Representations in Quantum Theory  

In General > s.a. wigner functions.
* Idea: The basic problem in the quantum theory of a physical system is choosing a complete set of observables that characterize the states one wants to describe, and find a representation of this set on a Hilbert space.
* Issues: How unique is the representation? Which functions on the basic operators can/should one represent?
* Relevant tools/results: The Stone-von Neumann theorem (see below), Van Hove theorem, GNS construction.
@ References: Shewell AJP(59) [operator ambiguities]; Wlodarz PLA(01) [phase-space-like]; de la Torre AJP(02)qp/02 [including aX+(1–a)P, (XP+PX)/2]; Halvorson SHPMP(04)qp/01 [and complementarity]; Vourdas JPA(06) [analytic, rev]; Bracci & Picasso AJP(07) [inequivalent, AB effect example].

Stone-Von Neumann Theorem
* Idea: Every irreducible regular representation of the canonical commutation relations in Weyl form for conventional quantum theory with configuration space Rn is unitarily equivalent to the Schrödinger representation on L2(Rn).
$ Def: All representations of the finite-dimensional Heisenberg algebra are unitarily equivalent.
@ References: von Neumann MA(31); Grosse & Pittner pr(87) [for supersymmetric quantum mechanics]; Cavallaro et al LMP(99) [non-regular representations].

Schrödinger Representation > s.a. in quantum field theory.
* Idea: The representation on L2(, d), where is the configuration space, in which states are t-dependent, observables t-independent.
* Use: Not convenient for the relativistic theory, since it treats time differently from the space coordinates.

Heisenberg Representation > s.a. in quantum field theory.
* Idea: The representation on L2(, d) in which states are t-independent, and observables t-dependent operators.
* Relationships: Compared to the Schrödinger representation,

H = exp{iHt/} S(t),   and   AH(t) = exp{iHt/} AS exp{–iHt/};

time evolution of the operators is given by i AH/t = [AH, H]; The Hamiltonian operator is the same.
* Advantages: Constant phase shifts in the operators due, e.g., to a constant potential are cancelled; The equations of motion for the operators are formally identical to the classical ones; It is convenient for relativistic theory.
* Disadvantages: It is difficult to solve practical problems with it.
@ Compared to Schrödinger representation: Faria et al PLA(02); Nikolic PLA(04)qp/03; de la Madrid qp/05-in [for unbounded operators, and rigged Hilbert space]; Solomon a0706 [in quantum field theory].

Bargmann-Segal (Coherent State) Representation > s.a. coherent states [including Segal-Bargmann transform].
* Idea: The holomorphic representation on L2(C, exp{– |z|2} dz dz*) ((z) is analytic), obtained from the usual L2(R,dx) using the heat kernel t by the transformation

(x) (Ct)(z):= dx t(z–x) (x) .

* Inner product, operators: If (z):= n=0infty (n!)–1/2 zn n|, with |a an eigenvector of a, then

|:= –1 *(z) (z) exp{–z*z} d2z ;
a (z) = z (z)   and   a (z) = (/z) (z) .

* Generalizations: The transform can be generalized to functions on groups.
* For the simple harmonic oscillator: Define the complex variable as z:= (m/2)1/2 q + i (2m)–1/2 p.
@ General references: Bargmann CPAM(61), PNAS(62); Hall qp/99-ln, CMP(02) [compact groups, geometric quantization]; Kowalski & Rembielinski JMP(01)qp/00 [S2]; Villegas-Blas JMP(02) [kernel of transform]; Ribeiro et al PRL(05) [conjugate]; Hübschmann m.DG/06 [and holomorphic Peter-Weyl theorem].
@ Other systems: Aldaya & Guerrero JPA(93) [relativistic oscillator]; Ashtekar et al JFA(96)gq/94 [spaces of connections]; Villegas-Blas JMP(06) [for L2(Sn)]; > s.a. oscillator, wigner function.

Interaction Representation
* Idea: Both states and operators are time-dependent, with evolutions governed by different parts of the Hamiltonian:

H'I(t) I(t) = i I(t)/t ,   and   i AI(t)/t = [AI(t), H0] ,

where H = H0 + H', a free and an interaction part, and the relationship with Schrödinger representation quantities is

I(t) = exp{iH0t/} S(t) ,   and   H'I(t) = exp{iH0t/} H'S exp{–iH0t/} .

* Advantage: Takes into account only the nontrivial evolution of states; The free H evolves the operators.
* Disadvantage: Does not exist in general for a relativistic theory.
* Remark: This representation is often implicitly used in ordinary quantum mechanics, when ignoring "the rest of the world".

Other Representations and Related Topics > s.a. formalism [operator ordering]; tests of quantum mechanics.
* Polymer representation: The name given to one of four related non-regular representations of the Heisenberg algebra, in which the spectrum of the configuration or the momentum variable is not continuous, and the corresponding infinitesimal generator is not defined.
@ Probability representation, tomography: Wootters FP(86); Man'ko et al PRA(98)qp [Green's functions], JPA(03) [identical particles], PLB(98)ht [in quantum field theory], PLA(06), qp/06-in [in abstract Hilbert spaces], qp/06/JMP; Weigert PRL(00)qp/99, qp/99 ["expectation-value representation" for spins, similar?]; Howard & March PLA(06) [and momentum density]; Caponigro et al FdP(06)qp; Man'ko et al qp/06 [bibliography]; > s.a. entropy, quantum states [reconstruction], in quantum field theory and in quantum gravity, wigner functions.
@ Polymer representation: Fredenhagen & Reszewski CQG(06)gq; Corichi et al CQG(07)gq/06, PRD(07)-a0704; Chiou CQG(07)gq/06 [and Galileo group]; Husain et al PRD(07)-a0707 [and Coulomb potential]; > s.a. fock space; gas; types of quantum field theories.
@ Other representations: Floyd qp/03-in [trajectory representation, high-energy limit]; Torres-Vega PRA(07) [energy-time].

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