Abstract

We present a photonic wave packet construction that is immune to the decoherence effects induced by the action of the Lorentz group. The amplitudes of a pure quantum state representing the wave packet remain invariant, irrespective of the reference frame into which the wave packet has been transformed. Transmitted information is encoded in the helicity degrees of freedom of two correlated momentum modes. The helicity encoding is considered to be particularly suitable for free-space communication. The integral part of the story is information retrieval on the receiver’s side. We employed probably the simplest possible helicity (polarization) projection measurement originally studied by Peres and Terno. Remarkably, the same conditions ensuring the invariance of the wave packet also guarantee perfect distinguishability in the process of measuring the helicity.

© 2011 Optical Society of America

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References

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  1. A. Peres and D. R. Terno, “Relativistic Doppler effect in quantum communication,” J. Mod. Opt. 50, 1165–1173 (2003).
    [CrossRef]
  2. Arvind and N. Mukunda, “Relativistic operator description of photon polarization,” Pramana 47, 347–359 (1996).
    [CrossRef]
  3. A. Aiello and J. P. Woerdman, “Intrinsic entanglement degradation by multimode detection,” Phys. Rev. A 70, 023808 (2004).
    [CrossRef]
  4. N. T. Lindner and D. Terno, “The effect of focusing on polarization qubits,” J. Mod. Opt. 52, 1177–1188 (2005).
    [CrossRef]
  5. T. Setlälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  6. P. Caban and J. Rembieliński, “Photon polarization and Wigner’s little group,” Phys. Rev. A 68, 042107 (2003).
    [CrossRef]
  7. P. M. Alsing and G. J. Stephenson, “The Wigner rotation for photons in an arbitrary gravitational field,” arXiv:quant-ph/0902.1399.
  8. M. Czachor and M. Wilczewski, “Relativistic Bennett–Brassard cryptographic scheme, relativistic errors, and how to correct them,” Phys. Rev. A 68, 010302 (2003).
    [CrossRef]
  9. P. Caban, “Einstein–Podolsky–Rosen correlations of photons: quantum-field-theory approach,” Phys. Rev. A 76, 052102 (2007).
    [CrossRef]
  10. E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Annals of Math 40, 149–204 (1939).
    [CrossRef]
  11. S. Weinberg, The Quantum Theory of Fields (Cambridge University, 1995).
  12. W.-K. Tung, Group Theory in Physics (World Scientific, 1985).
  13. F. R. Halpern, Special Relativity and Quantum Mechanics(Prentice-Hall, 1968).
  14. R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles (Springer, 2001).
    [CrossRef]
  15. R. M. Gingrich, A. J. Bergou, and C. Adami, “Entangled light in moving frames,” Phys. Rev. A 68, 042102 (2003).
    [CrossRef]
  16. S. D. Bartlett and D. R. Terno, “Relativistically invariant quantum information,” Phys. Rev. A 71, 012302 (2005).
    [CrossRef]
  17. A. Shaji and E. C. G. Sudarshan, “Who’s afraid of not completely positive maps?,” Phys. Lett. A 341, 48–54 (2005).
    [CrossRef]
  18. S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
    [CrossRef]
  19. S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Classical and quantum communication without a shared reference frame,” Phys. Rev. Lett. 91, 027901 (2003).
    [CrossRef] [PubMed]
  20. G. W. Mackey, “Infinite-dimensional group representations,” Bull. Am. Math. Soc. 69, 628–686 (1963).
    [CrossRef]
  21. L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum physics,” in Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981), Vol. 8.
  22. A. Aiello and J. P. Woerdman, “Notes on polarization measurements,” arXiv:quant-ph/0503124v1 (2005).

