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Journal articles on the topic 'Operatory Diraca'

Author: Grafiati

Published: 24 April 2022

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1

KHACHIDZE, TAMARI T., and ANZOR A. KHELASHVILI. "AN "ACCIDENTAL" SYMMETRY OPERATOR FOR THE DIRAC EQUATION IN THE COULOMB POTENTIAL." Modern Physics Letters A 20, no. 30 (September 28, 2005): 2277–81. http://dx.doi.org/10.1142/s0217732305018505.

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On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides with the Johnson and Lippmann operator and is closely connected to the familiar Laplace–Runge–Lenz vector. Our approach guarantees not only derivation of Johnson–Lippmann operator, but simultaneously commutativity with the Dirac Hamiltonian.

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2

AVRAMIDI, IVAN G. "DIRAC OPERATOR IN MATRIX GEOMETRY." International Journal of Geometric Methods in Modern Physics 02, no. 02 (April 2005): 227–64. http://dx.doi.org/10.1142/s0219887805000636.

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We review the construction of the Dirac operator and its properties in Riemannian geometry, and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also point out that the Einstein–Hilbert functional can be obtained as a linear combination of the first two spectral invariants of the Dirac operator. Next, we report on our previous attempts to generalize the notion of the Dirac operator to the case of Matrix Geometry, where, instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays a role of a "non-commutative" metric. We construct invariant first-order and second-order self-adjoint elliptic partial differential operators, which can be called "non-commutative" Dirac operators and non-commutative Laplace operators. We construct the corresponding heat kernel for the non-commutative Laplace type operator and compute its first two spectral invariants. A linear combination of these two spectral invariants gives a functional that can be considered as a non-commutative generalization of the Einstein–Hilbert action.

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3

Cojuhari, Petru, and Aurelian Gheondea. "Embeddings, Operator Ranges, and Dirac Operators." Complex Analysis and Operator Theory 5, no. 3 (April 13, 2010): 941–53. http://dx.doi.org/10.1007/s11785-010-0066-5.

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4

ARAI, ASAO. "HEISENBERG OPERATORS, INVARIANT DOMAINS AND HEISENBERG EQUATIONS OF MOTION." Reviews in Mathematical Physics 19, no. 10 (November 2007): 1045–69. http://dx.doi.org/10.1142/s0129055x07003206.

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An abstract operator theory is developed on operators of the form AH(t) := eitHAe-itH, t ∈ ℝ, with H a self-adjoint operator and A a linear operator on a Hilbert space (in the context of quantum mechanics, AH(t) is called the Heisenberg operator of A with respect to H). The following aspects are discussed: (i) integral equations for AH(t) for a general class of A; (ii) a sufficient condition for D(A), the domain of A, to be left invariant by e-itH for all t ∈ ℝ; (iii) a mathematically rigorous formulation of the Heisenberg equation of motion in quantum mechanics and the uniqueness of its solutions; (iv) invariant domains in the case where H is an abstract version of Schrödinger and Dirac operators; (v) applications to Schrödinger operators with matrix-valued potentials and Dirac operators.

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5

Aastrup, Johannes, and Jesper Møller Grimstrup. "The quantum holonomy-diffeomorphism algebra and quantum gravity." International Journal of Modern Physics A 31, no. 10 (April 6, 2016): 1650048. http://dx.doi.org/10.1142/s0217751x16500482.

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We introduce the quantum holonomy-diffeomorphism ∗-algebra, which is generated by holonomy-diffeomorphisms on a three-dimensional manifold and translations on a space of SU(2)-connections. We show that this algebra encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Furthermore, we show that semiclassical states exist on the holonomy-diffeomorphism part of the algebra but that these states cannot be extended to the full algebra. Via a Dirac-type operator we derive a certain class of unbounded operators that act in the GNS construction of the semiclassical states. These unbounded operators are the type of operators, which we have previously shown to entail the spatial three-dimensional Dirac operator and Dirac–Hamiltonian in a semiclassical limit. Finally, we show that the structure of the Hamilton constraint emerges from a Yang–Mills-type operator over the space of SU(2)-connections.

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6

MATSUTANI, SHIGEKI. "DIRAC OPERATOR ON A CONFORMAL SURFACE IMMERSED IN ℝ4: A WAY TO FURTHER GENERALIZED WEIERSTRASS EQUATION." Reviews in Mathematical Physics 12, no. 03 (March 2000): 431–44. http://dx.doi.org/10.1142/s0129055x00000149.

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In the previous report (J. Phys.A30 (1997) 4019–4029), I showed that the Dirac operator defined over a conformal surface immersed in ℝ3 by means of confinement procedure is identified with the differential operator of the generalized Weierstrass equation and the Lax operator of the modified Novikov–Veselov (MNV) equation. In this article, using the same procedure, I determine the Dirac operator defined over a conformal surface immersed in ℝ4, which is for a Dirac field confined in the surface. Then it is reduced to the Lax operators of the nonlinear Schrödinger and the MNV equations by taking appropriate limits. It means that the Dirac operator is related to the further generalized Weierstrass equation for a surface in ℝ4.

