Wave functions constructed from an invariant sum over histories satisfy constraints

Invariance of classical equations of motion under a group parametrized by functions of time implies constraints between canonical coordinates and momenta. In the Dirac formulation of quantum mechanics, invariance is normally imposed by demanding that physical wave functions are annihilated by the operator versions of these constraints. In the sum-over-histories quantum mechanics, however, wave functions are specified, directly, by appropriate functional integrals. It therefore becomes an interesting question whether the wave functions so specified obey the operator constraints of the Dirac theory. In this paper, we show for a wide class of theories, including gauge theories, general relativity, and first-quantized string theories, that wave functions constructed from a sum over histories are, in fact, annihilated by the constraints provided that the sum over histories is constructed in a manner which respects the invariance generated by the constraints. By this we mean a sum over histories defined with an invariant action, invariant measure, and an invariant class of paths summed over.

We use this result to give three derivations of the Wheeler-DeWitt equation for the wave function of the universe starting from the sum-over-histories representation of it. The first uses Becchi-Rouet-Stora-Tyutin methods and the explicit path-integral construction of Batalin, Fradkin, and Vilkovisky. The second is a direct derivation from the gauge-fixed Hamiltonian path integral. The third exploits the embedding variables introduced by Isham and Kuchař, in terms of which the connection with the constraints representing the four-dimensional diffeomorphism group is most clearly seen. In each case it is found that the symmetry leading to the Wheeler-DeWitt equation is not in fact four-dimensional diffeomorphism invariance; rather, it is the closely connected but slightly larger canonical symmetry of the Hamiltonian form of the action of general relativity. By allowing our path-integral construction to be either Euclidean or Lorentzian, we show that the consequent Wheeler-DeWitt equation is independent of which one is taken as a starting point. Our results are general, in that they do not depend on a particular representation of the sum over histories, but they are also formal, in that we do not address such issues as the operator ordering of the derived constraints. Instead, we isolate those general features of a sum over histories which define an invariant construction of a wave function and show that these imply the operator constraints.