Decoherence effects in Bose–Einstein condensate interferometry I. General theory

The present paper outlines a basic theoretical treatment of decoherence and dephasing effects in interferometry based on single component Bose–Einstein condensates in double potential wells, where two condensate modes may be involved. Results for both two mode condensates and the simpler single mode condensate case are presented. The approach involves a hybrid phase space distribution functional method where the condensate modes are described via a truncated Wigner representation, whilst the basically unoccupied non-condensate modes are described via a positive P representation. The Hamiltonian for the system is described in terms of quantum field operators for the condensate and non-condensate modes. The functional Fokker–Planck equation for the double phase space distribution functional is derived. Equivalent Ito stochastic equations for the condensate and non-condensate fields that replace the field operators are obtained, and stochastic averages of products of these fields give the quantum correlation functions that can be used to interpret interferometry experiments. The stochastic field equations are the sum of a deterministic term obtained from the drift vector in the functional Fokker–Planck equation, and a noise field whose stochastic properties are determined from the diffusion matrix in the functional Fokker–Planck equation. The stochastic properties of the noise field terms are similar to those for Gaussian–Markov processes in that the stochastic averages of odd numbers of noise fields are zero and those for even numbers of noise field terms are the sums of products of stochastic averages associated with pairs of noise fields. However each pair is represented by an element of the diffusion matrix rather than products of the noise fields themselves, as in the case of Gaussian–Markov processes. The treatment starts from a generalised mean field theory for two condensate modes, where generalised coupled Gross–Pitaevskii equations are obtained for the modes and matrix mechanics equations are derived for the amplitudes describing possible fragmentations of the condensate between the two modes. These self-consistent sets of equations are derived via the Dirac–Frenkel variational principle. Numerical studies for interferometry experiments would involve using the solutions from the generalised mean field theory in calculations for the stochastic fields from the Ito stochastic field equations.     

Stochastic models for lattice gauge theories

Stochastic equations are derived which describe the (Euclidean) time evolution of lattice field configurations, with and without fermions, on a three-dimensional space lattice. It is indicated how the drifts and transition functions may be obtained as asymptotic solutions of a differential equation or from a ground state ansatz. For non-Abelian gauge fields (without fermions) a ground state is constructed which is an exact eigenstate of a Hamiltonian with the same (naive) continuum limit as the Kogut-Susskind Hamiltonian. It is described how Euclidean correlations (like the Wilson loop) are obtained from the stochastic equations and how mass gaps may be obtained from the technique of exit times.     

HORIZONS AND GENERATING FUNCTIONALS FOR THE TEMPERATURE GREEN FUNCTIONS

The generating functional approach to Green functions in the thermal equilibrium is used to explore the geometrical origin of the temperatures of the quantum fields in the Rindler space-time and black hole spacetimes. It is shown that under the transformation from Minkowski space to the Rindler space the path integral representation for the Euclidean generating functionals of Green functions at zero temperature would transform into the corresponding ones of the quantum fields at a certain finite temperature, and the Minkowski vacuum state would have the same properties as that of the quantum mixed state at the same temperatfire. All thermal Green functions for the mixed state are given. Similar results would be obtained for the Schwarzschild, the Reissner-NordstrOm and the Kerr black holes and whereupon the Hawking temperature for the black holes would have geometrical origin as well as that in the Rindler spacetime. The various density operators of the mixed states at the Hawking temperature for the black hole sacetimes are specified.     

