On the symmetry of dynamical equations in quantum field theory

A study is made of two formulations of the requirement of invariance of a quantum field theory under a group $G$ whose generators include the Hamiltonian. In accordance with the first , all the generators of the group, expressed in terms of the seeond-quantized fields, must satisfy commutation relations that form the algebra of the generators of $G$. It is shown that this requirement does not impose any restrictions on the interaction Hamiltonian or the $S$ matrix. If we wish to restrict the treatment to Hamiltonians that lead to theories with an invariant matrix, it is sufficient to use the second formulation of the invariance requirement. The second formulation is obtained from the first by adding the following condition: all the generators must be expressed in terms of three-dirnensional integrals over the space of local field functions.