J. Zinn-Justin

Perturbation theory at large order. I. Theφ2Ninteraction

A new method for calculating the large orders of perturbation theory in quantum field theories has been discussed recently by Lipatov. We show that the same method applied to anharmonic oscillators in quantum mechanics allows one to rederive and generalize results previously obtained by Bender and Wu. We have also verified and generalized Lipatovs results to the case of an internal O(n) symmetry. These results show the divergence of the Wilson-Fisher epsilon expansion and indicate its Borel summability which is used for critical exponents. Similarly, the Callan-Symanzik functions for the phi/sup 4/ theory in three dimensions are characterized.

Generalized non-linear σ-models with gauge invariance

In these lectures we shall give a brief description of a family of models which generalize the non-linear σ-model, and possess in addition a local gauge invariance without containing explicitly a gauge field.

The hydrogen atom in strong magnetic fields: Summation of the weak field series expansion

Abstract Using the weak field expansion, we have calculated the ground-state energy of the hydrogen atom in a magnetic field for values of the field up to about 10 13 G. The perturbative expansion has been summed by an order-dependent mapping method. We compare our results with previous calculations. Our method allows us to obtain: first, more accurate values of the binding energy for a field up to 10 10 G, then good results in the transition region between the two sets of accurate calculations of the literature, and finally still reasonably accurate values up to 10 13 G.

Higher order contributions to critical exponents

Abstract The critical exponents γ and η have been calculated in powers of ϵ = 4- d , where d is the dimension of space, up to order ϵ 3 and ϵ 4 , respectively. The corrections to the previous orders are not small and the agreement with the results of the high temperature series is worse. This may indicate, as might have been conjectured from the approximate recursion formula of Wilson and Fisher, that the ϵ-expansion does not converge for ϵ = 1.

Universal ratios of critical amplitudes near four dimensions

Abstract The coefficients of the divergent susceptibilities and correlation lengths have universal ratios which are computed with the use of the Wilson-Fisher ϵ-expansion. The results are in satisfactory agreement with experiment or with the numbers derived from series expansions analysis.

The H2+ ion in an intense magnetic field: Improved adiabatic approximations

Abstract We study the ground state binding energy of the H 2 + ion in an intense magnetic field. Our calculations are based on improved forms of the adiabatic approximation. We calculate the binding energy, the equilibrium internuclear separation and the zero point energies of nuclear vibrations both parallel and perpendicular to the magnetic field. Comparison of our results with those obtained from the static adiabatic approximation and from variational calculations shows a substantial improvement.

Perturbation theory at large order. I. The Φ2N interaction

A new method for calculating the large orders of perturbation theory in quantum field theories has been discussed recently by Lipatov. We show that the same method applied to anharmonic oscillators in quantum mechanics allows one to rederive and generalize results previously obtained by Bender and Wu. We have also verified and generalized Lipatovs results to the case of an internal O( n ) symmetry. These results show the divergence of the Wilson-Fisher e expansion and indicate its Borel summability which is used for critical exponents. Similarly, the Callan-Symanzik functions for the Φ 4 theory in three dimensions are characterized.

Perturbation theory at large order. II. Role of the vacuum instability

We extend our previous study of large orders of perturbation series for nonrelativistic quantum mechanics and boson field theories to more complicated situations. It is shown that when perturbation theory is performed around an unstable vacuum and does not reveal any pathology at low orders the existence of real pseudoparticles, which are responsible for the tunneling to a more stable vacuum, also implies the divergence and the non-Borel-summability of the series. Conversely, large orders of perturbations around a stable vacuum are dominated by complex solutions to Euclidean field equations. They quantitatively characterize its behavior and indicate the Borel summability of the series. Thus the corresponding Greens functions are unambiguously determined by their perturbation series.

Perturbation theory at large order. I. The phi/sup 2//sup N/ interaction

A new method for calculating the large orders of perturbation theory in quantum field theories has been discussed recently by Lipatov. We show that the same method applied to anharmonic oscillators in quantum mechanics allows one to rederive and generalize results previously obtained by Bender and Wu. We have also verified and generalized Lipatovs results to the case of an internal O(n) symmetry. These results show the divergence of the Wilson-Fisher epsilon expansion and indicate its Borel summability which is used for critical exponents. Similarly, the Callan-Symanzik functions for the phi/sup 4/ theory in three dimensions are characterized.