An eikonal perturbation theory having applications in hadron dynamics. II. Massive quantum electrodynamics

An eikonal perturbation theory (EPT), derived in previous work for a superrenormalizable coupling, is here developed for massive quantum electrodynamics (MQED) involving scalar or spinor matter fields minimally coupled to neutral massive vector gluons. After summarizing the functional method, we present the EPT for the external field problem. In agreement with results known within ordinary perturbation theory (OPT) in the eikonal approximation (EA), from an exact eikonal equation derived here we show that the EPT for the external field problem provides an excellent approximation method for Green's functions at large momenta. We then discuss some general features of the EPT for MQED, and show that it leads to a renormalizable approximation method. Our approach is then illustrated by deriving explicit expressions for various renormalized Green's functions in lowest order EPT. We also discuss some asymptotic properties of such Green's functions and indicate how to proceed with calculations in higher orders. As in our previous work, we again find that the renormalization procedure in EPT bears close resemblance to the one for OPT. Contrary to what happens with the EA, the inclusion of self-interactions and of other field-theoretic effects does not spoil the virtues of the EPT as a far better high-momenta approximation than the OPT. As a typical example, if s is an energy parameter and g the coupling constant with $\text{g}{}^2\text{4π}\text{< 1}$, OPT to order g²ⁿ often fails to be a good approximation as soon as $\text{(}\text{g}{}^2\text{4π}\text{)}\text{ln}\text{s ～ 1}$, while in such cases EPT to order g²ⁿ is still a good approximation as long as $\text{(}\text{g}{}^2\text{4π}\text{)}{}^{\mathrm{n+1}}\text{ln}\text{s < 1}$. We also find that the EPT is superior to the EA in that, contrary to the EA, it provides a step-by-step rigorous and renormalizable iterative approximation method which can account for self-interactions and other field-theoretic effects. We emphasize that the EPT is much simpler and more general than other explicit approximate summation methods of classes of OPT Feynman graphs.In field theory, we consider the use of the EPT as a generalization of the EA for discussing, e.g. high-energy behaviors in MQED as well as infrared divergence and bound-state problems in the limit of massless gluons. It is also suggested that, in view of its nice field-theoretic and high-energy properties, the EPT for MQED might provide a useful laboratory where ideas and problems in hadron dynamics could be meaningfully investigated within a Lagrangian field theory.     

High orders of the perturbation theory in scalar electrodynamics

We calculate the asymptotically large order terms for the perturbative expansion of the Green functions in scalar electrodynamics.     

Two-dimensional quantum electrodynamics as a model in the constructive quantum field theory

We investigate a transformation of two-dimensional quantum electrodynamics ((QED)₂)-type models into sine-Gordon models in the constructive method.     

Quasiclassical perturbation theory for quantum K-systems

A quantum-dynamical system satisfying the condition of stochastic instability in the classical limit is studied. Behavioural properties of perturbation-theory series in powers of ¢ for such systems are discussed. The quantum corrections in the expressions for physical averages over the density matrix are shown to increase exponentially at small times.     

Quantum electrodynamic perturbation theory based on electrodynamics with an external field

A method is proposed for construction of mean values of quantum quantities in quantum optics and electrodynamics on the basis of exact solutions of the corresponding problems of electrodynamics with an external field. To illustrate the method the mean energy of a two-level atom in the Raby problem is calculated. It is shown that the mean value obtained with a specified accuracy coincides with the result of exact quantum electrodynamic calculation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 7–10, September, 1983.     

Alternative analysis to perturbation theory in quantum mechanics

We develop an alternative approach to the time independent perturbation theory in non-relativistic quantum mechanics. The method developed has the advantage to provide in one operation the correction to the energy and to the wave function; additionally we can analyze the time evolution of the system for any initial condition, which may be bothersome in the standard method. To verify our results, we apply our method to the harmonic oscillator perturbed by a quadratic potential. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained.     

Hopf algebra primitives in perturbation quantum field theory

The analysis of the combinatorics resulting from the perturbative expansion of the transition amplitude in quantum field theories, and the relation of this expansion to the Hausdorff series leads naturally to consider an infinite dimensional Lie subalgebra and the corresponding enveloping Hopf algebra, to which the elements of this series are associated. We show that in the context of these structures the power sum symmetric functionals of the perturbative expansion are Hopf primitives and that they are given by linear combinations of Hall polynomials, or diagrammatically by Hall trees. We show that each Hall tree corresponds to sums of Feynman diagrams each with the same number of vertices, external legs and loops. In addition, since the Lie subalgebra admits a derivation endomorphism, we also show that with respect to it these primitives are cyclic vectors generated by the free propagator, and thus provide a recursion relation by means of which the (n+1)-vertex connected Green functions can be derived systematically from the n-vertex ones.     

