Summary
Via the method of induced representations, all irreducible unitary projective representations of the recently introduced new relativistic dynamical group
are deduced and classified. An explicit form of the transformation law is given. The properties of the corresponding infinite-dimensional basis functions are studied. It is shown that in the limiting case ofl=∞ (corresponding to
) the infinite spin-tower representations become reducible and decompose into irreducible representations of the Poincaré group. The reduction of the direct product of two irreducible unitary ray representations of
is studied. The Clebsch-Gordan coefficients are computed. Finally, some comments on the physical interpretation of the results are given.
Riassunto
Tramite il metodo delle rappresentazioni indotte si deducono e classificano tutte le rappresentazioni proiettive unitarie irriducibili del nuovo gruppo dinamico relativistico
, introdotto di recente. Si dà un forma esplicita della legge di trasformazione. Si studiano le proprietà della corrispondente funzione di base ad infinite dimensioni. Si mostra che nel caso limitel=∞ (corrispondente a
) le infinite rappresentazioni della torre di spin diventano riducibili e si decompongono in rappresentazioni irriducibili del gruppo di Poincaré. Si studia la riduzione del prodotto diretto di due rappresentazioni radiali unitarie irreducibili di
. Si calcolano i coefficienti di Clebsch-Gordan. Infine si fanno alcuni commenti sull’interpretazione fisica dei risultati.
Реэюме
Испольэуя метод индуцированных представлений, выводятся и классифицируются все неприводимые унитарные проективные представления недавно введенной новой релятивистской динамической группы
. Приводится явная форма для эакона преобраэования. Исследуются свойства функций, соответствуюших бесконечномерному баэису. Покаэывается, что в предельном случаеl=∞ (соответствуюшем
) представления бесконечной спиновой бащни становятся приводимыми и раэлагаются на неприводимые представления группы Пуанкаре. Исследуется приведение прямого проиэведения двух неприводимых унитарных лучевых представлений
. Вьиисляются козффициенты Клебща-Гордана. В эаключение делаются некоторые эамечания относительно фиэической интерпретации полученных реэультатов.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. J. Aghassi, P. Roman andR. M. Santilli:Phys. Rev. D,1, 2753 (1970).
J. J. Aghassi, P. Roman andR. M. Santilli:Journ. Math. Phys.,11, 2297 (1970).
In a recent private communicationM. Noga (Purdue University) gave an alternative derivation of our group, emphasizing that it is actually the dynamical group of the standard equation of motion in relativistic mechanics. See also ref. (76).
See ref. (1), Appendix A.
See ref. (1), eqs. (3.7) through (3.12).
V. Bargmann:Ann. Math.,59, 1 (1954).
In this respect, see alsoJ. E. Johnson:Phys. Rev.,181, 1755 (1969);L. Castell:Nuovo Cimento,49 A, 285 (1967).
See ref. (1), Appendix C.
A very readable account of this powerful tool can be found inG. W. Mackey:Induced Representations of Groups and Quantum Mechanics (New York, 1968). A shorter, but more rigorous summary is given inG. W. Mackey:Group representations in Hilbert space, which is the Appendix inI. E. Segal:Mathematical Problems in Relativistic Physics (New York, 1963). The latter contains also a bibliography of original publications.
J. Voisin:Journ. Math. Phys.,6, 1519, 1822 (1965).
An alternative, somewhat more intuitive treatment of the ray representations of the nonrelativistic Galilei group was given byJ.-M. Lévy-Leblond:Journ. Math. Phys.,4, 776 (1963). Some parts of our calculations are analogous to those ofLévy-Leblond.
See Appendix A of ref. (1).
In his work on the ray representations of the nonrelativistic Galilei group,Voisin actually proceeds in a manner as now sketched, and obtains the orbitsE−p 2/2Mq 0 = = const (cf. eq. (14) of ref. (12), first paper), instead of the more desirableE−p 2/2M = const paraboloids. At a later point, he then setsq 0=1 which, even though it seems to be an artificial choice, apparently does not lead to loss of generality.
In view of (2.20), we are assured of the existence of such an element ofH. Actually,h r,p is not even unique.
See, for example,I. M. Gelfand, R. A. Minlos andZ. Ya. Shapiro:Representations of the Rotation and Lorentz Group (New York, 1963), especially p. 200 and p. 188. See alsoM. A. Naimark:Linear Representations of the Lorentz Group (New York, 1964).
Equation (3.6) has been already given, without proof and without detailed discussion, in Appendix C of our first paper, ref. (1).
Cf. footnote (18).
In ref. (1) we showed that the operatorsT μν generate anSL 2,C algebra.
For details, see, for example, our summary in Appendix B of ref. (1).
It is easy to check thatC 1 andC 2 commute now with all generators. That there are no other Casimir invariants can be checked by considering the contraction (see ref. (2)) of theISO 0(3,2) Casimir operators in the limitl → ∞.
An easily readable survey of the representations in question is given, for example, byH. Joos:Forts. Phys.,10, 65 (1962) or in the article byT. D. Newton: inThe Theory of Groups in Classical and Quantum Physics, Vol.1, edited byT. Kahan (Edinburgh, 1965).
This well-known result can be proved easily by using, for instance, the Ström basis, cf.S. Ström:Ark. Fys.,34, 215 (1967) and earlier papers quoted therein.
Cf. the article byP. Moussa andR. Stora: inLectures in Theoretical Physics, Vol.7 A (Boulder, Colo, 1965);H. Joos:Forts. Phys.,10, 65 (1962);A. J. MacFarlane:Journ. Math. Phys.,4, 490 (1962).
R. L. Anderson, R. Racka, M. A. Rashid andP. Winternitz: two papers inJourn. Math. Phys., in press (1970). We are much obliged to Dr.P. Winternitz for having made available to us the galley proofs of these papers prior to publication.
A simple review on tachyons, including references to the original literature, is given byO.-M. Bilaniuk andE. C. Sudarshan:Phys. Today,22, No. 5, 43 (1969). See also,Phys. Today,22, No. 12, 47 (1969).
M. Noga:Phys. Rev. D,2, 304 (1970).
Author information
Authors and Affiliations
Additional information
Work supported by the U.S. Air Force under Grant No. AFOSR-67-0385B.
Rights and permissions
About this article
Cite this article
Aghassi, J.J., Roman, P. & Santilli, R.M. Representation theory of a new relativistic dynamical group. Nuov Cim A 5, 551–590 (1971). https://doi.org/10.1007/BF02734565
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02734565