Canonical-covariant Wigner function in polar form

T. Hakioğlu Vol. 17, No. 12 / December 2000 / J. Opt. Soc. Am. A 2411 Canonical-covariant Wigner function in polar form T. Hakioğlu Department of Physics, Bilkent University, 06533 Ankara, Turkey, and High Energy Physics Division, Argonne National Laboratories, Argonne, Illinois 60439-4815 Received March 21, 2000; revised manuscript received August 30, 2000; accepted September 5, 2000 The two-dimensional Wigner function is examined in polar canonical coordinates, and covariance proper- ties under the action of affine canonical transformations are derived. © 2000 Optical Society of America [S0740-3232(00)03912-0] OCIS codes: 080.0080, 070.0070, 270.0270, 000.3860, 000.1600. 1. INTRODUCTION which an arbitrary operator F( p̂, q̂) as a function of the canonical phase-space operators ( p̂, q̂) can be invertibly Phase space is a remarkable concept facilitating the gen- mapped to a classical phase-space function f( p, q). The eralized understanding of the transition between the clas- crucial property is that, if the basis operators are sym- sical and the quantum formulations and is principally built on proper sets of independent dynamical variables metrically ordered, the WWGM correspondence is covari- (the canonical coordinates) describing the considered ant under the action of ACT’s between the transformed physical system and symmetry transformations between operator F ⬘ ( p̂, q̂) ⬅ F( p̂ ⬘ , q̂ ⬘ ) and its transformed sym- them. The transformations induced on the phase space bol f ⬘ ( p, q) ⬅ f( p ⬘ , q ⬘ ), where the phase-space variables are said to be canonical if the equations of motion are ( p, q) and the phase-space operators ( p̂, q̂) are trans- form invariant under their action. Although the canoni- formed under the same linear map. cal form derives its name from Hamilton for his historical The representations in quantum-mechanical phase work on the time evolution of quadratic systems, a gen- space and the distribution functions studied therein were eral frame on which a canonical structure can be built is largely limited until very recently to the linear canonical independent from any dynamical system considered. coordinate and momentum ( p, q). The Wigner function One simple feature of these systems initially considered W ␺ ( p, q), as the best example for such representations, by Hamilton is that the preservation of the canonical is well known to be covariant under ACT’s, and its time structure becomes identical to the covariance under time evolution under quadratic Hamiltonians is given by the evolution. Indeed, the simplicity afforded by quadratic classical Liouville equation. By contrast, the action of Hamiltonians is that they provide a natural transition be- the ACT’s on the linear canonical phase-space pair ( p, q) tween the linear (ray) optics and the phase-space repre- consists in the symmetry operations for such systems, sentation in mechanics in terms of the standard canonical whose dynamics are governed by quadratic Hamiltonians. phase-space pair, viz., linear coordinate and momentum. This implies that the time dependence of the Wigner func- If one replaces the time with the parameter defined along tion in quadratic systems can be represented by time- the optical axis, the equations of motion obtained for the parameterized trajectories that coincide with the classical phase-space variables are identical in terms of mechanics ones. This seems to be the ultimate limit to which one to those in the linear optics. The importance of the qua- can push the classical–quantum correspondence in the dratic Hamiltonians is not limited by the classical linear phase space. If we bear in mind that standard quadratic optics correspondence and extends far beyond the classi- systems represented in the linear canonical coordinate cal realm into the quantum world. The classical and the and momentum are not that numerous or that they are quantum versions of a quadratic system respect the same Gaussian approximations to the original ones, the utility phase-space symmetry transformations, viz., affine ca- of these results is limited. nonical transformations (ACT’s). For these systems, the The general canonical phase-space formulation of any equations of motion for the classical phase-space pair mechanical system (whether quantum or classical) is ex- ( p, q) and their quantum counterparts (p̂, q̂) are identi- pected to be independent of the choice of a particular ca- cal. nonical basis, which suggests that there can be more than One of the most important conceptual breakthroughs in one such basis doing the same job. In some cases a non- the phase-space representations of quantum systems was linear canonical transformation generated by, say, W ⬁ made by Weyl1 in 1927 and by Wigner2 in 1932 and later can be used to connect these two bases. It can be shown by Groenewold3 in 1946 and by Moyal3 in 1949. The that4 one WWGM correspondence scheme transforms Weyl–Wigner–Groenewold–Moyal (WWGM) correspon- noncovariantly to another such scheme under a general dence is based on the existence of an orthogonal and com- nonlinear canonical transformation. Within a particular plete operator basis [the Weyl–Heisenberg (WH) basis] in correspondence scheme the Weyl map is then expected to 0740-3232/2000/122411-11$15.00 © 2000 Optical Society of America 2412 J. Opt. Soc. Am. A / Vol. 17, No. 12 / December 2000 T. Hakioğlu maintain the covariance under ACT’s acting on the ca- in a representation-independent manner as nonical basis of the choice. In this paper we will consider the covariant phase-space formulation of two-dimensional W ⌿ 共 p, q兲 ⫽ 具 ⌿, ⌬ˆ , 共 p, q兲 ⌿ 典 . (3) (2D) systems in polar space coordinates and their canoni- The operator bases ⌬ˆ i , where i ⫽ x,y, are given by the cal momenta. The spectrum of the radial phase-space unitary displacement operator (WH basis) D̂ ␣ i , ␤ i as operator is shown to be the logarithmic variate of the clas- sical one, r 苸 R⫹, and in the operator language these two ⌬ˆ i ⫽ ⌬ˆ 共 p i , q i 兲 representations are connected by a Fourier–Mellin trans- formation. The logarithmic radial (log-radial) coordinate is itself a representation of r 苸 R⫹ in Cartesian r 苸 R, enabling a radial WWGM quantization as well as the cor- ⫽ 冕 冕 R d␣ i 2␲ d␤ i R 2␲ exp关 i 共 ␣ i q i ⫺ ␤ i p i 兲兴 D̂ ␣ i , ␤ i , responding radial