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7

Benameur, Moulay-Tahar, James L. Heitsch, and Charlotte Wahl. "An interesting example for spectral invariants." Journal of K-Theory 13, no. 2 (April 2014): 305–11. http://dx.doi.org/10.1017/is014002020jkt255.

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AbstractIn [HL99], the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then shown that the associated heat operator converges to the Chern character of the index bundle of the operator. In [BH08], this result was improved by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.

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8

DABROWSKI, LUDWIK, ANDRZEJ SITARZ, and ALESSANDRO ZUCCA. "DIRAC OPERATORS ON NONCOMMUTATIVE PRINCIPAL CIRCLE BUNDLES." International Journal of Geometric Methods in Modern Physics 11, no. 01 (December 16, 2013): 1450012. http://dx.doi.org/10.1142/s0219887814500121.

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We study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low-dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle [Formula: see text].

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9

Wang, Yong. "A Kastler-Kalau-Walze Type Theorem and the Spectral Action for Perturbations of Dirac Operators on Manifolds with Boundary." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/619120.

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We prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds.

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10

Akuamoah, Saviour Worlanyo, Aly R. Seadawy, and Dianchen Lu. "Energy and momentum operator substitution method derived from Schrödinger equation for light and matter waves." Modern Physics Letters B 33, no. 24 (August 30, 2019): 1950285. http://dx.doi.org/10.1142/s0217984919502853.

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In this paper, the energy and momentum operator substitution method derived from the Schrödinger equation is used to list all possible light and matter wave equations, among which the first light wave equation and relativistic approximation equation are proposed for the first time. We expect that we will have some practical application value. The negative sign pairing of energy and momentum operators are important characteristics of this paper. Then the Klein–Gordon equation and Dirac equation are introduced. The process of deriving relativistic energy–momentum relationship by undetermined coefficient method and establishing Dirac equation are mainly introduced. Dirac’s idea of treating negative energy in relativity into positrons is also discussed. Finally, the four-dimensional space-time representation of relativistic wave equation is introduced, which is usually the main representation of quantum electrodynamics and quantum field theory.

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11

TANIŞLI, MURAT, MUSTAFA EMRE KANSU, and SÜLEYMAN DEMİR. "SUPERSYMMETRIC QUANTUM MECHANICS AND EUCLIDEAN–DIRAC OPERATOR WITH COMPLEXIFIED QUATERNIONS." Modern Physics Letters A 28, no. 08 (March 12, 2013): 1350026. http://dx.doi.org/10.1142/s0217732313500260.

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We propose an alternative formulation of the supersymmetric quantum mechanics and Euclidean Dirac and Dirac–Yang–Mills (DYM) operators in terms of complexified quaternions. 4×4 matrix representations of the complexified quaternions are used to express the Euclidean–Dirac operator and Yang–Mills gauge field.

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12

BRACKX, F., D. EELBODE, T. RAEYMAEKERS, and L. VAN DE VOORDE. "TRIPLE MONOGENIC FUNCTIONS AND HIGHER SPIN DIRAC OPERATORS." International Journal of Mathematics 22, no. 06 (June 2011): 759–74. http://dx.doi.org/10.1142/s0129167x11007021.

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In the Clifford analysis context a specific type of solution for the higher spin Dirac operators [Formula: see text] is studied; these higher spin Dirac operators can be seen as generalizations of the classical Rarita–Schwinger operator. To that end subspaces of the space of triple monogenic polynomials are introduced and their algebraic structure is investigated. Also a dimensional analysis is carried out.

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13

Schmidt, Karl Michael. "Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 5 (1996): 1087–96. http://dx.doi.org/10.1017/s0308210500023271.

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For the one-dimensional Dirac operator, examples of electrostatic potentials with decay behaviour arbitrarily close to Coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum. A simple proof that the essential spectrum of one-dimensional Dirac operators with electrostatic potentials is never empty is given in the appendix.

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14

Kath, Ines, and Oliver Ungermann. "Spectra of Sub-Dirac Operators on Certain Nilmanifolds." MATHEMATICA SCANDINAVICA 117, no. 1 (September 28, 2015): 64. http://dx.doi.org/10.7146/math.scand.a-22237.

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We study sub-Dirac operators associated to left-invariant bracket-generating sub-Riemannian structures on compact quotients of nilpotent semi-direct products $G=\mathsf{R}^n\rtimes_A\mathsf{R}$. We prove that these operators admit an $L^2$-basis of eigenfunctions. Explicit examples of this type show that the spectrum of these operators can be non-discrete and that eigenvalues may have infinite multiplicity. In this case the sub-Dirac operator is neither Fredholm nor hypoelliptic.

