Wigner functions for curved spaces. I. On hyperboloids - UNAM

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 43, NUMBER 12

DECEMBER 2002

Wigner functions for curved spaces. I. On hyperboloids Miguel Angel Alonso,a) George S. Pogosyan,b) and Kurt Bernardo Wolf Centro de Ciencias Fı´sicas, Universidad Nacional Autońoma de Me´xico, Apartado Postal 48–3, Cuernavaca, Morelos 62251, Me´xico

共Received 25 June 2002; accepted 27 August 2002兲 We propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature, in this article on hyperboloids, which returns the correct marginals and has the covariance of the Shapiro functions under SO(D,1) transformations. To the free systems obeying the Laplace–Beltrami equation on the hyperboloid, we add a conic-oscillator potential in the hyperbolic coordinate. As an example, we analyze the one-dimensional case on a hyperbola branch, where this conic-oscillator is the Po¨schl–Teller potential. We present the analytical solutions and plot the computed results. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1518139兴

I. INTRODUCTION

In Hamiltonian systems which a have flat RD configuration space, among the phase space quasiprobability distribution functions the Wigner function1 is the only one covariant under Euclidean translations of phase space.2,3 The present article, and others that will follow it, aim to the construction of Wigner functions on configuration spaces that are conic surfaces, hyperboloids and spheres, which transform under the Lorentz and rotation groups respectively, and which reproduce the traditional Wigner function when the conic contracts to the plane. Quantum motion on spaces of constant curvature is of current interest in various fields of theoretical physics, such as quantum gravity and string theory,4 noncommutative geometry,5 and quantum chaos.6 Hamiltonian systems on conic manifolds have a natural kinetic energy given by the Laplace–Beltrami operator, and, moreover, on these conics also a natural oscillator ‘‘potential’’ can be proposed. In one dimension, this oscillator turns out to be one of the Po¨schl–Teller potentials.7 In this article, subtitled I, we propose a Wigner function on the D-dimensional hyperboloid D H⫹ , which generalizes the ordinary Wigner function on flat phase space. It displays the correct marginals, and returns the traditional form of the Wigner function under Ino¨nu¨ –Wigner contraction to the zero-curvature limit. The elements and background for this assertion are contained in Sec. II, including the Shapiro solutions ⌽ p(D) (x) to the Laplace–Beltrami equation.8 In Sec. III we present our proposed definition of Wigner function on the hyperboloid, and verify the properties of marginality and the contraction limit to flat phase space. Covariance remains an issue because the Wigner function that we propose here follows from the covariance of the basis of wavefunctions, between the argument x and the index p, as if they were canonically conjugate variables. In this context, we reexamine the interpretation of momentum coordinates. In Sec. IV we exemplify the D-dimensional theory with a one-dimensional sui generis oscillator on one branch of a hyperbola. This example may appear to be trivial, because the hyperbola is in most respects equivalent to a straight line. Nevertheless, the resulting Po¨schl–Teller potential is of particular interest because the wavefunctions are also the Clebsch–Gordan 共Wigner coupling兲 coefficients for the three-dimensional Lorentz algebra, so(2,1)⫽sp(2,R)⫽su(1,1). 9 We display a兲

Electronic mail: [email protected] Permanent address: Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia and International Center for Advanced Studies, Yerevan State University, Yerevan, Armenia.

b兲

0022-2488/2002/43(12)/5857/15/$19.00

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© 2002 American Institute of Physics

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J. Math. Phys., Vol. 43, No. 12, December 2002

Alonso, Pogosyan, and Wolf

the Wigner functions of some Po¨schl–Teller wavefunctions; these have not been examined before. Finally, in Sec. V we recapitulate the aim and offer the present outlook of our program. II. ELEMENTS OF PHASE SPACE AND HYPERBOLOIDS

In his fundamental article,1 Wigner proposed a distribution function to represent on phase space the wavefunctions of pure and of mixed states in quantum systems. In this section we recall the definition and properties that we shall generalize from flat to conic spaces, using the Laplace– Beltrami operator and the Shapiro functions. A. Wigner function on flat phase space

In D-dimensional flat configuration space x苸RD , the generalized Dirac basis of plane waves solves the free-space Schro¨dinger equation, which is identical to the Helmholtz equation ⫺⌬ ␾ 共 x兲 ⫽p 2 ␾ 共 x兲 ,

␾ p共 x兲 ⫽exp共 ip"x兲 ,

共1兲

p苸RD .

When we write p⫽⫹(p"p) 1/2, n⫽p/p, and call p⫽ pn the momentum or wavenumber vector, the functions ␾ p(x) represent plane waves in the direction of the unit vector n苸SD⫺1 in the (D ⫺1)-dimensional sphere manifold. In the quantum model with natural units ប⫽1, p has units of inverse length; in the wave optical model, p is the wavenumber of light. The basis of plane wave functions 共1兲 plays many roles: it provides the Fourier transform kernel which bridges the configuration and momentum realizations, it constitutes a basis for representations of the Euclidean group, and it serves for the construction of the R2D -Wigner function of wavefields f (x), g(x) through the equivalent expressions W RD 共 f ,g 兩 x,p兲 ⫽

1 共 2␲ 兲D

⫽

1 共 2␲ 兲D

冕冕

RD

冉冉

冊冊

冉

1 * 1 d D z f x⫺ z e ⫺ip"z g x⫹ z 2 2

1 * D z d z f x⫺ 2 RD

冉

1 ⫻e ⫹ip•(x⫺ 共 1/2兲 z) e ⫺ip•(x⫹ 共 1/2兲 z) g x⫹ z 2 ⫽

1 共 2␲ 兲D

冕

RD

d D x⬘

冉

冕

RD

冊

共2兲

冊

d D x⬙ f 共 x⬘ 兲 * g 共 x⬙ 兲

冊

1 ⫻ ␾ p共 x⬘ 兲 ␦ D x⫺ 共 x⬘ ⫹x⬙ 兲 ␾ p共 x⬙ 兲 * . 2

共3兲

This has the well-known properties of being sesquilinear in the functions, real for f ⫽g, with the marginal projections 兰 RD dpW⫽ f (x) * g(x), 兰 RD dxW⫽˜f (p) * ˜g (p) 共the tilde indicates ordinary Fourier transformation, F: f ⫽˜f ), and covariant under translations in coordinate and momentum spaces T a : f 共 x兲 ⫽ f 共 x⫺a兲 ⇒W RD 共 T a : f ,T a :g 兩 x,p兲 ⫽W RD 共 f ,g 兩 x⫺a,p兲 ,

共4兲

˜T b : f 共 x兲 ⫽e ib"x f 共 x兲 ⇒W RD 共 ˜T b : f ,T ˜ b :g 兩 x,p兲 ⫽W RD 共 f ,g 兩 x,p⫺b兲 ,

共5兲

F: f 共 x兲 ⫽˜f 共 x兲 ⇒W RD 共 F: f ,F:g 兩 x,p兲 ⫽W RD 共 f ,g 兩 p,⫺x兲 .

共6兲

The last intertwining by the Fourier transform was known when Garcıá–Calderoń and Moshinsky noticed that the Wigner function is covariant also under the larger group of Sp(2D,R) linear canonical transformations of phase space.10 This is exceptional in the sense that the Heisenberg– Weyl algebra 关whose generators are the phase space translations 共4兲 and 共5兲—and the unit that



Wigner functions for curved spaces. I.

