Functional Quantization Research Articles

Investigation of Green Functions and the Parisi-Wu Quantization Method in Background Stochastic Fields

In this review we present variational and stochastic method of evaluating functional integrals and quantization of different fields (scalar, electromagnetic and Yang-Mills gauge ones). These methods are applied to the study of the asymptotic behavior of the scalar particle Green functions in stochastic fields and to the construction of a finite quantum field theory by means of the nonlocal white noise-like background stochastic fields in the scheme due to Parisi and Wu. In our mathematical prescription the white noise-like field plays a double role, it controls the quantum behavior of the theory and at the same time it carries nonlocality in stochastic Langevin-type equations. We also introduce stochasticity of the space-time metric in the context of stochastic background field and discuss its consequences.

Extended GIPQ description of the RPA excitations

The work presents an application of the GIPQ quantization method, extended to include the wave function quantization. By explicit calculation on a particular proton-neutron system it is shown that the extended quantization gives the same excitation operator and energy as the RPA.

Diatomic predissociative widths and shifts from multichannel Siegert quantization and asymptotic analysis of resonance wave functions

Predissociative widths and shifts are studied for a diatomic molecule described by a Morse potential crossed by one or two repulsive exponential potentials. This defines two models with either two channels or three channels. The coupled channel equations are solved either with a complex rotated coordinate (in which case the boundary conditions are the same as for a bound state) or with explicit enforcement of Siegert outgoing wave only asymptotic boundary conditions. The widths and shifts derived from the complex quantized energies of the two channel model are compared to previous results for this model based on semiclassical, phase shift, or Green operator methods. In the weak coupling regime, significant improvement over previous phase shift based results is obtained. In the intermediate coupling case, Siegert quantization confirms the value given by the semiclassical approach to the coupling giving a resonance of zero width. This is in disagreement with the Green operator prediction. Additional support is given by an asymptotic analysis performed on the resonance wave function (A.A.R.W.) which gives widths in very good agreement with those derived from the complex quantized energies. The three channel model leads to the definition of a total width and of two partial widths. Asymptotic analysis of the three-channel resonance wave function gives directly partial widths which, in the weak width regime, sum accurately to the total width.

The Polyakov path integral over bordered surfaces (the open string amplitudes)

We use the Feynman functional quantization scheme adapted to the gauge theories with reparametrization invariance to the functional covariant first quantization of the open bosonic BDHP string in a position representation. The consistent functional integral representation of the open string propagator is derived and evaluated. This result is used as a starting point for two kinds of constructions of the off-shell multiloop open string amplitudes. The general idea of the presented approach is to consider the off-shell amplitudes as functionals on the space ℰ of contours endowed with an intrinsic metric or on the space ℰ/ℝ+.

Covariant functional quantization of a particle in general metric

In this paper we work out the quantum mechanics of a free particle in genral metric from the functional point of view. We want to obtain ageneral coordinate transformations (GCT) covariant definition of the path integral. To this end we redefine the Δx on the time lattice so that the «velocity» Δx/Δt behaves as a GCT vector at the quantum level. The resulting Hamiltonian is the canonical one plus a term proportional to the scalar curvature, in agreement with DeWitt classical paper. The case of a vector potential interaction is also worked out.

The metaplectic group within the Heisenberg–Weyl ring

The Heisenberg–Weyl ring contains the metaplectic group of canonical transforms acting unitarily on ℒ 2(ℛ). These ring elements are characterized through (i) the integral transform kernels, (ii) coset distributions, and (iii) classical functions under any quantization scheme. The isomorphism under group composition leads to several new relations involving twisted products and quantization of Gaussian classical functions. The Wigner inversion operator is a special central group element. It is shown that the only quantization scheme invariant under metaplectic transformations is the Weyl scheme. The structure studied here appears to be relevant to the study of wave optics with aberration.

Signal loss due to imperfect fringe rotation in VLBI correlators revisited

This not new problem is analysed in view of discrepancies met in the literature. Practical formulae for calculation of the signal-to-noise ratio losses caused by quantization of the lobe rotation function in digital VLBI correlators are derived. Examples are given for a few simpler rotation schemes.

Particle condensates in quantum field theory

A general equivalence between functional techniques and canonical quantization for a simple self-interacting scalar theory is proposed. Vacuum instability for the symmetric (〈φ〉=0) and non-symmetric cases is analysed in detail.

On the functional formulation of quantum field theory

This paper describes a functional formulation of Euclidean quantum field theory, based on a complete equivalence with classical statistical mechanics. One introduces an extra time variable and sets up a canonical scheme with new Lagrangian and Hamiltonian functions. The generating functional is then defined as the Gibbs average over the ensemble. This allows us, in particular, to control in a simple way the invariance properties of the integration measure. In several cases of physical interest it is seen that the invariance requirement leads to extra determinantal factors in the integration volume and, therefore, to a set of improved Feynman rules. In particular, enforcing dilatation invariance for the generating functional is shown to lead to a nonzero background,i.e. to spontaneous breaking of the symmetry. The application of the method to constrained systems is discussed in detail and in the case of Yang-Mills theories the Faddeev-Popov prescription for quantization is reproduced with remarkable simplicity. A discussion of the functional quantization of gravity is also offered.

