Representations in Quantum Theory| 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.
@ General 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].
@ Related topics: Fuss & Filinkov a1406 [periodic quantum systems, Colombeau algebra of generalized functions];
> s.a. Superseparability; Weyl Algebra.
Main types: see configuration-space based
representations [Schrödinger, Heisenberg, interaction pictures].
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) \(\mapsto\) (Ct ψ)(z):= ∫ dx ρt(z − x) 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.
Other Types of Representations
> s.a. fock space; momentum representation;
non-commutative physics; Polymer Representation.
* 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;
Korennoy & Man'ko IJTP(17)-a1610 [symplectic and optical joint probability distributions];
Man'ko et al a1905;
> s.a. entropy; quantum states [reconstruction];
in quantum field theory and in quantum gravity;
wigner functions.
@ Weil representation:
Gurevich & Hadani a0808 [in characteristic two].
@ Weyl representation:
Parthasarathy a1803 [and Lévy processes].
@ 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];
Aerts & Sassoli de Bianchi JMP(16)-a1504 [extended Bloch representation, interference and entanglement];
Chabaud et al PRL(20)-a1907 [stellar representation];
> s.a. formalism [operator ordering]; Stone-von Neumann
Theorem; tests of quantum mechanics.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 28 apr 2021