Second-quantized Theory Research Articles

Agreement of stochastic soliton formalism with second-quantized and configuration-space models

The stochastic theory presented by Drummond, Gardiner, and Walls [Phys. Rev. A 24, 914 (1981)] is an interesting approach to problems in quantum optics. In this theory, an exact, quantum evolution is written in terms of classical functions (not operators) driven by explicit, quantum noise. We examine the origin of uncertainty in the formalism through the simple example of a single, nonlinear oscillator. We then test the stochastic theory applied to the problem of soliton propagation. We extend the linearized stochastic model by computing analytically quantum uncertainties in the four basic soliton parameters: photon number, momentum, phase, and position. Agreement with second-quantized and configuration-space soliton theories verifies the stochastic formalism.

Can the linear operator in physics cause chaos?

In this paper it is shown that annihilation operator in the theory of second quantization and quantum fields theory in physics is linear chaotic operator.

N-quantum approach to the BCS theory of superconductivity

A method of general applicability to the solution of second-quantized field theories at finite temperature is illustrated using the BCS (Bardeen–Cooper–Schrieffer) model of superconductivity. Finite-temperature field theory is treated using the thermo field-theory formalism of Umezawa and collaborators. The solution of the field theory uses an expansion in thermal modes analogous to the Haag expansion in asymptotic fields used in the N-quantum approximation at zero temperature. The lowest approximation gives the usual gap equation.

Quantum-noise limits to matter-wave interferometry.

We derive the quantum limits for an atomic interferometer from a second-quantized theory in which the atoms obey either Bose-Einstein or Fermi-Dirac statistics. It is found that the limiting quantum noise is due to the uncertainty associated with the particle sorting between the two branches of the interferometer, and that this noise can be reduced in a sufficiently dense atomic beam by using fermions as opposed to bosons. As an example, the quantum-limited sensitivity of a generic matter-wave gyroscope is calculated and compared with that of a laser gyroscope.

CLOSURE OF THE SUPER-POINCARÉ ALGEBRA FOR LIGHT-CONE GAUGE HETEROTIC SUPERSTRINGS

Functional methods are used to give a complete closure of the second-quantized field theory nonlinear realization of the ten-dimensional super-Poincaré algebra for the heterotic superstring in the Green-Schwarz formalism. New terms in the fundamental generators JI−, J+− are constructed at quartic order in the fields so as to close the Poincare algebra; a quartic term in the Hamiltonian is also derived. No higher order terms are needed for any generators of the full algebra.

Excess noise in gain-guided amplifiers

A second-quantized theory of the radiation field is used to study the origin of the excess noise observed in gain-guided amplifiers. We find that the reduction of the signal-to-noise ratio is a function of the length of the amplifier, and thus the enhancement of the noise is a propagation effect arising from longitudinally inhomogeneous gain of the noise rather than from an excess of local spontaneous emission. We confirm this conclusion by showing that the microscopic rate of spontaneous emission into a given non-power-orthogonal cavity mode is not enhanced by the Petermann factor. In addition, we illustrate the difficulties associated with photon statistics for this and other open systems by showing that no acceptable family of photon-number operators corresponds to a set of non-power-orthogonal cavity modes.

The spin factor in knots and a relativistic treatment of the Bose-Fermi trasmutation in second-quantized theories

The transmutation of a charged scalar field into a Dirac field is studied. The self-energy of the scalar particle coupled with the Chern-Simons gauge field is evaluated without the loop-splitting regularization. A relation between the spin factor and the self-energy is clarified with the help of a topological feature of a knot. The propagation of a spinning particle in three dimensions is described in terms of the functional integral over random paths with the spin factor. It is shown that the partition function of the charged scalar field is exactly identical with that of a Dirac field in the language of the functional integral.

BRST quantization, IOSp ( D, 2|2) invariance and the PCT theorem

In this paper we formulate the BRST quantization of the spinless relativistic point particle without restricting the parameter space to one fundamental domain of the modular group. Modular invariance is instead enforced by a constraint on the physical subspace of the Hilbert space. The second-quantized theory has then a manifest IOSp( D, 2|2) invariance. Studying this symmetry we find that a special element of the SO( D, 2) subgroup of IOSp( D, 2|2) can be identified which induces the PCT transformation on the physical subspace. Using this, the PCT theorem is derived as a simple consequence of the IOSp( D, 2|2) invariance, thus confirming the relevant role of this graded symmetry. The subgroup SO( D, 2) seems to play a central role also in another respect: by requiring covariance of the BRST quantization with respect to it, and not just invariance with respect to the Lorentz group SO( D − 1,1) the overall IOSp( D, 2|2) symmetry is generated.

