diff --git a/Overview-paper/Physics.bib b/Overview-paper/Physics.bib index 1d58d8e..95df548 100644 --- a/Overview-paper/Physics.bib +++ b/Overview-paper/Physics.bib @@ -795,7 +795,39 @@ @book{ashcroft:2005a Publisher = {Harcourt College Publishers}, Title = {Solid {S}tate {P}hysics}, Year = {1976}, - Bdsk-Url-1 = {http://www.zentralblatt-math.org/zmath/en/search/?format=complete&q=an:1118.82001}} + Bdsk-Url-1 = {http://www.zentralblatt-math.org/zmath/en/search/?format=complete&q=an:1118.82001}} + +@Article{ashtekar_representation_1993, + author = {Ashtekar, Abhay and Lewandowski, Jerzy}, + title = {Representation {{Theory}} of {{Analytic Holonomy C}}* {{Algebras}}}, + year = {1993}, + month = nov, + archiveprefix = {arXiv}, + eprint = {gr-qc/9311010}, + eprinttype = {arxiv}, + keywords = {General Relativity and Quantum Cosmology,High Energy Physics - Theory,Mathematics - Algebraic Geometry}, + timestamp = {2016-06-09T20:19:03Z}, + urldate = {2016-06-09} +} + +@Article{ashtekar_representations_1992, + author = {Ashtekar, Abhay and Isham, C. J.}, + title = {Representations of the holonomy algebras of gravity and non-{{Abelian}} gauge theories}, + journal = {Classical and Quantum Gravity}, + year = {1992}, + volume = {9}, + number = {6}, + pages = {1433--1467}, + month = jun, + archiveprefix = {arXiv}, + doi = {10.1088/0264-9381/9/6/004}, + eprint = {hep-th/9202053}, + eprinttype = {arxiv}, + issn = {0264-9381, 1361-6382}, + keywords = {High Energy Physics - Theory}, + timestamp = {2016-06-09T20:18:50Z}, + urldate = {2016-06-09} +} @unpublished{aubrun:2010a, Author = {Aubrun, Guillaume and Szarek, Stanis{\l}aw and Werner, Elisabeth}, @@ -3218,6 +3250,25 @@ @unpublished{dittrich_decorated_2014 author = {Dittrich, Bianca and Mizera, Sebastian and Steinhaus, Sebastian}, year = {2014}, note = {arXiv:1409.2407}, +} + +@Article{dittrich_discrete_2012, + author = {Dittrich, Bianca}, + title = {From the discrete to the continuous - towards a cylindrically consistent dynamics}, + journal = {New Journal of Physics}, + year = {2012}, + volume = {14}, + number = {12}, + pages = {123004}, + month = dec, + archiveprefix = {arXiv}, + doi = {10.1088/1367-2630/14/12/123004}, + eprint = {1205.6127}, + eprinttype = {arxiv}, + issn = {1367-2630}, + keywords = {General Relativity and Quantum Cosmology,High Energy Physics - Lattice,Quantum Physics}, + timestamp = {2016-06-06T08:39:55Z}, + urldate = {2016-06-06} } @incollection{divincenzo:1998a, diff --git a/Overview-paper/ymoverview.tex b/Overview-paper/ymoverview.tex index 198b55d..6f37d3e 100644 --- a/Overview-paper/ymoverview.tex +++ b/Overview-paper/ymoverview.tex @@ -819,7 +819,7 @@ \section{The continuum limit} Conditioned on the validity of the previously mentioned conjectures, we've produced a sequence of states $|\Psi_m\rangle$ which tend, in the limit, to the (lattice) zero-coupling state. The important observation here is that since each term in our sequence is a MERA the correlation length of $|\Psi_m\rangle$ is given by $\xi_m = a_m\lambda^m$, for some $\lambda>0$. Given that the correlation length $\xi$ of pure Yang-Mills theory is only determined up to a constant which is ultimately fixed by experiment forces us to set the lattice spacing $a_m$ of the state $|\Psi_m\rangle$ to $a_m = a_0\lambda^{-m}$, where $a_0$ is a constant. Thus we have a sequence of states $|\Psi_m\rangle$ for lattices of ever finer discretisation. Each term in the sequence is the result of an isometry applied to a previous term. Further, we can compute all $n$-point functions for this sequence. It turns out that this is enough data to specify a continuum hilbert space and a canonical continuum ground state. -The continuum hilbert space we describe here is known as a \emph{direct limit} of hilbert spaces; in the context we use it here we call this direct limit the \emph{semicontinuous limit} \footnote{This terminology was suggested to us by Vaughan Jones.} to indicate that it doesn't quite correspond to what we might demand of a full continuous quantum Yang-Mills theory. Note that the direct limit is a basic categorical construction (you can read about it further in, e.g., \cite{lang:2002a}). The application of the direct limit to hilbert spaces has a very long history; one early proposal to use the direct limit to model continuum limits can be found in \cite{bimonte_lattices_1996}, but there are surely prior proposals. A recent fascinating attempt to use the direct limit to build continuum limits of lattice theories, in particular, conformal field theories, can be found in \cite{jones_unitary_2014}. +The continuum hilbert space we describe here is known as a \emph{direct limit} of hilbert spaces; in the context we use it here we call this direct limit the \emph{semicontinuous limit} \footnote{This terminology was suggested to us by Vaughan Jones.} to indicate that it doesn't quite correspond to what we might demand of a full continuous quantum Yang-Mills theory. Note that the direct limit is a basic categorical construction (you can read about it further in, e.g., \cite{lang:2002a}). The application of the direct limit to hilbert spaces has a very long history; one early proposal to use the direct limit to model continuum limits can be found in \cite{bimonte_lattices_1996}, but there are surely prior proposals. It is, for example, a standard technique in quantum gravity \cite{ashtekar_representations_1992, ashtekar_representation_1993, dittrich_discrete_2012}. A recent fascinating attempt to use the direct limit to build continuum limits of lattice theories, in particular, conformal field theories, can be found in \cite{jones_unitary_2014}. Let $\mathcal{D}$ be the directed set of regular partitions of $\mathbb{R}^d$ induced by integer lattices with lattice spacing $a$, i.e., $a\mathbb{Z}^d$. This set is directed by \emph{refinement}, i.e., a partition $Q$ is a \emph{refinement} of $P$, denoted $P \preceq Q$, if every element of $Q$ is a subset of an element of $P$. (A useful mnemonic to remember the ordering is that $Q$ has ``more'' elements than $P$.) We regard every lattice spacing $a$ as giving rise to a \emph{physically different} lattice.