Representations in Quantum Theory  

In General > s.a. wigner functions [phase-space representations].
* 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, Van Hove theorem, GNS construction.
@ References: Shewell AJP(59)jan [operator ambiguities]; de la Torre AJP(02)mar-qp/02 [including aX+(1–a)P, \(1\over2\)(XP+PX)]; Halvorson SHPMP(04)qp/01 [and complementarity]; Vourdas JPA(06) [analytic, rev]; Bracci & Picasso AJP(07)mar [inequivalent, Aharonov-Bohm effect example]; Blood a1310 [what kets represent]; Stepanian & Kohandel a1312 [unitarily inequivalent representations]; de Gosson a1404 [inequivalence of the Schrödinger and Heisenberg pictures].

Schrödinger / Configuration-Space Representation > s.a. quantum field theory.
* Idea: The representation on L2(\(\cal C\), dμ), where \(\cal C\) is the configuration space, in which states are time-dependent, observables time-independent.
* Use: It is 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(\(\cal C\), dμ) in which states are time-independent, and observables time-dependent operators.
* Relationships: Compared to the Schrödinger representation,

ψH = exp{iHt/\(\hbar\)} ψS(t),   and   AH(t) = exp{iHt/\(\hbar\)} AS exp{–iHt/\(\hbar\)};

time evolution of the operators is given by i\(\hbar\) ∂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); Nikolić PLA(04)qp/03; de la Madrid qp/05-conf [for unbounded operators, and rigged Hilbert space]; Solomon a0706 [in quantum field theory]; Aharonov a1303-fs [and non-local quantum phenomena]; Partovi a1305 [and Eulerian vs Lagrangian descriptions of fluid dynamics].

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

ψ(x) \(\mapsto\) (Ctψ)(z):= dx ρt(zx) ψ(x) .

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

\(\langle\)ψ|φ\(\rangle\):= π–1 ψ*(z) φ(z) exp{–z*z} d2z ;
a ψ(z) = z ψ(z)   and   a ψ(z) = (∂/∂z) ψ(z) .

* Generalizations: The transform can be generalized to functions on groups.
* For the simple harmonic oscillator: Define the complex variable as z:= (ωm/2\(\hbar\))1/2 q + i (2\(\hbar\)ωm)–1/2 p.
@ General references: Bargmann CPAM(61), PNAS(62); Hall CM-qp/99, CMP(02) [compact groups, geometric quantization]; Villegas-Blas JMP(02) [kernel of transform]; Hübschmann JGP(08)m.DG/06 [and holomorphic Peter-Weyl theorem]; Vourdas et al JPA(12)-a1111 [generalized]; Oeckl JMP(12)-a1109 [isomorphism with the Schrödinger representation, field theory]; Bergeron et al PLA(13) [equivalence to Weyl quantization].
@ Related representations: Ribeiro et al PRL(05), Ribeiro et al JPA(09)-a0809 [conjugate representation]; Parisio PTP(10)-a1003 [off-center coherent-state representation]; Viscondi et al a1510 [generalized coherent-state representation, semiclassical propagator].
@ Other systems: Kowalski & Rembieliński JMP(01)qp/00 [particle on S2]; 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. quantum oscillators; wigner functions.

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

H'I(t) ψI(t) = i\(\hbar\) ∂ψI(t)/∂t ,   and   i\(\hbar\) ∂AI(t)/∂t = [AI(t), H0] ,

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

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

* Advantage: Takes into account only the non-trivial 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".
@ In quantum field theory: Biswas a0807 [transformation to free fields not unitary].

Other Representations and Related Topics > s.a. fock space; non-commutative physics; Polymer Representation; tests of quantum mechanics.
* Tomography: A formulation of quantum mechanics without probability amplitudes, expressed entirely in terms of observable probabilities; In it quantum states are represented not by complex state vectors or density matrices, but by real "probability tables" or marginal distribution functions, whose time dependence is governed by a classical evolution equation.
@ 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-conf [in abstract Hilbert spaces], RPMP(08)qp/06; 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]; Kiukas et al PRA(09)-a0902; Ibort et al PS(09)-a0904 [intro]; Andreev et al JRLR-a0910 [for fermion fields]; Fuchs PiC-a1003, a1003 [quantum Bayesian viewpoint]; Ibort et al PLA(10)-a1004; Man'ko & Man'ko AIP(11)-a1102 [dynamical symmetries and entropic uncertainty relations]; Korennoy & Man'ko a1104 [propagator]; Man'ko & Ventriglia IJGMP(12)-a1111-conf [free particle motion, classical and quantum]; Ibort et al PS(11)-a1204 [C*-algebraic approach]; Man'ko & Man'ko AIP(12)-a1208 [tomographic entropic uncertainty relations]; Fedorov PLA(13) [Feynman integral and perturbation theory]; Aniello JPCS(13)-a1310 [evolution and semigroups]; Man'ko & Man'ko EPJWC(14)-a1403 [and Wigner functions]; Korennoy & Manko a1511 [gauge transformation of states]; López-Yela a1512-PhD; > s.a. entropy; quantum states [reconstruction]; in quantum field theory and in quantum gravity; wigner functions.
@ Weil representation: Gurevich & Hadani a0808 [in characteristic two].
@ Related topics: Floyd qp/03-proc [trajectory representation, high-energy limit]; Torres-Vega PRA(07) [energy-time]; Chmielowiec & Kijowski JGP(12)-a1002 [generalized, fractional Fourier transform]; Fuss & Filinkov a1406 [periodic quantum systems, Colombeau algebra of generalized functions]; Aerts & Sassoli de Bianchi a1504 [extended Bloch representation, interference and entanglement]; > s.a. formalism [operator ordering]; Stone-von Neumann Theorem; Superseparability; Weyl Algebra.

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