Weyl calculus

Weyl–Hörmander calculus

In Hamiltonian mechanics over a phase space , the Poisson bracket of two smooth observables and (cf. also Poisson brackets) is the new observable defined by

For a state of the phase space , the momentum vector is given by , while is the position vector. The Poisson brackets of the coordinate functions , are given by

By comparison, in quantum mechanics over , the position operators of multiplication by correspond to the classical momentum observables and the momentum operators corresponding to the coordinate observables are given by

The canonical commutation relations

hold for the commutator (cf. also Commutation and anti-commutation relationships, representation of).

In both classical and quantum mechanics, the position, momentum and constant observables span the Heisenberg Lie algebra over . The Heisenberg group corresponding to the Lie algebra is given on by the group law

Here, one writes for the dot product of with the -tuple of numbers or operators. Set and .

The mapping from to the group of unitary operators on formally defined by is a unitary representation of the Heisenberg group . The operator maps to the function .

If denotes the Fourier transform of a function , the Fourier inversion formula

retrieves from in the case that is also integrable.

Now suppose that is a function whose Fourier transform belongs to . Then the bounded linear operator is defined by

The Weyl functional calculus was proposed by H. Weyl [a27], Section IV.14, as a means of associating a quantum observable with a classical observable . Weyl's ideas were later developed by H.J. Groenewold [a11], J.E. Moyal [a18] and J.C.T. Pool [a22].

The mapping extends uniquely to a bijection from the Schwartz space of tempered distributions (cf. also Generalized functions, space of) to the space of continuous linear mappings from to . Moreover, the application defines a unitary mapping (cf. also Unitary operator) from onto the space of Hilbert–Schmidt operators on (cf. also Hilbert–Schmidt operator) and from into the space of compact operators on . For , the function is mapped by the Weyl calculus to the operator . The monomial terms in any polynomial are replaced by symmetric operator products in the expression . Harmonic analysis in phase space is a succinct description of this circle of ideas, which is exposed in [a9].

Under the Weyl calculus, the Poisson bracket is mapped to a constant times the commutator only for polynomials of degree less than or equal to two. Results of Groenewold and L. van Hove [a9], pp. 197–199, show that a quantization over a space of observables defined on a phase space and reasonably larger than the Heisenberg algebra is not possible. A general discussion of obstructions to quantization may be found in [a12].

In the theory of pseudo-differential operators, initiated by J.J. Kohn and L. Nirenberg [a16], one associates the symbol with the operator given by

so that if is a polynomial, differentiation always acts first (cf. also Pseudo-differential operator; Symbol of an operator). For singular integral operators (cf. also Singular integral), the product of symbols corresponds to the composition operators modulo regular integral operators. The symbolic calculus for pseudo-differential operators is studied in [a25], [a24], [a14]. The Weyl calculus has been developed as a theory of pseudo-differential operators by L.V. Hörmander [a13], [a14].

The Weyl functional calculus can also be formulated in an abstract setting. Suppose that is a -tuple of operators acting in a Banach space , with the property that for each , the operator is the generator of a -group of operators such that for some and , the bound holds for every . Then the bounded operator

is defined for every .

The operators do not necessarily commute with one another. Examples are -tuples of bounded self-adjoint operators (cf. also Self-adjoint operator) or, with , the system of unbounded position operators and momentum operators considered above (more accurately, one should use the closure of here).

By the Paley–Wiener–Schwartz theorem, the Weyl functional calculus is an operator-valued distribution with compact support if and only if there exists numbers such that

for all . For a -tuple of bounded self-adjoint operators, M. Taylor [a23] has shown that the choice , and is possible.

The Weyl calculus in this setting has been developed by R.F.V. Anderson [a1], [a2], [a3], E. Nelson [a20], and E. Albrecht [a6]. The last two authors provide the connection with the heuristic time-ordered operational calculus of R.P. Feynman [a10] developed in his study of quantum electrodynamics.

A combination of the Weyl and ordered functional calculi is studied in [a17] and [a19].

If the operators do not commute with each other, then the mapping need not be an algebra homomorphism and there may be no spectral mapping property, so the commonly used expression "functional calculus" is somewhat optimistic.

For the case of bounded operators, the Weyl functional calculus for analytic functions of real variables can also be constructed via a Riesz–Dunford calculus by replacing the techniques of complex analysis in one variable with Clifford analysis in real variables [a15].

