts to changing X µ to its charge conjugate state X 12) which is equivalent to changingn 2 to −n 2 . Indeed this is exactly what we need to induce the color reflection (4.8), because (n 1 , −n 2 , −n) now forms a right-handed orthonormal basis.…”

“…Even in the case of "simpler" QED, the integration of the effective action has been completed only recently [11,12]. Fortunately the integral for the pure electric (a = 0) and pure magnetic (b = 0) background can be correctly performed [9,13].…”

“…The massless gauge theories, however, have a very unique feature that the imaginary part of the one-loop effective action is propotional to g 2 , where g is the coupling constant. This is true in both QCD [7,8] and massless QED [11,12]. This enables us to calculate the imaginary part of the effective action with a perturbative method.…”