nLab reduced phase space



Symplectic geometry


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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


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.


Textbook account:

See also the references at quantization commutes with reduction.

Last revised on December 19, 2023 at 13:58:28. See the history of this page for a list of all contributions to it.