Abstract

The two-dimensional Wigner function is examined in polar canonical coordinates, and covariance properties under the action of affine canonical transformations are derived.

© 2000 Optical Society of America

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References

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  1. H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–39 (1927).
    [CrossRef]
  2. E. P. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–760 (1932).
    [CrossRef]
  3. H. J. Groenewold, “On the principles of elementary quantum mechanics,” Physica (Amsterdam) 12, 405–460 (1946);J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
    [CrossRef]
  4. A. Verçin, “Metaplectic covariance of the Weyl–Wigner–Groenewold–Moyal quantization and beyond,” Ann. Phys. (N.Y.) 266, 503–523 (1998);T. Dereli, A. Verçin, “W∞ covariance of the Weyl–Wigner–Groenewold–Moyal quantization,” J. Math. Phys. 38, 5515–5530 (1997).
    [CrossRef]
  5. M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984);N. L. Balazs, B. K. Jennings, “Wigner functions and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984);R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
    [CrossRef]
  6. P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, 1958).
  7. J. Twamley, “Quantum distribution functions for radial observables,” J. Phys. A 31, 4811–4819 (1998).
    [CrossRef]
  8. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, eds., Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vols. 1 and 2.
  9. M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971);“Canonical transformations and matrix elements,” 1780–1784 (1971); K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
    [CrossRef]
  10. N. M. Atakishiyev, S. M. Nagiyev, L. E. Vicent, K. B. Wolf, “Covariant discretization of axis-symmetric linear optical systems,” J. Opt. Soc. Am. A 17, 2301–2314 (2000).
    [CrossRef]
  11. R. Simon, K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
    [CrossRef]
  12. T. Curtright, D. Fairlie, C. Zachos, “Features of time-independent Wigner functions,” Phys. Rev. D 58, 025002-1–025002-14 (1998).
    [CrossRef]
  13. O. Bryngdahl, “Geometrical transforms in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974);M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  14. P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. London Ser. A 114, 243–265 (1927).
    [CrossRef]
  15. K. Fujikawa, L. C. Kwek, C. H. Oh, “q-deformed oscillator algebra and an index theorem for the photon phase operator,” Mod. Phys. Lett. A 10, 2543–2551 (1995);“A Schwinger term in q-deformed su(2) algebra,” Mod. Phys. Lett. A 12, 403–409 (1997).
    [CrossRef]
  16. T. Hakioğlu, “Finite dimensional Schwinger basis, deformed symmetries, Wigner function and an algebraic approach to quantum phase,” J. Phys. A 31, 6975–6994 (1998);“Linear canonical transformations and the quantum phase: unified canonical and algebraic approach,” J. Phys. A 32, 4111–4130 (1999); T. Hakioğlu, K. B. Wolf, “The canonical Kravchuk basis for discrete quantum mechanics,” J. Phys. A 33, 3313–3324 (2000).
    [CrossRef]
  17. T. Hakioğlu, E. Tepedelenlioğlu, “The action-angle Wigner function: a discrete and algebraic phase space formalism,” J. Phys. A 33, 6357–6383 (2000).
    [CrossRef]
  18. M. J. Bastiaans, P. G. J. van de Mortel, “Wigner distribution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996).
    [CrossRef]
  19. L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs., N.J., 1995); T. A. C. M. Claasen, W. F. G. 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 through lenses,” J. Opt. Soc. Am. 68, 1606–1610 (1978).
    [CrossRef]
  21. D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
    [CrossRef] [PubMed]

2000 (3)

1998 (4)

T. Hakioğlu, “Finite dimensional Schwinger basis, deformed symmetries, Wigner function and an algebraic approach to quantum phase,” J. Phys. A 31, 6975–6994 (1998);“Linear canonical transformations and the quantum phase: unified canonical and algebraic approach,” J. Phys. A 32, 4111–4130 (1999); T. Hakioğlu, K. B. Wolf, “The canonical Kravchuk basis for discrete quantum mechanics,” J. Phys. A 33, 3313–3324 (2000).
[CrossRef]

