T-invariance of relativistic first-order equations

We consider the time-inversion operation for relativistic first-order equations that are constructed without the use of multiple representations of the proper Lorentz group. It is proved that the matrix T of the antilinear time-inversion operation is symmetric (antisymmetric) for particles with integral (half-integral) spin. It is shown that for particles with integral (half-integral) spin there exists (does not exist) a linear transformation as a result of which the time-inversion operation reduces to the complex conjugate:(x)= _T ^*(x) where _T is a phase factor. A general expression for such a transformation is obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 5–9, December, 1988.     

Boundedness below for fermion model theories. Part II. The linear lower bound

The Generalised Yukawa Model (GY ₂), ± ^N+^2M, M>N>1, is bounded below, uniformly in a momentum cutoff, and linearly in the volume of a spatial cutoff.This article includes the major part of a doctoral dissertation submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in the Horace H. Rackham School of Graduate Studies at the University of Michigan.Junior Fellow, Michigan Society of Fellows.     

The collapse of the wave function

Probability distributions are seen to be observer dependent. The probability function can be put into an observer-dependent form. This eliminates the acausal behavior of the collapse of the wave function.     

Duality-invariant orthogonality relation between the magnetic monopole and associated quark charges

Duality invariance of the Dirac-Schwinger charge-symmetric theory for electromagnetism leads one to consider the complex-valued amplitudes ₁ and ₂ for the separation between the magnetic monopole and quarks in the logarithmic charge plane. It is observed that the orthogonality relation on the latter amplitudes, Re( ₁ ^* ₂)=0, is equivalent to the equation (ln 9 ^–1)(ln 2)=(1/2) ², which is indeed satisfied by the experimental value fora to within 0.027%. In addition to fixing the unit of electric charge at a primary physical value, the orientation of ₁, ₂ may also prescribe the Cabibbo angle to have the theoretical value 12.4438.     

Geometrical interpretation of the bilinear equations for the KP hierarchy

The Grassmann manifold approach to the KP hierarchy, in the spirit of Segal and Wilson, is used to define the wavefunction _W and its adjoint _W. From the fact that _W and _W are the orthogonal, we derive the bilinear equations. The modified equations are treated at the same time.     

First experiment to test whether space-time is flat. II. Effects of orbit

The gyroscope in an orbiting satellite will be acted on by additional gravitational fields due to the rotation of the earth and due to the orbital velocity of the satellite. According to special relativistic gravitational theory, we deduce _L ^(S) —the gyroscope's precession rate due to the orbital velocity—and _S ^(S) —the gyroscope's precession rate due to the earth's rotation in the polar orbit case. The results are _L ^(S) = (2/3) _L ^(G) , _S ^(S) = (3/2) cos (1 - sin² cos²)^1/2 _S ^(G) , where and are the gyroscope's polar angles, and _L ^(G) and _S ^(G) are the geodetic and frame-dragging precession rates predicted by general relativity, respectively.     

On the “Equivalence” of the Maxwell and Dirac Equations

It is shown that Maxwell's equation cannot be put into a spinor form that is equivalent to Dirac's equation. First of all, the spinor in the representation of the electromagnetic field bivector depends on only three independent complex components whereas the Dirac spinor depends on four. Second, Dirac's equation implies a complex structure specific to spin 1/2 particles that has no counterpart in Maxwell's equation. This complex structure makes fermions essentially different from bosons and therefore insures that there is no physically meaningful way to transform Maxwell's and Dirac's equations into each other.     

Yet another formulation of the Dirac equation

The 2-by-2 Pauli matrix algebra is used to write the 1-by-4 Dirac field in anequivalent 2-by-2 matrix . The current 4-vectors and *_µ are then compared and the latter is shown to not be easily interpretable as a probability density, and also tocontain .     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

Recovery of Dirac Equations from Their Solutions

We deal with quantum field theory in the restriction to external Bose fields. Let be the Dirac equation. We prove that a nonquantized Bose field is a functional of the Dirac field whenever this is canonical. Performing the verification for := m = const, which yields the free Dirac field, we also prepare the tedious verifications for all which are nonquantized and static. Such verifications must not be confused, however, with the proof of our formula, which is shown in detail.     

Are there more than five linearly-independent collision invariants for the Boltzmann equation?

The problem of finding the summational collision invariants for the Boltzmann equation is tackled with the aim of proving that the most general solution of the problem is not different from the standard one even when the equation defining a collision invariant is only satisfied almost everywhere inR ³×R ³×S ². The collision invariant is assumed to be in the Hilbert spaceH of the functions which are square integrable with respect to a Maxwellian weight.     

