64,801 research outputs found

### State Vector Reduction as a Shadow of a Noncommutative Dynamics

A model, based on a noncommutative geometry, unifying general relativity with
quantum mechanics, is further develped. It is shown that the dynamics in this
model can be described in terms of one-parameter groups of random operators. It
is striking that the noncommutative counterparts of the concept of state and
that of probability measure coincide. We also demonstrate that the equation
describing noncommutative dynamics in the quantum gravitational approximation
gives the standard unitary evolution of observables, and in the "space-time
limit" it leads to the state vector reduction. The cases of the spin and
position operators are discussed in details.Comment: 20 pages, LaTex, no figure

### Abelianization of Fuchsian Systems on a 4-punctured sphere and applications

In this paper we consider special linear Fuchsian systems of rank $2$ on a
$4-$punctured sphere and the corresponding parabolic structures. Through an
explicit abelianization procedure we obtain a $2-$to$-1$ correspondence between
flat line bundle connections on a torus and these Fuchsian systems. This
naturally equips the moduli space of flat $SL(2,\mathbb C)-$connections on a
$4-$punctured sphere with a new set of Darboux coordinates. Furthermore, we
apply our theory to give a complex analytic proof of Witten's formula for the
symplectic volume of the moduli space of unitary flat connections on the
$4-$punctured sphere.Comment: 23 pages, comments are welcom

### Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics
in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid
$\Gamma = E \times G$ where $E$ is the total space of a principal fibre bundle
over spacetime, and $G$ a suitable group acting on $\Gamma$. We show that
every $a \in {\cal A}$ defines a random operator, and we study the dynamics of
such operators. In the noncommutative regime, there is no usual time but, on
the strength of the Tomita-Takesaki theorem, there exists a one-parameter group
of automorphisms of the algebra ${\cal A}$ which can be used to define a state
dependent dynamics; i.e., the pair $({\cal A}, \phi)$, where $\phi$ is a state
on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions
are satisfied, the Connes-Nikodym-Radon theorem can be applied and the
dependence on $\phi$ disappears. In these cases, the usual unitary quantum
mechanical evolution is recovered. We also notice that the same pair $({\cal
A}, \phi)$ defines the so-called free probability calculus, as developed by
Voiculescu and others, with the state $\phi$ playing the role of the
noncommutative probability measure. This shows that in the noncommutative
regime dynamics and probability are unified. This also explains probabilistic
properties of the usual quantum mechanics.Comment: 13 pages, LaTe

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