From 147a19e84a743f1379f05bf2f444143b4afd7bd6 Mon Sep 17 00:00:00 2001 From: Yuchen Pei Date: Fri, 18 Jun 2021 12:58:44 +1000 Subject: Updated. --- ...015-07-01-causal-quantum-product-levy-area.html | 51 ++++++++++++++++++++++ 1 file changed, 51 insertions(+) create mode 100644 site-from-md/posts/2015-07-01-causal-quantum-product-levy-area.html (limited to 'site-from-md/posts/2015-07-01-causal-quantum-product-levy-area.html') diff --git a/site-from-md/posts/2015-07-01-causal-quantum-product-levy-area.html b/site-from-md/posts/2015-07-01-causal-quantum-product-levy-area.html new file mode 100644 index 0000000..57b4fcd --- /dev/null +++ b/site-from-md/posts/2015-07-01-causal-quantum-product-levy-area.html @@ -0,0 +1,51 @@ + + + + + On a causal quantum double product integral related to Lévy stochastic area. + + + + + +
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On a causal quantum double product integral related to Lévy stochastic area.

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Posted on 2015-07-01

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In this paper with Robin we study the family of causal double product integrals \[ \prod_{a < x < y < b}\left(1 + i{\lambda \over 2}(dP_x dQ_y - dQ_x dP_y) + i {\mu \over 2}(dP_x dP_y + dQ_x dQ_y)\right) \]

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where \(P\) 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 rotation-like operators in different planes due to a result in (Hudson-Pei2015). 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.

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