- Email: [email protected]
On marginalization of phase-space distribution functions
13 December 1999
Physics Letters A 264 Ž1999. 18–21 www.elsevier.nlrlocaterphysleta
On marginalization of phase-space distribution functions Joachim J. Włodarz
1
Chemical Physics, Department of Chemistry, Technical UniÕersity of Denmark, DTU–207, DK–2800 Lyngby, Denmark Department of Theoretical Chemistry, Silesian UniÕersity, Szkolna 9, PL–40006 Katowice, Poland 2 Received 8 September 1999; received in revised form 17 September 1999; accepted 25 October 1999 Communicated by P.R. Holland
Abstract We discuss marginalization procedures based on integration of quantum phase-space distribution functions over a family of phase-space manifolds. We show that under some conditions the resulting marginals are always nonnegative. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 03.65.-w; 03.65.Sq Keywords: Phase-space distribution functions; Quantum tomography; Quantization rules; Wigner distribution function
Phase-space representations of quantum mechanics have proven to be very useful in many areas, particularly in statistical physics and quantum optics Žfor review see, e.g., w1–3x.. The main feature of the phase-space approach is the possibility of expressing quantum-mechanical expectation values in the same way as it is done in classical statistical mechanics, employing integration over the phase space G : ² A: s
HG d p d q aŽ p,q . r Ž p,q .
Ž 1.
The functions aŽ p,q . and r Ž p,q . in the integrand represent the observable quantity A and the appropriate phase-space distribution function describing the state of the system considered, respectively.
1 2
E-mail: [email protected] Permanent address. E-mail: [email protected]
It is well-known that quantum phase-space distribution functions may assume negative values and hence they cannot be directly interpreted as probability densities in the usual meaning. To improve this situation, various ‘marginalization’ or ‘averaging’ procedures leading to proper probability densities have been suggested, especially for the Wigner distribution function w4x. The most obvious one seems to be the integrating out some of the variables, as it is done in probability theory Žcf., e.g., w5x.. Unfortunately, such marginals are not always positive andror correct from the physical point of view Žcf., e.g., w2x.. Moreover, even the correct marginal position and momentum distributions are not sufficient to reconstruct the quantum state w6x, therefore such simple marginalization is a one-way transformation only. Recently, there has been a renewed interest in this subject, in connection with the tomographic reconstruction of quantum states, proposed first in the
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 7 5 9 - 8
J.J. Włodarzr Physics Letters A 264 (1999) 18–21
context of the Wigner distribution function by Bertrand and Bertrand w7x and shortly after by Vogel and Risken w8x. In this case the respective marginals were defined as phase-space line integrals ŽRadon transforms. of the Wigner distribution function r W Ž p,q .: w Ž x ,u . s
1
Ž 2p .
2
HG d p d q d Ž x y qcos u y psin u .
=r W Ž p,q .
Ž 2.
For a recent review on this topic see, e.g. w9x, and w10x. In this Letter we will discuss marginalization of phase-space distributions through integration over a family of phase-space manifolds M Ž j . parametrized by j s Ž j 1 , j 2 , . . . , j n .
mŽ j . s
HM Ž j . d p d q r Ž p,q .
Ž 3.
Such transformations are generalizations of the Radon transform used in quantum state tomography w13x and were intensively studied in the field of integral geometry w11,12x. In the context of quantum phasespace distribution functions they were, to the best of our knowledge, first discussed by the present author w14x. We will perform our considerations within the framework of D-representations introduced by Srinivas and Wolf w15x and studied further by many authors Žsee, e.g., w16,17x.. This class contains all particular phase-space representations of quantum mechanics being of practical interest, including the Weyl–Wigner–Moyal one as the simplest and also most regular case. For simplicity sake, we restrict ourselves in the following to the case when G ' R 2 and when M Ž j . is a family of sufficiently regular curves in the phase-space plane, described by the equation w Ž p,q; j . s 0. We assume also that the phase-space function w Ž p,q; j . corresponds to a ‘quadrature observable’ F , represented alternatively by a quantum ˆ related to w Ž p,q; j . accordmechanical operator F, ing to the correspondence rules specific for the chosen D-representation, which we will express conD cisely by using the convenient s relation sign as D Fˆ s w Ž p,q; j ..
19
Under these assumptions, we could rewrite Eq. Ž3. in the following form:
mŽ j . s
HG d p d q d Ž w Ž p,q ; j . . r Ž p,q .
