Rotationally-Symmetric Model Field Theories

Abstract

A class of highly symmetric ,nonrelativistic, Euclidean invariant, model scalar field theories are examined assuming the existence of field and momentum operators that satisfy the canonical commutation relations (CCR). The high degree of symmetry that we assume permits explicit determination of every relevant CCR representation. These consist of a two-parameter family of unitarily inequivalent representations, some of which are irreducible while the others are reducible. It is demonstrated that only those models that are analogs of the free field can be encompassed within the irreducible representations. Hence every model with interaction—including an analog of the relativistic λφ4 theory—requires a reducible CCR representation. For the reducible representations, we determine every relevant Hamiltonian operator possessing the required high degree of symmetry. These Hamiltonians, as well as the generators of space translations, cannot be expressed (solely) as functions of the field and momentum operators, which is characteristic of any system with a unique ground state and reducible CCR representation. Nevertheless, it is demonstrated that these Hamiltonians, as well as the generators of space translations, fulfill the ``weak correspondence principle,'' in which the expectation value of a quantum generator, such as the Hamiltonian operator, in a suitably labeled overcomplete family of states is identified with the associated classical generator, such as the classical Hamiltonian. Our principal results depend on the existence and make extensive use of the countably infinite number of degrees of freedom existing in a field theory. Entirely analogous results apply to related models defined in a finite spacial volume since they still have an infinite number of degrees of freedom.