Singularity analysis of fourth-order Møller–Plesset perturbation theory

The usefulness of Møller–Plesset perturbation theory, a standard technique of quantum chemistry, is determined by singularities in the corresponding energy function in the complex plane of the perturbation parameter. A method is developed that locates singularities from fourth-order perturbation series, using quadratic approximants with bilinear conformal mappings.     

Bose-Einstein condensation beyond perturbation theory: Goldstone singularities and instanton solution

Temperature Green functions are applied to the analysis of Bose-condensation of weaklyinteracting gas. The character of Goldstone singularities of correlation functions isestablished to all orders in perturbation theory. These singularities are regularized bythe system volume. An anomalous volume dependence of the correlation functions isrevealed. Quantum-field perturbation series are studied in the framework of the instantonapproach. It is shown that there are no time-dependent instantons and that thetime-independent instanton solutions exhibit factorial growth in large orders of thequantum-field perturbation expansion.     

Lie transform perturbation theory for Hamiltonian systems

A review of the theory of Lie transform perturbation theory for Hamiltonian systems is presented. The operator theory of Dewar for continuous families of canonical transformations is discussed. It is then used to derive the perturbation method of Deprit. Two examples of the use of this method are provided. In addition, the more efficient perturbation method of Dragt and Finn is discussed.     

On Borel singularities in quantum field theory

We consider the effective one-loop lagrangian in a constant electric field. It is shown that perturbation theory behaves as n!, giving rise to singularities in the Borel plane. Comparing with the exact result we show how to integrate these singularities. We suggest that renormalons in QED and QCD should be integrated in a similar way. We make a speculation on a possible interpretation of this integration.     

Perturbation theory and the parton model in QCD

We prove that for any process which admits a parton-model interpretation, the naive parton model can be modified to include the effects of QCD interactions to all orders in perturbation theory. This requires that the mass singularities in quark and gluon inclusive cross sections factor into universal functions which renormalize the naive parton model distribution and decay functions. We prove that this factorization takes place for all leading and non-leading logs and thus check consistency of the parton model to all orders in perturbation theory.     

On general properties of Lorentz-invariant formulation of noncommutative quantum field theory

We study general properties of certain Lorentz-invariant noncommutative quantum field theories proposed in the literature. We show that causality in those theories does not hold, in contrast to the canonical noncommutative field theory with the light-wedge causality condition. This is the consequence of the infinite nonlocality of the theory getting spread in all spacetime directions. We also show that the time-ordered perturbation theory arising from the Hamiltonian formulation of noncommutative quantum field theories remains inequivalent to the covariant perturbation theory with usual Feynman rules even after restoration of Lorentz symmetry.     

Multimomentum hamiltonian formalism in quantum field theory

The familiar generating functional in quantum field theory fail to be true measures and make sense only in framework of perturbation theory. In our approach, generating functionals are defined strictly as the Fourier transforms of Gaussian measures in nuclear spaces of multimomentum canonical variables when field momenta correspond to derivatives of fields with respect to all world coordinates, not only to time.     

Degenerate perturbation theory and its application to the Hubbard model

Projection operators are incorporated in the canonical transformation to derive an analytical infinite series of a degenerate perturbation expansion. The theory is used to calculate the ground state energy of a half-filled s-band Hubbard model to seventh order.     

A perturbation density functional theory for hydrogen bonding cyclic molecules

Using the framework of Wertheim's thermodynamic perturbation theory, a new polyatomic density functional theory is developed to account for the intermolecular association of cyclic molecules in interfacial systems. To test the theory, Monte Carlo simulations in the canonical ensemble were performed for the specific case of an associating triatomic ring with one association site next to a hard wall. The theory and simulation results were found to be in good agreement.     

Semiclassical and quantum Liouville theory on the sphere

We solve the Riemann–Hilbert problem on the sphere topology for three singularities of finite strength and a fourth one infinitesimal, by determining perturbatively the Poincaré accessory parameters. In this way we compute the semiclassical four point vertex function with three finite charges and a fourth infinitesimal. Some of the results are extended to the case of n finite charges and m infinitesimal. With the same technique we compute the exact Green function on the sphere with three finite singularities. Turning to the full quantum problem we address the calculation of the quantum determinant on the background of three finite charges and the further perturbative corrections. The zeta function technique provides a theory which is not invariant under local conformal transformations. Instead by employing a regularization suggested in the case of the pseudosphere by Zamolodchikov and Zamolodchikov we obtain the correct quantum conformal dimensions from the one loop calculation and we show explicitly that the two loop corrections do not change such dimensions. We expect such a result to hold to all order perturbation theory.     

Quantization of non-local field theory

The scheme of quantisation of non-local field theory is formulated. An intermediate regularisation is introduced into the non-local Lagrangian of the classical scalar field in such a way that the procedure of the canonical quantisation leads to the appearance of additional ghost states with indefinite metrics. The ghost states disappear when the regularisation is removed but the propagator of the scalar particle becomes non-local and theS-matrix is finite, unitary, causal and covariant in each perturbation order.     

The production of soft dileptons in the quark-gluon plasma in resummed perturbation theory

The production rate of soft dileptons in a quarkgluon plasma is calculated for the general case where the intermediate virtual photons have non-zero momenta using Braaten-Pisarski resummed perturbation theory. It is found that the physics is more complicated and the structures are somewhat richer than the special case where the photons produced are stationary in the plasma. These structures are found to be determined, to a large extent, by the dispersion relations of the two propagating modes of the soft quarks. However, the Van Hove singularities present in the zero-momentum case are no longer there. The dependence of the rate on the total momentum of the dilepton pairs is not very strong, unless the photon mass shell is approached. Comparison with the results obtained from the one loop calculation shows that the resummed perturbation theory predicts more dileptons being produced from the plasma when the intermediate virtual photons are near their mass shell.     

Renormalisation and mass singularities

We emphasize the role of renormalization in the consistent treatment and identification of mass singularities. In this connection we present the treatment of a completely massless process: Compton scattering of massless electrons, to lowest non-trivial order in perturbation theory, including the calculation of all non-leading singularities.     

String theory and cosmological singularities

In general relativity space-like or null singularities are common: they imply that ‘time’ can have a beginning or end. Well-known examples are singularities inside black holes and initial or final singularities in expanding or contracting universes. In recent times, string theory is providing new perspectives of such singularities which may lead to an understanding of these in the standard framework of time evolution in quantum mechanics. In this article, we describe some of these approaches.      

Logarithmic operators in conformal field theory

Conformal field theories with correlation functions which have logarithmic singularities are considered. It is shown that those singularities imply the existence of additional operators in the theory which together with ordinary primary operators form the basis of the Jordan cell for the operator L₀. An example of the field theory possessing such correlation functions is given.