Remarks concerning the connection between properties of the 4-point-function and the Wilson-Zimmermann expansion

This paper concerns an investigation of the Wilson-Zimmermann (or “short distance”) expansion forA(x)A(y) withx →y whereA(x) is a real scalar field fulfilling Wightman's axioms. If one assumes that such an expansion exists, where the terms of the expansion are operators relatively local toA(x), then the singularities arising in the 4-point-function forx ₃ →x ₄ must control the singularities of then-point functions (n=4, 5, 6, ...) arising forx _j →x _x+1,j=1,2,...,n?1. A similar consequence can be drawn if the terms of the expansion are assumed to exist only as bilinear-forms (Section 2). For certain classes of fields one can show that this condition necessary for the short distance expansion is indeed fulfilled (Section 3). The result of the last section is that the above mentioned condition is also sufficient for the Wilson-Zimmermann expansion, interpreted as an expansion into bilinear forms, and also as an operator expansion in a somewhat modified sense.     

Simple explicit formulas for Gaussian path integrals with time-dependent frequencies

Quadratic fluctuations require an evaluation of ratios of functional determinants of second-order differential operators. We relate these ratios to the Green functions of the operators for Dirichlet, periodic and antiperiodic boundary conditions on a line segment. This permits us to take advantage of Wronski's construction method for Green functions without knowledge of eigenvalues. Our final formula expresses the ratios of functional determinants in terms of an ordinary 2 × 2 determinant of a constant matrix constructed from two linearly independent solutions of the homogeneous differential equations associated with the second-order differential operators. For ratios of determinants encountered in semiclassical fluctuations around a classical solution, the result can further be expressed in terms of this classical solution. In the presence of a zero mode, our method allows for a simple universal regularization of the functional determinants. For Dirichlet's boundary condition, our result is equivalent to Gelfand-Yaglom's. Explicit formulas are given for a harmonic oscillator with an arbitrary time-dependent frequency.     

A remark on the definition of superselection rules in terms of unbounded operators

The mathematical definition of superselection rules in the case when observables are described by unbounded operators in a fixed Hilbert space (for instance, in the frame of Wightman's axioms) is examined. The additional condition \(P_{H_q } D \subset D\) (whereD is the common domain of definition of the operators,H _q is theqth sector, and \(P_{H_q } \) is the projection onH _q ) is found to be sufficient in order to preserve-as in the case of bounded observables—the one-to-one correspondence between reducing subspacesH _q and projections \(P_{H_q } \) from the commutantA′ of the algebraA of observables. This additional condition is equivalent to the physical requirement that every physical vector state can be uniquely represented as a linear combination of physical states, each belonging to some sector.     

Genäherte 3-Punkt-Funktionen in der Heisenbergschen Theorie

Heisenberg's approximation for the wave function of a fermion is modified by a more extended account for the corresponding function of three field operators. This may also contain a certain type of form factor, although the calculations already finished are done without any dependence on the internal coordinates. The result depends remarkably on the ambiguous sign of the non-linear term inHeisenberg's 3 field equation.     

The nut solution as a gravitational dyon

Linearized theory suggests that the NUT solution of Einstein's equations corresponds to a source with both mass and dual mass i.e. to a gravitational dyon. This is born out by the striking identity between the Killing operators of the NUT solution and the ‘total angular momentum’ operators of the monopole. On this basis, Misner's periodic time condition is shown to be the analogue of the Dirac quantization, and results from the requirement that the generators integrate to a global Lie group. It is also shown that there are no bound states for a Klein-Gordon field in NUT space provided the field vanishes in the conventional way at the horizon. For this purpose a generalized ‘tortoise coordinate’ is introduced.     

Quantum Theory of Solid State Plasma Dielectric Response

We review the quantum mechanical derivation of the random phase approximation (RPA) for solid state plasmas, starting from the Hamilton equations for canonically paired “second quantized” creation and annhilation field operators of interacting quantum many‐body systems. Discussing variational differentiation, the coupled equations of motion for the quantum field operators are derived. The concept of Green's functions is reviewed and interpreted, first for retarded Green's functions, and their equations of motion are developed from the equations of motion for the field operators. Thermodynamic Green's functions are discussed, and their periodicity/antiperiodicity properties in imaginary time are carefully examined with discussion of Matsubara Fourier series and representation in terms of a spectral weight function. The analytic continuation from imaginary time to real time is treated. Finally, we define nonequilibrium Green's functions and discuss the linearized timedependent Hartree approximation leading to the random phase approximation. An interesting application to the case of Graphene in a perpendicular magnetic field is discussed in detail, along with applications to normal systems, in terms of attendant phenomenology involving electron‐hole pair excitations and plasmons (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularization of the quantum field theory of charges and monopoles

A gauge-invariant regularization procedure for quantum field theories of electric and magnetic charges based on Zwanziger's local formulation is proposed. The bare regularized full Green functions of gauge-invariant operators are shown to be Lorentz invariant. This would have as a consequence the Lorentz invariance of the finite Green functions that might result after any reasonable subtraction, if such a subtraction can be found.     

