Matrix Elements Of Field Operators Research Articles

Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas

The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number $N$ of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator $A$ for the total number of particles and the sum of a ``condensate wave function'' $\ensuremath{\xi}(\mathbf{x})$ and a phonon field operator $\ensuremath{\chi}(\mathbf{x})$ in the form $\ensuremath{\psi}(\mathbf{x})\ensuremath{\approx}A{\ensuremath{\xi}(\mathbf{x})+\ensuremath{\chi}(\mathbf{x})/\sqrt{N}}$ when the field operator acts on the $N$ particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of $\sqrt{N}$, and thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different $N$. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if $\ensuremath{\xi}(\mathbf{x})$ is a solution of the time-independent Gross-Pitaevskii equation for $N$ particles and any chemical potential $\ensuremath{\mu}$, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed $N$ is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of $\sqrt{N}$ for the equations of motion. In this expansion the condensate wave function has the time-dependent form $\ensuremath{\xi}(\mathbf{x},t)$, and the condition for the validity of the expansion is that $\ensuremath{\xi}(\mathbf{x},t)$ satisfies the time-dependent Gross-Pitaevskii equation $\ensuremath{\partial}\ensuremath{\xi}/\ensuremath{\partial}t=\ensuremath{-}({\ensuremath{\Elzxh}}^{2}/2m){\ensuremath{\nabla}}^{2}\ensuremath{\xi}+V\ensuremath{\xi}+Nu|\ensuremath{\xi}{|}^{2}\ensuremath{\xi}$. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as $\ensuremath{\chi}(\mathbf{x},t)={\ensuremath{\sum}}_{k}{\ensuremath{\xi}}_{k}(\mathbf{x},t){\ensuremath{\alpha}}_{k},$ where the operators ${\ensuremath{\alpha}}_{k}$ are time-independent annihilation operators, and the state vector has a time evolution described by a Schr\"odinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function $\ensuremath{\xi}(\mathbf{x},t)$.

Time-dependent Green functions in quantum cosmology

The aim of this article is twofold. First we examine from a new angle the question of the recovery of time in quantum cosmology. We construct Green functions for matter fields from the solutions of the Wheeler-DeWitt equation. For simplicity we work in a mini-superspace context. By evaluating these Green functions in a first-order development of the energy increment induced by matrix elements of field operators, we show that the background geometry is the solution of Einstein equations driven by the mean matter energy and that it is this background which determines the time lapses separating the field operators. Then, by studying higher-order corrections, we clarify the nature of the small dimensionless parameters which guarantee the validity of the approximations used. In this respect, we show that the formal expansion in the inverse Planck mass which is sometimes presented as the “standard procedure” is, in general, illegitimate. Secondly, by the present analysis of Green functions, we prepare the study of quantum matter transitions in quantum cosmology. In a next article, we show that the time parametrization of transition amplitudes appears for the same reasons that it appeared in this article. This proves that the background is dynamically determined by the transition under examination.

Non-perturbative analysis of the infrared properties of QED3

We analyse the Dyson-Schwinger equations of the photon and electron propagator of massless QED3. Although the perturbation expansion in this theory contains infrared divergences, the Dyson-Schwinger equations admit well-defined solutions. Perturbative infrared divergences are replaced, in the solutions of the Dyson-Schwinger equations, by new parameters, which are matrix elements of field operators. The full electron propagator is shown to depend crucially on the effective photon mass 〈AμAμ〉.We point out the relevance of our approach to the analysis of Dyson-Scwinger equations in other theories.

Zur Formulierung quantisierter Feldtheorien

A new formulation of quantized field theories is proposed. Starting from some general requirements we derive a set of equations which determine the matrix-elements of field operators and the S-Matrix. These equations contain no renormalization constants, but only experimental masses and coupling parameters. The main advantage over the conventional formulation is thus the elimination of all divergent terms in the basic equations. This means that no renormalization problem arises. The formulation is here restricted to theories which do not involve stable bound states. For simplicity we derive the equations for spin 0 particles, however the extension to other cases (e.g. quantum electrodynamics) is obvious. The solutions of the equations are discussed in a power-series expansion. They are then identical with the renormalized expressions of the conventional formulation. However, the equations set up here are not restricted to the application of perturbation theory. Si propone una nuova formulazione delle teorie di campo quantizzate. Partendo da alcuni requisiti generali, si deriva una serie di equazioni che determinano gli elementi di matrice degli operatori di campo e la matriceS. Queste equazioni non contengono costanti di rinormalizzazione ma solo masse sperimentali e parametri d’accoppiamento. Il principale vantaggio rispetto alla formulazione convenzionale e, pertanto, l’eliminazione di tutti i termini divergenti dalle equazioni fondamentali. Sono cosi eliminati i problemi di rinormalizzazione. La formulazione e, in questo lavoro, ristretta alle teorie che non considerano stati legati stabili. Per semplicita deriviamo le equazioni per le particelle con spin nullo; tuttavia, l’estensione ad altri casi (ad esempio, all’elettrodinamica quantistica) e immediata. Si discutono le soluzioni delle equazioni mediante uno sviluppo in serie di potenze. Esse sono allora identiche alle espressioni rinormalizzate della formulazione convenzionale. Tuttavia, l’applicazione delle equazioni date nel presente lavoro non e limitata alla teoria delle perturbazioni.