J. Twamley, “Quantum distribution functions for radial observables,” J. Phys. A 31, 4811–4819 (1998).
[CrossRef]

T. Curtright, D. Fairlie, C. Zachos, “Features of time-independent Wigner functions,” Phys. Rev. D 58, 025002-1–025002-14 (1998).
[CrossRef]

A. Verçin, “Metaplectic covariance of the Weyl–Wigner–Groenewold–Moyal quantization and beyond,” Ann. Phys. (N.Y.) 266, 503–523 (1998);T. Dereli, A. Verçin, “W∞ covariance of the Weyl–Wigner–Groenewold–Moyal quantization,” J. Math. Phys. 38, 5515–5530 (1997).
[CrossRef]

1996 (1)

1995 (1)

K. Fujikawa, L. C. Kwek, C. H. Oh, “q-deformed oscillator algebra and an index theorem for the photon phase operator,” Mod. Phys. Lett. A 10, 2543–2551 (1995);“A Schwinger term in q-deformed su(2) algebra,” Mod. Phys. Lett. A 12, 403–409 (1997).
[CrossRef]

1993 (1)

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

1984 (1)

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984);N. L. Balazs, B. K. Jennings, “Wigner functions and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984);R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

1978 (1)

1974 (1)

1971 (1)

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971);“Canonical transformations and matrix elements,” 1780–1784 (1971); K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
[CrossRef]

1946 (1)

H. J. Groenewold, “On the principles of elementary quantum mechanics,” Physica (Amsterdam) 12, 405–460 (1946);J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

1932 (1)

E. P. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–760 (1932).
[CrossRef]

1927 (2)

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–39 (1927).
[CrossRef]

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. London Ser. A 114, 243–265 (1927).
[CrossRef]

Atakishiyev, N. M.

Bastiaans, M. J.

Beck, M.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Bryngdahl, O.

Cohen, L.

L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs., N.J., 1995); T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution: a tool for time-frequency signal analysis,” Philips J. Res. 35, 217–250 (1980).

Curtright, T.

T. Curtright, D. Fairlie, C. Zachos, “Features of time-independent Wigner functions,” Phys. Rev. D 58, 025002-1–025002-14 (1998).
[CrossRef]

Dirac, P. A. M.

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. London Ser. A 114, 243–265 (1927).
[CrossRef]

P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, 1958).

Fairlie, D.

T. Curtright, D. Fairlie, C. Zachos, “Features of time-independent Wigner functions,” Phys. Rev. D 58, 025002-1–025002-14 (1998).
[CrossRef]

Faridani, A.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Fujikawa, K.

K. Fujikawa, L. C. Kwek, C. H. Oh, “q-deformed oscillator algebra and an index theorem for the photon phase operator,” Mod. Phys. Lett. A 10, 2543–2551 (1995);“A Schwinger term in q-deformed su(2) algebra,” Mod. Phys. Lett. A 12, 403–409 (1997).
[CrossRef]

Groenewold, H. J.

H. J. Groenewold, “On the principles of elementary quantum mechanics,” Physica (Amsterdam) 12, 405–460 (1946);J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

Hakioglu, T.

T. Hakioğlu, E. Tepedelenlioğlu, “The action-angle Wigner function: a discrete and algebraic phase space formalism,” J. Phys. A 33, 6357–6383 (2000).
[CrossRef]

T. Hakioğlu, “Finite dimensional Schwinger basis, deformed symmetries, Wigner function and an algebraic approach to quantum phase,” J. Phys. A 31, 6975–6994 (1998);“Linear canonical transformations and the quantum phase: unified canonical and algebraic approach,” J. Phys. A 32, 4111–4130 (1999); T. Hakioğlu, K. B. Wolf, “The canonical Kravchuk basis for discrete quantum mechanics,” J. Phys. A 33, 3313–3324 (2000).
[CrossRef]

Hillery, M.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984);N. L. Balazs, B. K. Jennings, “Wigner functions and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984);R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

Kwek, L. C.

K. Fujikawa, L. C. Kwek, C. H. Oh, “q-deformed oscillator algebra and an index theorem for the photon phase operator,” Mod. Phys. Lett. A 10, 2543–2551 (1995);“A Schwinger term in q-deformed su(2) algebra,” Mod. Phys. Lett. A 12, 403–409 (1997).
[CrossRef]

Moshinsky, M.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971);“Canonical transformations and matrix elements,” 1780–1784 (1971); K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
[CrossRef]

Nagiyev, S. M.

