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Functional Time Evolution, Anomaly Potentials, And The Geometric Phase, Charles G. Torre

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A free quantum field in 1+1 dimensions admits unitary Schrödinger picture dynamics along any foliation of spacetime by Cauchy curves. Kuchař showed that the Schrödinger picture state vectors, viewed as functionals of spacelike embeddings, satisfy a functional Schrödinger equation in which the generators of time evolution are the field energy-momentum densities with a particular normal-ordering and with a (nonunique) c-number contribution. The c-number contribution to the Schrödinger equation, called the “anomaly potential,” is needed to make the equation integrable in light of the Schwinger terms present in the commutators of the normal-ordered energy-momentum densities. Here we give a quantum geometric …