In yang-mills theory , the constraint algebra closes to form a lie algebra. Even string theory has a constraint algebra which closes to form a lie algebra. I wish to know if there are other cases where the constraint algebra doesn't close. What does it physically mean for the constraint algebra to not close ?

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    $\begingroup$ The canonical example of a theory with a constraint algebra that doesn't close is General Relativity. Its constraints are the spatial diffeomorphisms and refoliations (temporal diffeomorphisms). Their algebra only closes if the equations of motion are imposed. Physically, this means that the constraints are a mixture of first-class and second-class constraints. $\endgroup$ – Prof. Legolasov May 14 at 12:44
  • $\begingroup$ @Prof.Legolasov I see. So what is the procedure one must follow in order to quantize such theories ? $\endgroup$ – user44690 May 14 at 13:59

In the Hamiltonian formulation, the non-closure of the constraint algebra is typically associated with second-class constraints and/or anomalies. See also my related Phys.SE answer here.

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