Does there exist a sensible quantum theory of an "algebra-valued" scalar field?

Abstract

Consider a scalar field \ensuremath{\varphi} in Minkowski spacetime, but let \ensuremath{\varphi} be valued in an associative, commutative algebra openA rather than openR. One may view the resulting theory as describing a collection of coupled real scalar fields. At the classical level, theories of this type are completely well behaved and have a global symmetry group which is a nontrivial enlargement of the Poincar\'e group. (They are analogs of the new class of gauge theories for massless spin-2 fields found recently by one of us, whose gauge group is a nontrivial enlargement of the usual diffeomorphism group.) We investigate the quantization of such scalar field theories here by studying the case of a \ensuremath{\lambda}${\ensuremath{\varphi}}^{4}$ field, with \ensuremath{\varphi} valued in the two-dimensional algebra generated by an identity element e and a nilpotent element v satisfying ${v}^{2}$=0. The Coleman-Mandula theorem, which states that the symmetry group of a nontrivial quantum field theory cannot be a nontrivial enlargement of the Poincar\'e group, is evaded here because the finite ``extra'' symmetries of the classical theory fail to be implemented in the quantum theory by unitary operators and the infinitesimal symmetries (which can be represented in the quantum theory by quadratic forms) connect the one-particle Hilbert space to multiparticle states. Nevertheless, we find that the conventional Feynman rules for this theory lead to vacuum decay at the tree level and fail to yield a well-defined S matrix. Some alternative approaches are investigated, but these also appear to fail. Thus, although the classical theory is perfectly well behaved, it seems that there does not exist a sensible quantum theory of an algebra-valued scalar field.