Time-reversal symmetry violation and the oscillating universe

The expressions for the fractional number ofK ⁰'s and¯K ⁰'s in a neutral kaon beam are discussed with reference to time-reversal asymmetry. The suggested relation between the sign of Re ( is the Lee-WuT-violation parameter) and the cosmological arrow of time ifCPT is broken is further clarified.Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under AFOSR grant number EOOAR-68-0010, through the European Office of Aerospace Research.     

Distinguishable equivalent particles with symmetrized wave functions

The quantum formalism ofdistinguishable, yetequivalent particles (with symmetric or antisymmetric wave functions) is here worked out. The result is an entirely explicit formulation of the way in which classical mechanics emerges from quantum mechanics for such particles. Distinguishability is achieved at the cost of dynamical precision; the two are, in fact, complementary.Research sponsored in part by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. 557–67.     

A generalized Reissner-Nordström solution

A new class of solutions to the Einstein-Maxwell equations is presented; it stands in the same relation to the Robinson & Robinson (1969) metrics as the Reissner-Nordström solution to Schwarzschild's.This work was supported in part by the Aerospace Research Laboratories, Office of Aerospace Research, United States Air Force, under Contract F33615-68-C-1675; Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. 903-67; National Aeronautics and Space Administration under NASA Grant No. NGL 44-004-001; and National Science Foundation under Travel Grant GP-14505.     

Rigged Hilbert spaces in quantum mechanics

It is shown how rigged Hilbert spaces may be constructed in quantum mechanics, and the properties of the resulting spaces are derived. The theory is applied to non-relativistic quantum systems ofn interacting particles. The spectral theory in rigged Hilbert spaces is developed and the results necessary for the application to the Dirac formalism are derived.The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research under Grant AF EOAR 65–36, through the European Office of Aerospace Research (OAR), United States Air Force.     

Matrix elements of representations of non-compact groups in a continuous basis

Explicit formulas are obtained by a simple algebraic method for the representations of the finite group transformations ofO(2,1) in a continuous basis when a non-compact generator is diagonalized. Compact and non-compact cases are treated in a unified form and the nature of analytic continuation is determined. The transformation function between the discrete and the continuous bases is also given. These explicit formulas have not been obtained in the literature before.Supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under Grant No. AF-AFOSR- 30–67.     

The monodromy rings of a class of self-energy graphs

The monodromy rings of self-energy graphs, with two vertices and an arbitrary number of connecting lines, are determined.Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant 68-1365.     

Pulsating stars in general relativity

The equation governing radial pulsations of fully relativistic stars is derived and expressed in terms of quantities which are continuous even across density discontinuities which occur, e.g., in zero-temperature stellar models that undergo electron capture. When expressed in terms of these quantities, the pulsation equation can be integrated through density discontinuities without any special treatment of these points being necessary. Expressions for the adiabatic index and pulsation energy are derived in a simple way.Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant 70-1866.     

Vacuum metrics without symmetry

Vacuum metrics admitting null, geodetic, expanding, shear-free congruences are investigated. A new solution is derived which contains three holomorphic functions of a complex variable. It includes, as special cases, a number of well-known solutions such as those of Kerr and Taub-NUT. In general, however, it admits no Killing vectors.This work was supported in part by Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. 903-67 and National Aeronautics and Space Administration under NASA Grant No. NGL 44-004-001.     

A variational principle for boundary value problems in kinetic theory

A variational principle which applies directly to the integrodifferential form of the linearized Boltzmann equation is introduced. Extremely general boundary conditions and collision terms are allowed. For a class of interesting problems, the value of the functional to be varied is shown to be closely related to quantities of great physical interest. The formalism is applied to the treatment of plane Couette flow for different forms of the collision term (BGK model, rigid spheres, Maxwell's molecules).Research sponsored by the Air Force Office of Scientific Research under contract F 61(052)-68-C-0020, through the European Office of Aerospace Research, OAR, United States Air Force.     

