100 research outputs found
An obstruction for q-deformation of the convolution product
We consider two independent q-Gaussian random variables X and Y and a
function f chosen in such a way that f(X) and X have the same distribution. For
0 < q < 1 we find that at least the fourth moments of X + Y and f(X) + Y are
different. We conclude that no q-deformed convolution product can exist for
functions of independent q-Gaussian random variables.Comment: The proof of proposition 2 is corrected on 11 january 199
Symmetric Hilbert spaces arising from species of structures
Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces
over some `one particle space' \K are formed by certain symmetrization
procedures performed on the full Fock space. We investigate alternative ways of
symmetrization by building on Joyal's notion of a combinatorial species. Any
such species gives rise to an endofunctor \G_F of the category of Hilbert
spaces with contractions mapping a Hilbert space \K to a symmetric Hilbert
space \G_F(\K) with the same symmetry as the species . A general framework
for annihilation and creation operators on these spaces is developed, and
compared to the generalised Brownian motions of R. Speicher and M. Bo\.zejko.
As a corollary we find that the commutation relation
with admits a
realization on a symmetric Hilbert space whenever has a power series with
infinite radius of convergence and positive coefficients.Comment: 39 page
Generalised Brownian Motion and Second Quantisation
A new approach to the generalised Brownian motion introduced by M. Bozejko
and R. Speicher is described, based on symmetry rather than deformation. The
symmetrisation principle is provided by Joyal's notions of tensorial and
combinatorial species. Any such species V gives rise to an endofunctor F_V of
the category of Hilbert spaces with contractions. A generalised Brownian motion
is an algebra of creation and annihilation operators acting on F_V(H) for
arbitrary Hilbert spaces H and having a prescription for the calculation of
vacuum expectations in terms of a function t on pair partitions. The positivity
is encoded by a *-semigroup of "broken pair partitions" whose representation
space with respect to t is V. The existence of the second quantisation as
functor Gamma_t from Hilbert spaces to noncommutative probability spaces is
proved to be equivalent to the multiplicative property of the function t. For a
certain one parameter interpolation between the fermionic and the free Brownian
motion it is shown that the ``field algebras'' Gamma(K) are type II_1 factors
when K is infinite dimensional.Comment: 33 pages, 5 figure
Stochastic Schrodinger equations
A derivation of stochastic Schrodinger equations is given using quantum
filtering theory. We study an open system in contact with its environment, the
electromagnetic field. Continuous observation of the field yields information
on the system: it is possible to keep track in real time of the best estimate
of the system's quantum state given the observations made. This estimate
satisfies a stochastic Schrodinger equation, which can be derived from the
quantum stochastic differential equation for the interaction picture evolution
of system and field together. Throughout the paper we focus on the basic
example of resonance fluorescence.Comment: 24 page
Purification of quantum trajectories
We prove that the quantum trajectory of repeated perfect measurement on a
finite quantum system either asymptotically purifies, or hits upon a family of
`dark' subspaces, where the time evolution is unitary.Comment: 10 page
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