The use of time-dependent projection operators in the derivation of master equations

In this paper we generalize our previous work on the use of time-dependent projection operators for the derivation of master equations for general systems. Previously we had generalized the usual time-independent projection operator approach to include time-dependent projection operators, in which the relevant part of the full density operator is considered to be the uncorrelated part of the full density operator. The irrelevant part of the density operator was then the part describing the correlations between the coupled systems. In the present work we present new time-dependent projections operators which have the property that some correlations between the interacting subsystems are placed in the relevant part of the distribution function and the remaining correlations are placed in the irrelevant part of the distribution function.     

Beautiful Gauge Field Equations in Clifforms

By combining the tetrads of unholonomic frames of spacetime with the Dirac matrices to a one-form , we can reformulate not only the Dirac equation, but also the Einstein equations and supergravity in a very concise form. These Clifforms also shed some light on the chiral decomposition à la Ashtekar, the role of the axion as a dynamical degree of freedom dual to the torsion of the Einstein—Cartan theory, and the role of the Seiberg—Witten equation for S-duality.     

Quantum-Mechanical Uncertainty and the Stability of Incompatibility

In talking about the compatibility of quantum observables, discussions often center on the question of whether the corresponding operators commute—even though commutativity is a coarse-grained notion that largely fails to capture the salient nonclassical features of quantum theory. Often, too, such discussions involve the issue of whether the operators in question satisfy a Heisenberg-like inequality, of the form A·Br>0—even though such inequalities are specific to unbounded operators and (for this and other reasons) are typically not a useful way to discuss joint uncertainty in quantum mechanics. In the present paper we emphasize a simpler dichotomy, in which operator pairs (A, B) are classified according to whether or not states can be found with arbitrarily small dispersions in both A and B. If A and B cannot be made arbitrarily dispersionless simultaneously, then we call A and B an uncertainty pair. Otherwise, we call A and B a certainty pair. An interesting feature of uncertainty pairs in particular is that they are stable, in the sense that if A and B form an uncertainty pair, then slight enough perturbations of A and B must also form an uncertainty pair. This stability, obvious in the finite-dimensional case, follows in general from an operator inequality derived herein. A consequence of this inequality is that perturbed position and momentum operators X+X and P+P cannot share an eigenvector unless |X|·|P|/2. (Here vertical bars denote the operator norm.) This, despite the fact that (as we show) arbitrarily slight perturbations of X and P can fail to satisfy a Heisenberg inequality—a fact which raises interesting measurement issues in its own right.     

Eingenvalues of the Dirac operator on compact Kähler manifolds

Kählerian twistor operators are introduced to get lower bounds for the eigenvalues of the Dirac operator on compact spin Kähler manifolds. In odd complex dimensions, manifolds with the smallest eigenvalues are characterized by an over determined system of differential equations similar to the Riemannian case. In these dimensions, we show the existence of a unique natural Kählerian twistor operator. It is also proved that, on a Kähler manifold with nonzero scalar curvature, the space of Riemannian twistor-spinors is trivial.This work has been partially supported by the EEC programme GADGET Contract Nr. SC1-0105     

Relativistic scattering theory for long-range potentials of the nonelectrostatic type

We define a modified free-time evolution for the Dirac equation with long-range potentials (1/|x|), where is the Dirac matrix, and prove a strong asymptotic completeness of the corresponding wave operators. Our methods also work for the magnetic fields ·A(x).     

On stress-energy tensors

We show how to introduce the Noether Operator of a (possibly constrained) variational principle even when the Lagrangian contains spinor fields (and their derivatives to any finite order). After relating that operator to the so-called canonical and symmetric stress-energy tensors, we construct explicitly the divergence by which these differ. A brief appendix illustrates the method of dealing with spinors by calculating T^v for the Dirac equation.     

Algebraically Self-Consistent Quasiclassical Approximation on Phase Space

The Wigner–Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary * operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of Thomas–Fermi theory, and a new quantization rule which—unlike semiclassical quantization—is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism.     

Block Renormalization Group Approach for Correlation Functions of Interacting Fermions

Inspired by a decomposition of the lattice Laplacian operator into massive terms (coming from the use of the block renormalization group transformation for bosonic systems), we establish a telescopic decomposition of the Dirac operator into massive terms, with a property named orthogonality between scales. Making a change of Grassmann variables and writing the initial fields in terms of the eigenfunctions of the operators related to this decomposition, we propose a multiscale structure for the generating function of interacting fermions. Due to the orthogonality property we obtain simple formulas, establishing a trivial link between the correlation functions and the effective potential theories. In particular, for the infrared analysis of some asymptotically free models, the two point correlation function is written as a dominant term (decaying at large distances as the free propagator) plus a correction with faster decay, and the study of both terms is straightforward once the effective potential theory is controlled.     

The unitary bogoliubov transformation for a quantized scalar field in a friedman space-time

The time-dependent creation and annihilation operators for a complex scalar field, in a Friedmann space-time, defining particle states with respect to which the Hamiltonian is diagonal, are related by a Bogoliubov transformation to the creation and annihilation operators defined in strict analogy with the procedure carried out in Minkowski space. The Bogoliubov transformation is here written in terms of a unitary operator,U, and an expression for that operator is found via the generating functionF=i InU. The properties of the representation obtained by makingU act upon the state vector , to give a new state _U, are discussed. It is shown that the particle-number operator remains constant in such a picture so that the evolution of the system with time is clearly seen to depend upon the energy _k on the one hand, and upon the state vector _U on the other. Also, it is pointed out that this new representation permits the in and out states to be defined unambiguously.On leave of absence from Istituto de Fisica G. Galilei (Padova) and Istituto Nazionale di Fisica Nucleare (Sezione di Padova).     

