Free Relativistic Particle Research Articles

Representation of spacetime diffeomorphisms in canonical geometrodynamics under harmonic coordinate conditions

Using a method developed by Isham and Kuchar (1985), the Lie algebra of the spacetime diffeomorphism group is mapped homomorphically into the Poisson algebra of dynamical variables on the phase space of canonical geometrodynamics. To accomplish this the phase space must first be extended by including embedding variables and their conjugate momenta. Also, the relationship between the embeddings and the spacetime metric must be limited by coordinate conditions. This is done by adding a source term to the Hilbert action of general relativity in such a way that the standard vacuum Einstein equations result when harmonic conditions are imposed as a supplementary set of constraints. In the process the usual super-Hamiltonian and supermomentum constraints of geometrodynamics become temporarily altered, but they are restored to their familiar form at the end by imposing the harmonic constraints, constraints that are preserved in the dynamical evolution generated by the total Hamiltonian. It is shown that the same method can also be applied to other generally covariant systems, namely the free relativistic particle and the bosonic string.

On the Bose-Einstein condensation of free relativistic bosons with or without mass

The Bose-Einstein condensation of free relativistic particles [e=(M 2 c 4 +c 2 p 2 ) 1/2 −Mc 2 ] is studied rigorously. For massless bosons (e=cp), the condensation transition of third (second) order occurs in2 (3) dimensions (D). The molar heat capacity follows the T 2 (T 3 ) law below the condensation temperature Tc [k B Tc=(2πħ 2 c 2 n/1.645) 1/2 [(π 2 ħ 3 c 3 n/1.202) 1/3 ], reaches4.38 (10.8) R at T=Tc, and approaches the high-temperature-limit value2 (3) R with no jump (a jump equal to6.75R) in2 (3)D. For finite-mass (M) bosons, the phase transition occurs only in3D with the condensation temperature Tc always smaller than that of the corresponding nonrelativistic bosons [e=(2M) −1 p 2 ]. If the mass M is reduced to zero, the condensation temperature Tc grows monotonically and reaches eventually that of massless relativistic bosons. This mass-dependence of Tc is therefore distinct from the case of nonrelativistic bosons, where Tc grows to infinity as M →0. A brief discussion is given for a possible connection with the normal-to-super transition of the independently moving Cooper pairs (bosons).

Geometric construction of the measure: Minisuperspace quantum gravity and the relativistic particle compared.

We examine the Polyakov functional-integral quantization of a minisuperspace model for quantum gravity using the relativistic free particle as a guideline. While formally the two models are very similar, there are differences which we expect have a genuine physical significance.

Quantum mechanics of relativistic particles in multiply connected spaces and the Aharonov-Bohm effect

The authors consider the motion of free relativistic particles in multiply connected spaces. They show that if one of the spatial dimensions has the topology of a circle then the D-dimensional spacetime is compactified to D-1 dimensions and the particle mass increases by an amount which is proportional to a quantum phase factor and inversely proportional to the radius of the circle. This provides a new non-perturbative mechanism of mass generation. They also consider the relativistic Aharonov-Bohm effect and they show that the propagator is similar in form to the non-relativistic one. The propagators are calculated without any approximation in both situations.

Covariant canonical quantization of the relativistic free particle top.

Becchi-Rouet-Stora-Tyutin (BRST) quantization of the relativistic free particle top has been carried out as a first step to understand the theory of the string top, in which Weyl invariance is respected at the classical level. Unitarity and consistency of this quantum theory has been shown through the conservation and the nilpotency of the BRST charge. This statement is shown to be valid in any dimension of spacetime.

Symplectic and nonsymplectic realizations of groups for singular systems. An example

The actions of groups for singular systems are studied in the framework of the theory of canonical transformations for presymplectic systems. Symplectic realizations as well as nonsymplectic ones arise in a natural way. As a typical example we construct the Poincare realizations for the relativistic free massive particle.

Elementary particles as solutions of the Sivashinsky equation

Stationary spherically symmetric solutions of the Sivashinsky equation (1/ c 2) S 2 t −(∇ S) 2)= m 2 c 2+ α ℏ □ S+ ( β ℏ 3/ m 2 c 2)□ 2 S > are looked for. The problem is reduced to finding all admissible solutions y= y( r) of the ODE [ d 2/ dr 2+(2/ r) d/ dr+1]( d/ dr+2/ r) y=- y 2, i.e., solutions which are defined on the half-line 0≤ r≤ ∞ and satisfy y(0)= y(∞)=0. It is proved that for each y'(0) ϵ [-0.34, 0.33] the solution of the initial value problem with y(0)= y″ (0)=0 is admissible. In terms of the action function S these solutions are interpreted as free relativistic particles. The proof of the result employs analytic asymptotic expansion near r=0 and r=∞ and a new technique of rigorous estimates by a computer. The proved interval of existence y'(0) ϵ [-0.34, 0.33] nearly coincides with the one obtained by numerical experiments.

