Borel summation of perturbation series in quantum mechanics and field theory

The Borel summation of factorial divergent perturbation series is considered. The relation between the asymptotics of the coefficients in the series and the character of the exact function singularity is established. The applicability limits for the improved perturbation theory are obtained. The results are tested on a number of physical problems for which the exact solutions are available.     

On the summation of divergent perturbation series in quantum mechanics and field theory

The possibility of recovering the Gell-Mann–Low function in the asymptotic strong-coupling regime by known first-order perturbation-theory (PT) terms β_n and their asymptotics as \(\tilde \beta _n \) as n → ∞ is investigated. Conditions are formulated that are necessary for recovering the required function at the physical level of rigor: (1) a large number of PT coefficients are known whose asymptotics has already been established, and (2) there is no intermediate asymptotics. Higher orders of PT, their asymptotic behavior, and power corrections are calculated in quantum mechanical problems that involve divergent PT series (including series for a funnel potential, the ? ₍₀₎ ⁴ model, and the Stark effect in a strong field). The scalar field theory ? ₍₄₎ ⁴ is considered in the \(\overline {MS} \) and MOM regularization schemes. It is shown that one cannot make any definite conclusion about the asymptotics of the Gell-Mann–Low function as g → ∞ on the basis of information available for the above theory.     

Gauge invariance and gauge independence of the S-matrix in nonrelativistic quantum mechanics and relativistic quantum field theories

The gauge independence of transition rates as opposed to the gauge invariance of the equations of motion and gauge dependence of operators and state vectors is critically examined and explicitly demonstrated, both in nonrelativistic quantum mechanics and quantum field theory. Time independent as well as time dependent gauge transformations are explicitly analyzed using several techniques in order to clarify the physical content and significance of gauge independence and the conditions for its applicability.     

Perturbation theory at large order in reggeon field theory

We estimate the behaviour of the perturbation expansion and the ? expansion at large orders in reggeon field theory. The perturbation expansion is divergent but Borel summable for α₀ < 1. At fixed total rapidity, the Borel summability extends into α₀ > 1.     

The pion propagator in relativistic quantum field theories of the nuclear many-body problem

Pion interactions in the nuclear medium are studied using renormalizable relativistic quantum field theories. Previous studies using pseudoscalar πN coupling encountered difficulties due to the large strength of the πNN vertex. We therefore formulate renormalizable field theories with pseudovector πN coupling using techniques introduced by Weinberg and Schwinger. Calculations are performed for two specific models: the scalar-vector theory of Walecka, extended to include π and ρ mesons in a non-chiral fashion, and the linear σ-model with an additional neutral vector meson. Both models qualitatively reproduce low-energy πN phenomenology and lead to nuclear matter saturation in the relativistic Hartree formalism, which includes baryon vacuum fluctuations. The pion propagator is evaluated in the onenucleon-loop approximation, which corresponds to a relativistic random-phase approximation built on the Hartree ground state. Virtual $\text{N}\text{N}$ loops are included, and suitable renormalization techniques are illustrated. The local-density approximation is used to compare the threshold pion self-energy to the s-wave pion-nucleus optical potential. In the non-chiral model, s-wave pion-nucleus scattering is too large in both pseudoscalar and pseudovector calculations, indicating that additional constraints must be imposed on the lagrangian. In the chiral model, the threshold self-energy vanishes automatically in the pseudovector case, but does so for pseudoscalar coupling only if the baryon effective mass is chosen self-consistently. Since extrapolation from free space to nuclear density can lead to large effects, pion propagation in the medium can determine which πN coupling is more suitable for the relativistic nuclear many-body problem. Conversely, pion interactions constrain the model lagrangian and the nuclear matter equation of state. An approximately chiral model with pseudovector coupling is favored. The techniques developed here allow for a consistent treatment of these models using renormalizable relativistic quantum field theores.     

How to solve the measurement problem of quantum mechanics

A solution to the measurement problem of quantum mechanics is proposed within the framework of an intepretation according to which only quantum systems with an infinite number of degrees of freedom have determinate properties, i.e., determinate values for (some) observables of the theory. The important feature of the infinite case is the existence of many inequivalent irreducible Hilbert space representations of the algebra of observables, which leads, in effect, to a restriction on the superposition principle, and hence the possibility of defining (macro-) observables which commute with every observable. Such observables have determinate values which are not subject to quantum interference effects. A measurement process is schematized as an interaction between a microsystem and a macrosystem, idealized as an infinite quantum system, and it is shown that there exists a unitary transformation which transforms the initial pure state of the composite system in a finite time (the duration of the interaction) into the required mixture of disjoint states.     

Classical limit of quantum mechanics and the quantum billiards problem

Two limiting cases follow from an algebraic formulation of quantum mechanics: Hamiltonian mechanics and quantum mechanics. The results can be used to formulate a quantum billiards problem and to study it at a qualitative level.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 98–100, May, 1982.     

