Canonical Approach to Quantum Theory| 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 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 [curvilinear coordinates, invariant quantization procedure];
Gallone 15;
Cetto et al a2011 [operator formalism, physical basis].
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) | 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 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 AP(17)-a1704 [canonical transformations and functional integrals];
Mannheim a2001-proc
[light-front and time-instant quantization].
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];
Klauder a2006
[favored classical variables to promote to operators].
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];
Andrade e Silva & Jacobson a2011 [particle on S2 with magnetic flux]; > 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.
* Entanglement Hamiltonian: Given a composite
system S of two subsystems A and B, the entanglement Hamiltonian
for subsystem A is the operator HA
such that the reduced density matrix ρA
= trB ρS
can be written as ρA
= exp{−HA}.
@ 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 AP(14)-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];
Pourjafarabadi et al a2012
[entanglement Hamiltonian for interacting systems];
> s.a. Phase [for quantum states].
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