O’Connell, R. F.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984);N. L. Balazs, B. K. Jennings, “Wigner functions and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984);R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

Oh, C. H.

K. Fujikawa, L. C. Kwek, C. H. Oh, “q-deformed oscillator algebra and an index theorem for the photon phase operator,” Mod. Phys. Lett. A 10, 2543–2551 (1995);“A Schwinger term in q-deformed su(2) algebra,” Mod. Phys. Lett. A 12, 403–409 (1997).
[CrossRef]

Quesne, C.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971);“Canonical transformations and matrix elements,” 1780–1784 (1971); K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
[CrossRef]

Raymer, M. G.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Scully, M. O.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984);N. L. Balazs, B. K. Jennings, “Wigner functions and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984);R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

Simon, R.

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Tepedelenlioglu, E.

T. Hakioğlu, E. Tepedelenlioğlu, “The action-angle Wigner function: a discrete and algebraic phase space formalism,” J. Phys. A 33, 6357–6383 (2000).
[CrossRef]

Twamley, J.

J. Twamley, “Quantum distribution functions for radial observables,” J. Phys. A 31, 4811–4819 (1998).
[CrossRef]

van de Mortel, P. G. J.

Verçin, A.

A. Verçin, “Metaplectic covariance of the Weyl–Wigner–Groenewold–Moyal quantization and beyond,” Ann. Phys. (N.Y.) 266, 503–523 (1998);T. Dereli, A. Verçin, “W∞ covariance of the Weyl–Wigner–Groenewold–Moyal quantization,” J. Math. Phys. 38, 5515–5530 (1997).
[CrossRef]

Vicent, L. E.

Walther, A.

Weyl, H.

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–39 (1927).
[CrossRef]

Wigner, E. P.

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984);N. L. Balazs, B. K. Jennings, “Wigner functions and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984);R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

E. P. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–760 (1932).
[CrossRef]

Wolf, K. B.

Zachos, C.

T. Curtright, D. Fairlie, C. Zachos, “Features of time-independent Wigner functions,” Phys. Rev. D 58, 025002-1–025002-14 (1998).
[CrossRef]

Ann. Phys. (N.Y.) (1)

A. Verçin, “Metaplectic covariance of the Weyl–Wigner–Groenewold–Moyal quantization and beyond,” Ann. Phys. (N.Y.) 266, 503–523 (1998);T. Dereli, A. Verçin, “W∞ covariance of the Weyl–Wigner–Groenewold–Moyal quantization,” J. Math. Phys. 38, 5515–5530 (1997).
[CrossRef]

J. Math. Phys. (1)

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representations,” J. Math. Phys. 12, 1772–1780 (1971);“Canonical transformations and matrix elements,” 1780–1784 (1971); K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Phys. A (3)

J. Twamley, “Quantum distribution functions for radial observables,” J. Phys. A 31, 4811–4819 (1998).
[CrossRef]

T. Hakioğlu, “Finite dimensional Schwinger basis, deformed symmetries, Wigner function and an algebraic approach to quantum phase,” J. Phys. A 31, 6975–6994 (1998);“Linear canonical transformations and the quantum phase: unified canonical and algebraic approach,” J. Phys. A 32, 4111–4130 (1999); T. Hakioğlu, K. B. Wolf, “The canonical Kravchuk basis for discrete quantum mechanics,” J. Phys. A 33, 3313–3324 (2000).
[CrossRef]

T. Hakioğlu, E. Tepedelenlioğlu, “The action-angle Wigner function: a discrete and algebraic phase space formalism,” J. Phys. A 33, 6357–6383 (2000).
[CrossRef]

Mod. Phys. Lett. A (1)

K. Fujikawa, L. C. Kwek, C. H. Oh, “q-deformed oscillator algebra and an index theorem for the photon phase operator,” Mod. Phys. Lett. A 10, 2543–2551 (1995);“A Schwinger term in q-deformed su(2) algebra,” Mod. Phys. Lett. A 12, 403–409 (1997).
[CrossRef]

