Work Of Dirac Research Articles

A Matter of Principle: The Principles of Quantum Theory, Dirac’s Equation, and Quantum Information

This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be be addressed as well in view of their role in quantum electrodynamics and quantum field theory, specifically Dirac's work, which, in particular Dirac's derivation of his relativistic equation for the electron from the principles of relativity and quantum theory, is the main focus of this article. I shall, however, also consider Heisenberg's derivation of quantum mechanics, which inspired Dirac. I argue that Heisenberg's and Dirac's work alike was guided by their adherence to and confidence in the fundamental principles of quantum theory. The final section of the article discusses the recent work by G. M. D' Ariano and his coworkers on the principles of quantum information theory, which extends quantum theory and its principles in a new direction. This extension enabled them to offer a new derivation of Dirac's equation from these principles alone, without using the principles of relativity.

Validity of the ICFT R-matrix method: Be-like Al 9+ a case study

We have carried out 98-level configuration-interaction/close-coupling (CI/CC) intermediate coupling frame transformation (ICFT) and Breit–Pauli R-matrix calculations for the electron-impact excitation of Be-like Al 9+. The close agreement that we find between the two sets of effective collision strengths demonstrates the continued robustness of the ICFT method. On the other hand, a comparison of this data with previous 238-level CI/CC ICFT effective collision strengths shows that the results for excitation up to n = 4 levels are systematically and increasingly underestimated over a wide range of temperatures by R-matrix calculations whose CC expansion extends only to n = 4 (98-levels). Thus, we find to be false a recent conjecture that the ICFT approach may not be completely robust. The conjecture was based upon a comparison of 98-level CI/CC Dirac R-matrix effective collision strengths for Al 9+ with those from the 238-level CI/CC ICFT R-matrix calculations. The disagreement found recently is due to a lack of convergence of the CC expansion in the 98-level CI/CC Dirac work. The earlier 238-level CI/CC ICFT work has a superior target to the 98-level CI/CC Dirac one and provides more accurate atomic data. Similar considerations need to be made for other Be-like ions and for other sequences.

Resolution of the forbidden (J= 0 → 0) excitation puzzle in Mg-like ions

We have carried-out 98-level configuration-interaction / close-coupling (CI/CC) intermediate coupling frame transformation (ICFT) and Breit-Pauli R-matrix calculations for the electron-impact excitation of Be-like Al$^{9+}$. The close agreement that we find between the two sets of effective collision strengths demonstrates the continued robustness of the ICFT method. On the other hand, a comparison of this data with previous 238-level CI/CC ICFT effective collision strengths shows that the results for excitation up to n=4 levels are systematically and increasingly underestimated over a wide range of temperatures by R-matrix calculations whose close-coupling expansion extends only to n=4 (98-levels). Thus, we find to be false a recent conjecture that the ICFT approach may not be completely robust. The conjecture was based upon a comparison of 98-level CI/CC Dirac R-matrix effective collision strengths for Al$^{9+}$ with those from the 238-level CI/CC ICFT R-matrix calculations. The disagreement found recently is due to a lack of convergence of the close-coupling expansion in the 98-level CI/CC Dirac work. The earlier 238-level CI/CC ICFT work has a superior target to the 98-level CI/CC Dirac one and provides more accurate atomic data. Similar considerations need to be made for other Be-like ions and for other sequences.

Longitudinal (Electric/Scalar) Waves

One probably heard about Paul Dirac's work on Dirac sea in 1934. This is also called Delta-E field, Higgs field, Vacuum or just plain Coulomb charge field. Many people became concerned about the possibility of superluminal (faster than light) physics after faster than light experiments were claimed to be carried out in 2010 and earlier. This is now called superluminal physics. Maverick people in this field can be considered such as Paul Dirac, Nikola Tesla and lately Thomas L. Bearden, Jean L. Naudin and Konstantin Meyl. Please also see the links (papers and publications on the field) below.

