Skip to main content
SpringerLink
Log in

Algebraic identities for parafermi statistics of given order

Алгебраические тождества для парафермионной статистики заданного порядка

  • Published:
Il Nuovo Cimento A (1971-1996)

Summary

It is pointed out that the structure of parafermion algebras generated byN pairs of creation and annihilation operators is determined by the unique occurrence of certain representations ofU N in particular representations ofSO 2N andSO 2N+1, identified by the eigenvalues of their invariants. This observation is used to derive a variety of identities satisfied by the creation and annihilation operators, which are characteristic of the orderp of the parastatistics. The «statistical quantum number» of Ohnuki and Kamefuchi is identified with the invariant ofSO 2N and related in turn to the invariants ofU N. A simple method is stated for the systematic determination of special identities, some of which are already known.

Riassunto

Si mette in rilievo che la struttura delle algebre parafermioniche generate daN coppie di operatori di creazione ed annichilazione è determinata dal fatto che certe rappresentazioni diU N compaiono una sola volta in particolari rappresentazioni diSO 2N eSO 2N+1, identificate dagli autovalori dei loro invarianti. Si adopera questa osservazione per dedurre una varietà di identità soddisfatte dagli operatori di creazione ed annichilazione, che sono caratteristici dell'ordinep della parastatistica. Si identifica il «numero quantico statistico» di Ohnuki e Kamefuchi con l'invariante diSO 2N e lo si mette a sua volta in relazione con gli invarianti diU N. Si formula un semplice metodo per la determinazione sistematica di speciali identità, alcune delle quali sono già note.

Резюме

Отмечается, что структура парафермионных алгебр, образованныхN парами оператотов рождения и уничтожения, определяется с помоцью однозначного существования определенных представленийU N в специальных предствленияхSO 2N иSO 2N+1, которые идентифицируются через собственные значения их инвариантов. Этот результат используется для вывода многообразия тождеств, которым удовлетворяют операторы рождения и уничтожения и уничтожения, которые являются характерными для порядкаp парасттистики. «Статистическое квантовое число» Охнуки и Камефуки идентифицируется с инвариантномSO 2N и связано, в свою очередь, с инвариантамиU N. Формулируется простой метод для систематического определения специальных тождеств, некоторые из которых уже известны.

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

References

  1. H. S. Green:Phys. Rev.,90, 270 (1953).

    Article  ADS  Google Scholar 

  2. I. E. McCarthy:Proc. Camb. Phil. Soc.,51, 131 (1955);H. Scharfstein:Nuovo Cimento,30, 740 (1963);S. Kamefuchi andJ. Strathdee:Nucl. Phys.,42, 166 (1963);O. W. Greenberg andA. M. L. Messiah:Phys. Rev.,136, B 248 (1964);138, B 1155 (1965);Journ. Math. Phys.,6, 500 (1965);G. F. Dell'Antonio, O. W. Greenberg andE. C. G. Sudarshan: inGroup Theoretical Concepts and Methods in Elementary Particles Physics, edited byF. Gürsey (New York, 1961);P. V. Landshoff andH. P. Stapp:Ann. of Phys.,45, 72 (1967).

    Article  ADS  Google Scholar 

  3. Y. Ohnuki andP. V. Kamefuchi:Phys. Rev.,170, 1279 (1968);Ann. of Phys.,51, 337 (1969);57, 543 (1970); Tokyo preprint TUEP-70-29 (1970),On quantum mechanics of identical particles, II.

    Article  ADS  Google Scholar 

  4. K. V. Kademova:Nucl. Phys.,15 B, 350 (1970);Int. Journ. Theor. Phys.,3, 109, 295 (1970);K. V. Kademova andA. J. Kálnay:Int. Journ. Theor. Phys.,3, 115 (1970);K. V. Kademova andT. D. Palev:Int. Journ. Theor. Phys.,3, 337 (1970);K. V. Kademova andM. M. Kraev:Int. Journ. Theor. Phys.,3, 185 (1970);4, 159 (1971).

    Article  ADS  Google Scholar 

  5. S. Kamefuchi andY. Takahashi:Nucl. Phys.,36, 177 (1962);C. Ryan andE. C. G. Sudarshan:Nucl. Phys.,47, 207 (1963).

    Article  MathSciNet  Google Scholar 

  6. C. Alabiso, F. Duimio andJ. L. Redondo:Nuovo Cimento,61 A, 766 (1969);C. Alabiso andF. Duimio: Parma preprint (1971),On the paraboson representations of the Sp 4,R algebra and their reduction with respect to the O3,1 subalgebra.

    Article  ADS  MathSciNet  Google Scholar 

  7. H. S. Green: Adelaide preprint (1971),Parastatistics, leptons and the neutrino theory of light.

  8. O. W. Greenberg:Phys. Rev. Lett.,13, 598 (1964).

    Article  ADS  Google Scholar 

  9. A. J. Bracken andH. S. Green:Vector operators and a polynomial identity for SO n;H. S. Green:Characteristic identities for generators of GL n, On and Spn, both to appear inJourn. Math. Phys.,12, No. 10 (1971).

  10. F. D. Muranghan:The Theory of Group Representations (Baltimore, 1938);A. Pais:Rev. Mod. Phys.,38, 215 (1966).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Physics, University of Adelaide, Adelaide, S. Austr., South Australia

    A. J. Bracken & H. S. Green

Authors
  1. A. J. Bracken
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. S. Green
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Traduzione a cura della Redazione.

Перевебено ребакуией.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bracken, A.J., Green, H.S. Algebraic identities for parafermi statistics of given order. Nuov Cim A 9, 349–365 (1972). https://doi.org/10.1007/BF02789725

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02789725

Search

Navigation