Showing papers in "International Journal of Theoretical Physics in 1983"

Bell's Theorem and Backwards-in-Time Causality

Abstract: Bell has shown that quantum mechanics is incompatible with the notion of locality. The present paper begins by considering the possibility of invoking backwards-in-time causality to explain this violation of locality. This then leads to an examination of the possible relevance of backwards-in-time causality to measurement outcomes in general. It is found that the situations in which it could be involved are limited owing to causality paradoxes.

Algebraic Determination of the Metric from the Curvature in General Relativity

Abstract: The general solution for a symmetric second-order tensorX of the equationX e(a R e b cd=0 whereR is the Riemann tensor of a space-time manifold, andX is obtained in terms of the curvature 2-form structure ofR by a straightforward geometrical technique, and agrees with that given by McIntosh and Halford using a different procedure. Two results of earlier authors are derived as simple corollaries of the general theorem.

Second Quantization, Projective Modules, and Local Gauge Invariance

Abstract: Bundles and bundle structures have gained wide currency in modern approaches to certain topics in quantum physics, significant applications appearing in connection with gauge theories (e.g., Atiyah and Jones, 1978), theories of geometric quantization (e.g., Kostant, 1970; Sniatycki, 1974), and in numerous other contexts. In this paper we argue that such structures can already be discerned in the most elementary notions of second quantization, albeit in disguised form. An examination of the methods traditionally used by physicists in dealing quantum mechanically with systems exhibiting an infinite number of degrees of freedom reveals, almost from the outset, the implicit use of module structures over algebras of functions (Section 2). By making these structures explicit and adapting some results of perturbation theory we arrive at an association between bare particles and finitely generated projective modules (Sections 3 and 4). In particular, rank one modules emerge naturally, for algebraic reasons, as the appropriate descriptors of bosons in this association. (This provides a possible setting for the development of standard geometric quantization theory.) As a first application of the formalism we show the existence of phononlike excitations in general many-fermion systems. When these ideas are further specialized (local) gauge theoretical notions arise in a natural way out of a consideration of the bundles obtained via Swan's theorem. These theories emerge moreover equipped with an interpretation linked directly to the geometrical entities associated with the underlying bundles. Thus for example in the line bundle (or rank one) case,

Principle of Equivalence and Electromagnetism

Abstract: We show that the local expression of the electrostatic potential of a point charge suggested from the equivalence principle is different of the one resulting from the global consideration in the Schwarzschild space-time.

Vanishing of the Cosmological Constant in Non-Abelian Kaluza-Klein Theories

Abstract: We present a new approach to the unification of gravity and non-Abelian gauge fields in the framework of Kaluza-Klein theory. It consists in introducing a new connection on the (n + 4)-dimensional manifoldP (metrized principal fiber bundle). This connection is metrical, but with nonvanishing torsion. An enormous cosmological term in the Einstein equations vanishes due to this connection. The new connection simultaneously cancels Planck's mass term in the Dirac equation for the five-dimensional case. The usual interpretation of geodesic equations is still valid.

Minimal number of operators for observability ofN-level quantum systems

Abstract: This paper discusses the minimal numbernmin of operatorsA1,...,A n , whose expectation values at some instants determine the statistical state of anN-level quantum system. We assume that the macroscopic information about the system in question is given by the mean values Tr[ρ(t j )A i ]=m i (t j ) ofn self-adjoint operatorsA1,...,A n at some instantst1

Relativistic quantum logic

Abstract: On the basis of the well-known quantum logic and quantum probability a formal language of relativistic quantum physics is developed This language incorporates quantum logical as well as relativistic restrictions It is shown that relativity imposes serious restrictions on the validity regions of propositions in space-time By an additional postulate this relativistic quantum logic can be made consistent The results of this paper are derived exclusively within the formal quantum language; they are, however, in accordance with well-known facts of relativistic quantum physics in Hilbert space

Inverse problem in classical mechanics: Dissipative systems

Abstract: We show how the ambiguity of Lagrangian and Hamiltonian descriptions for conservative systems gives rise to an analogous ambiguity for dissipative systems. For a subclass of them we also give a Lagrangian description.

