Potential theory, path integrals and the Laplacian of the indicator
Abstract
This paper links the field of potential theory — i.e. the Dirichlet and Neumann problems for the heat and Laplace equation — to that of the Feynman path integral, by postulating the following seemingly illdefined potential: V(x):=∓ {{σ^2}}/2nabla_x^2{1_{{xin D}}} where the volatility is the reciprocal of the mass (i.e. m = 1/ σ ^{2}) and ħ = 1. The Laplacian of the indicator can be interpreted using the theory of distributions: it is the ddimensional analogue of the Dirac δ'function, which can formally be defined as partial_x^2{1_{x>0 }} . We show, first, that the path integral's perturbation series (or Born series) matches the classical single and double boundary layer series of potential theory, thereby connecting two hitherto unrelated fields. Second, we show that the perturbation series is valid for all domains D that allow Green's theorem (i.e. with a finite number of corners, edges and cusps), thereby expanding the classical applicability of boundary layers. Third, we show that the minus (plus) in the potential holds for the Dirichlet (Neumann) boundary condition; showing for the first time a particularly close connection between these two classical problems. Fourth, we demonstrate that the perturbation series of the path integral converges as follows:<Table Float="No" ID="Taba"> <tgroup align="left" cols="3"> <colspec align="left" colname="c1" colnum="1"/> <colspec align="left" colname="c2" colnum="2"/> <colspec align="left" colname="c3" colnum="3"/> <tbody> <row> <entry colname="c1"> <SimplePara> mode of convergence </SimplePara> </entry> <entry colname="c2"> <SimplePara>absorbed propagator</SimplePara> </entry> <entry colname="c3"> <SimplePara>reflected propagator</SimplePara> </entry> </row> <row> <entry colname="c1"> <SimplePara>convex domain</SimplePara> </entry> <entry colname="c2"> <SimplePara> alternating </SimplePara> </entry> <entry colname="c3"> <SimplePara> monotone </SimplePara> </entry> </row> <row> <entry colname="c1"> <SimplePara>concave domain</SimplePara> </entry> <entry colname="c2"> <SimplePara> monotone </SimplePara> </entry> <entry colname="c3"> <SimplePara> alternating </SimplePara> </entry> </row> </tbody> </tgroup> </Table> We also discuss the third boundary problem (which poses Robin boundary conditions) and discuss an extension to moving domains.
 Publication:

Journal of High Energy Physics
 Pub Date:
 November 2012
 DOI:
 10.1007/JHEP11(2012)032
 arXiv:
 arXiv:1302.0864
 Bibcode:
 2012JHEP...11..032L
 Keywords:

 Stochastic Processes;
 Integrable Equations in Physics;
 Boundary Quantum Field Theory;
 Exact SMatrix;
 Mathematical Physics;
 Quantum Physics;
 3102;
 31Axx;
 31Bxx;
 31Cxx;
 8102;
 81Uxx;
 30Gxx;
 45Fxx;
 35Jxx;
 35Kxx
 EPrint:
 46 pages, 2 figures