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On the space-time interpretation of classical canonical systems I: The general theory

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Abstract

We define a canonical system as a canonical manifoldM plus a canonical vectorfield onM. For such systems a unique kinematical interpretation is deduced from a set of Kinematical Axioms satisfied by the algebra of differentiable functions onM. This algebra is required to contain a subalgebra which is maximal commutative under the Poisson bracket.M is shown to be diffeomorphic to the cotangent bundle over its quotient manifold, which is defined by the given subalgebra. Canonical systems satisfying these axioms are then classified. If the “phase space interpretation” is adopted they are shown to describe the motion of masspoints in some configuration space under the influence of and interacting by arbitrary vector and scalar potentials.

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References

  1. Segal, I.E.: Postulates for general quantum mechanics. Ann. Math.48, 930–948 (1947)

    Google Scholar 

  2. Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys.5, 848–861 (1964)

    Google Scholar 

  3. Jost, R.: Poisson brackets (An unpedagogical lecture). Rev. Mod. Phys.36, 572–579 (1964)

    Google Scholar 

  4. Ehlers, J., Schild, A.: Geometry is a manifold with projective structure, Commun. math. Phys.32, 119–146 (1973)

    Google Scholar 

  5. Ehlers, J., Pirani, F.A.W., Schild, A.: General relativity, papers in honour of J.L. Synge (ed. L. O'Raifeartaigh) 63–84. Oxford: Clarendon Press 1972

    Google Scholar 

  6. Sniatycki, J., Tulczyjew, W.M.: Canonical dynamics of relativistic charged particles. Ann. Inst. H. Poincaré A,15, 177–187 (1971)

    Google Scholar 

  7. Kijowski, J.: A finite-dimensional canonical formulation in the classical field theory. Commun. math. Phys.30, 99–128 (1973)

    Google Scholar 

  8. Drühl, K.: Relativistic canonical systems, preprint Max-Planck-Institut zur Erforschung der Lebensbedingungen. Starnberg 1975

  9. Neumark, M.A.: Normierte Algebren. Berlin: VEB Deutscher Verlag der Wissenschaften 1959

    Google Scholar 

  10. Helgason, S.: Differential geometry and symmetric spaces. New York, London: Academic Press 1962

    Google Scholar 

  11. Hermann, R.: Differential geometry and the calculus of variations. New York, London: Academic Press 1968

    Google Scholar 

  12. Chevally, C.: Theory of Lie groups, 6th printing 1965. Princeton: Princeton University Press 1946

    Google Scholar 

  13. Drühl, K.: On the space-time interpretation of classical canonical systems. II: Relativistic canonical systems. Commun. math. Phys.49, 289–300 (1976)

    Google Scholar 

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  1. Max-Planck-Institut zur Erforschung der Lebensbedingungen der wissenschaftlich-technischen Welt, D-8130, Starnberg, Germany

    K. Drühl

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  1. K. Drühl
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Communicated by J. Ehlers

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Drühl, K. On the space-time interpretation of classical canonical systems I: The general theory. Commun.Math. Phys. 49, 277–288 (1976). https://doi.org/10.1007/BF01608732

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