Quantum mechanics and the local observer

Abstract: A notion of local observer inspired by the work of Segal is introduced here in the Hilbert space theory of quantum mechanics. The local observer finds a mathematical place in the Hilbert space through local negation or complementation. A logicomathematical theory of local negation is presented and its implications for quantum logic and the problem of measurement are discussed. The setting is constructivist mathematics and the main result of the paper states that the introduction of a local observer implies the nonorthocomplementability of the whole Hilbert space even in the finite-dimensional case. Making a mathematical place for the observer (the “projector”) thus modifies the structure of the observables or the system of the projections, in accordance with a nonclassical theory of quantum-mechanical measurement.

Coupling of Quantum Logics

Abstract: A quantum logic is a couple (L, M), whereL is a logic andM is a quite full set of states onL. A tensor product in the category of quantum logics is defined and a comparison with the definition of free orthodistributive product of orthomodular σ lattices is given. Several physically important cases are treated.

Hilbertian quantum theory as the theory of complementarity

Abstract: It is demonstrated that the notion of complementary physical quantities assumes the possibility of performing ideal first-kind measurements of such quantities. This then leads to an axiomatic reconstruction of the Hilbertian quantum theory based on the complementarity principle and on its connection with the measurement theoretical idealization known as the projection postulate. As the notion of complementary physical quantities does not presuppose the notion of probability, the given axiomatic reconstruction reveals complementarity as an essential reason for the irreducibly probabilistic nature of the quantum theory.

Analysis of the Einstein-Podolsky-Rosen experiment by relativistic quantum logic

Abstract: The EPR experiment is investigated within the abstract language of relativistic quantum physics (relativistic quantum logic). First we show that the principles of reality (R) and locality (L) contradict the validity principle (Q) of quantum physics. A reformulation of this argument is then given in terms of relativistic quantum logic which is based on the principlesR andQ. It is shown that the principleL must be replaced by a convenient relaxation ¯L, by which the contradiction can be eliminated. On the other hand this weak locality principle ¯L does not contradict Einstein causality and is thus in accordance with special relativity.

Reflections on quantum logic

Abstract: This is a short, self-contained summary of problems connected with the interpretation of state vectors in quantum mechanics. We discuss the reconstruction of the “ψ function” from statistical data, some related mathematical questions, the classical “paradoxes,” the probability interpretation of the state vectors, and, finally, quantum logic in relation to hidden variable theories and Hilbert space formalism, to build up a consistent framework for the indeterministic quantum picture of nature.

A Quantitative Occam's Razor

Abstract: Interpreting entropy as a prior probability suggests a universal but “purely empirical” measure of “goodness of fit.” This allows statistical techniques to be used in situations where the correct theory- and not just its parameters-is still unknown. As developed illustratively for least-squares nonlinear regression, the measure proves to be a transformation of theR2 statistic. Unlike the latter, however, it diminishes rapidly as the number of fitting parameters increases.

Causality between Preparation and Registration Processes in Relativistic Quantum Theory

Abstract: A causality postulate is considered which is based on the conception of systems that are prepared in some finite region of space-time and recorded in some other region. If these regions are spacelike separated, the recording apparatus should react as if no preparing apparatus were present, i.e., it should respond with at most some vacuum rate. The causality postulate is mathematically formulated within the framework of statistical theories. The connections with algebraic field theory are discussed and the relation between causality and spectral conditions is studied. General methods for constructing systems satisfying the causality postulate are given and applied in several examples.

Geometric dequantization and the correspondence problem

Abstract: On the way to settle a conjecture proposed by Mackey, we first present in detail a complete solution to the correspondence problem for systems whose configuration space isR n . We then indicate how this can be considered as a first step in the elaboration of a geometric dequantization program which would extend the results to more general manifolds.

Coherence and incompatability inWu* -algebraic quantum theory

Abstract: In the framework of generalized quantum theory using aW*-algebraic formalism, we introduce a completely symmetric coherence relation for states which is also applicable to nonpure states. Making use of lattice theoretic results the properties of this relation, especially its connection with incompatibility, are investigated. By means of algebraic decomposition theory the investigation is reduced to the case of factors where a complete classification of the coherence classes is given.

