Skip to main content
SpringerLink
Log in

Advanced-retarded polarization for relativistic Hamiltoniann-particle systems

«Опережающая-qzапаздывающая» поляризация для систем, систоящих изn релятивистских частиц

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

Summary

The space-time (evolution space) and possible-phase-space descriptions of a system ofn structureless point particles are discussed in the framework of instantaneous-action-at-a-distance theory (or predictive mechanics). The evolution spaceE is taken to be an open subset ofR 8n. The classical motions are described by integral manifolds of a particular type of 2n-dimensional differential system ℰ invariant under the natural action of the Poincaré group. Both as a first step towards quantization and as a restriction of the class of such interactions the 6n-dimensional phase space defined as the quotient setE/ℰ is required to possess a symplectic structure with a particular polarization (i.e. a selection of 3n «position» co-ordinates that have commuting Poisson brackets). In this paper it is shown that an earlier proposed «advanced-retarded» polarization for the two-particle system that admits nontrivial interactions is compatible with symmetry under particle permutations and generalizes ton>2.

Riassunto

Si discutono descrizioni dello spazio tempo (spazio di evoluzione) e di possibili spazi fase di un sistema din particelle puntiformi senza struttura nel contesto della teoria dell’azione istantanea a distanza (o meccanica predittiva). Si assume che lo spazio di evoluzioneE sia un sottogruppoR 8n. Si descrivono moti classici mediante multistrati integrali di un tipo particolare di sistema differenziale 2n dimensionale ℰ, invariante sotto l’azione naturale del gruppo di Poincaré. Sia come primo passo verso la quantizzazione che come restrizione della classe di queste interazioni si richiede che lo spazio fase 6n-dimensionale definito come il gruppo di quozientiE/ℰ possieda una struttura simplettica con una particolare polarizzazione (cioè una scelta di 3n coordinate di «posizione» che hanno parentesi di Poisson scambiabili). In questo lavoro si mostra che una polarizzazione «avanzata-ritardata», precedentemente proposta, per il sistema a due particelle che ammette interazioni non triviali è compatibile con la simmetria che tien conto di scambi di particelle, e generalizza an>2.

Резюме

Обсуждаются пространство-время и возможные описания фазового пространства системы изn бесструктурных точечных частиц в рамках теоии мгновенного действия на расстоянии. Считается, что пространство-времяE представляет открытую подсистемуR 8n. Классические движения описываются с оомощью интегральных множеств специального типа для 2n-мерной дифференциальной системыE инвариаитной относительно естественного действия группы Пуанкаре. При квантовании и при ограничении класса таких взаимодействий, для получения симплектической струры с частичной поляризацией (те. выбпр 3n координат «положения», которые имеют коммутирующие скобки. Пуассона) требуртся 6n-мерное фазовое пространство, определенное как система отношенийE/ℰ. В этой стттье показывается, что ранее предложенная «опережающая-запа-здывающая» поляризация для двух-частичной системы, которая допускает нетри-виальные взаимодействия, совметсима с симметрией отностельно перестановок частиц и обобщается на случайn>2.

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. P. A. M. Dirac:Rev. Mod. Phys.,21, 392 (1959).

    Article  MathSciNet  ADS  Google Scholar 

  2. L. Bel:Ann. Inst. H. Poincaré,14 A, 189 (1971).

    MathSciNet  Google Scholar 

  3. See,e.g.,E. C. G. Sudarshan:Action-at-a-distance (preprint, Austin, Tex., 1970).

  4. J. M. Souriau:Structures des systèmes dynamiques (Paris, 1970).

  5. B. Kostant:Lectures in Modern Analysis and Applications, Vol.3, edited byC. T. Taam (Berlin, 1970).

  6. B. Kostant:Géométrie symplectique et physique mathématique, inColloques Internationaux CNRS (Paris, 1975).

  7. D. J. Simms andN. M. J. Woodhouse:Lectures on Geometric Quantization (Berlin, 1976).

  8. J. Ehlers:General Relativity and Gravitation, edited byG. Shaviv andJ. Rosen (New York, N.Y., 1975).

  9. SeeP. Havas:Causality and Physical Theories, edited byW. B. Rolnick (New York, N.Y., 1974), for a good discussion of such questions of causality.

