# Super Weyl transform and some of its properties.

[section] 1. Introduction

The classical Weyl transform was first introduced in [6] by Hermann Weyl arising in quantum mechanics. The theory of Weyl transform is a vast subject of remarkable interest both in mathematical analysis and physics. In the theory of partial differential equations Weyl operators have been studied as a particular type of pseudo-differential operators. They have proved to be a useful technique in a quantity of problems like elliptic theory, spectral asymptotics, regularity problems, etc [7].

There is so many operators in theoretical mathematics that have powerful analytic methods for achieving the relationships and proposed which are require. But in the Theoretical physics (or applied mathematics!) we deal with functions or equations that do not abbey ordinary laws.

As operators acting on [L.sup.2]([R.sup.n]), Weyl operators have been deeply investigated mainly in the case where the symbol is a smooth function belonging to some special symbol classes [1-2].

In the microlocal analysis we deal with the space of symbols which are infinitely differentiable functions and make it into Frechet space by means of seminorms [5]. But in the Physics observations we often deal with functions which vary in respect of time and thus make nets of functions. In [4] introduced a class of symbols that vary in respect of time and are integrable with respect to an arbitrary measure. By means of this class of symbols we can generalize the classical theory with supersymbols and supersigular pseudodifferential operators.

[section] 2. Preliminaries

Suppose N is a fixed natural number. The pseudodifferential operator (abb. [psi]DO) generates from [S.sup.m] symbols as follow [4]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

in which [psi](x, [xi]) [member of] [S.sup.m], such that for all ([alpha], [beta], n) [member of] [Z.sup.N.sub.+] x [Z.sub.+],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One of the advantage operators which is used in the quantum meachanics is Wigner transform [7]. Let f and g be in the Schwartz space S([R.sup.N]). Then the W(f,g) on the [R.sup.2N], is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

which is called the Wigner transform of f and g.

In addition to usefulness of the Wigner transform, the other application of that, is its beautiful relationship with one of the most important operator in the quantum machanics, i. e. Weyl transform,

Definition 2.1. Suppose that [psi] lies in [S.sup.m]. Then the linear operator [W.sub.[psi]]defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

is the Weyl transform of the function f [member of] S([R.sup.N]).

Now we want to define a class of [psi]DOS such that be more general and applicable in physics phenomena.

Definition 2.2. Given an arbitrary measure [sigma] on [R.sup.N]. If [psi] : [R.sup.N] [right arrow] [S.sup.m] (m [member of] R), [psi] is said to be supersymbol if for all ([alpha], [beta], n) [member of] [Z.sup.N.sub.+] x [Z.sub.+],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The class of such [psi] is denoted by S[S.sup.m].

Each supersymbol, regarded together with the measure generates a supersingular pseudodifferential operator (abb. s[psi]DO) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As the trivial case when [sigma] is the unit measure [delta](t) supported at origin, T([psi], [sigma]) is the pseudodofferential operator OP[psi](0). As usual, notations [OPSS.sup.m]([sigma]) and [OPSS.sup.[infinity]] etc. Stand for the space of operators generated by the corresponding space of supersymbols, i.e. of [SS.sup.m]([sigma]), [SS.sup.-[infinity]] [equivalent to] [[intersection].sub.m] [SS.sup.m]. It is easy to check that for T([psi], [sigma]) we can rewrite it as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This technique is very useful for generalization of the differential operators to operators for an arbitrary measure space. So we can define a general Wigner transform that is integrated with an arbitrary measure [sigma] on RN as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where x,t,[xi] [member of][R.sup.N].

Now we are ready for introducing super Weyl transform that is more general and that will extend the classical theory and make a framework of operators that will be useful in theoretical physics and applied mathematics.

[section] 3. Super Weyl transform

Definition 3.1. Let [psi](t) be a super symbol and [sigma] be an arbitrary measure on [R.sup.N]. For both functions f and g in the Schwartz space, the integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

is called the super Weyl transform of f and g.

In the next theorem we will illuminate the relationship between the super Weyl transform and the generalized Wigner transform. The following lemma is needed.

Lemma 3.2. If [theta] is in [C.sup.[infinity].sub.0] ([R.sup.N]), such that [theta](0) = 1, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

exists and is independent of the choice of the function [theta]. Moreover, the convergence is uniform with respect to x on [R.sup.N].

Hint: Note that for any positive integer L,

[(1 - [[DELTA].sub.y]).sup.L]{[e.sup.i(x-y-t)x[xi]]} = (1 + [absolute value of [xi]]}] = [(1 + [[absolute value of [xi]].sup.2)].sup.L] [e.sup.i(x-y-t)x[xi]].

Theorem 3.3. Let [psi](t) [member of] S[S.sup.m], m [member of] R, and [sigma] be an finite measure on [R.sup.N]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Let [theta] be any function in [C.sup.[infinity].sub.0] ([R.sup.N]) such that [theta](0) = 1. Then, by the Lemma, Lebesgue dominated convergence theorem, and Fubini's theorem,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let u = x - t + y/2 and v = x - y/2 in the last term, by Lemma, Fubini's Theorem and the Lebesgue dominated convergence theorem,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In classica mechanics, the phase space used to describe the motion of a particle moving in [R.sup.N] is given by

[R.sup.2N] = {(x,[xi]); x, [xi] [member of] [R.sup.N]},

where the variables x and [xi] are used to denote the position and momentum of the particle, respectively. The observables of the motion are given by real-valued tempered distributions on [R.sup.2N]. The rules of quantization, with Planck's constant adjusted to 1, say that a quantum-mechanical mode of the motion can be set up using the Hilbert space [L.sup.2] ([R.sup.N]) for the phase space, the multiplication operator on [L.sup.2]([R.sup.N]) by the function [x.sub.j] for the position variable [x.sub.j], and the differential operator [D.sub.j] for the momentum variable [[xi].sub.j].

References

[1] P. Boggiatto, E. Buzano, L. Rodino, Global Ellipticity and Spectral Theory, Mathematical Research, Akademie-Verlag, 92(1996).

[2] P. Boggiatto, L. Rodino, Quantization and pseudo-differential operators, Cubo Mathematica Educational 1, 5(2003).

[3] N. Kasumov, Pseudodifferential calculus for oscilating symbols, Osaka J. Math, 32(1995), 919-940.

[4] M. A. Shubin, Pseudodifferential operator and spectral theory, Moscow, 1978, English transl., Springer-Verlag, 2010.

[5] M. Taylor, Pseudodifferential operator, Princeton University Press, Princeton, N. J, 1981.

[6] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1950.

[7] M. W. Wong, Weyl Transforms, Springer-Verlag, New York, 1998.