Wigner function formulations through D̂ ␣ i , ␤ i ⫽ exp关 i 共 ␣ i q̂ i ⫹ ␤ i p̂ i 兲兴 , (4) the standard Cartesian formalism. It must be empha- sized that, although the polar coordinate basis is the most where q̂ i , p̂ i are the canonical linear coordinate and mo- natural choice for systems with specific rotational symme- mentum operators satisfying 关 q̂ i , p̂ j 兴 ⫽ i ␦ i, j and ␣ i , ␤ i ; tries, the formulation is not limited in applications to q i , p i 苸 R for i ⫽ x, y. The properties of the standard them. Wigner function in two degrees of freedom given by Eq. Section 2 is devoted to the polar representation of the (3) are well known and have been examined in great de- Wigner function based on the log-radial spectrum. Sub- tail in a large number of publications.5 section 2.A discusses the log-radial and the angular ca- Our aim is to develop a covariant formalism for the 2D nonical bases and presents the respective Wigner func- Wigner function represented in terms of the polar canoni- tions. In each part therein, the properties of the Wigner cal momentum and coordinates p r , v r ; p ␪ , v ␪ , where, function are examined, and the covariance under ACT’s is respectively, p r ,v r are the radial and p ␪ , v ␪ are the an- discussed. There I also define a nonlinear canonical gular canonical momentum and coordinate pairs. In our transformation that basically undoes the effect of the log- case the domain of the radial phase-space variables is radial spectrum at the expense of losing most of the co- ⫺⬁ ⬍ v r , p r , ⬍⬁, and the angular ones are p ␪ 苸 Z and variances of the Wigner function. In Subsection 2.B the ⫺␲ ⭐ v ␪ ⬍ ␲ . The formalism will be based on a direct log-radial and the angular bases are combined in a prod- product form similar to that of Eq. (2) but in terms of the uct form, and the polar Wigner function is introduced. radial ⌬ˆ r ( p r , v r ) and angular ⌬ˆ ␪ ( p ␪ , v ␪ ) operator bases. Section 3 is a short and elementary example of the polar Wigner function. The validity of the canonical formalism 1. Radial Part presented here in generic mechanical as well as optical A log-radial Wigner function based on the idea of Dirac’s systems is implied by the absence of ប or by the reduced self-adjoint radial momentum operator6 p̂ r was recently wavelength ⑄ throughout the study. proposed.7 In these studies the radial momentum opera- tor p̂ r in the radial (r) coordinate representation is writ- ten by 2. CONTINUOUS POLAR REPRESENTATION Let us assume a 2D wave field in the x – y plane, with z representing the evolution parameter of the wave along the optical axis. If we follow the wave along the instan- p̂ r → ⫺i r 冉 ⳵ ⳵r ⫹␩ , 冊 ␩ 苸 R. (5) taneous direction of propagation by a screen normal to We will see below that ␩ is related to the dimensionality that direction, the wave field at the particular location z of the space. For the radial representations in a of the screen can be given by d-dimensional space, we have ␩ ⫽ d/2. If we write the radial position operator v̂ r ⫽ ln r̂, where ⌿ 共 r, ␾ ; z 兲 ⫽ 兺 ⌿̃ 共 r; z 兲 exp共 in ␾ 兲 . n苸Z n (1) v̂ r → ln r, in the radial coordinate representation, the Di- rac commutator of v̂ r and p̂ r yields Throughout the paper we will assume that the screen lo- 关 v̂ r , p̂ r 兴 ⫽ i. (6) cation is fixed. We will hence consider the z coordinate as implicit in all expressions. The eigenspace of p̂ r is spanned by A. Polar Canonical Basis ␸ ␭ 共 r 兲 ⫽ 共 1/冑2 ␲ 兲 r i␭⫺␩ , (7) Our main aim in this section is to introduce displacement where ␭ is the radial momentum eigenvalue and, for operators in polar representation in the form of an or- ␭ 苸 R, ␸ ␭ (r) is a complete and orthogonal basis for the thogonal and complete operator basis for the 2D WH harmonic analysis on the positive half-plane, viz., gener- group of polar canonical operators. The standard 2D alized positive Mellin transform.8,9 The function space is Wigner function W ⌿ (p, q), where p ⫽ ( p x , p y ) and q a Hilbert space defined by the inner product ⫽ (q x , q y ) are the canonical phase-space variables of the linear momentum and coordinate in the independent x and y directions, respectively, is written in terms of a 具␺, ␾典r ⬅ 冕 ⬁ 0 drr 2 ␩ ⫺1 ␺ * 共 r 兲 ␾ 共 r 兲 , ␺ , ␾ 苸 L共2␩ 兲 共 R⫹兲 complete and orthogonal operator basis, (8) ⌬ˆ 共 p, q兲 ⫽ ⌬ˆ x 共 p x , q x 兲 丢 ⌬ˆ y 共 p y , q y 兲 , (2) and by the dual orthogonality relations T. Hakioğlu Vol. 17, No. 12 / December 2000 / J. Opt. Soc. Am. A 2413 具 ␸ ␭⬘ , ␸ ␭典 r ⫽ 冕 0 ⬁ drr 2 ␩ ⫺1 ␸ ␭ ⬘ * 共 r 兲 ␸ ␭ 共 r 兲 ⫽ ␦ 共 ␭ ⬘ ⫺ ␭ 兲 , (9) Eqs. (9) and (10). The orthogonality of the basis is guar- anteed by Eq. (17), where Tr stands for the trace as ob- tained by 冕 ⬁ ⫺⬁ d␭ ␸ ␭ * 共 r 兲 ␸ ␭ 共 r ⬘ 兲 ⫽ ␦ 共 r ⫺ r ⬘ 兲 r ⫺2 ␩ ⫹1 . (10) Tr兵 D̂ r 共 ␣ r , ␤ r 兲 其 ⬅ 冕 ⫺⬁ ⬁ d␭ 具 ␸ ␭ , D̂ r 共 ␣ r , ␤ r 兲 ␸ ␭ 典 r . (19) In Eq. (8), L2(␩ ) (R⫹) denotes the Hilbert space of functions The composition law is stated by Eq. (18). These proper- with a finite norm, i.e., 储 ␺ 储 2 ⬅ 具 ␺ , ␺ 典 r ⬍ ⬁. It can be di- ties are translated for ⌬ˆ r ( p r , v r ) as rectly verified that p̂ r is self-adjoint in L2(␩ ) (R⫹) over the inner product defined by Eq. (8). In other words, ⌬ˆ r † 共 p r , v r 兲 ⫽ ⌬ˆ r 共 p r , v r 兲 , (20) 2␩ 具 ␺ , p̂ r ␾ 典 r ⫽ 具 p̂ r ␺ , ␾ 典 r ⫺ i ␺ * 共 r 兲 r ␾ 共 r 兲 兩 0⬁ , (11) 1 Tr兵 ⌬ˆ r 共 p r , v r 兲 其 ⫽ , (21) where, for all functions in L2(␩ ) (R⫹), the last term in Eq. 2␲ (11) vanishes, and hence p̂ r is self-adjoint. A specific case of Eqs. (9) and (10) is ␩ ⫽ 1/2, which corresponds to 1 Tr兵 ⌬ˆ r 共 p r , v r 兲 ⌬ˆ r 共 p ⬘r , v ⬘r 兲 其 ⫽ ␦ 共 p r ⫺ p ⬘r 兲 ␦ 共 v r ⫺ v ⬘r 兲 , the one-dimensional case in which the weight factors due 2␲ to r → ln r vanish and the basis in Eq. (7) becomes an iso- (22) morphic map between R and its nonnegative part R⫹, which is a more standard version of the Mellin transformation.9 Using the inner product in Eq. (8) and 冕 冕⬁ ⫺⬁ dv r ⬁ ⫺⬁ dp r ⌬ˆ r 共 p r , v r 兲 ⫽ Î, (23) the orthogonality relations in Eqs. (9) and (10), we can ex- pand an arbitrary function ␺ (r) in L2(␩ ) (R⫹) in the Mellin basis as 冕 ⬁ ⫺⬁ ˆ dv r ⌬ˆ r 共 p r , v r 兲 ⫽ P̃r 共 p r 兲 , (24) ␺ 共r兲 ⫽ 冕⫺⬁ ⬁ d␭A 共 ␭ 兲 ␸ ␭ 共 r 兲 , A共 ␭ 兲 ⫽ 具␸␭ , ␺典r . (12) 冕 ⬁ ⫺⬁ dp r ⌬ˆ r 共 p r , v r 兲 ⫽ P̂r 共 v r 兲 , (25) The inner product defined by Eq. (8) can be expressed in ˆ the radial momentum-␭ representation as where P̂r and P̃r are the radial projection operators as de- fined