Calculus on Complex Banach Spaces

Abstract: By using the techniques of modern functional analysis, a variety of new concepts have been developed and new results proved which extend considerably the new calculus on complex Banach spaces developed by Sharma and Rebelo. The distinguishing feature of the new calculus is that in this calculus the more general concept of additivity replaces that of linearity in the Frechet calculus. It is proved that the space of continuous additive maps between two complex Banach spaces is the direct sum of the spaces of linear and semilinear maps between the two spaces. The Hahn-Banach theorem and the open mapping theorem which in their standard versions are valid for continuous linear functionals and functions are shown to hold also for the additive case. The concepts of the adjoint of an additive map, of a new kind of orthogonal complement of a subset of a Banach space, and of a balanced additive map in which the norms of the linear and semilinear components are equal are developed. It is then proved that the orthogonal complement of the range of an additive map equals the null space of its adjoint and if the additive map is a functional on a complex Hilbert space and is balanced, then the orthogonal complement of the null space of the functional equals the range of the adjoint. A generalization of the inverse function theorem is proved by using our version of the open mapping theorem and then used to establish the Lagrange multiplier theorem in the new calculus. A number of related results are also proved. The applications of the new calculus to physics are briefly described.

Finite-dimensional quantum mechanics of a particle. III. The Weylian quantum mechanics of confined quarks

Abstract: A finite-dimensional analog of Weyl's formulation of quantum kinematics of a physical system through irreducible Abelian groups of unitary ray rotations in system space offers many possibilities for the quantum mechanics of confined particles. This paper is devoted to the expansion of the recently developed framework of such Weylian finite-dimensional quantum mechanics which may provide a new way of thinking about the characteristics of quark physics.

Elementary particle states based on the Clifford algebra C 7

Abstract: The lepton isodoublet (e−,v e ), the “bare” nucleon isodoublet (n,p), and their antiparticles are shown to constitute a basis of the irreducible representation of the Clifford algebraC7. The excited states of these doublets, i.e., (μ−,ν μ ), (τ−,ν τ ),..., and (s0,c+),(b0,t+) are generated by the products (e−,v e )⊗a and (n,p)⊗a, wherea≡2−1/2(e−e++v e v e ) has the same quantum numbers as the photon state. The bare baryonss, c, b, t carry the strangeness, charm, bottom, and top quantum numbers. These lepton and bare baryon states are in one-to-one correspondence with the integrally charged colored Han-Nambu quarks, and generate all the observedsu(3) andsu(4) hadron multiplets.

Affine Maczyński logics on compact convex sets of states

Abstract: Sets of affine functions satisfying Maczynski orthogonality postulate and defined on compact convex sets of states are examined. Relations between affine Maski logics and Boolean algebras when the set of states is a Bauer simplex (classical mechanics, some models of nonlinear quantum mechanics) are studied. It is shown that an affine Maczynski logic defined on a Bauer simplex is a Boolean algebra if it is a sublattice of a lattice consisting of all bounded affine functions defined on the simplex.

Conformally invariant gravitational waves in a modified gravitational theory

Abstract: An attempt is made to obtain a conformally invariant gravitational wave equation in an isotropic background universe by modifying the Einstein field equation through a correction term proposed in the Hilbert Lagrangian in the form of a series of finite terms inR (≡g ik R ik ). It is shown that only those waves which are described by Bessel functionJ 0(mη) in curved background can be transformed as classically periodic waves in flat background [without restricting the scale factora(η)].

Quantization and meaning of observables linear in momentum

Abstract: A detailed study is made of the observablesξ j(xk)pi+f(xk) linear in momentum on a Riemannian manifold: their quantization and (through quantum unitary transformations) physical meaning are discussed using geometrical methods.

