Weyl calculus

Weyl–Hörmander calculus

In Hamiltonian mechanics over a phase space $\mathbf{R} ^ { 2 n }$, the Poisson bracket $\\{ f , g \\}$ of two smooth observables $f : \mathbf{R} ^ { 2 n } \rightarrow \mathbf{R}$ and $g : \mathbf R ^ { 2 n } \rightarrow \mathbf R$ (cf. also Poisson brackets) is the new observable defined by

\begin{equation*} \\{ f , g \\} = \sum \left( \frac { \partial f } { \partial p _ { j } } \frac { \partial g } { \partial q _ { j } } - \frac { \partial f } { \partial q _ { j } } \frac { \partial g } { \partial p _ { j } } \right). \end{equation*}

For a state $( p , q )$ of the phase space $\mathbf{R} ^ { 2 n }$, the momentum vector is given by $p = ( p _ { 1 } , \dots , p _ { n } )$, while $q = ( q _ { 1 } , \dots , q _ { n } )$ is the position vector. The Poisson brackets of the coordinate functions ${\bf p}_j$, ${\bf q} _ { k }$ are given by

\begin{equation*} \\{ \mathbf{p} _ { j } , \mathbf{p} _ { k } \\} = \\{ \mathbf{q} _ { j } , \mathbf{q} _ { k } \\} = 0 , \quad \\{ \mathbf{p} _ { j } , \mathbf{q} _ { k } \\} = \delta _ { j k }. \end{equation*}

By comparison, in quantum mechanics over ${\bf R} ^ { n }$, the position operators $Q _ { j } = X _ { j }$ of multiplication by $\mathbf{q}_j$ correspond to the classical momentum observables $\mathbf{q}_j$ and the momentum operators $P _ { k }$ corresponding to the coordinate observables $\mathbf{p} _ { k }$ are given by

\begin{equation*} P _ { k } = \hbar D _ { k } = \frac { \hbar } { i } \frac { \partial } { \partial x _ { k } }. \end{equation*}

The canonical commutation relations

\begin{equation*} [ P _ { j } , P _ { k } ] = [ Q _ { j } , Q _ { k } ] = 0 , \quad [ P _ { j } , Q _ { k } ] = \frac { \hbar } { i } \delta _ { j k } I \end{equation*}

hold for the commutator $[ A , B ] = A B - B A$ (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 $\mathfrak{h} _ { n }$ over $\mathbf{R} ^ { 2 n + 1 }$. The Heisenberg group $\mathcal{H} _ { n }$ corresponding to the Lie algebra $\mathfrak{h} _ { n }$ is given on $\mathbf{R} ^ { 2 n + 1 }$ by the group law

\begin{equation*} ( p , q , t ) ( p ^ { \prime } , q ^ { \prime } , t ^ { \prime } ) = \end{equation*}

\begin{equation*} = \left( p + p ^ { \prime } , q + q ^ { \prime } , t + t ^ { \prime } + \frac { 1 } { 2 } ( p q ^ { \prime } - q p ^ { \prime } ) \right). \end{equation*}

Here, one writes $\xi a $ for the dot product $\sum \xi _ { j } a_ { j }$ of $\xi \in \mathbf{C} ^ { k }$ with the $k$-tuple $a = ( a _ { 1 } , \dots , a _ { k } )$ of numbers or operators. Set $\mathcal{D} = ( D _ { 1 } , \dots , D _ { n } )$ and $\mathcal{X} = ( X _ { 1 } , \dots , X _ { n } )$.

The mapping $\rho$ from $\mathcal{H} _ { n }$ to the group of unitary operators on $L ^ { 2 } ( \mathbf{R} ^ { n } )$ formally defined by $\rho ( p , q , t ) = e ^ { i ( p \mathcal D + q \mathcal X + t I ) }$ is a unitary representation of the Heisenberg group $\mathcal{H} _ { n }$. The operator $e ^ { i ( p {\cal D} + q {\cal X} + t I ) }$ maps $f \in L ^ { 2 } ( \mathbf{R} ^ { n } )$ to the function $x \mapsto e ^ { i t } e ^ { i p q / 2 } e ^ { i q x } f ( x + p )$.

