The Schrödinger representation and the metaplectic representation

2008 ◽  
pp. 161-189
Author(s):  
Ernst Binz ◽  
Sonja Pods
Keyword(s):  
Metaplectic Representation ◽  
Schrödinger Representation

METAPLECTIC REPRESENTATION, CONLEY–ZEHNDER INDEX, AND WEYL CALCULUS ON PHASE SPACE

2007 ◽  
Vol 19 (10) ◽  
pp. 1149-1188 ◽  
Author(s):  
MAURICE DE GOSSON
Keyword(s):  
Phase Space ◽  
Differential Operators ◽  
Heisenberg Algebra ◽  
Symplectic Space ◽  
Weyl Symbol ◽  
Metaplectic Representation ◽  
Weyl Calculus ◽  
Schrödinger Representation ◽  
Precise Calculation ◽  
Pseudo Differential Operators

We define and study a metaplectically covariant class of pseudo-differential operators acting on functions on symplectic space and generalizing a modified form of the usual Weyl calculus. This construction requires a precise calculation of the twisted Weyl symbol of a class of generators of the metaplectic group and the use of a Conley–Zehnder type index for symplectic paths, defined without restrictions on the endpoint. Our calculus is related to the usual Weyl calculus using a family of isometries of L2(ℝn) on closed subspaces of L2(ℝ2n) and to an irreducible representation of the Heisenberg algebra distinct from the usual Schrödinger representation.

scholarly journals Reproducing Groups for the Metaplectic Representation

2006 ◽  
pp. 227-244 ◽  
Author(s):  
E. Cordero ◽  
F. De Mari ◽  
K. Nowak ◽  
A. Tabacco
Keyword(s):  
Metaplectic Representation

scholarly journals Equivariant holomorphic maps into the Siegel disc and the metaplectic representation

1997 ◽  
Vol 62 (2) ◽  
pp. 160-174 ◽  
Author(s):  
Jean-Louis Clerc
Keyword(s):  
Symplectic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Invariant Subspaces ◽  
Holomorphic Maps ◽  
Harmonic Polynomials ◽  
Metaplectic Representation ◽  
Siegel Disc

AbstractWe restrict the metaplectic representation to subgroupsGof the symplectic group associated to equivariant holomorphic maps into the Siegel disc. We describe the invariant subspaces of the decomposition, and reduce the problem to the decomposition of a space of ‘harmonic’ polynomials under the action of the maximal compact subgroup ofG.

scholarly journals On the structure of analytic vectors for the Schrödinger representation

2011 ◽  
Vol 167 (1) ◽  
pp. 61-80 ◽  
Author(s):  
Rahul Garg ◽  
Sundaram Thangavelu
Keyword(s):  
Schrödinger Representation

scholarly journals Random products of automorphisms of Heisenberg nilmanifolds and Weil’s representation

2010 ◽  
Vol 31 (5) ◽  
pp. 1277-1286 ◽  
Author(s):  
BACHIR BEKKA ◽  
JEAN-ROMAIN HEU
Keyword(s):  
Probability Measure ◽  
Spectral Gap ◽  
Regular Representation ◽  
Metaplectic Representation ◽  
Matrix Coefficients ◽  
Left Regular Representation ◽  
Left Regular ◽  
Random Products ◽  
Image Position ◽  
Dense Set

AbstractForn≥1, letHbe the (2n+1)-dimensional real Heisenberg group, and let Λ be a lattice inH. Let Γ be the group of automorphisms of the corresponding nilmanifold Λ∖HandUthe associated unitary representation of Γ onL2(Λ∖H) . Denote byTthe maximal torus factor associated to Λ∖H. Using Weil’s representation (also known as the metaplectic representation), we show that a dense set of matrix coefficients of the restriction ofUto the orthogonal complement ofL2(T) inL2(Λ∖H) belong toℓ4n+2+ε(Γ) for every ε>0 . We give the following application to random walks on Λ∖Hdefined by a probability measureμon Aut (Λ∖H) . Denoting by Γ(μ) the subgroup of Aut (Λ∖H) generated by the support ofμand byU0andV0the restrictions ofUto, respectively, the subspaces ofL2(Λ∖H) andL2(T) with zero mean, we prove the following inequality:whereλis the left regular representation of Γ(μ) onℓ2(Γ(μ)) . In particular, the action of Γ(μ) on Λ∖Hhas a spectral gap if and only if the corresponding action of Γ(μ) onThas a spectral gap.

CRITICAL BEHAVIOR FROM SCHRÖDINGER REPRESENTATION

1992 ◽  
Vol 07 (22) ◽  
pp. 1975-1981 ◽  
Author(s):  
P. SURANYI
Keyword(s):  
Field Theory ◽  
Ising Model ◽  
Integral Equations ◽  
Critical Behavior ◽  
Transcendental Equation ◽  
Schrödinger Representation ◽  
Correlation Exponent ◽  
Truncation Scheme ◽  
Infinite Set ◽  
Good Agreement

The Schrödinger equation for Φ4 field theory is reduced to an infinite set of integral equations. A systematic truncation scheme is proposed and it is solved in second order to obtain the approximate critical behavior of the renormalized mass. The correlation exponent is given as a solution of a transcendental equation. It is in good agreement with the Ising model in all physical dimensions.

Sign in / Sign up

Export Citation Format

Share Document