2,939 research outputs found
Exponential Renormalization II: Bogoliubov's R-operation and momentum subtraction schemes
This article aims at advancing the recently introduced exponential method for
renormalisation in perturbative quantum field theory. It is shown that this new
procedure provides a meaningful recursive scheme in the context of the
algebraic and group theoretical approach to renormalisation. In particular, we
describe in detail a Hopf algebraic formulation of Bogoliubov's classical
R-operation and counterterm recursion in the context of momentum subtraction
schemes. This approach allows us to propose an algebraic classification of
different subtraction schemes. Our results shed light on the peculiar algebraic
role played by the degrees of Taylor jet expansions, especially the notion of
minimal subtraction and oversubtractions.Comment: revised versio
On the Exponentials of Some Structured Matrices
In this note explicit algorithms for calculating the exponentials of
important structured 4 x 4 matrices are provided. These lead to closed form
formulae for these exponentials. The techniques rely on one particular Clifford
Algebra isomorphism and basic Lie theory. When used in conjunction with
structure preserving similarities, such as Givens rotations, these techniques
extend to dimensions bigger than four.Comment: 19 page
Reciprocal relativity of noninertial frames: quantum mechanics
Noninertial transformations on time-position-momentum-energy space {t,q,p,e}
with invariant Born-Green metric ds^2=-dt^2+dq^2/c^2+(1/b^2)(dp^2-de^2/c^2) and
the symplectic metric -de/\dt+dp/\dq are studied. This U(1,3) group of
transformations contains the Lorentz group as the inertial special case. In the
limit of small forces and velocities, it reduces to the expected Hamilton
transformations leaving invariant the symplectic metric and the nonrelativistic
line element ds^2=dt^2. The U(1,3) transformations bound relative velocities by
c and relative forces by b. Spacetime is no longer an invariant subspace but is
relative to noninertial observer frames. Born was lead to the metric by a
concept of reciprocity between position and momentum degrees of freedom and for
this reason we call this reciprocal relativity.
For large b, such effects will almost certainly only manifest in a quantum
regime. Wigner showed that special relativistic quantum mechanics follows from
the projective representations of the inhomogeneous Lorentz group. Projective
representations of a Lie group are equivalent to the unitary reprentations of
its central extension. The same method of projective representations of the
inhomogeneous U(1,3) group is used to define the quantum theory in the
noninertial case. The central extension of the inhomogeneous U(1,3) group is
the cover of the quaplectic group Q(1,3)=U(1,3)*s H(4). H(4) is the
Weyl-Heisenberg group. A set of second order wave equations results from the
representations of the Casimir operators
Quantising on a category
We review the problem of finding a general framework within which one can
construct quantum theories of non-standard models for space, or space-time. The
starting point is the observation that entities of this type can typically be
regarded as objects in a category whose arrows are structure-preserving maps.
This motivates investigating the general problem of quantising a system whose
`configuration space' (or history-theory analogue) is the set of objects
\Ob\Q in a category \Q.
We develop a scheme based on constructing an analogue of the group that is
used in the canonical quantisation of a system whose configuration space is a
manifold , where and are Lie groups. In particular, we
choose as the analogue of the monoid of `arrow fields' on \Q. Physically,
this means that an arrow between two objects in the category is viewed as an
analogue of momentum. After finding the `category quantisation monoid', we show
how suitable representations can be constructed using a bundle (or, more
precisely, presheaf) of Hilbert spaces over \Ob\Q. For the example of a
category of finite sets, we construct an explicit representation structure of
this type.Comment: To appear in a volume dedicated to the memory of James Cushin
Positive-Operator-Valued Time Observable in Quantum Mechanics
We examine the longstanding problem of introducing a time observable in
Quantum Mechanics; using the formalism of positive-operator-valued measures we
show how to define such an observable in a natural way and we discuss some
consequences.Comment: 13 pages, LaTeX, no figures. Some minor changes, expanded the
bibliography (now it is bigger than the one in the published version),
changed the title and the style for publication on the International Journal
of Theoretical Physic
Projective Fourier Duality and Weyl Quantization
The Weyl-Wigner correspondence prescription, which makes large use of Fourier
duality, is reexamined from the point of view of Kac algebras, the most general
background for noncommutative Fourier analysis allowing for that property. It
is shown how the standard Kac structure has to be extended in order to
accommodate the physical requirements. An Abelian and a symmetric projective
Kac algebras are shown to provide, in close parallel to the standard case, a
new dual framework and a well-defined notion of projective Fourier duality for
the group of translations on the plane. The Weyl formula arises naturally as an
irreducible component of the duality mapping between these projective algebras.Comment: LaTeX 2.09 with NFSS or AMSLaTeX 1.1. 102Kb, 44 pages, no figures.
