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arXiv:0906.2532

Field theory with coherent states for many-body problems with specified particle- and symmetry- quantum numbers (Non-relativistic electrons in a central potential and an external magnetic field)

B. Mieck

Published Jun 14, 2009• 0 citations
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Abstract

Coherent state path integrals are applied to a many-body problem for non-relativistic electrons in a central potential and an external magnetic field; however, in comparison to previous coherent state path integrals, we definitely fix the symmetry quantum numbers to specific values with second quantized field operators which are restricted by delta functions of a Dirac identity in a trace representation of anti-commuting coherent states. We perform an anomalous doubling for the delta functions of the two-particle parts so that a Hubbard-Stratonovich transformation can be taken for corresponding self-energies with a coset decomposition. The given field theory can be extended to an ensemble average over the external magnetic field so that mean eigenvalue densities and eigenvalue correlations can be obtained.

DOI
10.48550/arxiv.0906.2532

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Coherent state path integrals are applied to a many-body problem for non-relativistic electrons in a central potential and an external magnetic field; however, in comparison to previous coherent state path integrals, we definitely fix the symmetry quantum numbers to specific values with second quantized field operators which are restricted by delta functions of a Dirac identity in a trace representation of anti-commuting coherent states. We perform an anomalous doubling for the delta functions of the two-particle parts so that a Hubbard-Stratonovich transformation can be taken for corresponding self-energies with a coset decomposition. The given field theory can be extended to an ensemble average over the external magnetic field so that mean eigenvalue densities and eigenvalue correlations can be obtained.
A coherent state path integral is considered for bosons with an ensemble average of a random potential and with an additional, repulsive interaction in the context of BEC under inclusion of specially prepared disorder. The essential normalization of the coherent state path integral, as a generating function of observables, is obtained from the non-equilibrium time contour for 'forward' and 'backward' propagation so that a time contour metric has to be taken into account in the ensemble average with the random potential. Therefore, the respective symmetries for the derivation of a nonlinear sigma model follow from the involved time contour metric which leads to a coset decomposition Sp(4)/U(2) x U(2) of the symplectic group Sp(4) with the subgroup U(2) for the unitary invariance of the density-related vacuum or ground state; the corresponding spontaneous symmetry breaking gives rise to anomalous- or 'Nambu'-doubled field degrees of freedom within self-energy matrices which are finally regarded by remaining coset matrices. The notion of a 'return probability', according to the original 'Anderson-localization', is thus naturally contained within coherent state path integrals of a non-equilibrium contour time for equivalent 'forward' and 'backward' propagation.
In a first part we give a brief presentation of general Fock space setting to describe quantum field theory. Bosons are quantum particles with integer spin and have symmetric wave functions; Fermions are quantum particles with half-integer spin and are represented by anti-symmetric wave functions. The functional setting is given by a symmetric or anti-symmetric tensor product of Hilbert spaces. We describe these spaces and transformations between these spaces. We shall follow the references [23, 42]. Coherent states are defined by translating the vacuum states with the Weyl operators. This is easily done here for bosons. We shall see in the next chapter how to deal with fermions. In a second part we give an interesting application of bosonic coherent states to the study of the classical limit as ℏ↘0\hbar \searrow 0ℏ↘0 of non-relativistic boson systems with two body interaction in the neighborhood of a solution of the classical system (here the Hartree equation). The classical limit corresponds here to the mean-field limit as the number of particles goes to infinity. As we have done for finite systems, we use here Hepp’s method, which is a linearization procedure of the quantum Hamiltonian around the classical field. The fluctuations around this solution are controlled by a purely quadratic Hamiltonian. In a series of several important papers [114–116] Ginibre and Velo have proven an asymptotic expansion and remainder estimates for these quantum fluctuations. Finally, following the paper [225] one can show that, in the limit ℏ→0\hbar \rightarrow 0ℏ→0, the marginal distribution of the time-evolved coherent states tends in trace-norm to the projector onto the solution of the classical fields equation (Hartree equation) with a uniform remainder estimates in time.
