42 research outputs found
Randomness in Classical Mechanics and Quantum Mechanics
The Copenhagen interpretation of quantum mechanics assumes the existence of
the classical deterministic Newtonian world. We argue that in fact the Newton
determinism in classical world does not hold and in classical mechanics there
is fundamental and irreducible randomness. The classical Newtonian trajectory
does not have a direct physical meaning since arbitrary real numbers are not
observable. There are classical uncertainty relations, i.e. the uncertainty
(errors of observation) in the determination of coordinate and momentum is
always positive (non zero).
A "functional" formulation of classical mechanics was suggested. The
fundamental equation of the microscopic dynamics in the functional approach is
not the Newton equation but the Liouville equation for the distribution
function of the single particle. Solutions of the Liouville equation have the
property of delocalization which accounts for irreversibility. The Newton
equation in this approach appears as an approximate equation describing the
dynamics of the average values of the position and momenta for not too long
time intervals. Corrections to the Newton trajectories are computed. An
interpretation of quantum mechanics is attempted in which both classical and
quantum mechanics contain fundamental randomness. Instead of an ensemble of
events one introduces an ensemble of observers.Comment: 12 pages, Late
Composite p-branes in various dimensions
We review an algebraic method of finding the composite p-brane solutions for
a generic Lagrangian, in arbitrary spacetime dimension, describing an
interaction of a graviton, a dilaton and one or two antisymmetric tensors. We
set the Fock--De Donder harmonic gauge for the metric and the "no-force"
condition for the matter fields. Then equations for the antisymmetric field are
reduced to the Laplace equation and the equation of motion for the dilaton and
the Einstein equations for the metric are reduced to an algebraic equation.
Solutions composed of n constituent p-branes with n independent harmonic
functions are given. The form of the solutions demonstrates the harmonic
functions superposition rule in diverse dimensions. Relations with known
solutions in D=10 and D=11 dimensions are discussed.Comment: 17 pages, no figures, latex. Contribution to the Proceedings of the
30th Ahrenshoop Symposium on the Theory of Elementary Particles, edited by D.
Lust, H.-J. Otto and G. Weigt, to appear in Nuclear Physics B, Proceedings
Supplemen
Non-Exponential decay for polaron model
A model of particle interacting with quantum field is considered. The model includes as particular cases the polaron model and non-relativistic quantum electrodynamics. We compute matrix elements of the evolution operator in the stochastic approximation and show that depending on the state of the particle one can get the non-exponential decay with the rate t^{-3/2}. In the process of computation a new algebra of commutational relations that can be considered as an operator deformation of quantum Boltzmann commutation relations is used
A White noise approach to stochastic calculus
During the past 15 years a new technique, called the stochastic limit of quantum theory,
has been applied to deduce new, unexpected results in a variety of traditional problems of quantum
physics, such as quantum electrodynamics, bosonization in higher dimensions, the emergence of
the noncrossing diagrams in the Anderson model, and in the large-N-limit in QCD, interacting
commutation relations, new photon statistics in strong magnetic fields, etc. These achievements
required the development of a new approach to classical and quantum stochastic calculus based
on white noise which has suggested a natural nonlinear extension of this calculus. The natural
theoretical framework of this new approach is the white-noise calculus initiated by T. Hida as a
theory of infinite-dimensional generalized functions. In this paper, we describe the main ideas of
the white-noise approach to stochastic calculus and we show that, even if we limit ourselves to the
first-order case (i.e. neglecting the recent developments concerning higher powers of white noise and
renormalization), some nontrivial extensions of known results in classical and quantum stochastic
calculus can be obtained
Airy functions over local fields
Airy integrals are very classical but in recent years they have been
generalized to higher dimensions and these generalizations have proved to be
very useful in studying the topology of the moduli spaces of curves. We study a
natural generalization of these integrals when the ground field is a
non-archimedean local field such as the field of p-adic numbers. We prove that
the p-adic Airy integrals are locally constant functions of moderate growth and
present evidence that the Airy integrals associated to compact p-adic Lie
groups also have these properties.Comment: Minor change
String Model with Dynamical Geometry and Torsion
A string model with dynamical metric and torsion is proposed. The geometry of
the string is described by an effective Lagrangian for the scalar and vector
fields. The path integral quantization of the string is considered
Conformal p-branes as a Source of Structure in Spacetime
We discuss a model of a conformal p-brane interacting with the world volume
metric and connection. The purpose of the model is to suggest a mechanism by
which gravity coupled to p-branes leads to the formation of structure rather
than homogeneity in spacetime. Furthermore, we show that the formation of
structure is accompanied by the appearance of a multivalued cosmological
constant, i.e., one which may take on different values in different domains, or
cells, of spacetime. The above results apply to a broad class of non linear
gravitational lagrangians as long as metric and connection on the p-brane
manifold are treated as independent variables.Comment: 10 pages, ReVTeX, no figure
The Hopf algebra of Feynman graphs in QED
We report on the Hopf algebraic description of renormalization theory of
quantum electrodynamics. The Ward-Takahashi identities are implemented as
linear relations on the (commutative) Hopf algebra of Feynman graphs of QED.
Compatibility of these relations with the Hopf algebra structure is the
mathematical formulation of the physical fact that WT-identities are compatible
with renormalization. As a result, the counterterms and the renormalized
Feynman amplitudes automatically satisfy the WT-identities, which leads in
particular to the well-known identity .Comment: 13 pages. Latex, uses feynmp. Minor corrections; to appear in LM
On A Superfield Extension of The ADHM Construction and N=1 Super Instantons
We give a superfield extension of the ADHM construction for the Euclidean
theory obtained by Wick rotation from the Lorentzian four dimensional N=1 super
Yang-Mills theory. In particular, we investigate the procedure to guarantee the
Wess-Zumino gauge for the superfields obtained by the extended ADHM
construction, and show that the known super instanton configurations are
correctly obtained.Comment: 22 pages, LaTeX, v2: typos corrected, references adde