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15

Lotfizadeh, M., and Ebrahim Nouri Asl. "Pseudo generalization of Ginsparg–Wilson algebra on the fuzzy EAdS2 including gauge fields." International Journal of Geometric Methods in Modern Physics 17, no. 03 (February 14, 2020): 2050046. http://dx.doi.org/10.1142/s0219887820500462.

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Using the gauged pseudo-Hermitian fuzzy Ginsparg–Wilson algebra, pseudo fuzzy Dirac and chirality operators on the fuzzy [Formula: see text] have been studied. Also, the spectrum of the gauged pseudo fuzzy Dirac operator in the instanton sector has been studied.

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16

Arai, Asao, and Dayantsolmon Dagva. "A Class of -Dimensional Dirac Operators with a Variable Mass." ISRN Mathematical Analysis 2013 (July 1, 2013): 1–13. http://dx.doi.org/10.1155/2013/913413.

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A class of d-dimensional Dirac operators with a variable mass is introduced (), which includes, as a special case, the 3-dimensional Dirac operator describing the chiral quark soliton model in nuclear physics, and some aspects of it are investigated.

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17

CHALUB, FABIO A. C. C. "On Huygens' principle for Dirac operators associated to electromagnetic fields." Anais da Academia Brasileira de Ciências 73, no. 4 (December 2001): 483–93. http://dx.doi.org/10.1590/s0001-37652001000400002.

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We study the behavior of massless Dirac particles, i.e., solutions of the Dirac equation with m = 0 in the presence of an electromagnetic field. Our main result (Theorem 1) is that for purely real or imaginary fields any Huygens type (in Hadamard's sense) Dirac operators is equivalent to the free Dirac operator, equivalence given by changes of variables and multiplication (right and left) by nonzero functions.

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18

Fatibene, Lorenzo, Raymond G. McLenaghan, and Giovanni Rastelli. "Symmetry operators and separation of variables for Dirac's equation on two-dimensional spin manifolds with external fields." International Journal of Geometric Methods in Modern Physics 12, no. 01 (December 28, 2014): 1550012. http://dx.doi.org/10.1142/s0219887815500127.

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The second-order symmetry operators that commute with the Dirac operator with external vector, scalar and pseudo-scalar potentials are computed on a general two-dimensional spin manifold. It is shown that the operator is defined in terms of Killing vectors, valence two Killing tensors and scalar fields defined on the background manifold. The commuting operator that arises from a non-trivial Killing tensor is determined with respect to the associated system of Liouville coordinates and compared to the second-order operator that obtained from the unique separation scheme associated with such operators. It is shown by the study of several examples that the operators arising from these two approaches coincide.

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19

FAN, HONGYI, and YUE FAN. "WEYL ORDERING FOR ENTANGLED STATE REPRESENTATION." International Journal of Modern Physics A 17, no. 05 (February 20, 2002): 701–8. http://dx.doi.org/10.1142/s0217751x02003257.

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We derive a new Weyl ordering operator formula which recasts given operators into Weyl ordering. In so doing, the Weyl ordering formulation of the entangled state representation is obtained, which turns out to be the Weyl ordered Dirac δ-operator functions. The Weyl ordering of the Wigner operator and squeezing operator in entangled state representation are also deduced.

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20

CAMPOS, RAFAEL G., J. L. LÓPEZ-LÓPEZ, and R. VERA. "LATTICE CALCULATIONS ON THE SPECTRUM OF DIRAC AND DIRAC–KÄHLER OPERATORS." International Journal of Modern Physics A 23, no. 07 (March 20, 2008): 1029–38. http://dx.doi.org/10.1142/s0217751x08038470.

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We use a lattice formulation to study the spectra of the Dirac and the Dirac–Kähler operators on the 2-sphere. This lattice formulation uses differentiation matrices which yield exact values for the derivatives of polynomials, preserving the Leibniz rule in subspaces of polynomials of low degree and therefore, this formulation can be used to study the fermion–boson symmetry on the lattice. In this context, we find that the free Dirac and Dirac–Kähler operators on the 2-sphere exhibit fermionic as well as bosonic-like eigensolutions for which the corresponding eigenvalues and the number of states are computed. In the Dirac case these solutions appear in doublets, except for the bosonic mode with zero eigenvalue, indicating the possible existence of a supersymmetry of the square of the Dirac operator.

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21

Branson, Thomas, and Oussama Hijazi. "Vanishing Theorems and Eigenvalue Estimates in Riemannian Spin Geometry." International Journal of Mathematics 08, no. 07 (November 1997): 921–34. http://dx.doi.org/10.1142/s0129167x97000433.

