Symplectic Radon Transform and the Metaplectic Representation.

We study the symplectic Radon transform from the point of view of the metaplectic representation of the symplectic group and its action on the Lagrangian Grassmannian. We give rigorous proofs in the general setting of multi-dimensional quantum systems. We interpret the Radon transform of a quantum state as a generalized marginal distribution for its Wigner transform; the inverse Radon transform thus appears as a "demarginalization process" for the Wigner distribution.

1. Introduction

The idea of using what is today called the “Radon transform” to reconstruct a function from partial data goes back to the 1917 work [1] by the Austrian mathematician Johann Radon. While Radon originally only considered two or three-dimensional systems (in which case it is called the “X-ray transform”), the theory has since then been generalized to arbitrary Euclidean spaces. The applications of the Radon transform to quantum mechanics and optics have been developing rapidly these last years (see the paper [2] by Vogel and Risken). Using an approach from Mancini, Man’ko, and Tombesi [3], several authors [4,5,6,7,8,9,10] study what they call the “symplectic Radon transform” of a mixed quantum state; in a recent paper [11], some of these results are extended to the framework of algebras. The aim of the present paper is to give a simple rigorous approach to the theory of the symplectic Radon transform in several degrees of freedom. For this, we will use systematically the theory of the metaplectic group as developed in our previous work [12], together with the elementary theory of Lagrangian subspaces of the standard symplectic space. Our main observation is the following: integration of the Wigner transform (of a function, or of a state) along the x and p coordinate planes give the correct probability distributions and in position and momentum space). However, these are not sufficient to reconstruct the state (this is an aspect of the “Pauli problem”, see Section 4). However, applying a metaplectic transform associated with a symplectic rotation U to the state transforms the Wigner transform following the rule (this is the “symplectic covariance property” of the Wigner transform, which is well-known in harmonic analysis, especially in the Weyl–Wigner–Moyal approach to quantum mechanics). This use of metaplectic transforms leads, by calculating the marginals of along the x and p coordinate planes to infinitely many probability distributions, and these allow the reconstruction of the state. More precisely, we will redefine the Radon transform as being given by the formula where and is the metaplectic operator associated with the symplectic rotation where are square matrices with and . We thereafter prove (under suitable conditions on ) the inversion formula which reduces for to the usual inversion formula found in the literature (the case is discussed in Section 2 to motivate the results in the general case). As an illustration, we apply our constructions to the generalized Gaussian states in Section 3, which gives us the opportunity to shortly discuss the Pauli problem for Gaussians, thus generalizing results in [13].

This paper is structured as follows:

In Section 2, we study the “easy case” of systems with one degree of freedom; this allows us to present the main ideas of this paper without using too much new terminology (for instance, any line through the origin of the phase plane is a Lagrangian subspace).

In Section 3, we deal with the multi-dimensional case and give a working definition of the symplectic Radon transform from the point of view of operator theory, with a particular emphasis on the metaplectic representation of the symplectic group. We prove an inversion formula allowing the reconstruction of a state from the knowledge of its Radon transform; in particular, the case of Gaussian states is discussed, which allows to give new insight in the old “Pauli problem”.

For the reader’s convenience, we have included two appendices where we shortly review the main properties of the metaplectic group (Appendix A) and of the Lagrangian Grassmannian (Appendix B).

2. The Case n = 1

Let be a mixed quantum state with one degree of freedom: is a positive-semidefinite trace class operator on with trace . In view of the spectral theorem, there exists a sequence with and a sequence of non-negative numbers with such that where is the orthogonal projection of onto the ray . By definition [14,15], the Wigner distribution of is the convex sum where is the usual Wigner transform of , defined for by

Recall [14,15,16] that the marginal properties make sense provided that and its Fourier transform are, in addition to being square integrable, absolutely integrable: .

In most texts studying the tomographic picture of quantum mechanics, the symplectic Radon transform of the quantum state is defined by the integral where a and b are real numbers, and it is claimed that the following essential reconstruction formula holds

The following result is simultaneously a rigorous restatement and a justification of these formulas. It will be extended to the case of quantum states with an arbitrary number n of freedom in the forthcoming sections. Among other things, we see that the inverse Radon transform can be viewed as a “demarginalization process” [17] for the Wigner distribution.

We will use the following notation: we set where ; clearly, is a rotation in the plane.

Let

The Radon transform where

The inverse Radon transform is given by the formula:

The Radon transform of ψ is given by the line integral where

It is sufficient to assume that is a pure state, that is, for some . (i) Let us make the change of variables in the integral (5). This leads to the expression

Since , this can be rewritten

In view of the symplectic covariance property [12,16,18] of the Wigner transform, we have where is any one of the two metaplectic operators (see the Appendix A) covering U and hence (12) yields hence Formula (7) using the marginal properties (4). (ii) Let us denote A the right-hand side of the equality (8). Using the first marginal property (4), we have

Replacing X with and using the symplectic covariance property (13), we get

Setting and (and hence ), we have so that

In view of the Fourier inversion formula, written formally as we thus have which was to be proven. (iii) It is sufficient to show that (9) holds for one parametrization. In view of Formula (11), we have that is, since , and replacing v with t,

Set now and . Then, and ; hence, (17) implies (9). □

The first part of the theorem above uses the physicist’s definition (

3. The Multivariate Case

For , we consider equipped with its standard symplectic structure, which is defined by

We denote by the symplectic group of and by its unitary representation of its double cover (the metaplectic group; see Appendix A).