Optimal quantization of standard distribution functions

Quantizer design for minimizing the mean square error has been developed independently by Lloyd and Max. They have tabulated the output and decision levels for Gaussian distribution function based on both uniform and nonuniform quantization. Subsequently this design has been extended to other standard distribution functions such as Rayleigh and Laplacian. Preliminary investigation of error analysis in image processing has shown that image reconstruction based on minimum MSE is not optimal. Quantizer design based on minimization of powers of quantization error other than two (MSE) yields less degraded images. This has led to the necessity of developing quantization tables for most frequently used distribution functions based on mean fourth power error (MFPE) and mean sixth power error (MSPE). The later criteria result in better quantizer design in regions of large luminance changes, contours, edges, etc. leading to subjectively higher quality images. This research focusses on optimal quantizer design for minimizing MFPE and MSPE. Quantization ranges and output levels based on both uniform and nonuniform spacing for Gaussian, Rayleigh and Laplacian distributions are developed. The tables, developed by numerical techniques are based on normalized standard deviation. All the computations are implemented with double precision accuracy (64 bits) with an algorithmic error range of 10 −6–10 −8 using IBM 370/155 digital computer. Plotting routines are implemented on Tektronix hardware (interactive graphics package)_using DEC-20 digital computer. Other relevant parameters such as distortion ratio (ratio of distortion of uniform to nonuniform quantizers), distortion and entropy as a function of quantization levels are illustrated. This tabular and graphical data will be useful in digital communications such as image processing.

On canonical and functional integral quantization of Yang-Mills theory

The canonical and functional integral quantization of Yang-Mills theory is discussed. The gauge-fixing «weak» conditionsAa0≈0,Aa3≈0 over phase space are found to be very convenient for any gauge group and in the presence of interactions. These conditions fix the gauge for arbitrary strong field and we obtain a description of Yang-Mills field in terms of physical degress of freedom only.

Error in determining the derivative in analog-digital measurements

1. In determining derivatives from the increments of the argument and function in analog-digital measuring systems, the error may attain large values, even for precise measurement of the variables. 2. When it is impossible to smooth out errors at the function quantization points, it is advisable to determine the derivative by first differentiating the function in analog form.

Quantization as deformation theory

We show that any rule for quantization of functions on the phase space of a dynamical system is equivalent to a deformation of the Poisson bracket, in the sense of Lie algebras. All Schrödinger-type quantizations are equivalent mathematically but not physically. We examine whether a quantization rule exists in which the Poisson bracket of the free Hamiltonian and any function not more than quadratic in momentum gets quantized into the commutator of the quantized versions of these two observables. We show that if the (Riemannian) configuration manifold is irreducible, this condition uniquely specifies the quantization of these functions, in the cases when it can be satisfied. However, it can only be satisfied for systems whose configuration manifolds are one-dimensional, or have vanishing Ricci tensor, or are spaces of constant curvature. Thus the condition will not serve as a general physical principle which could be invoked to fix a quantization rule.

Light-cone quantization of dual models and structure functions

The aim of this work is to study some of the light-cone aspects of the dual-resonance models and to show the agreement with the usual results of the light-cone analysis based on other models. To accomplish this program we quantize on the light-cone the Lagrangian formulation of the DRM. Our quantization procedure is based on the derivation of the completeness relation for thec-number solutions of the wave equation. An interesting result of this analysis is that thec-number solutions in configuration space form a complete set atx+=const only in the space of the functionsf(x) such that \( \mathop {\lim }\limits_{x^ - \to \pm \infty } \)f(x)=0. From the quantization of the field we obtain the current commutators and the explicit expressions for the bilocal operators. Using such expressions we calculate the structure functions of the deep inelastic scattering up to the second order in the interaction term which reproduces the dual amplitude. We find thatF2(ξ) has the correct Regge behaviour forξ→0.

An investigation of errors arising in photoelectric counters-aerosol particle analyser

The distorting influence of the errors of a photoelectric particle counting device on an aerosol size distribution is investigated. As a quantitative measure, a distortion function is taken which is equal to the ratio of the distorted function to the initial function. The gamma distribution is used as the initial function. Four main sources of errors are discussed. They are: (1) nonuniformity of illumination of the working volume in the device; (2) simultaneous penetration of more than one particle into the working volume; (3) photodetector noise; (4) quantization of a continuous distribution function when recorded by a pulse height analyzer.

EFFECTIVE MASS CHANGE OF ELECTRONS IN SILICON INVERSION LAYERS OBSERVED BY PIEZORESISTANCE

Piezoresistance coefficients were determined for n-type inversion layers at low gate fields on (100), (110), and (111) surfaces of silicon at room temperature. Results deviating from bulk values were found on the (110) and (111) planes. This anomaly can be understood in terms of changed effective masses of conduction electrons on the surface as a consequence of the quantization of the carrier wave function in the surface channel. Good agreement between theoretical results and experiment was found. It was observed that the quantization is efficient even at the threshold voltage (i.e., ∼2×104 V/cm) and occasions the anisotropic piezoresistance on the (110) surface.

Continuous-Representation Theory. III. On Functional Quantization of Classical Systems

The form of Schrödinger's equation in a continuous representation is indicated for general systems and analyzed in detail for elementary Bose and Fermi systems for which illustrative solutions are given. For any system, a natural continuous representation exists in which state vectors are expressed as continuous, bounded functions of the corresponding classical variables. The natural continuous representation is generated by a suitable set 𝔖 of unit vectors labeled by classical variables for which, for the system in question, the quantum action functional restricted to the domain 𝔖 is equivalent to the classical action. When a classical action is viewed in this manner it contains considerable information about the quantum system. Augmenting the classical action with some physical significance of its variables, we prove that the classical theory virtually determines the quantum theory for the Bose system, while it uniquely determines the quantum theory for the Fermi system.

The Wigner Distribution Function and Second Quantization in Phase Space

The Wigner Distribution Function and Second Quantization in Phase Space