Ž 4.
where d Ž w Ž p,q; j .. is the d-distribution concentrated on that curve. Using the integral representation of the d-distribution with the exponential expanded into power series we get:
mŽ j . s
1 2p
`
Ži x .
dx
HR Ý
k!
ks0
k
f Ž j ,k .
Ž 5.
where
f Ž j ,k . s
HG d p d q w
k
Ž p,q ; j . r Ž p,q .
Ž 6.
The last equation has the form an expectation value with the function w Ž p,q; j . in k th power, which in general case does not correspond to ² F k : s Tr Fˆ k rˆ , where rˆ is the density operator. It is caused by the fact that phase-space functions corresponding to operator products are nonlocal and noncommutative ‘wD-products’ of the respective components, not the ordinary pointwise function products, D D D ˆs ˆs ˆˆs i.e., if A aŽ p,q . and B bŽ p,q . then AB Ž awD b .Ž p,q .. Let us assume now that w Ž p,q; j . was chosen in such ingenious way that for any k the k-fold wDproduct for the given D-representation is in this particular case equal to the ordinary pointwise k-fold product. Let further rˆ s Ý i j c i c j) < i : ² j < be the expansion of the density operator into the dynamics < i : ² j < of Žorthonormal. eigenvectors < i : of the quadrature operator Fˆ corresponding to the eigenvalue l i . Then we have:
f Ž j ,k . s Tr Fˆ krˆ s Ý l ki < c i < 2
Ž 7.
i
Inserting Eq. Ž7. back into Eq. Ž5. we get, after some algebra, an explicitly nonnegative expression:
m Ž j . s Ý < c i < 2d Ž l i .
Ž 8.
i
Therefore, we have shown that the following theorem holds for any D –representation of quantum mechanics: Theorem. Let M Ž j . be a family of phase-space manifolds parametrized by j g R n , defined by the
J.J. Włodarzr Physics Letters A 264 (1999) 18–21
20
equation w Ž p,q; j . s 0, where w is a sufficiently smooth function. If for any k g N Ž9.
then the quantity
mŽ j . s
HM Ž j . d p d q r Ž p,q . D
is nonnegative for any phase-space distribution rDŽ p,q .. As an illustration, let us consider some examples evolving commonly employed phase-space representations. a. Weyl–Wigner–Moyal representation. We adopt here the following definition of the respective w W product w18x:
Ž aw W b . Ž p,q . s a Ž p,q . exp
"
§ ™ E E
2i
Ep Eq
½ž
§ ™ E E y
Eq Ep
/5
b Ž p,q .
d Ž j 1 p q j 2 q q j 3 . are the only d-shaped WDFs, which in turn indicates that there are no nonlinear curves for which Eq. Ž4. is nonnegative for all possible WDFs. On the other hand, it is evident from the form of Eq. Ž10. that any nonlinear term in x Ž p,q; j . will break Eq. Ž9.. The invariance of WDFs under affine symplectic phase-space transformations Žcf., e.g., w26,27x. and the respective equivariance of the w W -product enables one to simplify the things considerably. Namely, by using such transformations the quantity x Ž p,q; j . in Eq. Ž11. could be expressed as a multiple of the new momentum pX Žor the new position qX . which trivializes the evaluation of Eq. Ž4.. Moreover, the equality x w W . . . w W x s x k can be then transformed to pXw W . . . w W pX s pX k Žor to qXw W . . . w W qX s qX k . which is fulfilled for the Weyl–Wigner–Moyal and some other phase-space representations w28x, including the standard and antistandard representations discussed below. b. Standard and Antistandard representations. The respective w-products can be defined here as follows w24x:
Ž aw S b . Ž p,q .
§ ™ E E
Ž 10 . It could be verified by a direct calculation that the function:
x Ž p,q ; j . s j 1 p q j 2 q q j 3
Ž 11 .
where j i g R, fulfills the requirement contained in Eq. Ž9.. This particular case corresponds exactly to the so-called ‘symplectic tomography’, introduced and investigated extensively by Mancini, Man’ko and Tombesi w19–23x in the context of various approaches to quantum tomography. The respective integral transform is invertible and therefore the resulting marginal distribution could be even used to establish a classical-like description of quantum mechanics w23x. The nonnegativity of the integral transform m Ž j . for all Wigner distribution functions ŽWDFs. implies Žcf., e.g. w2x, par. 2.1., that d Ž x Ž p,q; j .. could be regarded as a Žgeneralized. WDF Žcf w1x. pp. 366–367 for another derivation and discussion.. It could be shown Žcf., e.g., Appendix C in Ref. w25x. that
½ ž /5
s a Ž p,q . exp yi"
Ep Eq
b Ž p,q .