If $\hat { f } ( \xi ) = \int _ { \mathbf{R} ^ { 2 n }} e ^ { - i x \xi } f ( x ) d x$ denotes the Fourier transform of a function $f \in L ^ { 1 } ( {\bf R} ^ { 2 n } )$, the Fourier inversion formula

\begin{equation*} f ( x ) = ( 2 \pi ) ^ { - 2 n } \int _ { {\bf R} ^ { 2 n } } e ^ { i x \xi } \hat { f } ( \xi ) d \xi \end{equation*}

retrieves $f$ from $\hat { f }$ in the case that $\hat { f }$ is also integrable.

Now suppose that $\sigma : \mathbf{R} ^ { 2 n } \rightarrow \mathbf{C}$ is a function whose Fourier transform $\hat { \sigma }$ belongs to $L ^ { 1 } ( {\bf R} ^ { 2 n } )$. Then the bounded linear operator $\sigma( \mathcal {D , X} )$ is defined by

\begin{equation*} ( 2 \pi ) ^ { - 2 n } \int _ { \mathbf{R} ^ { 2 n } } \rho ( p , q , 0 ) \hat { \sigma } ( p , q ) d p d q = \end{equation*}

\begin{equation*} = ( 2 \pi ) ^ { - 2 n } \int _ { \mathbf{R} ^ { 2 n } } e ^ { i ( p \mathcal{D} + q \mathcal{X} ) } \hat { \sigma } ( p , q ) d p d q. \end{equation*}

The Weyl functional calculus $\sigma \mapsto \sigma (\mathcal{D} , \mathcal{X} )$ was proposed by H. Weyl [a27], Section IV.14, as a means of associating a quantum observable $\sigma( \mathcal {D , X} )$ with a classical observable $\sigma$. Weyl's ideas were later developed by H.J. Groenewold [a11], J.E. Moyal [a18] and J.C.T. Pool [a22].

The mapping $\sigma \mapsto \sigma (\mathcal{D} , \mathcal{X} )$ extends uniquely to a bijection from the Schwartz space $\mathcal{S} ^ { \prime } ( \mathbf{R} ^ { 2 n } )$ of tempered distributions (cf. also Generalized functions, space of) to the space of continuous linear mappings from $\mathcal{S} ( \mathbf{R} ^ { n } )$ to $\mathcal S ^ { \prime } ( \mathbf R ^ { n } )$. Moreover, the application $\sigma \mapsto \sigma (\mathcal{D} , \mathcal{X} )$ defines a unitary mapping (cf. also Unitary operator) from $L ^ { 2 } ( \mathbf{R} ^ { 2 n } )$ onto the space of Hilbert–Schmidt operators on $L ^ { 2 } ( \mathbf{R} ^ { n } )$ (cf. also Hilbert–Schmidt operator) and from $L ^ { 1 } ( {\bf R} ^ { 2 n } )$ into the space of compact operators on $L ^ { 2 } ( \mathbf{R} ^ { n } )$. For $a , b \in \mathbf{C} ^ { n }$, the function $\sigma ( \xi , x ) = ( a \xi + b x ) ^ { k }$ is mapped by the Weyl calculus to the operator $\sigma ( \mathcal{D} , \mathcal{X} ) = ( a \mathcal{D} + b \mathcal{X} ) ^ { k }$. The monomial terms in any polynomial $\sigma ( \xi , x )$ are replaced by symmetric operator products in the expression $\sigma( \mathcal {D , X} )$. 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 $\sigma ( \xi , x )$ 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 $\mathbf{R} ^ { 2 n }$ and reasonably larger than the Heisenberg algebra $\mathfrak{h} _ { n }$ 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 $\sigma$ with the operator $\sigma ( \mathcal{D} , \mathcal{X} ) _ { \operatorname{KN} }$ given by