2007 (2)

P. Caban, “Einstein–Podolsky–Rosen correlations of photons: quantum-field-theory approach,” Phys. Rev. A 76, 052102 (2007).
[CrossRef]

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
[CrossRef]

2005 (3)

S. D. Bartlett and D. R. Terno, “Relativistically invariant quantum information,” Phys. Rev. A 71, 012302 (2005).
[CrossRef]

A. Shaji and E. C. G. Sudarshan, “Who’s afraid of not completely positive maps?,” Phys. Lett. A 341, 48–54 (2005).
[CrossRef]

N. T. Lindner and D. Terno, “The effect of focusing on polarization qubits,” J. Mod. Opt. 52, 1177–1188 (2005).
[CrossRef]

2004 (1)

A. Aiello and J. P. Woerdman, “Intrinsic entanglement degradation by multimode detection,” Phys. Rev. A 70, 023808 (2004).
[CrossRef]

2003 (5)

R. M. Gingrich, A. J. Bergou, and C. Adami, “Entangled light in moving frames,” Phys. Rev. A 68, 042102 (2003).
[CrossRef]

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Classical and quantum communication without a shared reference frame,” Phys. Rev. Lett. 91, 027901 (2003).
[CrossRef] [PubMed]

A. Peres and D. R. Terno, “Relativistic Doppler effect in quantum communication,” J. Mod. Opt. 50, 1165–1173 (2003).
[CrossRef]

P. Caban and J. Rembieliński, “Photon polarization and Wigner’s little group,” Phys. Rev. A 68, 042107 (2003).
[CrossRef]

M. Czachor and M. Wilczewski, “Relativistic Bennett–Brassard cryptographic scheme, relativistic errors, and how to correct them,” Phys. Rev. A 68, 010302 (2003).
[CrossRef]

2002 (1)

T. Setlälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

1996 (1)

Arvind and N. Mukunda, “Relativistic operator description of photon polarization,” Pramana 47, 347–359 (1996).
[CrossRef]

1963 (1)

G. W. Mackey, “Infinite-dimensional group representations,” Bull. Am. Math. Soc. 69, 628–686 (1963).
[CrossRef]

1939 (1)

E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Annals of Math 40, 149–204 (1939).
[CrossRef]

Adami, C.

R. M. Gingrich, A. J. Bergou, and C. Adami, “Entangled light in moving frames,” Phys. Rev. A 68, 042102 (2003).
[CrossRef]

Aiello, A.

A. Aiello and J. P. Woerdman, “Intrinsic entanglement degradation by multimode detection,” Phys. Rev. A 70, 023808 (2004).
[CrossRef]

A. Aiello and J. P. Woerdman, “Notes on polarization measurements,” arXiv:quant-ph/0503124v1 (2005).

Alsing, P. M.

P. M. Alsing and G. J. Stephenson, “The Wigner rotation for photons in an arbitrary gravitational field,” arXiv:quant-ph/0902.1399.

Arvind,

Arvind and N. Mukunda, “Relativistic operator description of photon polarization,” Pramana 47, 347–359 (1996).
[CrossRef]

Bartlett, S. D.

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
[CrossRef]

S. D. Bartlett and D. R. Terno, “Relativistically invariant quantum information,” Phys. Rev. A 71, 012302 (2005).
[CrossRef]

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Classical and quantum communication without a shared reference frame,” Phys. Rev. Lett. 91, 027901 (2003).
[CrossRef] [PubMed]

Bergou, A. J.

R. M. Gingrich, A. J. Bergou, and C. Adami, “Entangled light in moving frames,” Phys. Rev. A 68, 042102 (2003).
[CrossRef]

Biedenharn, L. C.

L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum physics,” in Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981), Vol. 8.

Caban, P.

P. Caban, “Einstein–Podolsky–Rosen correlations of photons: quantum-field-theory approach,” Phys. Rev. A 76, 052102 (2007).
[CrossRef]

P. Caban and J. Rembieliński, “Photon polarization and Wigner’s little group,” Phys. Rev. A 68, 042107 (2003).
[CrossRef]

Czachor, M.

