The classical mechanics for bose-fermi systems | Il Nuovo Cimento A (1971-1996) Skip to main content
Springer Nature Link
Log in

The classical mechanics for bose-fermi systems

Классическая механика для Боэе и Ферми систем

  • Published:
Il Nuovo Cimento A (1965-1970)

Summary

In this paper we study in a systematic way the classical mechanics of systems described byc-number variables and by Grassmann variables. We derive the general form of the nonrelativistic action and we study the theory of canonical transformations. For a general action, we show that the Jacobian matrices of the canonical transformations acting onN Grassmann variables form a groupO N, N . This group becomesO N for the nonrelativistic action, due to the presence of second-class constraints. We study some examples which give rise to a correct classical description of the spin. Considering a relativistic extension of one of these models, we get a first quantized «substratum» for the superfield theories.

Riassunto

In questo lavoro si studia in maniera sistematica la meccanica classica di sistemi descritti da variabili numeroc e da variabili di Grassmann. Si deduce la forma generale dell’azione non relativistica e si studia la teoria delle trasformazioni canoniche. Si mostra che, per un’azione generica, le matrici jacobiane delle trasformazioni canoniche agenti suN variabili di Grassmann formano un gruppoO N,N . Questo gruppo si riduce adO N per l’azione non relativistica, a causa della presenza di vincoli di seconda classe. Si studiano alcuni esempi che danno luogo ad una corretta descrizione classica dello spin. Estendendo relativisticamente uno di questi modelli, si ottiene il « substratum » di prima quantizzazione delle teorie di supercampo.

Реэюме

В зтой работе мы исследуем систематическим обраэом классическую механику систем, которые описываются с помошью переменных Грассмана и пере-менных, представляюших с числа. Мы выводим обшую форму для нерелятивист-ского действия и исследуем теорию канонических преобраэований. Для обшего действия мы покаэываем, что Якобианы канонических преобраэований, действуюших на N переменных Грассмана, обраэуют группу О N,N . Эта группа преврашается в О N для нерелятивистского действия вследствие наличия ограничений второго класса. Мы исследуем некоторые примеры, которые приводят к правильному классическому описанию спина. Рассматривая релятивистское обобшение одной иэ зтих моделей, мы получаем первично квантованную «основу» для теорий суперполей

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literatur

  1. R. Casalbuoni:On the quantization of systems with anticommuting variables, Florence preprint (1975).

  2. D. V. Volkov andV. P. Akulov:Phys. Lett.,46 B, 109 (1973);J. Wess andB. Zumino:Nucl. Phys.,70 B, 39 (1974);A. Salam andJ. Strathdee:Nucl. Phys.,76 B, 477 (1974). See alsoB. Zumino:Proceedings of the XVII International Conference on High-Energy Physics (London, 1974), Rutherford Laboratory Chilton, Didcot (1974), p. I-254.

    Article  ADS  Google Scholar 

  3. J. L. Martin:Proc. Roy. Soc.,251 A, 543 (1959).

    Article  ADS  Google Scholar 

  4. A. Neveu andJ. H. Schwarz:Nucl. Phys.,31 B, 86 (1971);P. Ramond:Phys. Rev. D,3, 2415 (1971).

    Article  ADS  Google Scholar 

  5. Y. Iwasaki andK. Kikkawa:Phys. Rev. D,8, 440 (1973);L. N. Chang, K. Macrae andF. Mansouri:Phys. Lett.,57 B, 59 (1975). See alsoM. Ademollo, L. Brink, A. d’Adda, R. d’Auria, E. Napolitano, S. Sciuto, E. Del Giudice, P. Di Vecchia, S. Ferrara, F. Gliozzi, R. Musto andR. Pettorino:Supersymmetric string and colour confinement, CERN preprint TH. 2097 (1975).

    Article  ADS  Google Scholar 

  6. B. Zumino:Supersymmetry, CERN preprint TH. 2120 (1976).

  7. F. J. Belinfante, D. J. Caplan andW. L. Kennedy:Rev. Mod. Phys.,29, 518 (1957), and references therein. See also, for more recent attempts related with super-symmetries,L. N. Chang, K. Macrae andF. Mansouri:Phys. Lett.,57 B, 59 (1975);Y. M. Cho andP. G. O. Freund: University of Chicago preprint, EFI 75-15 (1975);P. Nath andR. Arnowitt:Phys. Lett.,56 B, 171 (1975);B. Zumino:Supersymmetry, CERN preprint TH. 2120 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. J. L. Martin:Proc. Roy. Soc.,251 A, 536 (1959);F. A. Berezin andM. S. Marinov:JETP Lett.,21, 321 (1975).

    Article  ADS  Google Scholar 

  9. R. Casalbuoni:Relativity and supersymmetries, Florence preprint (1975).

  10. P. A. M. Dirac:Lectures on Quantum Mechanics (New York, N. Y., 1964).

  11. M. H. L. Pryce:Proc. Roy. Soc.,195 A, 62 (1948); see also, for a recent attempt,A. J. Hanson andT. Regge:Ann. of Phys.,87, 498 (1974), and references therein.

    Article  MathSciNet  ADS  Google Scholar 

  12. D. J. Almond:Ann. Inst. H. Poincaré,19, 105 (1973);Z. Haba:Nuovo Cimento,30 A, 567 (1975).

    Google Scholar 

  13. J. Schwinger:Quantum Kinematics and Dynamics (New York, N. Y., 1970).

  14. F. A. Berezin:The Method of Second Quantization (New York, N. Y., and London, 1966).

  15. We follow essentially the exposition byH. C. Corben andP. Stehle:Classical Mechanics, 2nd Edition (New York, N. Y., London and Sidney, 1965).

  16. R. Arnowitt, P. Nath andB. Zumino:Phys. Lett.,56 B, 81 (1975).

    Article  MathSciNet  ADS  Google Scholar 

  17. See, for instance,W. H. Greub:Linear Algebra (Berlin, 1967).

  18. H. Boerner:Representations of Groups (Amsterdam, 1963).

  19. See, for instance,R. Casalbuoni, J. Gomis andG. Longhi:Nuovo Cimento,24 A, 249 (1974).

    Article  ADS  Google Scholar 

Download references

Author information

Author notes

  1. R. Casalbuoni

    Present address: Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Largo E. Fermi 2, 50125, Firenze

Authors and Affiliations

  1. CERN, Geneva

    R. Casalbuoni

Authors
  1. R. Casalbuoni
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Work supported in part by the U.S. Energy Research and Development Administration, under contract E(11-1)3285. Report No. COO 3285-27.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Casalbuoni, R. The classical mechanics for bose-fermi systems. Nuov Cim A 33, 389–431 (1976). https://doi.org/10.1007/BF02729860

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02729860

Search

Navigation