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A hierarchy of compatibility and comeasurability levels in quantum logics with unique conditional probabilities
In the quantum mechanical Hilbert space formalism, the probabilistic
interpretation is a later ad-hoc add-on, more or less enforced by the
experimental evidence, but not motivated by the mathematical model itself. A
model involving a clear probabilistic interpretation from the very beginning is
provided by the quantum logics with unique conditional probabilities. It
includes the projection lattices in von Neumann algebras and here probability
conditionalization becomes identical with the state transition of the Lueders -
von Neumann measurement process. This motivates the definition of a hierarchy
of five compatibility and comeasurability levels in the abstract setting of the
quantum logics with unique conditional probabilities. Their meanings are: the
absence of quantum interference or influence, the existence of a joint
distribution, simultaneous measurability, and the independence of the final
state after two successive measurements from the sequential order of these two
measurements. A further level means that two elements of the quantum logic
(events) belong to the same Boolean subalgebra. In the general case, the five
compatibility and comeasurability levels appear to differ, but they all
coincide in the common Hilbert space formalism of quantum mechanics, in von
Neumann algebras, and in some other cases.Comment: 12 page
Two-dimensional models as testing ground for principles and concepts of local quantum physics
In the past two-dimensional models of QFT have served as theoretical
laboratories for testing new concepts under mathematically controllable
condition. In more recent times low-dimensional models (e.g. chiral models,
factorizing models) often have been treated by special recipes in a way which
sometimes led to a loss of unity of QFT. In the present work I try to
counteract this apartheid tendency by reviewing past results within the setting
of the general principles of QFT. To this I add two new ideas: (1) a modular
interpretation of the chiral model Diff(S)-covariance with a close connection
to the recently formulated local covariance principle for QFT in curved
spacetime and (2) a derivation of the chiral model temperature duality from a
suitable operator formulation of the angular Wick rotation (in analogy to the
Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational
chiral theories. The SL(2,Z) modular Verlinde relation is a special case of
this thermal duality and (within the family of rational models) the matrix S
appearing in the thermal duality relation becomes identified with the
statistics character matrix S. The relevant angular Euclideanization'' is done
in the setting of the Tomita-Takesaki modular formalism of operator algebras.
I find it appropriate to dedicate this work to the memory of J. A. Swieca
with whom I shared the interest in two-dimensional models as a testing ground
for QFT for more than one decade.
This is a significantly extended version of an ``Encyclopedia of Mathematical
Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section
Generalized Non-Commutative Inflation
Non-commutative geometry indicates a deformation of the energy-momentum
dispersion relation for massless particles.
This distorted energy-momentum relation can affect the radiation dominated
phase of the universe at sufficiently high temperature. This prompted the idea
of non-commutative inflation by Alexander, Brandenberger and Magueijo (2003,
2005 and 2007). These authors studied a one-parameter family of
non-relativistic dispersion relation that leads to inflation: the
family of curves . We show here how the
conceptually different structure of symmetries of non-commutative spaces can
lead, in a mathematically consistent way, to the fundamental equations of
non-commutative inflation driven by radiation. We describe how this structure
can be considered independently of (but including) the idea of non-commutative
spaces as a starting point of the general inflationary deformation of
. We analyze the conditions on the dispersion relation that
leads to inflation as a set of inequalities which plays the same role as the
slow roll conditions on the potential of a scalar field. We study conditions
for a possible numerical approach to obtain a general one parameter family of
dispersion relations that lead to successful inflation.Comment: Final version considerably improved; Non-commutative inflation
rigorously mathematically formulate
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