Ž 12 .
for the standard representation, and
Ž awA b . Ž p,q .
§ ™ E E
½ ž /5
s a Ž p,q . exp i "
Eq Ep
b Ž p,q .
Ž 13 .
for the antistandard one. It could be easily verified that the function x Ž p,q; j . defined in Eq. Ž11. above may fulfill the requirement stated in Eq. Ž9. only when one of the parameters, j 1 or j 2 , is equal to zero. The resulting integral transforms are here rather trivial and noninvertible, e.g.:
mS Ž j . s s
HG d p d q d Ž j HG d p r
S
ž
2 qqj3
p,y
j3 j2
/
. r S Ž p,q . Ž 14 .
J.J. Włodarzr Physics Letters A 264 (1999) 18–21
which is, up to rescaling, equivalent to integrating out one of the variables in the respective distribution function r Ž p,q ..
Acknowledgements It is a pleasure to thank Professor Jens Peder Dahl for extremely warm hospitality during my stays in Lyngby and comments on drafts of this paper. Constructive comments received from one of the anonymous referees and the support from the Danish National Research Council is also acknowledged.
References w1x N.L. Balazs, B.K. Jennings, Phys. Rep. 104 Ž1984. 347. w2x M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner, Phys. Rep. 106 Ž1984. 121. w3x H-W. Lee, Phys. Rep. 259 Ž1995. 147. w4x E.P. Wigner, Phys. Rev. 40 Ž1932. 749. w5x W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York, 1968. w6x A. Vogt, Position and Momentum Distributions do not Determine the Quantum State, in: A.R. Marlow ŽEd.., Mathematical Foundations of Quantum Theory, Academic Press, New York, 1978.
21
w7x J. Bertrand, P. Bertrand, Found. of Phys. 17 Ž1987. 397. w8x K. Vogel, H. Risken, Phys. Rev. A 40 Ž1989. 2847. w9x W.P. Schleich, M. Raymer ŽEds.., Quantum state preparation and measurement, J. Mod. Opt. 44, special issues 11 and 12. w10x I.A. Walmsley, L. Waxler, J. Phys. B 31 Ž1998. 1825. w11x I.M. Gelfand, M.I. Graev, N.Ya. Vilenkin, Generalized Functions, vol. 5: Integral Geometry and Representation Theory, Academic Press, New York, 1966. w12x S. Helgason, The Radon Transform, Birkhauser, Boston, ¨ 1980. w13x A. Wunsche, J. Mod. Opt. 44 Ž1997. 2293. ¨ w14x J.J. Włodarz, Ph.D. Thesis, 1990. w15x M.D. Srinivas, E. Wolf, Phys. Rev. D 11 Ž1975. 1477. w16x R. Groblicki, J. Math. Phys. 24 Ž1983. 841. w17x J.J. Włodarz, Int. J. Quantum Chem. 51 Ž1994. 123. w18x M.D. de Groot, L.G. Suttorp, Foundations of Electrodynamics, North Holland, Amsterdam, 1972. w19x S. Mancini, V.I. Man’ko, P. Tombesi, Quantum Semiclass. Opt. 7 Ž1995. 615. w20x S. Mancini, V.I. Man’ko, P. Tombesi, Quantum Semiclass. Opt. 8 Ž1996. 1017. w21x S. Mancini, V.I. Man’ko, P. Tombesi, Quantum Semiclass. Opt. 9 Ž1997. 987. w22x S. Mancini, V.I. Man’ko, P. Tombesi, J. Mod. Opt. 44 Ž1997. 2281. w23x S. Mancini, V.I. Man’ko, P. Tombesi, Found. Phys. 27 Ž1997. 801. w24x C.L. Mehta, J. Math. Phys. 5 Ž1964. 677. w25x N.L. Balazs, Physica A 102 Ž1980. 236. w26x R.G. Littlejohn, Phys. Rep. 138 Ž1986. 193. w27x F.J. Narcowich, J. Math. Phys. 31 Ž1990. 354. w28x S.P. Misra, T.S. Shankara, J. Math. Phys. 9 Ž1968. 299.
