This is the first in a series of papers dealing with four-dimensional quantum electrodynamics on a finite-element lattice. We begin by studying the canonical structure of the theory without interactions. This tells us how to construct momentum expansions for the field operators. Next we examine the interaction term in the Dirac equation. We construct the transfer matrix explicitly in the temporal gauge, and show that it is unitary. Therefore, fermion canonical anticommutation relations hold at each lattice site. Finally, we expand the interaction term to second order in the temporal-lattice spacing and deduce the magnetic moment of the electron in a background field, consistent with the continuum value of g = 2.
We apply the finite-element lattice equations of motion for quantum electrodynamics to an examination of anomalies in the current operators. By taking explicit lattice divergences of the vector and axial-vector currents we compute the vector and axialvector anomalies in two and four dimensions. We examine anomalous commutators of the currents to compute divergent and finite Schwinger terms. And, using free lattice propagators, we compute the vacuum polarization in two dimensions and hence the anomaly in the Schwinger model. This paper summarizes the status of the finite-element approach to gauge theories, in which the Heisenberg operator equations of motion are converted to operator difference equations consistent with unitarity. A review of the entire program, from quantum mechanics to quantum field theory, is given in Ref. [1].
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