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extended affine symplectic group

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Group Theory

Symplectic geometry

Geometric quantization

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Definition

Given a symplectic vector space (V,ω)(V,\omega) one may consider the restriction of its quantomorphism group to the affine symplectic group ASp(V,ω)ASp(V,\omega) (Robbin-Salamon 93, corollary 9.3)

ESp(V,ω) QuantMorph(V,ω) ASp(V,ω) HamSympl(V,ω) \array{ ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega) \\ \downarrow && \downarrow \\ ASp(V,\omega) &\hookrightarrow& HamSympl(V,\omega) }

Sometimes (e.g. Robbin-Salamon 93, p. 30) this ESp(V,ω)ESp(V,\omega) is called the extended symplectic group, but maybe to be more specific one should at the very least say “extended affine symplectic group” or “extended inhomogeneous symplectic group” (ARZ 06, prop. V.1).

Notice that the further restriction to VV regarded as the translation group over itself is the Heisenberg group Heis(V,ω)Heis(V,\omega)

Heis(V,ω) ESp(V,ω) QuantMorph(V,ω) V ASp(V,ω) HamSympl(V,ω) \array{ Heis(V,\omega) &\hookrightarrow& ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega) \\ \downarrow && \downarrow && \downarrow \\ V &\hookrightarrow& ASp(V,\omega) &\hookrightarrow& HamSympl(V,\omega) }

The group ESp(V,ω)ESp(V,\omega) is that of those quantomorphisms which come from Hamiltonians that are quadratic Hamiltonians. Those elements covering elements in the symplectic group instead of the affine symplectic group come from homogeneously quadratic Hamiltonians (e.g. Robbin-Salamon 93, prop. 10.1). In fact ESpESp is the semidirect product of the metaplectic group Mp(V,ω)Mp(V,\omega) with the Heisenberg group (ARZ 06, prop. V.1, see also Low 12)

ESp(V,ω)Heis(V,ω)Mp(V,ω). ESp(V,\omega) \simeq Heis(V,\omega) \rtimes Mp(V,\omega) \,.

Example

Let (V,ω)=( 2,dpdq)(V,\omega) = (\mathbb{R}^2, \mathbf{d}p\wedge \mathbf{d}q) be the 2-dimensional symplectic vector space.

Write

q,p: 2 q,p \colon \mathbb{R}^2 \longrightarrow \mathbb{R}

for its two canonical coordinates functions (the “canonical coordinates and canonical momenta”).

Write

i: 2 i \colon \mathbb{R}^2 \longrightarrow \mathbb{R}

for the constant function with value 1.

The Poisson bracket is

{p.q}=i. \{p.q\} = i \,.

Any smooth function H: 2H \colon \mathbb{R}^2 \to \mathbb{R} we may call a Hamiltonian. Given a Hamiltonian HH, its Hamiltonian flow is the flow given by the vector field (the Hamiltonian vector field) corresponding to the derivation {H,}\{H,-\} on C ( 2)C^\infty(\mathbb{R}^2).

Those Hamiltonians whose Hamiltonian flows are linear functions on 2\mathbb{R}^2 are precisely the homogeneously quadratic Hamiltonians:

exp(t{12p 2,}):[q p][q+tp p] \exp(t\{\tfrac{1}{2}p^2,-\}) \colon \left[ \array{ q \\ p } \right] \mapsto \left[ \array{ q +t p \\ p } \right]
exp({12q 2,}):[q p][q p+tq] \exp(\{\tfrac{1}{2}q^2,-\}) \colon \left[ \array{ q \\ p } \right] \mapsto \left[ \array{ q \\ p + t q } \right]
exp(t{qp,}):[q p][e tq e tp] \exp(t \{q p,-\}) \colon \left[ \array{ q \\ p } \right] \mapsto \left[ \array{ e^t q \\ e^{-t} p } \right]

The general element of the metaplectic group Mp( 2,dqdp)Mp(\mathbb{R}^2,\mathbf{d}q \wedge \mathbf{d}p) is hence

exp(t 112p 2+t 212q 2+t 3qp) \exp(t_1 \tfrac{1}{2}p^2 + t_2 \tfrac{1}{2}q^2 + t_3 q p)

By differentiating this by tt at t=0t = 0 we obtain a basis for the Lie algebra 𝔰𝔭(V,ω)\mathfrak{sp}(V,\omega) of, both, the symplectic group Sp(V,ω)Sp(V,\omega) as well as its metaplectic group Mp(V,ω)Mp(V,\omega)

[0 0 1 0],[0 1 0 0],[1 0 0 1] \left[ \array{ 0 & 0 \\ 1 & 0 } \right] \,, \;\;\; \left[ \array{ 0 & 1 \\ 0 & 0 } \right] \,, \;\;\; \left[ \array{ 1 & 0 \\ 0 & -1 } \right]

The Hamiltonians that generate translations are precisely the homogeneously linear Hamiltonians:

exp(t{p,}):[q p][q+t p] \exp(t\{p,-\}) \colon \left[ \array{ q \\ p } \right] \mapsto \left[ \array{ q + t \\ p } \right]
exp(t{q,}):[q p][q pt] \exp(t\{q,-\}) \colon \left[ \array{ q \\ p } \right] \mapsto \left[ \array{ q \\ p - t } \right]

This is the key point where the extension appears: While these two linear translation operations themselves (i.e. the underlying symplectomorphisms) of course commute with each other, their generating Hamiltonians do not Poisson commute but instead form the Heisenberg algebra extension of the translation group.

The general element of the extended affine symplectic group ESp( 2,dqdp)ESp(\mathbb{R}^2, \mathbf{d}q \wedge \mathbf{d}p) is

exp(t 112p 2+t 212q 2+t 3qp+t 4p+t 5q+t 6i). \exp(t_1 \tfrac{1}{2}p^2 + t_2 \tfrac{1}{2}q^2 + t_3 q p + t_4 p + t_5 q + t_6 i) \,.

References

  • Joel Robbin, Dietmar Salamon, Feynman path integrals on phase space and the metaplectic representation, Math. Z. 221 (1996), no. 2, 307–-335, (MR98f:58051, doi, pdf), also in Dietmar Salamon (ed.), Symplectic Geometry, LMS Lecture Note series 192 (1993)

  • Sergio Albeverio, J. Rezende and J.-C. Zambrini, Probability and Quantum Symmetries. II. The Theorem of Noether in quantum mechanics, Journal of Mathematical Physics 47, 062107 (2006) (pdf)

  • Stephen G. Low, Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group, J. Math. Phys. 55, 022105 (2014) (arXiv:1207.6787)

Last revised on June 9, 2015 at 13:03:38. See the history of this page for a list of all contributions to it.