arXiv subjects

Dmitri Vassiliev

Publications and source records attributed to Dmitri Vassiliev.

At least 37 records · Page 2Linked to original sources

arXiv2015

Spectral theoretic characterization of the massless Dirac action

We consider an elliptic self-adjoint first order differential operator L acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of the operator L is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Given a positive scalar weight function, we study the weighted eigenvalue problem for the operator L. The corresponding counting function (number of eigenvalues between zero and a positive lambda) is known to admit, under appropriate assumptions on periodic trajectories, a two-term asymptotic expansion as lambda tends to plus infinity and we have recently derived an explicit formula for the second asymptotic coefficient. The purpose of this paper is to establish the geometric meaning of the second asymptotic coefficient. To this end, we identify the geometric objects encoded within our eigenvalue problem - metric, nonvanishing spinor field and topological charge - and express our asymptotic coefficients in terms of these geometric objects. We prove that the second asymptotic coefficient of the counting function has the geometric meaning of the massless Dirac action.

math.SP↗

arXiv2015

Analysis of first order systems of partial differential equations

The paper deals with a formally self-adjoint first order linear differential operator acting on m-columns of complex-valued half-densities over an n-manifold without boundary. We study the distribution of eigenvalues in the elliptic setting and the propagator in the hyperbolic setting, deriving two-term asymptotic formulae for both. We then turn our attention to the special case of a two by two operator in dimension four. We show that the geometric concepts of Lorentzian metric, Pauli matrices, spinor field, connection coefficients for spinor fields, electromagnetic covector potential, Dirac equation and Dirac action arise naturally in the process of our analysis.

math.SP↗

arXiv2015

Analysis as a source of geometry: a non-geometric representation of the Dirac equation

Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4-manifold without boundary. We examine the geometric content of such an operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients for spinor fields and an electromagnetic covector potential. This observation allows us to give a simple representation of the massive Dirac equation as a system of four scalar equations involving an arbitrary two-by-two matrix operator as above and its adjugate. The point of the paper is that in order to write down the Dirac equation in the physically meaningful 4-dimensional hyperbolic setting one does not need any geometric constructs. All the geometry required is contained in a single analytic object - an abstract formally self-adjoint first order linear differential operator acting on pairs of complex-valued scalar fields.

math.AP↗

arXiv2013

Spectral theoretic characterization of the massless Dirac operator

We consider an elliptic self-adjoint first order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive lambda. We derive an explicit two-term asymptotic formula for the spectral function as lambda tends to plus infinity, expressing the second asymptotic coefficient via the trace of the subprincipal symbol and the geometric objects encoded within the principal symbol - metric, torsion of the teleparallel connection and topological charge. We then address the question: is our operator a massless Dirac operator on half-densities? We prove that it is a massless Dirac operator on half-densities if and only if the following two conditions are satisfied at every point of the manifold: a) the subprincipal symbol is proportional to the identity matrix and b) the second asymptotic coefficient of the spectral function is zero.

math.SP↗

arXiv2013

Spectral asymmetry of the massless Dirac operator on a 3-torus

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.

math.SP↗

arXiv2012

The spectral function of a first order elliptic system

We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of complex-valued half-densities over a connected compact manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semi-bounded. We study the following objects: the propagator (time-dependent operator which solves the Cauchy problem for the dynamic equation), the spectral function (sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive lambda) and the counting function (number of eigenvalues between zero and a positive lambda). We derive explicit two-term asymptotic formulae for all three. For the propagator "asymptotic" is understood as asymptotic in terms of smoothness, whereas for the spectral and counting functions "asymptotic" is understood as asymptotic with respect to lambda tending to plus infinity.

math.SP↗

arXiv2012

The spectral function of a first order system

We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of m complex-valued half-densities over a connected compact n-dimensional manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semi-bounded. We study the spectral function, i.e. the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive lambda. We derive a two-term asymptotic formula for the spectral function as lambda tends to plus infinity. We then restrict our study to the case when m=2, n=3, the operator is differential and has trace-free principal symbol, and address the question: is our operator a massless Dirac operator? We prove that it is a massless Dirac operator if and only if the following two conditions are satisfied at every point of the manifold: a) the subprincipal symbol is proportional to the identity matrix and b) the second asymptotic coefficient of the spectral function is zero.

math.AP↗

arXiv2010

Rotational elasticity

We consider an infinite 3-dimensional elastic continuum whose material points experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose the coframe and a density. We write down the general dynamic variational functional for our rotational theory of elasticity, assuming our material to be physically linear but the kinematic model geometrically nonlinear. Allowing geometric nonlinearity is natural when dealing with rotations because rotations in dimension 3 are inherently nonlinear (rotations about different axes do not commute) and because there is no reason to exclude from our study large rotations such as full turns. The main result of the paper is an explicit construction of a class of time-dependent solutions which we call plane wave solutions; these are travelling waves of rotations. The existence of such explicit closed form solutions is a nontrivial fact given that our system of Euler-Lagrange equations is highly nonlinear. In the last section we consider a special case of our rotational theory of elasticity which in the stationary setting (harmonic time dependence and arbitrary dependence on spatial coordinates) turns out to be equivalent to a pair of massless Dirac equations.