  10. L. Bel:Ann. Inst. H. Poincaré,18 A, 57 (1973).

    MathSciNet  Google Scholar 

  11. L. Bel andJ. Martin:Ann. Inst. H. Poincaré,22 A, 173 (1975).

    MathSciNet  Google Scholar 

  12. E. H. Kerner (Editor):The Theory of Action-at-a-Distance in Particle Dynamics, Reprint collection (New York, N.Y., 1972).

  13. H. P. Künzle:Symposia Mathematica,14, 53 (1974).

    Google Scholar 

  14. H. P. Künzle:Journ. Math. Phys.,15, 1033 (1974).

    Article  ADS  Google Scholar 

  15. R. Arens:Journ. Math. Phys.,16, 1191 (1975);F. Coester andP. Havas:Phys. Rev. D,14, 2556 (1976);M. Pauri andG. M. Prosperi:Journ. Math. Phys.,17, 1468 (1976);C. Piron andF. Reuse:The realitivistic two-body problem, preprint, Genève (1975).

    Article  MathSciNet  ADS  Google Scholar 

  16. L. Bel andX. Fustero:Ann. Inst. H. Poincaré,25 A, 411 (1976).

    Google Scholar 

  17. T. F. Jordan:Phys. Rev. D,11, 2807, 3035 (1975).

    Article  MathSciNet  ADS  Google Scholar 

  18. D. G. Currie, T. F. Jordan andE. C. G. Sudarshan:Rev. Mod. Phys.,35, 350 (1963);D. G. Currie:Journ. Math. Phys.,4, 1470 (1963);J. T. Cannon andT. F. Jordan:Journ. Math. Phys.,5, 299 (1964);H. Leutwyler:Nuovo Cimento,37, 556 (1965). For more recent proofs using differential geometric methods see alsoL. Bel:Ann. Inst. H. Poincaré,12 A, 307 (1970);J. Martin andJ. L. Sanz:No interaction theorem of Currie, Jordan and Sudarshan. Erpansion in c −1 (preprint, 1977).

    Article  MathSciNet  ADS  Google Scholar 

  19. E. H. Kerner:Journ. Math. Phys.,6, 1218 (1965);R. N. Hill:Journ. Math. Phys.,8, 1756 (1967).

    Article  MathSciNet  ADS  Google Scholar 

  20. H. P. Künzle:Int. Journ. Theor. Phys.,11, 395 (1974). In the meantime, it has turned out that this example of an interaction corresponds exactly to the electromagnetic interaction, made «predictive» in the sense of Bel(21). Similarly the interaction via a massless scalar field fits into this scheme. These results will be published elsewhere.

    Article  Google Scholar 

  21. L. Bel, A. Salas andJ. M. Sanchez:Phys. Rev. D,1, 1099 (1973);L. Bel andJ. Martin:Phys. Rev. D,8, 4347 (1973);9, 2760 (1974).

    Article  ADS  Google Scholar 

  22. P. Droz-Vincent:Rep. Math. Phys.,8, 79 (1975).

    Article  MathSciNet  ADS  Google Scholar 

  23. (x α k ,v α k ) is a fibre co-ordinate system of\(E - T\bar V\);k=1, 2, …,n; α, β=0, 1, 2, 3;\(\partial \begin{array}{*{20}c} k \\ \alpha \\ \end{array} : = \partial /\partial x\begin{array}{*{20}c} k \\ \alpha \\ \end{array} ; \partial \begin{array}{*{20}c} k \\ \alpha \\ \end{array} : = \partial /\partial \upsilon \begin{array}{*{20}c} \alpha \\ k \\ \end{array} \) and the summation convention is applied except for lower-case Latin indices.

  24. This definition expresses in homogenesous space-time form the Newtonian idea that then-particle system is described byn ordinary second-order differential equations.

  25. H. P. Künzle:Ann. Inst. H. Poincaré,17, 337 (1972).

    Google Scholar 

  26. This is a strong global condition. It means in practice that the evolution space must in general be restricted to some open subset of\(T\bar V\) ifV was originally chosen diffeomorphic toR 4. We will not be concerned with such global questions here. Indeed, it is quite doubtful whether this is the right framework in which to ask them, since even for a case as simple as the Galilei-invariant harmonic oscillator ℰ turns out to be differentiable only on an open subset ofE≃R 16.

