First-class constraint
[1] First- and second-class constraints were introduced by Dirac (1950, p. 136, 1964, p. 17) as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate.Consider a Poisson manifold M with a smooth Hamiltonian over it (for field theories, M would be infinite-dimensional).Suppose we have some constraints for n smooth functions These will only be defined chartwise in general.Look at the orbits of the constrained subspace under the action of the symplectic flows generated by the f 's.This gives a local foliation of the subspace because it satisfies integrability conditions (Frobenius theorem).It turns out if we start with two different points on a same orbit on the constrained subspace and evolve both of them under two different Hamiltonians, respectively, which agree on the constrained subspace, then the time evolution of both points under their respective Hamiltonian flows will always lie in the same orbit at equal times.For most "practical" applications of first-class constraints, we do not see such complications: the quotient space of the restricted subspace by the f-flows (in other words, the orbit space) is well behaved enough to act as a differentiable manifold, which can be turned into a symplectic manifold by projecting the symplectic form of M onto it (this can be shown to be well defined).A global anomaly is a barrier to defining a quantum gauge theory discovered by Witten in 1980.Another complication is that Δf might not be right invertible on subspaces of the restricted submanifold of codimension 1 or greater (which violates the stronger assumption stated earlier in this article).First of all, we will assume the action is the integral of a local Lagrangian that only depends up to the first derivative of the fields.Consider the dynamics of a single point particle of mass m with no internal degrees of freedom moving in a pseudo-Riemannian spacetime manifold S with metric g. Assume also that the parameter τ describing the trajectory of the particle is arbitrary (i.e. we insist upon reparametrization invariance).Consider now the case of a Yang–Mills theory for a real simple Lie algebra L (with a negative definite Killing form η) minimally coupled to a real scalar field σ, which transforms as an orthogonal representation ρ with the underlying vector space V under L in (d − 1) + 1 Minkowski spacetime.Note that the A here differs from the A used by physicists by a factor of i and g. This agrees with the mathematician's convention.where the second term is a formal shorthand for pretending the Lie bracket is a commutator, D is the covariant derivative and α is the orthogonal form for ρ.Well, first, we have to split A noncovariantly into a time component φ and a spatial part A→.The Hamiltonian, The last two terms are a linear combination of the Gaussian constraints and we have a whole family of (gauge equivalent)Hamiltonians parametrized by f. In fact, since the last three terms vanish for the constrained states, we may drop them.In a constrained Hamiltonian system, a dynamical quantity is second-class if its Poisson bracket with at least one constraint is nonvanishing.Start with the action describing a Newtonian particle of mass m constrained to a spherical surface of radius R within a uniform gravitational field g. When one works in Lagrangian mechanics, there are several ways to implement a constraint: one can switch to generalized coordinates that manifestly solve the constraint, or one can use a Lagrange multiplier while retaining the redundant coordinates so constrained.In this case, the particle is constrained to a sphere, therefore the natural solution would be to use angular coordinates to describe the position of the particle instead of Cartesian and solve (automatically eliminate) the constraint in that way (the first choice).For pedagogical reasons, instead, consider the problem in (redundant) Cartesian coordinates, with a Lagrange multiplier term enforcing the constraint.We are here treating •λ as a shorthand for a function of the symplectic space which we have yet to determine and not as an independent variable.In other words, the constraints must not evolve in time if they are going to be identically zero along the equations of motion.We keep turning the crank, demanding this new constraint have vanishing Poisson bracket We might despair and think that there is no end to this, but because one of the new Lagrange multipliers has shown up, this is not a new constraint, but a condition that fixes the Lagrange multiplier: Plugging this into our Hamiltonian gives us (after a little algebra)Now that there are new terms in the Hamiltonian, one should go back and check the consistency conditions for the primary and secondary constraints.In particular, The above Poisson bracket does not just fail to vanish off-shell, which might be anticipated, but even on-shell it is nonzero.Here, we have a symplectic space where the Poisson bracket does not have "nice properties" on the constrained subspace.Examination of the above Hamiltonian shows a number of interesting things happening.Since φ1 is a first-class primary constraint, it should be interpreted as a generator of a gauge transformation.Therefore, that λ dropped out of the Hamiltonian, that u1 is undetermined, and that φ1 = pλ is first-class, are all closely interrelated.Note that it would be more natural not to start with a Lagrangian with a Lagrange multiplier, but instead take r² − R² as a primary constraint and proceed through the formalism: The result would the elimination of the extraneous λ dynamical quantity.