Phys. Rep. (1)

M. Hillery, R. F. O’Connell, M. O. Scully, E. P. Wigner, “Distribution functions in physics: fundamentals,” Phys. Rep. 106, 121–167 (1984);N. L. Balazs, B. K. Jennings, “Wigner functions and other distribution functions in mock phase spaces,” Phys. Rep. 104, 347–391 (1984);R. G. Littlejohn, “Semiclassical evolution of wave packets,” Phys. Rep. 138, 193–291 (1986).
[CrossRef]

Phys. Rev. (1)

E. P. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Phys. Rev. 40, 749–760 (1932).
[CrossRef]

Phys. Rev. D (1)

T. Curtright, D. Fairlie, C. Zachos, “Features of time-independent Wigner functions,” Phys. Rev. D 58, 025002-1–025002-14 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Physica (Amsterdam) (1)

H. J. Groenewold, “On the principles of elementary quantum mechanics,” Physica (Amsterdam) 12, 405–460 (1946);J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. London Ser. A 114, 243–265 (1927).
[CrossRef]

Z. Phys. (1)

H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1–39 (1927).
[CrossRef]

Other (3)

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, eds., Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vols. 1 and 2.

P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, 1958).

L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs., N.J., 1995); T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution: a tool for time-frequency signal analysis,” Philips J. Res. 35, 217–250 (1980).

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Figures (1)

Fig. 1
Fig. 1

Radial part of the Wigner function [WΨ˜(CA)(pr, vr) in relations (112)] for the circular aperture of unit radius versus the phase-space variables pr, vr. The Wigner function vanishes for vr0.

Equations (148)

Equations on this page are rendered with MathJax. Learn more.

