2024-03-09   51033

Constrained Scalar Quantum Fields

This article uses the path integral formulation to construct toy models of multi-component scalar quantum fields whose equations of motion are implicitly nonlinear because of a constraint on the values of the field variables. The constraint can lead to interesting phenomena, like spontaneous symmetry breaking and asymptotic freedom, even though the equation of motion looks very simple.

This article constructs a few families of such models, using a path integral formulation that treats spacetime as a discrete lattice. Each family is characterized by a different type of target space, the space of possible values of the scalar field at each point in spacetime. In the O(N) models, the target space is a sphere SN-1. In the Zn models or clock models, the target space consists of n equally-space points around a circle. In the principal chiral models, the target space is a Lie group.

For some of these models (the O(2) model, Zn models, and principal chiral models), this article also explains how to derive a corresponding hamiltonian formulation. This is especially interesting in the case of the Zn models, because deriving a hamiltonian formulation requires taking a limit as time becomes continuous, but the field variables themselves are constrained to a discrete set of values. This article explains how a useful hamiltonian formulation can still be derived by taking a special kind of continuous-time limit. For the Z2 model, the result is the hamiltonian formulation of the quantum Ising model, whose phase structure is studied in article 81040.

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www.cphysics.org updated 2024-05-22