p-form electromagnetism on discrete spacetimes

Details p-form electromagnetism on discrete spacetimes p-form electromagnetism on discrete spacetimes

Sep 2006

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006cqgra..23.5129w&link_type=abstract

Classical and Quantum Gravity, Volume 23, Issue 17, pp. 5129-5176 (2006).

Mathematics

Logic

8

Scientific paper

We investigate p-form electromagnetism—with the Maxwell and Kalb Ramond fields as lowest-order cases—on discrete spacetimes, including not only the regular lattices commonly used in lattice gauge theory, but also more general examples. After constructing a maximally general model of discrete spacetime suitable for our purpose—a chain complex equipped with an inner product on (p + 1)-cochains—we study both the classical and quantum versions of the theory, with either {\bb R} or U(1) as gauge group. We find results—such as a 'p-form Bohm Aharonov effect'—that depend in interesting ways on the cohomology of spacetime. We quantize the theory via the Euclidean path integral formalism, where the natural kernels in the U(1) theory are not Gaussians but theta functions. As a special case of the general theory, we show that p-form electromagnetism in p + 1 dimensions has an exact solution which reduces when p = 1 to the Abelian case of 2D Yang Mills theory as studied by Migdal and Witten. Our main result describes p-form electromagnetism as a 'chain field theory'—a theory analogous to a topological quantum field theory, but with chain complexes replacing manifolds. This makes precise a notion of time evolution in the context of discrete spacetimes of arbitrary topology.
A somewhat less technical, more expository version of this material appeared as gr-qc/0510033 [38].

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