by 具␺, ␾典r ⫽ 冕 ⫺⬁ ⬁ d␭A * 共 ␭ 兲 B 共 ␭ 兲 , A 共 ␭ 兲 , B 共 ␭ 兲 苸 L2 共 R兲 , P̂r 共 v r 兲 P̂r 共 v r⬘ 兲 ⫽ ␦ 共 v r ⫺ v ⬘r 兲 P̂r 共 v r 兲 , where L2 (R) is the usual Hilbert space of square- (13) 冕 dv r P̂r 共 v r 兲 ⫽ Î, integrable functions on the real line. In close analogy with Eqs. (4), the radial canonical operator basis can now P̂r 共 p r 兲 P̂r 共 p ⬘r 兲 ⫽ ␦ 共 p r ⫺ p ⬘r 兲 P̂r 共 p r 兲 , 冕 be established as ˆ dp r P̃r 共 p r 兲 ⫽ Î, 冕 冕 (26) ⬁ d␣ r ⬁ d␤ r ⌬ˆ r 共 p r , v r 兲 ⫽ ⫺⬁ 2␲ ⫺⬁ 2␲ with ⫻ exp关 ⫺i 共 ␣ r v r ⫹ ␤ r p r 兲兴 D̂ r 共 ␣ r , ␤ r 兲 , 具 ␺ , P̂r 共 v r 兲 ␾ 典 r ⬅ exp共 2 ␩ v r 兲 ␺ * 共 exp v r 兲 ␾ 共 exp v r 兲 , (27) D̂ r 共 ␣ r , ␤ r 兲 ⫽ exp关 i 共 ␣ r v̂ r ⫹ ␤ r p̂ r 兲兴 , (14) ˆ 具 ␺ , P̃r 共 p r 兲 ␾ 典 ␭ ⬅ A * 共 p r 兲 B 共 p r 兲 , (28) where v r , p r 苸 R are the log-radial phase-space vari- where A( p r ) and B( p r ) are the Mellin transforms of ␺ (r) ables. The properties of the log-radial canonical basis and ␾ (r) as calculated by Eq. (12). ⌬ˆ r ( p r , v r ) follow from those of D̂ r ( ␣ r , ␤ r ), which are Log-radial Wigner function and its properties. The representation-independent form of the log-radial Wigner D̂ r 共 0, 0 兲 ⫽ Î, (15) function for a state ␺ will be defined as D̂ r † 共 ␣ r , ␤ r 兲 ⫽ D̂ r ⫺1 共 ␣ r , ␤ r 兲 ⫽ D̂ r 共 ⫺␣ r ,⫺␤ r 兲 , (16) W ␺ 共 p r , v r 兲 ⫽ 具 ␺ , ⌬ˆ r 共 p r , v r 兲 ␺ 典 r . (29) Tr兵 D̂ r 共 ␣ r , ␤ r 兲 其 ⫽ 2 ␲ ␦ 共 ␣ r 兲 ␦ 共 ␤ r 兲 , (17) Equation (29) is represented in the radial coordinate ba- sis as D̂ r 共 ␣ r , ␤ r 兲 D̂ r 共 ␣ ⬘r , ␤ ⬘r 兲 ⫽ exp关 ⫺i 共 ␣ r ␤ ⬘r ⫺ ␤ r ␣ ⬘r 兲 /2兴 W ␺共 p r , v r 兲 ⫽ 1 2␲ 冕 ⫺⬁ ⬁ d␤ r exp共 ⫺i ␤ r p r 兲 exp共 2 ␩ v r 兲 ⫻ D̂ r 共 ␣ r ⫹ ␣ ⬘r , ␤ r ⫹ ␤ ⬘r 兲 . (18) ⫻ ␺ * 关 exp共 v r ⫹ ␤ r /2兲兴 ␺ 关 exp共 v r ⫺ ␤ r /2兲兴 Equation (15) defines the unit element. Equation (16) is (30) the statement of unitarity and inversion guaranteed by the self-adjointness of p̂ r and v̂ r over the inner product in and in the radial momentum basis as 2414 J. Opt. Soc. Am. A / Vol. 17, No. 12 / December 2000 T. Hakioğlu W A ␺共 p r , v r 兲 ⫽ 1 2␲ 冕 ⬁ ⫺⬁ d␣ r exp共 i ␣ r v r 兲 A * where g is in the group Sp(2, R) of 2 ⫻ 2 symplectic ma- trices. The three one-parameter subgroups will be iden- tified in the conventional way by ⫻ 共 p r ⫹ ␣ r /2兲 A 共 p r ⫺ ␣ r /2兲 . The static properties of the radial Wigner function fol- (31) g1 ⫽ 冋 cos ␴ sin ␴ ⫺sin ␴ cos ␴ 册 , g2 ⫽ 冋 cosh ␶ ⫺sinh ␶ ⫺sinh ␶ cosh ␶ 册 , 冋 册 low directly from Eqs. (29)–(31). These are exp共 ⫺␹ 兲 0 (1) W ␺ ( p r , v r ) is real, namely, g3 ⫽ , (40) 0 exp共 ␹ 兲 W ␺共 p r , v r 兲 ⫽ W ␺*共 p r , v r 兲 , (32) with ⫺␲ ⭐ ␴ ⬍ ␲ , ⫺⬁ ⬍ ␶ ⬍ ⬁, and ⫺⬁ ⬍ ␹ ⬍ ⬁. We which follows directly from Eq. (20). now examine the action of each subgroup by considering (2) The integral of W ␺ ( p r , v r ) with respect to one of g ⫽ g i for i ⫽ 1, 2, 3 independently. The representation the phase-space variables yields the marginal probability T̂ g of the transformation in the operator basis ⌬ˆ r is given with respect to the other variable: by 冕 ⫺⬁ ⬁ ˆ dv r W ␺ 共 p r , v r 兲 ⫽ 具 ␺ , P̃共 p r 兲 ␺ 典 r ⫽ 兩 A 共 p r 兲 兩 2 , (33) T̂ g : ⌬ˆ r 共 p r , v r 兲 ⫽ T̂ g ⌬ˆ r 共 p r , v r 兲 T̂ g ⫺1 ⬅ ⌬ˆ r 共 p r⬘ , v ⬘r 兲 . (41) 冕 We expand T̂ g in the complete and orthogonal radial ⬁ dp r W ␺ 共 p r , v r 兲 ⫽ 具 ␺ , P̂共 v r 兲 ␺ 典 r WH basis as 冕 冕 ⫺⬁ ⬁ ⬁ T̂ g ⫽ d␥ r d␦ r C 共r g 兲 共 ␥ r , ␦ r 兲 D̂ r 共 ␥ r , ␦ r 兲 , (42) ⫽ exp共 2 ␩ v r 兲 兩 ␺ 共 exp v r 兲 兩 , 2 (34) ⫺⬁ ⫺⬁ which follow directly from Eqs. (24) and (25). where the coefficients C (r g ) characterize the transforma- (3) Static covariance properties: The standard p, q tion. More generally, D̂ r (or, alternatively, its Fourier Wigner function is known to be covariant under ACT’s. transform ⌬ˆ r ) is an operator basis for any Hilbert– For a system with one degree of freedom, the ACT is a five Schmidt operator. Using the unitarity of D̂ r ’s as stated parameter group of which three are the parameters of the in Eq. (16) and demanding the unitarity of T̂ g ’s we can de- group of linear canonical transformations (LCT’s). The remaining two are the parameters of the Galilean trans- rive a condition on the coefficients as 关 C (r g ) ( ␥ r , ␦ r ) 兴 * ⫺1 formations. A similar construction can also be made for ⫽ C (r g ) ( ␥ r , ␦ r ). Through Eq. (41) the coefficients also the log-radial Wigner function. Below we will examine satisfy the covariance under ACT’s within each subgroup inde- C 共r g 兲 共 ⑀ ⫺ ␣ r , v ⫺ ␤ r 兲 pendently. In paragraphs (a) and (b) the log-radial ana- logs of the Galilean transformations will be studied; in ⫽ exp兵 i 关 ⑀ 共 ␤ r ⫹ ␤ ⬘r 兲 ⫺ v 共 ␣ r ⫹ ␣ ⬘r 兲兴 /2其 paragraphs (c)–(e) the LCT’s will be studied. Radial analogs of the Galilean transformations. These ⫻ C 共gr 兲 共 ⑀ ⫺ ␣ r⬘ , v ⫺ ␤ ⬘r 兲 (43) are as follows. for all ⑀, v, ␣ r , ␤ r , where 冉 冊 冉 冊 (a) Covariance under radial dilations. We define a map from a wave function ␺ (r) to ␺ ⬘ (r) by ␣ ⬘r ␣r ⫽g . (44) ␤ ⬘r ␤r ␺ ⬘ 共 r 兲 ⬅ exp共 i ␤ ⬘r p̂ r 兲 ␺ 共 r 兲 ⫽ exp共 ␩ ␤ ⬘r 兲 ␺ 关 exp共 ␤ ⬘r 兲 r 兴 . (35) Although a general solution to Eq. (44) can be given as Inserting Eq. (33) into Eqs. (29) and (30), we find that C 共r g 兲 共 ␣ r , ␤ r 兲 ⫽ N exp关 i 共 U ␣ 2r ⫹ V ␤ 2r ⫹ W ␣ r ␤ r 兲兴 , (45) W ␺ 共 p r , v r 兲 ⫽ W ␺ ⬘ 共 p r , v r ⫹ ␤ ⬘r 兲 , (36) where U, V, W, and N are functions of the parameters of which states the covariance of the Wigner function under g, it is more illuminating to give the solutions for each radial dilations in Eq. (35). subgroup in Eqs. (40) separately. Using Eqs. (39) in Eq. (b) Covariance under local phase shifts. We now de- (43), we find that fine a map from ␺ (r) to ␺ ⬘ (r) as 共 g1兲 exp共 i ␲ /2兲 ␺ ⬘ 共 r 兲 ⬅ exp共 ⫺i ␣ ⬘r v̂ r 兲 ␺ 共 r 兲 ⫽ r ⫺i ␣ r⬘ ␺ 共 r 兲 . (37) Cr 共␣r , ␤r兲 ⫽ 关 sin共 ␴ /2兲兴 ⫺1 4␲ Inserting Eq. (37) into Eqs. (29) and (30), we find that ⫻ exp关 ⫺共 i/4兲 cot共 ␴ /2兲共 ␣ 2r ⫹ ␤ 2r 兲兴 , W ␺ 共 p r , v r 兲 ⫽ W ␺ ⬘ 共 p r ⫺ ␣ r⬘ , v r 兲 , (38) 共 g2兲 1 which states the covariance of the Wigner function under Cr 共␣r , ␤r兲 ⫽ 兩 sinh共 ␶ /2兲 兩 ⫺1 4␲ Galilean transformations on radial momentum p r . (c) Covariance under radial linear canonical transfor- ⫻ exp关 ⫺共 i/4兲 coth共 ␶ /2兲共 ␣ 2r ⫺ ␤ 2r 兲兴 , mations. A general LCT acting on the radial phase space p r ,v r will be defined by the map 共 g3兲 1 Cr 共␣r , ␤r兲 ⫽ 兩 sinh共 ␹ /2兲 兩 ⫺1 冉 冊 冉 冊 p ⬘r v ⬘r ⫽g pr vr , g⫽ 冉 冊 a c b d , det g ⫽ 1, (39) 4␲ ⫻ exp关 ⫺共 i/2兲 coth共 ␹ /2兲 ␣ r ␤ r 兴 , (46) T. Hakioğlu Vol. 17, No. 12 / December 2000 / J. Opt. Soc. Am. A 