Robin Hudson's pathless path to quantum stochastic calculus

Applebaum, David

doi:10.31390/cosa.4.4.02

Cited by 2 publications

(1 citation statement)

References 72 publications

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“…This allows a very natural variant of the area to be constructed using the minimal onedimensional version of the calculus. It may be regarded as a response to the call [App10] for a study in this quantum context of some of the deeper properties of Brownian motion, as well as a furtherance of the theory of quantum stochastic product integrals [Hud07a,Hud07b,HP81].…”

Section: The Lévy Stochastic Areamentioning

confidence: 99%

On a causal quantum stochastic double product integral related to Lévy area

Hudson

Pei

2018

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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We study the family of causal double product integralswhere P and Q are the mutually noncommuting momentum and position Brownian motions of quantum stochastic calculus. The evaluation is motivated heuristically by approximating the continuous double product by a discrete product in which infinitesimals are replaced by finite increments. The latter is in turn approximated by the second quantisation of a discrete double product of rotationlike operators in different planes due to a result in [HP15]. The main problem solved in this paper is the explicit evaluation of the continuum limit W of the latter, and showing that W is a unitary operator. The kernel of W is written in terms of Bessel functions, and the evaluation is achieved by working on a lattice path model and enumerating linear extensions of related partial orderings, where the enumeration turns out to be heavily related to Dyck paths and generalisations of Catalan numbers.In this paper we begin the explicit construction of the unitary causal double product integralas the second quantisation Γ (W ) of a unitary operator W which differs from the identity operator I by an integral operator on the Hilbert space L 2 ([a, b)) whose kernel will be found explicitly.

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Section: The Lévy Stochastic Areamentioning

confidence: 99%