Canonical Proper-Time Formulation of Relativistic Particle Dynamics. II

Our purpose in this paper is to provide theframework for a generalization of classical mechanicsand electrodynamics, including Maxwell's theory, whichis simple, technically correct, and requires noadditional work for the quantum case. We first show thatthere are two other definitions of proper-time, eachhaving equal status with the Minkowski definition. Weuse the first definition, called the proper-velocity definition, to construct a transformationtheory which fixes the proper-time of a given physicalsystem for all observers. This leads to a new invariancegroup and a generalization of Maxwell's equations left covariant under the action of this group.The second definition, called the canonical variablesdefinition, has the unique property that it isindependent of the number of particles. This definition leads to a general theory of directlyinteracting relativistic particles. We obtain theLorentz force for one particle (using its proper-time),and the Lorentz force for the total system (using theglobal proper-time). Use of the global proper-time tocompute the force on one particle gives the Lorentzforce plus a dissipative term corresponding to thereaction of this particle back on the cause of itsacceleration (Newton's third law). The wave equation derivedfrom Maxwell's equations has an additional term, firstorder in the proper-time. This term arisesinstantaneously with acceleration. This shows explicitly that the longsought origin of radiationreaction is inertial resistance to changes in particlemotion. The field equations carry intrinsic informationabout the velocity and acceleration of the particles in the system. It follows that our theory isnot invariant under time reversal, so that the existenceof radiation introduces an arrow for the (proper-time ofthe) system.     

Canonical variables for the Dirac theory

A new canonical structure for Dirac's theory is proposed. The new configuration space A is a real, four-dimensional subbundle of the spinor bundle. A Lagrangian defined on Q describes a theory equivalent to the Dirac one. In this way we obtain a theory without second-type constraints.     

Dirac Canonical Quantization of Composite Fermions QED

Composite Fermions QED is quantized by using the Dirac’s canonical formalism for constrained systems. As a strategy, we first work out the constraints (including primary and secondary constraints), combine two first-class constraints, introduce Coulomb gauge and its stationary as gauge conditions, and then quantize, replacing the Dirac brackets with quantum commutators.     

Proper-Time Classical Electrodynamics

We construct a generalization of Maxwell's equations associated with the proper-time of the source which accounts for radiation reaction without any assumptions concerning the nature of the source. The theory leads to a new invariance group, related to the Lorentz group, which leaves the proper-time of the source fixed for all observers.     

Proper-Time Formulation of Relativistic Dynamics

It will be argued that Minkowski's implementation of distances is inconsistent. An alternative implementation will be proposed. In the new model the proper time of an object is taken as its fourth coordinate. Distances will be measured according to a four dimensional Euclidean metric. In the present approach mass is a constant of motion. A mass can therefore be ascribed to photons and neutrinos. Mechanics and dynamics will be reformulated in close correspondence with classical physics. Of particular interest is the equation of motion for the proper time momentum. In the classical limit it reduces to the classical law of conservation of (kinetic+potential) energy. In the relativistic limit it is similar to the conservation of energy of the theory of relativity. The conservation of proper time momentum allows for an alternative explanation for Compton scattering and pair annihilation. On the basis of the proper time formulation of electrodynamics also an alternative explanation will be offered for the spectra of hydrogenic atoms. The proper time formulation of gravitational dynamics leads to the correct predictions of gravitational time dilation, the deflection of light and the precession of the perihelia of planets. For this no curvature will be needed. That is, spacetime is flat everywhere, even in the presence of sources of gravitation. Some cosmological consequences will be discussed. The present approach gives a new notion to energy, antiparticles and the structure of spacetime. The contents of the present paper will have important implications for the foundations of physics in general.     

Ambiguity of Perturbative Dirac Theory

Degeneracy of parity even and odd electron solutions of the free Dirac equation may cause uncertainties in first order calculation of the perturbatiove energy.Choosing the even parity solution to start perturbation is though direct,not theoretically well supported.The arbitrariness in choosing lowest order electron wave functions causes nucertainties in the Foldy-Wouthuysen transformations and reduction of the Pauli equation from the Dirac equation.     

Energy Translation and Proper-Time Eigenstates

The usual quantum mechanics describes the mass eigenstates. To describe the proper-time eigenstates, a duality theory of the usual quantum mechanics was developed. The time interval is treated as an operator on an equal footing with the space interval, and the quantization of the spacetime intervals between events is obtained. As a result, one can show that there exists a zero-point time interval.     

Simplectic Geometry and the Canonical Variables for Dirac–Nambu–Goto and Gauss–Bonnet System in String Theory

Using a strongly covariant formalism given by Carter for the deformations dynamics of p-branes in a curved background and a covariant and gauge invariant geometric structure constructed on the corresponding Witten's phase space, we identify the canonical variables for Dirac–Nambu–Goto (DNG) and Gauss–Bonnet (GB) system in string theory. Future extensions of the present results are outlined.     

Proper-Time Relativistic Dynamics on Hyperboloid

The dynamics of a point-like relativistic particle with respect of the proper time is formulated on the hyperboloid p ² ₀–p ²=M ² c ². The Hamilton-Jacobi equations on the hyperboloid are derived for the particles with mass (M ²=m ², m>0), for the particles with zero-mass (M=0, m>0), and for the neutrino. It is shown, in a certain factorization of the momentum, the model can be identified with Nambu's three-dimensional phase space formalism. A first quantized version of the model is formulated according to a canonical scheme of quantization (Schrödinger quantization scheme).     

