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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 new file mode 100644 index 0000000..2d05b8e --- /dev/null +++ b/site/posts/2015-07-01-causal-quantum-product-levy-area.html @@ -0,0 +1,29 @@ +<!doctype html> +<html lang="en"> + <head> + <meta charset="utf-8"> + <title>On a causal quantum double product integral related to Lévy stochastic area.</title> + <link rel="stylesheet" href="../assets/css/default.css" /> + <script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script> + </head> + <body> + <header> + <span class="logo"> + <a href="../blog.html">Blog</a> + </span> + <nav> + <a href="../index.html">About</a><a href="../postlist.html">All posts</a><a href="../blog-feed.xml">Feed</a> + </nav> + </header> + + <div class="main"> + <div class="bodyitem"> + <h2> On a causal quantum double product integral related to Lévy stochastic area. </h2> + <p>Posted on 2015-07-01</p> + <p>In <a href="https://arxiv.org/abs/1506.04294">this paper</a> with <a href="http://homepages.lboro.ac.uk/~marh3/">Robin</a> we study the family of causal double product integrals \[ \prod_{a < x < y < b}\left(1 + i{\lambda \over 2}(dP_x dQ_y - dQ_x dP_y) + i {\mu \over 2}(dP_x dP_y + dQ_x dQ_y)\right) \]</p> +<p>where <span class="math inline"><em>P</em></span> and <span class="math inline"><em>Q</em></span> 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 <a href="http://www.actaphys.uj.edu.pl/findarticle?series=Reg&vol=46&page=1851">(Hudson-Pei2015)</a>. The main problem solved in this paper is the explicit evaluation of the continuum limit <span class="math inline"><em>W</em></span> of the latter, and showing that <span class="math inline"><em>W</em></span> is a unitary operator. The kernel of <span class="math inline"><em>W</em></span> 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.</p> + + </div> + </div> + </body> +</html> |