A variation of Dirac equation based on SO(2,1) group is suggested for treating low dimensional systems in the three dimensional x,y,t space. Non-unitary representations are developed in an analogous way to those used in the ordinary Dirac equations and quantum field theory is developed for the present SO(2,1) relativistic equation. The theory is applied for low dimensional systems including especially holes-electrons pairs, or other two-particle states, in the low dimensional space.
Dirac algorithm allows to construct Hamiltonian systems for singular systems, and so contributing to its successful quantization. A drawback of this method is that the resulting quantized theory does not have manifest Lorentz invariance. This motivated the quest of alternatives to the usual Hamiltonian theory on the space of sections; a particular instance of this search is the so called restricted Hamiltonian theory, where the equations of motion of a field theory are formulated by means of a multisymplectic structure, a kind of generalization of the symplectic form to the multidimensional co…
We introduce some discrete analogues of the Dirac magnetic monopole on a unit sphere S^2 and explain how to compute the corresponding spectrum using the representation theory of finite groups. The main examples are certain magnetic Laplacians on the regular polyhedral graphs, coming from induced representations of the binary polyhedral groups.
In this work we study some symplectic submanifolds in the cotangent bundle of a factorizable Lie group defined by second class constraints. By applying the Dirac method, we study many issues of these spaces as fundamental Dirac brackets, symmetries, and collective dynamics. This last item allows to study integrability as inherited from a system on the whole cotangent bundle, leading in a natural way to the AKS theory for integrable systems.
We identify the category of integrable lowest-weight representations of the loop group LG of a compact Lie group G with the linear category of twisted, conjugation-equivariant curved Fredholm complexes on the group G: namely, the twisted, equivariant matrix factorizations of a super-potential built from the loop rotation action on LG. This lifts the isomorphism of K-groups of [FHT1,2, 3] to an equivalence of categories. The construction uses families of Dirac operators.
We show that the exterior algebra bundle over a curved spacetime can be used as framework in which both the Dirac and the Einstein equations can be obtained. These equations and their coupling follow from the variational principle applied to a Lagrangian constructed from natural geometric invariants. We also briefly indicate how other forces can potentially be incorporated within this geometric framework.
Tetrad based equation for Dirac-Kähler particle is solved in spherical coordinates in the flat Minkocski space-time. Spherical solutions of boson type (J =0,1,2,...) are constructed. After performing a special transformation over spherical boson solutions of the Dirac-Kähler equation, 4 \times 4-matrices U(x) \Longrightarrow V(x), simple linear expansions of the four rows of new representativeof the Dirac--Kähler field V(x) in terms of spherical fermion solutions Ψ_{i}(x) of the four ordinary Dirac equations have been derived. However, this fact cannot be interpreted as the possibility not to …
The three first sections contain an updated, not-so-short account of a partly original approach to spinor geometry and field theories introduced by Jadczyk and myself; it is based on an intrisic treatment of 2-spinor geometry in which the needed background structures do not need to be assumed, but rather arise naturally from a unique geometric datum: a vector bundle with complex 2-dimensional fibres over a real 4-dimensional manifold. The two following sections deal with Dirac algebra and 4-spinor groups in terms of two spinors, showing various aspects of spinor geometry from a different persp…
The Dirac monopole problem is studied in details within the framework of infinite-dimensional representations of the rotation group, and a consistent pointlike monopole theory with an arbitrary magnetic charge is deduced.
We calculate the torsion free spin connection on the quantum group B q [SU 2 ] at the fourth root of unity. From this we deduce the covariant derivative and the Riemann curvature. Next we compute the Dirac operator of this quantum group and we give numerical approximations of its eigenvalues.