\begin{equation*} ( 2 \pi ) ^ { - 2 n } \int _ { \mathbf{R} ^ { 2 n } } e ^ { i q \mathcal{X} } e ^ { i p \mathcal{D} } \hat { \sigma } ( p , q ) d p d q, \end{equation*}

so that if $\sigma$ 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 ${\cal A} = ( A _ { 1 } , \dots , A _ { k } )$ is a $k$-tuple of operators acting in a Banach space $X$, with the property that for each $\xi \in \mathbf{R} ^ { k }$, the operator $i \xi A$ is the generator of a $C _ { 0 }$-group of operators such that for some $C > 0$ and $s \geq 0$, the bound $\| e ^ { i \xi A } \| \leq C ( 1 + | \xi | ) ^ { s }$ holds for every $\xi \in \mathbf{R} ^ { k }$. Then the bounded operator

\begin{equation*} f ( \mathcal{A} ) = ( 2 \pi ) ^ { - k } \int _ { \mathbf{R} ^ { k } } e^ { i \xi \mathcal{A} } \hat { f } ( \xi ) d \xi \end{equation*}

is defined for every $f \in S ( \mathbf{R} ^ { k } )$.

The operators $A _ { 1 } , \dots , A _ { k }$ do not necessarily commute with one another. Examples are $k$-tuples of bounded self-adjoint operators (cf. also Self-adjoint operator) or, with $k = 2 n$, the system of unbounded position operators $Q_{j}$ and momentum operators $P_{l}$ considered above (more accurately, one should use the closure $i \overline { \xi \mathcal{A} }$ of $i \xi A$ here).

By the Paley–Wiener–Schwartz theorem, the Weyl functional calculus $f \mapsto f ( \mathcal{A} )$ is an operator-valued distribution with compact support if and only if there exists numbers $C ^ { \prime } , s ^ { \prime } , r \geq 0$ such that

\begin{equation*} \| e ^ { i \zeta \cal A } \| \leq C ^ { \prime } ( 1 + | \zeta | ) ^ { s ^ { \prime } } e ^ { r | \operatorname { Im } \zeta | } \end{equation*}

for all $\zeta \in \mathbf{C} ^ { k }$. For a $k$-tuple $\mathcal{A}$ of bounded self-adjoint operators, M. Taylor [a23] has shown that the choice $C ^ { \prime } = 1$, $s ^ { \prime } = 0$ and $r ^ { 2 } = \sum \| A _ { j } \| ^ { 2 }$ 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 $A _ { 1 } , \dots , A _ { k }$ do not commute with each other, then the mapping $f \mapsto f ( \mathcal{A} )$ 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 $f \mapsto f ( \mathcal{A} )$ for analytic functions $f$ of $k$ 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 $k + 1$ real variables [a15].

Given a $k$-tuple $\mathcal{A}$ of matrices for which the matrix $\xi A$ has real eigenvalues for each $\xi \in \mathbf{R} ^ { k }$, the distribution $f \mapsto f ( \mathcal{A} )$ is actually the matrix-valued fundamental solution $E ( x , t )$ of the symmetric hyperbolic system

\begin{equation*} \left( I \frac { \partial } { \partial t } + \sum A _ { j } \frac { \partial } { \partial x _ { j } } \right) E = I \delta \end{equation*}

at time $t = 1$. 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[edit]

[a1] |  R.F.V. Anderson, "The Weyl functional calculus" J. Funct. Anal. , 4 (1969) pp. 240–267 MR0635128 Zbl 0191.13403

