I. Kanatchikov
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.
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We analyse the behaviour of the MacDowell-Mansouri action with internal symmetry group $\mathrm{SO}(4,1)$ under the covariant 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 covariant formulation. The decomposition of the internal algebra $\mathfrak{so}(4,1)\simeq\mathfrak{so}(3,1)\oplus\mathbb{R}^{3,1}$ allows the symmetry breaking $\mathrm{SO}(4,1)\to\mathrm{SO}(3,1)$, which reduces the original action to the Palatini action without the topologi…
We compute the linearized Weyl-Weyl correlator using a new solution for the graviton propagator on de Sitter background in de Donder gauge. The result agrees exactly with a previous computation in a noncovariant gauge. We also use dimensional regularization to compute the one loop expectation value of the square of the Weyl tensor.
The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) de Donder-Weyl (dDW) one. One of the main points is the fact that the Legendre transform in the dDW approach is replaced by a Legendre correspondence in the LP theory (This correspondence behaves differently: ignoring the singularities whenever the Lagrangian is dege…
We show that the covariant analytic mechanics (CAM) is closely related to the De Donder-Weyl (DW) theory. To treat space and time on an equal footing, the DW theory introduces $D$ conjugate fields ($D$ is the dimension of space-time) for each field and the CAM regards the differential forms as the basic variables. The generalization of the canonical equations is called the DW equations. Although one of the DW equations is not correct for the gauge field and the gravitational field, we show the way to improve it. By rewriting the canonical equations of the CAM, which are manifestly general coor…
The analogue of the Poisson bracket for the De Donder-Weyl (DW) Hamiltonian formulation of field theory is proposed. We start from the Hamilton- Poincaré-Cartan (HPC) form of the multidimensional variational calculus and define the bracket on the differential forms over the space-time (=horizontal forms). This bracket is related to the Schouten-Nijenhuis bracket of the multivector fields which are associated with the horizontal forms by means of the "polysymplectic form". The latter is given by the HPC form and generalizes the symplectic form to field theory. We point out that the algebra of f…
The De Donder-Weyl (DW) covariant Hamiltonian formulation of Palatini first-order Lagrangian of vielbein (tetrad) gravity and its precanonical quantization are presented. No splitting into the 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 Schrö…
We discuss a field theoretical extension of the basic structures of classical analytical mechanics within the framework of the De Donder--Weyl (DW) covariant Hamiltonian formulation. The analogue of the symplectic form is argued to be the {\em polysymplectic} form of degree $(n+1)$, where $n$ is the dimension of space-time, which defines a map between multivector fields or, more generally, graded derivation operators on exterior algebra, and forms of various degrees which play a role of dynamical variables. The Schouten-Nijenhuis bracket on multivector fields induces the graded analogue of the…
Canonical Lifts in Multisymplectic De Donder-Weyl Hamiltonian Field Theories
本文定义了向量场到De Donder-Weyl哈密顿场论多动量丛及约束子流形的典范提升，用于研究正则与奇异场论中的自然Noether对称性及守恒量，并深入分析了Klein-Gordon场、玻色弦和Einstein-Cartan引力，给出了Virasoro约束的新几何解释。
Elements of the quantization in field theory based on the covariant polymomentum Hamiltonian formalism (the De Donder-Weyl theory), a possibility of which was originally discussed in 1934 by Born and Weyl, are developed. The approach is based on a recently proposed graded Poisson bracket on differential forms in field theory (see e.g. hep-th/9709229). A covariant analogue of the Schrödinger equation for a hypercomplex wave function on the space of field and space-time variables is put forward. It is shown to lead to the De Donder-Weyl Hamilton-Jacobi equations in quasiclassical limit. A possib…
The (pre)multisymplectic geometry of the De Donder--Weyl formalism for field theories is further developed for a variety of field theories including a scalar field theory from the canonical Klein-Gordon action, the electric and magnetic Carrollian scalar field theories, bosonic string theory from the Nambu-Goto action, and $2+1$ gravity as a Chern-Simons theory. The Lagrangians for the scalar field theories and for $2+1$ Chern-Simons gravity are found to be singular in the De Donder--Weyl sense while the Nambu-Goto Lagrangian is found to be regular. Furthermore, the constraint structure of the…
We 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 beta-matrices (5 X 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 so-called ``k-symplectic'' (k=4) structure. We also briefly discuss what could be viewed as the covariant Poisson bracket given by beta-matrices and the correspo…
In this short note we study unimodular gravity in Weyl-De Donder formalism. We find corresponding Hamiltonian and study consequence of the unimodular constraint on the conjugate covariant momenta. We also find covariant Hamiltonian for Henneaux-Teitelboim unimodular action and study corresponding equations of motion.
We analyze the De Donder-Weyl covariant field equations for the topologically massive Yang-Mills theory. These equations are obtained through the Poisson-Gerstenhaber bracket described within the polysymplectic framework. Even though the Lagrangian defining the system of our interest is singular, we show that by appropriately choosing the polymomenta one may obtain an equivalent regular Lagrangian, thus avoiding the standard analysis of constraints. Further, our simple treatment allows us to only consider the privileged $(n-1)$-forms in order to obtain the correct field equations, in oppositio…
Deformation quantization is a formal deformation of the algebra of smooth functions on some manifold. In the classical setting, the Poisson bracket serves as an initial conditions, while the associativity allows to proceed to higher orders. Some applications to string theory require deformation in the direction of a quasi-Poisson bracket (that does not satisfy the Jacobi identity). This initial condition is incompatible with associativity, it is quite unclear which restrictions can be imposed on the deformation. We show that for any quasi-Poisson bracket the deformation quantization exists and…
We present the BRST formalism of a Weyl conformal gravity in Weyl geometry. Choosing the extended de Donder gauge-fixing condition (or harmonic gauge condition) for the general coordinate invariance and the new scalar gauge-fixing for the Weyl invariance we find that there is a Poincar${\rm{\acute{e}}}$-like $\IOSp(10|10)$ supersymmetry as in a Weyl invariant scalar-tensor gravity in Riemann geometry. We also point out that there is a gravitational conformal symmetry in quantum gravity although there is a massive Weyl gauge field as a result of spontaneous symmetry breakdown of Weyl gauge symm…
The spectrum of masses of the colorless (color neutral) states of the DW (De Donder-Weyl) Hamiltonian operator of quantum SU(2) Yang-Mills field theory on $\mathbb{R}^D$ obtained via the precanonical quantization is shown to be purely discrete and bounded from below. The scale of the mass gap is estimated to be of the order of magnitude of the scale of the ultra-violet parameter $\varkappa$ introduced by precanonical quantization on dimensional grounds.
