The inversion operators on a lattice in finite phase plane are used for building a complete set of mutually orthogonal Hermitian operators. The lattice is given by tc in the x direction and by $s\hbar \frac{2\pi }{Mc}$ in the p-direction; c is an arbitrary length constant and M is the dimension of the space; s and t assume the values from 0 to M − 1. For M odd the M2 inversion operators on the lattice form a complete set of mutually orthogonal operators. For M even we assign a sum of 4 inversion operators (a quartet) to each site of the lattice (t, s). We prove that these quartets for t, s = 0, 1, …, M − 1 form a mutually orthogonal set of M2 Hermitian operators.
REFERENCES
1.
2.
J.
Schwinger
, Proc. Natl. Acad. Sci. U.S.A.
46
, 570
(1960
).3.
E. P.
Wigner
, Phys. Rev.
40
, 749
(1932
).4.
W. K.
Wootters
, Ann. Phys.
176
, 1
(1987
).5.
D.
Cohendet
, Ph.
Combe
, M.
Sirugue
, and M.
Sirugue-Collin
, J. Phys. A
1
, 2875
(1988
).6.
M.
Horibe
, A.
Takami
, F.
Hashimoto
, and A.
Hayashi
, Phys. Rev. A
65
, 032105
(2002
).7.
U.
Leonhardt
, Phys. Rev. A
53
, 2998
(1996
).8.
C.
Miquel
, J. P.
Paz
, and M.
Saraceno
, Phys. Rev. A
65
, 062309
(2002
).9.
J.
Zak
, J. Phys. A: Math. Theor.
44
, 345305
(2011
).10.
A.
Vourdas
, Rep. Prog. Phys.
67
, 267
(2004
).11.
G.
Bjork
, A. B.
Klimov
, and L. L.
Sanchez-Soto
, Prog. Opt.
51
, 469
(2008
).12.
T.
Durt
, B. G.
Englert
, I.
Bengtsson
, and K.
Zyczkowski
, Int. J. Quantum Inf.
8
, 535
(2010
).13.
A.
Grossmann
, Commun. Math. Phys.
48
, 191
(1976
).14.
A.
Royer
, Phys. Rev. A
15
, 449
(1977
).15.
A. M. F.
Rivas
and A. M.
Ozorio de Almeida
, Ann. Phys.
276
, 223
(1999
).16.
R. J.
Glauber
, Phys. Rev.
131
, 2766
(1963
).17.
H. J.
Groenewold
, Physica
12
, 405
(1946
).18.
J. E.
Moyal
, Proc. Cambridge Philos. Soc.
41
, 99
(1949
).19.
C. L.
Metha
, J. Math. Phys.
5
, 677
(1964
).20.
J. G.
Kirkwood
, Phys. Rev.
44
, 31
(1933
).21.
A. N.
Rihaczek
, IEEE Trans. Inf. Theory
14
, 369
(1968
).22.
J. H.
Hannay
and M. V.
Berry
, Physica D
1
, 267
(1980
).23.
O.
Agam
and N.
Brenner
, J. Phys. A
28
, 1345
(1995
).24.
A.
Arguelles
and T.
Dittrich
, Physica A
356
, 72
(2005
).© 2012 American Institute of Physics.
2012
American Institute of Physics
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