2415 where the normalizations are determined by the identity resentations of p̂ r and derive the infinitesimal generators transformation limit such that in terms of v̂ r , p̂ r . Kernels such as those in Eqs. (49)– 共 gi兲 (51) were studied in detail in Ref. 9. The generators of lim C r 共 ␣r , ␤r兲 ⫽ ␦ 共 ␣r兲␦ 共 ␤r兲, infinitesimal LCT are given by g i →I 1 lim T̂ g i ⫽ Î, i ⫽ 1, 2, 3. (47) T̂ g 1 ⫽ exp共 i2 ␴ K̂ 1 兲 , K̂ 1 ⫽ 4共 p̂ 2r ⫹ v̂ 2r 兲 , g i →I 1 It is also possible to show that a general group element T̂ g 2 ⫽ exp共 i2 ␶ K̂ 2 兲 , K̂ 2 ⫽ 4共 p̂ 2r ⫺ v̂ 2r 兲 , can be obtained through T̂ g ⬅ T̂ g 3 T̂ g 2 T̂ g 1 , where T g is not 1 exactly a group representation but a projective (ray) one9 T̂ g 3 ⫽ exp共 i2 ␹ K̂ 3 兲 , K̂ 3 ⫽ 4共 p̂ r v̂ r ⫹ v̂ r p̂ r 兲 . (53) satisfying T̂ g T̂ g ⬘ ⫽ ⌳T̂ gg ⬘ , where ⌳ is an overall phase The action of the group elements T̂ g i on the functions in factor that depends on the parameters of g, g ⬘ . L2 (R⫹) can now be very easily found, since the log-radial T g acts in the function space as a linear canonical inte- gral transform. The expressions in the radial momen- coordinate representations of the operators K̂ i are known. tum representation are much simpler than those in the Their counterparts in terms of the linear momentum and radial coordinate representations, which are defined by coordinate are known in the theory of integral transforms,9 and they define the Sp(2, R) algebra: T̂ g A 共 ␭ 1 兲 ⫽ 冕 ⫺⬁ ⬁ d␭ 2 c 共r g 兲 共 ␭ 1 , ␭ 2 兲 A 共 ␭ 2 兲 , (48) 关 K̂ 1 , K̂ 2 兴 ⫽ iK̂ 3 , 关 K̂ 1 , K̂ 3 兴 ⫽ ⫺iK̂ 2 , where the kernel of the integral transform c (r g ) can be 关 K̂ 2 , K̂ 3 兴 ⫽ ⫺iK̂ 1 , (54) found, for each subgroup g i (i ⫽ 1, 2, 3), to be with the central element in this case being K̂ 2 ⫽ ⫺K̂ 1 2 共 g1兲 cr 共␭1 , ␭2兲 ⫽ exp共 i ␲ /4兲 冑2 ␲ sin ␴ exp ⫺ 再 2 sin ␴ i ⫺ K̂ 2 2 ⫹ K̂ 3 2 ⫽ 3/16. The important observation here is that the log-radial self-adjoint generators in Eq. (53) are represented in quadratic functions (in exactly the same ⫻ 关 cos ␴ 共 ␭ 1 2 ⫹ ␭ 2 2 兲 ⫺ 2␭ 1 ␭ 2 兴 , 冎 (49) form as their Cartesian ones) of p̂ r , v̂ r , which themselves are self-adjoint in L2(␩ ) (R⫹). However, their algebraic counterparts in the radial (nonlogarithmic) Hankel basis 共 g2兲 cr 共␭1 , ␭2兲 ⫽ exp共 i ␲ /4兲 冑2 ␲ sinh ␶ exp ⫺ i 2 sinh ␶ 再 were also identified10 as generators of certain linear opti- cal transformations induced by thin lenses, magnifiers, and free-space propagators [i.e., Ĵ i (i ⫽ 0, 1, 2) in Eqs. 冎 (26)–(30) in that reference]. However, unlike the case ⫻ 关 cosh ␶ 共 ␭ 1 2 ⫹ ␭ 2 2 兲 ⫺ 2␭ 1 ␭ 2 兴 , (50) above, the linear canonical generators (let us denote them by P̂ r , r̂) on the half-line R⫹ are not self-adjoint, nor do 共 g3兲 they have known extensions as such. This implies that cr 共 ␭ 1 , ␭ 2 兲 ⫽ exp共 ⫺␹ /2兲 ␦ 关 ␭ 2 ⫺ exp共 ⫺␹ 兲 ␭ 1 兴 . (51) these radial elements P̂ r , r̂ do not support unitary WH In Eqs. (49)–(51) the identity transformation is recovered representations of the type shown in Eq. (42), which can in the appropriate limit as shown in Eq. (52). be summarized in the following diagram: C 共x g 兲 D̂ x ⇔ c 共x g 兲 m m 关 C 共x g 兲 D̂ x 兴关 C 共y g 兲 D̂ y 兴 ⇔ c 共x g 兲 c 共y g 兲 ⇑ ⇑ ? no! D̂ x D̂ y ⫽ 丣m D̂ 共rm 兲 丢 D̂ 共␪m 兲 共 x, y 兲 ↔ 共 r, ␪ 兲 yes ⇓ ⇓ ⇔ 共 g 兲共 m 兲 ? 丣 mc r exp共 im ␪ 兲 . (55) 共 gi兲 lim c r 共 ␭1 , ␭2兲 ⫽ ␦ 共 ␭1 ⫺ ␭2兲. (52) On the left-hand side of correspondence (55) we have g i →I what is essentially the unitary representation shown in Eq. (42). On the corresponding right-hand side we have The log-radial coordinate representations can be found the integral operator representation shown in Eq. (48). by calculation of the Mellin transform of Eq. (48). But On extension of the scheme to two or more Cartesian di- there is an easier way. We continue to use the eigenrep- mensions the correspondence is manifested, as expected, 2416 J. Opt. Soc. Am. A / Vol. 17, No. 12 / December 2000 T. Hakioğlu by a direct product in the WH basis and an ordinary prod- a nonlinear canonical transformation12 from W ␺ ( p r , v r ) uct in the function space. The integral operator repre- to another Wigner function ␻ ␺ (P r , r) on the basis of the sentations on the right can be written in terms of the he- more desirable canonical pair P r , r without being blocked licity (m) expansion of the wave field in the non- by the nonexisting radial (nonlogarithmic) Weyl corre- logarithmic radial Hankel basis, as shown in Ref. 10. spondence. One can partially achieve this by first devis- They are structurally different from the log-radial coordi- ing a canonical transformation generator from p r , v r nate representation of those shown in Eqs. (49)–(51). ⫽ ln r to P r ⫽ exp(⫺vr)pr , r ⫽ exp(vr). The former (nonlogarithmic Hankel) ones do not have WH Let us now consider the following Fourier–Mellin operator kernels [via correspondence (55)], whereas the transform ˜␺ (v r ) of a radial signal ␺ (r) as log-radial representations of the WH kernels of Eqs. (49)– ˜␺ 共 v 兲 ⫽ 共 F : ␺ 兲共 v 兲 (51) exist and are given by Eqs. (46). r M r 冕 From the optics point of view it is desirable to formu- ⬁ d␭ late a Wigner function covariant under the action of lin- ⫽ exp共 ⫺i␭v r 兲共 ␸ ␭ , ␺ 兲 r ear optical devices. For convenience, let us call the latter ⫺⬁ 冑2 ␲ the linear optical covariance. This covariance arises in the Spx (2, R) 丢 Spy (2, R) subgroup decomposition of the ⫽ exp共 ␩ v r 兲 ␺ 共 exp v r 兲 . (58) group of LCT in two Cartesian dimensions. This sub- Equation (58) is a unitary transformation between group further decomposes10 into an infinite helicity (m) functions in the radial r representation and functions in sum of the actions of Sp(rm ) (2, R⫹), each acting irreducibly the radial v r ⫽ ln r representation. The impulse in the definite helicity (m) subspace for integer m. More- response13 corresponding to this coordinate transforma- general ones for Sp(4, R) have also been reported.11 tion is What the diagram in correspondence (55) then says is that, within a logarithmic or nonlogarithmic radial coor- g v r 共 r 兲 ⫽ exp共 ⫺␩ v r 兲 ␦ 共 v r ⫺ ln r 兲 , dinate representation achieving linear optical covariance ˜␺ 共 v 兲 ⫽ 共 g , ␺ 兲 ⫽ exp共 ␩ v 兲 ␺ 共 exp v 兲 . (59) and canonicality simultaneously—in the context of WH r vr r r r representations—may be difficult. Using Eqs. (59), we find that Eq. (30), as expected, adopts However, in the log-radial representation, it is still pos- the