Higher bracket structure of density operators in Weyl fermion systems and topological insulators
本文研究了Weyl费米子体系与拓扑绝缘体中电子密度算符的高阶括号结构，发现其与量子霍尔效应的代数平行对应，并揭示了其与拓扑不变量循环上闭链及非交换几何的联系。
Hamiltonian for Weyl Transverse Gravity
本文推导Weyl横向引力的哈密顿量，给出三级约束及其Poisson括号，并证明规范固定后可得到幺模引力的哈密顿量。
The covariant canonical method of quantization based on the De Donder-Weyl covariant canonical formalism is used to formulate a world-sheet covariant quantization of bosonic strings. To provide the consistency with the standard non-covariant canonical quantization, it is necessary to adopt a Bohmian deterministic hidden-variable equation of motion. In this way, string theory suggests a solution to the problem of measurement in quantum mechanics.
Higher spin dynamics in Weyl reference frame
本研究在Weyl参考系中推导了有限距离类空超曲面上的$w_{1+\infty}$代数，解决了高自旋电荷括号的非局域问题，为有限距离记忆效应提供新视角，可推广至渐近(A)dS$_4$时空。
Quantization of constraint systems within the Weyl-Wigner-Groenewold-Moyal framework is discussed. Constraint dynamics of classical and quantum systems is reformulated using the skew-gradient projection formalism. The quantum deformation of the Dirac bracket is generalized to match smoothly the classical Dirac bracket in and outside of the constraint submanifold in the limit $\hbar \to 0$.
The Wigner-Weyl- Moyal approach to Quantum Mechanics is recalled, and similarities to classical probability theory emphasised. The Wigner distribution function is generalised and viewed as a construction of a bosonic object, a target space co-ordinate, for example, in terms of a bilinear convolution of two fermionic objects, e.g. a quark antiquark pair. This construction is essentially non-local, generalising the idea of a local current. Such Wigner functions are shown to solve a BPS generalised Moyal-Nahm equation.
A generalization of canonical quantization which maps a dynamical operator to a dynamical superoperator is suggested. Weyl quantization of dynamical operator, which cannot be represented as Poisson bracket with some function, is considered. The usual Weyl quantization of observables is a specific case of suggested quantization. This approach allows to define consistent quantization procedure for non-Hamiltonian and dissipative systems. Examples of the harmonic oscillator with friction (generalized Lorenz-Rossler-Leipnik-Newton equation), the Fokker-Planck-type system and Lorenz-type system are…
A brief pedagogical survey of the star product is provided, through Groenewold's original construction based on the Weyl correspondence. It is then illustrated how simple Landau orbits in a constant magnetic field, through their Dirac Brackets, define a noncommutative structure since these brackets exponentiate to a star product---a circumstance rarely operative for generic Dirac Brackets. The geometric picture of the star product based on its Fourier representation kernel is utilized in the evaluation of chains of star products. The intuitive appreciation of their associativity and symmetries…
The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) de Donder-Weyl (dDW) one. For instance the Legendre transform in the dDW approach is replaced by a Legendre correspondence in the LP theory, while ignoring singularities whenever the Lagrangian is degenerate. Moreover we show that there exist two different definition…
The Weyl-Wigner-Moyal formalism for Dirac second class constrained systems has been proposed recently as the deformation quantization of Dirac bracket. In this paper, after a brief review of this formalism, it is applied to the case of the relativistic free particle. Within this context, the Stratonovich-Weyl quantizer, Weyl correspondence, Moyal $\star$-product and Wigner function in the constrained phase space are obtained. The recent Hamiltonian treatment for constrained systems, whose constraints depend explicitly on time, are used to perform the deformation quantization of the relativisti…
The Moyal--Weyl description of quantum mechanics provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in $\hbar$. Its semiclassical expansion `coefficients,' acting on symbols that represent observables, are simple, globally defined differential operators constructed in terms of the classical flow. Two methods of constructing this expansion are discussed. The first introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold's formula for the Weyl product of s…
Classical dynamics from QFT via Stratonovich-Weyl correspondence
该研究通过Stratonovich-Weyl对应开发从S矩阵提取相对论经典动力学的形式体系，结合S矩阵指数形式得到含Magnus振幅的嵌套括号，可应用于引力波物理。
Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction to grade one regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra $\G\otimes{\bf C}[λ,λ^{-1}]$ are studied. The graded Heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the Weyl group ${\bf W}(\G)$ of the simple Lie algebra $\G$. A representative $w\in {\bf W}(\G)$ of a regular conjugacy class can be lifted to an inner automorphism of $\G$ given by $\hat w=\exp\left(2iπ{\rm ad I_0}/m\right)$, where $I_0$ is the defining vector of an $sl_2$ suba…
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