Função de Wigner-80 anos e as origens da geometria não-comutativa

O conceito de espaços não-comutativos tem origem com a formulação de Wigner da mecânica quântica no espaço de fase, em 1932. Em paralelo, Heisenberg foi o primeiro a propor relações de não-comutação entre as componentes do operador de posição. Essa possibilidade ganha formulação matemática com Snyder, ao estudar representações do grupo de De Sitter (4+1). Uma síntese desses trabalhos é o conceito de geometria não-comutativa, estabelecida a partir do produto de Moyal, que aparece no formulismo de Wigner. Além disso, este tipo de não-comutatividade é reencontrada em certos limites das teorias de cordas, gerando expectativas da mensuração da não-comutatividade espacial na física de altas energias. Neste trabalho, apresentamos uma revisão pedagógica sobre teorias físicas em espaços não-comutativos, a partir de uma perpectiva histórica. Destacamos as teorias de representação de grupos de simetria no espaço de fase, apontando dois aspectos de interesse, mas pouco conhecidos: (a) a noção de amplitude de probabilidade e a representação da equação de Schrödinger no espaço de fase (usualmente a representação no espaço de fase da mecânica quântica é construída através da matrix de densidade e da equação de Liouville-von Neumann); e (b) um trabalho de Dirac de 1930, onde foi introduzida pela primeira vez uma formulação da física quântica no espaço de fase.

Normal eulerian clique-covering and hamiltonicity

A connected graph is hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is hamiltonian is known as an NP-complete problem and no satisfactory characterization for these graphs has been found. Since the seminal work of Dirac in 1952 many sufficient conditions were found. In 1974, Goodman and Hedetniemi gave such a condition based on the existence of a clique-covering of the graph. This condition was recently generalized using the notion of eulerian clique-covering. In addition, an algorithm able to find a normal eulerian clique-covering for a large class of graphs was also introduced. A normal clique-covering has additional properties, making the search for such a covering easier than in the general case. In this article, we prove several properties of normal clique-coverings. In particular we prove that there exists an eulerian clique-covering of a graph if and only if there exists a normal one. Using this result, the search for an eulerian clique-covering as a sufficient condition for hamiltonicity can be reduced to the normal case.

Léon Rosenfeld's pioneering steps toward a quantum theory of gravity

In an article published in 1930 Léon Rosenfeld invented a general Hamiltonian formalism that purported to realize general coordinate, local Lorentz, and U(1) symmetries as canonical phase space transformations. He applied the formalism to a q-number version of tetrad gravity in interaction with both the electromagnetic field and a spinorial Dirac electron matter field. His procedure predated by almost two decades the algorithms of Dirac and Bergmann, and with regard to internal (non-spacetime) symmetries is fully equivalent to them. Dirac was in fact already in 1932 familiar with Rosenfelds work, although as far as I can tell he never acknowledged in print his perhaps unconscious debt to Rosenfeld. I will review the general formalism, comparing and contrasting with the work of Dirac, Bergmann and his associates. Although Rosenfeld formulated a correct prescription for constructing the vanishing Hamiltonian generator of time evolution, he evidently did not succeed in carrying out the construction. Nor did he have the correct phase space generators of diffeomorphism-induced symmetry variations. He did not take into account that some of the Lagrangian symmetries are not projectable under the Legendre transformation to phase space.

Wigner–Weyl correspondence in quantum mechanics for continuous and discrete systems—a Dirac-inspired view

Drawing inspiration from Dirac's work on functions of non-commuting observables, we develop an approach to phase-space descriptions of operators and the Wigner-Weyl correspondence in quantum mechanics, complementary to standard formulations. This involves a two-step process: introducing phase-space descriptions based on placing position dependences to the left of momentum dependences (or the other way around); then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators. The method works uniformly for both continuous Cartesian degrees of freedom and for systems with finite-dimensional state spaces. It is interesting that the kernel encountered is naturally expressible in terms of geometric phases, and its removal involves extracting its square root in a suitable manner.