On the BRS’s

After an analysis of the concept of Lagrangian gauge fixing, it is shown that the arbitrariness in the parametrization of gauge transformations gives rise to a whole family of classical BRS transformations. This is explicitly shown for the free-relativistic particle. Two inequivalent classes of BRS Lagrangians are defined. While the former generates a Kato–Ogawa-like Hamiltonian BRS formalism, the latter gives rise to the Batalin–Fradkin–Vilkovisky theory. A comparison is made between these Hamiltonian theories, the multitemporal description of 1st-class constraints, the Konstant–Sternberg and Loll approaches, and the Bonora–Cotta Ramusino interpretation of ghosts. The relevance of an equivariance condition for the BRS observables is shown. The quantum BRS theory is briefly discussed.

Antibracket—antifield formalism for constrained hamiltonian systems

Abstract The detailed comparison between the hamiltonian approach to the BRST symmetry and the antibracket—antifield formalism is carried out in the case of first order action principles. This is done by explicitl expressing the solution of the master equation in terms of the hamiltonian BRST charge. Complete agreement is found between both formalisms. This conclusion holds for irreducible and reducible constrained systems. The example of the free relativistic particle is considered.

BRST quantization of free massless relativistic particles of arbitrary spin

A BRST quantization of free massless relativistic particles of arbitrary spin is performed. Explicit covariant wave equations are derived for any spin, which are gauge invariant equations. The spinning particles are described by pseudoclassical gauge models in which the spin variables are odd Grassmann numbers. For a spin s particle the gauge group is an O(2 s) extended supersymmetrized reparametrization group. The model has several special features for s⩾1: It cannot be canonically quantized, which makes the solutions to the BRST condition noncanonical. This in turn allows for the peculiarity that there are two equivalent solutions with different ghost numbers.

The propagator of a free relativistic particle in a generic gauge dλ/dτ=f(λ)

The authors compute the propagator of a free spinless relativistic particle in a generic relativistic gauge d lambda d tau =f( lambda ). A criterion is given to decide whether a particular gauge fixing is valid or not.

Conformal O(3,2) symmetry of the two-dimensional inverse square potential

It is shown that the dynamical systems describing a point mass moving in a repulsive inverse square potential in a plane and a free relativistic massless particle are isomorphic to each other. The obvious conformal invariance of the massless particle appears as a hidden dynamical symmetry of the inverse square potential.

Berry's phase in the relativistic theory of spinning particles.

We show that the change of the phase of the wave function of a free relativistic spinning particle under the action of the Poincar\'e group is described by a certain universal connection in momentum space. This connection determines the parallel transport of the spin during the motion of the particle through external fields. In the special case of photons we reproduce the formula for Berry's phase calculated recently by Chiao and Wu.

Path integrals in parametrized theories: The free relativistic particle.

The connection between the canonical and the path-integral formulations of the quantum mechanics of a free relativistic particle is discussed as a model of theories in which time is one of the dynamical variables (parametrized theories). Several features central to the canonical formulation, such as the choice of Hilbert space, are reflected in the measure of the sum over paths and especially in the range of integration.

Toward the variational treatment of spin‐orbit and other relativistic effects for heavy atoms and molecules

AbstractObservation of trends in computed spin‐orbit splittings for relatively light molecules leads to the conclusion that relativistic corrections to the electronic charge distribution are important when treating molecules containing heavy atoms (Z > 18). In order to preserve the nature of the successful computational techniques currently applied to light molecules in so far as possible, particularly to allow for the treatment of correlation effects in an efficient CI procedure on an equal footing with relativistic effects, emphasis is placed on the development of a two‐component formalism for this purpose. A first attempt in this direction consists of formulating a spin‐free quantum mechanical operator that reflects relativistic kinematics. The mass‐velocity term in the Breit‐Pauli Hamiltonian is not appropriate for a variational treatment, however, since it drastically alters the spectrum and gives results that are not bounded from below. To avoid this problem the relativistic free particle energy has been used directly for the representation of the kinetic energy, and in addition the Darwin term has been included as a correction to the potential energy. This approach can be justified with reference to the Foldy–Wouthuysen reduction of the Dirac equation, but the class of basis functions used in a variational procedure with this Hamiltonian must be restricted to avoid the formation of a node in the wavefunction at the nucleus; the same problem is circumvented in the Cowan–Griffin method by imposing Dirac boundary conditions on the wavefunction. With this method, accurate spin‐orbit splittings have been computed for Br, I, Xe+, CBr, and XeF, but the resulting total energies are found to be overly sensitive to the representation of the inner shells of these systems. Improved results for both valence and inner shells are then shown to follow from the use of the no‐pair equation, which provides a variationally tractable two‐component method employing a momentum dependent potential that gives a realistic description of relativistic effects for atoms and molecules over a suitably large range of Z.