Ψ(r, ϕ; z)=nΨ˜n(r; z)exp(inϕ).
Δˆ(p, q)=Δˆx(px, qx)Δˆy(py, qy),
WΨ(p, q)=Ψ, Δˆ, (p, q)Ψ.
Δˆi=Δˆ(pi, qi)=dαi2πdβi2πexp[i(αiqi-βi pi)]Dˆαi,βi,
Dˆαi,βi=exp[i(αiqˆi+βi pˆi)],
pˆr-irr+η,η.
[vˆr, pˆr]=i.
φλ(r)=(1/2π)riλ-η,
ψ, ϕr0drr2η-1ψ*(r)ϕ(r),ψ, ϕL2(η)(+)
φλ, φλr=0drr2η-1φλ*(r)φλ(r)=δ(λ-λ),
-dλφλ*(r)φλ(r)=δ(r-r)r-2η+1.
ψ, pˆrϕr=pˆrψ, ϕr-iψ*(r)r2ηϕ(r)|0,
ψ(r)=-dλA(λ)ϕλ(r),A(λ)=φλ, ψr.
ψ, ϕr=-dλA*(λ)B(λ),A(λ), B(λ)L2(),
Δˆr(pr, vr)=-dαr2π-dβr2π×exp[-i(αrvr+βrpr)]Dˆr(αr, βr), Dˆr(αr, βr)=exp[i(αrvˆr+βrpˆr)],
Dˆr(0, 0)=Iˆ,
Dˆr(αr, βr)=Dˆr-1(αr, βr)=Dˆr(-αr,-βr),
Tr{Dˆr(αr, βr)}=2πδ(αr)δ(βr),
Dˆr(αr, βr)Dˆr(αr, βr)
=exp[-i(αrβr-βrαr)/2]×Dˆr(αr+αr, βr+βr).
Tr{Dˆr(αr, βr)}-dλφλ, Dˆr(αr, βr)φλr.
Δˆr(pr, vr)=Δˆr(pr, vr),
Tr{Δˆr(pr, vr)}=12π,
Tr{Δˆr(pr, vr)Δˆr(pr, vr)}=12πδ(pr-pr)δ(vr-vr),
-dvr-dprΔˆr(pr, vr)=Iˆ,
-dvrΔˆr(pr, vr)=P˜ˆr(pr),
-dprΔˆr(pr, vr)=Pˆr(vr),
Pˆr(vr)Pˆr(vr)=δ(vr-vr)Pˆr(vr),
dvrPˆr(vr)=Iˆ,
Pˆr(pr)Pˆr(pr)=δ(pr-pr)Pˆr(pr),
dprP˜ˆr(pr)=Iˆ,
ψ, Pˆr(vr)ϕrexp(2ηvr)ψ*(exp vr)ϕ(exp vr),
ψ, P˜ˆr(pr)ϕλA*(pr)B(pr),
Wψ(pr, vr)=ψ, Δˆr(pr, vr)ψr.
Wψ(pr, vr)=12π-dβr exp(-iβrpr)exp(2ηvr)×ψ*[exp(vr+βr/2)]ψ[exp(vr-βr/2)]
WAψ(pr, vr)=12π-dαr exp(iαrvr)A*×(pr+αr/2)A(pr-αr/2).
Wψ(pr, vr)=Wψ*(pr, vr),
-dvrWψ(pr, vr)=ψ, P˜ˆ(pr)ψr=|A(pr)|2,
-dprWψ(pr, vr)=ψ, Pˆ(vr)ψr=exp(2ηvr)|ψ(exp vr)|2,
ψ(r)exp(iβrpˆr)ψ(r)=exp(ηβr)ψ[exp(βr)r].
Wψ(pr, vr)=Wψ(pr, vr+βr),
ψ(r)exp(-iαrvˆr)ψ(r)=r-iαrψ(r).
Wψ(pr, vr)=Wψ(pr-αr, vr),
prvr=gprvr,g=abcd,det g=1,
g1=cos σ-sin σsin σcos σ,g2=cosh τ-sinh τ-sinh τcosh τ,
g3=exp(-χ)00exp(χ),
Tˆg:Δˆr(pr, vr)=TˆgΔˆr(pr, vr)Tˆg-1Δˆr(pr, vr).
Tˆg=-dγr-dδrCr(g)(γr, δr)Dˆr(γr, δr),
Cr(g)(-αr, v-βr)
=exp{i[(βr+βr)-v(αr+αr)]/2}
×Cg(r)(-αr, v-βr)
αrβr=gαrβr.
Cr(g)(αr, βr)=N exp[i(Uαr2+Vβr2+Wαrβr)],
Cr(g1)(αr, βr)=exp(iπ/2)4π[sin(σ/2)]-1×exp[-(i/4)cot(σ/2)(αr2+βr2)],