Dirac and the dispensability of mathematics

In this paper, I examine the role of the delta function in Dirac's formulation of quantum mechanics (QM), and I discuss, more generally, the role of mathematics in theory construction. It has been argued that mathematical theories play an indispensable role in physics, particularly in QM [Colyvan, M. (2001). The indispensability of mathematics. Oxford University Press: Oxford]. As I argue here, at least in the case of the delta function, Dirac was very clear about its dispensability. I first discuss the significance of the delta function in Dirac's work, and explore the strategy that he devised to overcome its use. I then argue that even if mathematical theories turned out to be indispensable, this wouldn’t justify the commitment to the existence of mathematical entities. In fact, even in successful uses of mathematics, such as in Dirac's discovery of antimatter, there's no need to believe in the existence of the corresponding mathematical entities. An interesting picture about the application of mathematics emerges from a careful examination of Dirac's work.

Exact solution of the relativistic dynamics of a spin‐½ particle moving in a homogeneous magnetic field

AbstractThe relativistic dynamics of one spin‐½ particle moving in a uniform magnetic field is described by the Hamiltonian \documentclass{article}\pagestyle{empty}\begin{document}$\mathbf{h}^{0}_{D}(\pi)=c\alpha\cdot\pi+\beta mc^{2}$\end{document}. The discrete (and semidiscrete) eigenvalues and the corresponding eigenspinors are in principle known from the work of Dirac, Rabi, and Bloch. These are extensively reviewed here. Next, exact solutions are worked out for the recoil dynamics in relative coordinates, which involves the Hamiltonian \documentclass{article}\pagestyle{empty}\begin{document}$\mathbf{h}^{0}_{D}(-\mathbf{k})=-c\alpha\cdot\mathbf{k}+\beta mc^{2}$\end{document}. Exact solutions are also explicitly calculated in the case where the spin‐½ particle has an anomalous magnetic moment such that its Hamiltonian is given by \documentclass{article}\pagestyle{empty}\begin{document}$\mathbf{h}_{D}(\pi)=\mathbf{h}^{0}_{D}(\pi)-\beta\mu_{\mathrm{ano}}\sigma\cdot\mathbf{B}$\end{document}. Similar exact solutions are derived here when the recoiling particle has an anomalous magnetic moment, that is, the eigenvalues and eigenspinors of the Hamiltonian \documentclass{article}\pagestyle{empty}\begin{document}$\mathbf{h}_{D}(-\mathbf{k})=\mathbf{h}^{0}_{D}(-\mathbf{k})-\beta\mu_{\mathrm{ano}}\sigma\cdot\mathbf{B}$\end{document} are explicitly obtained. The diagonalized and separable form of the Hamiltonian hD(π), written as \documentclass{article}\pagestyle{empty}\begin{document}$\tilde{\mathbf{h}}_{D}(\pi)$\end{document}, has exceedingly simple forms of eigenspinors. Similarly, the diagonalized and separable form of the operator hD(−k), written as \documentclass{article}\pagestyle{empty}\begin{document}$\tilde{\mathbf{h}}_{D}(-\mathbf{k})$\end{document}, has very simple eigenspinors. The importance of these exact solutions is that the eigenspinors can be used as bases in a calculation involving many spin‐½ particles placed in a uniform magnetic field. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 209–217, 2001