M. Czachor and M. Wilczewski, “Relativistic Bennett–Brassard cryptographic scheme, relativistic errors, and how to correct them,” Phys. Rev. A 68, 010302 (2003).
[CrossRef]

Friberg, A. T.

T. Setlälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Gingrich, R. M.

R. M. Gingrich, A. J. Bergou, and C. Adami, “Entangled light in moving frames,” Phys. Rev. A 68, 042102 (2003).
[CrossRef]

Halpern, F. R.

F. R. Halpern, Special Relativity and Quantum Mechanics(Prentice-Hall, 1968).

Kaivola, M.

T. Setlälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Lindner, N. T.

N. T. Lindner and D. Terno, “The effect of focusing on polarization qubits,” J. Mod. Opt. 52, 1177–1188 (2005).
[CrossRef]

Louck, J. D.

L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum physics,” in Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981), Vol. 8.

Mackey, G. W.

G. W. Mackey, “Infinite-dimensional group representations,” Bull. Am. Math. Soc. 69, 628–686 (1963).
[CrossRef]

Mukunda, N.

Arvind and N. Mukunda, “Relativistic operator description of photon polarization,” Pramana 47, 347–359 (1996).
[CrossRef]

Peres, A.

A. Peres and D. R. Terno, “Relativistic Doppler effect in quantum communication,” J. Mod. Opt. 50, 1165–1173 (2003).
[CrossRef]

Rembielinski, J.

P. Caban and J. Rembieliński, “Photon polarization and Wigner’s little group,” Phys. Rev. A 68, 042107 (2003).
[CrossRef]

Rudolph, T.

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
[CrossRef]

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Classical and quantum communication without a shared reference frame,” Phys. Rev. Lett. 91, 027901 (2003).
[CrossRef] [PubMed]

Setlälä, T.

T. Setlälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Sexl, R. U.

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles (Springer, 2001).
[CrossRef]

Shaji, A.

A. Shaji and E. C. G. Sudarshan, “Who’s afraid of not completely positive maps?,” Phys. Lett. A 341, 48–54 (2005).
[CrossRef]

Shevchenko, A.

T. Setlälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Spekkens, R. W.

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
[CrossRef]

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Classical and quantum communication without a shared reference frame,” Phys. Rev. Lett. 91, 027901 (2003).
[CrossRef] [PubMed]

Stephenson, G. J.

P. M. Alsing and G. J. Stephenson, “The Wigner rotation for photons in an arbitrary gravitational field,” arXiv:quant-ph/0902.1399.

Sudarshan, E. C. G.

A. Shaji and E. C. G. Sudarshan, “Who’s afraid of not completely positive maps?,” Phys. Lett. A 341, 48–54 (2005).
[CrossRef]

Terno, D.

N. T. Lindner and D. Terno, “The effect of focusing on polarization qubits,” J. Mod. Opt. 52, 1177–1188 (2005).
[CrossRef]

Terno, D. R.

S. D. Bartlett and D. R. Terno, “Relativistically invariant quantum information,” Phys. Rev. A 71, 012302 (2005).
[CrossRef]

A. Peres and D. R. Terno, “Relativistic Doppler effect in quantum communication,” J. Mod. Opt. 50, 1165–1173 (2003).
[CrossRef]

Tung, W.-K.

W.-K. Tung, Group Theory in Physics (World Scientific, 1985).

Urbantke, H. K.

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles (Springer, 2001).
[CrossRef]

Weinberg, S.

S. Weinberg, The Quantum Theory of Fields (Cambridge University, 1995).

Wigner, E. P.

E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Annals of Math 40, 149–204 (1939).
[CrossRef]

Wilczewski, M.

M. Czachor and M. Wilczewski, “Relativistic Bennett–Brassard cryptographic scheme, relativistic errors, and how to correct them,” Phys. Rev. A 68, 010302 (2003).
[CrossRef]

Woerdman, J. P.