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generates a commuting phase factor兴 has the outer automorphism group Sp(2D,R). This accident does not occur for Lorentz algebras, so we should not expect similar covariances of the Wigner function under groups larger than SO(D,1). B. Laplace–Beltrami operator on the hyperboloid

The purpose of this article is to generalize the expression of the Wigner function 共2兲 and 共3兲 with functions on a D-dimensional space x苸RD of constant curvature. This manifold can be seen in an ‘‘ambient’’ space of D⫹1 dimensions as a hyperboloid, with vectors x⫽(x 0 ,x)苸RD⫹1 . D 傺RD⫹1 of hyperbolic radius Consider the upper sheet of the two-sheeted hyperboloid H ⫹ R⬎0, 兩 x 兩 2 ⫽x 20 ⫺x2 ⫽R 2 ,

2 x2 ⫽x 21 ⫹x 22 ⫹¯⫹x D .

共7兲

↑ In this ambient Minkowski space, the isometry group is the Poincare´ group ISO(D,1) ⫹ , in place of the Euclidean group ISO(D) ⫹ of flat space. The Lie algebra so(D,1) has then the standard realization

M j,k ⫽x j ⳵ x k ⫺x k ⳵ x j ,

M 0,k ⫽x 0 ⳵ x k ⫹x k ⳵ x 0 ,

j,k⫽1,2,...,D.

共8兲

↑ , is The second-order Casimir operator, C, which is an invariant under the group SO(D,1) ⫹ D 2 (⫺R times兲 the Laplace–Beltrami operator on H ⫹ , namely

1 1 2 C⫽⫺⌬ LB⫽ 2 R R

冉

兺

1⭐ j⬍k⭐D

M 2j,k ⫺

兺

1⭐k⭐D

冊

共9兲

2 M 0,k .

This operator replaces the Laplacian in the Scho¨dinger equation for hyperbolic curved space. Thus, the Schro¨dinger equation on this space with a potential V(x) is

冉

冊

⫺1 ⌬ ⫹R 2 V 共 x兲 f 共 x兲 ⫽R 2 E f 共 x兲 . 2 ␮ LB

共10兲

In quantum mechanics ␮ ⫽m/ប 2 , where m is the particle mass. For application in paraxial wave optics, we recall the interpretation where the extra term characterizes the refractive index anomaly of the medium, n ⴰ↔ ␮ , n 共 x兲 ⫽n ⴰ ⫺ ␯ 共 x兲 ,

n ⴰ ⫽n 共 0 兲 ,

共11兲

␯ 共 x兲 ↔V 共 x兲 . First we consider the case when the potential is identically zero, V(x)⫽0; a nonzero oscillator potential will be introduced in Sec. II E. For the free case, in the unitary irreducible representation spaces of the D-dimensional Lorentz group belonging to the most degenerate continuous series indicated by p, 11 the operator 共9兲 has a real lower-bound spectrum, as does 共1兲. The wavefunctions of the free system on the hyperboloid are the solutions to the equation ⌬ LB f 共 x兲 ⫽⫺

冋冉冊册 D⫺1 2R

2

⫹ p 2 f 共 x兲 ⫽⫺

p苸R⫹ 0 ⫽[0,⬁),

␭ 共 ␭⫹D⫺1 兲 f 共 x兲 , R2

1 ␭⫽⫺ 共 D⫺1 兲 ⫺ipR. 2

共12兲

Any wavefield of a given wavenumber p is a solution of this equation.


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C. Shapiro functions

A privileged basis for the solutions of the Laplace–Beltrami equation 共12兲 was given by Gel’fand, Graev and Shapiro8 in the form of D-dimensional plane waves of momentum p⫽ pn, with positive wavenumber p and in the direction of a unit vector on the sphere n苸S D⫺1 , ⌽ p(D) 共 x 兲 ⫽

冉

x 0 ⫺n"x R

冊

⫺ 共 1/2兲 (D⫺1)⫺ipR

⫽ 共 cosh ␹ ⫺n"␰ sinh ␹ 兲 ⫺ 共 1/2兲 (D⫺1)⫺ipR ,

共13兲

D (x 2 ⫽R 2 ) will be denoted, according to convewhere functions f (x) on the hyperboloid x苸H ⫹ nience, by

x 0 ⫽⫹ 冑R 2 ⫹x2 ⫽R cosh ␹ ⭓R, f 共 x 兲 ⫽ f 共 x 0 ,x兲 ⫽ f 共 x兲 ,

共14兲 x⫽R ␰ sinh ␹ 苸R , D

␹ 苸R⫹ 0 ,

␰苸S

D⫺1

.

The Shapiro functions 共13兲 are a Dirac basis for functions on the hyperboloid, which are orthogonal and complete over x- and p-spaces: R 共 2␲ 兲D

冕

d D x (D) (D) ⌽ p 共 x 兲 * ⌽ p⬘ 共 x 兲 ⫽N (D) 共 p 兲 ␦ D 共 p⫺p⬘ 兲 , x苸RD x 0

1 共 2␲ 兲D

冕

d Dp ⌽ (D) 共 x 兲 * ⌽ p(D) 共 x ⬘ 兲 ⫽ ␦ D 共 x,x ⬘ 兲 , (D) 共p兲 p p苸RD N

共15兲共16兲

with the measure and Dirac ␦ under 兰 H D d D x⫽R 兰 RD d D x/x 0 , ⫹

冏冉

N (D) 共 p 兲 ⫽

␦ D 共 x,x ⬘ 兲 ⫽

⌫ 共 ipR 兲

1 ⌫ 共 D⫺1 兲 ⫹ipR 2

x0 D ␦ 共 x⫺x⬘ 兲 ⫽ R

冑

冊冏

1⫹

2

共 pR 兲 D⫺1 ,

共17兲

x2 D ␦ 共 x⫺x⬘ 兲 . R2

共18兲

In particular, N (1) (p)⫽1, N (2) (p)⫽coth(pR), and N (3) (p)⫽1. ↑ The Ino¨nu–Wigner contraction limit of the Lorentz to the Euclidean group SO(D,1) ⫹ →ISO(D) ⫹ is the limit R→⬁ in our expressions for vectors with x 0 ⬇R, x2 ⰆR 2 , and p⫽ pn as before, i.e., lim ⌽ p(D) 共 x 兲 ⫽ lim R→⬁

R→⬁

冉

x 0 ⫺x"n R

冊

⫺ 共 1/2兲 (D⫺1)⫺ipR

冉

⬇ lim 1⫺ R→⬁

x"n R

冊

⫺ipR

⫽exp共 ix"p兲 .