Given a -tuple of matrices for which the matrix has real eigenvalues for each , the distribution is actually the matrix-valued fundamental solution of the symmetric hyperbolic system

at time . The study of the support of the Weyl calculus for matrices is intimately related to the theory of lacunas of hyperbolic differential operators and techniques of algebraic geometry [a21], [a7], [a8], [a4], [a5], [a26].

References

[a1]	R.F.V. Anderson, "The Weyl functional calculus" J. Funct. Anal. , 4 (1969) pp. 240–267
[a2]	R.F.V. Anderson, "On the Weyl functional calculus" J. Funct. Anal. , 6 (1970) pp. 110–115
[a3]	R.F.V. Anderson, "The multiplicative Weyl functional calculus" J. Funct. Anal. , 9 (1972) pp. 423–440
[a4]	M. Atiyah, R. Bott, L. Gårding, "Lacunas for hyperbolic differential operators with constant coefficients I" Acta Math. , 124 (1970) pp. 109–189
[a5]	M. Atiyah, R. Bott, L. Gårding, "Lacunas for hyperbolic differential operators with constant coefficients II" Acta Math. , 131 (1973) pp. 145–206
[a6]	E. Albrecht, "Several variable spectral theory in the non-commutative case" , Spectral Theory , Banach Centre Publ. , 8 , PWN (1982) pp. 9–30
[a7]	J. Bazer, D.H.Y. Yen, "The Riemann matrix of -dimensional symmetric hyperbolic systems" Commun. Pure Appl. Math. , 20 (1967) pp. 329–363
[a8]	J. Bazer, D.H.Y. Yen, "Lacunas of the Riemann matrix of symmetric-hyperbolic systems in two space variables" Commun. Pure Appl. Math. , 22 (1969) pp. 279–333
[a9]	G.B. Folland, "Harmonic analysis in phase space" , Princeton Univ. Press (1989)
[a10]	R.P. Feynman, "An operator calculus having applications in quantum electrodynamics" Phys. Rev. , 84 (1951) pp. 108–128
[a11]	H.J. Groenewold, "On the principles of elementary quantum mechanics" Physica , 12 (1946) pp. 405–460
[a12]	M.J. Gotay, H.B. Grundling, G.M. Tuynman, "Obstruction results in quantization theory" J. Nonlinear Sci. , 6 (1996) pp. 469–498
[a13]	L. Hörmander, "The Weyl calculus of pseudodifferential operators" Commun. Pure Appl. Math. , 32 (1979) pp. 359–443
[a14]	L. Hörmander, "The analysis of linear partial differential operators" , III , Springer (1985)
[a15]	B. Jefferies, A. McIntosh, "The Weyl calculus and Clifford analysis" Bull. Austral. Math. Soc. , 57 (1998) pp. 329–341
[a16]	J.J. Kohn, L. Nirenberg, "An algebra of pseudodifferential operators" Commun. Pure Appl. Math. , 18 (1965) pp. 269–305
[a17]	V.P. Maslov, "Operational methods" , Mir (1976)
[a18]	J.E. Moyal, "Quantum mechanics as a statistical theory" Proc. Cambridge Philos. Soc. , 45 (1949) pp. 99–124
[a19]	V.E. Nazaikinskii, V.E. Shatalov, B.Yu. Sternin, "Methods of noncommutative analysis" , Studies Math. , 22 , W. de Gruyter (1996)
[a20]	E. Nelson, "A functional calculus for non-commuting operators" F.E. Browder (ed.) , Functional Analysis and Related Fields: Proc. Conf. in Honor of Professor Marshal Stone (Univ. Chicago, May (1968) , Springer (1970) pp. 172–187
[a21]	I. Petrovsky, "On the diffusion of waves and lacunas for hyperbolic equations" Mat. Sb. , 17 (1945) pp. 289–368 (In Russian)
[a22]	J.C.T. Pool, "Mathematical aspects of the Weyl correspondence" J. Math. Phys. , 7 (1966) pp. 66–76
[a23]	M.E. Taylor, "Functions of several self-adjoint operators" Proc. Amer. Math. Soc. , 19 (1968) pp. 91–98
[a24]	M.E. Taylor, "Pseudodifferential operators" , Princeton Univ. Press (1981)
[a25]	F. Treves, "Introduction to pseudodifferential and Fourier integral operators" , I , Plenum (1980)
[a26]	V.A. Vassiliev, "Ramified integrals, singularities and lacunas" , Kluwer Acad. Publ. (1995)
[a27]	H. Weyl, "The theory of groups and quantum mechanics" , Methuen (1931) (Reprint: Dover, 1950)