A. Aiello and J. P. Woerdman, “Intrinsic entanglement degradation by multimode detection,” Phys. Rev. A 70, 023808 (2004).
[CrossRef]

A. Aiello and J. P. Woerdman, “Notes on polarization measurements,” arXiv:quant-ph/0503124v1 (2005).

Annals of Math (1)

E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Annals of Math 40, 149–204 (1939).
[CrossRef]

Bull. Am. Math. Soc. (1)

G. W. Mackey, “Infinite-dimensional group representations,” Bull. Am. Math. Soc. 69, 628–686 (1963).
[CrossRef]

J. Mod. Opt. (2)

A. Peres and D. R. Terno, “Relativistic Doppler effect in quantum communication,” J. Mod. Opt. 50, 1165–1173 (2003).
[CrossRef]

N. T. Lindner and D. Terno, “The effect of focusing on polarization qubits,” J. Mod. Opt. 52, 1177–1188 (2005).
[CrossRef]

Phys. Lett. A (1)

A. Shaji and E. C. G. Sudarshan, “Who’s afraid of not completely positive maps?,” Phys. Lett. A 341, 48–54 (2005).
[CrossRef]

Phys. Rev. A (6)

R. M. Gingrich, A. J. Bergou, and C. Adami, “Entangled light in moving frames,” Phys. Rev. A 68, 042102 (2003).
[CrossRef]

S. D. Bartlett and D. R. Terno, “Relativistically invariant quantum information,” Phys. Rev. A 71, 012302 (2005).
[CrossRef]

P. Caban and J. Rembieliński, “Photon polarization and Wigner’s little group,” Phys. Rev. A 68, 042107 (2003).
[CrossRef]

M. Czachor and M. Wilczewski, “Relativistic Bennett–Brassard cryptographic scheme, relativistic errors, and how to correct them,” Phys. Rev. A 68, 010302 (2003).
[CrossRef]

P. Caban, “Einstein–Podolsky–Rosen correlations of photons: quantum-field-theory approach,” Phys. Rev. A 76, 052102 (2007).
[CrossRef]

A. Aiello and J. P. Woerdman, “Intrinsic entanglement degradation by multimode detection,” Phys. Rev. A 70, 023808 (2004).
[CrossRef]

Phys. Rev. E (1)

T. Setlälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Classical and quantum communication without a shared reference frame,” Phys. Rev. Lett. 91, 027901 (2003).
[CrossRef] [PubMed]

Pramana (1)

Arvind and N. Mukunda, “Relativistic operator description of photon polarization,” Pramana 47, 347–359 (1996).
[CrossRef]

Rev. Mod. Phys. (1)

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609 (2007).
[CrossRef]

Other (7)

S. Weinberg, The Quantum Theory of Fields (Cambridge University, 1995).

W.-K. Tung, Group Theory in Physics (World Scientific, 1985).

F. R. Halpern, Special Relativity and Quantum Mechanics(Prentice-Hall, 1968).

R. U. Sexl and H. K. Urbantke, Relativity, Groups, Particles (Springer, 2001).
[CrossRef]

P. M. Alsing and G. J. Stephenson, “The Wigner rotation for photons in an arbitrary gravitational field,” arXiv:quant-ph/0902.1399.

L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum physics,” in Encyclopedia of Mathematics and Its Applications (Addison-Wesley, 1981), Vol. 8.

A. Aiello and J. P. Woerdman, “Notes on polarization measurements,” arXiv:quant-ph/0503124v1 (2005).

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Figures (2)

Fig. 1
Fig. 1

Four fixed points corresponding to Eqs. (18a, 18b, 18c, 18d) are indicated by points a, b, c, and d. Curves corresponding to a different rotation angle ϖ about the y axis will always meet in these points. The fixed points satisfy Eq. (13). One can notice several gaps on every curve. These are singularities of Eq. (13).

Fig. 2
Fig. 2

Here we demonstrate how the position of fixed points changes for a different momentum direction. The fixed points satisfy Eq. (14), so in contrast to Fig. 1, there is a global (trivial) fixed point in the center that is constant for all ϑ 1 , ϕ 1 . One can notice several gaps on every curve. These are singularities of Eq. (14).