共19兲

Correspondingly, limR→⬁ N (D) (p)⫽1 and ␦ D (x,x ⬘ )→ ␦ D (x⫺x⬘ ). D. Momentum space for the hyperboloid

The Shapiro functions 兵 ⌽ p(D) (x) 其 p苸RD in 共13兲 serve as the integral transform kernel between functions of x on the hyperboloid, f (x), and conjugate functions of p, that has the interpretation of momentum or wavenumber space, and is indicated ˜f (p). Using 共14兲 for x,p苸RD , one writes ˜f 共 p兲 ⫽

R 共 2 ␲ 兲 D/2

冕

d D x (D) ⌽ p 共 x兲 * f 共 x兲 , x苸RD x 0

共20兲



f 共 x兲 ⫽

1 2 ␲ 共兲 D/2


冕

d Dp ⌽ (D) 共 x兲 ˜f 共 p兲 . 共p兲 p p苸RD N (D)

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共21兲

This Shapiro transform has been used as a relativistic analog of the Fourier transform 共the physical context here, though, is not that of space–time relativity, as we shall clarify below兲, and is a vector form of one of the two branches of the bilateral Mellin transform.12 Here the Shapiro transform replaces the traditional Fourier transform in the definition of a momentum space p苸RD , canonically conjugate with respect to this basis, to a configuration space of constant curvature. The corresponding Parseval relation is R

冕

冕

d Dx f 共 x兲 * g 共 x兲 ⫽ 共 f , g 兲 H D ⫽ ⫹ x苸RD x 0

d Dp ˜f 共 p兲 * ˜g 共 p兲 . (D) 共p兲 p苸RD N

共22兲

The manifold of momentum p⫽pn苸RD (p苸R⫹ and n苸S D⫺1 ) can be placed also in a (D⫹1)-dimensional ‘‘ambient’’ space, where it occupies the cone ␼⫽( p,p)苸∨ ⫹ . The momentum thus defined by the Shapiro functions has certain features, however, which do not correspond to those of a standard relativistic momentum vector. If f (x) is a monochromatic wavefield with a definite value of p, this wavenumber will not change under SO(D,1) translations of the hyperboloid 共‘‘boosts’’兲, because it is the invariant value of the Casimir operator 共9兲–共12兲. Only the direction of momentum, n, can shift over the sphere; it will do so following the well-known Bargmann deformation of the circle,13 where the colatitude angle ‘‘boosts’’ as tan 21␾哫e⫺␨tan 21␾ for rapidity ␨ 苸R. Quotation marks are used for ‘‘boost’’ because here we mean a translation in the hyperboloid, and not the well-known relativistic acceleration. E. Oscillators on conics

The Laplace–Beltrami equation 共9兲–共12兲 provides the free fields 共whose energy is purely kinetic兲 on the hyperboloid. In Eq. 共10兲 we allowed a potential energy term as in the Schro¨dinger equation of quantum mechanics, by adding a function of position V(x). 7,14,15 A straightforward and useful generalization of the SO(D)-isotropic harmonic oscillator potential from flat to conic D-dimensional configuration space is14 兩 x兩 2 1 1 1 V 共 x兲 ⫽ ␮ ␻ 2 R 2 2 ⫽ ␮ ␻ 2 R 2 tanh2 ␹ ⫽ ␮ ␻ 2 R 2 共 1⫺sech2 ␹ 兲 , 2 2 2 x0

共23兲

¨ where ␹ 苸R⫹ 0 is the hyperbolic angle coordinate defined in Eqs. 共14兲. This is the Poschl–Teller ‘‘secant-hyperbolic-squared’’ trough. III. WIGNER FUNCTION ON THE HYPERBOLOID

With the Shapiro basis of wavefunctions of the free system, we construct now our proposed Wigner function following the double-integral form in Eq. 共3兲 for two wavefunctions, f (x) and g(x), by means of integrals on two hyperboloids, 兩 x ⬘ 兩 ⫽R and 兩 x ⬙ 兩 ⫽R. A. Definition

With the measures in Eqs. 共15兲 and the Shapiro functions in 共13兲, we define the Wigner function on the hyperboloid by W H共 f ,g 兩 x,p兲 ⫽

R2 共 2␲ 兲D

冕

x⬘ 苸RD

d D x⬘ x ⬘0

冕

x⬙ 苸RD

d D x⬙ x ⬙0

f 共 x⬘兲* g共 x⬙兲

⫻⌽ p(D) 共 x ⬘ 兲 ⌬ D 共 x;x ⬘ ,x ⬙ 兲 ⌽ p(D) 共 x ⬙ 兲 * ,

共24兲

where ⌬ D (x;x ⬘ ,x ⬙ ) takes the place of the Dirac delta ␦ D (x⫺ 21 (x⬘ ⫹x⬙ )) on flat space, Eq. 共3兲, and which will be detailed below.


5862



The crucial property that we must require of this ‘‘binding-⌬’’ in 共24兲 is that it should guarantee that x be the midpoint of the geodesic between x ⬘ and x ⬙ , so that all three points lie on the D . We achieve this in the following way:16 given x苸RD⫹1 in the upper sheet of a hyperboloid H ⫹ ˜ D of the same two-sheeted hyperboloid, we build any y苸RD⫹1 on a one-sheeted hyperboloid H radius R, such that it be Minkowski-orthogonal to x, 兩 y 兩 2 ⫽y 20 ⫺y2 ⫽⫺R 2 ,

y⫽ 共 y 0 ,y兲 ,

x 0 y 0 ⫺x"y⫽0.

共25兲

Then, we can express x ⬘ and x ⬙ as vectors obtained from x and y as follows: x ⬘ ⫽x cosh 21 ␶ ⫺y sinh 21 ␶ ,

x ⬙ ⫽x cosh 21 ␶ ⫹y sinh 21 ␶ ,

共26兲

D for all ␶ 苸R. Also, it is easy to show that where x ⬘ ,x ⬙ 苸H ⫹

x ⬘0 x ⬙0 ⫺x⬘ •x⬙ ⫽R 2 cosh ␶ ,

x 0 x ⬘0 ⫺x•x⬘ ⫽x 0 x ⬙0 ⫺x•x⬙ ⫽R 2 cosh 21 ␶ ,

共27兲

i.e., the geodesic distance between x ⬘ and x ⬙ is R ␶ , while x is at 21 R ␶ from both x ⬘ and x ⬙ . The arguments x ⬘ and x ⬙ in the expression 共24兲 thus emulate the arguments x⫾ 21 z in 共2兲 with the parameter 21 ␶ . Using 共18兲 and the parameter ␶ in 共27兲, we propose the binding-⌬ in 共24兲 to be ⌬ D 共 x;x ⬘ ,x ⬙ 兲 ⫽

冉

冊

x0 D x⬘ ⫹x⬙ ␦ x⫺ . R 2 cosh 21 ␶

共28兲

This will yield the correct marginals 共to be seen below兲 due to its properties ⌬ D 共 x;x ⬘ ,x ⬘ 兲 ⫽

x0 D ␦ 共 x⫺x⬘ 兲 , R

R

冕

d Dx D ⌬ 共 x;x ⬘ ,x ⬙ 兲 ⫽1. x苸RD x 0

共29兲

B. Integral forms

The 2D-fold integral form of the Wigner function in 共24兲 contains Dirac ␦ ’s; it can therefore ˜ D leaves the freedom of be brought to a (D⫹1)-fold integral noting that the definition of y苸H D⫺1 . When we change variables from x ⬘ and x ⬙ to x and y rotating y around x on a sphere S according to 共26兲, we reduce the integration to y and ␶ while keeping Minkowski-orthogonality. The proposed Wigner function 共24兲 then becomes W H共 f ,g 兩 x,p兲 ⫽