requires subeqnarray.sty, amssymb.sty, amsfonts.sty. Final version with text
improvements and crucial typos correction
Fourier Duality as a Quantization Principle
The Weyl-Wigner prescription for quantization on Euclidean phase spaces makes
essential use of Fourier duality. The extension of this property to more
general phase spaces requires the use of Kac algebras, which provide the
necessary background for the implementation of Fourier duality on general
locally compact groups. Kac algebras -- and the duality they incorporate -- are
consequently examined as candidates for a general quantization framework
extending the usual formalism. Using as a test case the simplest non-trivial
phase space, the half-plane, it is shown how the structures present in the
complete-plane case must be modified. Traces, for example, must be replaced by
their noncommutative generalizations - weights - and the correspondence
embodied in the Weyl-Wigner formalism is no more complete. Provided the
underlying algebraic structure is suitably adapted to each case, Fourier
duality is shown to be indeed a very powerful guide to the quantization of
general physical systems.Comment: LaTeX 2.09 with NFSS or AMSLaTeX 1.1. 97Kb, 43 pages, no figures.
requires subeqnarray.sty, amssymb.sty, amsfonts.sty. Final version with (few)
text and (crucial) typos correction
Strong nonlocality: A trade-off between states and measurements
Measurements on entangled quantum states can produce outcomes that are
nonlocally correlated. But according to Tsirelson's theorem, there is a
quantitative limit on quantum nonlocality. It is interesting to explore what
would happen if Tsirelson's bound were violated. To this end, we consider a
model that allows arbitrary nonlocal correlations, colloquially referred to as
"box world". We show that while box world allows more highly entangled states
than quantum theory, measurements in box world are rather limited. As a
consequence there is no entanglement swapping, teleportation or dense coding.Comment: 11 pages, 2 figures, very minor change
The Physical Origins of Entropy Production, Free Energy Dissipation and their Mathematical Representations
A complete mathematical theory of nonequilibrium thermodynamics of stochastic
systems in terms of master equations is presented. As generalizations of
isothermal entropy and free energy, two functions of states play central roles:
the Gibbs entropy and the relative entropy , which are related via the
stationary distribution of the stochastic dynamics. satisfies the
fundamental entropy balance equation with entropy production
rate and heat dissipation rate , while . For
closed systems that satisfy detailed balance: . For open system
one has where the housekeeping heat
was first introduced in the phenomenological nonequilibrium steady state
thermodynamics. Entropy production consists of free energy dissipation
associated with spontaneous relaxation, , and active energy pumping that
sustains the open system . The amount of excess heat involved in the
relaxation .Comment: 4 pages, no figure
Rare Events and Scale--Invariant Dynamics of Perturbations in Delayed Dynamical Systems
We study the dynamics of perturbations in time delayed dynamical systems.
Using a suitable space-time coordinate transformation, we find that the time
evolution of the linearized perturbations (Lyapunov vector) can be mapped to
the linear Zhang surface growth model [Y.-C. Zhang, J. Phys. France {\bf 51},
2129 (1990)], which is known to describe surface roughening driven by power-law
distributed noise. As a consequence, Lyapunov vector dynamics is dominated by
rare random events that lead to non-Gaussian fluctuations and multiscaling
properties.Comment: Final version to appear in Phys. Rev. Lett., 4 pages, 3 eps fig
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