nLab Weyl functional calculus

Idea

What is called Weyl quantization is a method of quantization applicable to symplectic manifolds which are symplectic vector spaces or quotients of these by discrete groups (tori).

In Weyl quantization of the flat space $\mathbf{R}^n$, the classical observables of the form $f(x,p)$ are replaced by suitable operators which in the case when $f$ is a polynomial correspond to writing $f$ with $x$ and $p$ replaced by noncommutative variables $x$ and $i h\frac{\partial}{\partial x}$ in symmetric or Weyl ordering. This means that all possible orderings between $x$ and $i h\frac{\partial}{\partial x}$ are summed with an equal weight. More generally, one can extend this rule to more general functions via integral formulas due Weyl and Wigner. This is also useful in fundations of the theory of pseudodifferential operators.

Related concepts

• free field theory

References

• Lars Hörmander, The Weyl calculus of pseudodifferential operators, Comm. Pure Appl. Math. 32 (1979), no. 3, 360–444. MR80j:47060, doi

• Robert F. V. Anderson, The Weyl functional calculus, J. Functional Analysis 4:240-267, 1969, MR635128;

• On the Weyl functional calculus, J. Functional Analysis 6:110–115, 1970, MR262857

• E. M. Stein, Harmonic analysis: real variable methods, orthogonality, and oscillatory integrals, Princeton University Press 1993

• M. W. Wong, Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator, Annals Global Anal. Geom. 28 (2005) 271–283

• Thomas L. Curtright, David B. Fairlie, Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific (2014) [doi:10.1142/8870]

Discussion of quantization of Chern-Simons theory in terms of Weyl quantization is in

• Jørgen Andersen, Deformation quantization and geometric quantization of abelian moduli spaces, Commun. Math. Phys., 255 (2005), 727–745

• Razvan Gelca, Alejandro Uribe, The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same, Commun.Math.Phys. 233 (2003) 493-512 (arXiv:math-ph/0201059)

• Razvan Gelca, Alejandro Uribe, From classical theta functions to topological quantum field theory (arXiv:1006.3252, slides pdf)

• Razvan Gelca, Alejandro Uribe, Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$ (arXiv:1007.2010)

Discussion of the generalization to BV-quantization is in

• Owen Gwilliam, Rune Haugseng, Linear Batalin-Vilkovisky quantization as a functor of ∞-categories (arXiv:1608.01290)