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The above Poisson bracket does not just fail to vanish off-shell, which might be anticipated, but ''even on-shell it is nonzero''. Therefore, {{math| ''φ''<sub>2</sub>}} and {{math| ''φ''<sub>3</sub>}} are '''second class constraints''' while {{math| ''φ''<sub>1</sub>}} is a first class constraint. Note that these constraints satisfy the regularity condition.
Here, we have a symplectic space where the Poisson bracket does not have "nice properties" on the constrained subspace. However, [[Paul Dirac|Dirac]] noticed that we can turn the underlying [[differential manifold]] of the [[symplectic manifold|symplectic space]] into a [[Poisson manifold]] using his eponymous modified bracket, called the [[Dirac bracket]], such that this ''Dirac bracket of any (smooth) function with any of the second class constraints always vanishes''. Effectively, these brackets, illustrated for this spherical surface in the [[Dirac
If one then wished to canonically quantize this system, then one need promote the canonical Dirac brackets<ref>{{Cite journal | last1 = Corrigan | first1 = E. | last2 = Zachos | first2 = C. K. | doi = 10.1016/0370-2693(79)90465-9 | title = Non-local charges for the supersymmetric σ-model | journal = Physics Letters B | volume = 88 | issue = 3–4 | pages = 273 | year = 1979 | pmid = | pmc = |bibcode = 1979PhLB...88..273C }}</ref>, ''not'' the canonical Poisson brackets to commutation relations.
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