Equations (60)

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| ψ = d μ ( p ) f ( p ) ( α p | p , + + β p | p , ) ,
| ψ = α d μ ( p ) f ( p ) | p , + + β d μ ( p ) f ( p ) | p , .
U ( R ( ϑ , ϕ ) ) | ψ = α d μ ( q ) f ( q ) exp [ i θ p , q ] | q , + + β d μ ( q ) f ( q ) exp [ i θ p , q ] | q , ,
| + k = ( 1 0 0 ) , | k = ( 0 1 0 ) .
S = 1 2 ( 1 i 0 1 i 0 0 0 2 ) .
R ˜ ( ϑ , ϕ ) = 1 2 ( ( cos ϑ + 1 ) exp ( i ϕ ) ( cos ϑ 1 ) exp ( i ϕ ) 2 sin ϑ exp ( i ϕ ) ( cos ϑ 1 ) exp ( i ϕ ) ( cos ϑ + 1 ) exp ( i ϕ ) 2 sin ϑ exp ( i ϕ ) 2 sin ϑ 2 sin ϑ 2 cos ϕ ) ,
S = 1 2 ( 1 1 0 i i 0 0 0 2 ) .
| Φ ( p 1 , p 2 ) ± = 1 2 ( | p 1 , + | p 2 , + ± | p 1 , | p 2 , ) ,
| Ψ ( p 1 , p 2 ) ± = 1 2 ( | p 1 , + | p 2 , + ± | p 1 , | p 2 , ) ,
| + p | p , + = 1 2 ( ( cos ϑ + 1 ) exp ( i ϕ ) ( cos ϑ 1 ) exp ( i ϕ ) 2 sin ϑ ) ,
| p | p , = 1 2 ( ( cos ϑ 1 ) exp ( i ϕ ) ( cos ϑ + 1 ) exp ( i ϕ ) 2 sin ϑ ) .
U ( R z ( λ ) ) : | Φ ( p 1 , p 2 ) ± | Φ ( R z p 1 , R z p 2 ) ± ,
U ( R z ( λ ) ) : | Ψ ( p 1 , p 2 ) ± | Ψ ( R z p 1 , R z p 2 ) ± .
θ p = arctan [ sin ϖ sin ϕ sin ϖ cos ϑ cos ϕ + cos ϖ sin ϑ ] .
sin ϖ sin ϕ 1 sin ϖ cos ϑ 1 cos ϕ 1 + cos ϖ sin ϑ 1 = sin ϖ sin ( ϕ 1 + x ) sin ϖ cos ( ϑ 1 + y ) cos ( ϕ 1 + x ) + cos ϖ sin ( ϑ 1 + y ) ,
sin ϖ sin ϕ 1 sin ϖ cos ϑ 1 cos ϕ 1 + cos ϖ sin ϑ 1 = sin ϖ sin ( ϕ 1 + x ) sin ϖ cos ( ϑ 1 + y ) cos ( ϕ 1 + x ) + cos ϖ sin ( ϑ 1 + y ) .
tan ϖ [ sin ϕ 1 cos ( ϑ 1 + y ) cos ( ϕ 1 + x ) + sin ( ϕ 1 + x ) cos ϑ 1 cos ϕ 1 ) = sin ( ϕ 1 + x ) sin ϑ 1 sin ϕ 1 sin ( ϑ 1 + y ) .
cos ( ϑ 1 + y ) cos ( ϕ 1 + x ) sin ( ϕ 1 + x ) = cos ϑ 1 cos ϕ 1 sin ϕ 1 ,
sin ( ϑ 1 + y ) sin ( ϕ 1 + x ) = sin ϑ 1 sin ϕ 1 .
ϕ 2 = ϕ 1 , ϑ 2 = ϑ 1 ,