  27. A closed 2-form of maximal rank,i.e. a skew symmetric tensor field ωαβ that has an inverse ωαβ and satisfies\(\partial _{[\alpha ^\omega \beta \gamma ]} = 0\).

  28. See,e.g.,R. Abraham andJ. E. Marsden:Foundations of Mechanics (New York, N.Y., 1967);S. Sternberg:Lectures on Differential Geometry (Englewood Cliffs, N. J., 1964), which will be the main reference also for differential geometrical notation.

  29. «┘» denotes contraction:X f ┘ ω=df⇔(Y, X t )=Y┘df≡(df)(Y)=Y(f) for any vector fieldY.

  30. If ϕ:M→N is a differentiable map, then ϕ*:T D M→T ϕ(D) N denotes the tangent map, ϕ*:T *(p)ϕ N→T * p M its dual.

  31. Always locally. If μ exists globally onM, then ψ M is called a strongly symplectic action, see ref. (4,5)J. M. Souriau:Structures des systèmes dynamiques (Paris, 1970).B. Kostant:Lectures in Modern Analysis and Applications, Vol.3, edited byC. T. Taam (Berlin, 1970).

  32. See,e.g.,R. Hermann:Vector Bundles in Mathematical Physics, II (New York, N.Y., 1970).

  33. For a nontechnical discussion seeJ. M. Souriau:Géométrie symplectique et physique mathématique, preprint, Marseille (1975).

  34. E.g.,P. Havas:Problems in the Foundations of, Physics, edited byM. Bunge (New York, N.Y., 1970); and ref. (11)L. Bel andJ. Martin:Ann. Inst. H. Poincaré 22 A, 173 (1975).

  35. εαβγδ is the alternating symbol and indices are raised by means of ηαβ, also forv=0.

  36. See,e.g.,R. M. F. Houtappel,H. Van Dam andE. P. Wigner:Rev. Mod. Phys.,37, 595 (1965); the problems are the same as those encountered in scattering theory of several particles. For a recent example seeA. Gheorghe, D. B. Ion andEl. Mihul:Ann. Inst. H. Poincaré,22, 131 (1975).

    Article  ADS  Google Scholar 

  37. Upper-case Latin indices range from 1 to 3.

  38. We consider here Σ as a fixed manifold (which will soon be identified with a subset ofR 8n), but the mappingi as somewhat arbitrary.

  39. Loca in the sense of the manifold of motions: some exceptional types of motion might escape the description. Global studies of the manifold of all motions can only be undertaken if many more assumptions are made about the class of interactions considered.

  40. So far Σ was an abstract manifold diffeomorphic toi(Σ). Identification of Σ withi o(Σ) gives some meaning to its co-ordinates.

  41. See ref. (18); In these papers the phase space is not the tangent boundle of\(R^{3n} = \left\{ {\bar x\begin{array}{*{20}c} A \\ k \\ \end{array} ,\bar \upsilon \begin{array}{*{20}c} A \\ k \\ \end{array} } \right\}\), but the cotangent bundle\(R^{3n} = \left\{ {\bar x\begin{array}{*{20}c} A \\ k \\ \end{array} ,\bar \upsilon \begin{array}{*{20}c} A \\ k \\ \end{array} } \right\}\). The action of the group onR 3n (on the position variables) is, however, the same as ours and this is all that must be specified.

    Article  MathSciNet  ADS  Google Scholar 

  42. P. Droz-Vincent:Ann. Inst. H. Poincaré,20 A, 269 (1974).

    MathSciNet  Google Scholar 

  43. P. Droz-Vincent:Nuovo Cimento,12 B 1 (1972).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Alberta, T6G 2G1, Edmonton, Canada

    H. P. Künzle

Authors
  1. H. P. Künzle
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported in part by the National Research Council of Canada.

Traduzione a cura della Redazione.

Переведено редакцией.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Künzle, H.P. Advanced-retarded polarization for relativistic Hamiltoniann-particle systems. Nuov Cim B 43, 87–109 (1978). https://doi.org/10.1007/BF02728292

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02728292

Search

Navigation