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We use the representation theory of the structure group Spin (n), together with the theory of conformally covariant differential operators, to generalize results estimating eigenvalues of the Dirac operator to other tensor-spinor bundles, and to get vanishing theorems for the kernels of first-order differential operators.

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22

GINOUX, NICOLAS, and BERTRAND MOREL. "ON EIGENVALUE ESTIMATES FOR THE SUBMANIFOLD DIRAC OPERATOR." International Journal of Mathematics 13, no. 05 (July 2002): 533–48. http://dx.doi.org/10.1142/s0129167x0200140x.

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We give lower bounds for the eigenvalues of the submanifold Dirac operator in terms of intrinsic and extrinsic curvature expressions. We also show that the limiting cases give rise to a class of spinor fields generalizing that of Killing spinors. We conclude by translating these results in terms of intrinsic twisted Dirac operators.

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23

Monakhov, Vadim. "The Dirac Sea, T and C Symmetry Breaking, and the Spinor Vacuum of the Universe." Universe 7, no. 5 (May 1, 2021): 124. http://dx.doi.org/10.3390/universe7050124.

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We have developed a quantum field theory of spinors based on the algebra of canonical anticommutation relations (CAR algebra) of Grassmann densities in the momentum space. We have proven the existence of two spinor vacua. Operators C and T transform the normal vacuum into an alternative one, which leads to the breaking of the C and T symmetries. The CPT is the real structure operator; it preserves the normal vacuum. We have proven that, in the theory of the Dirac Sea, the formula for the charge conjugation operator must contain an additional generalized Dirac conjugation operator.

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24

LOYA, PAUL, SERGIU MOROIANU, and RAPHAËL PONGE. "ON THE SINGULARITIES OF THE ZETA AND ETA FUNCTIONS OF AN ELLIPTIC OPERATOR." International Journal of Mathematics 23, no. 06 (May 6, 2012): 1250020. http://dx.doi.org/10.1142/s0129167x11007616.

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Let P be a self-adjoint elliptic operator of order m > 0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form [Formula: see text], where k ranges over all nonzero integers ≤ n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions become singular at all points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, self-adjoint first-order differential operators and self-adjoint elliptic pseudodifferential operators. As consequences, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a purely analytical proof of a well-known result of Branson–Gilkey [Residues of the eta function for an operator of Dirac type, J. Funct. Anal. 108(1) (1992) 47–87], which was obtained by invoking Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of [R. Ponge, Spectral asymmetry, zeta functions and the noncommutative residue, Int. J. Math. 17 (2006) 1065–1090]. Corrections to that statement are given in this paper.

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25

Monakhov, Vadim. "Vacuum and Spacetime Signature in the Theory of Superalgebraic Spinors." Universe 5, no. 7 (July 2, 2019): 162. http://dx.doi.org/10.3390/universe5070162.

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A new formalism involving spinors in theories of spacetime and vacuum is presented. It is based on a superalgebraic formulation of the theory of algebraic spinors. New algebraic structures playing role of Dirac matrices are constructed on the basis of Grassmann variables, which we call gamma operators. Various field theory constructions are defined with use of these structures. We derive formulas for the vacuum state vector. Five operator analogs of five Dirac gamma matrices exist in the superalgebraic approach as well as two additional operator analogs of gamma matrices, which are absent in the theory of Dirac spinors. We prove that there is a relationship between gamma operators and the most important physical operators of the second quantization method: number of particles, energy–momentum and electric charge operators. In addition to them, a series of similar operators are constructed from the creation and annihilation operators, which are Lorentz-invariant analogs of Dirac matrices. However, their physical meaning is not yet clear. We prove that the condition for the existence of spinor vacuum imposes restrictions on possible variants of the signature of the four-dimensional spacetime. It can only be (1, − 1 , − 1 , − 1 ), and there are two additional axes corresponding to the inner space of the spinor, with a signature ( − 1 , − 1 ). Developed mathematical formalism allows one to obtain the second quantization operators in a natural way. Gauge transformations arise due to existence of internal degrees of freedom of superalgebraic spinors. These degrees of freedom lead to existence of nontrivial affine connections. Proposed approach opens perspectives for constructing a theory in which the properties of spacetime have the same algebraic nature as the momentum, electromagnetic field and other quantum fields.

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26

Ciubotaru, Dan, and Marcelo De Martino. "Dirac Induction for Rational Cherednik Algebras." International Mathematics Research Notices 2020, no. 17 (July 5, 2018): 5155–214. http://dx.doi.org/10.1093/imrn/rny153.

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Abstract We introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex reflection group acting on a finite-dimensional vector space $\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the graded $G$-character of the module, the Euler–Poincaré pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for $\mathsf{H}_{t,c}(G,\mathfrak{h})$ constructed from finite-dimensional $G$-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function $c$. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl–Opdam operators.