Two-time physics in field theory

A field theory formulation of two-time physics in d+2 dimensions is obtained from the covariant quantization of the constraint system associated with the OSp(n|2) worldline gauge symmetries of two-time physics. Interactions among fields can then be included consistently with the underlying gauge symmetries. Through this process a relation between Dirac's work in 1936 on conformal symmetry in field theory and the more recent worldline formulation of two-time physics is established while providing a worldline gauge symmetry basis for the field equations in d+2 dimensions. It is shown that the field theory formalism goes well beyond Dirac's goal of linearizing conformal symmetry. In accord with recent results in the worldline approach of two-time physics, the d+2 field theory can be brought down to diverse d dimensional field theories by solving the subset of field equations that correspond to the ``kinematic'' constraints. This process embeds the one ``time'' in d-dimensions in different ways inside the d+2 dimensional spacetime. Thus, the two-time d+2 field theory appears as a more fundamental theory from which many one-time d dimensional field theories are derived. It is suggested that the hidden symmetries and relations among computed quantities in certain d-dimensional interacting field theories can be taken as the evidence for the presence of a higher unifying structure in a d+2 dimensional spacetime. These phenomena have similarities with ideas such as dualities, AdS-CFT correspondence and holography.

Is general relativity an "already parametrized" theory?

Beginning with the work of Dirac and Arnowitt, Deser, Misner in the late fifties and early sixties, and then after subsequent development by Kucha\v r, the canonical dynamical structure of general relativity has often been viewed as that of a parametrized field theory in which the many-fingered spacetime variables are hidden amongst the geometrodynamical field variables. This paradigm of general relativity as an ``already parametrized theory'' forms the basis for one of the most satisfactory resolutions of the problems of time and observables in classical and quantum gravity. However, despite decades of effort, no identification of many-fingered spacetime variables has ever been satisfactorily obtained for vacuum general relativity. We point out that there is an obstruction to identifying the constraint surface of general relativity (for the case of a closed universe) with that of any parametrized theory. Therefore, strictly speaking, general relativity cannot be viewed as a parametrized field theory. We discuss implications for the canonical quantization program.

Bohr entre Einstein et Dirac

SUMMARY. — The idea of complementarity may be viewed as the core of Bohr's strategy, in 1927, to save both observation and definition. To this end, Bohr sought to incorporate, within the new quantum theory, the usual concepts of space and time, as well as the invariance principles and their associated conservation laws. At the time, Bohr saw himself as following in the footsteps of Einstein, who, in 1905, used a similar strategy to resolve the apparent contradiction between the constancy of the velocity of light and galilean invariance. By clarifying Bohr's deliberately vague notions, Dirac made the search for invariance transformations the aim of quantum physics. Dirac's work thus illustrates the importance of Bohr's of 1927 for the future development of theoretical physics.

New generalization of Dirac's positive-energy relativistic wave equation

The relativistic wave equation for a spinless massive particle, suggested by Dirac ten years ago, has only positive-energy solutions. However, this new system of wave equations becomes inconsistent when electromagnetic interactions are included. This is a defect. In this paper, Dirac's work is generalized for cases in which the symmetry of interchangeability among the subparticles in each group contained in a particle is satisfied. The inconsistency of Dirac's positive-energy wave equation is avoided either by having a more complex internal structure for a particle or by increasing the degrees of freedom of motion for the internal subparticles involved in this same particle.

On Connectivity in Matroids and Graphs

In this paper we derive several results for connected matroids and use these to obtain new results for -connected graphs. In particular, we generalize work of Murty and Seymour on the number of two-element cocircuits in a minimally connected matroid, and work of Dirac, Plummer and Mader on the number of vertices of degree two in a minimally $2$-connected graph. We also solve a problem of Murty by giving a straightforward but useful characterization of minimally connected matroids. The final part of the paper gives a matroid generalization of Dirac and Plummer’s result that every minimally $2$-connected graph is $3$-colourable.

On connectivity in matroids and graphs

In this paper we derive several results for connected matroids and use these to obtain new results for -connected graphs. In particular, we generalize work of Murty and Seymour on the number of two-element cocircuits in a minimally connected matroid, and work of Dirac, Plummer and Mader on the number of vertices of degree two in a minimally $2$-connected graph. We also solve a problem of Murty by giving a straightforward but useful characterization of minimally connected matroids. The final part of the paper gives a matroid generalization of Dirac and Plummer’s result that every minimally $2$-connected graph is $3$-colourable.