standard form sible to approximate the effective action of some optical elements in certain regions of the radial space by use of the combinations of the log-radial Galilean and the Sp(2, R) generators. The first example is exp(i␤ p̂r) as W ␺共 p r , v r 兲 ⫽ 2␲ 1 冕 ⬁ ⫺⬁ d␤ r exp共 ⫺i ␤ r p r 兲 ˜␺ * 共 v r ⫹ ␤ r /2兲 a dilation generator, whose effect is a magnification ⫻ ˜␺ 共 v r ⫺ ␤ r /2兲 . (60) of the initial wave field as exp(i ln sp̂r): ␺ (r)/ 冑r ␩ ⫺1/2 → 冑s ␺ (sr)/ 冑r ␩ ⫺1/2 after one accounts for the appropriate Consider a new pseudo Wigner function of the form 冕 weight factor in the denominators. The second example 1 is the multiplication by a Gaussian phase, whose effect is ␻ ␺共 P r , r 兲 ⫽ ds s ⫺irP r ⫺1 r 2 ␩ ␺ * 共 冑sr 兲 ␺ 共 r/ 冑s 兲 , generated by thin lenses. By direct inspection of Eq. (37) 2␲ R⫹ in the range 兩 1⫺r 兩 Ⰶ 1 (remember that r is in units of the ␺ 共 r 兲 苸 L共2␩ 兲 共 R⫹兲 . (61) optical wavelength ⑄), we can observe that the local phase shift is effectively approximated by a Gaussian and is ex- Using Eqs. (58) and (59), we can relate Eqs. (60) and (61) pressed in terms of the generators K̂ 1 , K̂ 2 and v̂ r as through exp关 i ␣ 共 r 2 ⫺ 1 兲兴 ␺ 共 r 兲 ⯝ 兵 exp关 i2 ␣ 共 v̂ r ⫹ v̂ 2r 兲兴 其 ␺ 共 r 兲 ⫽ exp共 i2 ␣ v̂ r 兲 W ␺共 p r , v r 兲 ⫽ 冕 dP r drT 共 p r , v r ; P r , r 兲 w ␺ 共 P r , r 兲 , T ⫽ ␦ 共 rP r ⫺ p r 兲 ␦ 共 v r ⫺ ln r 兲 , (62) ⫻ exp关 i4 ␣ 共 K̂ 1 ⫺ K̂ 2 兲兴 ␺ 共 r 兲 which does correspond to a canonical transformation; i.e., (56) ( p r , v r ) → 关 P r ⫽ exp(⫺vr)pr , r ⫽ exp(vr)兴. Some of the up to terms O关 exp(i␣v̂3r )兴 on the right-hand side, provided properties of the pseudo Wigner function read as follows: that 兩 ln r兩 Ⰶ 1. (1) The pseudo Wigner function is real. (4) The inner product property reads as follows: (2) Its normalization is given by 兰 dp r dv r W ␺ 冕 冕 ⬁ ⫺⬁ dv r ⬁ ⫺⬁ dp r W ␺ 共 p r , v r 兲 W ␾ 共 p r , v r 兲 ⫽ 兰 dP r dr ␻ ␺ ⫽ 1. Essentially, ␻ ␺共 P r , r 兲 ⫽ W ␺共 p r , v r 兲 兩 pr⫽ r Pr . (63) 冏冕 冏 v r ⫽ln r 1 ⬁ 2 ⫽ dv exp共 2 ␩ v 兲 ␺ * 共 exp v 兲 ␾ 共 exp v 兲 (3) The marginal probability for r is obtained, as ex- 2␲ ⫺⬁ pected, as ⫽ 1 2␲ 兩共 ␺ , ␾ 兲 r兩 2. (57) 冕 dP r ␻ ␺ 共 P r , r 兲 ⫽ r 2 ␩ ⫺1 兩 ␺ 共 r 兲 兩 2 . (64) Radial Wigner function in a noncovariant form. Our (4) Under scale changes induced by the operator purpose in this section is to learn whether one can define exp(i␤p̂r) in Eq. (35), one has T. Hakioğlu Vol. 17, No. 12 / December 2000 / J. Opt. Soc. Am. A 2417 exp共 i ␤ p̂ r 兲 p ␪ 苸 Z. The geometry of the semidiscrete limit is visual- ␺ → ␺ ⬘ ⇒ ␻ ␺ 共 P r , r 兲 ⫽ ␻ ␺ ⬘ (exp共 ␤ 兲 P r , exp共 ⫺␤ 兲 r). ized as a cylinder of rings of unit radius. Each ring is (65) separated from the other by a unit of angular momentum, Hence the covariance is manifest under radial dilations. with the rings corresponding to the boundaries of the cyl- inder along the axis located at ⫾⬁. A point in the phase If one considers 冑r 2 ␩ ⫺1 ␺ (r) 苸 L2(1/2) (R⫹), one has the space is then defined by an angular variable (the phase Hankel-type normalization used in Ref. 10. Expression v ␪ ) parameterizing the ring and by a discrete number (the (66), then, states the covariance of the pseudo Wigner angular momentum or the helicity factor, ⫺⬁ ⬍ p ␪ ⬍ ⬁) function under the scaling generator Ĵ 2(m ) therein. By parameterizing which ring, along the axis, that it is re- contrast, the pseudo Wigner function is not covariant un- ferred to. der log-radial Sp(2, R) or under any local phase shift in- The rigorous definition of the angular Wigner function duced by the operator exp关i␣g(r̂)兴, where g is any function. requires this specific limiting procedure from a fully dis- It is also not covariant under SP(rm ) (2, R⫹) other than un- crete to a semidiscrete form, as described above. For der the scaling transformation. It is expected that the clarity here we will start from the semidiscrete formalism Sp(2, R) covariance would be lost through the transforma- and refer to Refs. 16 and 17 for details. tion in Eqs. (62). The log-radial conjugate coordinates The semidiscrete angular kernel, as the angular analog can mix under the action of the off-diagonal elements of of Eqs. (14), basically amounts to construction of a semi- the LCT because they have the same domain, which is discrete WH operator basis D̂ ␪ (n, ␨ ), with n 苸 Z and ␨ simply R. The off-diagonal LCT’s on the other radial pair 苸 关 ⫺␲ , ␲ ), whose action on functions F( ␾ ) on the unit (P r , r) are forbidden. This is because Eq. (62) implies circle is defined by that P r 苸 R and that r 苸 R⫹. Hence the conjugate coor- dinates in the new pair cannot covariantly mix with each D̂ ␪ 共 n, ␨ 兲 F 共 ␾ 兲 ⬅ exp共 in ␨ /2兲 exp共 in ␾ 兲 F 共 ␾ ⫹ ␨ 兲 , other. Indeed, the scaling generators K̂ 3 of the log-radial Sp(2, R) in Eqs. (53) and the scaling generator Ĵ 2(m ) in ⫺␲ ⭐ ␾ ⬍ ␲ . (67) Sp(rm ) (2, R⫹) are related to each other by the same canoni- For construction of the angular Wigner function, it will cal transformation as in Eq. (62). They also have no off- also be necessary to know the action of D̂ ␪ (n, ␨ ) on the diagonal elements. Hence they leave both Wigner func- Fourier transform of F. This Fourier transform is de- tions covariant. Similarly, we will observe in Subsection fined by 2.A.2, on the angular part, that the standard LCT covari- ance is absent in the angular Wigner function inasmuch as the domains of the angular and the angular- momentum variables are quite distinct from each other. fm ⫽ 1 冑2 ␲ 冕 ⫺⬁ ⬁ d␾ F 共 ␾ 兲 exp共 ⫺im ␾ 兲 , ⫺⬁ ⬍ m ⬍ ⬁, (68) With regard to the fact that the P r distribution is rep- resented by where m must have the same domain as does n in rela- 冕 tions (67). For f m , we find that dr ␻ ␺ 共 P r , r 兲 , (66) R⫹ D̂ ␪ 共 n, ␨ 兲 f m ⫽ exp共 ⫺in ␨ 兲 exp共 im ␨ 兲 f m⫺n . (69) we can say only that it is real by construction of ␻ ␺ (P r , r) It can be seen that, if F and G are two functions on the and that it is normalized to unity. Beyond this trivial re- unit circle, D̂ ␪ (n, ␨ ) is unitary, sult, it should also be determined whether it is nonnega- tive for acceptability as a distribution. 