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27

FUJIKAWA, KAZUO. "GENERALIZED GINSPARG–WILSON ALGEBRA AND INDEX THEOREM ON THE LATTICE." International Journal of Modern Physics B 16, no. 14n15 (June 20, 2002): 1931–41. http://dx.doi.org/10.1142/s0217979202011652.

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Recent studies of the topological properties of a general class of lattice Dirac operators are reported. This is based on a specific algebraic realization of the Ginsparg-Wilson relation in the form γ5 (γ5D) + (γ5D)γ5 = 2a2k+1(γ5D)2k+2 where k stands for a non-negative integer. The choice k = 0 corresponds to the commonly discussed Ginsparg-Wilson relation and thus to the overlap operator. It is shown that local chiral anomaly and the instanton-related index of all these operators are identical. The locality of all these Dirac operators for vanishing gauge fields is proved on the basis of explicit construction, but the locality with dynamical gauge fields has not been fully established yet.

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28

Inskeep, Warren H. "On Electromagnetic Spinors and Electron Theory." Zeitschrift für Naturforschung A 44, no. 4 (April 1, 1989): 327–28. http://dx.doi.org/10.1515/zna-1989-0414.

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Abstract The relationship between the Dirac theory and electromagnetic spinors is extended to the case of finite mass. Certain products of the electromagnetic fields give rise to the Dirac differential operator upon the usual subsitutions for the energy and momentum. By placing mass in the proper place for the wave mechanical approach to quantum theory, the algebra of the fields, interpreted as quantum operators, may be deduced.

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29

Takaesu, Toshimitsu. "Essential Self-Adjointness of Anticommutative Operators." Journal of Mathematics 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/265349.

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The self-adjoint extensions of symmetric operators satisfying anticommutation relations are considered. It is proven that an anticommutative type of the Glimm-Jaffe-Nelson commutator theorem follows. Its application to an abstract Dirac operator is also considered.

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30

Leach, P. G. L. "DIRAC AND HAMILTON." Acta Polytechnica 54, no. 2 (April 30, 2014): 127–29. http://dx.doi.org/10.14311/ap.2014.54.0127.

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Dirac devised his theory of Quantum Mechanics and recognised that his operators resembled the canonical coordinates of Hamiltonian Mechanics. This gave the latter a new lease of life. We look at what happens to Dirac’s Quantum Mechanics if one starts from Hamiltonian Mechanics.

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31

Petitjean, Michel. "Chirality of Dirac Spinors Revisited." Symmetry 12, no. 4 (April 14, 2020): 616. http://dx.doi.org/10.3390/sym12040616.

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We emphasize the differences between the chirality concept applied to relativistic fermions and the ususal chirality concept in Euclidean spaces. We introduce the gamma groups and we use them to classify as direct or indirect the symmetry operators encountered in the context of Dirac algebra. Then we show how a recent general mathematical definition of chirality unifies the chirality concepts and resolve conflicting conclusions about symmetry operators, and particularly about the so-called chirality operator. The proofs are based on group theory rather than on Clifford algebras. The results are independent on the representations of Dirac gamma matrices, and stand for higher dimensional ones.

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32

Simulik, V. M., and I. O. Gordievich. "Symmetries of Relativistic Hydrogen Atom." Ukrainian Journal of Physics 64, no. 12 (December 9, 2019): 1148. http://dx.doi.org/10.15407/ujpe64.12.1148.

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The Dirac equation in the external Coulomb field is proved to possess the symmetry determined by 31 operators, which form the 31-dimensional algebra. Two different fermionic realizations of the SO(1,3) algebra of the Lorentz group are found. Two different bosonic realizations of this algebra are found as well. All generators of the above-mentioned algebras commute with the operator of the Dirac equation in an external Coulomb field and, therefore, determine the algebras of invariance of such Dirac equation. Hence, the spin s = (1, 0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended here for the Dirac equation interacting with an external Coulomb field. A relativistic hydrogen atom is modeled by such Dirac equation. We are able to prove for the relativistic hydrogen atom both the fermionic and bosonic symmetries known from our papers in the case of a non-interacting spinor field. New symmetry operators are found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which are known from our recent papers as well. Hidden symmetries were found both in the canonical Foldy–Wouthuysen and covariant Dirac representations. The found symmetry operators, which are pure matrix ones in the Foldy–Wouthuysen representation, become non-local in the Dirac model.

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33

Morimoto, Hiroshi. "Infinite dimensional cycles associated to operators." Nagoya Mathematical Journal 127 (September 1992): 1–14. http://dx.doi.org/10.1017/s0027763000004074.

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A family of operators defined on infinite dimensional spaces gives rise to interesting cycles (or subvarieties) of infinite dimension which represent a topological or non-topological feature of operator families. In this paper we will give a general theory of these cycles, and give some estimates among them. We will apply this theory, in the final section, to cycles derived from Dirac operators.

