10,965 research outputs found
Quantum Dynamics, Minkowski-Hilbert space, and A Quantum Stochastic Duhamel Principle
In this paper we shall re-visit the well-known Schr\"odinger and Lindblad
dynamics of quantum mechanics. However, these equations may be realized as the
consequence of a more general, underlying dynamical process. In both cases we
shall see that the evolution of a quantum state has the not
so well-known pseudo-quadratic form
where
is a vector operator in a complex Minkowski space and the pseudo-adjoint
is induced by the Minkowski metric . The
interesting thing about this formalism is that its derivation has very deep
roots in a new understanding of the differential calculus of time. This
Minkowski-Hilbert representation of quantum dynamics is called the
\emph{Belavkin Formalism}; a beautiful, but not well understood theory of
mathematical physics that understands that both deterministic and stochastic
dynamics may be `unraveled' in a second-quantized Minkowski space. Working in
such a space provided the author with the means to construct a QS (quantum
stochastic) Duhamel principle and known applications to a Schr\"odinger
dynamics perturbed by a continual measurement process are considered. What is
not known, but presented here, is the role of the Lorentz transform in quantum
measurement, and the appearance of Riemannian geometry in quantum measurement
is also discussed
Higher-Dimensional Twistor Transforms using Pure Spinors
Hughston has shown that projective pure spinors can be used to construct
massless solutions in higher dimensions, generalizing the four-dimensional
twistor transform of Penrose. In any even (Euclidean) dimension d=2n,
projective pure spinors parameterize the coset space SO(2n)/U(n), which is the
space of all complex structures on R^{2n}. For d=4 and d=6, these spaces are
CP^1 and CP^3, and the appropriate twistor transforms can easily be
constructed. In this paper, we show how to construct the twistor transform for
d>6 when the pure spinor satisfies nonlinear constraints, and present explicit
formulas for solutions of the massless field equations.Comment: 17 pages harvmac tex. Modified title, abstract, introduction and
references to acknowledge earlier papers by Hughston and other
Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy
We consider two-component integrable generalizations of the dispersionless
2DTL hierarchy connected with non-Hamiltonian vector fields, similar to the
Manakov-Santini hierarchy generalizing the dKP hierarchy. They form a
one-parametric family connected by hodograph type transformations. Generating
equations and Lax-Sato equations are introduced, a dressing scheme based on the
vector nonlinear Riemann problem is formulated. The simplest two-component
generalization of the dispersionless 2DTL equation is derived, its differential
reduction analogous to the Dunajski interpolating system is presented. A
symmetric two-component generalization of the dispersionless elliptic 2DTL
equation is also constructed.Comment: 10 pages, the text of the talk at NEEDS 09. Notations clarified,
references adde
On a vector-valued Hopf-Dunford-Schwartz lemma
In this paper, we state as a conjecture a vector-valued Hopf-Dunford-Schwartz
lemma and give a partial answer to it. As an application of this powerful
result, we prove some Fe fferman-Stein inequalities in the setting of Dunkl
analysis where the classical tools of real analysis cannot be applied
Fractional generalizations of filtering problems and their associated fractional Zakai equations
In this paper we discuss fractional generalizations of the filtering problem. The âfractionalâ nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and LĂ©vy process
Finite W_3 Transformations in a Multi-time Approach
Classical {\W} transformations are discussed as restricted diffeomorphism
transformations (\W-Diff) in two-dimensional space. We formulate them by using
Riemannian geometry as a basic ingredient. The extended {\W} generators are
given as particular combinations of Christoffel symbols. The defining equations
of \W-Diff are shown to depend on these generators explicitly. We also consider
the issues of finite transformations, global transformations and
\W-Schwarzians.Comment: 10 pages, UB-ECM-PF 94/20, TOHO-FP-9448, QMW-PH-94-2
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