De Donder-Weyl Hamiltonian Research Articles

Canonical lifts in multisymplectic De Donder–Weyl Hamiltonian field theories

We define canonical lifts of vector fields to the multisymplectic multimomentum bundles of De Donder–Weyl Hamiltonian (first-order) field theories and to the appropriate premultisymplectic embedded constraint submanifolds on which singular field theories are studied. These new canonical lifts are used to study the so-called natural Noether symmetries present in both regular and singular Hamiltonian field theories along with their associated conserved quantities obtained from Noether’s theorem. The Klein–Gordon field, the Polyakov bosonic string, and Einstein–Cartan gravity in 3+1 dimensions are analyzed in depth as applications of these concepts; as a peripheral result obtained in the analysis of the bosonic string, we provide a new geometrical interpretation of the well-known Virasoro constraint.

Massive propagating modes of torsion

The dynamics of the torsion field is analyzed in the framework of the Covariant Canonical Gauge Theory of Gravity (CCGG), a De Donder–Weyl Hamiltonian formulation of gauge gravity. The action is quadratic in both, the torsion and the Riemann–Cartan tensor. Since the latter adds the derivative of torsion to the equations of motion, torsion is no longer identical to spin density, as in the Einstein–Cartan theory, but an additional propagating degree of freedom. As torsion turns out to be totally anti-symmetric, it can be parametrised via a single axial vector. It is shown in this paper that, in the weak torsion limit, the axial vector obeys a wave equation with an effective mass term which is partially dependent on the scalar curvature. The source of torsion is thereby given by the fermion axial current which is the net fermionic spin density of the system. Possible measurable effects and approaches to experimental analysis are addressed. For example, neutron star mergers could act as a dipoles or quadrupoles for torsional radiation, and an analysis of radiation of pulsars could lead to a detection of torsion wave background radiation.

On Clifford multivector-valued actions, generalized Dirac equation and quantization of branes

We explore the construction of a generalized Dirac equation via the introduction of the notion of Clifford multivector-valued actions, which was inspired by the work of Kanatchikov [Kanatchikov IV. De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory. Rept Math Phys. 1999;43:157–170; Kanatchikov IV. De Donder-Weyl Hamiltonian formulation and precanonical quantization of Vielbein gravity. J Phys Conf Ser. 2013;442:012041] on the De Donder-Weyl formulation of field theory. Crucial in this construction is the evaluation of the exponentials of multivectors associated with Clifford (hypercomplex) analysis. Exact matrix solutions (instead of spinors) of the generalized Dirac equation in D = 2 , 3 space–time dimensions are found. This formalism can be extended to curved space–time backgrounds. We conclude by proposing a wave-functional equation governing the quantum dynamics of branes living in C-spaces (Clifford spaces), and which is based on the De Donder–Weyl Hamiltonian formulation of field theory.

Torsion driving cosmic expansion

We study a cosmological model based on the canonical Hamiltonian transformation theory. Using a linear-quadratic approach for the free gravitational De Donder–Weyl Hamiltonian H_textrm{Gr}, the model contains terms describing a deformation of an AdS spacetime and a fully anti-symmetric torsion in addition to Einstein’s theory. The resulting extension of the Einstein–Cartan theory depends on two initially unknown constants, Omega _textrm{g} and Omega _textrm{s}. Given an appropriate choice of these parameters resulting from the analysis of asymptotics, numerical calculations were performed with Omega _mathrm {Lambda } = 0. Values from the Planck Collaboration (Aghanim in Cosmol Parameters 641, 6, 2020) were used for all other required cosmological parameters. In this way, it is shown that torsion can explain phenomena commonly attributed to dark energy, and thus can replace Einstein’s cosmological constant.

Clifford Algebraic Approach to the De Donder–Weyl Hamiltonian Theory

The Clifford algebraic formulation of the Duffin-Kemmer-Petiau (DKP) algebras is applied to recast the De Donder-Weyl Hamiltonian (DWH) theory as an algebraic description independent of the matrix representation of the DKP algebra. We show that the DWH equations for antisymmetric fields arise out of the action of the DKP algebra on certain invariant subspaces of the Clifford algebra which carry the representations of the fields. The matrix representation-free formula for the bracket associated with the DKP form of the DWH equations is also derived. This bracket satisfies a generalization of the standard properties of the Poisson bracket

Polysymplectic formulation for BF gravity with Immirzi parameter

The polysymplectic formulation of the CMPR action, which is a BF-type formulation of general relativity that involves an arbitrary Immirzi parameter, is performed. We implement a particular scheme within this covariant Hamiltonian approach to analyze the constraints that characterize the CMPR model. By means of the privileged -forms and the Poisson–Gerstenhaber bracket, inherent to the polysymplectic framework, the BF field equations associated to the CMPR action are obtained and, in consequence, the Einstein equations naturally emerge by solving the simplicity constraints of the theory. Further, from the polysymplectic analysis of the CMPR action the De Donder–Weyl Hamiltonian formulation of the Holst action is recovered, which is consistent with the Lagrangian analysis of this model as reported in the literature.

Schrödinger wave functional of a quantum scalar field in static space-times from precanonical quantization

The functional Schrödinger representation of a scalar field on an [Formula: see text]-dimensional static space-time background is argued to be a singular limiting case of the hypercomplex quantum theory of the same system obtained by the precanonical quantization based on the space-time symmetric De Donder–Weyl Hamiltonian theory. The functional Schrödinger representation emerges from the precanonical quantization when the ultraviolet parameter [Formula: see text] introduced by precanonical quantization is replaced by [Formula: see text], where [Formula: see text] is the time-like tangent space Dirac matrix and [Formula: see text] is an invariant spatial [Formula: see text]-dimensional Dirac’s delta function whose regularized value at [Formula: see text] is identified with the cutoff of the volume of the momentum space. In this limiting case, the Schrödinger wave functional is expressed as the trace of the product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration and the canonical functional derivative Schrödinger equation is derived from the manifestly covariant Dirac-like precanonical Schrödinger equation which is independent of a choice of a codimension-one foliation.