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34

GUENTNER, ERIK, and NIGEL HIGSON. "A NOTE ON TOEPLITZ OPERATORS." International Journal of Mathematics 07, no. 04 (August 1996): 501–13. http://dx.doi.org/10.1142/s0129167x9600027x.

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We study Toeplitz operators on Bergman spaces using techniques from the analysis of Dirac-type operators on complete Riemannian manifolds, and prove an index theorem of Boutet de Monvel from this point of view. Our approach is similar to that of Baum and Douglas [2], but we replace boundary value theory for the Dolbeaut operator with much simpler estimates on complete manifolds.

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35

HARIKUMAR, E. ""SCHWINGER MODEL" ON THE FUZZY SPHERE." Modern Physics Letters A 25, no. 37 (December 7, 2010): 3151–67. http://dx.doi.org/10.1142/s0217732310034079.

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In this paper, we construct a model of spinor fields interacting with specific gauge fields on the fuzzy sphere and analyze the chiral symmetry of this "Schwinger model". In constructing the theory of gauge fields interacting with spinors on the fuzzy sphere, we take the approach that the Dirac operator Dq on the q-deformed fuzzy sphere [Formula: see text] is the gauged Dirac operator on the fuzzy sphere. This introduces interaction between spinors and specific one-parameter family of gauge fields. We also show how to express the field strength for this gauge field in terms of the Dirac operators Dq and D alone. Using the path integral method, we have calculated the 2n-point functions of this model and show that, in general, they do not vanish, reflecting the chiral non-invariance of the partition function.

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36

Bögli, Sabine, and Marco Marletta. "Essential numerical ranges for linear operator pencils." IMA Journal of Numerical Analysis 40, no. 4 (November 22, 2019): 2256–308. http://dx.doi.org/10.1093/imanum/drz049.

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Abstract We introduce concepts of essential numerical range for the linear operator pencil $\lambda \mapsto A-\lambda B$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=\lambda x$ into the pencil problem $BTx=\lambda Bx$ for suitable choices of $B$, we can obtain nonconvex spectral enclosures for $T$ and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of nonselfadjoint Schrödinger operators which has not been possible to treat with existing methods.

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37

Dzhunushaliev, Vladimir, and Vladimir Folomeev. "Nonperturbative QED on the Hopf Bundle." Physical Sciences Forum 2, no. 1 (July 22, 2021): 43. http://dx.doi.org/10.3390/ecu2021-09286.

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We consider the Dirac equation and Maxwell’s electrodynamics in ℝ×S3 spacetime, where a three-dimensional sphere is the Hopf bundle S3→S2. The method of nonperturbative quantization of interacting Dirac and Maxwell fields is suggested. The corresponding operator equations and the infinite set of the Schwinger–Dyson equations for Green’s functions is written down. To illustrate the suggested scheme of nonperturbative quantization, we write a simplified set of equations describing some physical situation. Additionally, we discuss the properties of quantum states and operators of interacting fields.

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38

Trisetyarso, Agung. "Dirac four-potential tunings-based quantum transistor utilizing the Lorentz force." Quantum Information and Computation 12, no. 11&12 (November 2012): 989–1010. http://dx.doi.org/10.26421/qic12.11-12-7.

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We propose a mathematical model of \textit{quantum} transistor in which bandgap engineering corresponds to the tuning of Dirac potential in the complex four-vector form. The transistor consists of $n$-relativistic spin qubits moving in \textit{classical} external electromagnetic fields. It is shown that the tuning of the direction of the external electromagnetic fields generates perturbation on the potential temporally and spatially, determining the type of quantum logic gates. The theory underlying of this scheme is on the proposal of the intertwining operator for Darboux transfomations on one-dimensional Dirac equation amalgamating the \textit{vector-quantum gates duality} of Pauli matrices. Simultaneous transformation of qubit and energy can be accomplished by setting the $\{\textit{control, cyclic}\}$-operators attached on the coupling between one-qubit quantum gate: the chose of \textit{cyclic}-operator swaps the qubit and energy simultaneously, while \textit{control}-operator ensures the energy conservation.

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39

Ben-Artzi, Matania, and Tomio Umeda. "Spectral theory of first-order systems: From crystals to Dirac operators." Reviews in Mathematical Physics 33, no. 05 (January 16, 2021): 2150014. http://dx.doi.org/10.1142/s0129055x21500148.

