Consider the sigma model in 2 dimensions with $N$ sigma fields $$ \mathcal{L}=\frac{N}{2f}\partial_{\mu}\sigma^a\partial^{\mu}\sigma^a. $$ We want these fields to obey the constraints $$ \sigma^a\sigma^a=1. $$ In order to do this a Lagrange multiplier field $\alpha$ is introduced in the Lagrangian such that the action is $$ S=\frac{1}{2}\int d^2x\bigg[\partial_{\mu}\sigma^a\partial^{\mu}\sigma^a-\frac{\alpha}{N^{1/2}}(\sigma^a\sigma^a-\frac{N}{f})\bigg]. $$ I understand that classically this imposes a constraint in the equations of motion giving the desired constraint (up to some nontrivial normalization). Nonetheless, what about the quantum theory? I mean this constraint is well motivated in classical field theory, but the role it plays in quantum dynamics is not obvious to me. What is the effect, in say, the correlation functions of the theory of introducing this classically constraint setting term?

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    $\begingroup$ There could in principle be non-zero higher quantum average moments of quantum fluctuations in the classically forbidden constrained direction. $\endgroup$ – Qmechanic Apr 27 '18 at 14:23

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