Abstract
It is emphasized that for interactions with derivative couplings, the Ward Identity (WI) securing the preservation of a global symmetry should be modified. Scalar QED is taken as an explicit example. More precisely, it is rigorously shown in scalar QED that the naive WI and the improved Ward Identity (‘Master Ward Identity’, MWI) are related to each other by a finite renormalization of the time-ordered product (‘T-product’) for the derivative fields, and we point out that the MWI has advantages over the naive WI—in particular with regard to the proof of the MWI. We show that the MWI can be fulfilled in all orders of perturbation theory by an appropriate renormalization of the T-product, without conflict with other standard renormalization conditions. Relations with other recent formulations of the MWI are established.
Highlights
In spinor QED, the Master Ward Identity (MWI) expressing global U (1)symmetry contains all information that is needed for a consistent perturbative BRST-construction of the model, see [8] or [6, Chap. 5]
We prove that the validity of the MWI for T is equivalent to the validity of naive WI for T
The original references for the MWI are [7,9] and [2]. It is a universal formulation of symmetries; it can be understood as the straightforward generalization to QFT of the most general classical identity for local fields that can be obtained from the field equation and the fact that classical fields may be multiplied pointwise
Summary
In spinor QED, the Master Ward Identity (MWI) expressing global U (1)symmetry contains all information that is needed for a consistent perturbative BRST-construction of the model, see [8] or [6, Chap. 5]. This ‘QED-MWI’ is a renormalization condition on T -products to be satisfied to all orders of perturbation theory. 4.2, we prove, in the perturbative approach to scalar QED, that the MWI is equivalent to the so-called ‘unitary MWI’. The latter is an identity, conjectured by Fredenhagen [4], which seems to be well-suited for the formulation of symmetries in the Buchholz–Fredenhagen quantum algebra [5]. All proofs are given to all orders of perturbation theory
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