De Donder–Weyl Hamiltonian formalism of MacDowell–Mansouri gravity

We analyse the behaviour of the MacDowell–Mansouri action with internal symmetry group under the De Donder–Weyl Hamiltonian formulation. The field equations, known in this formalism as the De Donder–Weyl equations, are obtained by means of the graded Poisson–Gerstenhaber bracket structure present within the De Donder–Weyl formulation. The decomposition of the internal algebra allows the symmetry breaking , which reduces the original action to the Palatini action without the topological term. We demonstrate that, in contrast to the Lagrangian approach, this symmetry breaking can be performed indistinctly in the polysymplectic formalism either before or after the variation of the De Donder–Weyl Hamiltonian has been done, recovering Einstein’s equations via the Poisson–Gerstenhaber bracket.

Vorticity and symplecticity in multi-symplectic, Lagrangian gas dynamics

The Lagrangian, multi-dimensional, ideal, compressible gas dynamic equations are written in a multi-symplectic form, in which the Lagrangian fluid labels, mi (the Lagrangian mass coordinates) and time t are the independent variables, and in which the Eulerian position of the fluid element and the entropy are the dependent variables. Constraints in the variational principle are incorporated by means of Lagrange multipliers. The constraints are: the entropy advection equation St = 0, the Lagrangian map equation where is the fluid velocity, and the mass continuity equation which has the form where is the Jacobian of the Lagrangian map in which and is the specific volume of the gas. The internal energy per unit volume of the gas corresponds to a non-barotropic gas. The Lagrangian is used to define multi-momenta, and to develop de Donder–Weyl Hamiltonian equations. The de Donder–Weyl equations are cast in a multi-symplectic form. The pullback conservation laws and the symplecticity conservation laws are obtained. One class of symplecticity conservation laws give rise to vorticity and potential vorticity type conservation laws, and another class of symplecticity laws are related to derivatives of the Lagrangian energy conservation law with respect to the Lagrangian mass coordinates mi. We show that the vorticity–symplecticity laws can be derived by a Lie dragging method, and also by using Noether’s second theorem and a fluid relabelling symmetry which is a divergence symmetry of the action. We obtain the Cartan–Poincaré form describing the equations and we discuss a set of differential forms representing the equation system.

Multisymplectic formulation of vielbein gravity: I. De Donder–Weyl formulation, Hamiltonian (n − 1)-forms

We consider the De Donder–Weyl (DW) Hamiltonian formulation of the Palatini action of vielbein gravity formulated in terms of the solder form and spin connection, which are treated as independent variables. The basic geometrical constructions necessary for the DW Hamiltonian theory of vielbein gravity are presented. We reproduce the DW Hamilton equations in the multisymplectic and pre-multisymplectic formulations. We also give basic examples of Hamiltonian -forms and related Poisson brackets.

De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity

The De Donder-Weyl (DW) covariant Hamiltonian formulation of the Palatini first-order Lagrangian of vielbein (tetrad) gravity and its precanonical quantization are presented. No splitting into space and time is required in this formulation. Our recent generalization of Dirac brackets is used to treat the second class primary constraints appearing in the DW Hamiltonian formulation and to find the fundamental brackets. Quantization of the latter yields the representation of vielbeins as differential operators with respect to the spin connection coefficients and the Dirac-like precanonical Schrödinger equation on the space of spin connection coefficients and space time variables. The transition amplitudes on this space describe the quantum geometry of space-time. We also discuss the Hilbert space of the theory, the invariant measure on the spin connection coefficients, and point to the possible quantum singularity avoidance built in in the natural choice of the boundary conditions of the wave functions on the space of spin connection coefficients.

Covariant canonical quantization

We present a manifestly covariant quantization procedure based on the de Donder–Weyl Hamiltonian formulation of classical field theory. This procedure agrees with conventional canonical quantization only if the parameter space is d=1 dimensional time. In d>1 quantization requires a fundamental length scale, and any bosonic field generates a spinorial wave function, leading to the purely quantum-theoretical emergence of spinors as a byproduct. We provide a probabilistic interpretation of the wave functions for the fields, and we apply the formalism to a number of simple examples. These show that covariant canonical quantization produces both the Klein–Gordon and the Dirac equation, while also predicting the existence of discrete towers of identically charged fermions with different masses. Covariant canonical quantization can thus be understood as a “first” or pre-quantization within the framework of conventional QFT.

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

Precanonical quantization of pure Yang-Mills fields, which is based on the covariant De Donder-Weyl (DW) Hamiltonian formulation, and its connection with the functional Schrödinger representation in the temporal gauge are discussed. The mass gap problem is related to the finite-dimensional spectral problem for a generalized Clifford-valued magnetic Schrödinger operator which represents the DW Hamiltonian operator.

On the Duffin-Kemmer-Petiau formulation of the covariant Hamiltonian dynamics in field theory

We show that the De Donder-Weyl (DW) covariant Hamiltonian field equations of any field can be written in universal 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 to be the covariant Poisson bracket given by β-matrices and the corresponding Poisson bracket formulation of DW Hamiltonian field equations.

Canonical structure of classical field theory in the polymomentum phase space

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 the 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.