Triangle Seminars
Tuesday, 8 Jun 2021
Towards the path integral for gravity
Neil Turok
(Perimeter Institute and University of Edinburgh)
Abstract:
We show how Feynman's path integral for quantum mechanics may be defined without a Wick rotation to imaginary time.
Instead, we employ analytic continuation (and Cauchy's theorem) in the complexified space of paths being integrated over. We outline an existence proof and describe applications to both nonrelativistic quantum mechanics and to interference patterns due to gravitational microlensing in radio astronomy.
[please email a.held@imperial.ac.uk for zoom link or password]
We show how Feynman's path integral for quantum mechanics may be defined without a Wick rotation to imaginary time.
Instead, we employ analytic continuation (and Cauchy's theorem) in the complexified space of paths being integrated over. We outline an existence proof and describe applications to both nonrelativistic quantum mechanics and to interference patterns due to gravitational microlensing in radio astronomy.
[please email a.held@imperial.ac.uk for zoom link or password]
Posted by: IC
Thursday, 10 Jun 2021
Dually weighted graphs and 2d quantum gravity
Vladimir Kazakov
(ENS Paris)
Abstract:
Dually weighted graphs (DWG) are planar Feynman graphs bearing two sets of couplings: one set of usual couplings \(t_n\) attached to the vertices of valence \(n\), and another set of dual couplings \(t_n^*\) attached to the faces (dual vertices) of valence \(n\). Such couplings allow a deep control on possible shapes of planar graphs. For example, if one turns on only the couplings \(t_4\) and \(t_4^*\) the graph takes a ''fishnet form'' of a regular square lattice. The problem of counting of such graphs can be formulated as a modified hermitian one matrix model with an extra constant matrix. The partition function can be then represented in terms of the ''character expansion'' over Young tableaux, solvable by the saddle point approximation. I will review old results on DWG from my papers with M.Staudacher and Th.Wynter, including the techniques of computing Schur characters of a large Young tableau and deriving the elliptic algebraic curve for counting of planar quadrangulations. Then I will present new results from our ongoing work with F.Levkovich-Maslyuk where we count the disc quadrangulations with large, macroscopic area and boundary. This allows to extract interesting continuous limit of fluctuating 2d geometry, interpolating between the ''almost'' flat disc with a few dynamical conical defects and the disc partition function for pure 2d quantum gravity, generalizing old results for the spherical topology. –– Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.
Dually weighted graphs (DWG) are planar Feynman graphs bearing two sets of couplings: one set of usual couplings \(t_n\) attached to the vertices of valence \(n\), and another set of dual couplings \(t_n^*\) attached to the faces (dual vertices) of valence \(n\). Such couplings allow a deep control on possible shapes of planar graphs. For example, if one turns on only the couplings \(t_4\) and \(t_4^*\) the graph takes a ''fishnet form'' of a regular square lattice. The problem of counting of such graphs can be formulated as a modified hermitian one matrix model with an extra constant matrix. The partition function can be then represented in terms of the ''character expansion'' over Young tableaux, solvable by the saddle point approximation. I will review old results on DWG from my papers with M.Staudacher and Th.Wynter, including the techniques of computing Schur characters of a large Young tableau and deriving the elliptic algebraic curve for counting of planar quadrangulations. Then I will present new results from our ongoing work with F.Levkovich-Maslyuk where we count the disc quadrangulations with large, macroscopic area and boundary. This allows to extract interesting continuous limit of fluctuating 2d geometry, interpolating between the ''almost'' flat disc with a few dynamical conical defects and the disc partition function for pure 2d quantum gravity, generalizing old results for the spherical topology. –– Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant. The link will be emailed on Tuesday.
Posted by: andrea