math-ph↗

arXiv2010

Modeling the electron with Cosserat elasticity

We suggest an alternative mathematical model for the electron in dimension 1+2. We think of our (1+2)-dimensional spacetime as an elastic continuum whose material points can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose a coframe and a density. We then add an extra (third) spatial dimension, extend our coframe and density into dimension 1+3, choose a conformally invariant Lagrangian proportional to axial torsion squared, roll up the extra dimension into a circle so as to incorporate mass and return to our original (1+2)-dimensional spacetime by separating out the extra coordinate. The main result of our paper is the theorem stating that our model is equivalent to the Dirac equation in dimension 1+2. In the process of analyzing our model we also establish an abstract result, identifying a class of nonlinear second order partial differential equations which reduce to pairs of linear first order equations.

math-ph↗

arXiv2010

Modelling the neutrino in terms of Cosserat elasticity

The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pauli matrices or covariant differentiation. We think of our 3-dimensional space as an elastic continuum and assume that material points of this continuum can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points of the space continuum are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose the coframe and a density. We choose a particular potential energy which is conformally invariant and then incorporate time into our action in the standard Newtonian way, by subtracting kinetic energy. The main result of our paper is the theorem stating that in the stationary setting our model is equivalent to a pair of Weyl equations. The crucial element of the proof is the observation that our Lagrangian admits a factorisation.

math-ph↗

arXiv2010

The stationary Weyl equation and Cosserat elasticity

The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pauli matrices or covariant differentiation. We think of our 3-dimensional space as an elastic continuum and assume that material points of this continuum can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points of the space continuum are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose the coframe and a density. We choose a particular potential energy which is conformally invariant and then incorporate time into our action in the standard Newtonian way, by subtracting kinetic energy. The main result of our paper is the theorem stating that in the stationary setting our model is equivalent to a pair of Weyl equations. The crucial element of the proof is the observation that our Lagrangian admits a factorization.

gr-qc↗

arXiv2009

Massless Dirac equation as a special case of Cosserat elasticity

We suggest an alternative mathematical model for the massless neutrino. Consider an elastic continuum in 3-dimensional Euclidean space and assume that points of this continuum can experience no displacements, only rotations. This framework is a special case of the so-called Cosserat theory of elasticity. Rotations of points of the continuum are described by attaching to each point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose a coframe and a density. We write down a potential energy which is conformally invariant and then incorporate time in the standard Newtonian way, by subtracting kinetic energy. Finally, we rewrite the resulting nonlinear variational problem in terms of an unknown spinor field. We look for quasi-stationary solutions, i.e. solutions that harmonically oscillate in time. We prove that in the quasi-stationary setting our model is equivalent to a pair of massless Dirac equations. The crucial element of the proof is the observation that our Lagrangian admits a factorisation.

gr-qc↗

arXiv2009

Weyl's Lagrangian in teleparallel form

The main result of the paper is a new representation for the Weyl Lagrangian (massless Dirac Lagrangian). As the dynamical variable we use the coframe, i.e. an orthonormal tetrad of covector fields. We write down a simple Lagrangian - wedge product of axial torsion with a lightlike element of the coframe - and show that this gives the Weyl Lagrangian up to a nonlinear change of dynamical variable. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. Our result assigns a variational meaning to the tetrad representation of the Weyl equation suggested by J. B. Griffiths and R. A. Newing.

gr-qc↗

arXiv2008

Dirac equation as a special case of Cosserat elasticity

We suggest an alternative mathematical model for the electron in which the dynamical variables are a coframe (field of orthonormal bases) and a density. The electron mass and external electromagnetic field are incorporated into our model by means of a Kaluza-Klein extension. Our Lagrangian density is proportional to axial torsion squared. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. We prove that in the special case with no dependence on the third spatial coordinate our model is equivalent to the Dirac equation. The crucial element of the proof is the observation that our Lagrangian admits a factorisation.

gr-qc↗

arXiv2007

A teleparallel representation of the Weyl Lagrangian

The main result of the paper is a new representation of the Weyl Lagrangian (massless Dirac Lagrangian). As the dynamical variable we use the coframe, i.e. an orthonormal tetrad of covector fields. We write down a simple Lagrangian - wedge product of axial torsion with a lightlike element of the coframe - and show that variation of the resulting action with respect to the coframe produces the Weyl equation. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. Our result assigns a variational meaning to the tetrad representation of the Weyl equation suggested by J.B.Griffiths and R.A.Newing.

gr-qc↗

arXiv2006

A teleparallel model for the neutrino

The main result of the paper is a new representation for the Weyl Lagrangian (massless Dirac Lagrangian). As the dynamical variable we use the coframe, i.e. an orthonormal tetrad of covector fields. We write down a simple Lagrangian - wedge product of axial torsion with a lightlike element of the coframe - and show that variation of the resulting action with respect to the coframe produces the Weyl equation. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. Our result assigns a variational meaning to the tetrad representation of the Weyl equation suggested by J.B.Griffiths and R.A.Newing.

gr-qc↗

arXiv2005

PP-waves with torsion and metric-affine gravity

A classical pp-wave is a 4-dimensional Lorentzian spacetime which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. We generalise this definition to metric compatible spacetimes with torsion and describe basic properties of such spacetimes. We use our generalised pp-waves for constructing new explicit vacuum solutions of quadratic metric-affine gravity.

gr-qc↗

arXiv2003

Quadratic metric-affine gravity

We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is quadratic in curvature and study the resulting system of Euler-Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi-Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with metric of a pp-wave and parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non-Riemannian solutions. We define the notion of a "Weyl pseudoinstanton" (metric compatible spacetime whose curvature is purely Weyl) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non-Riemannian solution which is a wave of torsion in Minkowski space. We discuss the possibility of using this non-Riemannian solution as a mathematical model for the graviton or the neutrino.

gr-qc↗