arXiv subjects
Publications and source records attributed to Dan Radu Grigore.
We generalise a result of Aste and Scharf saying that, under some reasonable assumptions, consistent with renormalisation theory, the non-Abelian gauge theories describes the only possibility of coupling the gluons. The proof is done using Epstein-Glaser approach to renormalisation theory.
We present here a possible generalisation of the Poincaré-Cartan form in classical field theory in the most general case: arbitrary dimension, arbitrary order of the theory and in the absence of a fibre bundle structure. We use for the kinematical description of the system the $(r,n)$-Grassmann manifold associated to a given manifold $X$, i.e. the manifold of $r$-contact elements of $n$-dimensional submanifolds of $X$. The idea is to define globally a $n+1$ form on this Grassmann manifold, more precisely its class with respect to a certain subspace and to write it locally as the exterior derivative of a $n$ form which is the Poincaré-Cartan form. As an important application we obtain a new proof for the most general expression of a variationally trivial Lagrangian.
The geometric Lagrangian theory (of arbitrary order) is based on the analysis of some basic mathematical objects such as: the contact ideal, the (exact) variational sequence, the existence of Euler-Lagrange and Helmholtz-Sonin forms, etc. In this paper we give new and much simpler proofs for the whole theory using Fock space methods. Using these results we give the most general expression for a variationally trivial Lagrangian and the generic expression for a locally variational system of partial differential equation.
The Lagrangian formalism on a arbitrary non-fibrating manifold is considered. The kinematical description of this generic situation is based on the concept of (higher-order) Grassmann manifolds which is the factorization of the regular velocity manifold to the action of the differential group. Here we introduce in this context the basic concepts of the Lagrangian formalism as Lagrange, Euler-Lagrange and Helmholtz-Sonin forms. These objects come in pairs, namely we have homogeneous objects (defined on the regular velocity manifold) and non-homogeneous objects (defined on the Grassmann manifold). We will establish the connection between the homogeneous objects and their non-homogeneous counterparts. As a result we will conclude that the generic expressions for a variationally trivial Lagrangian and for a locally variational differential equation remain the same as in the fibrating case.
An $(r,n)$-velocity is an $r$-jet with source at $0 \in \R^n$, and target in a manifold $Y$. An $(r,n)$-velocity is said to be regular, if it has a representative which is an immersion at $0 \in \R^{n}$. The manifold $T^{r}_{n}Y$ of $(r,n)$-velocities as well as its open, $L^{r}_{n}$-invariant, dense submanifold $\Imm T^{r}_{n}Y$ of regular $(r,n)$-velocities, are endowed with a natural action of the differential group $L^{r}_{n}$ of invertible $r$-jets with source and target $0 \in \R^{n}$. In this paper, we describe all continuous, $L^{r}_{n}$-invariant, real-valued functions on $T^{r}_{n}Y$ and $\Imm T^{r}_{n}Y$. We find local bases of $L^{r}_{n}$-invariants on $\Imm T^{r}_{n}Y$ in an explicit, recurrent form. To this purpose, higher order Grassmann bundles are considered as the corresponding quotients $P^{r}_{n}Y = \Imm T^{r}_{n}Y/L^{r}_{n}$, and their basic properties are studied. We show that nontrivial $L^{r}_{n}$-invariants on $\Imm T^{r}_{n}Y$ cannot be continuously extended onto $T^{r}_{n}Y$.
Trivial second-order Lagrangians are studied and a complete description of the dependence on the second-order derivatives is given. This extends previous work of Olver and others. In particular, this description involves some polynomial expressions called hyper-Jacobians. There exists some linear dependencies between these polynomials which are elucidated for the (second-order) hyper-Jacobians.
We continue the study of symmetries in the Lagrangian formalism of arbitrary order with the help of the so-called Anderson-Duchamp-Krupka equations. For the case of second-order equations and arbitrary vector fields we are able to establish a polynomial structure in the second-order derivatives. This structure is based on the some linear combinations of Olver hyper-Jacobians. We use as the main tools Fock space techniques and induction. This structure can be used to analyze Lagrangian systems with groups of Noetherian symmetries. As an illustration we analyze the case of Lagrangian equations with Abelian gauge invariance.
We provide a new proof of a important theorem in the Lagrangian formalism about necessary and sufficient conditions for a second-order variational system of equations to follow from a first-order Lagrangian.
We apply Mackey procedure of classifying projective systems of imprimitivity to a thorough study of the projective unitary irreducible representations of the Galilei group in 1+3 and 1+2 dimensions.
A general discussion of equations with universal invariance for a scalar field is provided in the framework of Lagrangian theory of first-order systems.
We give a definition for the notion of statistics in the lattice-theoretical (or propositional) formulation of quantum mechanics of Birchoff, von Neumann and Piron. We show that this formalism is compatible only with two types of statistics: Bose-Einstein and Fermi-Dirac. Some comments are made about the connection between this result and the existence of exotic statistics (para-statistics, infinite statistics, braid statistics).
A geometric global formulation of the higher-order Lagrangian formalism for systems with finite number of degrees of freedom is provided. The formalism is applied to the study of systems with groups of Noetherian symmetries.
We give a complete analysis of the projective unitary irreducible representations of the Poincaré group in 1+2 dimensions applying Mackey theorem and using an explicit formula for the universal covering group of the Lorentz group in 1+2 dimensions. We provide explicit formulae for all representations.
It is shown that the quantum position operator of Newton and Wigner for non-zero mass systems is uniquely determined if one imposes a quantum ''manifest covariance'' condition of the same type as the similar condition of Currie, Jordan and Sudarshan in the the framework of the Hamiltonian formalism.