Abstract We consider bosonic atoms with a repulsive contact interaction in a trap potential for a Bose‐Einstein condensation (BEC) and additionally include a random potential. The ensemble averages for two models of static (I) and dynamic (II) disorder are performed and investigated in parallel. The bosonic many body systems of the two disorder models are represented by coherent state path integrals on the Keldysh time contour which allow exact ensemble averages for zero and finite temperatures. These ensemble averages of coherent state path integrals therefore present alternatives to replica field theories or super‐symmetric averaging techniques. Hubbard‐Stratonovich transformations (HST) lead to two corresponding self‐energies for the hermitian repulsive interaction and for the non‐hermitian disorder‐interaction. The self‐energy of the repulsive interaction is absorbed by a shift into the disorder‐self‐energy which comprises as an element of a larger symplectic Lie algebrasp(4M) the self‐energy of the repulsive interaction as a subalgebra (which is equivalent to the direct product ofM×sp(2); ‘M’ is the number of discrete time intervals of the disorder‐self‐energy in the generating function ). After removal of the remaining Gaussian integral for the self‐energy of the repulsive interaction, the first order variations of the coherent state path integrals magnified image result in the exact mean field or saddle point equations, solely depending on the disorder‐self‐energy matrix. These equations can be solved by continued fractions and are reminiscent to the `Nambu‐Gorkov' Green function formalism in superconductivity because anomalous terms or pair condensates of the bosonic atoms are also included into the selfenergies. The derived mean field equations of the models with static (I) and dynamic (II) disorder are particularly applicable for BEC ind= 3 spatial dimensions because of the singularity of the density of states at vanishing wavevector. However, one usually starts out from restricted applicability of the mean field approach ford= 2; therefore, it is also pointed out that one should consider different HST's ind= 2 spatial dimensions with the block diagonal densities as ‘hinge’ functions and that one has to introduce a coset decompositionSp(4M)\U(2M) into densities and anomalous terms of the total disorder‐self‐energysp(4M) for deriving a nonlinear sigma model.
Generating functions with graded variables are mainly used for the description of disordered systems. Instead of averaging over many random samples, the averaging procedure is performed only once with a generating function containing anticommuting variables. An approach is described for the calculation of correlation functions by transfer matrix methods including the aspect of anticommuting variables. The transfer matrix of the generating function of a disordered many body system is constructed in coherent state representation and transformed to a differential eigenvalue equation. According to the presence of anticommuting numbers, this general eigenvalue equation separates into sets of differential eigenvalue systems of commuting variables. These characteristic eigenvalue systems are reduced with operators and their commutation relations to an ordinary Sturm Liouville eigenvalue problem whose lowest eigenvalues contain the information about the averaged, asymptotic critical values of the correlation function. The eigensolutions for the U(N) symmetry preserving problem can be obtained from a recursion process for eigenstates with an arbitrary number M of anticommuting pairs χ*χ, (1 ≤ j ≤ M) which is proved by induction. The given approach, avoiding statistical averaging techniques, can be generalized to problems with U(N) symmetry breaking terms.
We describe here the coherent formulation of electromagnetism in the nonrelativistic quantummechanical many-body theory. We use the mathematical frame of the field theory and its quantization in the spirit of the QED. This is necessary because of the manifold of misinterpretations emerging from the hystorical development of quantum mechanics, starting from the Schrödinger equation of a single particle in the presence of given electromagnetic fields, followed by the many-body theories of many charged identical particles having just Coulomb interactions inspired from the classical electromagnetic theory of point-like charges. However, this later is known to be inconsistent due to the self-interaction. This way could not be continued further to include properly the magnetic forces between the charged particles and lead to a lot of confusion about the interpretation of the magnetic field in the Hamiltonian, as well as about the gauge invariance. We emphasize the importance of the distinction between the applied (external fields) and the field in the matter. All these problems are length properly solved within the non-relativistic QED, nevertheless the confusion dominates in all the problems related to the magnetic properties of the solid state.
The Galilei-covariant fermionic field theories are quantized by using the path-integral method and five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. Firstly we review the five-dimensional approach to the Galilean Dirac equation, which leads to the Lévy-Leblond equations, and define the Galilean generating functional and Green’s functions for positive- and negative-energy/mass solutions. Then, as an example of interactions, we consider the quartic self-interacting potential λ(ΨΨ) 2, and we derive expressions for the 2- and 4-point Green’s functions. Our results are compatible with those found in the literature on non-relativistic many-body systems. The extended manifold allows for compact expressions of the contributions in (3 + 1) space-time. This is particularly apparent when we represent the results with diagrams in the extended (4 + 1) manifold, since they usually encompass more diagrams in Galilean (3 + 1) space-time. 1