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Let [Formula: see text] be a constant coefficient first-order partial differential system, where the matrices [Formula: see text] are Hermitian. It is assumed that the homogeneous part is strongly propagative. In the non-homogeneous case it is assumed that the operator is isotropic. The spectral theory of such systems and their potential perturbations is expounded, and a Limiting Absorption Principle is obtained up to thresholds. Special attention is given to a detailed study of the Dirac and Maxwell operators. The estimates of the spectral derivative near the thresholds are based on detailed trace estimates on the slowness surfaces. Two applications of these estimates are presented: Global spacetime estimates of the associated evolution unitary groups, that are also commonly viewed as decay estimates. In particular, the Dirac and Maxwell systems are explicitly treated. The finiteness of the eigenvalues (in the spectral gap) of the perturbed Dirac operator is studied, under suitable decay assumptions on the potential perturbation.

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40

Dębowska, Kamila, and Leonid P. Nizhnik. "Direct and inverse spectral problems for Dirac systems with nonlocal potentials." Opuscula Mathematica 39, no. 5 (2019): 645–73. http://dx.doi.org/10.7494/opmath.2019.39.5.645.

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The main purposes of this paper are to study the direct and inverse spectral problems of the one-dimensional Dirac operators with nonlocal potentials. Based on informations about the spectrum of the operator, we find the potential and recover the form of the Dirac system. The methods used allow us to reduce the situation to the one-dimensional case. In accordance with the given assumptions and conditions we consider problems in a specific way. We describe the spectrum, the resolvent, the characteristic function etc. Illustrative examples are also given.

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41

Dhanuk, B. B., K. Pudasainee, H. P. Lamichhane, and R. P. Adhikari. "Dirac Delta Function from Closure Relation of Orthonormal Basis and its Use in Expanding Analytic Functions." Journal of Nepal Physical Society 6, no. 2 (December 31, 2020): 158–63. http://dx.doi.org/10.3126/jnphyssoc.v6i2.34872.

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One of revealing and widely used concepts in Physics and mathematics is the Dirac delta function. The Dirac delta function is a distribution on real lines which is zero everywhere except at a single point, where it is infinite. Dirac delta function has vital role in solving inhomogeneous differential equations. In addition, the Dirac delta functions can be used to explore harmonic information’s imbedded in the physical signals, various forms of Dirac delta function and can be constructed from the closure relation of orthonormal basis functions of functional space. Among many special functions, we have chosen the set of eigen functions of the Hamiltonian operator of harmonic oscillator and angular momentum operators for orthonormal basis. The closure relation of orthonormal functions used to construct the generator of Dirac delta function which is used to expand analytic functions log(x + 2),exp(-x2) and x within the valid region of arguments.

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42

Reis, João Alfíeres Andrade de Simões dos, and Marco Schreck. "Formal Developments for Lorentz-Violating Dirac Fermions and Neutrinos." Symmetry 11, no. 10 (September 24, 2019): 1197. http://dx.doi.org/10.3390/sym11101197.

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The current paper is a technical work that is focused on Lorentz violation for Dirac fermions as well as neutrinos, described within the nonminimal Standard-Model Extension. We intend to derive two theoretical results. The first is the full propagator of the single-fermion Dirac theory modified by Lorentz violation. The second is the dispersion equation for a theory of N neutrino flavors that enables the description of both Dirac and Majorana neutrinos. As the matrix structure of the neutrino field operator is very involved for generic N, we will use sophisticated methods of linear algebra to achieve our objectives. Our main finding is that the neutrino dispersion equation has the same structure in terms of Lorentz-violating operators as that of a modified single-fermion Dirac theory. The results will be valuable for phenomenological studies of Lorentz-violating Dirac fermions and neutrinos.

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43

Dzhunushaliev, Vladimir, and Vladimir Folomeev. "Nonperturbative Quantization Approach for QED on the Hopf Bundle." Universe 7, no. 3 (March 11, 2021): 65. http://dx.doi.org/10.3390/universe7030065.

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We consider the Dirac equation and Maxwell’s electrodynamics in R×S3 spacetime, where a three-dimensional sphere is the Hopf bundle S3→S2. In both cases, discrete spectra of classical solutions are obtained. Based on the solutions obtained, the quantization of free, noninteracting Dirac and Maxwell fields is carried out. The method of nonperturbative quantization of interacting Dirac and Maxwell fields is suggested. The corresponding operator equations and the infinite set of the Schwinger–Dyson equations for Green’s functions is written down. We write a simplified set of equations describing some physical situations to illustrate the suggested scheme of nonperturbative quantization. Additionally, we discuss the properties of quantum states and operators of interacting fields.

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44

ARAI, ASAO. "NON-RELATIVISTIC LIMIT OF A DIRAC–MAXWELL OPERATOR IN RELATIVISTIC QUANTUM ELECTRODYNAMICS." Reviews in Mathematical Physics 15, no. 03 (May 2003): 245–70. http://dx.doi.org/10.1142/s0129055x0300162x.

