Bose-Einstein condensation and condensation of qq-particles in equilibrium and non equilibrium thermodynamics: a new approach

In the setting of the principle of local equilibrium which asserts that the temperature is a function of the energy levels of the system, we exhibit plenty of steady states describing the condensation of free Bosons which are not in thermal equilibrium. The surprising facts are that the condensation can occur both in dimension less than 3 in configuration space, and even in excited energy levels. The investigation relative to non equilibrium suggests a new approach to the condensation, which allows an unified analysis involving also the condensation of $q$-particles, $-1\leq q\leq 1$, where $q=\pm1$ corresponds to the Bose/Fermi alternative. For such $q$-particles, the condensation can occur only if $0<q\leq1$, the case 1 corresponding to the standard Bose-Einstein condensation. In this more general approach, completely new and unexpected states exhibiting condensation phenomena naturally occur also in the usual situation of equilibrium thermodynamics. The new approach proposed in the present paper for the situation of $2^\text{nd}$ quantisation of free particles, is naturally based on the theory of the Distributions, which might hopefully be extended to more general casesComment: It is a preprint of July 201

The quadratic Fock functor

We construct the quadratic analogue of the boson Fock functor. While in the first order case all contractions on the 1--particle space can be second quantized, the semigroup of contractions that admit a quadratic second quantization is much smaller due to the nonlinearity. Within this semigroup we characterize the unitary and the isometric elements.Comment: 2

A Generalization of Grover's Algorithm

We investigate the necessary and sufficient conditions in order that a unitary operator can amplify a pre-assigned component relative to a particular basis of a generic vector at the expense of the other components. This leads to a general method which allows, given a vector and one of its components we want to amplify, to choose the optimal unitary operator which realizes that goal. Grover's quantum algorithm is shown to be a particular case of our general method. However the general structure of the unitary we find is remarkably similar to that of Grover's one: a sign flip of one component combined with a reflection with respect to a vector. In Grover's case this vector is fixed; in our case it depends on a parameter and this allows optimization.Comment: 20 pages, no figure

Chameleon effect, the range of values hypothesis and reproducing the EPR-Bohm correlations

We present a detailed analysis of assumptions that J. Bell used to show that local realism contradicts QM. We find that Bell's viewpoint on realism is nonphysical, because it implicitly assume that observed physical variables coincides with ontic variables (i.e., these variables before measurement). The real physical process of measurement is a process of dynamical interaction between a system and a measurement device. Therefore one should check the adequacy of QM not to ``Bell's realism,'' but to adaptive realism (chameleon realism). Dropping Bell's assumption we are able to construct a natural representation of the EPR-Bohm correlations in the local (adaptive) realistic approach.Comment: To be published in Proceedings of International Conference Foudations of Probability and Physics-4, June 2006, Vaxjo, Swede

A note on noncommutative unique ergodicity and weighted means

In this paper we study unique ergodicity of $C^*$-dynamical system (\ga,T), consisting of a unital $C^*$-algebra \ga and a Markov operator T:\ga\mapsto\ga, relative to its fixed point subspace, in terms of Riesz summation which is weaker than Cesaro one. Namely, it is proven that (\ga,T) is uniquely ergodic relative to its fixed point subspace if and only if its Riesz means {equation*} \frac{1}{p_1+...+p_n}\sum_{k=1}^{n}p_kT^kx {equation*} converge to $E_T(x)$ in \ga for any x\in\ga, as $n\to\infty$, here $E_T$ is an projection of \ga to the fixed point subspace of $T$. It is also constructed a uniquely ergodic entangled Markov operator relative to its fixed point subspace, which is not ergodic.Comment: 11 pages. submitted. Linear Alg. Applications (to appear

The Quantum Black-Scholes Equation

Motivated by the work of Segal and Segal on the Black-Scholes pricing formula in the quantum context, we study a quantum extension of the Black-Scholes equation within the context of Hudson-Parthasarathy quantum stochastic calculus. Our model includes stock markets described by quantum Brownian motion and Poisson process.Comment: Has appeared in GJPAM, vol. 2, no. 2, pp. 155-170 (2006