Weyl's Gauge Field and Its Behavior

Using a manifestly gauge-invariant Lagrangian density of a system in which a real scalar field (matter field) is interacting with itself and with Weyl's gauge field, we shall study equations of the real scalar field and of Weyl's gauge field, and discuss the self-interacting term of the real scalar field. For a special self-interacting term, we shall obtain an equation of only Weyl's gauge field which plays an important role in solving the equation of Weyl's gauge field interacting with the real scalar field. By making use of the above mentioned equation we shall obtain a rigorous solution for Weyl's gauge field. Next, combining the equation of only Weyl's gauge field with the condition in Weyl's gauge field that the length scale of any vector changes under parallel transfer, we shall obtain a nonlinear equation for the length scale of Weyl's gauge field, which may be important in mathematical physics and is shown to have meron-type solution. By making use of the same techniques being used above, we shall study solution of equation of gradient Weyl's gauge field and as a result, obtain a nonlinear equation of the same type as being found above. Finally we shall study relation between local gauge transformation and symmetric connection in space-time. As a result, we can partly make clear relation between the change in the measure of length scale of a vector due to an infinitesimal parallel transfer and the coefficients of affine connection of Weyl's geometry.     

Energy levels of terbium(III) in the elpasolite Cs2NaTbBr6. II. A correlation crystal field analysis

A set of more than 100 electronic energy levels in Cs₂NaTbBr₆, extending from the ground state to ⁵H₄, is used to test different models of the correlation crystal field (CCF). These are based on Judd's orthogonal g_iQ ^(k) two-electron operators, and more specifically on contributions due to spin-correlation, or ligand polarization. Similar data from Cs₂NaTbCl₆ and Cs₂NaTbF₆ has also been analysed. Only fourth-rank operators make clear improvements to the quality of the fit in deviant multiplets. Empirically the g₇ ⁽⁴⁾ and g₉ ⁽⁴⁾ operators are found to be the most effective. Although fourth-rank operators achieve modest success in correcting the calculated spread of the multiplets, no single operator has a significant impact on the shortcomings of the one-body crystal field. This result is discussed in terms of the limitations of an effective-operator Hamiltonian.     

Another derivation of Weinberg's formula

To investigate how quantum effects might modify special relativity, we will study a Lorentz transformation between classical and quantum reference frames and express it in terms of the four-dimensional (4D) momentum of the quantum reference frame. The transition from the classical expression of the Lorentz transformation to a quantum-mechanical one requires us to symmetrize the expression and replace all its dynamical variables with the corresponding operators, from which we can obtain the same conclusion as that from quantum field theory (given by Weinberg's formula): owing to the Heisenberg's uncertainty relation, a particle (as a quantum reference frame) can propagate over a spacelike interval.     

An Extension of Gleason's Theorem for Quantum Computation

We develop a synthesis of Turing's paradigm of computation and von Neumann's quantum logic to serve as a model for quantum computation with recursion, such that potentially non-terminating computation can take place, as in a quantum Turing machine. This model is based on the extension of von Neumann's quantum logic to partial states, defined here as sub-probability measures on the Hilbert space, equipped with the natural point-wise partial ordering. The sub-probability measures allow a certain probability for the non-termination of the computation. We then derive an extension of Gleason's theorem and show that, for Hilbert spaces of dimension greater than two, the partial order of sub-probability measures is order isomorphic with the collection of partial density operators, i.e. trace class positive operators with trace between zero and one, equipped with the usual partial ordering induced from positive operators. We show that the expected value of a bounded observable with respect to a partial state can be defined as a closed bounded interval, which extends the classical definition of expected value.     

The operator product expansion for minimally subtracted operators

The operator product expansion is proved for minimally subtracted operators using an extension of a simplified version of Zimmermann's momentum subtraction proof. An explicit algorithm for calculating the coefficient functions is obtained, and it is shown that they are analytic in masses and can therefore be calculated perturbatively in asymptotically free theories. This result, which is implicitly assumed in many papers, has only recently been proved by Tkachov et al. who used a very different method.     

Proof for Gauge Independence of the Energy-Momentum Tensor in Quantum Electrodynamics

Proof is given for gauge independence of the (Belinfante's) symmetric energy-momentum tensor in QED. Under the covariant LSZ-formalism it is shown that expectation values, supplemented with physical state conditions, of the energy-momentum tensor are gauge independent to all orders of the purturbation theory (the loop expansion). A study is also made, in terms of the gauge invariant operators of electron (known as the Dirac's or Steinmann's electron) and photon, in expectation of gauge invariant result without any restriction. It is, however, shown that singling out gauge invariant quantities is merely synonymous to fixing a gauge, then there needs again a use of the asymptotic condition to obtain gauge independent results.     

Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics

Given the local observables in the vacuum sector fulfilling a few basic principles of local quantum theory, we show that the superselection structure, intrinsically determined a priori, can always be described by a unique compact global gauge group acting on a field algebra generated by field operators which commute or anticommute at spacelike separations. The field algebra and the gauge group are constructed simultaneously from the local observables. There will be sectors obeying parastatistics, an intrinsic notion derived from the observables, if and only if the gauge group is non-Abelian. Topological charges would manifest themselves in field operators associated with spacelike cones but not localizable in bounded regions of Minkowski space. No assumption on the particle spectrum or even on the covariance of the theory is made. However the notion of superselection sector is tailored to theories without massless particles. When translation or Poincaré covariance of the vacuum sector is assumed, our construction leads to a covariant field algebra describing all covariant sectors.Research supported by Ministero della Pubblica Istruzione and CNR-GNAFA     

The influence of insulating and conductive ellipsoidal objects on the impedance and permittivity of media

For ellipsoidal objects, the complex conductivity of the suspension depends on the objects' axis ratio and orientation. It can be described by analytical equations that were derived by combining the influential radius approach with the mixing equation of Maxwell and Wagner. Here, we consider conductive or insulating homogeneous spheroids, with their symmetry axes being oriented in parallel, in perpendicular or at random with respect to the external field. Considerations show that the field-induced orientations of both nonconductive and conductive objects will result in a reduction of the suspension's impedance and an increased dissipation of electrical energy.     

On the weak‐field approximation of nonlinear vacuum electrodynamics of the Plebański class

Plebański's class of nonlinear vacuum electrodynamics is considered, which is for several reasons of interest at the present time. In particular, the question is answered under which circumstances Maxwell's original field equations are recovered approximately and which ‘post‐Maxwellian’ effects could arise. To this end, a weak field approximation method is developed, allowing to calculate ‘post‐Maxwellian’ corrections up to Nth order. In some respect, this is analogue of determining ‘post‐Newtonian’ corrections from relativistic mechanics by a low velocity approximation. As a result, we got a series of linear field equations that can be solved order by order. In this context, the solutions of the lower orders occur as source terms inside the higher order field equations and represent a ‘post‐Maxwellian’ self‐interaction of the electromagnetic field, which increases order by order. It becomes apparent that one has to distinguish between problems with and without external source terms because without sources also high frequency solutions can be approximately described by Maxwell's original equations. The higher order approximations, which describe ‘post‐Maxwellian’ effects, can give rise to experimental tests of Plebańksi's class. Finally, two boundary value problems are discussed to have examples at hand.     

Theory of intensity and phase fluctuations of a homogeneously broadened laser

We extend our previous quantum mechanical nonlinear treatment of laser noise to the following problem: We consider a set of atoms each with three levels, which support laser action of one or several modes. The laser action can take place either between the upper or the lower two levels. The atomic line is assumed to be homogeneously broadened. The broadening can be caused by the decay into the nonlasing modes, by the pumping process, lattice vibrations and other, non specified sources. The fluctuations of the atomic variables (or operators) are taken into account in a quantum mechanically consistent way using results of previous papers byHaken andWeidlich as well asSchmid andRisken. The laser modes are coupled to the thermal resonator noise usingSenitzky's method. In the first part of the present paper, we treat quite generally multimode laser action. It is shown, that each light mode chooses a specificcollective atomic “mode” to interact with. We introduce a set of suitable collective atomic “modes”, which leads to a simplification of the equations of motion for theHeisenberg operators of the light field and the atomic operators. From the new equations we can eliminate all atomic operators. We are then left with a set of coupled nonlinear, integro-differential equations for the light field operators alone. These equations, which are completely exact and valid both for running and standing waves, represent a considerable simplification of the original problem. In the second part of this paper, these equations are specialized to single mode operation, which is studied above laser threshold. In the vicinity of the threshold the laser equation can be simplified to an operator-equation, whose classical analogue is vander-Pol's equation with a noisy driving force. With increasing inversion, the full equation must be treated, however. Using the method of our previous paper, we decompose the light amplitude into a phase-factor and a real amplitude, which is expanded around its stable value. We determine the Fourier-transform of the intensity correlation function and the total intensity of the fluctuating part of the amplitude. Somewhat above threshold this intensity drops down with the inverse of the photon output power,P, while the inherent relaxation frequency increases withP. The noise intensity stems in this region from the off-diagonal elements of the noise operators and not from the diagonal elements, which are responsible for the shot noise. This result is insofar remarkable, as a rate equation treatment would include only the latter ones. Under certain conditions the intensity fluctuations can show resonances with increasing output power,P. At high inversion the vacuum fluctuations of the light field are dominant, while the other noise sources give rise to contributions which vanish with the inverse of the output power. As a by-product our treatment yields the following formula for the linewidth (half width at half power) which is caused by phase fluctuations:

$$\Delta \nu = \frac{{\gamma _{3 2}^2 \kappa ^2 }}{{(\kappa + \gamma _{3 2} )^2 }}\frac{{\hbar \omega }}{P}\left( {\frac{1}{2}\frac{{(N_3 + N_2 )}}{{N_3 - N_2 }} + n_{Th} + \frac{1}{2}} \right)$$