Covariant derivatives of a spinor in finsler geometry

The aim of the author in the present article is to introduce covariant derivatives of a spinor into Finsler geometry and to show that such an operation is not only interesting from the purely mathematical point of view but also from the physical one since it is thus possible to extend the spinor theory of Heisenberg [1] to the Finsler space [2] (and, in particular, to the Riemann space [3]). The covariant form of the spinor theory of matter is essentially a nonlinear theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 16, No. 2, pp. 43–49, February, 1973.     

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. PACS 04.62.+v; 11.10.Ef; 12.10.Kt     

Covariant superfluid mechanics

A relativistic extension of the Landau two-fluid model is developed.     

Dirac operators on Lagrangian submanifolds

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge–de Rham Laplacian provided the complex structure identifies the spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.     

Covariant coordinate transformations on noncommutative space

We show how to define gauge-covariant coordinate transformations on a noncommuting space. The construction uses the Seiberg-Witten equation and generalizes similar results for commuting coordinates.     

Covariant derivative on non-linear fiber bundles

A gauge field is usually described as a connection on a principal bundle. It induces a covariant derivative on associated vector bundles, sections of which represent matter fields. In general, however, it is not possible to define a covariant derivative on non-linear fiber bundles, i.e. on those which are not vector bundles. We definelogarithmic covariant derivatives acting on two special non-linear fiber bundles — on the principal bundle and on the local gauge group bundle. The logarithmic derivatives map from sections of these bundles to the sections of the local gauge algebra bundle. Some properties of the logarithmic derivatives are formulated.     

A note on singular lagrangian submanifolds

Classification of singular lagrangian submanifolds which appear as images of a regular one under a symplectic relation, is considered from the point of view of standard singularity theory. The classification is carried out in small dimensions and restricted to special types of symplectic objects. Normal forms for singular pullbacks and pushforwards are given using an appropriate symplectic equivalence group. It is shown that the general classification problem reduces to the classification problem for appropriate mapping diagrams. An approach to the classical theories of phase transition is given based on the geometry of singular lagrangian images. The variational open swallowtails and regularly intersecting pairs of holonomic components are resolved using an appropriate reduction relation. Examples are given of singularities encountered in physics.     

Generally Covariant Dirac Hamiltonian

The generally covariant Dirac equation reduces to a mathematically simple form with a clear physical meaning that is best demonstrated by the Hamiltonian formalism. The resulting Hamiltonian is compared to the corresponding classical Hamiltonian, and assumes a particularly simple form for the Robertson-Walker metric of the Standard Cosmological Model when that metric is expressed in a coordinate system referred to as Einstein coordinates because it is based on a coordinate constraint recommended by Einstein in his original paper on general relativity. A kinematic definition of distance, based on the form of the classical Hamiltonian, is used in discussing the physical meaning of the results.     

Covariant holographic entanglement negativity

We propose a covariant holographic conjecture for the entanglement negativity of bipartite mixed states in \((1+1)\)-dimensional conformal field theories dual to bulk non static \(AdS_{3}\) configurations. Application of our conjecture to \((1+1)\)-dimensional conformal field theories dual to bulk non extremal and extremal rotating BTZ black holes exactly reproduce the corresponding entanglement negativity obtained through the replica technique, in the large central charge limit. We briefly discuss the issue of the generalization of our conjecture to higher dimensions.     

The Covariant Stark Effect

This paper examines the Stark effect, as a first order perturbation of manifestly covariant hydrogen-like bound states. These bound states are solutions to a relativistic Schrödinger equation with invariant evolution parameter, and represent mass eigenstates whose eigenvalues correspond to the well-known energy spectrum of the nonrelativistic theory. In analogy to the nonrelativistic case, the off-diagonal perturbation leads to a lifting of the degeneracy in the mass spectrum. In the covariant case, not only do the spectral lines split, but they acquire an imaginary part which is linear in the applied electric field, thus revealing induced bound state decay in first order perturbation theory. This imaginary part results from the coupling of the external field to the non-compact boost generator. In order to recover the conventional first order Stark splitting, we must include a scalar potential term. This term may be understood as a fifth gauge potential, which compensates for dependence of gauge transformations on the invariant evolution parameter.     

Covariant open string field theory on multiple Dp-branes

We study covariant open bosonic string field theories on multiple Dp-branes by using the deformed cubic string field theory, which is equivalent to string field theory in the proper-time gauge. Constructing the Fock space representations of the three-string vertex and the four-string vertex on multiple Dp-branes, we obtain the field theoretical effective action in the zero-slope limit. On multiple D0-branes, the effective action reduces to the Banks-Fishler-Shenker-Susskind(BFSS) matrix model. We also discuss the relation between open string field theory on multiple D-instantons in the zero-slope limit and the Ishibashi-Kawai-Kitazawa-Tsuchiya(IKKT) matrix model.The covariant open string field theory on multiple Dp-branes could be useful to study the non-perturbative properties of quantum field theories in(p+1)-dimensions in the framework of the string theory. The non-zero-slope corrections may be evaluated systematically by using covariant string field theory.     

Covariant quantization of string based on BRS invariance

Covariant canonical quantization of the bosonic string is performed, based on the BRS invariance principle. Defining a physical state by a modified form of Kugo-Ojima's subsidiary condition Q_B|phys〉 = 0 (Q_B = BRS charge), we prove a no-ghost theorem in a simple manner. There an interesting relation between critical dimension and BRS symmetry is noticed; namely, the conditions D = 26 and α₀ = 1 for the space-time dimension D and the zero-intercept α₀ of leading trajectory are required by the nilpotency Q_B² = 0 of the BRS charge. In addition, ghost number fractionization is observed in this system.     

Covariant string theory Feynman amplitudes

The covariant string amplitudes on a strip with the associated Feynman rules are considered. The contribution of the Faddeev-Popov ghost is evaluated. The overlap vertex (coming from a δ-function interaction) is constructed. The corresponding scattering diagrams are shown to give the standard Veneziano model answer. The ghosts provide the necessary contribution to the measure.