ϕ 2 = π ϕ 1 , ϑ 2 = ϑ 1 ,
ϕ 2 = π + ϕ 1 , ϑ 2 = π ϑ 1 ,
ϕ 2 = ϕ 1 , ϑ 2 = ϑ 1 + π .
ϕ 2 = ϕ 1 , ϑ 2 = ϑ 1 + π ,
ϕ 2 = π ϕ 1 , ϑ 2 = π ϑ 1 ,
ϕ 2 = π + ϕ 1 , ϑ 2 = π ϑ 1 ,
ϕ 2 = ϕ 1 , ϑ 2 = ϑ 1 .
U ( R y ( ϖ ) ) : | Φ ( p 1 , p 2 ) ± | Φ ( R y p 1 , R y p 2 ) ± .
U ( R y ( ϖ ) ) : | Ψ ( p 1 , p 2 ) ± | Ψ ( R y p 1 , R y p 2 ) ±
Π k : | p , + p , + = 1 N ( ( cos ϑ + 1 ) exp ( i ϕ ) ( cos ϑ 1 ) exp ( i ϕ ) ) ,
Π k : | p , p , = 1 N ( ( cos ϑ 1 ) exp ( i ϕ ) ( cos ϑ + 1 ) exp ( i ϕ ) ) ,
Π k ( 1 ) Π k ( 2 ) : | Φ ( p 1 , p 2 ) + Φ ( p 1 , p 2 ) + = ( 1 exp ( 2 i ϕ 1 ) cos 2 ϑ 1 1 cos 2 ϑ 1 + 1 exp ( 2 i ϕ 1 ) cos 2 ϑ 1 1 cos 2 ϑ 1 + 1 1 ) normalization a 1 Φ + + a 2 Ψ + + i a 3 Ψ ,
Π k ( 1 ) Π k ( 2 ) : | Φ ( p 1 , p 2 ) Φ ( p 1 , p 2 ) = 1 1 + cos 2 ϑ 1 ( 2 cos ϑ 1 0 0 2 cos ϑ 1 ) normalization Φ ,
1 + cos 2 ϑ 1 2 1 + cos 4 ϑ 1 ,
1 + cos 2 ϑ 1 8 cos ϑ 1 ,
Φ ( p 1 , p 2 ) + | Φ ( p 1 , p 2 ) = 0 Π k ( 1 ) Π k ( 2 ) Φ ( p 1 , p 2 ) + Φ ( p 1 , p 2 ) = 0 .
Π k ( 1 ) Π k ( 2 ) : | Φ ( B z p 1 , B z p 2 ) + Φ ( B z p 1 , B z p 2 ) + ,
Π k ( 1 ) Π k ( 2 ) : | Φ ( B z p 1 , B z p 2 ) Φ ( B z p 1 , B z p 2 ) .
| Ω 12 = d μ ( p 1 ) d μ ( p 2 ) f ( p 1 , p 2 ) ( α p 1 , 2 | Φ ( p 1 , p 2 ) + + β p 1 , 2 | Φ ( p 1 , p 2 ) ) .
| 12 = U ( R z ( λ ) R y ( ϖ ) B z ( η ) ) | Ω 12 = α d μ ( q 1 ) d μ ( q 2 ) f ( q 1 , q 2 ) | Φ ( q 1 , q 2 ) + + β d μ ( q 1 ) d μ ( q 2 ) f ( q 1 , q 2 ) | Φ ( q 1 , q 2 ) ,
Γ 1 = Φ Φ , Γ 2 = 1 Γ 1 .
Π k ( 1 ) Π k ( 2 ) : | p i , σ i | p j , σ j p i , σ i p j , σ j ,
1 N = 1 ( 1 + cos 2 ϑ 1 ) ( 1 + cos 2 ϑ 2 ) ,