Physics Letters A 264 Ž1999. 18–21 www.elsevier.nlrlocaterphysleta
On marginalization of phase-space distribution functions Joachim J. Włodarz
1
Chemical Physics, Department of Chemistry, Technical UniÕersity of Denmark, DTU–207, DK–2800 Lyngby, Denmark Department of Theoretical Chemistry, Silesian UniÕersity, Szkolna 9, PL–40006 Katowice, Poland 2 Received 8 September 1999; received in revised form 17 September 1999; accepted 25 October 1999 Communicated by P.R. Holland
Abstract We discuss marginalization procedures based on integration of quantum phase-space distribution functions over a family of phase-space manifolds. We show that under some conditions the resulting marginals are always nonnegative. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 03.65.-w; 03.65.Sq Keywords: Phase-space distribution functions; Quantum tomography; Quantization rules; Wigner distribution function
Phase-space representations of quantum mechanics have proven to be very useful in many areas, particularly in statistical physics and quantum optics Žfor review see, e.g., w1–3x.. The main feature of the phase-space approach is the possibility of expressing quantum-mechanical expectation values in the same way as it is done in classical statistical mechanics, employing integration over the phase space G : ² A: s
HG d p d q aŽ p,q . r Ž p,q .
Ž 1.
The functions aŽ p,q . and r Ž p,q . in the integrand represent the observable quantity A and the appropriate phase-space distribution function describing the state of the system considered, respectively.
1 2
E-mail: [email protected] Permanent address. E-mail: [email protected]
It is well-known that quantum phase-space distribution functions may assume negative values and hence they cannot be directly interpreted as probability densities in the usual meaning. To improve this situation, various ‘marginalization’ or ‘averaging’ procedures leading to proper probability densities have been suggested, especially for the Wigner distribution function w4x. The most obvious one seems to be the integrating out some of the variables, as it is done in probability theory Žcf., e.g., w5x.. Unfortunately, such marginals are not always positive andror correct from the physical point of view Žcf., e.g., w2x.. Moreover, even the correct marginal position and momentum distributions are not sufficient to reconstruct the quantum state w6x, therefore such simple marginalization is a one-way transformation only. Recently, there has been a renewed interest in this subject, in connection with the tomographic reconstruction of quantum states, proposed first in the
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 7 5 9 - 8
J.J. Włodarzr Physics Letters A 264 (1999) 18–21
context of the Wigner distribution function by Bertrand and Bertrand w7x and shortly after by Vogel and Risken w8x. In this case the respective marginals were defined as phase-space line integrals ŽRadon transforms. of the Wigner distribution function r W Ž p,q .: w Ž x ,u . s
1
Ž 2p .
2
HG d p d q d Ž x y qcos u y psin u .
=r W Ž p,q .
Ž 2.
For a recent review on this topic see, e.g. w9x, and w10x. In this Letter we will discuss marginalization of phase-space distributions through integration over a family of phase-space manifolds M Ž j . parametrized by j s Ž j 1 , j 2 , . . . , j n .
mŽ j . s
HM Ž j . d p d q r Ž p,q .
Ž 3.
Such transformations are generalizations of the Radon transform used in quantum state tomography w13x and were intensively studied in the field of integral geometry w11,12x. In the context of quantum phasespace distribution functions they were, to the best of our knowledge, first discussed by the present author w14x. We will perform our considerations within the framework of D-representations introduced by Srinivas and Wolf w15x and studied further by many authors Žsee, e.g., w16,17x.. This class contains all particular phase-space representations of quantum mechanics being of practical interest, including the Weyl–Wigner–Moyal one as the simplest and also most regular case. For simplicity sake, we restrict ourselves in the following to the case when G ' R 2 and when M Ž j . is a family of sufficiently regular curves in the phase-space plane, described by the equation w Ž p,q; j . s 0. We assume also that the phase-space function w Ž p,q; j . corresponds to a ‘quadrature observable’ F , represented alternatively by a quantum ˆ related to w Ž p,q; j . accordmechanical operator F, ing to the correspondence rules specific for the chosen D-representation, which we will express conD cisely by using the convenient s relation sign as D Fˆ s w Ž p,q; j ..
19
Under these assumptions, we could rewrite Eq. Ž3. in the following form:
mŽ j . s
HG d p d q d Ž w Ž p,q ; j . . r Ž p,q .