In quantum information context, the groups generated by Pauli spin matrices, and Dirac gamma matrices, are known as the single qubit Pauli group P, and two-qubit Pauli group P2, respectively. It has been found [M. Socolovsky, Int. J. Theor. Phys. 43, 1941 (2004)] that the CPT group of the Dirac equation is isomorphic to P. One introduces a two-qubit entangling orthogonal matrix S basically related to the CPT symmetry. With the aid of the two-qubit swap gate, the S matrix allows the generation of the three-qubit real Clifford group and, with the aid of the Toffoli gate, the Weyl group W(E8) is …
A kind of Dirac-Connes operator defined in the framework of Connes' NCG is introduced on discrete abelian groups; it satisfies a Junk-free condition, and bridges the NCG composed by Dimakis, Müller-Hoissen and Sitarz and the NCG of Connes. Then we apply this operator to d-dimensional lattices.
We discuss generalized Dirac strings associated with a given Lie group. They live in r-dimensional complex space (r being the rank of the group). Such strings show up in the effective Born-Oppenheimer Hamiltonian for 3d supersymmetric Yang-Mills-Chern-Simons theories, brought up by the gluon loops. We calculate accurately the number of the vacuum states in the effective Hamiltonian associated with these strings. We also show that these states are irrelevant for the final SYMCS vacuum counting. The Witten index of SYMCS theories depends thus only on the strings generated by fermion loops and ca…
We study the restriction to a family of second class constrained submanifolds in the cotangent bundle of a double Lie group, equipped with a 2-cocycle extended symplectic form, building the corresponding Dirac brackets. It is shown that, for a 2-cocycle vanishing on each isotropic subspaces of the associated Manin triple, the Dirac bracket contains no traces of the cocycle. We also investigate the restriction of the left translation action of the double Lie group on its cotangent bundle, where it fails in to be a symmetry a canonical transformation. However, the hamiltonian symmetry is restore…
We show that the $(2+1)$-dimensional Dirac-Moshinsky oscillator coupled to an external magnetic field can be treated algebraically with the $SU(1,1)$ group theory and its group basis. We use the $su(1,1)$ irreducible representation theory to find the energy spectrum and the eigenfunctions. Also, with the $su(1,1)$ group basis we construct the relativistic coherent states in a closed form for this problem.
We introduce a Dirac operator $D$ for the quantum group $U_q(\mathfrak{sl}_2)$, as an element of the tensor product of $U_q(\mathfrak{sl}_2)$ with the Clifford algebra on two generators. We study the properties of $D$, including an analogue of Vogan's conjecture. We compute the cohomology of $D$ acting on various $U_q(\mathfrak{sl}_2)$-modules.
This paper develops the theory of discrete Dirac reduction of discrete Lagrange-Dirac systems with an abelian symmetry group acting on the configuration space. We begin with the linear theory and, then, we extend it to the nonlinear setting using retraction compatible charts. We consider the reduction of both the discrete Dirac structure and the discrete Lagrange-Pontryagin principle, and show that they both lead to the same discrete Lagrange-Poincaré-Dirac equations. The coordinatization of the discrete reduced spaces relies on the notion of discrete connections on principal bundles. At last,…
By following the Foldy-Wouthuysen (FW) transformation of the Dirac equation, we work out the exact analytic expressions up to the $1/M^4$ order for the general cases in the covariant density functional theory. These results are further compared with the corresponding ones derived from another novel non-relativistic expansion method, the similarity renormalization group (SRG). Based on that, the origin of the difference between the results obtained by the FW transformation and SRG method is explored.
Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for a principal bundle $M\to M/H$, integrations of a Dirac structure on $M/H$ to $H$-admissible integrations of its pullback Dirac structure on $M$ by pre-symplectic groupoids. Our construction gives a distinguished class of explicit real or holomorphic pre-symplectic and symplectic groupoids over semi-simple Lie groups and some of their homogeneou…
Starting from the representation of the $(n-1)+n-$dimensional Lorentz pseudo-sphere on the projective space $\mathbb{P}\mathbb{R}^{n,n}$, we propose a method to derive a class of solutions underlying to a Dirac-Kähler type equation on the lattice. We make use of the Cayley transform $φ({\bf w})=\dfrac{1+{\bf w}}{1-{\bf w}}$ to show that the resulting group representation arise from the same mathematical framework as the conformal group representation in terms of the {\it general linear group} $GL\left(2,Γ(n-1,n-1)\cup\{ 0\}\right)$. That allows us to describe such class of solutions as a commu…
Using the standard representation of the Dirac equation we show that, up to signs, there exist only TWO SETS of consistent solutions for the matrices of charge conjugation (C), parity (P), and time reversal (T). In both cases, P^2=-1, and then two succesive applications of the parity transformation to spin 1/2 fields NECESSARILY amounts to a 2πrotation. Each of these sets generates a non abelian group of sixteen elements, G_1 and G_2, which are non isomorphic subgroups of the Dirac algebra, which, being a Clifford algebra, gives a geometric nature to the generators, in particular to C. It turn…
Transformation properties of Dirac equation correspond to Spin(3,1) representation of Lorentz group SO(3,1), but group GL(4,R) of general relativity does not accept a similar construction with Dirac spinors. On the other hand, it is possible to look for representation of GL(4,R) in some bigger space, there Dirac spinors are formally situated as some "subsystem." In the paper is described construction of such representation, using Clifford and Grassmann algebras of 4D space.
In this paper we show the existence of infinitely many symmetric solutions for a cubic Dirac equation in two dimensions, which appears as effective model in systems related to honeycomb structures. Such equation is critical for the Sobolev embedding and solutions are found by variational methods. Moreover, we prove also prove smoothness and exponential decay at infinity.
We initiate the study of an algebra of symmetries for the 3D Dirac-Dunkl operator associated with the Weyl group of the exceptional root system $G_2$. For this symmetry algebra, we give both an abstract definition and an explicit realisation. We then construct ladder operators, using an intermediate result we prove for the Dirac-Dunkl symmetry algebra associated with arbitrary finite reflection group acting on a three-dimensional space.
We study the solutions to the Dirac equation for the massive spinor field in the universal covering space of two-dimensional anti-de Sitter space. For certain values of the mass parameter, we impose a suitable set of boundary conditions which make the spatial component of the Dirac operator self-adjoint. Then, we use the transformation properties of the spinor field under the isometry group of the theory, namely, the universal covering group of $\mathrm{SL}(2,\mathbb{R})$, in order to determine which self-adjoint boundary conditions are invariant under this group. We identify the corresponding…
By following the conventional similarity renormalization group (SRG) expansion of the Dirac equation developed in [J.-Y. Guo, Phys. Rev. C \textbf{85}, 021302 (2012)], we work out the analytic expression of the ${1}/{M^4}$ order and verify the convergence of this method. As a step further, the reconstituted SRG method is proposed by using the re-summation technique. The speed of convergence of the reconstituted SRG becomes much faster than the conventional one, and the single-particle densities with the reconstituted SRG are also almost identical to the exact values.
We generalize the method of obtaining the fundamental linear partial differential equations such as the diffusion and Schrodinger equation, Dirac and telegrapher's equation from a simple stochastic consideration to arrive at certain nonlinear form of these equations. The group classification through one parameter group of transformation for two of these equations is also carried out.
