From d4d048e66b16a3713caec957e94e8d7e80e39368 Mon Sep 17 00:00:00 2001 From: Yuchen Pei Date: Sun, 3 Jun 2018 22:22:43 +0200 Subject: fixed mathjax conversion from md --- site/posts/2015-07-01-causal-quantum-product-levy-area.html | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'site/posts/2015-07-01-causal-quantum-product-levy-area.html') diff --git a/site/posts/2015-07-01-causal-quantum-product-levy-area.html b/site/posts/2015-07-01-causal-quantum-product-levy-area.html index cda8121..3fdaa72 100644 --- a/site/posts/2015-07-01-causal-quantum-product-levy-area.html +++ b/site/posts/2015-07-01-causal-quantum-product-levy-area.html @@ -22,7 +22,7 @@

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

Posted on 2015-07-01

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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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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