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 qs and ps 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 {(qi, pi) | iI}.
(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 L2(\(\cal C\), dμ), 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.
@ 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 T2 in B field]; Bojowald & Strobl JMP(00)qp/99 [S1 × \(\mathbb R\)+]; Brau JPA(99)qp [and H atom]; Benavides & Reyes-Lega in(10)-a0806 [particle on S2 and projective plane]; Aldaya et al RPMP(09) [non-linear sigma model, particle on S2]; > 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 [intro]; > s.a. approaches to quantum field theory.
@ 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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