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The non-relativistic (scaling) limit of a particle-field Hamiltonian H, called a Dirac–Maxwell operator, in relativistic quantum electrodynamics is considered. It is proven that the non-relativistic limit of H yields a self-adjoint extension of the Pauli–Fierz Hamiltonian with spin 1/2 in non-relativistic quantum electrodynamics. This is done by establishing in an abstract framework a general limit theorem on a family of self-adjoint operators partially formed out of strongly anticommuting self-adjoint operators and then by applying it to H.

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45

Cecchini, Simone. "Callias-type operators in C∗-algebras and positive scalar curvature on noncompact manifolds." Journal of Topology and Analysis 12, no. 04 (November 16, 2018): 897–939. http://dx.doi.org/10.1142/s1793525319500687.

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A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert [Formula: see text]-module bundles and prove an index theorem for such operators. As an application, we derive an obstruction to the existence of complete Riemannian metrics of positive scalar curvature on noncompact spin manifolds in terms of closed submanifolds of codimension one. In particular, when [Formula: see text] is a closed spin manifold, we show that if the cylinder [Formula: see text] carries a complete metric of positive scalar curvature, then the (complex) Rosenberg index on [Formula: see text] must vanish.

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46

Nikitin, A. G. "Nonstandard Dirac equations for nonstandard spinors." International Journal of Modern Physics D 23, no. 14 (December 2014): 1444007. http://dx.doi.org/10.1142/s0218271814440076.

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Generalized Dirac equation with operator mass term is presented. Its solutions are nonstandard spinors (NSS) which, like eigenspinoren des Ladungskonjugationsoperators (ELKO), are eigenvectors of the charge conjugation and dual-helicity operators. It is demonstrated that in spite of their noncovariant nature the NSS can serve as a carrier space of a representation of Poincaré group. However, the corresponding boost generators are not manifestly covariant and generate nonlocal momentum dependent transformations, which are presented explicitly. These results can present a new look on group-theoretical grounds of ELKO theories.

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47

Liu, Q. H., and S. F. Xiao. "A self-adjoint decomposition of the radial momentum operator." International Journal of Geometric Methods in Modern Physics 12, no. 03 (February 27, 2015): 1550028. http://dx.doi.org/10.1142/s0219887815500280.

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With acceptance of the Dirac's observation that the canonical quantization entails using Cartesian coordinates, we examine the operator erPr rather than Pr itself and demonstrate that there is a decomposition of erPr into a difference of two self-adjoint but noncommutative operators, in which one is the total momentum and another is the transverse one. This study renders the operator Pr indirectly measurable and physically meaningful, offering an explanation of why the mean value of Pr over a quantum mechanical state makes sense and supporting Dirac's claim that Pr "is real and is the true momentum conjugate to r".

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48

KERLER, WERNER. "CHIRAL FERMION OPERATORS ON THE LATTICE." International Journal of Modern Physics A 18, no. 15 (June 20, 2003): 2565–90. http://dx.doi.org/10.1142/s0217751x03013910.

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We only require generalized chiral symmetry and γ5-hermiticity, which leads to a large class of Dirac operators describing massless fermions on the lattice, and use this framework to give an overview of developments in this field. Spectral representations turn out to be a powerful tool for obtaining detailed properties of the operators and a general construction of them. A basic unitary operator is seen to play a central rôle in this context. We discuss a number of special cases of the operators and elaborate on various aspects of index relations. We also show that our weaker conditions lead still properly to Weyl fermions and to chiral gauge theories.

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49

García, A. G., and M. A. Hernández-Medina. "On an integral transform associated with the regular Dirac operator." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 6 (December 2001): 1357–70. http://dx.doi.org/10.1017/s0308210500001438.

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In this paper we deal with a linear integral transform, defined on a vectorial L2-space, whose kernel arises from a one-dimensional system of Dirac operators. Unlike the regular Sturm–Liouville transform, which is associated with a regular Sturm–Liouville problem, the range of this transform is a whole Paley–Wiener space. As a consequence, some results for the Paley–Wiener space are derived; in particular, the sampling formula associated with a regular Dirac operator. Finally, we obtain an inversion formula by means of a continuous measure for suitable Sobolev spaces in the initial L2-space.

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50

Hughes, Daniel, and Karl Michael Schmidt. "Absolutely continuous spectrum of Dirac operators with square-integrable potentials." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 144, no. 3 (May 16, 2014): 533–55. http://dx.doi.org/10.1017/s0308210512001187.

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We show that the absolutely continuous part of the spectral function of the one-dimensional Dirac operator on a half-line with a constant mass term and a real, square-integrable potential is strictly increasing throughout the essential spectrum (−∞, −1] ∪ [1, ∞). The proof is based on estimates for the transmission coefficient for the full-line scattering problem with a truncated potential and a subsequent limiting procedure for the spectral function. Furthermore, we show that the absolutely continuous spectrum persists when an angular momentum term is added, thus also establishing the result for spherically symmetric Dirac operators in higher dimensions.

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