Cosmology as a science

In recent years, observational techniques at cosmological distances have been sufficiently improved that cosmology has become an empirical science, rather than a field for unchecked speculation. There remains the fact that its object, the whole universe, exists only once; hence, we are unable to separate general features from particular aspects of our universe. This might not be a serious drawback if we were justified in the belief that presently accepted laws of nature remain valid on the cosmological scale. In the author's opinion, however, there are grounds for doubting that belief. The three arguments presented are (1) the possibility that apparent constants of nature may, during cosmological times, turn out to vary (Dirac conjecture); (2) the effective breakdown of the principle of relativity caused by the effects of the cosmological environment on local experiments; and (3) the fact that present theory leads to field singularities at the early stages of the expanding universe, which might be a signal that currently accepted theoretical concepts are inadequate for an understanding of highly condensed matter.Presented at the Annual Meeting of the American Association for the Advancement of Science, 26–31 December 1969 at Boston, Massachusetts. The author's work has been supported by the Air Force Office of Scientific Research, under Grant AF AFOSR 789-66, and by Aerospace Research Laboratories, US Air Force, under Contract F 33615-70-C-1110.     

Anisotropy of the electron mean free path in the thermal conductivity of very pure gallium

In order to determine the anisotropy of the bulk mean free pathl _b, we have measured the thermal conductivity of 6–9's pure gallium single crystals as a function of size and crystal orientation between 1.4 and 4.2°K. The results were analyzed by adapting the existing theory for the electrical conductivity of thin wires to the thermal case. The results show that the reciprocal of the bulk mean free path can be expressed as a linear function ofT ³ for all three orientations. The residual contributions to this quantity are highly anisotropic, as expected. The electron-phonon contributions are, on the other hand, nearly equal in magnitude and exhibit a ratio of 3:4:4 for the A:B:C axis, respectively. In addition, an unusual dependence upond ⁻², whered is the specimen diameter, was observed for the C-axis. It is possibly related to the fact that for this direction only 6% of the entire free electron Fermi surface takes part in conduction which, as a consequence, may be largely dominated by Umklapp scattering. Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force under Grant No. AF-AFOSR 1051-66.     

Local fields with the wrong connection between spin and statistics

It is proved that there exist free field operators which satisfy local commutativity and which transform according to certain unitary representations of the homogeneous Lorentz group. The fields satisfy axioms similar to the Wightman axioms, and give rise to local algebras of observables obeying postulates similar to those suggested byHaag. They describe a tower of particles with spins 1/2, 1 1/2, 2 1/2, ..., but commute at space-like separation, giving rise to Bose statistics for the particles. This shows that the well-known theorem on spin and statistics cannot be extended to general theories of local observables; it also shows that the assumptions made in S-matrix theory do not hold for theS-matrix of a theory of interacting infinite fields.The research reported in this document has been sponsored in part by the Air Force Office ofScientific Research OAR through the European Office Aerospace Research U.S. Air Force.     

The representations of the oscillator group

Using the Mackey theory of induced representations all the unitary continuous irreducible representations of the 4-dimensional Lie groupG generated by the canonical variables and a positive definite quadratic hamiltonian are found. These are shown to be in a one to one correspondence with the orbits underG in the dual spaceG to the Lie algebraG ofG, and the representations are obtained from the orbits by inducing from one-dimensional representations provided complex subalgebras are admitted. Thus a construction analogous to that ofKirillov andBernat gives all the representations of this group.The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research OAR through the European Office Aerospace Research, United States Air Force.     

On the Šilov boundary of the vertex function

The ilov boundary of the vertex function is computed without resource to analytic completion techniques.The research reported in this document has been supported in part by the Aerospace Research Laboratories under Grant No. AF EOAR 65–80, through the European Office of Aerospace Research (OAR), U.S. Air Force.This work is part of a thesis submitted to the Senate of the Technion-Israel Institute of Technology, as partial fulfilment of the requirements for a Ph.D. degree.     