3.1. Definitions

Let be two real square matrices with such that and . Setting and noting that is invertible, we have the factorization where is a symplectic rotation. Note that its inverse is

The symplectic Radon transform of ψ is the transformation

defined, for where

is any of the two elements of

That maps into is clear, since is unitary on .

When , the metaplectic operator is defined by after one has made a choice of the integer m modulo 4 (see Appendix A).

Recall that in the multi-dimensional case, the Wigner transform of is given by the integral and that the marginal properties generalizing (4) hold for ; the Fourier transform is here given by

3.2. The Radon Inversion Formula

We have the following straightforward generalization of the inversion result (ii) in Theorem 1:

Viewing A and B as elements of where

It goes exactly as the proof of Theorem 1 (ii). Let us denote by W the integral in the right-hand side of (26). We have, using the first marginal condition (24), that is, by Formula (19):

Setting and , we have and hence

Integration of the exponential with respect to the variables A and B yields and hence which was to be proven. □

3.3. Interpretation as Generalized Marginals

Let us return to the definition (20) of the Radon transform:

If we choose and , this reduces to the formula similarly, if and , we get definition (20) reduces to the formula showing that the Radon transform is essentially a margin property for the “rotated” Wigner transform of . In fact, we can view as the surface integral of the Wigner transform on the (affine) Lagrangian plane (see Appendix B)

In view of the marginal properties and the symplectic covariance of the Wigner transform, followed by the change of variables , we have that is, explicitly,

Using the multi-parametrization we have we have , so we can interpret the formula above as a surface integral where is the Lebesgue measure on .

4. Radon Transform of Generalized Gaussians

4.1. Generalized Gaussians

By generalized (centered) Gaussian, we mean a function where V and W are real symmetric matrices with (i.e., positive definite). Such centered Gaussians are generalizations of the usual squeezed coherent states appearing in the physical literature; see [12,18]. The function is normalized to unity: and its Wigner transform is given by where G is the symmetric and symplectic matrix

That easily follows from the observation that where clearly is symplectic. Let be the set of all centered Gaussians (28); a central (and often implicitly used) result is that the metaplectic group acts transitively on : we have an action which is totally described by the action of on the fiducial coherent state [18] whose Wigner transform is

Using the symplectic covariance of the Wigner transform, it is not difficult to prove that if covers the symplectic matrix , then where and

4.2. The Radon Transform of

Let us calculate the Radon transform of using Formula (20). For this, we have to determine , where covers the symplectic rotation

The most natural (and easiest) way to determine is to use the symplectic covariance formula of the Wigner transform; it yields, taking (29) into account and using the relation

To explicitly determine is a rather lengthy (though straightforward) calculation; however, since we have an action (32), we know (by (30)) that there will exist and such that corresponding to the Gaussian

Now, since it is only the (squared) modulus of which appears in (35), we will have so that it suffices to determine , whose inverse is the lower right block of . This is easily done using the relation and one finds, after a few calculations and simplifications,

Summarizing, we have, after insertion in (35),

Suppose for instance is a “squeezed coherent state”; then, and is

Its Radon transform is then

4.3. Application: The Pauli Problem

When , the Formula (38) takes the simple form (replacing with scalars ) where

Let us apply this formula to the “Pauli problem”. This problem goes back to the question Pauli asked in [19], whether the probability densities and uniquely determine the wavefunction . The answer is “no”: consider in fact the Gaussian wavepacket whose Fourier transform of is given by where is an unessential constant real phase. Thus, and these relations imply the knowledge of and of but not of the covariance (the latter can actually be determined up to a sign using the fact that saturates the Robertson–Schrödinger uncertainty principle: we have ). Let us calculate the Radon transform of using Formula (43). We have here , ; hence, Formula (43) becomes

Notice that, as expected, and, using the relation ,

These relations show why we cannot recover the state using the two radon transforms and : none of them allow us to determine the covariance . However, it suffices with one clever choice of the parameters a and b in Formula (47). Suppose indeed that we have measured, for some values of the parameters a and b, the positive quantity appearing in the exponent of (47). Assuming that the variances and are known, we can find the covariance as follows: viewing (48) as a quadratic equation in the unknown , we demand that this equation has exactly one real root. This imposes the relation and reduces (48) to the equation from which is unambiguously determined.

5. Concluding Remarks

In this paper, we outlined a novel approach to the symplectic Radon transform, which we believe is conceptually very simple once one has realized the fundamental role played in quantum mechanics by the metaplectic representation of the symplectic group. As we have discussed elsewhere [20] a few years ago, this is the shortest bridge between classical (Hamiltonian) mechanics, and its refinement, quantum mechanics. This being said, our approach is somewhat sketchy, since we have not characterized the classes of functions (or states) to which we can apply the radon transform, limiting ourselves, for simplicity, to the square integrable case. It is however well-known (at least by people belonging to the harmonic analysis community) that there are function spaces invariant under metaplectic transformations which are larger than the space of square integrable functions. We are, among other possibilities, thinking about Feichtinger’s modulation spaces [21], which are a very flexible tool for creating quantum states, and which can be used together with Shubin’s pseudodifferential calculus [22] to extend the theory (one could, for instance, envisage the reconstruction of general observables along these lines).

We will definitely come back to these topics in a near future.