The additivity of the η-invariant. The case of a singular tangential operator

We prove the decomposition formula for the -invariant of the compatible Dirac operator on a closed manifoldM which is a sum of two submanifolds with common boundary.Research partially supported by NSF     

The Phase Transition in Statistical Models Defined on Farey Fractions

We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as spin chains, with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle–Perron–Frobenius operator), which is defined using the maps (presentation functions) generating the Farey tree. The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form C [ ln² ]^–1. The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition.     

A Velocity Field and Operator for Spinning Particles in (Nonrelativistic) Quantum Mechanics

Starting from the formal expressions of the hydrodynamical (or local) quantities employed in the applications of Clifford algebras to quantum mechanics, we introduce—in terms of the ordinary tensorial language—a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (nonrelativistic) velocity operator for a spin-1/2 particle. This operator appears as the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass), and of a second part associated with the socalled Zitterbewegung, which is the spin internal motion observed in the center-of-mass from (CMF). This spin component of the velocity operator is nonzero not only in the Pauli theoretical framework, i.e., in the presence of external electromagnetic fields with a nonconstant spin function, but also in the Schrödinger case, when the wavefunction is a spin eigenstate. Thus, one gets even in the latter case a decomposition of the velocity field for the Madelung fluid into two distinct parts, which constitutes the nonrelativistic analogue of the Gordon decomposition for the Dirac current. Explicit calculations are presented for the velocity field in the particular cases of the hydrogen atom, of a spherical well potential, and of an electron in a uniform magnetic field. We find, furthermore, that the Zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In the presence of a nonuniform spinvector (Pauli case) we have, besides the component of the local velocity normal to the spin (present even in the Schrödinger theory), also a component which is parallel to the curl of the spin vector.     

Dirac Equation in Lemaįtre—Tolman—Bondi Models

The Dirac equation is studied for a sufficiently large class of Lematre—Tolman—Bondi cosmological models. While the angular equation (whose solution is known) separates directly, the spatial and temporal dependence de-couples only after a suitable separation procedure. The separated time equation is integrated by series. The separated spatial equation still depends on an arbitrary function relative to the integration of the cosmological model.     

On the geometric origin of the equation ϕ,11 — ϕ,22 = eϕ — e-2ϕ

It is shown that the equation ,11 — ,22 = e — e^-2 determines the intrinsic geometry of the two-dimensional affine sphere in the three-dimensional unimodular affine space like the sine-Gordon equation describes the metric on the surface of a constant negative curvature in the three-dimensional Euclidean space. The linear equations that determine the moving frame on the affine sphere are the Lax operators to the equation ,11 — ,22 = e — e^-2.     

MSA for segregating liquid mixtures: Crossover from mean-spherical to mean-field exponents

The critical exponents , , and of a binary mixture of equal-sized hard spheres with repulsive Yukawa interactions for unlike-atom pairs and attractive Yukawa interactions for like-atom pairs are calculated from analytical solutions in the mean spherical approximation. For strong to moderate screening we find the expected mean spherical values of the critical exponents; for weak screening the system shows a mean field behaviour. For an intermediate range of the potential, the system displays a crossover from mean-field behaviour to mean-spherical behaviour in the immediate vicinity of the critical point. The domain over which the mean-spherical exponents describe the critical behaviour shrinks as the range of the interatomic potential increases.     

“Square root” of the Dirac equation in extended supersymmetry

We discuss the possibility of extracting the square root from the Dirac equation in N-extended supersymmetry, with the aim of constructing a more fundamental dynamical theory. Although a square root of the Dirac operator can be defined in N-extended superspace for N2, it is not possible to construct with its help a new dynamical model that meets the standard requirements imposed on the theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 5–9, March, 1990.     

A Bound on Binding Energies and Mass Renormalization in Models of Quantum Electrodynamics

We study three models of matter coupled to the ultraviolet cutoff, quantized radiation field and to the Coulomb potential of arbitrarily many nuclei. Two are nonrelativistic: the first uses the kinetic energy (p+eA(x))² and the second uses the Pauli–Fierz energy (p+eA(x))²+eB(x). The third, no-pair model, is relativistic and replaces the kinetic energy with the Dirac operator D(A), but restricted to its positive spectral subspace, which is the electron subspace. In each case we are able to give an upper bound to the binding energy–as distinct from the less difficult ground state energy. This implies, for the first time we believe, an estimate, albeit a crude one, of the mass renormalization in these theories.     

Convergence of observables on quantum logics

We define two types of convergence for observables on a quantum logic which we call M-weak and uniform M-weak convergence. These convergence modes correspond to weak convergence of probability measures. They are motivated by the idea that two (in general unbounded) observables are close if bounded functions of them are close. We show that M-weak and uniform M-weak convergence generalize strong resolvent and norm resolvent convergence for self-adjoint operators on a Hilbert space. Also, these types of convergence strengthen the weak operator convergence and operator norm convergence of bounded self-adjoint operators on a Hilbert space. Finally, we consider spectral perturbation by showing that the spectra of approximating observables approach the spectrum of the limit in a certain sense.     

One massless particle equals two Dirac singletons

The remarkable representations of the 3+2 de Sitter group, discovered by Dirac, later called singleton representations and here denoted Di and Rac, are shown to possess the following truly remarkable property: Each of the direct products Di Di, Di Rac, and Rac Rac decomposes into a direct sum of unitary, irreducible representations, each of which admits an extension to a unitary, irreducible representation of the conformal group SO(4, 2). Therefore, in de Sitter space, every state of a free, massless particle may be interpreted as a state of two free singletons — and vice versa. The term massless is associated with a set of particle-like representations of SO(3, 2) that, besides the noted conformal extension, exhibit other phenomena typical of masslessness, especially gauge invariance.