Quantum theory of amplified spontaneous emission: scaling properties

We formulate a second-quantized theory of propagation in laser-active media and apply it to the description of amplified spontaneous emission for the case of homogeneously broadened three-level atoms in a rodlike geometry with arbitrary Fresnel number. The electromagnetic field is treated in paraxial approximation by an ad hoc quantization scheme, and spontaneous emission into off-axis modes is described by noise operator methods. We show by a scaling (dimensional) analysis how to derive the characteristic fields and lengths for superfluorescence and amplified spontaneous emission. The results show that dye media can be used as experimental analogs for x-ray lasers.

Second quantization of a spinless relativistic particle and quantum supersymmetry

A second-quantized theory of a relativistic particle with one-dimensional linear supersymmetry is obtained, starting from the Batalin-Fradkin-Vilkovisky (BFV) first-quantization method.

BRST ghosts as worldsheet parameters in gauge-invariant string field theory

The closed and open bosonic strings are quantized in the Tomonaga-Schwinger-Dirac (TSD) formalism. This leads to a gauge-invariant second-quantized free string field theory. The worldsheet parameters are dynamical variables which, in the quantum theory, are represented by anticommuting operators. This TSD quantization is seen to be formally identical to the BRST quantization and provides a geometrical interpretation of the anticommuting BRST ghosts as quantized worldsheet parameters.

Interaction of chemical bonds: Strictly localized wave functions in orthogonal basis

A second-quantized theory is presented with the aim to study the nature and interactions of well-localizable chemical bonds in molecules. The basis set is partitioned by assigning each basis function to a chemical bond possessing two electrons. The Schr\"odinger equation within each limited basis subset is solved exactly for each bond, leading to correlated, strictly localized, intrabond wave functions. The total many-electron wave function $\ensuremath{\Psi}$ is defined as the antisymmetrized product of these strictly localized geminals. The second-quantized Hamiltonian $\stackrel{^}{H}$ is partitioned as $\stackrel{^}{H}={\stackrel{^}{h}}^{0}+\stackrel{^}{W}$, where the zeroth-order Hamiltonian ${\stackrel{^}{h}}^{0}$ contains all terms which contribute to the energy of the strictly localized wave function $\ensuremath{\Psi}$, while the expectation value of $\stackrel{^}{W}$, as calculated by $\ensuremath{\Psi}$, is zero. Since ${\stackrel{^}{h}}^{0}$ is not simply the sum of intrabond Hamiltonians, the strictly localized wave function accounts for certain interbond interactions (inductive effects). The concept of bond creation and annihilation operators is introduced which formally shows Bose-type behavior since they refer to a two-electron composite system.

Second-quantized theory of Anderson localization ind=2+ε

Renormalized perturbation-theory methods with dimensional regularization are applied to the localization transition of electrons in random potentials within a field theory where the generating functional is the configurational averaged ${〈{Z}^{n}〉}_{\mathrm{av}}$ for $Z$ the vacuum-to-vacuum amplitude expressed as a functional integral over Grassman fields, in the replica limit $n=0$. The bare parameters of the theory are the Fermi level ${E}_{0}$ and the variance of the random potential ${W}_{0}$. Power counting says that at dimensionality $d=2+\ensuremath{\epsilon}$ the theory is super-renormalizable. Renormalization of the inverse propagator by the definition of renormalized Fermi level and interaction, ${E}_{F}$ and $u=(\frac{\ensuremath{\pi}{W}_{0}}{{E}_{F}}){\ensuremath{\kappa}}^{d\ensuremath{-}2}$, respectively, in order to cancel the single dimensional pole, leads to the same Wilson function $\ensuremath{\beta}(u)$ as for the compact nonlinear $\ensuremath{\sigma}$ model when the scale parameter $\ensuremath{\kappa}$ is varied at constant $\frac{{W}_{0}}{{E}_{0}}$. The conductivity is calculated in a perturbation expansion in ${(\ensuremath{\tau}{E}_{F})}^{\ensuremath{-}1}$, with ${\ensuremath{\tau}}^{\ensuremath{-}1}=\ensuremath{\pi}{W}_{0}{E}_{F}^{\frac{d}{2}\ensuremath{-}1}$ being the inverse lifetime, at $d=2+\ensuremath{\epsilon}$. It is explicitly shown that in ultraviolet-divergent $d$-dimensional loop integrals over advanced and retarded propagators, the leading term is regular while the dimensional pole occurs in the next-to-leading term. Then to leading order the conductivity ${\ensuremath{\sigma}}_{0}(\ensuremath{\omega})$ is regular while the first correction that includes the diffusion modes has a dimensional pole with residue $\ensuremath{\approx}{(\frac{i\ensuremath{\omega}}{{\ensuremath{\kappa}}^{2}})}^{\frac{\ensuremath{\epsilon}}{2}}$. To cancel this pole a renormalized inverse conductance $t(u)$ is defined, and the new Wilson function $\ensuremath{\beta}(t)$ obtained by varying $\ensuremath{\kappa}$ at constant "bare" interaction $\frac{{W}_{0}}{{E}_{F}}$ coincides with the scaling theory of Abrahams, Anderson, Licciardello, and Ramakrishnan. Scaling laws are derived from the solution of the renormalization-group equation for the conductivity.