Feynman integral and perturbation theory in quantum tomography

We present a definition for tomographic Feynman path integral as representation for quantum tomograms via Feynman path integral in the phase space. The proposed representation is the potential basis for investigation of Path Integral Monte Carlo numerical methods with quantum tomograms. Tomographic Feynman path integral is a representation of solution of initial problem for evolution equation for tomograms. The perturbation theory for quantum tomograms is constructed.     

On the stationary perturbation theory in quantum mechanics

A simple method is proposed for solving the Shcrödinger equation in the presence of a perturbation. Formally exact expressions are obtained in the form of infinite series for the energies and wave functions without assuming that the perturbation is small. Under the additional assumption that it is small (in accordance with a well-defined criterion of smallness) the obtained results yield directly the results of Schrödinger's perturbation theory. The developed approach contains in compact form various formulations of perturbation theory. The calculations use neither the complex technique of projection operators nor the complicated formalism of Green's functions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 33–36, November, 1980.     

Constant-density thermodynamic perturbation theory for quantum fluids

A perturbation series is given which expresses the equilibrium thermodynamic quantities of a quantal fluid in terms of that of another (unperturbed) fluid having the same temperature and number density ρ. The general term of each series can be given; it is the integral of a product involving either the grand-canonical correlation functions of the unperturbed fluid or its fundamental probability distribution W^c_N(1,…,N) for sets of free particles and the modified Ursell-functions of the quantal fluid; each integral can be conveniently represented in the language of graphs.     

“Minus c” symmetry in classical electrodynamics and quantum theory

A symmetry associated with the inversion of the speed of light is considered.     

Convergent perturbation expansions for Euclidean quantum field theory

Mayer perturbation theory is designed to provide computable convergent expansions which permit calculation of Greens functions in Euclidean quantum field theory to arbitrary accuracy, including nonper-turbative contributions from large field fluctuations. Here we describe the expansions at the example of 3-dimensional ⁴-theory (in continuous space). They are not essentially more complicated than standard perturbation theory. Then ^th order term is expressed in terms ofO(n)-dimensional integrals, and is of order ^k if 4k–3n4k.Dedicated to the memory of Kurt Symanzik     

Infrared problems in quantum electrodynamics

We present a self-contained treatment of the infrared problem in Quantum Electrodynamics. Our program includes a derivation and proof of finiteness of modified reduction formulae for scattering in Coulomb potentials and unitary extensions of the relativistic Coulomb amplitudes in the forward direction. The renormalization structure of the theory is discussed in connection with the infrared problem and the renormalization group is reconsidered and shown to be inadequate for the “improvement” of perturbation theoretic results. However, simple forms of the renormalization group equations are easily established, which allow for a simple discussion of the renormalization structure and the extraction of physical quantities out of Green functions normalized at an arbitrary mass μ < m (m is the fermion mass). As an example of such a quantity we consider the construction of a renormalized and infrared finite mass-operator in presence of external fields. Scattering theory in Quantum Electrodynamics is elaborated in the context of the coherent state formulation of the asymptotic condition. Dimensional regularization techniques are systematically used for the reduction of coherent states and the construction of S-matrix elements and the cross-section formulae. The latter are obtained in a relatively simple form, which allows for a direct comparison with the exact cross-section formulae derived in the traditional context. This establishes the equivalence of the two approaches at the cross-section level. Various applications illustrate the techniques presented here and relative topics are discussed.     

Critical charge in quantum electrodynamics

The critical nuclear charge Z _cr and the critical distance R _cr in the system of two colliding heavy nuclei—they are defined as those at which the ground-state level of the electron spectrum descends to the boundary of the lower continuum, with the result that beyond them (that is, for Z>Z _cr or R<R _cr) spontaneous positron production from a vacuum becomes possible—are important parameters in the quantum electrodynamics of ultrastrong Coulomb fields. Various methods for calculating Z _cr and R _cr are considered, along with the dependence of these quantities on the screening of the Coulomb field of a nucleus by the electron shell of the atom, on an external magnetic field, on the particle mass and spin, and on some other parameters of relevance. The effective-potential method for the Dirac equation and the application of the Wentzel-Kramers-Brillouin method to the Coulomb field for Z>137 and to the two-body Salpeter equation for the quark-antiquark system are discussed. Some technical details in the procedure for calculating the critical distance R _cr in the relativistic problem of two Coulomb centers are described.     

Stationary perturbation theory in the probability representation of quantum mechanics

In the probability representation of quantum mechanics, the eigenvalue problems in Hilbert space appear as *-genvalue equations. We show the possibility of employing the nondegenerate stationary perturbation method in the probability representation of quantum mechanics. The perturbed eigentomograms and the eigenvalues of energy are shown to be computed ab initio in terms of tomographic symbols of the operators involved.