Date: 2024-03-09 | Article ID: 51033 |

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 S

For some of these models (the O(2) model, Z

**2024-03-09**(clarified the relationship between two equations that use different notation)**2024-03-08**(made a footnote about Wick rotation more careful, fixed a notation inconsistency)**2023-10-26**(removed a dubious-and-unnecessary statement)**2023-04-17**(first version)

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