R2 共 2␲ 兲D

冉

冕

⬁

0

共 sinh ␶ 兲 D⫺1 d ␶

冕

˜D y苸H

d D y ␦ 共 x 0 y 0 ⫺x"y兲

冊冉冊冉

1 1 * 1 1 ⫻ f x cosh ␶ ⫺y sinh ␶ g x cosh ␶ ⫹y sinh ␶ 2 2 2 2

冉

⫻⌽ p(D) x cosh

冊

冊

1 1 1 1 * ␶ ⫺y sinh ␶ ⌽ p(D) x cosh ␶ ⫹y sinh ␶ . 2 2 2 2

共30兲

The Dirac ␦ remaining in 共30兲 can be used to find a third alternative form of the Wigner function. This is obtained with the parametrization of the ambient-space vectors given by x⫽ 共 x 0 ,x兲 ⫽R 共 cosh ␹ , ␰ sinh ␹ 兲 ,

y⫽ 共 y 0 ,y兲 ⫽R 共 sinh ␻ , ␩ cosh ␻ 兲 ,

共31兲

where ␰ and ␩ are unit vectors on the sphere S D⫺1 and ␹ , ␻ 苸R⫹ 0 . The Dirac ␦ in Eq. 共30兲 is then



␦ 共 x 0 y 0 ⫺x"y兲 ⫽

1 cosh ⍀ ␦ 共 ␻ ⫺⍀ 兲 , R 2 cosh ␹


5863

tanh ⍀⫽ ␰"␩ tanh ␹ .

共32兲

with

D ˜⫹ becomes R D (cosh ␻)D⫺1d␻ dD⫺1␩, so the Wigner The differential d D y of the integral in y苸H function 共24兲 becomes a D-fold integral with the structure of 共2兲, viz.,

W H共 f ,g 兩 x,p兲 ⫽

1 共 2␲ 兲D

冉

冕

⬁

0

共 sinh ␶ 兲 D⫺1 d ␶

⫻ f x cosh

冉

冕

兩 y兩 D D⫺1 d ␩ S D⫺1 cosh ␹

冊冉冊冉

1 1 * 1 1 ␶ ⫺y sinh ␶ g x cosh ␶ ⫹y sinh ␶ 2 2 2 2

⫻⌽ p(D) x cosh

冊

冊

1 1 1 1 * ␶ ⫺y sinh ␶ ⌽ p(D) x cosh ␶ ⫹y sinh ␶ , 2 2 2 2

共33兲

with y⫽R 共 sinh ⍀, ␩ cosh ⍀ 兲 ⫽R

共 ␰"␩ tanh ␹ , ␩兲

冑1⫺ 共 ␰"␩ tanh ␹ 兲 2

.

共34兲

C. Marginal projections

The marginal projections obtained by integrating the proposed Wigner function 共24兲 over momentum and configuration space should yield, respectively, f (x) * g(x) and ˜f (p) *˜g (p) as defined in 共20兲 and 共21兲. The two marginals follow from the orthogonality and completeness relations of the Shapiro functions, Eqs. 共15兲 and 共16兲. The integration of the Wigner function over RD momentum space with the measure 1/N (D) (p) in 共17兲 is

冕

d Dp W 共 f ,g 兩 x,p兲 M H共 f ,g 兩 x兲 ⫽ (D) 共p兲 H p苸RD N

冕冕

⫽R 2 ⫽R

d D x⬘ x 0⬘

x⬘ 苸RD

d D x⬘

x⬘ 苸RD

x ⬘0

冕

x⬙ 苸RD

d D x⬙ x 0⬙

f 共 x ⬘ 兲 * g 共 x ⬙ 兲 ⌬ D 共 x;x ⬘ ,x ⬙ 兲 ␦ D 共 x ⬘ ,x ⬙ 兲

f 共 x ⬘ 兲 * g 共 x ⬘ 兲 ⌬ D 共 x;x ⬘ ,x ⬘ 兲 ⫽ f 共 x兲 * g 共 x兲 ,

共35兲

where we used 共16兲, 共18兲, and the first property of the binding-⌬ in 共29兲. Similarly, the integration over RD configuration space with the measure R/x 0 in 共18兲 is M H共 f ,g 兩 p兲 ⫽R ⫽

冕

d Dx

x苸RD

R2 共 2␲ 兲D

冑R 2 ⫹x2

冕

x⬘ 苸RD

W H共 f ,g 兩 x,p兲

d D x⬘ x ⬘0

f 共 x ⬘ 兲 * ⌽ p(D) 共 x ⬘ 兲

冕

x⬙ 苸RD

d D x⬙ x ⬙0

g 共 x ⬙ 兲 ⌽ p(D) 共 x ⬙ 兲 * ⫽˜f 共 p兲 * ˜g 共 p兲 , 共36兲

where we used the second property of the binding-⌬ in 共29兲 and the Shapiro transform 共20兲. Finally, integrating over the whole of phase space with the appropriate measures in the Parseval relation 共22兲, we have the total probability


5864


R

冕

d Dx M H共 f ,g 兩 x兲 ⫽ 共 f , g 兲 H D ⫽ ⫹ x苸RD x 0


冕

d Dp M 共 f ,g 兩 p兲 . 共p兲 H p苸RD N (D)

共37兲

D. Covariance under rotations and conic translations

Under rotations R苸SO(D), wavefunctions f (x 0 ,x) transform through T 共 R兲 : f 共 x 兲 ⫽ f 共 x 0 ,R⫺1 x兲 .

共38兲

In particular, the basis of Shapiro functions 共13兲 transforms as (D) ⫺1 x兲 ⫽⌽ (D) T 共 R兲 :⌽ (D) p n 共 x 0 ,x 兲 ⫽⌽ p n 共 x 0 ,R p Rn共 x 0 ,x 兲 .

共39兲

Applying rotations T(R) to the wavefields f and g in the Wigner function 共24兲, we next change ¯ ⬘ and x⬙ ⫽Rx ¯ ⬙ 共the ambient x 0 -components behave as scalars兲, then use 共39兲 variables to x⬘ ⫽Rx for R⫺1 , noting that the binding-⌬ in 共28兲 is invariant, ⌬ D (x ¯ ;x ¯ ⬘ ,x ¯ ⬙ )⫽⌬ D (x;x ⬘ ,x ⬙ ) for ¯x ⫺1 D D ⫽R x, and so are the measures d x⬘ ⫽d ¯x⬘ . It thus follows that the Wigner function 共24兲 is covariant under rotations, fulfilling W H共 T 共 R兲 : f ,T 共 R兲 :g 兩 x,p兲 ⫽W H共 f ,g 兩 R⫺1 x,R⫺1 p兲 .