Recent developments in conformal invariant quantum field theory

A review of the recent results concerning the kinematics of conformal fields, the analysis of dynamical equations and dynamical derivation of the operator product expansion is given.The classification and transformational properties of fields which are transformed according to the representations of the universal covering group of the conformal group have been considered. A derivation of the partial wave expansion of Wightman functions is given. The analytical continuation to the Euclidean domain of coordinates is discussed. As shown, in the Euclidean space the partial wave expansion can be applied either to one-particle irreducible vertices or to the Green functions, depending on the dimensions of the fields.The structure of Green functions, which contain a conserved current and the energy-momentum tensor, has been studied. Their partial wave expansion has been obtained. A solution of the Ward identity has been found. Special cases are discussed.The program of the construction of exact solution of dynamical equations is discussed. It is shown, that integral dynamical equations for vertices (or Green's functions) can be diagonalized by means of the partial wave expansion. The general solution of these equations is obtained. The equations of motion for renormalized fields are considered. The way to define the product of renormalized fields at coinciding points (arising on the right-hand side) is discussed. A recipe for calculating this product is presented. It is shown, that this recipe necessarily follows from the renormalized equations.The role of bare term and of canonical commutation relations (for unrenormalized fields) is discussed in connection with the problem of the field product determination at coinciding points. As a result an exact relation between fundamental field dimensions is found for various three-linear interactions (section 16 and Appendix 6). The problem of closing the infinite system of dynamical equations is discussed.Al above said results are demonstrated using Thirring model as an example. A new approach to its solving is developed.The program od closing the infinite system of dynamical equations is discussed. The Thirring model is considered as an example. A new approach to the solution of this model is discussed.Methods are developed for the approximate calculation of dimensions and coupling constants in the 3-vertex and 5-vertex approximations. The dimensions are calculated in the γ_?³ theory in 6-dimensional space.The problem of calculating the critical indices in statistics (3-dimensional Euclidean space) is considered. The calculation of the dimension is carried out in the framework of the γ_?⁴ model. The value of the dimension and the critical indices thus obtained coincide with the experimental ones.     

Approximate solutions of the Liouville equation. III. Variational principles and projection operators

Aspects of stationary variational principles for the Laplace-transformed Liouville equation are discussed. Projection techniques are used to derive new stationary principles applicable to the space orthogonal to the space spanned by functions occurring in the conservation laws. As a result, any trial function automatically leads to results satisfying the conservation laws. The procedure is also applied to the parity-even and parity-odd distributions which obey equations governed by the square of the Liouville operator. The technique is extended to eliminate the one-body additive contribution to the solution exactly. Finally, the ideas of the moment method, which leads to the continued-fraction representation of autocorrelation functions, are applied to variational principles. We find continued-fraction variational principles such that a zero trial function yields the usual representation. However, a trial function representing noninteracting particles contains the results of the moment method and in addition yields the exact analytic behavior for free particles.Work supported by a grant from the National Science Foundation.     

An approximate method for solving radiation transport problems in temporally and spatially stochastic media

A new approach has been developed to deal with stochastic transport problems in three-dimensional media. It is assumed that the medium consists of randomly distributed lumps of material embedded in a background matrix and in each lump the properties may vary randomly with time. The coefficients for scattering and absorption are represented mathematically by members of a random characteristic set function, which depend on space and time. Different physical situations can be described by different forms and combinations of these set functions. In order to effect a solution of the resulting stochastic transport equation, which may be for photons or neutrons, we make the, a priori, assumption that the functional form for the solution of the transport equation, i.e. the stochastic flux, can be represented by the same mathematical form as the scattering and absorption coefficients (or cross sections), i.e. we introduce a stochastic ansatz. This procedure leads to a set of deterministic equations from which the mean and variance of the flux in space and time can be obtained. For the case of a two-phase medium, either two or four coupled integro-differential equations are obtained for the deterministic functions that arise (depending on the problem) and expressions are given for the mean and variance of the angular flux. There is a close relationship between these equations and those from the Levermore-Pomraning (LP) theory, but the new equations offer an opportunity to deal with more general forms of stochastic processes and combine simultaneously time and space fluctuations. The stochastic characteristics of the medium are defined by the correlation functions which appear in the equations and, by making plausible assumptions about the functional form of these autocorrelation functions, different physical situations can be simulated, according to the structure of the medium. The main contribution of the present work is to include space and time fluctuations simultaneously as a pseudo-dichotomic Markov process.     

Axially symmetric stationary electrovac solutions

Perjes and Israel and Wilson have given independently a new class of solutions of the sourcefree Einstein-Maxwell equations, which can be interpreted as the external gravitational and electromagnetic fields of a spinning source with unit specific charge. Starting from Zipoy's solutions in oblate and prolate spheroidal coordinates for the source-free gravitational field we generate some axially symmetric stationary solutions of the source-free Einstein-Maxwell equations by using Perjes' method. All these solutions become Euclidean at infinity. The asymptotic behavior and the singularity of the solutions are studied in order to gain some insight into the nature of the source. The solution in prolate spheroidal coordinates is found to contain closed timelike lines.     

