Skip to main content
SpringerLink
Log in

Group Theoretical Derivation of Consistent Massless Particle Theories

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

Current theories of massless free particle assume unitary space inversion and anti-unitary time reversal operators. In so doing robust classes of possible theories are discarded. In the present work theories of massless systems are derived through a strictly deductive development from the principle of relativistic invariance, so that a kind of space inversion or time reversal operator is ruled out only if it causes inconsistencies. As results, new classes of consistent theories for massless isolated systems are explicitly determined. On the other hand, the approach determines definite constraints implied by the invariance principle; they were ignored by some past investigations that, as a consequence, turn out to be not consistent with the invariance principle. Also the problem of the localizability for massless systems is reconsidered within the new theoretical framework, obtaining a generalization and a deeper detailing of previous results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The commutativity condition \([Q_j,Q_k]={{\textsf {I}}\hbox {O}}\) establishes the possibility of performing a measurement that yields all three values of the position coordinates. The nonexistence of commutative position operators in certain circumstances [3] prompted to search for non-commutative [20,21,22,23] or unsharp position operators (see [24] and references therein). In the present work we are interested in the commutative concept of position only.

References

  1. Wigner, E.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40, 149 (1939)

    Article  ADS  MathSciNet  Google Scholar 

  2. Bargmann V., Wigner E.P.: Group theoretical discussion of relativistic wave equations. Proc. Natl. Acad. Sci. 34 211 (1948)

  3. Wightman, A.S.: On the localizability of quantum mechanical systems. Rev. Mod. Phys. 34, 845 (1962)

    Article  ADS  MathSciNet  Google Scholar 

  4. Costa, G., Fogli, G.: Lecture Notes in Physics, vol. 823. Springer, New York (2012)

    MATH  Google Scholar 

  5. Wigner, E.P.: Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra. Academic Press, Cambridge, MA (1959)

    MATH  Google Scholar 

  6. Weinberg, S.: Feynman rules for any spin II. Massless particles. Phys. Rev. 134, B882 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  7. Weinberg, S.: The Quantum Theory of Fields, vol. I. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  8. Jordan, T.F.: Simple proof of no position operator for quanta with zero mass and nonzero helicity. J. Math. Phys. 19, 1382 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  9. Niederer, U.H., O’Raifeartaigh, L.: Realizations of the unitary representations of the inhomogeneous space-time groups. II. Covariant realizations of the Poincarè group. Fortsch. Phys. 22, 131 (1974)

    Article  ADS  Google Scholar 

  10. Nisticò, G.: 2020: group theoretical derivation of consistent free particle theories. Found. Phys. 50, 977–1007 (2020). https://doi.org/10.1007/s10701-020-00364-2

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Klein, O.: Quantentheorie und Fünfdimensionale Relativitätstheorie. Z. Phys. 37, 895 (1926)

    Article  ADS  Google Scholar 

  12. Fock, V.: Zur Schrödingerschen Wellenmechanik. Z. Phys. 37, 242 (1926)

    Article  ADS  Google Scholar 

  13. Gordon, W.: Der Comptoneffekt Nach der Schrödingerschen Theorie. Z. Phys. 40, 117 (1926)

    Article  ADS  Google Scholar 

  14. Newton, T.D., Wigner, E.P.: Localized states for elementary systems. Rev. Mod. Phys. 21, 400 (1949)

    Article  ADS  Google Scholar 

  15. Amrein, W.O.: Localizability for particles of mass zero. Helv. Phys. Acta 42, 149 (1969)

    MATH  Google Scholar 

  16. Barut, A.O., Racza, R.: Theory of Group Repesentations and Applications. World Scientific, Singapore (1986)

    Book  Google Scholar 

  17. Nisticò G.: Group theoretical derivation of the minimal coupling principle. Proc. R. Soc. A. (2017) https://doi.org/10.1098/rspa.2016.0629

  18. Simon B.: In: Lieb, E.H, Simon, B., Wightman, A.S. (eds.) Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann. Princeton: Princeton Un. Press, pp. 327–350 (1976)

  19. Foldy, L.L.: Synthesis of covariant particle equations. Phys. Rev. 102, 568 (1956)

    Article  ADS  MathSciNet  Google Scholar 

  20. Kàlnay, A.J.: In: Bunge, M. (ed) Problems in the Foundations of Physics. Springer, Berlin (1971)

  21. Jadzyck, A.Z., Jancewicz, B.: Maximal localizability of photons. Bull. Ac. Sci. Polon XXI:477 (1972)

  22. Bacry, H.: The Poincare group, the Dirac Monopole and photon localisation. J. Phys. A 14, L73 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  23. Bacry, H.: Localizability and Space in Quantum Physics. Lecture Notes in Physics, vol. 308. Springer, Berlin (1988)

    Book  Google Scholar 

  24. Schroeck, E.F.: Quantum Mechanics on Phase Space. Kluwer Academic Publishers, Dordrecht (1996)

    Book  Google Scholar 

  25. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, New York (1972)

    MATH  Google Scholar 

  26. Nisticò, G.: New representations of Poincare group for consistent relativistic particle theories. J. Phys. 1275, 12034 (2019)

    Google Scholar 

  27. Raman, C.V., Bhagavantam, S.: Experimental proof of the spin of the photon. Indian J. Phys. 6, 353 (1931)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università della Calabria, Rende, Italy

    Giuseppe Nisticò

  2. INFN – gruppo collegato di Cosenza, Cosenza, Italy

    Giuseppe Nisticò

Authors
  1. Giuseppe Nisticò
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuseppe Nisticò.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nisticò, G. Group Theoretical Derivation of Consistent Massless Particle Theories. Found Phys 51, 112 (2021). https://doi.org/10.1007/s10701-021-00494-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10701-021-00494-1

Keywords

Search

Navigation