Π k ( 1 ) Π k ( 2 ) : | Φ ( p 1 , p 2 ) + Φ ( p 1 , p 2 ) + = 1 N ( ( cos ϑ 1 cos ϑ 2 + 1 ) exp ( i ( ϕ 1 + ϕ 2 ) ) ( cos ϑ 1 cos ϑ 2 1 ) exp ( i ( ϕ 1 ϕ 2 ) ) ( cos ϑ 1 cos ϑ 2 1 ) exp ( i ( ϕ 1 ϕ 2 ) ) ( cos ϑ 1 cos ϑ 2 + 1 ) exp ( i ( ϕ 1 + ϕ 2 ) ) ) ,
Π k ( 1 ) Π k ( 2 ) : | Φ ( p 1 , p 2 ) Φ ( p 1 , p 2 ) = 1 N ( ( cos ϑ 1 + cos ϑ 2 ) exp ( i ( ϕ 1 + ϕ 2 ) ) ( cos ϑ 1 cos ϑ 2 ) exp ( i ( ϕ 2 ϕ 1 ) ) ( cos ϑ 1 cos ϑ 2 ) exp ( i ( ϕ 2 ϕ 1 ) ) ( cos ϑ 1 + cos ϑ 2 ) exp ( i ( ϕ 1 + ϕ 2 ) ) ) ,
Π k ( 1 ) Π k ( 2 ) : | Ψ ( p 1 , p 2 ) + Ψ ( p 1 , p 2 ) + = 1 N ( ( cos ϑ 1 cos ϑ 2 1 ) exp ( i ( ϕ 1 + ϕ 2 ) ) ( cos ϑ 1 cos ϑ 2 + 1 ) exp ( i ( ϕ 2 ϕ 1 ) ) ( cos ϑ 1 cos ϑ 2 + 1 ) exp ( i ( ϕ 2 ϕ 1 ) ) ( cos ϑ 1 cos ϑ 2 1 ) exp ( i ( ϕ 1 + ϕ 2 ) ) ) ,
Π k ( 1 ) Π k ( 2 ) : | Ψ ( p 1 , p 2 ) Ψ ( p 1 , p 2 ) = 1 N ( ( cos ϑ 1 cos ϑ 2 ) exp ( i ( ϕ 1 + ϕ 2 ) ) ( cos ϑ 1 + cos ϑ 2 ) exp ( i ( ϕ 1 ϕ 2 ) ) ( cos ϑ 1 + cos ϑ 2 ) exp ( i ( ϕ 1 ϕ 2 ) ) ( cos ϑ 1 cos ϑ 2 ) exp ( i ( ϕ 1 + ϕ 2 ) ) ) .
P μ | p , σ = p μ | p , σ .
U ( { Λ ˜ , b } ) U ( { Λ , a } ) = exp { ( i ϒ ) } U ( { Λ ˜ Λ , Λ ˜ a + b } ) .
U ( Λ ) | p , σ = σ D σ , σ ( S k ) | p , σ ,
S k = L 1 ( Λ p ) Λ L ( p ) ,
J 3 | k , σ = σ | k , σ .
U ( R ( ϑ , ϕ ) ) J 3 U 1 ( R ( ϑ , ϕ ) ) | p , σ = σ | p , σ ,
D σ , σ ( R z ( θ ) ) = exp [ i σ θ ] δ σ σ ,
U ( Λ ) | p , σ = exp [ i σ θ p , q ] | q , σ .
Π p i j = σ = ± ϵ p , σ i ϵ ¯ p , σ j = δ i j p i p j p k p k ,
Π k = σ = ± | σ σ | k = ( 1 0 0 0 1 0 0 0 0 ) , | + k = ( 1 0 0 ) | + k = ( 0 1 0 ) ,
ϱ eff = 1 N d μ ( p ) | f ( p ) | 2 Π k [ ( α p | + p + β p | p ) ( α ¯ p + | p + β ¯ p | p ] Π k ,
ϱ = 1 3 + i = 1 8 s i su ( 3 ) λ i su ( 3 ) ,
Coulomb gauge: ϵ q , σ ϵ q , σ g q , σ q ,

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