Contents

Contents

Idea

In physics, a local Lagrangian induces a covariant phase space equipped with a canonical presymplectic form. The quotient of this by symmetries that, in good cases, make the pre-symplectic form a genuine symplectic form, is called the reduced phase space.

Generally, given a symplectic manifold or presymplectic manifold or Poisson manifold regarded as a phase space equipped with a suitable (Hamiltonian-) action by a Lie group, the corresponding symplectic reduction or presymplectic reduction or Poisson reduction is, if it exists, the corresponding reduced phase space.

Reductions of (pre-)symplectic manifolds:

symplectic geometryphysics
presymplectic manifoldcovariant phase space
$\downarrow$ gauge reduction$\downarrow$ quotient by gauge symmetry
symplectic manifoldreduced phase space
$\downarrow$ symplectic reduction$\downarrow$ quotient by global symmetry
symplectic manifoldreduced phase space

Details

For local Lagrangians

Given a local Lagrangian (we display it in codimension 1 mechanics for simplicity of notation)

$L \;\colon\; \Omega^{0,1}\left(\mathbf{Fields}(I) \times I\right)$

the corresponding covariant phase space is the space of solutions of the Euler-Lagrange equations

$\{EL = 0\}$

equipped with the presymplectic form

$\omega = \delta \frac{\delta L}{\delta \dot \phi} \wedge \delta \phi$

which is exact, with potential

$\theta = \frac{\delta L}{\delta \dot \phi} \wedge \delta \phi$

as discussed in detail at covariant phase space.

Together, this is a prequantum bundle, hence a circle bundle with connection whose curvature is $\omega$. It so happens that the underlying U(1)-principal bundle of this is trivial, and hence the connection is given by the globally defined differential 1-form $\theta$.

But this trivialility is only superficial: the symmetry group $G$ of the Lagrangian is supposed to act by Hamiltonian flows and the prequantum connection $\theta$ is to be equipped with $G$-equivariant connection structure for it to count as a connection on the reduced phase space.

Another way to say this, using the higher differential geometry of smooth groupoids: the above prequantum bundle is modulated by a map $\theta \;\colon\; \{EL = 0\} \longrightarrow \mathbf{B}U(1)_{conn}$ to the smooth moduli stack of circle bundles with connection, and the $G$-Hamiltonian action induced the action groupoid $\{EL = 0\} \longrightarrow \{EL = 0\}//G$; and the construction of the reduced prequantization is the construction of the diagonal morphism $\nabla_{red}$ in the following diagram (of smooth groupoids)

$\array{ \{EL = 0\} &\stackrel{\theta}{\longrightarrow}& \mathbf{B}U(1)_{conn} \\ \downarrow & \nearrow_{\nabla_{red}} \\ \{EL = 0\}//G } \,.$

For extended local Lagrangians

For $n$-dimensional field theory, the local Lagrangian of the above dsicussion arises as the transgression of an $n$-form Lagrangian down in codimension $n$, a refinement discussed in more detail at local prequantum field theory. Such a Lagrangian may analogously be understood as being the higher connection on a prequantum n-bundle which, in turn, maps to the ordinary prequantum bundle under transgression. Hence one may ask for a universal or extended equivariant structure already on this prequantum n-bundle which is such that under transgression (to any Cauchy surface) it induces an equivariant structure and hence a reduced phase space as above.

Examples of such extended reduced phase space structures include:

and other examples discussed at local prequantum field theory.

See the references at quantization commutes with reduction.