Dirac's conjecture

An integral invariant for a singular Lagrangian which depends on time explicitly is deduced. The connection between this invariant and Dirac's conjecture is discussed. An example shows that Dirac's conjecture fails.     

Cawley's counterexample to Dirac's conjecture as a curved spacetime

Cawley's counterexample Lagrangian to Dirac's conjecture on dynamical systems is modified to a line element in curved spacetime, and the energy-momentum tensor corresponding to such a spacetime is found. The spacetime obtained satisfies the Einstein field equations and describes a three-dimensional matterfilled universe. It is further shown that such a universe cannot be filled up with other sources, such as a perfect fluid, a scalar field, or an electromagnetic field, without violating the Einstein field equations.     

From Dirac's Contact Transformation to Dirac's Conjecture

The finite Dirac's contact transformation according to Dirac's physical idea is obtained. It is shown that its generators belong to the smallest evolution closed subalgebra of the first class constraints but do not exhaust all the first class constraints in general. Then the extended Dirac's conjecture is proposed and is applied to Cawley's counter example of Dirac's conjecture successfully.     

A conjecture about two-dimensional hydrodynamics

The 6 ¹P₁ state of ¹⁷³Yb has been investigated by the level-crossing technique with additional optical pumping of the ground state. With the results of this experiment a long existing uncertainty in the hfs coupling constants could be removed.     

Remarks about the thin sandwich conjecture

The thin sandwich conjecture in the initial value problem of general relativity is investigated by means of differential geometrical methods, in connection to the definition of arc length in a Wheeler superspace. The conjecture is proved to be false by good physical reasons. Its failure on a flat space is related to the “functional dimension” of the superspace and to the behaviour in the linearized theory of quantum gravitation.     

On Dirac's Conjecture

The extended canonical Noether identities and canonical first Noether theorem derived from an extended action in phase space for a system with a singular Lagrangian are formulated. Using these canonical Noether identities, it can be shown that the constraint multipliers connected with the first-class constraints may not be independent, so a query to a conjecture of Dirac is presented. Based on the symmetry properties of the constrained Hamiltonian system in phase space, a counterexample to a conjecture of Dirac is given to show that Dirac's conjecture fails in such a system. We present here a different way rather than Cawley's examples and other's ones in that there is no linearization of constraints in the problem. This example has a feature that neither the primary first-class constraints nor secondary first-class constraints are generators of the gauge transformation.     

A conjecture about Bäcklund transformations

It has been suggested that given a constant of motion for an evolution equation which comes from a Bäcklund transformation one can obtain further constants of motion from the one by further use of the Bäcklund transformation. Our conjecture is that no new constants can be so obtained. A number of examples verifying the conjecture are given.     

Four-space formulation of Dirac's equation

Dirac's equation is reviewed and found to be based on nonrelativistic ideas of probability. A 4-space formulation is proposed that is completely Lorentzinvariant, using probability distributions in space-time with the particle's proper time as a parameter for the evolution of the wave function. This leads to a new wave equation which implies that the proper mass of a particle is an observable, and is sharp only in stationary states. The model has a built-in arrow of time, which is associated with a restriction to positive-energy solutions. The usual solution for a Coulomb field is retained, though it now implies a slightly different charge distribution. The conventional nonstationary solutions become invalid. The new formulation appears to offer a resolution of difficulties that have been associated with Dirac's equation. It also predicts the occurrence of virtual pairs at a level that may be experimentally testable, and suggests a mechanism for self-cancellation of the vacuum energy.     

On Dirac's quantization condition

The consequences of Dirac's quantization condition are discussed. Observing that this condition generally leads to contradiction and that the normal explanation to resolve the contradiction seems unsatisfactory, we propose a modified quantization condition. It is shown that operator ordering ambiguity could be represented as the arbitrariness of the distribution functions in the modified condition.     