On the Quantization of Physical Space-time Operators

§ 1. IntrDduction In the conventional field theory, space-time coordinates are treated as c-number parameters, and field operators (consequently energy-momentum density, current density and such quantities are expressed as the products of field operators also) are represented as their functions. On the other hand, many authorsl) investigated the possibility to regard space-time coordinates as operators, or to quantize them, in order to get rid of the so-called divergence difficulties. However, no promising guiding principle to quantize space-time coordinates has been found. Among many attempts, a series of investigations 2 )-4) which was begun by the famous Dirac's work 5 ) has an exceptional attraction for the present author, although these works were developed regardless of the divergence difficulties. For their quantization of the physical coordinate operators seems to express the uncertainty principle in the measurements, as will be seen in the following discussions. The purpose of this article is to make the physical meaning of the quantization clear and to analyse the nature of the extended Hilbert space. The meaning of the uncertainty principle in quantum mechanics is as follows. As long as the particle, for which the uncertainty relation between its position and momentum holds, is used in the observation of the motion of some particle, the position and the momentum of the object also cannot be measured simultaneously and exactly. If any particle which is free from the uncertainty principle does exist, we would be able to use this particle to make the simultaneous measurement of the position and the momentum of the object, whether they are elementary or not. It may in fact be possible to take the standpoint that the uncertainty relation is derived from such a theory that introduces some hidden variable free from the uncertainty principle, as Bohm

Radiation damping in a gravitational field

The validity of the principle of equivalence is examined from the point of view of a charged mass point moving in an externally given gravitational field. The procedure is a covariant generalization of Dirac's work on the classical radiating electron. Just as Dirac's calculation was kept Lorentz invariant throughout, so the present calculation is maintained generally covariant throughout. With the aid of bi-tensors, which are nonlocal generalizations of ordinary local tensors, the manifest general covariance of each step is achieved in an elegant and useful way. The Green's functions for the scalar and vector wave equations in a curved manifold are obtained and applied to the derivation of the covariant Liénard-Wiechert potentials. The computation of energy-momentum balance across a world tube of infinitesimal radius surrounding the particle world-line then leads to the ponderomotive equations including radiation damping. Because of the nonlocal electromagnetic field which a charged particle carries with itself, its use as a device to distinguish locally between gravitational and inertial fields is really not allowable. One should be prepared to find an explicit occurrence of the Riemann tensor in the ponderomotive equations, leading to the result that acceleration by a “true” gravitational field can produce bremsstrahlung, thereby causing a reactive force in addition to the force of inertia. It is remarkable, however, that such an explicit occurrence does not happen. The particle tries its best to satisfy the equivalenc principle in spite of its charge. It is only prevented from doing so (i.e., from following a geodetic path) because of the fact that, contrary to the case of flat space-time, the electromagnetic Green's function in a curved spacetime does not generally vanish inside the light cone, but gives rise to a “tail” on any initially sharp pulse of radiation. The ponderomotive equations have exactly the same form as Dirac found for the flat-space-time case except for the addition of an integral over the entire past history of the particle, representing the effect of the “tail.”

An Interpretation of the Variable Mass in Wessel's Theory of the Electron

The work of Dirac, Bopp, Hönl, and Wessel on the theory of the electron is briefly reviewed. The equations of motion, with a variable rest mass, are integrated to obtain the velocities. The initial condition in this development can be so chosen that one obtains a motion which is analogous to the “zitterbewegung” first derived by Schrödinger. The connection between spin and radiation reaction force is clarified.

The physical interpretation of wave mechanics

The object of this paper is to give an account of the fundamental principles of wave mechanics in a manner which shall make clear the physical significance of all the quantities and processes involved. The principles are illustrated by discussions of the propagation of free electric waves in uniform electromagnetic fields and of bound electric waves in the hydrogen atom. The paper concludes with relativistic wave mechanics (prior to the work of Dirac and Darwin) and a short account of the Compton effect.