Ž 4.
where d Ž w Ž p,q; j .. is the d-distribution concentrated on that curve. Using the integral representation of the d-distribution with the exponential expanded into power series we get:
mŽ j . s
1 2p
`
Ži x .
dx
HR Ý
k!
ks0
k
f Ž j ,k .
Ž 5.
where
f Ž j ,k . s
HG d p d q w
k
Ž p,q ; j . r Ž p,q .
Ž 6.
The last equation has the form an expectation value with the function w Ž p,q; j . in k th power, which in general case does not correspond to ² F k : s Tr Fˆ k rˆ , where rˆ is the density operator. It is caused by the fact that phase-space functions corresponding to operator products are nonlocal and noncommutative ‘wD-products’ of the respective components, not the ordinary pointwise function products, D D D ˆs ˆs ˆˆs i.e., if A aŽ p,q . and B bŽ p,q . then AB Ž awD b .Ž p,q .. Let us assume now that w Ž p,q; j . was chosen in such ingenious way that for any k the k-fold wDproduct for the given D-representation is in this particular case equal to the ordinary pointwise k-fold product. Let further rˆ s Ý i j c i c j) < i : ² j < be the expansion of the density operator into the dynamics < i : ² j < of Žorthonormal. eigenvectors < i : of the quadrature operator Fˆ corresponding to the eigenvalue l i . Then we have:
f Ž j ,k . s Tr Fˆ krˆ s Ý l ki < c i < 2
Ž 7.
i
Inserting Eq. Ž7. back into Eq. Ž5. we get, after some algebra, an explicitly nonnegative expression:
m Ž j . s Ý < c i < 2d Ž l i .
Ž 8.
i
Therefore, we have shown that the following theorem holds for any D –representation of quantum mechanics: Theorem. Let M Ž j . be a family of phase-space manifolds parametrized by j g R n , defined by the
J.J. Włodarzr Physics Letters A 264 (1999) 18–21
20
equation w Ž p,q; j . s 0, where w is a sufficiently smooth function. If for any k g N Ž9.
then the quantity
mŽ j . s
HM Ž j . d p d q r Ž p,q . D
is nonnegative for any phase-space distribution rDŽ p,q .. As an illustration, let us consider some examples evolving commonly employed phase-space representations. a. Weyl–Wigner–Moyal representation. We adopt here the following definition of the respective w W product w18x:
Ž aw W b . Ž p,q . s a Ž p,q . exp
"
§ ™ E E
2i
Ep Eq
½ž
§ ™ E E y
Eq Ep
/5
b Ž p,q .
d Ž j 1 p q j 2 q q j 3 . are the only d-shaped WDFs, which in turn indicates that there are no nonlinear curves for which Eq. Ž4. is nonnegative for all possible WDFs. On the other hand, it is evident from the form of Eq. Ž10. that any nonlinear term in x Ž p,q; j . will break Eq. Ž9.. The invariance of WDFs under affine symplectic phase-space transformations Žcf., e.g., w26,27x. and the respective equivariance of the w W -product enables one to simplify the things considerably. Namely, by using such transformations the quantity x Ž p,q; j . in Eq. Ž11. could be expressed as a multiple of the new momentum pX Žor the new position qX . which trivializes the evaluation of Eq. Ž4.. Moreover, the equality x w W . . . w W x s x k can be then transformed to pXw W . . . w W pX s pX k Žor to qXw W . . . w W qX s qX k . which is fulfilled for the Weyl–Wigner–Moyal and some other phase-space representations w28x, including the standard and antistandard representations discussed below. b. Standard and Antistandard representations. The respective w-products can be defined here as follows w24x:
Ž aw S b . Ž p,q .
§ ™ E E
Ž 10 . It could be verified by a direct calculation that the function:
x Ž p,q ; j . s j 1 p q j 2 q q j 3
Ž 11 .
where j i g R, fulfills the requirement contained in Eq. Ž9.. This particular case corresponds exactly to the so-called ‘symplectic tomography’, introduced and investigated extensively by Mancini, Man’ko and Tombesi w19–23x in the context of various approaches to quantum tomography. The respective integral transform is invertible and therefore the resulting marginal distribution could be even used to establish a classical-like description of quantum mechanics w23x. The nonnegativity of the integral transform m Ž j . for all Wigner distribution functions ŽWDFs. implies Žcf., e.g. w2x, par. 2.1., that d Ž x Ž p,q; j .. could be regarded as a Žgeneralized. WDF Žcf w1x. pp. 366–367 for another derivation and discussion.. It could be shown Žcf., e.g., Appendix C in Ref. w25x. that
½ ž /5
s a Ž p,q . exp yi"
Ep Eq
b Ž p,q .