Proper-Time Relativistic Dynamics and the Fushchych-Shtelen Transformation

Abstract

We report on a new formulation of classical relativistic Hamiltonian mechanics which is based on a proper-time implementation of special relativity using a transformation from observer proper-time, which is not invariant, to system proper-time which is a canonical contact transformation on extended phase-space. This approach does not require the use of time as a fourth coordinate and so we prove that it satisfies the two postulates of special relativity. In the free particle case, our transformation theory generates a Poincaré group which fixes time (system proper-time). We prove that the Fushchych-Shtelen transformation is an element of our group, which fixes time for Maxwell’s equations. In the interaction case, our transformation theory allows us to avoid the no-interaction theorem. We show that the Santilli Isotopes appear naturally when interaction is turned on.     

Behavioral Capital Theory via Canonical Quantization

We show how a behavioral form of capital theory can be derived using canonical quantization. In particular, we introduce quantum cognition into capital theory by applying Dirac’s canonical quantization approach to Weitzman’s Hamiltonian formulation of capital theory, the justification for the use of quantum cognition being the incompatibility of questions encountered in the investment decision-making process. We illustrate the utility of this approach by deriving the capital-investment commutator for a canonical dynamic investment problem.     

Dirac Quantum Field Theory in Rindler Spacetime

The dynamical properties of Dirac particles inRindler spacetime are investigated. It is shown that thevacuum state of the Dirac field in Minkowski spacetimeappears to be a thermal state for a Rindler observer, and the usual thermal equilibriumstate of the Dirac field in Minkowski spacetime is aquasithermal equilibrium state, which is timeindependent and characterized by two quasi-temperatureparameters for a Rindler observer.     

5D Dirac Equation in Induced-Matter Theory

According to an induced-matter approach, Liu and Wesson obtained the rest mass of a typical particle from the reduction of a 5D Klein–Gordon equation to a 4D one. Introducing an extra-dimension momentum operator identified with the rest mass eigenvalue operator, we consider a way to generalize the 4D Dirac equation to 5D. An analogous normal Dirac equation is gained when the generalization reduces to 4D. We find the rest mass of a particle in curved space varies with spacetime coordinates and check this for the case of exact solitonic and cosmological solution of the 5D vacuum gravitational field equations.     

PT-Symmetry, Canonical Decomposition, Perturbation Theory

Let H be any PT-symmetric Schrödinger operator of the type H=- ² +x ² +igW(x), where W is a real polynomial, odd under reflection of all coordinates, gR, acting on L ² ( R ^d ). The proof is outlined of the following statements: PH is self-adjoint and its eigenvalues coincide with the eigenvalues of (H*H). Moreover the eigenvalues of (H*H), known as the singular values of H, can be computed via perturbation theory by Borel summability.     

Dirac Formalism for Arbitrary Spin in S-Matrix Theory

The S-matrix formalism is built up from the beginning, for particles with and without mass, through systematic use of the representation D_s,0 D_0,s of the homogeneous Lorentz group. Only concepts referring to space and time are considered. Within the framework of the formalism, the close interdependence between CPT, spin and statistics, crossing symmetry and unitarity is explicitly emphasized. The concepts of intrinsic parity, spin-flip and particle conjugation are re-examined. A comparison is made with conventional field-theoretic formalism.     

Theory of superconductivity for Dirac electrons in graphene

Phonon-exchange-induced superconducting pairing of effectively ultrarelativistic electrons in graphene is investigated. The Eliashberg equation obtained for describing pairing in the Cooper channel with allowance for delayed interaction are matrix equations with indices corresponding to the valence and conduction bands. The equations are solved in the high doping limit, in which pairing is effectively a single-band process, and in the vicinity of a critical quantum point of underdoped graphene for a value of the coupling constant for which pairing is an essentially multiband process. For such cases, analytic estimates are obtained for the superconducting transition temperature of the system. It is shown that the inclusion of dynamic effects makes it possible to determine the superconducting transition temperature, as well as the critical coupling constant for underdoped graphene, more accurately than in the static approximation of the BCS type. Estimates of the constants of electron interaction with the scalar optical phonon mode in graphene indicate that an appreciable superconducting transition temperature can be attained under a high chemical doping level of graphene.     

带边界条件复标量场的正则量子化

研究了带边界条件有质量复标量场的量子化. 与把边界条件当作Dirac约束方法不同, 我们在经典解空间研究这个问题, 利用Fadeev-Jackiw(FJ)方法获得所有傅里叶模的对易关系, 避免用Dirac方法而产生的问题.     

On the Grand Canonical Gibbs Ensemble in Euclidean Field Theory

By generalizing the integration by parts formula on the function space, the isomorphism between the general class of selfinteracting Euclidean, Bose fields and classical gases is proven in a nonperturbative way. Rigorous, non-perturbative derivations of the Mayer equations and Bogoliubov-Born-Kirkwood-Grenn-Yvone hierarchical equations are given. Expansions of several physical quantities are discussed in terms of the linear graphs, for some general classes of Euclidean fields.     

The Quantum R-Matrix via Canonical Quantization in the Liouville Theory

Starting from the classical Liouville theory, we study its quantum theory through canonical quantization. We find that if the Poisson bracket relations between two vectors are,dominated by the classical r-matrix in the classical case, their quantum analogue is replaced by the exchange relations dominated by the quantum R-matrix. The quantum group structure in the quantum LiouviUe theory is studied and the central charge of the quantum Liouville theory is also obtained.     

Unambiguous Quantization from the Maximum Classical Correspondence that Is Self-consistent: The Slightly Stronger Canonical Commutation Rule Dirac Missed

Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which can manifest negative densities in phase space.