Canonical Approach to Quantum Theory  

In General > s.a. formulations of quantum theory.
* Physically: Translate into quantum mechanics language the common distinction between initial conditions and evolution laws.
* Postulates: 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 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 (a pure state if = | |); The expectation value of an observable A is A = tr(A)/tr(); Usually normalized to tr = 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, 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) + Q k+l–1, (2) [Q(W), Q(Z)] = i Q({W, Z}) + 2 Q k+l–2, with k and l the degrees of W and Z (corrections are necessary – see 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 Howe theorem].

Steps > s.a. geometric quantization.
(1) Identify the classical manifold of states or phase space (e.g., the cotangent bundle T* of a classical configuration space ), and the regions of in which the system can be localized (e.g., Borel sets (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) | 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 = T*, the usual choice is L2(, d), for some measure ; Otherwise, can use densities of weight 1/2 on phase space, with a choice of polarization; In the -dimensional case, needs to be extended to a suitable quantum configuration space 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 , 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 to something else, that gives it a continuous spectrum);
(7) Find spectra of operators and interpret probabilistically.

Specific Theories > see klein-gordon quantum field theory; quantum gauge theory; quantum gravity: supergravity.

Technical Issues > s.a. first-class and second-class constraints; formulations; Parity; representation.
* 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, that 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.
@ Operator ordering: Cohen JMP(70), Dowker JMP(76) [for Hamiltonian, and path integrals]; Crehan JPA(89); issue RNC(88)#11.
@ Other ambiguities: Calogero & Degasperis AJP(04)sep [classically equivalent H's].
@ 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 a0705-JFA [based on resolvent algebra].
@ Examples: Zainuddin PRD(89) [particle on T2 in B field]; Bojowald & Strobl JMP(00)qp/99 [S1 × R+]; Brau JPA(99)qp [and H atom]; Benavides & Reyes-Lega a0806 [particle on S2 and projective plane]; > s.a. canonical quantum gravity; parametrized theories; quantum black holes; quantum gauge theories; sigma models.

Other Variations > s.a. Phase [for quantum states]; deformation quantization; geometric quantization; modified quantum mechanics.
@ Overviews: Doebner et al RVMP(01)mp [topological aspects, and Borel quantization]; Arbatsky mp/05 [intro].
@ Discretizations: Husain & Winkler CQG(04)gq/03 [consistent].
@ Related topics: Casalbuoni NCA(76) [anticommuting variables]; Balachandran et al NPB(87) [wave functions as functions on U(1) bundle over ]; Fukuyama & Kamimura PRD(90) [complex action]; Bojowald & Strobl JMP(00)qp/99, IJMPD(03)qp/99 [projection quantization]; 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].

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