Green functions in stochastic field theory

Functional representations are reviewed for the generating function of Green functions of stochastic problems stated either with the use of the Fokker-Planck equation or the master equation. Both cases are treated in a unified manner based on the operator approach similar to quantum mechanics. Solution of a second-order stochastic differential equation in the framework of stochastic field theory is constructed. Ambiguities in the mathematical formulation of stochastic field theory are discussed. The Schwinger-Keldysh representation is constructed for the Green functions of the stochastic field theory which yields a functional-integral representation with local action but without the explicit functional Jacobi determinant or ghost fields.     

Renormalized Normal Product and equations of motion of Φ4-coupling

The notion of a Renormalized Normal Product (RNP) in Euclidean space of 1 ≤ r ≤ 4 dimensions is introduced for a Φ⁴-model in a nonperturbative approach. The essential ingredients used for this purpose are the composite operators defined in perturbation theory and the renormalized G-convolution product constructed in the axiomatic field theory framework in Euclidean momentum space. Convergent equations of motion for the connected Green's functions are established where the interaction term is represented by the RNP. The corresponding renormalization constants are defined as boundary values of the RNP by imposing “physical” renormalization conditions. In the special case of 2-dimensions it is proved that these equations conserve analyticity and algebraic properties (in complex Minkowski space of 2-momenta) coming from the first principles of general local field theory, together with properties of asymptotic behaviour at infinity (in Euclidean space of 2-momenta).     

From Euclidean to Minkowski space with the Cauchy–Riemann equations

We present an elementary method to obtain Green’s functions in non-perturbative quantum field theory in Minkowski space from Green’s functions calculated in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes often is unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore, we suggest to use the Cauchy–Riemann equations, which perform the analytical continuation without assuming global information on the function in the entire complex plane, but only in the region through which the equations are solved. We use as example the quark propagator in Landau gauge quantum chromodynamics, which is known from lattice and Dyson–Schwinger studies in Euclidean space. The drawback of the method is the instability of the Cauchy–Riemann equations against high-frequency noise,which makes it difficult to achieve good accuracy. We also point out a few curious details related to the Wick rotation.     

The Continual Approach in the Yang-Mills Theory in the External Non-Abelian Field

The theory of Yang-Mills field in interaction with matter fields is considered in the presence of external gauge field. A closed expression for the generating functional of the Green functions is obtained, and a detailed analysis of the Green functions of the scalar, spinor, ghost and Yang-Mills fields is performed. The path-integral solution for all these Green functions is obtained, which includes the functional averaging over the classical trajectories in the space of commuting and anticommuting variables, the latter being anociated with the particle spin and isospin. For illustration an arbitrary Abelian-like external field is considered, as well as non-Abelian-like constant external field.     

Exactly solvable models of conformal-invariant quantum field theory in D-dimensional space

The method for exact solution of a certain class of models of conformal quantum field theory in D-dimensional Euclidean space is proposed. The method allows one to derive closed differential equations for all the Green functions and also algebraic equations to scale dimensions of all field. A scalar field P of a scale dimension d_p = D − 2 is needed for nontrivial solutions to exist. At D ≠ 2 this field is converted to a constant that coincides with the central charge of two-dimensional theories. A new class of D = 2 models has been obtained, where the infinite-parametric symmetry is not manifest. The two-dimensional Wess-Zumino model is used to illustrate the method of solution.     

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

The nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter. We show that on a suitable coarse scale this map generates an oscillating parameter-dependent diffusion coefficient, similarly to hyperbolic maps, whose asymptotic functional form can be understood in terms of simple random walk approximations. On finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter. By using a Green–Kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations. Starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized Takagi functions obeying generalized de Rham-type functional recursion relations. We finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive, and that for normal diffusion it increases by increasing the parameter value.     