共40兲

↑ in the direction of Now consider translations by ␨ 共‘‘boosts’’ of rapidity ␨兲 Bm( ␨ )苸SO(D,1) ⫹ D⫺1 unit m苸S , which transform the ambient space vectors preserving the constant-curvature D for each radius R⬎0. We denote by x储 m and x⬜m the projections of x parallel subspaces x苸H ⫹ and perpendicular to the direction of m, so that x⫽x储 m⫹x⬜m . Then, wavefunctions on the hyperboloid transform as

T 共 Bm共 ␨ 兲兲 : f

冉冊冉

冊

x0 x 0 cosh ␨ ⫺m"x sinh ␨ x储 m ⫽ f x储 m cosh ␨ ⫺x 0 m sinh ␨ . x⬜m x⬜m

共41兲

When this transformation is applied to the plane-wave basis of Shapiro functions, their directions n on the sphere change, and they acquire a multiplier factor: ⫺ 共 1/2兲 (D⫺1)⫺ipR ⌽ p n⬘ 共 x 0 ,x兲 , T 共 Bm共 ␨ 兲兲 :⌽ (D) p n 共 x 0 ,x 兲 ⫽ 共 cosh ␨ ⫹m"n sinh ␨ 兲 (D)

共42兲

where the components of n⬘ 苸S D⫺1 that are orthogonal and parallel to m are

⬘ ⫽ n⬜m

n⬜m , cosh ␨ ⫹n"m sinh ␨

n⬘储 m⫽

n"m cosh ␨ ⫹sinh ␨ m, cosh ␨ ⫹n"m sinh ␨

共43兲

↑ irreducible representation characterized by the invariant wavenumber, within the same SO(D,1) ⫹ p. If the angle from m to n is ␾, it will transform through the well-known Bargmann SO共2,1兲 map of the circle. The expression in the multiplier factor of Eq. 共42兲,

␮ 共 m, ␨ ;n兲 ⫽cosh ␨ ⫹m"n sinh ␨ ,

共44兲

is, not coincidentally, p ⬘ /p—if the (D⫹1)-vector ␼⫽( p,p)苸∨ ⫹ were allowed to transform as a ‘‘lightlike’’ vector in relativity, i.e., without being constrained to its p-sphere. Under the inner product 共16兲, the SO(D,1) boost with the multiplier 共42兲 is unitary nonetheless, because the measure in p-space is d D p⫽ p D dpd N⫺1 n, and while p is invariant, from 共43兲 it follows that d D⫺1 n⫽ ␮ (m, ␨ ;n) D⫺1 d D⫺1 n⬘ . This cancels the absolute square of the multiplier 共44兲 in the




5865

Shapiro functions 共41兲. This type of covariance modulo a multiplier function is determined by the Shapiro function basis; we may call Eqs. 共41兲 and 共42兲 the Shapiro covariance between the conjugate transformations in x and p. When the wavefields in the Wigner function 共24兲 are translated within the hyperboloid by 共41兲, the ambient vectors x are multiplied by a (D⫹1)⫻(D⫹1) 共‘boost’兲 matrix Bm( ␨ ), to x ⬘ 哫x ¯ ⬘ ⫽Bm( ␨ ) ⫺1 x ⬘ and x ⬙ 哫x ¯ ⬙ ⫽Bm( ␨ ) ⫺1 x ⬙ . Under this transformation, the measures are D D ¯ 0⬘ , etc., and so is ⌬ D (x;x ⬘ ,x ⬙ )⫽⌬ D (x ¯ ; ¯x ⬘ ,x ¯ ⬙ ); hence, ¯x again invariant, d x⬘ /x 0⬘ ⫽d ¯x⬘ /x ⫺1 ⫽Bm( ␨ ) x will appear in the first argument of the transformed Wigner function. But the corresponding transformation of p in each of the two Shapiro functions, Eqs. 共42兲 and 共44兲, yields a multiplier factor. The imaginary exponents of ␮ (m, ␨ ;n) cancel, and there remains a positive net multiplier factor: W H共 T 关 Bm共 ␨ 兲兴 : f , T 关 Bm共 ␨ 兲兴 :g 兩 x, p n兲 ⫽ 共 ␮ 共 m, ␨ ;n兲兲 ⫺D⫹1 W H共 f ,g 兩 B共 ␨ 兲 ⫺1 :x, p Bm共 ␨ 兲 ⫺1 :n兲 , 共45兲 where n⬘ ⫽Bm( ␨ ) ⫺1 :n is given by 共43兲. Note that in the D⫽1-dimensional case, the multiplier factor is 1. Covariance of the Wigner function is usually understood in the simple form it has under rotations, as given by 共40兲. Under these transformations, the hyperboloid in the ambient x-space rotates on its axis, and in the momentum plane the circles n of all radii p rotate in synchrony. Translations within the hyperboloid 共45兲, on the other hand, deform the ambient and projected space vectors, x and x, through 共41兲; momentum space is concurrently squeezed in the direction of the translation so that its points move on constant-p circles and with a common Bargmann deformation of the angle. Since areas are not conserved in momentum space, a multiplier function of p is necessary for the Wigner function to ensure the total conservation of probability 共37兲.

E. Contraction limit

We now show that, when f (x) and g(x) are significantly different from zero only within a small, essentially flat patch of the hyperboloid, the definition of the Wigner function in Eq. 共33兲 reduces to the standard Wigner function for flat space in Eq. 共3兲. In 共33兲, the integrand for ␩ 关recall Eqs. 共31兲 and 共34兲兴 will be significant only when RD norms of the vectors fulfill

冏

再

冏

兩 x兩 cosh 21 ␶ ⰆR 1 1 x cosh ␶ ⫾y sinh ␶ ⰆR⇒ 2 2 兩 y兩 sinh 21 ␶ ⰆR

⇒x⬇R 共 1,␹ ␰兲 ,

⇒

sinh ␹ Ⰶ1

⇒

sinh ␶ Ⰶ1

共46兲

y⬇R 共 ␹ ␰"␩, ␩兲 .

共47兲

Also, using the limit in 共19兲, and approximating sinh ␶⬇␶ and cosh 21␶⬇cosh ␹⬇cosh ␻⬇1, the Wigner function in Eq. 共33兲 reduces to W H共 f ,g 兩 x,p兲 ⫽

RD 共 2␲ 兲D

冉

冕␶ ⬁

d␶

D⫺1

0

冕

冊

S D⫺1

d D⫺1 ␩

冉

冊

* 1 1 ⫻ f x 0 ,x⫺ R ␶ ␩ exp共 ⫺iR ␶ ␩"p兲 g x 0 ,x⫹ R ␶ ␩ . 2 2

共48兲

Finally, changing variables to z⫽R ␶ ␩ and noticing that

冕

RD

d D z ¯⫽R D

冕␶ ⬁

0

D⫺1

d␶

冕

S D⫺1

d D⫺1 ␩ ¯ ,

共49兲

we see that 共48兲 reduces to 共2兲.


5866



F. Special case of one dimension

In the case D⫽1, the Wigner function 共33兲 actually coincides in form with the corresponding one-dimensional standard flat space form 共2兲, as we now proceed to show. First, notice that the unit vectors n and ␰ in Eq. 共13兲 are now the unit scalars n, ␰ ⫽⫾1, and that the Shapiro functions become simple exponentials: ⫺ipR ⫽exp共 in ␰ pR ␹ 兲 . ⌽ (1) p 共 R ␰ cosh ␹ ,R ␰ sinh ␹ 兲 ⫽ 关 exp共 ⫺n ␰␹ 兲兴

共50兲

We can now let np哫p and ␰␹ 哫 ␹ with p, ␹ 苸(⫺⬁,⬁), and recognize that 共50兲 is a 1:1 function of only one variable, x 1 苸R, ⌽ (1) p 共 x 1 兲 ⫽exp共 i ␹ pR 兲 ,

x 1 ⫽R sinh ␹ .

