Precanonical Quantization and the Schroedinger Wave Functional

A relation between the Schroedinger wave functional and the Clifford-valued wave function which appears in what we call precanonical quantization of fields and fulfills a Dirac-like generalized covariant Schroedinger equation on the space of field and space-time variables is discussed. The Schroedinger 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 Schroedinger equation is derived as a consequence of the Dirac-like covariant Schroedinger equation.Comment: 16pp, LaTeX2e. v2: minor changes in the presentation, misprints in eqs. 4.21, 4.24 and unnumbered after eq. 4.8 fixed, sect. 4.1 partly rewritten, the conclusions section expanded, references added, LaTeX format changed; to appear in Phys. Lett.

On Field Theoretic Generalizations of a Poisson Algebra

A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an $(n+1)$-ary bracket on functions are considered. The Poisson bracket on differential forms gives rise to various generalizations of a Gerstenhaber algebra: the noncommutative (in the sense of Loday) and the higher-order (in the sense of the higher order graded Leibniz rule). The $(n+1)$-ary bracket fulfills the properties of the Nambu bracket including the ``fundamental identity'', thus leading to the Nambu-Poisson algebra. We point out that in the field theory context the Nambu bracket with a properly defined covariant analogue of Hamilton's function determines a joint evolution of several dynamical variables.Comment: 10 pages, LaTeX2e. Missprint in Ref. 1 is corrected (hep-th/9709229 instead of ...029

Precanonical quantization of Yang-Mills fields and the functional Schroedinger representation

Precanonical quantization of pure Yang-Mills fields, which is based on the covariant De Donder-Weyl (DW) Hamiltonian formalism, and its connection with the functional Schrodinger representation in the temporal gauge are discussed. The YM mass gap problem is related to a finite dimensional spectral problem for a generalized Clifford-valued magnetic Schr\"odinger operator in the space of gauge potentials which represents the DW Hamiltonian operator.Comment: LaTeX2e, 11pages. v2: 13 pages, minor changes, references added, sect. 5 extended, to appear in Rep. Math. Phy