Interaction in the geometro-differential conception of extended particles and the Galilei semigroup of trajectories

Along the lines of a previous work, the geometrical structure of Hibert bundles describing extended quantum free particles is repeated with Galilei external and internal independent symmetries. Then, in order to introduce the interaction, this structure is extended by replacing configuration and momentum spaces by the socelled spaces of trajectories and extended velocity boosts, respectively. These provide representations giving the probability amplitudes for the particle to follow certain trajectories. The interaction can be introduced in the transformation law from functions on the space of trajectories (free dynamics) to functions on spacetime (intracting dynamics). This transformation law, which makes use of a universal distribution, is seen as a functional in our work according to a quantum functional theory which generalizes the ideas of de Broglie. Intertwining of induced representations gives the free propagator in the space of trajectories and, henceforth, the propagator with interaction in space-time for the extended particle.     

THE REPRESENTATION THEORY in QUANTUM MECHANICS with NEGATIVE METRIC

In this paper the coherent state for the negative metric boson is established, which, originating from its particle number representation provides a basis for constructing its coordinate and nomenturn representations in the Euclidean space. However,it is demonstrated by us that these two representations do not exist in the Minkowski space. Thus this formulation provides a physical foundation for the path integral quantiration theory of the negative metric boson in the Euclidean space.     

Thermal effects for an accelerating observer

An observer accelerating through empty Minkowski space appears to find himself in a heat bath. To establish this we translate a Minkowski propagator into his language and find a temperature propagator. An interpretation in terms of path integrals is given.     

Renormalization and scaling behavior of non-Abelian gauge fields in curved spacetime

In this article we discuss the one loop renormalization and scaling behavior of non-Abelian gauge field theories in a general curved spacetime. A generating functional is constructed which forms the basis for both the perturbation expansion and the Ward identities. Local momentum space representations for the vector and ghost particles are developed and used to extract the divergent parts of Feynman integrals. The one loop diagram for the ghost propagator and the vector-ghost vertex are shown to have no divergences not present in Minkowski space. The Ward identities insure that this is true for the vector propagator as well. It is shown that the above renormalizations render the three- and four-vector vertices finite. Finally, a renormalization group equation valid in curved spacetimes is derived. Its solution is given and the theory is shown to be asymptotically free as in Minkowski space.     

On the spectra of noncommutative 2D harmonic oscillator

The spectra and wave functions of the 2-dimensional harmonic oscillator in a noncommutative plane are revised by using the path integral formulation in coordinate space and momentum space, respectively. We perform the path integral formulation in coordinate space first. Then we study this problem in momentum space. The propagator is computed both in coordinate space and in momentum space. The modification due to noncommutativity of eigenvalues and eigenfunctions is studied. Both the small and large noncommutative parameter limits are discussed. PACS 11.10.Ef     

On External and Internal Properties of Extended Elementary Objects

The physical interpretation of induced representation intertwining as a process of materialization or localization is extrapolated to mappings (which are not intertwinings) between configuration and momentum representations. Propagation of extended particles composed of an external and an internal mode is a combination of two generalized materializations and two generalized localizations. Our aim is to submit, in the spinless case, the idea that mappings from external representations to internal ones are possible alternatives, probability amplitudes of which must be summed up in the whole propagator. However, it turns out that they lead to different final states and then correspond to 64 different propagators and not to alternatives of the same propagation. In addition to being transitions from one location to another, these propagations contain processes of interconversion between internal and external modes which are responsible for creation and annihilation of the extended particle. The physical interpretations are striking, while the mathematical stricture is poor. Inconsistencies of the model are outlined.     

Relativistic quantum mechanics of supersymmetric particles

The quantum mechanics of an electron in an external field is developed by Hamiltonian path integral methods. The electron is described classically by an action which is invariant under gauge supersymmetry transformations as well as worldline reparametrizations. The simpler case of a spinless particle is first reviewed and it is pointed out that a strictly canonical approach does not exist. This follows formally from the gauge invariance properties of the action and physically it corresponds to the fact that particles can travel backwards as well as forward in coordinate time. However, appropriate application of path integral techniques yields directly the proper time representation of the Feynman propagator. Next we extend the formalism to systems described by anticommuting variables. This problem presents some difficulty when the dimension of the phase space is odd, because the holomorphic representation does not exist. It is shown, however, that the usual connection between the evolution operator and the path integral still holds provided one indludes in the action the boundary term that makes the classical variational principle well defined. The path integral for the relativistic spinning particle is then evaluated and it is shown to lead directly to a representation for the Feynman propagator in terms of two proper times, one commuting, the other anticommuting, which appear in a symmetric manner. This representation is used to derive scattering amplitudes in an external field. In this step the anticommuting proper time is integrated away and the analysis is carried in terms of one (commuting) proper time only, just as in the spinless case. Finally, some properties of the quantum mechanics of the ghost particles that appear in the path integral for constrained systems are developed in an appendix.     

Solving the BSE with an Integral Representation in Minkowski Space

Special method of solution of BSE in Minkowski space is presented. Perturbation theory integral representation is introduced for BSE and the applicability is discussed beyond the simplest kernel and constituent propagator approximation.     

