Canonical Approach to Quantum Theory |

**In General** > s.a. formulations of
quantum theory; hilbert space.

* __Idea__: The Hilbert-space representation;
Translates into quantum mechanics language the distinction between initial conditions
and evolution laws.

* __Postulates__: They can be
reduced to (1) A dynamical variable corresponds to a linear operator on a
Hilbert space; The spectrum of the operator gives its possible
measured values, and the operator must be Hermitian if the observable is real; and (2)
A state corresponds to a Hermitian, non-negative operator or density matrix *ρ*
(a pure state if *ρ* = |*ψ*\(\rangle\langle\)*ψ*|);
The expectation value of an observable *A* is \(\langle\)*A*\(\rangle\) =
tr(*ρA*) (if tr *ρ* is normalized to 1).

* __Based on Heisenberg algebra__:
We need, for each degree of freedom of the system, a pair of operators, e.g., *p* and *q*,
such that [*q*,* p*] = –i\(\hbar\),
and self-adjoint operators associated with observables in the algebra generated
by *p* and *q*; However, the *q*s
and *p*s do not have to be observables themselves (although in many
common cases they are) – think of fermion fields, etc.

* __General formulation__:
Given an *n*-graded Poisson algebra *A* of
(gauge-invariant) observables, a map *Q* to an associative filtered
algebra *A*, such that (1)
*Q*(*W*) *Q*(*Z*) = *Q*(*WZ*)
+ \(\hbar\) *Q*^{ k+l–1},
(2) [*Q*(*W*),* Q*(*Z*)]
= i\(\hbar\) *Q*({*W*, *Z*})
+ \(\hbar\)^{2}* Q*^{ k+l–2},
with *k* and *l* the
degrees of *W* and *Z* (corrections are necessary – see
the Groenewold-van Hove theorem).

* __Problem__: Some states can be prepared, but we can never measure completely
any state, even approximately.

* __Result__: If *A* has positive support, its canonically conjugate
variable does not have a self-adjoint extension.

@ __References__: Landsman mp/01 [quantization as a functor];
Giulini LNP(03)qp [Groenewold-van Hove theorem];
Gudder a1011 [and decoherence functionals];
Błaszak & Domański AP(13)-a1305 [in curvilinear coordinates, invariant quantization procedure];
Gallone 15.

**Steps** > s.a. geometric quantization.

(1) Identify the classical manifold Γ of
states or phase space (e.g., the cotangent bundle T*\(\cal C\) of
a classical configuration space \(\cal C\)),
and the regions of Γ in
which the system can be localized (e.g., Borel sets
\(\cal B\)(*M*)).

(2) Choose a complete set of elementary observables or functions on *M*,
closed under Poisson brackets (commutation relations); For example, a complete
set of canonically conjugate pairs {(*q*^{i}, *p*_{i})
| *i* ∈ *I*}.

(3) Find a representation of the Poisson algebra on a complex vector space,
in which states are unit rays; This may require factor ordering and regularization;
If Γ = T*\(\cal C\),
the usual choice is L^{2}(\(\cal C\),
d*μ*), for some measure *μ*;
Otherwise, can use densities of weight 1/2 on phase space, with a choice of polarization;
In the infinite-dimensional case, \(\cal C\) needs
to be extended to a suitable quantum configuration space \(\cal C\)_{Q};
However, for a linear field theory, one usually
bypasses this by using a Fock space, and demands a unique Poincaré-invariant
ground state and compatibility of observables with the symmetries of the theory.

(4) If there are constraints which have not been eliminated by a reduced
phase space approach, define them as operators on this vector space, etc (> see constraint
quantization); Physical states are then those in the kernel.

(5) Define an inner product that makes the space of physical states into
a Hilbert space \(\cal H\),
such that real physical observables act as self-adjoint operators; Notice that
not all unit rays are always physical states (e.g.,
if there are superselection rules, or if some have infinite energy);

(6) Dynamics: Substitute the appropriate operators in the Hamiltonian *H*,
to get the Schrödinger equation (done by calculating the point spectrum of *H* and
enlarging \(\cal H\) to something
else, that gives it a continuous spectrum);

(7) Find spectra of operators and interpret probabilistically.

**For Quantum Field Theories** > s.a. klein-gordon
quantum field theory; quantum gauge theory;
quantum gravity:
supergravity.