Cr(g2)(αr, βr)=14π|sinh(τ/2)|-1×exp[-(i/4)coth(τ/2)(αr2-βr2)],
Cr(g3)(αr, βr)=14π|sinh(χ/2)|-1×exp[-(i/2)coth(χ/2)αrβr],
limgiI Cr(gi)(αr, βr)=δ(αr)δ(βr),
limgiI Tˆgi=Iˆ,i=1, 2, 3.
TˆgA(λ1)=-dλ2cr(g)(λ1, λ2)A(λ2),
cr(g1)(λ1, λ2)=exp(iπ/4)2π sin σexp-i2 sin σ×[cos σ(λ12+λ22)-2λ1λ2],
cr(g2)(λ1, λ2)=exp(iπ/4)2π sinh τexp-i2 sinh τ×[cosh τ(λ12+λ22)-2λ1λ2],
cr(g3)(λ1, λ2)=exp(-χ/2)δ[λ2-exp(-χ)λ1].
limgiI cr(gi)(λ1, λ2)=δ(λ1-λ2).
Tˆg1=exp(i2σKˆ1),Kˆ1=14(pˆr2+vˆr2),
Tˆg2=exp(i2τKˆ2),Kˆ2=14(pˆr2-vˆr2),
Tˆg3=exp(i2χKˆ3),Kˆ3=14(pˆrvˆr+vˆrpˆr).
[Kˆ1, Kˆ2]=iKˆ3,[Kˆ1, Kˆ3]=-iKˆ2,
[Kˆ2, Kˆ3]=-iKˆ1,
Cx(g)Dˆxcx(g)[Cx(g)Dˆx][Cy(g)Dˆy]cx(g)cy(g)no!DˆxDˆymDˆr(m)Dˆθ(m)(x, y)?(r, θ)yes?mcr(g)(m) exp(imθ).
exp[iα(r2-1)]ψ(r){exp[i2α(vˆr+vˆr2)]}ψ(r)=exp(i2αvˆr)×exp[i4α(Kˆ1-Kˆ2)]ψ(r)
-dvr-dprWψ(pr, vr)Wϕ(pr, vr)
=12π-dv exp(2ηv)ψ*(exp v)ϕ(exp v)2
=12π|(ψ, ϕ)r|2.
ψ˜(vr)=(FM: ψ)(vr)=-dλ2πexp(-iλvr)(φλ, ψ)r=exp(ηvr)ψ(exp vr).
gvr(r)=exp(-ηvr)δ(vr-ln r),
ψ˜(vr)=(gvr, ψ)r=exp(ηvr)ψ(exp vr).
Wψ(pr, vr)=12π-dβr exp(-iβrpr)ψ˜*(vr+βr/2)×ψ˜(vr-βr/2).
ωψ(Pr, r)=12π+dss-irPr-1r2ηψ*(sr)ψ(r/s),
ψ(r)L2(η)(+).
Wψ(pr, vr)=dPrdrT(pr, vr; Pr, r)wψ(Pr, r),
T=δ(rPr-pr)δ(vr-ln r),
ωψ(Pr, r)=Wψ(pr, vr)| vr=ln rpr=r Pr.
dPrωψ(Pr, r)=r2η-1|ψ(r)|2.
ψexp(iβpˆr)ψωψ(Pr, r)=ωψ(exp(β)Pr, exp(-β)r).
+drωψ(Pr, r),
Dˆθ(n, ζ)F(ϕ)exp(inζ/2)exp(inϕ)F(ϕ+ζ),
-πϕ<π.
fm=12π-dϕF(ϕ)exp(-imϕ),-<m<,
Dˆθ(n, ζ)fm=exp(-inζ)exp(imζ)fm-n.
F, Dˆθ(n, ζ)Gθ=Dˆθ(n, ζ)F, Gθ=Dˆθ-1(n, ζ)F, Gθ,
F, Gθ=-ππdϕF*(ϕ)G(ϕ)
F, Gθ=m=-fm* gm,
Dˆθ(0, 0)=Iˆ,
Dˆθ(n, ζ)=Dˆθ-1(n, ζ)=Dˆθ(-n,-ζ),
Tr[Dˆθ(n, ζ)]=2πδ(ζ)δn,0,
Dˆθ(n, ζ)Dˆθ(n, ζ)=exp[-i(nζ-ζn)/2]×Dˆθ(n+n, ζ+ζ)
Tr[Dˆθ(n, ζ)]m=-Lm, Dˆθ(n, ζ)Lmθ,
Lm(ϕ)=12πexp(imϕ).
Δˆθ(pθ, vθ)=12πn=--ππdζ2π×exp[-i(nvθ+ζpθ)]Dˆθ(n, ζ),
Δˆθ(pθ, vθ)=Δˆθ(pθ, vθ),
Tr{Δˆθ(pθ, vθ)}=12π,
Tr{Δˆθ(pθ, vθ)Δˆθ(pθ, vθ)}=12πδpθ, pθδ(vθ-vθ),
-ππdvθpθ=-Δˆθ(pθ, vθ)=Iˆ,
-ππdvθΔˆθ(pθ, vθ)=P˜ˆθ(pθ),
