Abstract
The Schrödinger-Robertson inequality for relativistic position and momentum operators X μ, P ν, μ, ν = 0, 1, 2, 3, is interpreted in terms of Born reciprocity and ‘non-commutative’ relativistic position-momentum space geometry. For states which saturate the Schrödinger-Robertson inequality, a typology of semiclassical limits is pointed out, characterised by the orbit structure within its unitary irreducible representations, of the full invariance group of Born reciprocity, the so-called ‘quaplectic’ group U(3, 1) #x2297;s H(3, 1) (the semi-direct product of the unitary relativistic dynamical symmetry U(3, 1) with the Weyl-Heisenberg group H(3, 1)). The example of the ‘scalar’ case, namely the relativistic oscillator, and associated multimode squeezed states, is treated in detail. In this case, it is suggested that the semiclassical limit corresponds to the separate emergence of spacetime and matter, in the form of the stress-energy tensor, and the quadrupole tensor, which are in general reciprocally equivalent.
Similar content being viewed by others
References
1. X. Bekaert and N. Boulanger, “Mixed symmetry gauge fields in a flat background,” hep-th/0310209, Talk given at the International Workshop on Supersymmetries and Quantum Symmetries (SQS03), Dubna, 2003.
2. M. Born, “Elementary particles and the principle of reciprocity,” Nature 163, 207–208 (February 1949).
3. M. Born, “Reciprocity theory of elementary particles,” Rev. Mod. Phys. 21(3), 463–473 (July 1949).
4. A Connes, Noncommutative Geometry (Academic, New York, 1994).
5. P A M Dirac, “Unitary representations of the Lorentz group,” Proc. Roy. Soc. (London) A 183, 284 (1945).
6. G. B. Folland, Harmonic Analysis on Phase Space (University Press, Princeton, 1989).
7. H. S. Green, “Quantized field theories and the principle of reciprocity,” Nature 163, 208–9 (February 1949).
8. Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics: Group Theoretical Approach (World Scientific, Singapore, 1991).
9. Stephen G. Low, “Representations of the canonical group, (the semidirect product of the unitary and Weyl-Heisenberg groups), acting as a dynamical group on noncommutative extended phase space,” J. Phys. A 35, 5711–5729 (2002).
10. Stephen G. Low, “Poincaré and Heisenberg quantum dynamical symmetry: Casimir invariant field equations of the quaplectic group,” math-ph/0502018, 2005.
11. Stephen G. Low, “Reciprocal relativity of noninertial frames and the quaplectic group,” Found. Phys. 36 (7) (2006); math-ph/0506031, 2005.
12. S. Majid, “Hopf algebras for physics at the Planck scale,” Class. Quant. Grav. 5, 1587–1606 (1993).
13. H. R. Robertson, “An indeterminacy relation for several observables and its classical interpretation,” Phys. Rev. 46, 794 (1934).
14. E. Schrödinger, Sitzungsber. Preuss. Acad. Wiss. 24, 296 (1930).
15. J. L. Synge, “Geometrical approach to the Heisenberg uncertainty relation and its generalisation,” Proc. Roy. Soc. Lond. A 325, 151–156 (1971).
16. D. A. Trifonov, “Robertson intelligent states,” J. Phys. A 30, 594 (1997).
17. V. G. Turaev, “Algebras of loops on surfaces, algebras of knots, and quantisation,” in C. N. Yang and M. L. Ge, eds., Braid Group, Knot Theory and Statistical Mechanics (World Scientific, Singapore, 1991), pp. 59–95.
18. J. Williamson, “On the algebraic problem concerning the normal forms of linear dynamical systems,” Am. J. Math. 58, 141 (1936).
19. J. A. Wolf, “Representations of certain semidirect product groups,” J. Func. Anal. 19, 339–372 (1975).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jarvis, P., Morgan, S. Born Reciprocity and the Granularity of Spacetime. Found Phys Lett 19, 501–517 (2006). https://doi.org/10.1007/s10702-006-1006-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10702-006-1006-5