no. 19-20
Mathematical Physics
Study of a self-adjoint operator indicating the direction of time within standard quantum mechanics
[Étude dʼun opérateur auto-adjoint qui indique la direction du temps en mécanique quantique]

Présenté par : Peter D. Lax

Strauss, Yossef1 ; Silman, Jonathan2 ; Machnes, Shai2 ; Horwitz, Lawrence P.2, 3, 4
1 Einstein Institute of Mathematics, Edmond J. Safra campus, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
2 School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
3 Physics Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
4 Department of Physics, The Ariel University Center of Samaria, Ariel 40700, Israel
Comptes Rendus. Mathématique, Tome 349 (2011) no. 19-20, pp. 1117-1122.

Dans [Y. Strauss, Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schrödinger evolution, J. Math. Phys. 51 (2010) 022104], un opérateur auto-adjoint a été introduit ayant la propriété dʼindiquer la direction du temps dans le formalisme standard de la mécanique quantique, au sens où sa valeur moyenne décroit de façon monotone avec le temps pour tout état initial. Dans cet article, nous étudions les propriétés de cet opérateur. En particulier, nous dérivons son spectre et ses vecteurs propres généralisés et traitons en détail lʼexample de la particule libre

In [Y. Strauss, Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schrödinger evolution, J. Math. Phys. 51 (2010) 022104] a self-adjoint operator was introduced that has the property that it indicates the direction of time within the framework of standard quantum mechanics, in the sense that as a function of time its expectation value decreases monotonically for any initial state. In this paper we study some of this operatorʼs properties. In particular, we derive its spectrum and generalized eigenstates, and treat the example of the free particle.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.09.007
Strauss, Yossef 1 ; Silman, Jonathan 2 ; Machnes, Shai 2 ; Horwitz, Lawrence P. 2, 3, 4

1 Einstein Institute of Mathematics, Edmond J. Safra campus, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
2 School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
3 Physics Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
4 Department of Physics, The Ariel University Center of Samaria, Ariel 40700, Israel 
@article{CRMATH_2011__349_19-20_1117_0,
     author = {Strauss, Yossef and Silman, Jonathan and Machnes, Shai and Horwitz, Lawrence P.},
     title = {Study of a self-adjoint operator indicating the direction of time within standard quantum mechanics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1117--1122},
     publisher = {Elsevier},
     volume = {349},
     number = {19-20},
     year = {2011},
     doi = {10.1016/j.crma.2011.09.007},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.09.007/}
}
TY  - JOUR
AU  - Strauss, Yossef
AU  - Silman, Jonathan
AU  - Machnes, Shai
AU  - Horwitz, Lawrence P.
TI  - Study of a self-adjoint operator indicating the direction of time within standard quantum mechanics
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1117
EP  - 1122
VL  - 349
IS  - 19-20
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.09.007/
DO  - 10.1016/j.crma.2011.09.007
LA  - en
ID  - CRMATH_2011__349_19-20_1117_0
ER  - 
%0 Journal Article
%A Strauss, Yossef
%A Silman, Jonathan
%A Machnes, Shai
%A Horwitz, Lawrence P.
%T Study of a self-adjoint operator indicating the direction of time within standard quantum mechanics
%J Comptes Rendus. Mathématique
%D 2011
%P 1117-1122
%V 349
%N 19-20
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2011.09.007/
%R 10.1016/j.crma.2011.09.007
%G en
%F CRMATH_2011__349_19-20_1117_0
Strauss, Yossef; Silman, Jonathan; Machnes, Shai; Horwitz, Lawrence P. Study of a self-adjoint operator indicating the direction of time within standard quantum mechanics. Comptes Rendus. Mathématique, Tome 349 (2011) no. 19-20, pp. 1117-1122. doi : 10.1016/j.crma.2011.09.007. http://www.numdam.org/articles/10.1016/j.crma.2011.09.007/

[1] Galapon, E.A. Pauliʼs theorem and quantum canonical pairs: the consistency of a bounded, self adjoint time operator canonically conjugate to a Hamiltonian with nonempty point spectrum, Proc. R. Soc. Lond. Ser. A, Volume 458 (2002), pp. 451-472

[2] Holevo, A.S. Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam, 1982

[3] Mackey, G.W., University of Chicago Press, Chicago, 1976 (See, for example The Theory of Unitary Group Representations)

[4] Misra, B.; Prigogine, I.; Courbage, M. Lyapounov variable: entropy and measurement in quantum mechanics, Proc. Natl. Acad. Sci. USA, Volume 76 (1979), pp. 4768-4772

[5] Paley, R.E.A.C.; Wiener, N. Fourier Transforms in the Complex Domain, Amer. Math. Soc., Providence, RI, 1934

[6] Pauli, W.E. Handbuch der Physik, Springer-Verlag, Berlin, 1926

[7] Strauss, Y. Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schrödinger evolution, J. Math. Phys., Volume 51 (2010), p. 022104

[8] Strauss, Y. Transition representations of quantum evolution with application to scattering resonances, J. Math. Phys., Volume 52 (2011), p. 032106

[9] Strauss, Y.; Silman, J.; Machnes, S.; Horwitz, L.P. Transition decomposition of quantum mechanical evolution, Int. J. Theor. Phys., Volume 50 (2011), pp. 2179-2190 (Special issue in honour of Prof. M. Gadella on the occasion of his 60th birthday)

[10] Unruh, W.G.; Wald, R.M. Time and the interpretation of canonical quantum gravity, Phys. Rev. D, Volume 40 (1989), p. 25982614

Cité par Sources :