nLab canonical quantization

Idea

In quantum physics, the term canonical quantization refers, somewhat broadly, to quantization of (possibly constrained) classical mechanical systems regarded in their Hamiltonian formulation, via symplectic geometry of phase spaces. It proceeds, ideally, by construction of Hilbert spaces of quantum states on which algebras of quantum observables, $\hbar$-deforming the Poisson algebra of classical observables, are represented by linear operators (the Schrödinger picture of quantum mechanics).

Beware that the use of “canonical” in “canonical quantization” is quite different from the use of the word in mathematics, where it refers to preferred choices. In contrast, deformation quantization of Poisson algebras is generally beset with operator ordering ambiguities and (beyond purely formal deformation quantization) with obstructions known as quantum anomalies, whence the saying that …quantization is a mystery… (cf. here).

Instead, “canonical” here refers to the older terminology of canonical coordinates $q$ and canonical momenta $p$ in classical Hamiltonian mechanics (which of course are not really canonical, either), with their Poisson bracket $\{q,p\} = 1$ defined up to canonical transformations: In canonical quantization these are to be promoted to (multiplication) operators $\widehat{q}$ and (differentiation) operators $\widehat{p} (= -\mathrm{i}\hbar\partial/\partial q)$ satisfying the canonical commutation relation $[\widehat{q}, \widehat{p}] = \mathrm{i}\hbar$.

But in particular when applied to classical field theory, the existence of a Hamiltonian formulation crucially depends on a foliation of spacetime with spacelike leaves (or lightlike leaves for light cone quantization). While this exists at least on globally hyperbolic spacetimes, such a choice breaks manifest local Lorentz-symmetry and general covariance. Therefore canonical quantization is sometimes referred to in physics jargon as non-covariant.

This is in contrast primarily to methods of quantization based on the Lagrangian formulation of classical mechanics, such as notably path integral methods and their derivatives, like the construction of scattering amplitude S-matrices via Feynman perturbation series used traditionally in perturbative quantum field theory.

Indeed, canonical quantization lends itself to the more elusive ambition of non-perturbative quantum field theory.

The archetypical non-trivial example of non-perturbative canonical quantization of non-free field theories is the quantization of Chern-Simons theory specifically by geometric quantization of its phase space of flat connections, producing a modular functor worth of Hilbert spaces of states.

References

General

The origin of the idea of canonical quantization, deforming Poisson brackets to operator commutators, is due to

• P. A. M. Dirac: The Fundamental Equations of Quantum Mechanics, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 109, No. 752 (Dec. 1, 1925), pp. 642-653 [jstor:94441, pdf]

as recalled in:

• P. A. M. Dirac; around p. 124 of: Recollections of an Exciting Era, in: Proceedings of the International School of Physics “Enrico Fermi”, Course LVII, 1972— History of Twentieth Century Physics, Academic Press (1977) 109-146 [pdf]

For application in classical/quantum mechanics see most references listed there.

For instance:

• Moo Youn-Han: Canonical Quantization, chapter 3 in: From Photons to Higgs (2014) 17-21 [doi:10.1142/9789814579964_0004]

See also:

• Wikipdia: Canonical quantization

Canonical quantization of gauge theories

On canonical quantization of gauge theories with Gauss law constraint (predominantly of Yang-Mills theory/QCD and Maxwell theory/QED – for the quantization of Chern-Simons theory see also there):

• P. Hasenfratz, P. Hraskó: Canonical quantization of gauge theories, Phys. Rev. D 13 (1976) 2235 [doi:10.1103/PhysRevD.13.2235, pdf]

• Taichiro Kugo, Izumi Ojima, Manifestly Covariant Canonical Formulation of the Yang-Mills Field Theories. I: General Formalism, Progress of Theoretical Physics 60 6 (1978) 1869–1889 [doi:10.1143/PTP.60.1869]

• John L. Friedman, Nicholas J. Papastamatiou, On the canonical quantization of Yang-Mills theories, Nuclear Physics B 219 1 (1983) 125-142 [doi:10.1016/0550-3213(83)90431-5]

• A. Bassetto, I. Lazzizzera, R. Soldati, Yang-Mills theories in space-like axial and planar gauges, Nuclear Physics B 236 2 (1984) 319-335 [doi:10.1016/0550-3213(84)90538-8]

• D. M. Gitman, S. L. Lyakhovich & I. V. Tyutin, Canonical quantization of the Yang-Mills Lagrangian with higher derivatives, Soviet Physics Journal 28 (1985) 554–556 [doi:10.1007/BF00896182]

• Kurt Haller: Yang-Mills theory and quantum chromodynamics in the temporal gauge, Phys. Rev. D 36 (1987) 1839 [doi:10.1103/PhysRevD.36.1839]

• Marc Henneaux, Claudio Teitelboim; §19.1.1 in: Quantization of Gauge Systems, Princeton University Press (1992) [doi:10.2307/j.ctv10crg0r]

• P. E. Haagensen, On The Exact Implementation Of Gauss’ Law In Yang-Mills Theory [arXiv:hep-ph/9307319]

• Sarada G. Rajeev, O. T. Turgut, Poisson Algebra of Wilson Loops in Four-Dimensional Yang-Mills Theory, Int. J. Mod. Phys. A 10 (1995) 2479 [doi:10.1142/S0217751X95001194, arXiv:hep-th/9410053]

  (in light-front formalism)

• Jonathan Dimock: Canonical Quantization of Yang-Mills on a circle, Reviews in Mathematical Physics 08 01 (1996) 85-102 [doi:10.1142/S0129055X96000044]

• Alejandro Corichi, Introduction to the Fock Quantization of the Maxwell Field, Rev. Mex. Fis. 44 4 (1998) 402-412 [arXiv:physics/9804018]

• F. Lenz, K. Ohta, K. Yazaki: Canonical quantization of gauge fields in static space-times with applications to Rindler spaces, Phys. Rev. D 78 (2008) 065026 [doi:10.1103/PhysRevD.78.065026, arXiv:0803.2001]

• Daniel N. Blaschke, François Gieres, On the canonical formulation of gauge field theories and Poincaré transformations, Nuclear Physics B 965 (2021) 115366 [doi:10.1016/j.nuclphysb.2021.115366, arXiv:2004.14406]

• Aldo Riello, Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back, SciPost Phys. 10 125 (2021) [doi:10.21468/SciPostPhys.10.6.125]