Within the context of infinite-dimensional representations of the rotation group the Dirac monopole problem is studied in details. Irreducible infinite-dimensional representations, being realized in the indefinite metric Hilbert space, are given by linear unbounded operators in infinite-dimensional topological spaces, supplied with a weak topology and associated weak convergence. We argue that an arbitrary magnetic charge is allowed, and the Dirac quantization condition can be replaced by a generalized quantization rule yielding a new quantum number, the so-called topological spin, which is re…
In this work, we use the Sternberg phase space (which may be considered as the classical phase space of particles in gauge fields) in order to explore the dynamics of such particles in the context of Hamilton-Dirac systems and their associated Hamilton-Pontryagin variational principles. For this, we develop an analogue of the Pontryagin bundle in the case of the Sternberg phase space. Moreover, we show the link of this new bundle to the so-called magnetized Tulczyjew triple, which is an analogue of the link between the Pontryagin bundle and the usual Tulczyjew triple. Taking advantage of the s…
An equation, we call Dirac gamma-equation, is introduced with the help of the mathematical tools connected with the Clifford algebra. This equation can be considered as a generalization of the Dirac equation for the electron. Some features of Dirac gamma- equation are investigated (plane waves, currents, canonical forms). Furthermore, on the basis of local gauge invariance regarding unitary group, a system of equations is introduced consisting of Dirac gamma-equation and the Yang-Mills or Maxwell equations. This system of equations describes a Dirac's field interacting with the Yang-Mills or M…
The massless field of spin 1 is defined on the eight-dimensional configuration space; this space is a direct product of Minkowski space and of a two-dimensional complex sphere. Field equations for the spin-one field are derived from a Dirac-like Lagrangian separately for the translation group and Lorentz group parts. It is shown that a Dirac form of Maxwell equations (so-called Majorana-Oppenheimer formulation of electrodynamics) follows directly from the field equations of translation group part. The photon field is realized via Biedenharn type functions on the Poincaré group. This allows us …
Symplectic unitary representations for the Poincaré group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Klein-Gordon and Dirac equations are derived in phase space. As an application, we study the Dirac equation with electromagnetic interaction in phase space.
We introduce a Banach Lie group $G$ of unitary operators subject to a natural trace condition. We compute the homotopy groups of $G$, describe its cohomology and construct an $S^1$-central extension. We show that the central extension determines a non-trivial gerbe on the action Lie groupoid $G\ltimes \mathfrak{k}$, where $\mathfrak{k}$ denotes the Hilbert space of self-adjoint Hilbert-Schmidt operators. With an eye towards constructing elements in twisted K-theory, we prove the existence of a cubic Dirac operator $\mathbb{D}$ in a suitable completion of the quantum Weil algebra $\mathcal{U}(\…
We describe an averaging procedure on a Dirac manifold, with respect to a class of compatible actions of a compact Lie group. Some averaging theorems on the existence of invariant realizations of Poisson structures around (singular) symplectic leaves are derived. We show that the construction of coupling Dirac structures (invariant with respect to locally Hamiltonian group actions) on a Poisson foliation is related with a special class of exact gauge transformations.
Discrete versions of the Laplace and Dirac operators haven been studied in the context of combinatorial models of statistical mechanics and quantum field theory. In this paper we introduce several variations of the Laplace and Dirac operators on graphs, and we investigate graph-theoretic versions of the Schrödinger and Dirac equation. We provide a combinatorial interpretation for solutions of the equations and we prove gluing identities for the Dirac operator on lattice graphs, as well as for graph Clifford algebras.
The continuous and discrete symmetries of the Dirac-type operators produced by particular Killing-Yano tensors are studied in manifolds of arbitrary dimensions. The Killing-Yano tensors considered are covariantly constant and realize certain square roots of the metric tensor. Such a Killing-Yano tensor produces simultaneously a Dirac-type operator and the generator of a one-parameter Lie group connecting this operator with the standard Dirac one. The Dirac operators are related among themselves through continuous or discrete transformations. It is shown that the groups of the continuous symmet…
We solve for spectrum, obtain explicitly and study group properties of eigenfunctions of Dirac operator on the Riemann sphere $S^2$. The eigenvalues $λ$ are nonzero integers. The eigenfunctions are two-component spinors that belong to representations of SU(2)-group with half-integer angular momenta $l = |λ| - \half$. They form on the sphere a complete orthonormal functional set alternative to conventional spherical spinors. The difference and relationship between the spherical spinors in question and the standard ones are explained.
