J. Phys. A: Math. Theor. 41 (2 May 2008) 175303 (9pp) doi: 10.1088/1751-8113/41/17/175303
Derivation of the relativistic 'proper-time' quantum evolution equations from canonical invariance
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.ilAbstract.
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 
, the proper-time rest-energy transformation matrices, are given as aexp(-iEsts/
), from which we obtain the proper-time rest-energy evolution equation 
. For special relativistic situations this equation can be reduced to the usual 
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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