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
L^{2}(\(\cal C\), d*μ*),
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
L^{2}(\(\cal C\),
d*μ*) in which states are time-independent, and observables time-dependent operators.

* __Relationships__: Compared to the Schrödinger representation,

*ψ*_{H} =
exp{i*Ht*/\(\hbar\)}
*ψ*_{S}(*t*), and *A*_{H}(*t*)
= exp{i*Ht*/\(\hbar\)} *A*_{S} exp{–i*Ht*/\(\hbar\)};

time evolution of the operators is given by i\(\hbar\) ∂*A*_{H}/∂*t* =
[*A*_{H}, *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 L^{2}(\(\mathbb C\),
exp{– |*z*|^{2}}
d*z* d*z**) (*ψ*(*z*)
is analytic), obtained from the usual L^{2}(\(\mathbb R\),d*x*)
using the heat kernel *ρ*_{t} by the transformation

*ψ*(*x*) \(\mapsto\) (*C*_{t}*ψ*)(*z*):=
∫ d*x* *ρ*_{t}(*z*–*x*)
*ψ*(*x*) .

* __Inner product, operators__: If *ψ*(*z*):=
∑_{n=0}^{∞}
(*n*!)^{–1/2}* z*^{n} \(\langle\)*n*|*ψ*\(\rangle\),
with |*a*\(\rangle\) an eigenvector of *a*, then

\(\langle\)*ψ*|*φ*\(\rangle\):= π^{–1}∫ *ψ**(*z*) *φ*(*z*)
exp{–*z***z*} d^{2}*z* ;

*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*:=
(*ω**m*/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 S^{2}]; Aldaya & Guerrero JPA(93)
[relativistic oscillator]; Ashtekar et al JFA(96)gq/94 [spaces
of connections];
Villegas-Blas JMP(06)
[for L^{2}(S^{n})]; > 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*) *ψ*_{I}(*t*)
= i\(\hbar\) ∂*ψ*_{I}(*t*)/∂*t*
, and i\(\hbar\) ∂*A*_{I}(*t*)/∂*t*
= [*A*_{I}(*t*),* H*_{0}]
,

where *H* = *H*_{0} + *H'*,
a free and an interaction part, and the relationship with Schrödinger
representation quantities is

*ψ*_{I}(*t*)
= exp{i*H*_{0}*t*/\(\hbar\)} *ψ*_{S}(*t*)
, and *H'*_{I}(*t*)
= exp{i*H*_{0}*t*/\(\hbar\)}
*H'*_{S} exp{–i*H*_{0}*t*/\(\hbar\)}
.

* __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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