Physical Constants as Cosmological Constraints

Abstract: A solution to the primary “missing mass” problem is found in the context of accounting for the coincidence of large dimensionless numbers first noticed by Weyl, Eddington, and Dirac. This solution entails (1) a log2 relation between the electromagnetic and gravitational coupling constants; (2) setting the maximum radius of curvature at the gravitational radius, 2GM/c2; (3) a changing gravitational parameterG, which varies as an inverse function of the universal radius of curvature. These features motivate the development of a neo-Friedmann formalism, which employs a function,e(χ). governing the change from Euclidian to non-Euclidian volumes. Observational consequences include (1) a universal density of 7.6×10−31g cm−3, (2) a Hubble parameter of 15 km s−1 Mpc−1, (3) an age of the universe of 32×109 yr, (4) a gravitational parameter diminishing at a current rate of 2.2×10−12 yr−1, and (5) a deceleration parameter of 1.93. Moreover, it is shown that for a Friedmann-type (λ=0) cosmology (whether open or closed) any deceleration parameter will be represented by a straight line in the (log-log) red shift: luminosity-distance space of the Hubble diagram. The major claim of this paper is that we have devised a model in which the large-scale structure of the universe is completely determined by the values of the fundamental physical constants:c, h, e, andme setting the scale, andG selecting the epoch.

Electromagnetic and Weak Interactions in Stochastic Space-Time: A Review

Abstract: A scheme for a nonlocal theory of quantized fields based on the hypothesis of stochastic space is proposed. Within this scheme the gauge-invariant quantum electrodynamics of particles with spin 0, 1/2, 1 and four-fermion weak interactions are constructed, and nonlocal corrections to the anomalous magnetic moments of leptons and to the Lamb shift are calculated. Some consequences of the neutrino oscillations and the electromagnetic properties of neutrinos are considered in detail. Further the rare decayK L 0 →Μ+Μ− and the mass difference ofK L 0 andK S 0 mesons are investigated in this model. It is shown that the parameter of nonlocality (elementary lengthl) of weak interactions which can characterize a domain of unification of weak and electromagnetic interactions is ∼10−16 cm. The low-energy experiments imply that quantum electrodynamics is valid up to distances of order ∼10−15 cm.

Large Numbers Hypothesis. II. Electromagnetic Radiation

Abstract: This paper develops the theory of electromagnetic radiation in the units covariant formalism incorporating Dirac's large numbers hypothesis (LNH). A direct field-to-particle technique is used to obtain the photon propagation equation which explicitly involves the photon replication rate. This replication rate is fixed uniquely by requiring that the form of a free-photon distribution function be preserved, as required by the 2.7 K cosmic radiation. One finds that with this particular photon replication rate the units covariant formalism developed in Paper I actually predicts that the ratio of photon number to proton number in the Universe varies ast1/4 precisely in accord with LNH. The cosmological red-shift law is also derived and it is shown to differ considerably from the standard form ofνR = const.

Statistical interpretation of the Planck constant and the uncertainty relation

Abstract: A statistically founded derivation of the quanta of energy is presented, which yields the Planck formula for the mean energy of the blackbody radiation without making use of the quantum postulate. The derivation presupposes an ensemble of particles and leads to a statistical interpretation of the Planck constant, which is defined and discussed. By means of the proposed interpretation ofh and as an application of it, the quantum uncertainty relation is derived classically and results as a statistical inequality. On the whole this paper is compatible with the statistical ensemble interpretation of quantum mechanics.

A finite-dimensional quark model

Abstract: A description of quarks is given in terms of a finite-dimensional Hilbert space model. Color and flavor observables are defined and the corresponding motion and energy observables are constructed using the methods of finite-dimensional quantum mechanics. It is shown that a fundamental color condition implies that quarks can only combine as mesons, baryons, antibaryons, and collections of these. Baryon and meson Hamiltonians are proposed and various masses are computed.

Geometric quantization in the presence of an electromagnetic field

Abstract: Some aspects of the formalism of geometric quantization are described emphasizing the role played by the symmetry group of the quantum system which, for the free particle, turns out to be a central extensionG(m) of the Galilei groupG. The resulting formalism is then applied to the case of a particle interacting with the electromagnetic field, which appears as a necessary modification of the connection 1-form of the quantum bundle when its invariance group is generalized to alocal extension ofG. Finally, the quantization of the electric charge in the presence of a Dirac monopole is also briefly considered.

Does the electromagnetic mass of an electron depend on where it is

Abstract: The question of whether the electromagnetic mass of the electron depends upon its electromagnetic environment is discussed in connection with previous theoretical and experimental work. When the quantization of the field in other than free space is properly understood, it is evident that the true electromagnetic mass of the electron is unaltered.