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

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