Uniqueness of the Hamiltonian in quantum field theories

In most quantum field theories, one defines the Hamiltonian (energy) operatorH as a limit of cutoff operators . (The operatorH _s would be the correct Hamiltonian for a world in which all momenta are smaller thans.) Since the cutoff operators seldom converge in any of the standard operator topologies, it is often necessary to invent more subtle notions of convergence. For some of the these, it is not obvious that the limit operatorH is unique. In this note we point out that for one such method of obtaining convergence, the limit operator isnot unique. In fact, (under mild assumptions about the operatorsH _s ), ifH _s converges toH, thenH _s also converges toH+R, whereR is an arbitrary bounded positive operator.     

Integrable quantum field theories and conformal field theories from lattice models in the light-cone approach

The light-cone lattice approach to two-dimensional quantum field theories is generalized to a large class of vertex models with any number of possible states per link and any simple Lie group of symmetry. Starting from a given lattice model, different scaling limits are defined leading to conformal field theories or to massive integrable quantum field theories, for which the lattice hamiltonian, momentum and currents are constructed. For a large set of models, the complete mass spectrum is also exhibited. Our approach applies equally well to chiral fermionic theories (like the chiral Gross-Neveu) and to bosonic models like the principal chiral model.     

Quantum logical solution to the measurement problem of quantum mechanics

In this paper I propose a reformulation and solution of the measurement problem of quantum mechanics. The reformulation depends on a quantum logical interpretation of quantum mechanics, broadly construed. The solution depends on a theorem about partial Boolean algebras which is proved here.     

The problem of objectification in quantum mechanics

The hypotheses of weak and strong objectification of quantum mechanical observables, as well as theoretical arguments and experimental evidence against these hypotheses, are systematically reviewed.     

Degrees of symmetry in quantum field theories

A hierarchy of possible symmetries in quantum field theory is defined, which reaches from a purely mathematical invariance to the conventional physical invariance, including the commonly discussed type of spontaneously broken symmetry (SBS). It is shown that one type of SBS, which is usually not considered, naturally leads to theories with an algebra of non-conserved currents and a non-linearly transforming phenomenological Lagrangian. An exactly solvable model is given and some general remarks are made.     

Finiteness of quantum field theories and supersymmetry

The consequences of finiteness are studied for a general renormalizable quantum field theory by analysing the finiteness conditions resulting from the requirement of absence of divergent contributions to the renormalizations of the parameters of an arbitrary gauge theory. In all cases considered, the well-known two-loop finite supersymmetric theories prove to be the unique solution of the finiteness criterion.     

Extension of the electrokinematics theorem to the electromagnetic field and quantum mechanics

Summary A recent electrokinematics theorem leads to a general equation that, through an arbitrary irrotational fieldF, connects the motion of the electric-charge carriers, the internal potential and the dielectric properties of a physical system with its external currents, voltages and powers. It has been proved for quasi-electrostatic fields,i.e. when the vector potential may be disregarded, and on the basis of classical mechanics. Here the theorem is extended to any type of electromagnetic field and to quasi-relativistic quantum mechanics, in the case of many-particle systems for which, moreover, the probability current density is suitably computed. The new equation so obtained, throughF, connects the external currents again with the internal electric permittivity and the scalar potential, in the same way as in the preceding approach, and with the carrier velocity that, however, has to be computed according to quantum mechanics. Moreover, it contains two new contributions, one deriving from the vector potential and the other from a current density arising from the electron spin. By means of proper choices ofF, new expressions of the external currents of the system are determined as functions of the motion of its internal carriers. In particular, the electrokinematics theorem is exploited to compute the output current in two-terminal nanoelectronic devices in which, owing to the small sizes, quantum effects cannot be disregarded. Finally, such results, when they are applied to the double-barrier tunnelling structures, allow us to show the splitting of the electron pulse into two uncorrelated pulses, and as a consequence, to obtain a possible shot noise suppression, up to fifty per cent of the full shot noise.     

Measurement problem in quantum mechanics: Characteristics of an apparatus

The problem of measurement in Quantum Mechanics will be briefly reviewed. Since the measurement process involves a macroscopic apparatus, the attention is focussed on the dynamics of a pointer-like variable of the apparatus when it interacts with a quantum system. It is argued that since the measurement process requires an apparent collapse of the wave function in a certain basis, and collapse is an irreversible process, understanding of irreversibility in a quantum macroscopic system is crucial. The chief characteristics of an apparatus that are important in understanding measurement process are (a) its closely spaced energy levels and (b) its interaction with environment. The coupling with the environment drives the density matrix of the apparatus to diagonal form, but to have persistent correlations between system and apparatus states, it seems necessary to have a pointer variable that has a classical limit     

Star products and quantum groups in quantum mechanics and field theory

Hopf algebras and quantum groups have recently been applied to the analysis of the combinatorics of Feynman graphs in relativistic quantum field theory. On the other hand, in accordance with the program of deformation quantization, the relation between star products and the perturbative expansion in field theory has also been the subject of intensive study. In the present work we clarify the relation between these two approaches. We show how these techniques can be applied in a unified way to quantum systems with a finite number of degrees of freedom and to quantum field theories. In particular, we find that the time-ordered product of quantum fields is the Weyl transform of a certain twisted product. We also show that one can pass from systems involving bosons to systems with fermions, essentially just by replacing the symmetric algebra of the relevant vector space by its exterior algebra.