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 \(\langle A \rangle = {\rm tr}(\rho A)\) (if tr\(\rho\) 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 AP(17)-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];
Müller a1903 [on the von Neumann rule].

@ __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.

* __Coherent quantization__: A generalization
of group quantization, in which groups are not assumed to be compact, locally compact, or
finite-dimensional, which makes this setting suitable for application to quantum field 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 ,
a1811-proc [intro, examples];
> s.a. Affine Quantization;
quantum field theory approaches.

@ __Covariant canonical quantization__:
Basu PRD(05) [and perturbation theery];
Liebrich et al a1907 [finite vacuum energy].

@ __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];
Neumaier & Farashahi a1809 [coherent quantization];
> 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];
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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