共51兲

The arguments x⫽(x 0 ,x 1 ) of the functions f and g in 共33兲 then simplify, in components, to

冉

冉冊

冊

cosh共 ␹ ⫾ 21 ␩ ␶ 兲 1 1 x0 x⫽ 哫x cosh ␶ ⫾ysinh ␶ ⫽R . x1 2 2 sinh共 ␹ ⫾ 21 ␩ ␶ 兲

共52兲

For short, we indicate f (R cosh ␹,R sinh ␹)⫽f(␹). The unit vector ␩ in 共33兲 and 共34兲 also becomes a unit scalar, ␩ ⫽⫾1, and the integral extends over y⫽ ␩ R(sinh ␹,cosh ␹), and ⍀⫽ ␩ ␹ 关see Eq. 共32兲兴. Finally, the integral over ␩ reduces to a sum over ␩ ⫽⫾1, and for ␶ 哫 ␩ ␶ 苸(⫺⬁,⬁), the Wigner function 共33兲 becomes W H共 f ,g 兩 x, p 兲 ⫽

R 2␲

⫽

1 2␲

冉冕冉冕

⬁

冊冊

冉冉

冊冊

1 * 1 d ␶ f ␹ ⫺ ␶ e ⫺ipR ␶ g ␹ ⫹ ␶ 2 2 ⫺⬁ ⬁

1 * 1 d ␷ ˜f p⫺ ␷ e ⫹iR ␷ x ˜g p⫹ ␷ . 2 2 ⫺⬁

共53兲

共54兲

The last expression is the usual flat-space Wigner function in terms of the conjugate wavefunctions on momentum space. Finally, note that for D⫽1, the net multiplier which appears under ‘‘boost’’ transformations in 共45兲 is unity, so standard and Shapiro covariances coincide.

IV. EXAMPLE: OSCILLATOR ON THE HYPERBOLA

We consider the open one-dimensional space which is the upper branch of a hyperbola of fixed radius R⬎0, 1 H⫹ ⫽ 兵共 x 0 ,x 1 兲苸R2 兩 x 20 ⫺x 21 ⫽R 2 其 ,

共55兲

parametrized as usual by the hyperbolic angle ␹ 苸R. A. Laplace–Beltrami operator and the oscillator

When the potential V(x) is a constant 共corresponding to a homogeneous optical medium兲, the D⫽1 Schro¨dinger equation 共10兲 is the free wave equation, and its Shapiro solutions are simply the oscillating exponentials 共51兲, with energy E⫽ p 2 /2␮ ⭓0. Since the Laplace–Beltrami operator on the hyperbola 共55兲 is ⌬ LB⫽R ⫺2 d 2 /d ␹ 2 , the Schro¨dinger equation for the conic oscillator 共23兲 is

冉

冊

⫺1 d 2 ⫺R 2 E 0 sech2 ␹ f 共 ␹ 兲 ⫽R 2 共 E⫺E 0 兲 f 共 ␹ 兲 , 2␮ d␹2

1 E 0⫽ ␮ ␻ 2R 2, 2

共56兲




5867

x 21 1 V 共 ␹ 兲 ⫽ ␮ ␻ 2 R 2 2 ⫽ 冑s 共 s⫹1 兲共 1⫺sech2 ␹ 兲 , 2 x0

共57兲

s⫽⫺ 21 ⫹ 冑共 ␮ ␻ R 2 兲 2 ⫹ 41 .

共58兲

The bound solutions are17

␺ sn 共 ␹ 兲 ⫽ ⫽

2 ⫺(s⫺n) ⌫ 共 s⫺n⫹1 兲

冑

冑

冉

冏

共 s⫺n 兲 ⌫ 共 2s⫺n⫹1 兲 ⫺n, 2s⫺n⫹1 1⫺tanh ␹ sechs⫺n ␹ 2 F 1 s⫺n⫹1 n! 2

冉

冊

1 共 s⫺n 兲 n! ⌫ s⫺n⫹ 共 2 sech ␹ 兲 s⫺n C s⫺n⫹1/2 共 tanh ␹ 兲 , n ␲ ⌫ 共 2s⫺n⫹1 兲 2

冊共59兲

where n is a non-negative integer bounded by s⫹1, and C n␣ ( ␰ ) are the Gegenbauer 共or ultraspherical兲 polynomials18 for ␣ ⬎⫺ 21 . The corresponding quantized values of the energy are quadratic in n, and counted from the lowest level up by

E sn ⫽

␮ ␻ 2R 2 1 ⫺ 共 n⫺s 兲 2 , 2 2␮R2

共60兲

n⫽0,1,2,...⬍s⫹1.

As a check on our concepts, we verify that the contraction limit R→⬁ of this system, when the radius of the hyperbola grows without bound, is the harmonic oscillator on flat space. Since the coefficient s correspondingly grows as s⬇ ␮ ␻ R 2 , the linear-quadratic spectrum of energies in Eq. 共60兲 becomes the linear spectrum of the quantum harmonic oscillator E n ⫽ ␻ (n⫹ 21 ). To implement this limit on the wavefunctions 共59兲, it is convenient to use the following forms for the Gegenbauer polynomials in ␰ ⫽tanh ␹:

C ␣n 共 ␰ 兲 ⫽

冦

共 ⫺1 兲共 1/2兲 n 共 ⫺1 兲

共 1/2兲 (n⫺1)

⌫ 共 ␣ ⫹ 21 n 兲共 21 n 兲 ! ⌫( ␣ )

⌫ 共 ␣ ⫹ 21 (n⫹1) 兲共 21 (n⫺1) 兲 ! ⌫ 共 ␣ 兲

2F 1

冉冉

2 ␰ 2F 1

⫺ 21 n, 12 n⫹ ␣ 1 2

冏冊

␰2 ,

⫺ 21 共 n⫺1 兲 , 21 共 n⫹1 兲 ⫹ ␣ 3 2

n even,

冏冊

␰2 ,

n odd. 共61兲

Then, for ␣ ⫽s⫺n⫹ 12 →⬁, the hypergeometric polynomials simplify: for n even, 2 F 1 (⫺ 21 n, ␣ ; 21 ; ␰ 2 )⬇ 1 F 1 (⫺ 21 n; 21 ; ␣ ␰ 2 )⫽H n ( 冑␣ ␰ ), and similarly for n odd. Replacing this into Eq. 共59兲, with ␰ ⫽tanh ␹⬇sinh ␹⫽x1 /R and cosh ⫺s⫹n␹⬇exp(⫺s tanh2 21␹), we obtain the harmonic oscillator wavefunctions on flat space, 1

冑R

␺ sn 共 ␹ 兲 ⬇

1

冑2 n n! 冑␲ / ␮ ␻

e ⫺ ␮ ␻ x 1 /2 H n 共冑␮ ␻ x 1 兲 . 2

共62兲

The factor 冑R restores the proper normalization on the x 1 axis. In addition to the bound states, the sech-trough Po¨schl–Teller potential also has free states with energy above the asymptotic value of the potential lim␹ →⫾⬁ V( ␹ )⫽ 21 ␮ ␻ 2 R 2 . These scattering solutions contain associated Legendre polynomials of imaginary upper index:


5868



p⫽R 冑2 ␮ E⫺ ␮ 2 ␻ 2 R 2 ⬎0,

␺ p共 ␹ 兲 ⫽

兩 ⌫ 共 1⫺ip 兲兩 ip P ␴ 共 tanh ␹ 兲 , 2␲

共63兲

␴ ⫽ 12 ⫾ 冑␻ 2 ␮ 2 R 4 ⫹ 41 .

