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| author | Yuchen Pei <me@ypei.me> | 2018-06-03 22:22:43 +0200 |
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| committer | Yuchen Pei <me@ypei.me> | 2018-06-03 22:22:43 +0200 |
| commit | d4d048e66b16a3713caec957e94e8d7e80e39368 (patch) | |
| tree | 1aa7c6640d56de3741f23073bb5d6f1e3db61e17 /site/posts/2015-07-01-causal-quantum-product-levy-area.html | |
| parent | 2e38d28086714175d680f9d4541c735ca793d2b7 (diff) | |
fixed mathjax conversion from md
Diffstat (limited to 'site/posts/2015-07-01-causal-quantum-product-levy-area.html')
| -rw-r--r-- | site/posts/2015-07-01-causal-quantum-product-levy-area.html | 2 |
1 files changed, 1 insertions, 1 deletions
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 @@ <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> +<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> |