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Thursday, 28 May 2015

New article "Polysymplectic Hamiltonian field theory"

My new article "Polysymplectic Hamiltonian field theoryarXiv: 1505.01444


Applied to field theory, the familiar symplectic technique leads to instantaneous Hamiltonian formalism on an infinite-dimensional phase space. A true Hamiltonian partner of first order Lagrangian theory on fibre bundles Y->X is covariant Hamiltonian formalism in different variants, where momenta correspond to derivatives of fields relative to all coordinates on X. We follow polysymplectic (PS) Hamiltonian formalism on a Legendre bundle over Y provided with a polysymplectic TX-valued form. If X=R, this is a case of time-dependent non-relativistic mechanics. PS Hamiltonian formalism is equivalent to the Lagrangian one if Lagrangians are hyperregular. A non-regular Lagrangian however leads to constraints and requires a set of associated Hamiltonians. We state comprehensive relations between Lagrangian and PS Hamiltonian theories in a case of semiregular and almost regular Lagrangians. Quadratic Lagrangian and PS Hamiltonian systems, e.g. Yang - Mills gauge theory are studied in detail. Quantum PS Hamiltonian field theory can be developed in the frameworks both of familiar functional integral quantization and quantization of the PS bracket.

  • First order Lagrangian formalism on fibre bundles
  • Cartan and Hamilton - De Donder equations
  • Polysymplectic structure
  • PS bracket
  • Hamiltonian forms
  • Covariant Hamilton equations
  • Hamiltonian time-dependent mechanics
  • Iso-PS structure
  • Associated Hamiltonian and Lagrangian systems
  • Lagrangian and Hamiltonian conservation laws
  • Lagrangian and Hamiltonian Jacobi fields
  • Quadratic Lagrangian and Hamiltonian systems
  • PS Hamiltonian gauge theory
  • Affine Lagrangian and Hamiltonian systems
  • Functional integral quantization
  • Algebraic quantization. Quantum PS bracket

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