These wavefunctions are Dirac-orthonormal.

B. Momentum representation of the wavefunctions

The bound wavefunctions of the Po¨schl–Teller sech-trough potential in the momentum representation, ˜␺ sn (p), are found through the ordinary Fourier transform 关Eqs. 共20兲 and 共21兲 for D ⫽1 and 共51兲兴 of the wavefunctions ␺ sn ( ␹ ) found in Eq. 共59兲. The result is ˜␺ sn 共 p 兲 ⫽

⫽

冑冕冑 R 2␲

R 2

⬁

⫺⬁

d ␹ exp共 ⫺ipR ␹ 兲 ␺ sn 共 ␹ 兲

冉

冏冊

⌫ 共 2s⫺n⫹1 兲兩 ⌫ 共 2 共 s⫺n⫺ipR 兲兲兩 2 ⫺n, 2s⫺n⫹1, 21 共 s⫺n⫺ipR 兲 F 1 3 2 ␲ 共 s⫺n 兲 n! ⌫ 共 s⫺n 兲 2 s⫺n⫹1,s⫺n 1

共64兲 ⫽

共 ⫺i 兲 n R

2 冑␲

冑共 s⫺n 兲 n! ⌫ 共 2s⫺n⫹1 兲

冉

⌫ 共 s 兲 ⌫ 共 s⫹1 兲

冏冉 ⌫

1 共 s⫺n⫺ipR 兲 2

冊冏

2

冊

1 1 1 1 1 ⫻R n ⫺ ipR; 共 s⫺n 兲 , 共 s⫺n 兲 , 共 s⫺n 兲 , 共 s⫺n 兲 ⫹1 , 2 2 2 2 2

共65兲

where R n (z; ␣ , ␤ , ␥ , ␦ ) are the continuous Hahn polynomials.19 On the other hand, the unbound solutions 共63兲 are not square-integrable, so their Fourier transform must be performed allowing for the phase difference between asymptotic incoming and outgoing waves, as determined by the scattering properties of this Po¨schl–Teller potential. We shall not further detail the free states here; they can be found in Ref. 9 among the coupling coefficients between the D ⫹ ⫻D ⫺ → 兺 D ⫹ ⫹ 兰 C irreducible representations series of SO共2,1兲.

C. Wigner function for the oscillator eigenstates on the hyperbola 1 On the one-dimensional hyperbola H ⫹ , the Wigner function 共24兲 collapses to 共53兲 and 共54兲, its usual form in Eqs. 共2兲 and 共3兲 for D⫽1. f 兩 p,⫺x) for the We are interested in the single-function form W( f 兩 x,p)⬅W( f , f 兩 x,p)⫽W(˜f ,˜ Po¨schl–Teller wavefunctions, whose explicit form is in Eq. 共59兲 for ␺ sn (x), and in Eq. 共64兲 for ˜␺ sn (p); we find the latter more amenable to analytic solution. We change the integration to a contour along the imaginary axis, and find

W 共 ␺ sn 兩 ␹ , p 兲 ⫽

s⫺n R2 8 ␲ 2 n! ⌫ 共 2s⫺n⫹1 兲 ⫻

n

兺 m,l⫽0

共 ⫺n 兲 m 共 ⫺n 兲 l ⌫ 共 s⫺n⫹m 兲 ⌫ 共 s⫺n⫹l 兲

⌫ 共 2s⫺n⫹m⫹1 兲 ⌫ 共 2s⫺n⫹l⫹1 兲 1 s I 共 ␹ ,p 兲 , ⌫ 共 s⫺n⫹m⫹1 兲 ⌫ 共 s⫺n⫹l⫹1 兲 l!m! n

共66兲

where



I sn 共 ␹ , p 兲 ⫽⫺

4i R

冕

i⬁

⫺i⬁

⫻e ⫺4 ␹ z ⌫

dz ⌫

冉

冉


冊冉

1 1 1 1 共 s⫺n 兲 ⫺ ipR⫺z ⌫ 共 s⫺n 兲 ⫹ ipR⫹z⫹m 2 2 2 2

冊冉

冊

5869

冊

1 1 1 1 共 s⫺n 兲 ⫹ ipR⫺z ⌫ 共 s⫺n 兲 ⫺ ipR⫹z⫹l . 2 2 2 2

共67兲

The last integral can be computed in the complex plane straightforwardly, and leads to a pair of complex conjugate 2 F 1 -functions of e ⫺4 ␹ . We thus finally have ⌫ 共 2s⫺n⫹m⫹1 兲 ⌫ 共 2s⫺n⫹l⫹1 兲 2R 共 s⫺n 兲 n! e ⫺2 ␹ (s⫺n) ␲ ⌫ 共 2s⫺n⫹1 兲 m,l⫽0 ⌫ 共 s⫺n⫹l⫹1 兲 ⌫ 共 s⫺n⫹m⫹1 兲 ⌫ 共 s⫺n⫹l 兲 n

W 共 ␺ sn 兩 ␹ , p 兲 ⫽

兺

⫻

再

共 ⫺1 兲 l⫹m Re ⌫ 共 ipR 兲 ⌫ 共 s⫺n⫹l⫺ipR 兲 e 2ipR ␹ l!m! 共 n⫺m 兲 ! 共 n⫺l 兲 !

⫻ 2F 1

冉

冏冊冎

s⫺n⫹m,s⫺n⫹l⫺ipR, ⫺4 ␹ e 1⫺ipR

.

共68兲

We note that this expression is suitable for numerical computation only for ␹ ⬎0 because it converges fast, but it holds everywhere analytically, with the reflection symmetries W( ␺ sn 兩 ␹ ,p) ⫽W( ␺ sn 兩 ⫺ ␹ , p)⫽W( ␺ sn 兩 ␹ ,⫺p). The Wigner functions, together with their marginal projections 兩 ␺ sn (x) 兩 2 and 兩 ˜␺ sn (p) 兩 2 , are shown in Fig. 1, for n⫽0,1,2,3, and for the potential depth parameter s⫽4 and 30. It can be appreciated that the Wigner function of the most tightly bound states resemble the familiar Gaussian-bell form of the harmonic oscillator ground state. According to 共60兲, for s⫽4 there are only five bound states (n⫽0,...,4), and as the energy of the state approaches the binding energy, the wavefunction stretches in space with ever-smaller momentum in a neighborhood of the classical turning point. The contraction limit can be appreciated in the s⫽30 column, corresponding to a large binding energy; the Wigner functions for the eigenstates in Eq. 共59兲 approach the familiar Laguerre–Gaussian form that corresponds to the Wigner function of the harmonic oscillator wavefunctions on flat space. V. CONCLUDING REMARKS