Asymptotic solutions for Mathieu instability under random parametric excitation and nonlinear damping

A theoretical analysis is presented of the response of a lightly and nonlinearly damped mass-spring system in which the spring constant contains a small randomly fluctuating component. Damping is represented by a combination of linear and nonlinear power-law damping. System response to some initial disturbance at time zero is described by a sinusoidal wave whose amplitude and phase vary slowly and randomly with time. Leading order formulations for the equations of amplitude and phase are obtained through the application of methods of stochastic averaging of Stratonovich. The equations of amplitude and phase are given in two versions: Fokker-Planck equations for transient probability and Langevin equations for response in the time-domain. Solutions in closed-form of these equations are derived by methods of mathematical and theoretical physics involving higher transcendental functions. They are used to study the behavior of system response for ever increasing time applying asymptotic methods of analysis such as the method of steepest descent or saddle-point method. It is found that system behavior depends on the power density of the parametric excitation at twice the natural frequency and on the magnitude and form of the damping. Depending on these parameters different types of system behavior are found to be possible: response which decays exponentially to zero, response which leads to a stationary state of random behavior, and response which can either grow unboundedly or which approaches zero in a finite time.     

Renormalization constants and asymptotic behaviour in quantum electrodynamics

Using dimensional regularization, a field theory contains at least one parameter less than usual with the dimension of mass. The Callan-Symanzik equations for the renormalization constants then become solvable entirely in terms of the coefficient functions. Explicit expressions are obtained for all the renormalization constants in quantum electrodynamics. At non-exceptional momenta the infrared behaviour and the three leading terms in the asymptotic expansion of any Green function are controlled by the Callan-Symanzik equations. For the propagators the three leading terms are computed explicitly. The gauge dependence of the asymptotic electron propagator in momentum space is calculated in all orders of perturbation theory.     

Stochastic quantization of order two parafermi fields

The method of stochastic quantization of Parisi–Wu is extended to include spinor fields obeying the generalized statistics of order two consistent with the weak locality requirement. Appropriate Langevin and Fokker–Planck equations are constructed using paragrassmann variables, which give rise to two fields with different masses in the equilibrium limit, in agreement with the results of the canonical quantization procedure. The connection between the stochastic quantization method and conventional Euclidean field theory is established through Klein transformations. Received: 14 November 2001 / Published online: 8 February 2002     

Bound-state perturbation method via SVZ sum rules in quantum mechanics

The perturbation method for bound states within the framework of the Shifman-Vainshtein-Zakharov sum rule method is studied on simple systems (linear harmonic oscillator, hydrogen atom) in external electric fields. It is pointed out that for stronger fields reasonable results for the ground-state energy can only be achieved when sum rules are written for the correction to the Euclidean Green function caused by the external field. Moreover, if the system is bound by a singular (Coulomb) potential, one needs to sum higher perturbative corrections to the Green function and to find a realistic approximation of the continuum contribution to the sum rules. The results are of relevance e.g. for calculations of nucleon magnetic moments and toponium properties via SVZ sum rules in QCD.     

Maxwell equations in conformal invariant electrodynamics

We consider a conformal invariant formulation of quantum electrodynamics. Conformal invariance is achieved with a specific mathematical construction based on the indecomposable representations of the conformal group associated with the electromagnetic potential and current. As a corollary of this construction modified expressions for the 3-point Green functions are obtained which both contain transverse parts. They make it possible to formulate a conformal invariant skeleton perturbation theory. It is also shown that the Euclidean Maxwell equations in conformal electrodynamics are manifestations of its kinematical structure: in the case of the 3-point Green functions these equations follow (up to constants) from the conformal invariance while in the case of higher Green functions they are equivalent to the equality of the kernels of the partial wave expansions. This is the manifestation of the mathematical fact of a (partial) equivalence of the representations associated with the potential, current and the field tensor.     

Exploring correlations in the stochastic Yang–Mills vacuum

The correlation lengths of nonperturbative-nonconfining and confining stochastic background Yang–Mills fields are obtained by means of a direct analytic path-integral evaluation of the Green functions of the so-called one- and two-gluon gluelumps. Numerically, these lengths turn out to be in a good agreement with those known from the earlier, Hamiltonian, treatment of such Green functions. It is also demonstrated that the correlation function of nonperturbative-nonconfining fields decreases with the deviation of the path in this correlation function from the straight-line one.