Klein-Gordon propagator via first quantization

We quantize the classical mechanics of a free spinless relativistic point particle via Feyman's path integral prescription to obtain the Klein-Gordon propagator.

Equations of minimal manifolds.

The equations of the minimal submanifolds embedded in a higher-dimensional space are presented. As is well known, the familiar relativistic free particle and the dual string, which provides a dynamical model of hadrons, are described by this sort of equation. We show that the equations constitute a vector normal to the submanifold and reduce to a form universal irrespective of the dimension of the submanifolds. In particular, for embedding into a flat space, an interesting theorem is proved concerning the independent variables of the equations.

Theory of optical transitions in inversion layers of narrow-gap semiconductors

The theory of electric sub-bands in MOS structures involving InSb-type narrow-gap semiconductors is developed and used to resolve a long-standing problem concerning a possibility of inter-sub-band resonance excitations with light polarisation parallel to the interface. k.p p theory is formulated for electrons in inversion layers, taking into account the main features of the InSb-type band structure and assuming a triangular model potential near the interface. It is shown that the interband k.p coupling and the small value of the gap make it possible to excite inter-sub-band optical transitions with light polarisation parallel to the interface. The transition probability is proportional to k/sub ///2, where k/sub /// is the electron wavevector parallel to the interface. Numerical estimations are carried out using variational functions for electric sub-bands. For typical experimental conditions in InSb MOS structures one obtains the optical transition probability for parallel light polarisation to be a few percent of that for the transverse one. Both the k/sub ///2-dependence and the numerical values agree with existing experiments. The possibility of excitation with parallel light polarisation is interpreted in terms of an analogy between free relativistic particles and electrons in narrow-gap semiconductors.

Generating functionalZ0for the one-wormhole sector

We proceed in constructing a quantum theory of wormholes by adapting the method of Gervais, Jevicki, and Sakita. We calculate the transition amplitude for one-wormhole processes and separate the motion of the wormhole in the tree approximation: it is the motion of a free relativistic particle with the mass $\frac{m}{G}$, the classical mass of the wormhole. The quantum corrections can be calculated from the generating functional $Z$ for the Green's functions. We derive a formula for the zero approximation ${Z}_{0}$ to this functional, find a suitable gauge which simplifies the formula, and show that this gauge is compatible with our boundary conditions. The problem of zero modes is solved and a regular propagator is shown to exist. Finally, we estimate the cross section for the capture of photons and gravitons by a wormhole and find that it increases with energy reaching the value ${(4m)}^{2}$ asymptotically.

The mass-shell and gauge-fixing conditions for the free relativistic massive quantum particle

Working in the context of the Weyl group, which describes off-mass-shell relativistic particles, we impose “gauge-fixing” constraints involvingR0,R+, andD as matrix element conditions to be satisfied by the on-mass-shell states of a massive particle. We evaluate the matrix elements inp-space using five sets of co-ordinates: (p2,p), (p2,p+,pT), (p2,p−,pT), (p2,π), and (p2,π+,πT) where\(\pi ^\mu \equiv p^\mu /(p^2 )^{\tfrac{1}{2}} \). We find that, only in the case ofR0 with (p2,p) coordinates,R+ with (p2,p+,pT) coordinates, andD with (p2, π) or (p2,π+,πT) coordinates, can the condition be satisfied by arbitrary on-mass-shell states. In all other cases, the condition can be satisfied only by states belonging to a subset of subspaces of the on-mass-shell Hilbert space, i.e it forces a violation of the superposition principle. These results constitute thep-space quantum version of Shanmugadhasan's theorem for constrained classical systems which states that there exists, at least locally in phase space, a canonical transformation to a set of variables in which the second-class constraints become canonical pairs equal to zero with the other canonical coordinates independent of the second-class constraints.