Ž 12 .
for the standard representation, and
Ž awA b . Ž p,q .
§ ™ E E
½ ž /5
s a Ž p,q . exp i "
Eq Ep
b Ž p,q .
Ž 13 .
for the antistandard one. It could be easily verified that the function x Ž p,q; j . defined in Eq. Ž11. above may fulfill the requirement stated in Eq. Ž9. only when one of the parameters, j 1 or j 2 , is equal to zero. The resulting integral transforms are here rather trivial and noninvertible, e.g.:
mS Ž j . s s
HG d p d q d Ž j HG d p r
S
ž
2 qqj3
p,y
j3 j2
/
. r S Ž p,q . Ž 14 .
J.J. Włodarzr Physics Letters A 264 (1999) 18–21
which is, up to rescaling, equivalent to integrating out one of the variables in the respective distribution function r Ž p,q ..
Acknowledgements It is a pleasure to thank Professor Jens Peder Dahl for extremely warm hospitality during my stays in Lyngby and comments on drafts of this paper. Constructive comments received from one of the anonymous referees and the support from the Danish National Research Council is also acknowledged.
References w1x N.L. Balazs, B.K. Jennings, Phys. Rep. 104 Ž1984. 347. w2x M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner, Phys. Rep. 106 Ž1984. 121. w3x H-W. Lee, Phys. Rep. 259 Ž1995. 147. w4x E.P. Wigner, Phys. Rev. 40 Ž1932. 749. w5x W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York, 1968. w6x A. Vogt, Position and Momentum Distributions do not Determine the Quantum State, in: A.R. Marlow ŽEd.., Mathematical Foundations of Quantum Theory, Academic Press, New York, 1978.
21
w7x J. Bertrand, P. Bertrand, Found. of Phys. 17 Ž1987. 397. w8x K. Vogel, H. Risken, Phys. Rev. A 40 Ž1989. 2847. w9x W.P. Schleich, M. Raymer ŽEds.., Quantum state preparation and measurement, J. Mod. Opt. 44, special issues 11 and 12. w10x I.A. Walmsley, L. Waxler, J. Phys. B 31 Ž1998. 1825. w11x I.M. Gelfand, M.I. Graev, N.Ya. Vilenkin, Generalized Functions, vol. 5: Integral Geometry and Representation Theory, Academic Press, New York, 1966. w12x S. Helgason, The Radon Transform, Birkhauser, Boston, ¨ 1980. w13x A. Wunsche, J. Mod. Opt. 44 Ž1997. 2293. ¨ w14x J.J. Włodarz, Ph.D. Thesis, 1990. w15x M.D. Srinivas, E. Wolf, Phys. Rev. D 11 Ž1975. 1477. w16x R. Groblicki, J. Math. Phys. 24 Ž1983. 841. w17x J.J. Włodarz, Int. J. Quantum Chem. 51 Ž1994. 123. w18x M.D. de Groot, L.G. Suttorp, Foundations of Electrodynamics, North Holland, Amsterdam, 1972. w19x S. Mancini, V.I. Man’ko, P. Tombesi, Quantum Semiclass. Opt. 7 Ž1995. 615. w20x S. Mancini, V.I. Man’ko, P. Tombesi, Quantum Semiclass. Opt. 8 Ž1996. 1017. w21x S. Mancini, V.I. Man’ko, P. Tombesi, Quantum Semiclass. Opt. 9 Ž1997. 987. w22x S. Mancini, V.I. Man’ko, P. Tombesi, J. Mod. Opt. 44 Ž1997. 2281. w23x S. Mancini, V.I. Man’ko, P. Tombesi, Found. Phys. 27 Ž1997. 801. w24x C.L. Mehta, J. Math. Phys. 5 Ž1964. 677. w25x N.L. Balazs, Physica A 102 Ž1980. 236. w26x R.G. Littlejohn, Phys. Rep. 138 Ž1986. 193. w27x F.J. Narcowich, J. Math. Phys. 31 Ž1990. 354. w28x S.P. Misra, T.S. Shankara, J. Math. Phys. 9 Ž1968. 299.