We have generalized the Wigner quasiprobability distribution function by replacing the oscillating exponential functions of the standard version, which are Dirac solutions of the free Schro¨dinger equation on flat space, by the Shapiro functions, because they are solutions to the Laplace– Beltrami equation on a simply connected hyperbolic space, while respecting the midpoint condition through an appropriate Dirac-like ⌬ D (x;x ⬘ ,x ⬙ ). The role of the Fourier transform in the standard version is transfered by the Shapiro functions to a Mellin-like transform between the position and momentum coordinates of phase space. Indeed, the relation between what were called position and momentum variables is actually defined by the argument and index of the basis of Shapiro functions. Thus built, the proposed Wigner function is covariant under the group SO(D,1) of motions of the hyperbola, with the hyperbolic translations extracting a multiplier factor. The correct marginals are found and the contraction limit to flat space returns the standard Wigner function. The transformations of the momentum direction under translations of the hyperbola are the 共unique兲 action of SO(D,1) on the sphere S D . 13 There are several models of Hamiltonian systems where momentum is restricted to a sphere, such as geometric and Helmholtz 共monochromatic兲 optics.20,21 In the first model, a ‘‘Descartes sphere’’ of momentum vectors p, which is of radius 兩 p兩 ⫽n(x) 共the refractive index兲, is associated to each point x in space. This sphere of momentum vectors has been subject to Bargmann’s ‘‘boost’’ transformation in Ref. 22 共which is unique for Lorentz groups acting on spheres兲 and here given by 共43兲, with a canonically conjugate transformation of the ray positions at a plane screen. The resulting phenomenon has been called relativ-


5870



FIG. 1. Wigner functions of the Po¨schl–Teller eigenfunctions ␺ sn ( ␹ ) on a quadrant of phase space 共axes are position ␹ 冑s and momentum pR/ 冑s; the quadrants have reflection symmetry across the axes兲. Rows are numbered by the mode n ⫽0,1,2,3. Left: s⫽4 共so only five states, n⫽0,...,4, are bound兲; right: s⫽30 共so states are bound up to n⫽30). White is the maximum, black is the minimum; the shade at the upper right corner corresponds to zero. From each Wigner function we project up the marginal distribution of position 兩 ␺ sn ( ␹ ) 兩 2 , and right the marginal of momentum 兩 ˜␺ sn (p) 兩 2 .

istic coma, because the aberration in the images projected on moving screens in vacuum is comatic. The Doppler effect is of course absent, so this transformation cannot be called physically relativistic in the Helmholtz case, which involves a configuration space with a nonlocal metric. Indeed, the difference between the various models consists in their space of positions; here, it is the hyperboloid. A Wigner function on spheres will be examined in part II of this title; it is expected to clarify further the use of function bases to define conjugate variables as a simile or substitute for phase space. The context of this work has a wider significance as a model for phase space with noncom-




5871

mutative geometry, although here the noncommutativity is restricted to momentum space only. In one dimension this argument does not apply, but the example of the one-dimensional Wigner function of Po¨schl–Teller wavefunctions is of interest on its own for the traditional quantum mechanical model, and also for the paraxial propagation of light along shallow nonharmonic waveguides whose index of refraction has a sech2 profile, as given by Eq. 共11兲. One of the manifestations of higher symmetry is the appearance of ‘‘closed-form’’ wavefunctions expressed in terms of well-known 共and some not-so-well-known兲 special functions. Generally, the Fourier transforms and Wigner functions of such wavefunctions are again known special functions because symmetry, when it occurs, is displayed in phase space. ACKNOWLEDGMENTS

We are indebted with Natig M. Atakishiyev for his interest and comments on our work, and to Rufat Mir-Kasimov for a crucial remark on the Shapiro basis. We thank the support of the Direccioń General de Asuntos del Personal Acade´mico, Universidad Nacional Autońoma de Me´xico 共 DGAPA-UNAM兲 Grant No. IN112300 Optica Matema´tica. G.S.P. acknowledges the Consejo Nacional de Ciencia y Tecnologıá 共Me´xico兲 for a Ca´tedra Patrimonial Nivel II, and the Armenian National Science and Engineering Foundation for Grant No. PS124-01. E. Wigner, Phys. Rev. 40, 749 共1932兲. H.-W. Lee, Phys. Rep. 259, 147 共1995兲; C. Brif and A. Mann, Phys. Rev. A 59, 971 共1999兲. See, e.g., Feature Issue Wigner Distributions and Phase Space in Optics, edited by G. W. Forbes, V. I. Man’ko, H. M. Ozaktas, R. Simon, and K. B. Wolf, J. Opt. Soc. Am. A 17 共2000兲. 4 M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory I & II 共Cambridge University Press, Cambridge, 1988兲. 5 H. S. Snyder, Phys. Rev. 71, 38 共1947兲; C. N. Yang, ibid. 72, 874 共1947兲. 6 M. C. Gutzwiller, Phys. Scr. 9, 184 共1985兲. 7 C. Grosche, G. S. Pogosyan, and A. N. Sissakian, Phys. Part. Nucl. 27, 244 共1996兲; Fortschr. Phys. 43, 453 共1995兲. 8 I. M. Gel’fand, M. I. Graev, and I. I. Pyatetskiı˘ -Shapiro, Representation Theory and Automorphic Functions 共Saunders, Philadelphia, 1969兲. 9 D. Basu and K. B. Wolf, J. Math. Phys. 24, 478 共1983兲, Eqs. 共6.9兲–共6.12兲; A. Frank and K. B. Wolf, Phys. Rev. Lett. 52, 1737 共1984兲; J. Math. Phys. 26, 973 共1985兲. 10 G. Garcıá-Calderoń and M. Moshinsky, J. Phys. A 13, L185 共1980兲. 11 I. M. Gel’fand and M. A. Naı˘mark, Izv. Akad. Nauk SSSR 共Ser. Mat.兲 11, 411 共1947兲; I. M. Gel’fand, R. A. Minlos, and Z. Y. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications 共Pergamon, New York, 1963兲. 12 K. B. Wolf, Integral Transforms in Science and Engineering 共Plenum, New York, 1979兲, Sec. 8.2 13 V. Bargmann, Ann. Math. 48, 568 共1947兲. 14 P. W. Higgs, J. Phys. A 12, 309 共1979兲; H. I. Leemon, ibid. 12, 489 共1979兲. 15 E. Schro¨dinger, Proc. R. Irish Soc. 46, 9 共1941兲; 46, 183 共1941兲; 47, 53 共1941兲. 16 M. A. Alonso, J. Opt. Soc. Am. A 18, 910 共2001兲. 17 L. D. Landau and E. Lifshitz, Quantum Mechanics 共Addison-Wesley, Reading, MA, 1968兲. 18 A. Erde´lyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. 2 共McGraw-Hill, New York, 1953兲; formula 10 on p. 110. 19 W. Hahn, Math. Nachr. 2, 4 共1949兲. 20 K. B. Wolf and G. Kro¨tzsch, Eur. J. Phys. 16, 14 共1995兲; K. B. Wolf, ‘‘Elements of Euclidean optics,’’ in Lie Methods in Optics, Lecture Notes in Physics, Vol. 352 共Springer Verlag, Heidelberg, 1989兲, Chap. 6, pp. 115–162. 21 K. B. Wolf, M. A. Alonso, and G. W. Forbes, J. Opt. Soc. Am. A 16, 2476 共1999兲. 22 N. M. Atakishiyev, W. Lassner, and K. B. Wolf, J. Math. Phys. 30, 2457 共1989兲; 30, 2463 共1989兲; K. B. Wolf, J. Opt. Soc. Am. A 10, 1925 共1993兲; E. V. Kurmyshev and K. B. Wolf, Phys. Rev. A 47, 3365 共1993兲. 1 2 3


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