Relativistic 'proper-time' quantum evolution equations from canonical invariance

doi:10.1088/1751-8113/41/17/175303

Derivation of the relativistic 'proper-time' quantum evolution equations from canonical invariance

Moshe Shapiro 2008 J. Phys. A: Math. Theor. 41 175303 (9pp) doi: 10.1088/1751-8113/41/17/175303

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Moshe Shapiro
Department of Chemistry and Physics, University of British Columbia, Vancouver, BC V6T 1Z3, Canada
and
Department of Chemical Physics, The Weizmann Institute, Rehovot 76100, Israel
E-mail: mshapir@chem.ubc.ca and moshe.shapiro@weizmann.ac.il

Abstract. Based on (1) the spectral resolution of the energy operator; (2) the linearity of correspondence between physical observables and quantum self-adjoint operators; (3) the definition of conjugate coordinate–momentum variables in classical mechanics; and (4) the fact that the physical point in phase space remains unchanged under (canonical) transformations between one pair of conjugate variables to another, we are able to show that $\bra{t_s}E_s\rangle$ , the proper-time rest-energy transformation matrices, are given as aexp(−iE_st_s/ planck ), from which we obtain the proper-time rest-energy evolution equation ${\rm i}\hbar{\partial\over \partial t_s}\ket{\Psi}={\skew3\hat{E}_s}\ket{\Psi}$ . For special relativistic situations this equation can be reduced to the usual ${\rm i}\hbar{\partial\over \partial t}\ket{\Psi}=\skew3\hat{E}\ket{\Psi}$ dynamical equations, where t is the 'reference time' and E is the total energy. Extension of these equations to accelerating frames is then provided.

PACS number: 03.65.Ta

Print publication: Issue 17 (2 May 2008)
Received 8 February 2008, in final form 10 March 2008
Published 15 April 2008

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Derivation of the relativistic 'proper-time' quantum evolution equations from canonical invariance

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