Canonical structure of classical field theory in n dimensions is studied within the covariant polymomentum Hamiltonian formulation of De Donder-Weyl (DW). The bi-vertical (n + 1)-form, called polysymplectic, is put forward as a generalization of the symplectic form in mechanics. Although not given in intrinsic geometric terms differently than a certain coset it gives rise to an invariantly defined map between horizontal forms playing the role of dynamical variables and the so-called vertical multivectors generalizing Hamiltonian vector fields. The analogue of the Poisson bracket on forms is defined which leads to the structure of Z-graded Lie algebra on the so-called Hamiltonian forms for which the map above exists. A generalized Poisson structure appears in the form of what we call a "higher-order" and a right Gerstenhaber algebra. The equations of motion of forms are formulated in terms of the Poisson bracket with the DW Hamiltonian n-form H vol ( vol is the space-time volume form, H is the DW Hamiltonian function) which is found to be related to the operation of the total exterior differentiation of forms. A few applications and a relation to the standard Hamiltonian formalism in field theory are briefly discussed. *

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A quantization of field theory based on the De Donder-Weyl (DW) covariant Hamiltonian formulation is discussed. A hypercomplex extension of quantum mechanics, in which the space-time Clifford algebra replaces that of the complex numbers, appears as a result of quantization of Poisson brackets of differential forms put forward for the DW formulation earlier. The proposed covariant hypercomplex Schrödinger equation is shown to lead in the classical limit to the DW Hamilton-Jacobi equation and to obey the Ehrenfest principle in the sense that the DW canonical field equations are satisfied for the expectation values of properly chosen operators. *

Abstract. Polymomentum canonical theories, which are manifestly covariant multi-parameter generalizations of the Hamiltonian formalism to field theory, are considered as a possible basis of quantization. We arrive at a multi-parameter hypercomplex generalization of quantum mechanics to field theory in which the algebra of complex numbers and a time parameter are replaced by the spacetime Clifford algebra and space-time variables treated in a manifestly covariant fashion. The corresponding covariant generalization of the Schrödinger equation is shown to be consistent with several aspects of the correspondence principle such as a relation to the De Donder-Weyl Hamilton-Jacobi theory in the classical limit and the Ehrenfest theorem. A relation of the corresponding wave function (over a finite dimensional configuration space of field and space-time variables) with the Schrödinger wave functional in quantum field theory is examined in the ultra-local approximation.

A relation between the Schrödinger wave functional and the Clifford-valued wave function which appears in what we call precanonical quantization of fields and fulfils a Dirac-like generalized covariant Schrödinger equation on the space of field and space-time variables is discussed. The Schrödinger wave functional is argued to be the trace of the positive frequency part of the continual product over all spatial points of the values of the aforementioned wave function restricted to a Cauchy surface. The standard functional differential Schrödinger equation is derived as a consequence of the Dirac-like covariant Schrödinger equation. *

The elements of the contrained dynamics algorithm in the De Donder-Weyl (DW) Hamiltonian theory for degenerate Lagrangian theories are discussed. A generalization of the Dirac bracket to the DW Hamiltonian theory with second class constraints (defined in the text) is presented.

FSU TPI 11/99 hep-th/9911175We show that the De Donder-Weyl (DW) covariant Hamiltonian field equations of any field can be written in Duffin-Kemmer-Petiau (DKP) matrix form. As a consequence, the (modified) DKP matrices β µ (5×5 in four space-time dimensions) are of universal significance for all fields admitting the DW Hamiltonian formulation, not only for a scalar field, and can be viewed as field theoretic analogues of the symplectic matrix, leading to the "k-symplectic" (k=4) structure. We also briefly discuss what could be viewed as the covariant Poisson brackets given by β-matrices and the corresponding Poisson bracket formulation of DW Hamiltonian field equations.In the end of the thirties is was observed [1] (see [2] for historical details) that the first order form of the Klein-Gordon and the Proca field equations can be represented in the Dirac-like matrix formwhere β-matrices fulfill the relationwhich defines the so-called Duffin-Kemmer-Petiau (DKP) algebra (for a textbook introduction see, e.g., [3,4]). Later on similar considerations have been extended to higher-spin theories [5][6][7], gauge fields [9,8], curved space-time [10], and even to the Einstein gravity [9,11] and arbitrary non-linear equations (see [4,9,11] and the references therein). From a more mathematical point of view DKP algebras have been considered, e.g., in [7,[12][13][14]. *

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