Propagational effects in an ab initio theory of super-radiance from extended systems

The authors report a comprehensive ab initio all-mode second-quantized theory of super-radiance from extended systems. Spatially-dependent chirps during the cooperative motion are not consistent with the assumption of a single or symmetrical two-mode solution. Numerical integration of an associated set of c-number equations shows that super-radiance is characterized in general by a train of pulses in agreement with the two experiments. Propagational effects have an important effect on the cooperative radiative emission.

Bose quantization of a new spin-1/2 equation

We second quantize a relativistic Schrodinger equation involving a HamiltonianH that describes free spin-1/2 particles and that depends on a parameterG. We require a positive definite metric and a positive definite energy in the Fock space in which the field ψ(x,t) and its adjoint operate. IfG=±i, one obtains the usual second-quantized Dirac theory, but for real values ofG one has Bose statistics. Whereas the anticommutator [ψ(x,t), ψ * (x′,t′)]+ vanishes for a Dirac field when the interval between (x,t) and (x′,t′) lies outside the light cone, whenG is real the commutator [ψ (x,t), ψ * (x′,t′)− vanishes for such points.

The exponential potential and the Klein-Gordon equation

We analytically solve thes-wave radial Klein-Gordon equation with an exponential potential. We study in detail the behaviour of particle and antiparticle bound states and phase shifts as a function of the range and strength of the potential. We find our results to be qualitatively similar to those found for the square-well potential. As in the nonrelativistic case for the exponential potential, redundant poles appear in theS-matrix. Some difficulties connected with the interpretation of these results in the framework of a second-quantized theory are briefly considered.

Parastatistics with variable numbers of particles

We examine the properties of two possible first-quantized theories for a variable number of identical paraparticles. In one (the « ray » theory), states are described by vectors in a Fock space constructed as the direct sum of the fixed-particle « ray » Hilbert spaces of Hartle and Taylor; in the other (the « generalized-ray » theory), the Fock space is constructed from the « generalized ray » Hilbert spaces of Messiah and Greenberg. The «ray » theory, which is equivalent to the second-quantized parafield theory, appears to be consistent with all reasonable physical requirements. However, we show that the « generalized-ray » theory cannot consistently describe systems in which the particle numbers can vary. This is achieved by showing that an absolute selection rule derived by Messiah and Greenberg in the « generalized ray » framework prohibits the creation and destruction of particles.

Correspondence between the first- and second-quantized theories of paraparticles

It is shown that the parabosons and parafermions of the second-quantized parafield theory are equivalent to the finite-order paraparticles of the first-quantized theory. We follow the prescription of Hartle and Taylor which eliminates the redundancy of the generalized ray from the first-quantized formalism. For each statistical type of particle this allows us to establish an isomorphism between the state spaces of the two theories, in such a way that all corresponding observables have identical matrix elements for all states. This means that any system of finite-order particles can equally well be described by either the first- or the second-quantized formalism, in the same sense as applies to ordinary bosons and fermions. We analyze the different conclusions reached by other authors and explain the reasons for our disagreement.

Classification of Paraparticles

Hartle and Taylor have shown that the cluster law imposes certain restrictions on the allowed symmetries of first-quantized systems of identical particles. We extend their results to show that all particles which are neither bosons nor fermions and which obey the cluster law can be divided into two classes, those of finite order, and those of infinite order. For every positive integer $p$ there are two types of finite-order particle, which we call parabosons and parafermions of order $p$. Ordinary bosons and fermions can be fitted into this scheme as particles of order 1. We conjecture that the finite-order particles can be identified with the parafermions and parabosons of the second-quantized, parafield theory. Infinite-order particles would seem to have no analog in the second-quantized theory, as presently formulated.

Asymptotic states in quantum electrodynamics

The time evolution of a system of charged particles due to the long-range part of the Coulomb interaction and the interaction with the soft part of the transverse radiation field is studied within the frame-work of frist-quantized theory of the charged particles and second-quantized theory of the electromagnetic field.States are defined which may serve as asymptotic states in a scattering theory. They differ from the free states by a phase factor (generalization of the Coulomb phase) and by the presence of a coherent soft radiation field. The time development of the phase factor and of the wave function that describes the coherent field is studied in detail and expressed in terms of elementary functions.An outline is made of an S-matrix theory based on the asymptotic states thus defined.