@ __References__: Blasone et al a1704 [canonical transformations and functional integrals].

**Technical Issues** > s.a. first-class
and second-class constraints;
formulations; parity; symmetries [including reduction].

* __Ambiguities, pictures,
representations__: To do calculations in ordinary quantum mechanics one chooses
some "picture" and representation,
but these are examples of ambiguities, beginning with which algebra of observables
is the primary one; In classical theories canonical transformations lead to equivalent descriptions of the dynamics, but in quantum theory for infinite-dimensional Hilbert spaces –as in the case of field theories– the corresponding changes of representations in general lead to inequivalent theories;
The choice between them depends on the physical questions one wants
to ask, and figuring out how this dependence works is not that straightforward; > s.a. representations; Stone-Von Neumann Theorem.

@ __General references__: Lin & Jiang a1408 [decomposition of *H* into state-preserving + state-varying parts]; López et al a1608 [different quantum dynamical behavior from classically equivalent Hamiltonians].

@ __Operator ordering__: Cohen JMP(70),
Dowker JMP(76)
[for Hamiltonian, and path integrals]; Crehan JPA(89);
issue RNC(88)#11; Tagirov a1805.

@ __Related topics__: Calogero & Degasperis AJP(04)sep [classically equivalent Hamiltonians];
Bernatska & Messina PS(12) [Hamiltonians from time evolutions].

@ __And path integrals__: Mayes & Dowker JMP(73); Klauder AP(88); Gollisch & Wetterich
PRL(01).

@ __Choice of variables / algebra__: Kastrup PRA(06)qp/05 [angle-angular
momentum].

**Other Algebras, Group Quantization** > s.a. anomalies.

* __Idea__: The elementary variables
to promote to basic operators are not necessarily canonically conjugate pairs.

@ __General references__: Isham & Kakas CQG(84),
CQG(84);
Isham NPPS(87);
Rovelli NCB(87) [with constraints];
Navarro-Salas & Klauder CQG(90) + refs;
Navarro et al JMP(96)ht/95,
JMP(97)ht/96;
Varadarajan PRD(00)gq [quantum field theory, holonomies];
Buchholz & Grundling JFA-a0705 [based on resolvent algebra].

@ __Examples__: Zainuddin PRD(89) [particle on T^{2} in *B* field];
Bojowald & Strobl JMP(00)qp/99 [S^{1} × \(\mathbb R\)^{+}];
Brau JPA(99)qp [and H atom];
Benavides & Reyes-Lega in(10)-a0806 [particle on S^{2} and projective plane];
Aldaya et al RPMP(09) [non-linear sigma model, particle on S^{2}];
> s.a. canonical quantum gravity; parametrized theories;
quantum black holes; quantum gauge theories;
sigma models.

**Variations** > s.a.
deformation quantization; geometric quantization;
modified quantum mechanics; Precanonical Quantization;
relativistic quantum theory.

@ __Overviews__: Doebner et al RVMP(01)mp [topological aspects, and Borel quantization];

Arbatsky mp/05 [intro].

@ __Discretizations__: Husain & Winkler CQG(04)gq/03 [consistent].

@ __Enhanced quantization__: Klauder JPA(12)-a1204,
MPLA(14)-a1211,
a1308-conf,
15 [canonical and affine];
Klauder a1611,
a1702-proc,
a1710-conf [intro, examples];
> s.a. Affine Quantization;
quantum field theory approaches.

@ __Other variations__: Bojowald & Strobl JMP(00)qp/99,
IJMPD(03)qp/99 [projection quantization];
Gazeau & Bergeron a1308 [integral quantization];
Nisticò a1411-conf [group theoretical approach];
> s.a. quantum systems.

@ __Related topics__: Casalbuoni NCA(76) [anticommuting variables];
Balachandran et al NPB(87)
[wave functions as functions on a U(1) bundle over configuration space];
Fukuyama & Kamimura PRD(90) [complex action];
Tymczak et al PRL(98) [inner product];
Mauro PLA(03)qp [and Koopman-von Neumann classical mechanics];
Isidro ht/03 [projective – complex compact phase space];
Basu PRD(05) [covariant, and perturbations];
Bergeron et al a1102 [equivalent to coherent-state quantization];
Kauffmann FP(11)
[unambiguous quantization from maximum classical correspondence];
> s.a. Phase [for quantum states].

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