Quantum mechanics in Dirac's front form

In 1949 Dirac published a paper in which he proposed various ways to combine special relativity with the Hamiltonian formulation of dynamics; these were referred toas forms and three differentforms, theinstant, thepoint, and thefront forms, were discussed. Dirac considered thefront form to be mathematically the most interesting. Despite this, thefront form appears to have been the least explored. This paper presents the results of a study of quantum mechanics in thefront form.     

Dirac's aether in relativistic quantum mechanics

The introduction by Dirac of a new aether model based on a stochastic covariant distribution of subquantum motions (corresponding to a vacuum state alive with fluctuations and randomness) is discussed with respect to the present experimental and theoretical discussion of nonlocality in EPR situations. It is shown (1) that one can deduce the de Broglie waves as real collective Markov processes on the top of Dirac's aether; (2) that the quantum potential associated with this aether's modification, by the presence of EPR photon pairs, yields a relativistic causal action at a distance which interprets the superluminal correlations recently established by Aspect et al.; (3) that the existence of the Einstein-de Broglie photon model (deduced from Dirac's aether) implies experimental predictions which conflict with the Copenhagen interpretation in certain specific testable interference experiments.     

Four-space formulation of Dirac's equation

Dirac's equation is reformulated in a way which, while preserving the usual energy levels, leads to predictions of vacuum polarization that may be experimentally testable.     

Dirac's Claim and the Chemists

In 1929 Paul A. M. Dirac claimed that “the underlying physical laws necessary for the mathematical theory of ... the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” This sentence of Dirac's is cited frequently by historians and philosophers of chemistry in the context of discussions on the hypothetical reduction of chemistry to physics. But how did chemists themselves react to Dirac's claim? Did they feel threatened by physicists who felt they could do their job better than themselves? Did they feel indifferent, or did they simply not care? Was Dirac's paper often cited by chemists? Why was it cited? In this paper, I provide answers to these questions on the basis of an analysis of citations to Dirac's 1929 paper in the Science Citation Index. RID="*" ID="*"Ana Sim?es teaches history of science at the University of Lisbon. Her research is in the history of quantum chemistry and in the history of the sciences at the European peripheries.     

Intertial forces in Dirac's vacuum

In Dirac's version of classical electrodynamics with the gauge A_µ A^µ = k², the vector potential may, following Dirac's suggestion, be interpreted as the velocity 4-vector of an electromagnetic ether. It is shown that the inertial forces generated by this ether are identical with the Lorentz force acting on a charged particle.     

Rotational symmetry and Dirac's monopole

The field of a Dirac monopole is constructed in the framework of symplectic mechanics by imposing rotational and time translation invariance on the motion of a test particle. Quantization is achieved by the geometric method of Kostant and Souriau, which allows for an elegant solution of the quantum symmetry problem. Space-reflection symmetry is studied in addition.     

Spin of Dirac's relativistic membrane

The Hamiltonian of Dirac's relativistic membrane is linearized by means of a gauge transformation. It is pointed out that the membrane spin is +/- (planck)/2. Furthermore, in a quantum relativistic framework neutrinolike particles are obtained when the membrane charge vanishes.     

Quaternionic Electron Theory: Dirac's Equation

We perform a one-dimensional complexifiedquaternionic version of the Dirac equation based oni-complex geometry. The problem of the missing complexparameters in quaternionic quantum mechanics withi-complex geometry is overcome by a nicetrick which allows us to avoid the Diracalgebra constraints in formulating our relativisticequation. A brief comparison with other quaternionicformulations is also presented.     

Dirac's cosmology with varying cosmological constant

The phenomenological approach to investigating the decay of the vacuum energy density is generalized in the spirit of Dirac's large number hypothesis. Different arguments are outlined to justify an universal dependence of the type =H ² ( is a pure number andH is the Hubble parameter). Such time-varying A does change the predictions of the earlier Dirac's cosmology. The deceleration parameter (q ₀=(2–)/(1+)) and the age parameter (H ₀t₀=(1+)/3) are now compatible with the observations. The model also allows a power-law inflationary phase and a rate of matter creation smaller than the one present in the steady state universe.