Physical symmetries in field theory

Results obtained previously by the author [1] [Commun. Math. Phys., 12, 80 (1969); 14, 158 (1969)] are extended to the more general case of several fields with an arbitrary transformation law under the Poincare group with several nonvanishing masses. Bound states are also taken into account. A number of locally conserved tensor currents corresponding to physical symmetries are considered. The main result is the proof of the assertion that the free fields obtained by a symmetry transformation from the asymptotic fields and the asymptotic fields are connected by a linear transformation. This assertion can be interpreted as a proof that nonlinear $n$-dimensional realizations of a symmetry group for which there do not exist linear $n$-dimensional representations of the group cannot be realized if there is a mass gap in a theory with interaction. This result indicates that nonlinear realizations are intimately connected with the presence of particles of vanishing mass subject to the condition that the fields and currents remain local and the charges are translationally invariant.