具 F, D̂ ␪ 共 n, ␨ 兲 G 典 ␪ ⫽ 具 D̂ ␪ † 共 n, ␨ 兲 F, G 典 ␪ ⫽ 具 D̂ ␪ ⫺1 共 n, ␨ 兲 F, G 典 ␪ , (70) 2. Angular Part The angular phase-space representations have been one over the inner product in the angle representation 冕 of the long-standing problems since the 1920’s because of ␲ their connection with one of the fundamental anomalies 具 F, G 典 ␪ ⫽ d␾ F * 共 ␾ 兲 G 共 ␾ 兲 (71) in quantum mechanics.14,15 The standard canonical co- ⫺␲ ordinates with unbounded (continuous or discrete) spec- or in the discrete angular-momentum representation tra are not in the trace class, and their standard commu- ⬁ tation rule violates a fundamental trace identity, which prevents a well-defined unitary phase operator to exist.15 具 F, G 典 ␪ ⫽ 兺 m⫽⫺⬁ * gm , fm (72) The resolution of this problem requires a different start- ing point than the standard continuous phase space: a and it satisfies discrete and finite-dimensional phase space with periodic boundaries, which is effectively a discrete torus.16 One D̂ ␪ 共 0, 0 兲 ⫽ Î, (73) then defines the standard quantum-mechanical phase space in a semidiscrete limit17 in which one increases the D̂ ␪ † 共 n, ␨ 兲 ⫽ D̂ ␪ ⫺1 共 n, ␨ 兲 ⫽ D̂ ␪ 共 ⫺n, ⫺␨ 兲 , (74) number of discrete points in both directions in the phase Tr关 D̂ ␪ 共 n, ␨ 兲兴 ⫽ 2 ␲ ␦ 共 ␨ 兲 ␦ n,0 , (75) space to infinity in such a way that one of the discrete co- ordinates approaches a continuous and bounded phase D̂ ␪ 共 n, ␨ 兲 D̂ ␪ 共 n ⬘ , ␨ ⬘ 兲 ⫽ exp关 ⫺i 共 n ␨ ⬘ ⫺ ␨ n ⬘ 兲 /2兴 variable, ⫺␲ ⭐ v ␪ ⬍ ␲ , and the other one remains dis- crete as its conjugate partner (the angular momentum), ⫻ D̂ ␪ 共 n ⫹ n ⬘ , ␨ ⫹ ␨ ⬘ 兲 (76) 2418 J. Opt. Soc. Am. A / Vol. 17, No. 12 / December 2000 T. Hakioğlu Equation (75), where Tr stands for the trace of the matrix elements of D̂ ␪ (n, ␨ ), guarantees that the angular WH basis is orthogonal. To calculate Eq. (75) we consider the W F共 p ␪ , v ␪ 兲 ⫽ 1 2␲ 冕 ⫺␲ ␲ d␨ exp共 ⫺i ␨ p ␪ 兲 F * 共 v ␪ ⫺ ␨ /2兲 simplest complete and orthonormal basis functions on the ⫻ F 共 v ␪ ⫹ ␨ /2兲 . (89) unit circle as the Fourier basis L m ( ␾ ) by which the trace is defined. This definition is written as The angular-momentum representation of Eq. (88) re- ⬁ quires the use of a fractionally shifted angular- Tr关 D̂ ␪ 共 n, ␨ 兲兴 ⬅ 兺 m⫽⫺⬁ 具 L m , D̂ ␪ 共 n, ␨ 兲 L m 典 ␪ , momentum spectrum.16,17 In our discussions here we shall construct the Wigner function in the angular coordi- nate representations to avoid this sort of abstraction. 1 The static properties of the angular Wigner function fol- L m共 ␾ 兲 ⫽ exp共 im ␾ 兲 . (77) 冑2 ␲ low directly from Eq. (89). These are as follows: By use of Eqs. (71) and (77) one can derive Eq. (75). The (1) W F ( p ␪ , v ␪ ) is real, namely, construction of the angular kernel follows by direct anal- ogy with Eqs. (14). We introduce the angular kernel W F共 p ␪ , v ␪ 兲 ⫽ W F*共 p ␪ , v ␪ 兲 , (90) ⌬ˆ ␪ ( p ␪ , v ␪ ) as 冕 ⬁ which follows directly from Eq. (79). 1 ␲ d␨ ⌬ˆ ␪ 共 p ␪ , v ␪ 兲 ⫽ 兺 2 ␲ n⫽⫺⬁ ⫺␲ 2␲ (2) The integral (sum) of W F ( p ␪ , v ␪ ) with respect to one of the phase-space variables p ␪ ,v ␪ yields the mar- ginal probability with respect to the other variable: ⫻ exp关 ⫺i 共 nv ␪ ⫹ ␨ p ␪ 兲兴 D̂ ␪ 共 n, ␨ 兲 , (78) where ⫺␲ ⭐ v ␪ ⬍ ␲ and p ␪ 苸 Z. These properties of D̂ ␪ translate to those of the angular kernel ⌬ˆ ␪ ( p ␪ , v ␪ ) as 冕 ⫺␲ ␲ dv ␪ W F 共 p ␪ , v ␪ 兲 ⫽ 兩 f p ␪ 兩 2 , (91) ⌬ˆ ␪ 共 p ␪ , v ␪ 兲 ⫽ ⌬ˆ ␪ † 共 p ␪ , v ␪ 兲 , (79) ⬁ Tr兵 ⌬ˆ ␪ 共 p ␪ , v ␪ 兲 其 ⫽ 1 , (80) 兺 p ␪ ⫽⫺⬁ W F共 p ␪ , v ␪ 兲 ⫽ 兩 F 共 v ␪ 兲兩 2, (92) 2␲ 1 which follow directly from Eqs. (79) and (80). Tr兵 ⌬ˆ ␪ 共 p ␪ , v ␪ 兲 ⌬ˆ ␪ 共 p ⬘␪ , v ⬘␪ 兲 其 ⫽ ␦ p ␪ , p ⬘ ␦ 共 v ␪ ⫺ v ⬘␪ 兲 , (81) (3) Static covariance properties: Unlike the radial 2␲ ␪ part, the angular Wigner function is not covariant under 冕 ⬁ ␲ the action of the LCT’s. This is because v ␪ , with a finite ⫺␲ dv ␪ 兺 p ␪ ⫽⫺⬁ ⌬ˆ ␪ 共 p ␪ , v ␪ 兲 ⫽ Î, (82) and continuous support 兵i.e., v ␪ 苸 关 ⫺␲ , ␲ ) 其 and p ␪ , with an infinite and discrete one (i.e., p ␪ 苸 Z), do not mix. 冕 ␲ ⫺␲ ˆ dv ␪ ⌬ˆ ␪ 共 p ␪ , v ␪ 兲 ⫽ P̃␪ 共 p ␪ 兲 , (83) For this reason, below we consider only the Galilean transformations for the angular part. Angular Galilean transformations. These are per- ⬁ formed as follows. 兺 p ␪ ⫽⫺⬁ ⌬ˆ ␪ 共 p ␪ , v ␪ 兲 ⫽ P̂␪ 共 v ␪ 兲 , (84) (a) Define a new function F ⬘ ( ␾ ) as ˆ where the angular projection operators P̂␪ (v ␪ ) and P̃␪ ( p ␪ ) F 共 ␾ 兲 ⫽ exp共 i ␨ ⬘ p̂ ␪ 兲 F ⬘ 共 ␾ 兲 ⫽ F ⬘ 共 ␾ ⫹ ␨ ⬘ 兲 , ␨ ⬘ 苸 R. are defined in a manner similar to that of the radial ones (93) in Eqs. (26) and (28) as Inserting Eq. (93) into Eqs. (88) and (89), we find that P̂␪ 共 v ␪ 兲 P̂␪ 共 v ⬘␪ 兲 ⫽ ␦ 共 v ␪ ⫺ v ␪⬘ 兲 P̂␪ 共 v ␪ 兲 , 冕 dv ␪ P̂共 v ␪ 兲 ⫽ Î, W F共 p ␪ , v ␪ 兲 ⫽ W F ⬘共 p ␪ , v ␪ ⫹ ␨ ⬘ 兲 . (94) ⬁ ˆ ˆ ˆ P̃␪ 共 p ␪ 兲 P̃␪ 共 p ␪⬘ 兲 ⫽ ␦ p ␪ , p ⬘ P̃␪ 共 p ␪ 兲 , ␪ 兺 p ␪ ⫽⫺⬁ ˆ P̃␪ 共 p ␪ 兲 ⫽ Î, (b) We now define the new function F ⬘ ( ␾ ) as (85) F 共 ␾ 兲 ⫽ exp共 il ␪ˆ 兲 F ⬘ 共 ␾ 兲 ⫽ exp共 il ␪ 兲 F ⬘ 共 ␾ 兲 , l 苸 Z. 具 F, P̂␪ 共 v ␪ 兲 G 典 ␪ ⬅ F * 共 v ␪ 兲 G 共 v ␪ 兲 , (86) (95) ˆ Inserting Eq. (95) into Eqs. (88) and (89), we find that 具 F, P̃␪ 共 p ␪ 兲 G 典 ␪ ⬅ f p*␪ g p ␪ . (87) Angular Wigner Function and Its Properties. The W F 共 p ␪ , v ␪ 兲 ⫽ W F ⬘ 共 p ␪ ⫺ l, v ␪ 兲 . (96) representation-independent form of the angular Wigner function can be defined as Equations (94) and (96) describe the covariance of the an- gular Wigner function under Galilean transformations in W F 共 p ␪ , v ␪ 兲 ⫽ 具 F, ⌬ˆ ␪ 共 p ␪ , v ␪ 兲 F 典 ␪ , (88) the angular coordinate space. which is represented in the angular coordinate basis as (4) The inner product property reads as follows: T. Hakioğlu Vol. 17, No. 12 / December 2000 / J. Opt. Soc. Am. A 2419 冕 ⫺␲ ␲ dv ␪ 兺 ⬁ p ␪ ⫽⫺⬁ W F共 p ␪ , v ␪ 兲 W G共 p ␪ , v ␪ 兲 ⌿̃m 共 r 兲 ⫽ 冕 ⬁ ⫺⬁ d␭ A m 共 ␭ 兲 ␸ ␭ 共 r 兲 , A n 共 ␭ 兲 ⫽ 具 ␸ ␭ ,⌿̃n 典 r , (103) ⫽ 1 2␲ 冏冕 ⫺␲ ␲ dvF * 共 v 兲 G 共 v 兲 冏 2 ⫽ 1 2␲ 兩 具 F, G 典 ␪ 兩 2 . (97) where we have used the orthogonality relations (9) and (10). From Eqs. (103) the radial part in Eq. (100) be- comes B. Polar Representation of the Wigner Function We now demand that the 2D kernel ⌬ˆ (p, q) in the Carte- 具 ⌿̃n , ⌬ˆ r 共 p r , v r 兲 ⌿̃m 典 r ⫽ 冕 ⫺⬁ ⬁ d␭ A n* 共 ␭ 兲 冕⫺⬁ ⬁ d␭ ⬘ A m 共 ␭ ⬘ 兲 sian representation be equivalent to ⫻ 具 ␸ ␭ ,⌬ˆ r 共 p r , v r 兲 ␸ ␭ ⬘ 典 r . (104) ⌬ˆ 共 p, q兲 ⫽ ⌬ˆ r 共 p r , v r 兲 丢 ⌬ˆ ␪ 共 p ␪ , v ␪ 兲 (98) The radial part in Eq. (104) is given in the radial coor- in the polar representation. We use the log-radial repre- dinate representation by sentation for the radial part in Eq. (98). It is clear that the radial and the angular kernels in Eq. (98), as well as 具 ␸ ␭ , ⌬ˆ r 共 p r , v r 兲 ␸ ␭ ⬘ 典 r 冕 their arguments ( p r , v r ) and ( p ␪ , v ␪ ), respectively, are ⬁ 1 independent of each other. The radial representation of ⫽ d␤ r exp共 ⫺i ␤ r p r 兲 the 2D Wigner function is then given by 2␲ ⫺⬁ W ⌿ 共 p r , v r ; p ␪ , v ␪ 兲 ⫽ 具 ⌿, ⌬ˆ r 共 p r , v r 兲 丢 ⌬ˆ ␪ 共 p ␪ , v ␪ 兲 ⌿ 典 r, ␪ , ⫻ exp共 nv r 兲 ␸ ␭ * 共 v r ⫺ ␤ r /2兲 ␸ ␭ ⬘ 共 v r ⫹ ␤ r /2兲 where ⌬ˆ r and ⌬ˆ ␪ independently act on the radial and the (99) ⫽ 1 2␲ exp关 ⫺iv r 共 ␭ ⫺ ␭ ⬘ 兲兴 ␦ p r ⫺ ␭ ⫹ ␭⬘ 2 冉 . 