Spin manifolds,killing spinors and universality of the Hijazi inequality

In terms of the Dirac operator P, we introduce on any field a first-order operator D and show that the operator (–) on the spinors (=(n/4(n–1))R; dim W=n) is positive. By means of a universal formula, we show that, on a compact spin manifold of dimension 3, the Hijazi inequality [8] holds for every spinor field such that (P, P) = ²(, ) (=const.). In the limiting case, the manifold admits a Killing spinor which can be evaluated in terms of . Different properties of spin manifolds admitting Killing spinors are proved. D is nothing but the twistor operator.     

Universal Spin Structure

Any Dirac spin structure on a world manifold xis a subbundle of the composite spinor bundle , where is a bundle oftetrad gravitational fields. The bundle S admits generalcovariant transformations that enable us to discover theenergy-momentum conservation law in gauge gravitationtheory.     

On Relativistic Collisional Invariants

Two simple proofs of the result that a relativistic summational invariant is a linear combination of the momentum four-vector p are given by assuming that is a continuous and differentiable function of class C ². The results can be extended to the case when is just assumed to be a generalized function.     

Dynamical Analysis of Schrödinger Operators with Growing Sparse Potentials

We consider discrete half-line Schrödinger operators H with potentials of the form V(n)=S(n)+Q(n). Here Q is any compactly supported real function, if n=L_N and S(n)=0 otherwise, where (0,1) and L_N is a very fast growing sequence. We study in a rather detailed manner the time-averaged dynamics exp(–itH) for various initial states . In particular, for some we calculate explicitly the intermittency function ^–(p) which turns out to be nonconstant. The dynamical results obtained imply that the spectral measure of H has exact Hausdorff dimension for all boundary conditions, improving the result of Jitomirskaya and Last.Acknowledgement I would like to thank F. Germinet for useful discussions.     

MPT versus: A manifestly covariant presentation of motion reversal and particle-antiparticle exchange

We show that particle-antiparticle exchange and covariant motion reversal are two physically different aspects of the same mathematical transformation, either in the prequantal relativistic equation of motion of a charged point particle, in the general scheme of second quantization, or in the spinning wave equations of Dirac and of Petiau-Duffin-Kemmer. While, classically, charge reversal and rest mass reversal are equivalent operations, in the wave mechanical case mass reversal must be supplemented by exchange of the two adjoint equations, implying .Denoting by M the rest mass reversal, P the parity reversal, T the Racah time reversal, and Z the exchange, the connection with the usual scheme of charge conjugation, parity reversal, and Wigner motion reversal, is with, of course,     

Percolation of level sets for two-dimensional random fields with lattice symmetry

Let (x),x², be a random field, which may be viewed as the potential of an incompressible flow for which the trajectories follow the level lines of . Percolation methods are used to analyze the sizes of the connected components of level sets {x:(x)=h} and sets {x:(x)h} in several classes of random fields with lattice symmetry. In typical cases there is a sharp transition at a critical value ofh from exponential boundedness for such components to the existence of an unbounded component. In some examples, however, there is a nondegenerate interval of values ofh where components are bounded but not exponentially so, and in other cases each level set may be a single infinite line which visits every region of the lattice.     

Surface channeling of protons and electrons on reflection by a single crystal

Computer modeling is used to investigate the reflection of 0.67-MeV protons and 1-MeV electrons by the (110) face of an Au single crystal for angles of divergence < _cr of the incident beam with the direction 110 lying in the surface. The angular distributions and reflection coefficients of protons and electrons are compared. Several maxima are found in the angular distributions. In additon to the specular-reflection maximum at = there is a maximum at > for protons and a maximum at < for electrons. The maxima in the angular distributions are attributed to multiple scattering at atomic chains of the crystal surface. An analytic approximation of the reflection coefficient for protons K_cor() is given. The great contribution to the reflected flux from particles channeled in the surface layer allows it to be used for structural analysis of crystal surfaces.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 85–90, March, 1979.     

Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation

A rigorous analysis is given of the dynamics of the renormalization map associated to a discrete Schrödinger operatorH onl ²(), defined byH(n)=(n+1)+(n–1)+Vf(n)(n), whereV is a real parameter,f is a certain discontinuous period-1 function, and is the golden mean. The renormalization map forH is a diffeomorphism,T, of ³, preserving a cubic surfaceS _V . ForV8 we prove that the non-wandering set of the restriction ofT toS _v is a hyperbolic set, on whichT is conjugate to a subshift on six symbols. It follows from results in dynamical systems theory that the optimally approximating periodic operators toH have spectra which obey a global scaling law. We also define a set which we call the pseudospectrum of the operatorH. We prove it to be a Cantor set of measure zero, and obtain bounds on its Hausdorff dimension. It is an open question whether the pseudospectrum coincides with the spectrum ofH.