Restriction on Dirac's Conjecture

First class constraints in a canonical formalism of a gauge theory might generate transformations which map a state to its physically equivalent state. This is called Dirac's conjecture. There are two examples which may be candidates of counter-example of the conjecture. One is the toy model found by Cawley, and another is the bilocal model proposed by the author. A quantum analysis of the bilocal model shows that the model has the critical dimension of spacetime, which is surprisingly equal to four. The derivation, however, is based on the assumption that true symmetry of the system is generated by the first class constraints, which holds if Dirac's conjecture is satisfied. In the present paper we give detailed and mathematically rigorous analysis of Dirac's conjecture in general gauge theories, which involves new concept like semi-gauge invariance. We find the condition for the conjecture to hold. This is a set of equations for the generating function of the transformation, expressed in terms of Poisson brackets and M-brackets introduced in the paper. The above condition reduces the range of gauge theories where Dirac's conjecture holds. Along with the general prescription described in the paper we find that the bilocal model satisfies the above condition with some exceptions. Some examples are used to illustrate our method.