Mass Operator and Dynamical Implementation of Mass Superselection Rule

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• 作者：Eleonora Annigoni (1)
  Valter Moretti (1)
  
• 刊名：Annales Henri Poincare
• 出版年：2013
• 出版时间：May 2013
• 年：2013
• 卷：14
• 期：4
• 页码：893-924
• 全文大小：426KB
• 参考文献：1. Bargmann V.: On unitary ray representations of continuous groups. Ann. Math. 59, 1-6 (1954) CrossRef
  2. Barut A.O., Raczka R.: Theory of Group Representations and Applications, 2nd revised edn. World Scientific, Singapore (1986)
  3. Busch P., Grabowski M., Lahti P.J.: Operational Quantum Physics. Springer, Berlin (1995)
  4. Blanchard, Ph., Giulini, D., Joos, E., Kiefer, C., Stamatescu, I.-O. (Eds.): Decoherence: Theoretical, Experimental, and Conceptual Problems. Lecture Notes in Physics. Springer-Verlag, Berlin (2000)
  5. Giulini D.: On Galilei invariance in quantum mechanics and the Bargmann superselection rule. Ann. Phys. 249, 222-35 (1996) quant-ph/9508002 CrossRef
  6. Giulini D.: Decoherence: a dynamical approach to superselection rules? Lect. Notes Phys. 559, 67-2 (2000) quant-ph/0010090 CrossRef
  7. Reed M., Simon B.: Methods of Modern Mathematical Physics. Vol. I. Academic Press, New York (1980)
  
• 作者单位：Eleonora Annigoni (1)
  Valter Moretti (1)
  
  1. Department of Mathematics, Faculty of Science, University of Trento, via Sommarive 14, 38050, Povo, Trento, Italy
  
• ISSN：1424-0661

文摘

We start reviewing Giulini’s dynamical approach to Bargmann superselection rule proposing some improvements. First of all we discuss some general features of the central extensions of the Galilean group used in Giulini’s programme, in particular focussing on the interplay of classical and quantum picture, without making any particular choice for the multipliers. Preserving other features of Giulini’s approach, we modify the mass operator of a Galilei invariant quantum system to obtain a mass spectrum that is (i) positive and (ii) discrete, so giving rise to a standard (non-continuous) superselection rule. The model results to be invariant under time reversal but a further degree of freedom appears that can be interpreted as describing an internal conserved charge of the system (however, adopting a POVM approach, the unobservable degrees of freedom can be pictured as a generalized observable automatically gaining a positive mass operator without assuming the existence of such a charge). The effectiveness of Bargmann rule is shown to be equivalent to an averaging procedure over the unobservable degrees of freedom of the central extension of Galileian group. Moreover, viewing the Galileian invariant quantum mechanics as a non-relativistic limit, we prove that the above-mentioned averaging procedure giving rise to Bargmann superselection rule is nothing but an effective de-coherence phenomenon due to time evolution if assuming that real measurements includes a temporal averaging procedure. It happens when the added term Mc 2 is taken in due account in the Hamiltonian operator since, in the dynamical approach, the mass M is an operator and cannot be trivially neglected as in classical mechanics. The presented results are quite general and rely upon the only hypothesis that the mass operator has point spectrum. These results explicitly show the interplay of the period of time of the averaging procedure, the energy content of the considered states, and the minimal difference of the mass operator eigenvalues.