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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 deleted file mode 100644 index 3fdaa72..0000000 --- a/site/posts/2015-07-01-causal-quantum-product-levy-area.html +++ /dev/null @@ -1,30 +0,0 @@ -<!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> - <script src="../assets/js/analytics.js" type="text/javascript"></script> - </head> - <body> - <header> - <span class="logo"> - <a href="../blog.html">Yuchen's 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">\(P\)</span> and <span class="math inline">\(Q\)</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">\(W\)</span> of the latter, and showing that <span class="math inline">\(W\)</span> is a unitary operator. The kernel of <span class="math inline">\(W\)</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> |