nLab
reduced phase space

Context

Symplectic geometry

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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 (-):

\downarrow \downarrow quotient by
\downarrow \downarrow quotient by

Details

For local Lagrangians

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

L:Ω 0,1(Fields(I)×I) 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} \{EL = 0\}

equipped with the presymplectic form

ω=δδLδϕ˙δϕ \omega = \delta \frac{\delta L}{\delta \dot \phi} \wedge \delta \phi

which is exact, with potential

θ=δLδϕ˙δϕ \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 ω\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 θ\theta.

But this trivialility is only superficial: the symmetry group GG of the Lagrangian is supposed to act by Hamiltonian flows and the prequantum connection θ\theta is to be equipped with GG-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 θ:{EL=0}BU(1) conn\theta \;\colon\; \{EL = 0\} \longrightarrow \mathbf{B}U(1)_{conn} to the smooth moduli stack of circle bundles with connection, and the GG-Hamiltonian action induced the action groupoid {EL=0}{EL=0}//G\{EL = 0\} \longrightarrow \{EL = 0\}//G; and the construction of the reduced prequantization is the construction of the diagonal morphism red\nabla_{red} in the following diagram (of smooth groupoids)

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

For extended local Lagrangians

For nn-dimensional field theory, the local Lagrangian of the above dsicussion arises as the transgression of an nn-form Lagrangian down in codimension nn, 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.

References

See the references at quantization commutes with reduction.

Created on September 16, 2013 at 00:10:52. See the history of this page for a list of all contributions to it.