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[a2] |  R.F.V. Anderson, "On the Weyl functional calculus" J. Funct. Anal. , 6 (1970) pp. 110–115 MR0262857 Zbl 0196.14302
[a3] |  R.F.V. Anderson, "The multiplicative Weyl functional calculus" J. Funct. Anal. , 9 (1972) pp. 423–440 MR0301541 Zbl 0239.47010
[a4] |  M. Atiyah, R. Bott, L. Gårding, "Lacunas for hyperbolic differential operators with constant coefficients I" Acta Math. , 124 (1970) pp. 109–189 MR0470499 MR0470500 Zbl 0191.11203
[a5] |  M. Atiyah, R. Bott, L. Gårding, "Lacunas for hyperbolic differential operators with constant coefficients II" Acta Math. , 131 (1973) pp. 145–206 MR0470499 MR0470500 Zbl 0266.35045
[a6] |  E. Albrecht, "Several variable spectral theory in the non-commutative case" , Spectral Theory , Banach Centre Publ. , 8 , PWN (1982) pp. 9–30 MR738273
[a7] |  J. Bazer, D.H.Y. Yen, "The Riemann matrix of $( 2 + 1 )$-dimensional symmetric hyperbolic systems" Commun. Pure Appl. Math. , 20 (1967) pp. 329–363 MR240452
[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 MR0509838 Zbl 0167.10003
[a9] |  G.B. Folland, "Harmonic analysis in phase space" , Princeton Univ. Press (1989) MR0983366 Zbl 0682.43001
[a10] |  R.P. Feynman, "An operator calculus having applications in quantum electrodynamics" Phys. Rev. , 84 (1951) pp. 108–128 MR0044379 Zbl 0044.23304
[a11] |  H.J. Groenewold, "On the principles of elementary quantum mechanics" Physica , 12 (1946) pp. 405–460 MR0018562 Zbl 0060.45002
[a12] |  M.J. Gotay, H.B. Grundling, G.M. Tuynman, "Obstruction results in quantization theory" J. Nonlinear Sci. , 6 (1996) pp. 469–498 MR1411344 Zbl 0863.58030
[a13] |  L. Hörmander, "The Weyl calculus of pseudodifferential operators" Commun. Pure Appl. Math. , 32 (1979) pp. 359–443 MR0517939
[a14] |  L. Hörmander, "The analysis of linear partial differential operators" , III , Springer (1985) MR1540773 MR0781537 MR0781536 Zbl 0612.35001 Zbl 0601.35001
[a15] |  B. Jefferies, A. McIntosh, "The Weyl calculus and Clifford analysis" Bull. Austral. Math. Soc. , 57 (1998) pp. 329–341 MR1617328 Zbl 0915.47015
[a16] |  J.J. Kohn, L. Nirenberg, "An algebra of pseudodifferential operators" Commun. Pure Appl. Math. , 18 (1965) pp. 269–305 MR176362
[a17] |  V.P. Maslov, "Operational methods" , Mir (1976) MR0512495 Zbl 0449.47002
[a18] |  J.E. Moyal, "Quantum mechanics as a statistical theory" Proc. Cambridge Philos. Soc. , 45 (1949) pp. 99–124 MR0029330 Zbl 0031.33601
[a19] |  V.E. Nazaikinskii, V.E. Shatalov, B.Yu. Sternin, "Methods of noncommutative analysis" , Studies Math. , 22 , W. de Gruyter (1996) MR1460489 Zbl 0876.47015
[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 MR0412857 Zbl 0239.47011
[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 MR0204049 Zbl 0139.45903
[a23] |  M.E. Taylor, "Functions of several self-adjoint operators" Proc. Amer. Math. Soc. , 19 (1968) pp. 91–98 MR0220082 Zbl 0164.16604
[a24] |  M.E. Taylor, "Pseudodifferential operators" , Princeton Univ. Press (1981) MR0618463 Zbl 0453.47026
[a25] |  F. Treves, "Introduction to pseudodifferential and Fourier integral operators" , I , Plenum (1980) MR0597145 MR0597144 Zbl 0453.47027
[a26] |  V.A. Vassiliev, "Ramified integrals, singularities and lacunas" , Kluwer Acad. Publ. (1995) MR1336145 Zbl 0935.32026
[a27] |  H. Weyl, "The theory of groups and quantum mechanics" , Methuen (1931) (Reprint: Dover, 1950) MR0889677 Zbl 58.1374.01