On the einstein-murphy interaction

This paper is a first attempt to reconcile the two great concepts of twentieth century physics: Einstein's theory of general relativity, and Murphy's law.This work has been supported in part by the Office of Aerospace Research, United States Air Force, on contract No. AF. AFOSR-$3.14159265.Refuses to divulge present address.     

On subalgebras which survive contraction

We prove the following theorem: if a subalgebraB of an algebraG is spanned by root vectors, then ifX is a regular element ofG, the limit {ie160-1} Ad exptX(B) exists and is isomorphic toB, i.e.B survives contraction withX. The algebraSL(2C) is considered as an example. In particular it is shown thatSL(2C) itself survives and applications to relativistic scattering theory are indicated.Research supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, under Grant No. AF-AFOSR-232-66.On leave of absence from the Central Research Institute for Physics. Address after September 15, 1967: Central Research Institute for Physics, Budapest, Hungary.N.A.T.O. post-doctoral fellow.     

Shock profile solutions of the Boltzmann equation

Shock waves in gas dynamics can be described by the Euler Navier-Stokes, or Boltzmann equations. We prove the existence of shock profile solutions of the Boltzmann equation for shocks which are weak. The shock is written as a truncated expansion in powers of the shock strength, the first two terms of which come exactly from the Taylor tanh (x) profile for the Navier-Stokes solution. The full solution is found by a projection method like the Lyapunov-Schmidt method as a bifurcation from the constant state in which the bifurcation parameter is the difference between the speed of soundc ₀ and the shock speeds.Research supported in part by the National Science Foundation, the Army Research Office, the Air Force of Scientific Research, the Office of Naval Research, and the Department of Energy     

Poincaré cycles and ergodic behavior of a linear diatomic chain

The ergodic behavior of a linear diatomic chain is shown to be analogous to that of a linear monatomic chain. Starting with the expressions for the time-relaxed correlation functions between any two particles in the chain, we show that the existence of Poincaré cycles is not inconsistent with the development of an equilibrium state. Also, we show that those dynamical variables that are ergodic for the linear monoatomic chain remain ergodic in the diatomic chain. It is shown that the autocorrelation functions for particles with equal or different masses decay in time ast ^–1/2 .This work was completed when the author was associated with the Institute for Fluid Dynamics and Applied Mathematics, University of Maryland and supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under contract No. AF18(600)1315.     

Positivity and self adjointness of theP(φ)2 Hamiltonian

We give a new proof that the locally correct HamiltonianH(g) is self adjoint, and that the vacuum energyE(g)=inf spectrumH(g) satisfies –O(D)E(g), where 0g1 andD=diam.supp.g.Supported in part by Air Force Office of Scientific Research, Contract AF 49(638) -1719.On leave at Princeton University. Alfred P. Sloan Foundation Fellow. Supported in part by Air Force Office of Scientific Research, Contract F44620-70-C-0030.     

Proper-connected-diagram expansion of electrical conductivity for electron-impurity system

The straightforward expansion of the electric conductivity for an electron-impurity system in powers of the impurity densityn _s is known to diverge term by term. It is shown that this difficulty which arises from the average potential field of impurities, can be removed by using the expansion in terms of proper connected diagrams. This expansion, however, leads to a complicated dependence of onn _s . The analytical behavior of(n _s ) in the neighborhood ofn _s =0 is investigated qualitatively with the assumption that the interaction potential between electron and impurity is arbitrarily strong but has a finite range. It is found that this investigation doesnot support the argument recently advanced by several authors that(n _s ) should contain in its asymptotic expansion terms in the combination ofn _s and lnn _s The present theory starts with Kubo's current correlation function formula for and proceeds with the aid of connected diagrams. It is developed for a quantum system but also applies for a classical system with few modifications in definitions.Research sponsored in part by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. 854-65 and by the National Science Foundation, NSF No. GP 9040.