Feynman's approach to quantum mechanics: Trajectories,commutation relations and uncertainty principle

Summary In this paper the meaning of trajectory for a quantum-mechanical particle is discussed, starting from the path integral expression of the propagator. By a direct method the trajectories mostly contributing to the total amplitude are found, but it seems impossible to interpret them as paths in the physical space-time; on the contrary, the position-momentum commutation relations directly follow. Moreover, we show that the Heisenberg uncertaint principle can be obtained from the path integral approach. In order to give a better understanding of the characteristic quantum-mechanical features and of the difference from the classical problems, the diffusion equation for a Brownian particle is considered in the first part of the paper. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

Worldline path integrals for a Dirac particle in a weak gravitational plane wave

The problem of a relativistic spinning particle interacting with a weak gravitational plane wave in (3+1) dimensions is formulated in the frame work of covariant supersymmetric path integrals. The relative Green function is expressed through a functional integral over bosonic trajectories that describe the external motion and fermionic variables that describe the spin degrees of freedom. The (3+1) dimensional problem is reduced to the (1+1) dimensional one by using an identity. Next, the relative propagator is exactly calculated and the wave functions are extracted. PACS 04.30.-w; 03.65.Ca; 03.65.Db; 03.65.Pm     

Path Integral Formulation of Fractionally Perturbed Lagrangian Oscillators on Fractal

Fractional path integration and particles trajectories in fractional dimensional space are motivating issues in quantum mechanics and kinetics. In this paper, a fractional path integral characterized by a fractional propagator is developed based on the framework of the fractional action-like variational approach. A fractional generalization of the free particle problem is found, the corresponding fractional Schrödinger equation is derived and a fractional path integral formulation of harmonic oscillators characterized by a perturbed Lagrangian is constructed after reducing the fractional action to an integral action on fractal. The new fractal-like path integral offers a number of motivating features which are discussed and analyzed. The main outcome is connected to the possibility of constructing on a fractal a path integral for the oscillators characterized by modified ground energy. In particular for low-temperature case, the fractional perturbed oscillator is characterized by a free energy larger than the standard value \( E_{0} = {{\hbar \omega } \mathord{\left/ {\vphantom {{\hbar \omega } 2}} \right. \kern-0pt} 2}.\) Such an increase in the ground energy generalizes the uncertainty principle without involving differentiable paths or even invoking new phenomenological theories based on deformed algebra.     

Distributions on Minkowski space and their connection with analytic representations of the conformal group

Unitary analytic representations of the conformal group are relized on Hilbert spaces of holomoprhic or antiholomorphic functions over a tube domain in complex Minkowski space. The distributional boundary values of these functions are tempered distributions on real Minkowski space. The representations are characterized by an integral scale dimension labeln and two spin labelsj ₁ andj ₂. The connection between the dimensionn and the degree of singularity of the tempered distribution is investigated. We propose an application to inclusive reactions of elementary particles.     

Study of the gluon propagator in the axial gauge

It is shown that a non-linear integral equation for the gluon propagator in the axial gauge (Baker et al.) can be simplified considerably. A comparison is made with an approximate equation for the gluon propagator in the Landau gauge (Mandelstam). Both equations have polynomial kernels where the argument is the divisor of the internal and external momenta. A solution which behaves as a double pole for low momenta remains consistent.     

The Polyakov path integral over bordered surfaces

Following the general scheme of the convariant path integral quantization of gauge systems, two alternative formulations of the first quantized closed bosonic string in a position representation are presented. In both approaches the covariant path integral representations of the propagator and of the higher order off-shell amplitudes are constructed. For a wide class of gauges the explicit formulae for off-shell amplitudes are obtained. This paper is the continuation of our previous work where the corresponding problems in the open string case were considered [20].     

Exact propagator of the harmonic oscillator with a time-dependent mass

By finding a space and time transformation, the exact evaluation of the propagator for the harmonic oscillator with a time-dependent mass by the path integral method becomes possible. We then derive the wavefunctions from the propagator obtained. Finally, the propagator beyond and at caustics is evaluated by using its modified semi-group property and is confirmed by investigating the classical paths with two fixed end-positions.     

Implications of Analyticity to the Solution of Schwinger-Dyson Equations in Minkowski Space

We discuss some recent developments in nonperturbative studies of quantum field theory (QFT) using the Schwinger-Dyson equations formulated directly in Minkowski space. We begin with the introduction of essential ideas of the integral representation in QFT and a discussion of the renormalization in this approach. The technique based on the integral representation of Green’s functions is exploited to solve Schwinger-Dyson equations in several models of quantum field theory, e.g., in scalar models and in strong coupling QED₃₊₁ in the quenched and in the unquenched approximation. The phenomenon of dynamical chiral symmetry breaking in regularized theory is touched. In QCD, the analyticity of the gluon propagator on the complex momentum square plane is exploited to continue some recent lattice data to the timelike momentum axis. We find a contribution to the non-positive absorptive part in the Landau-gauge gluon propagator which is in agreement with some other new recent analyses.     

Semiclassical propagator of the Wigner function

Propagation of the Wigner function is studied on two levels of semiclassical propagation: one based on the Van Vleck propagator, the other on phase-space path integration. Leading quantum corrections to the classical Liouville propagator take the form of a time-dependent quantum spot. Its oscillatory structure depends on whether the underlying classical flow is elliptic or hyperbolic. It can be interpreted as the result of interference of a pair of classical trajectories, indicating how quantum coherences are to be propagated semiclassically in phase space. The phase-space path-integral approach allows for a finer resolution of the quantum spot in terms of Airy functions.     

The propagator of the Calogero-Moser system in an external quadratic potential

We obtain the Hamilton operator of the Calogero-Moser quantum system in an external quadratic potential by conjugating the radial part for the action of SO(n) by conjugacy of the Hamilton operator of the quantum harmonic oscillator on the Euclidean vector space of real symmetric matrices. Then, with Mehler's formula, we derive the propagator of the problem. We also investigate some schemes to change the interaction constant. For two-particle systems, we obtain explicit formulae, whereas for many-particle systems, we reduce the computation of the propagator to finding a definite integral. We give also the short time approximation, the energy levels and the trace of the propagation operator.