pθ=-Δˆθ(pθ, vθ)=Pˆθ(vθ),
 Pˆθ(vθ)Pˆθ(vθ)=δ(vθ-vθ)Pˆθ(vθ),dvθPˆ(vθ)=Iˆ,
P˜ˆθ(pθ)P˜ˆθ(pθ)=δpθ, pθP˜ˆθ(pθ),pθ=-P˜ˆθ(pθ)=Iˆ,
F, Pˆθ(vθ)GθF*(vθ)G(vθ),
F, P˜ˆθ(pθ)Gθfpθ* gpθ.
WF(pθ, vθ)=F, Δˆθ(pθ, vθ)Fθ,
WF(pθ, vθ)=12π-ππdζ exp(-iζpθ)F*(vθ-ζ/2)×F(vθ+ζ/2).
WF(pθ, vθ)=WF*(pθ, vθ),
-ππdvθWF(pθ, vθ)=|fpθ|2,
pθ=-WF(pθ, vθ)=|F(vθ)|2,
F(ϕ)=exp(iζpˆθ)F(ϕ)=F(ϕ+ζ),ζ.
WF(pθ, vθ)=WF(pθ, vθ+ζ).
F(ϕ)=exp(ilθˆ)F(ϕ)=exp(ilθ)F(ϕ),l.
WF(pθ, vθ)=WF(pθ-l, vθ).
-ππdvθpθ=-WF(pθ, vθ)WG(pθ, vθ)
=12π-ππdvF*(v)G(v)2=12π|F, Gθ|2.
Δˆ(p, q)=Δˆr(pr, vr)Δˆθ(pθ, vθ)
WΨ(pr, vr; pθ, vθ)=Ψ, Δˆr(pr, vr)Δˆθ(pθ, vθ)Ψr,θ,
WΨ(pr, vr; pθ, vθ)=2πn,mLn,Δˆθ(pθ, vθ)Lmθ×Ψ˜n, Δˆr(pr, vr)Ψ˜mr,
WΨ(pr, vr; pθ, vθ)
=12πn,m exp[-ivθ(n-m)]
×-ππdζ2πexp{-iζ[pθ-(n+m)/2]}
×Ψ˜n, Δˆr(pr, vr)Ψ˜mr.
Ψ, Φr,θ=ddrΨ*(r)Φ(r).
Ψ˜m(r)=-dλ Am(λ)φλ(r),An(λ)=φλ,Ψ˜nr,
Ψ˜n, Δˆr(pr, vr)Ψ˜mr=-dλ An*(λ)-dλAm(λ)
×φλ,Δˆr(pr, vr)φλr.
φλ, Δˆr(pr, vr)φλr
=12π-dβr exp(-iβrpr)×exp(nvr)φλ*(vr-βr/2)φλ(vr+βr/2)=12πexp[-ivr(λ-λ)]δpr-λ+λ2.
Ψ˜n, Δˆr(pr, vr)Ψ˜mr
=12π-dλ exp(-iλvr)An*(pr+λ/2)Am(pr-λ/2),
WΨ(pr, vr; pθ, vθ)
=1(2π)3n,m exp[-ivθ(n-m)]×-ππdζ exp{-iζ[pθ-(n+m)/2]}×-dλ exp(-iλvr)An*(pr+λ/2)Am(pr-λ/2).
Wψ(pr, vr; pθ, vθ)=dprpθWρ(pr-pr, vr; pθ-pθ, vθ)Wψ(pr, vr; pθ, vθ)=dvrpθWρ(pr, vr-vr; pθ-pθ, vθ)Wψ(pr, vr; pθ, vθ)=, etc.
ρ(CA)(r)=1πa2Θ(r-a)=1/πa2ifra0elsewhereρ˜m(CA)(r)=2aΘ(r-a)δm,0,
Wρ(CA)(pr, vr; pθ, vθ)=12πδpθ,0Wρ˜(CA)(pr, vr).
Wρ˜(CA)(pr, vr)
 =1πa2-dβr exp(-iβrpr)exp(2vr)
×Θ[a-exp(vr-βr/2)]Θ[a-exp(vr+βr/2)],
Wρ(CA)(pr, vr; pθ, vθ)=1π2a2δpθ,0 exp(2vr)1pr×sin[2pr(ln a-vr)]ifvrln a
Dr(CA)(vr)=pθ=--ππdvθ-dprWρ(CA)(pr, vr; pθ, vθ)=2a2exp(2vr)Θ[a-exp(vr)],
Dr(CA)(pr)=pθ=--ππdvθ-dvrWρ(CA)(pr, vr; pθ, vθ)=1π11+pr2,
Dθ(CA)(vθ)=pθ=--dvr-dprWρCA(pr, vr; pθ, vθ)=12π,
Dθ(CA)(pθ)=-ππdvθ-dvr-dprWρ(CA)(pr, vr; pθ, vθ)=δpθ,0,

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