1,721,015 research outputs found
On the central extensions of the Heisenberg algebra
Full text linkWe describe the nontrivial central extensions CE(Heis) of the Heisenberg algebra and their
representation as sub–algebras of the Schroedinger algebra. We also present the characteristic and moment
generating functions of the random variable corresponding to the self-adjoint sum of the generators of
CE(Heis)
The Schrodinger Fock Kernel and the no-go Theorem for the First Order and Renormalized Square of White Noise Lie Alegbras
Full text linkUsing the non-positive definiteness of the Fock kernel associated
with the Schr ̈odinger algebra we prove the impossibility of a joint Fock representation
of the first order and Renormalized Square of White Noise Lie
algebras with the convolution type renormalization δ2(t−s) = δ(s) δ(t−s) for
the square of the Dirac delta function. We show how the Schr ̈odinger algebra
Fock kernel can be reduced to a positive definite kernel through a restriction of
the set of exponential vectors. We describe how the reduced Schr ̈odinger kernel
can be viewed as a tensor product of a Renormalized Square of White Noise
(sl(2)) and a First Order of White Noise (Heisenberg) Fock kernel. We also
compute the characteristic function of a stochastic process naturally associated
with the reduced Schr ̈odinger kernel
The emergence of the Virasoro and algebras through the renormalized higher powers of quantum white noise
No full textWe introduce a new renormalization for the powers of the Dirac delta function.
We show that this new renormalization leads to a second quantized version of the Virasoro
sector of the extended conformal algebra with infinite symmetries
of Conformal Field Theory ( \cite{4a}-\cite{4d}, \cite{ketov}, \cite{7}, \cite{8}). In particular
we construct a white noise (boson) representation of the generators and commutation relations and of their second quantization
Commutators associated with the renormalized powers of quantum white noise
Full text linkLet denote the Dirac delta function. We show how, when the renormalization constant in is large or approaches , the commutation relations for the Renormalized Powers of Quantum White Noise (RPQWN) can be truncated to yield either the Heisenberg Canonical Commutation Relations (CCR) or the Renormalized Square of White Noise (RSWN) commutation relations of \cite{3}, parametrized by the order of the white noise functionals. The, still open, problem of choosing a renormalization of the powers of the delta function that will lead to a Fock representation of the RPQWN commutation relations is described
The Quantum Black-Scholes equation
Full text linkMotivated by the work of Segal and Segal in \cite{5} 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
CONTRACTIONS AND CENTRAL EXTENSIONS OF QUANTUM WHITE NOISE LIE ALGEBRAS
Full text linkWe show that the Renormalized Powers of Quantum White Noise Lie algebra RPQWN, with the convolution type renormalization δnt − s = δsδt − s of the n≥­ 2 powers of the Dirac delta function, can be obtained through a contraction of the Renormalized Powers of Quantum White Noise Lie algebra RPQWNc with the scalar renormalization δnt = cn−1δt, c > 0. Using this renormalization, we also obtain a Lie algebra W∞c which contains the w∞ Lie algebra of Bakas and the Witt algebra as contractions. Motivated by the W∞ algebra of Pope, Romans and Shen, we show that W∞c can also be centrally extended in a non-trivial fashion. In the case of the Witt subalgebra of W∞, the central extension coincides with that of the Virasoro algebra.We show that the Renormalized Powers of Quantum White Noise Lie algebra RPQWN, with the convolution type renormalization δnt − s = δsδt − s of the n≥­ 2 powers of the Dirac delta function, can be obtained through a contraction of the Renormalized Powers of Quantum White Noise Lie algebra RPQWNc with the scalar renormalization δnt = cn−1δt, c > 0. Using this renormalization, we also obtain a Lie algebra W∞c which contains the w∞ Lie algebra of Bakas and the Witt algebra as contractions. Motivated by the W∞ algebra of Pope, Romans and Shen, we show that W∞c can also be centrally extended in a non-trivial fashion. In the case of the Witt subalgebra of W∞, the central extension coincides with that of the Virasoro algebra
Control of elementary quantum flows
Full text linkWe consider the problem of controlling the size of an elementary quantum
stochastic flow generated by a unitary stochastic evolution driven by first
order white noise
The Centrally extended Heisenberg algebra and its connection with Schrodinger, Galilei and renormalized higher powers of quantum white noise Lie algebra
Full text linkIn previous papers we have shown that the one mode Heisenberg algebra Heis(1) admits a unique non-trivial central extensions CeHeis(1) which can be realized as a sub-Lie-algebra of the Schrödinger algebra, in fact the Galilei Lie algebra. This gives a natural family of unitary representations of CeHeis(1) and allows an explicit determination of the associated group by exponentiation. In contrast with Heis(1), the group law for CeHeis(1) is given by nonlinear (quadratic) functions of the coordinates. The vacuum characteristic and moment generating functions of the classical random variables canonically associated to CeHeis(1) are computed. The second quantization of CeHeis(1) is also considered
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