冊 (105) angular parts, respectively, of the wave function in Eq. Inserting Eq. (105) into Eq. (104), we find that (1). Using Eq. (1) in Eq. (99), we find that 具 ⌿̃n , ⌬ˆ r 共 p r , v r 兲 ⌿̃m 典 r 兺 具 L n ,⌬ˆ ␪ 共 p ␪ , v ␪ 兲 L m 典 ␪ 冕 W ⌿共 p r , v r ; p ␪ , v ␪ 兲 ⫽ 2 ␲ ⬁ n,m苸Z 1 ⫽ d␭ exp共 ⫺i␭v r 兲 A n * 共 p r ⫹ ␭/2兲 A m 共 p r ⫺ ␭/2兲 , 2␲ ⫺⬁ ⫻ 具 ⌿̃n , ⌬ˆ r 共 p r , v r 兲 ⌿̃m 典 r , (100) (106) where ⌿̃n represents the radial part of the wave function which we use in Eq. (101). Finally, an explicit form can ⌿ and L n is the Fourier basis, as given by Eqs. (77). The be given by radial and the angular inner products ( , ) r and ( , ) ␪ in Eq. (100) are defined in Eqs. (8) and (70), respectively. W ⌿共 p r , v r ; p ␪ , v ␪ 兲 Performing the calculations in the angular part, we can 1 present Eq. (100) in a more explicit form as ⫽ 共 2 ␲ 兲 3 n,m苸Z 兺 exp关 ⫺iv ␪ 共 n ⫺ m 兲兴 冠冕 冡 W ⌿共 p r , v r ; p ␪ , v ␪ 兲 ␲ ⫻ d␨ exp兵 ⫺i ␨ 关 p ␪ ⫺ 共 n ⫹ m 兲 /2兴 其 1 ⫽ 兺 2 ␲ n,m苸Z exp关 ⫺iv ␪ 共 n ⫺ m 兲兴 ⫺␲ 冕 ⬁ ⫻ 冠冕 ⫺␲ ␲ d␨ 2␲ exp兵 ⫺i ␨ 关 p ␪ ⫺ 共 n ⫹ m 兲 /2兴 其 冡 ⫻ ⫺⬁ d␭ exp共 ⫺i␭v r 兲 A n * 共 p r ⫹ ␭/2兲 A m 共 p r ⫺ ␭/2兲 . (107) ⫻ 具 ⌿̃n , ⌬ˆ r 共 p r , v r 兲 ⌿̃m 典 r . (101) 3. APPLICATION We now shift our attention to the radial part in Eq. (100). An arbitrary wave function ⌿(r) in L2 (R) can be Although some very specific results exist, an explicitly ca- expanded in the polar representation (r, ␾ ) of r, as in Eq. nonical formulation of the polar (hence radial) Wigner (1). We define an inner product in this space as function has not, to the author’s knowledge, previously been tackled. The study that has most closely ap- 具 ⌿, ⌽ 典 r, ␪ ⫽ 冕 Rd dr⌿ * 共 r兲 ⌽ 共 r兲 . (102) proached this goal is the recent work of Bastiaans and van de Mortel,18 whose research on the Wigner function of a circular aperture was based on an approximate Carte- Comparing the radial part of Eq. (102) with the radial in- sian method specific to the model that they used. How- ner product given in Eq. (8), we find that ␩ ⫽ d/2, where ever, it has been shown here that it is possible to con- d is the dimension of the space. Here we are interested struct a generalized Wigner function formalism directly, in d ⫽ 2 only; hence ␩ ⫽ 1. starting from the radial (log or nonlog) coordinates for Here the most natural representation of the radial part wave functions, which can be represented in a polar ex- is the Mellin basis ␸ ␭ (r), given in Eq. (7), in which we ex- pansion of the form given in Eq. (1). Although it has pand ⌿̃m (r) as been shown here that a Weyl correspondence for this 2420 J. Opt. Soc. Am. A / Vol. 17, No. 12 / December 2000 T. Hakioğlu transformation, relating the right-hand sides of Eqs. (2) where we have used ␩ ⫽ 1. Clearly, the radial Wigner and (98), may not exist, the question whether a coordinate function above vanishes if ln a ⭐ vr . A simple calcula- transformer can be found in the phase space within the tion yields general context of Eq. (62) is very relevant from both the W ␳ 共 CA兲共 p r , v r ; p ␪ , v ␪ 兲 linear optics and the quantum mechanics points of view. The advantage of the Cartesian method, if it can be 1 1 handled exactly and with sufficient generality, over the ⫽ ␦ p ␪ ,0 exp共 2v r 兲 ␲ a 2 2 pr radial one is that the transformation under the action of linear optical systems coincides with the covariance ⫻ sin关 2p r 共 ln a ⫺ v r 兲兴 if v r ⭐ ln a (112) transformations of the Wigner function. This is not the and zero elsewhere. Equation (112) is depicted in Fig. 1 case in the direct radial (logarithmic or nonlogarithmic) for the unit aperture radius a ⫽ 1. situation, as we have already seen. In contrast, the ad- One can obtain the marginal probability distributions vantage of the radial Wigner function is that it becomes for the phase-space variables by integrating (summing) favorable if the initial wave field is more appropriately all other variables as 冕 冕 represented in the angular-momentum (m) expansion, as ⬁ ␲ ⬁ in Eq. (1). If the action of a linear optical device is rep- resented by the function ␳ (r) multiplying the radial field D共rCA兲 共 v r 兲 ⫽ 兺 p ␪ ⫽⫺⬁ ⫺␲ dv ␪ ⫺⬁ dp r W ␳ 共 CA兲共 p r , v r ; p ␪ , v ␪ 兲 ␺ (r), the corresponding Wigner function goes through a 2 convolution similar to that of the Cartesian one,18 which ⫽ exp共 2v r 兲 ⍜ 关 a ⫺ exp共 v r 兲兴 , (113) can be written in one of the four equivalent ways between a2 冕 冕 radial and angular coordinates and momenta as ⬁ ␲ ⬁ 兺 冕 D共rCA兲 共 pr兲 ⫽ dv ␪ dv r W ␳ 共 CA兲共 p r , v r ; p ␪ , v ␪ 兲 W ␺ ⬘ 共 p r⬘ , v r⬘ ; p ␪ , v ␪⬘ 兲 ⫽ dp r 兺 W 共 p⬘ ⫺ p p␪ ␳ r r , v ⬘r ; p ␪⬘ p ␪ ⫽⫺⬁ ⫺␲ ⫺⬁ 1 1 ⫺ p ␪ , v ⬘␪ 兲 W ␺ 共 p r , v ⬘r ; p ␪ , v ⬘␪ 兲 ⫽ , (114) ␲ 1 ⫹ p r2 ⫽ 冕 dv r 兺 W 共 p⬘ , v⬘ ⫺ v p␪ ␳ r r r ; p ␪⬘ D␪ 共 CA兲 共v␪兲 ⫽ 兺 ⬁ p ␪ ⫽⫺⬁ 冕 冕 ⬁ ⫺⬁ dv r ⬁ ⫺⬁ dp r W ␳ CA共 p r , v r ; p ␪ , v ␪ 兲 ⫺ p ␪ , v ⬘␪ 兲 W ␺ 共 p ⬘r , v r ; p ␪ , v ⬘␪ 兲 1 ⫽ ..., etc. (108) ⫽ , (115) 2␲ Below we calculate the Wigner function of the circular aperture, using the polar formalism developed here. We describe the wave function ␳ ( CA) (r) of the circular aper- D共␪CA兲 共 p ␪ 兲 ⫽ 冕 冕 冕 ⫺␲ ␲ dv ␪ ⫺⬁ ⬁ dv r ⫺⬁ ⬁ dp r W ␳ 共 CA兲共 p r , v r ; p ␪ , v ␪ 兲 ture as18 ⫽ ␦ p ␪ ,0 , (116) 1 ␳ 共 CA兲 共 r兲 ⫽ ⍜共 r ⫺ a 兲 冑␲ a 2 ⫽ 再 1/冑␲ a 2 0 if r ⭐ a elsewhere ⇒ ˜␳ 共mCA兲 共 r 兲 冑2 ⫽ ⍜ 共 r ⫺ a 兲 ␦ m,0 , (109) a where we have used Eq. (1). Inserting relation (109) into Eq. (101), we find that 1 W ␳ 共 CA兲共 p r , v r ; p ␪ , v ␪ 兲 ⫽ ␦ p ␪ ,0W ˜␳ 共 CA兲共 p r , v r 兲 . (110) 2␲ Using Eq. (30), we calculate the radial part in Eq. (110) from W ˜␳ 共 CA兲共 p r , v r 兲 ⫽ 1 ␲a2 冕 ⫺⬁ ⬁ d␤ r exp共 ⫺i ␤ r p r 兲 exp共 2v r 兲 Fig. 1. Radial part of the Wigner function [W ⌿̃ ( CA) ( p r , v r ) in re- lations (112)] for the circular aperture of unit radius versus the ⫻ ⍜ 关 a ⫺ exp共 v r ⫺ ␤ r /2兲兴 ⍜ 关 a ⫺ exp共 v r ⫹ ␤ r /2兲兴 , phase-space variables p r , v r . The Wigner function vanishes for (111) v r ⭐ 0. T. Hakioğlu Vol. 17, No. 12 / December 2000 / J. Opt. Soc. Am. A 2421 where all marginal distributions are normalized to unity. tum mechanics,’’ Physica (Amsterdam) 12, 405–460 (1946); One can also equivalently represent Eq. (111) in terms of J. E. Moyal, ‘‘Quantum mechanics as a statistical theory,’’ the pseudo Wigner function given in Eq. (61), using Eq. Proc. Cambridge Philos. Soc. 45, 99–124 (1949). (63). 4. A. Verçin, ‘‘Metaplectic covariance of the Weyl–Wigner– Groenewold–Moyal quantization and beyond,’’ Ann. Phys. (N.Y.) 266, 503–523 (1998); T. Dereli and A. Verçin, ‘‘W ⬁ co- variance of the Weyl–Wigner–Groenewold–Moyal quanti- 4. CONCLUSIONS zation,’’ J. Math. Phys. 38, 5515–5530 (1997). 5. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Formulation of physical systems in the phase space by ‘‘Distribution functions in physics: fundamentals,’’ Phys. use of Wigner functions has become a powerful tool in the Rep. 106, 121–167 (1984); N. L. Balazs and B. K. Jennings, application of the fundamental phase-space concepts,5 ‘‘Wigner functions and other distribution functions in mock particularly to signal processing19 and to classical18,20 as phase spaces,’’ Phys. Rep. 104, 347–391 (1984); R. G. Littlejohn, ‘‘Semiclassical evolution of wave packets,’’ Phys. well as quantum optics.21 The existence of a complete or- Rep. 138, 193–291 (1986). thogonal and unitary Weyl–Heisenberg operator basis, 6. P. A. M. Dirac, The Principles of Quantum Mechanics (Clar- given, for example, by the expressions presented in Eqs. endon, Oxford, 1958). (4), for the Cartesian basis, or by those presented in Eqs. 7. J. Twamley, ‘‘Quantum distribution functions for radial ob- (14), for the radial one, is the crucial element for the servables,’’ J. Phys. A 31, 4811–4819 (1998). 8. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Wigner function formulation of the phase space. Tricomi, eds., Tables of Integral Transforms (McGraw-Hill, It is desirable to adopt the symmetries of the physical New York, 1954), Vols. 1 and 2. system in its representations on the phase space. Per- 9. M. Moshinsky and C. Quesne, ‘‘Linear canonical transfor- haps the action-angle basis, as built on the idea of repre- mations and their unitary representations,’’ J. Math. Phys. senting a physical system by its maximum number of 12, 1772–1780 (1971); ‘‘Canonical transformations and ma- trix elements,’’ 1780–1784 (1971); K. B. Wolf, Integral symmetry generators, does this in the most natural Transforms in Science and Engineering (Plenum, New way.16,17 The polar canonical phase-space representa- York, 1979). tions adopted in this study are expected to be important 10. N. M. Atakishiyev, S. M. Nagiyev, L. E. Vicent, and K. B. for paraxial optical systems as well as other systems in Wolf, ‘‘Covariant discretization of axis-symmetric linear op- which a rotational symmetry around a particular axis is tical systems,’’ J. Opt. Soc. Am. A 17, 2301–2314 (2000). 11. R. Simon and K. B. Wolf, ‘‘Structure of the set of paraxial present. A simple example from classical electromagne- optical systems,’’ J. Opt. Soc. Am. A 17, 342–355 (2000). tism is presented in Section 3. 12. T. Curtright, D. Fairlie, and C. Zachos, ‘‘Features of time- Other immediate areas of application of the polar independent Wigner functions,’’ Phys. Rev. D 58, 025002-1– Wigner function are expected to be in the field of atomic 025002-14 (1998). and condensed-matter physics. Specifically, studies on 13. O. Bryngdahl, ‘‘Geometrical transforms in optics,’’ J. Opt. Soc. Am. 64, 1092–1099 (1974); M. J. Bastiaans, ‘‘The quantum wires and dots, as well as studies on Bose– Wigner distribution function applied to optical signals and Einstein phase-space condensation of atomic systems un- systems,’’ Opt. Commun. 25, 26–30 (1978). der external potentials with certain rotational symmetry 14. P. A. M. Dirac, ‘‘The quantum theory of the emission and properties, can be facilitated by use of the polar Wigner absorption of radiation,’’ Proc. R. Soc. London Ser. A 114, function formalism. 243–265 (1927). 15. K. Fujikawa, L. C. Kwek, and C. H. Oh, ‘‘q-deformed oscil- lator algebra and an index theorem for the photon phase operator,’’ Mod. Phys. Lett. A 10, 2543–2551 (1995); ‘‘A ACKNOWLEDGMENTS Schwinger term in q-deformed su(2) algebra,’’ 12, 403–409 (1997). The author is particularly grateful to K. B. Wolf (Centro 16. T. Hakioğlu, ‘‘Finite dimensional Schwinger basis, de- Internacional de Ciencias/Cuernavaca, Mexico), for pro- formed symmetries, Wigner function and an algebraic ap- viding a copy of Ref. 10 prior to its publication and for proach to quantum phase,’’ J. Phys. A 31, 6975–6994 (1998); ‘‘Linear canonical transformations and the quantum stimulating conversations, and to H. Özaktaş (Bilkent phase: unified canonical and algebraic approach,’’ 32, University), for informing the author about Ref. 18. Dis- 4111–4130 (1999); T. Hakioğlu and K. B. Wolf, ‘‘The canoni- cussions with L. Barker (Bilkent University), A. Verçin cal Kravchuk basis for discrete quantum mechanics,’’ J. (Ankara University), and C. Zachos at Argonne National Phys. A 33, 3313–3324 (2000). Laboratories, where parts of this manuscript were writ- 17. T. Hakioğlu and E. Tepedelenlioğlu, ‘‘The action-angle Wigner function: a discrete and algebraic phase space for- ten, are also gratefully acknowledged. This work was malism,’’ J. Phys. A 33, 6357–6383 (2000). supported in part by the U.S. Department of Energy, Di- 18. M. J. Bastiaans and P. G. J. van de Mortel, ‘‘Wigner distri- vision of High Energy Physics, under contract W-31-109- bution function of a circular aperture,’’ J. Opt. Soc. Am. A Eng-38. 13, 1698–1703 (1996). 19. L. Cohen, Time-Frequency Analysis (Prentice-Hall, Engle- T. Hakioğlu can be reached by e-mail at hakioglu wood Cliffs., N.J., 1995); T. A. C. M. Claasen and W. F. G. @theory.hep.anl.gov. Mecklenbräuker, ‘‘The Wigner distribution: a tool for time-frequency signal analysis,’’ Philips J. Res. 35, 217–250 (1980). 20. A. Walther, ‘‘Propagation of the generalized radiance REFERENCES through lenses,’’ J. Opt. Soc. Am. 68, 1606–1610 (1978). 1. H. Weyl, ‘‘Quantenmechanik und Gruppentheorie,’’ Z. Phys. 21. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, 46, 1–39 (1927). ‘‘Measurement of the Wigner distribution and the density 2. E. P. Wigner, ‘‘On the quantum corrections for thermody- matrix of a light mode using optical homodyne tomography: namic equilibrium,’’ Phys. Rev. 40, 749–760 (1932). Application to squeezed states and the vacuum,’’ Phys. Rev. 3. H. J. Groenewold, ‘‘On the principles of elementary quan- Lett. 70, 1244–1247 (1993).