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\title{Computational High Energy Physics}
\author{Daniel Doro Ferrante and Gerald S. Guralnik}
\institute{%
  \texttt{http://chep.het.brown.edu/} \\
  Brown University, BOX 1843}
\email{\{danieldf,gerry\}@het.brown.edu}
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\begin{document}
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\maketitle
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\begin{multicols}{3}
  \section{Source-Galerkin}
  %
  \subsection{The Galerkin Technique}
  The Source-Galerkin technique consists of the application of the Galerkin technique to
  try to solve the Schwinger-Dyson equations. The Galerkin technique can be easily
  understood from the following example: Let $L[u] = 0$, where $L$ is a linear operator
  and $u$ is unknown. We try to solve for $u$ selecting a set of basis functions,
  $(u_1,\dotsc,u_n,\dotsc)$, such that $u \approx \sum_i c_i\, u_i$. Since this set of
  functions is arbitrarily chosen, what we want is to minimize the following expression:
  $c_i\, L[\sum_i u_i] = \mathrm{Res}(u)$, \textit{i.e.}, we want to find the set of
  $u_i$'s that minimizes the residues $\mathrm{Res}(u_i)$. In principle, if we are able to
  find $u_i$'s such that $\mathrm{Res}(u_i) = 0$, then $u \equiv \sum_i c_i\, u_i$. \\

  This method has the advantage of not having to deal with fermionic integrals, treating
  both, bosons and fermions, on a symmetrical footing. Furthermore, phases are easily
  handled. \\

  \subsection{Applications to Quantum Field Theory}
  In the context of QFT's, this takes the following form: once all the information of the
  theory is contained on its Generating Functional, $\mathcal{Z}[J]$, we want to use the
  above technique to solve Schwinger-Dyson's equation,
  \begin{equation*}
    \frac{\delta S[-i\, \delta/\delta J]}{\delta\phi}\mathcal{Z}[J] + J(x)\,
      \mathcal{Z}[J] = 0 \; .
  \end{equation*}
  (That is, this is equivalent to substituting $\phi \mapsto -i\, \delta/\delta J$ in the
  action and to expanding the generating functional, requiring the equations of motion to
  be the solutions that extremizes the action.) \\

  Our ansätze will be based on our knowledge of the Green's functions. So, if we are
  looking for a first-order try, we will use first order Green's functions, and so on and
  so forth. The guide to construct these trial solutions is to use the symmetries of the
  theory: Lorentz, gauge, etc. That is, we parameterize the ansätz via Feynman graphs with
  arbitrary internal topology, weight and mass, spanning the Poincaré-invariant space of
  the given theory.\\

  It is known from mathematics that the Galerkin technique produces and approximate
  solution that converges to the exact solution as the number of terms goes to
  infinity. In practice, the series will have to be truncated and it is likely that, for
  nontrivial theories, calculations of the 8th-order will be very hard to compute. Thus,
  rapid convergence and high accuracy must be must be ensured (even with only a few terms
  in the ansätz). Once this convergence is in the mean (afterall, we are projecting the
  solution in the space of the $u_i$'s), the accuracy will depend on the choice of inner
  product. \\

  \subsection{Dimensionally Deconstructing Quantum Field Theories (just to build them
  again)}
  Analyzing the dimensionally deconstructed $\phi^4$ theory, (\textit{a.k.a} ultra-local
  $\phi^4$; 0 spacetime dimensions), a good choice of $u_i$'s is given by [modified]
  Hermite polynomials (which carry an exponentially decreasing factor --- Gaussian --- in
  the inner product measure). Note that, from this 0-dimensional example, we are expected
  to obtain 3 solutions, given that the equation of motion is of the 3rd-order. However,
  using standard perturbative techniques, only one of these solutions can be obtained,
  namely that which is regular when $g \rightarrow 0$. The other two symmetry breaking
  solutions are not accessible via perturbation theory, making the use of non-perturbative
  techniques very desirable. \\

  For higher dimensions, there is a ``simple'' generalization of the [modified] Hermite
  polynomials, given by:
  \begin{equation*}
    H_n(\xi_1,\dotsc,\xi_n) = e^{1/2\, \xi^{\text{T}}\xi}\, (-1)^n\,
      \tfrac{\partial^n}{\partial\xi_1\dotsb\partial\xi_n}\, e^{-1/2\, \xi^{\text{T}}\xi}
      \; ;
  \end{equation*}
  where our n-th trial function is given by, $\mathcal{Z}^* = \sum_{i=1}^n c_i\, H_i$,
  where the $c_i$'s are given by sums of Feynman graphs up to order $n$.
  %
  \columnbreak
  %%
  \section{Mollifying Quantum Field Theory}
  %
  \subsection{Lattice Quantum Field Theory}
  Motivated by the question \emph{``Is it possible to do Lorentzian Lattice QFT
    ($\exp\bigl\{i\, S[\phi]\bigr\}$)?''}, we engaged in the study of convolutions (used
  as filters) in order to try to smooth out the integrand above. \\

  Note that, in Euclidian Lattice QFT, the object of study is,
  \begin{gather*}
    \langle\mathcal{O}[\phi]\rangle = \varint \mathcal{O}[\phi]\, \exp\bigl\{i\,
      S[\phi]\bigr\}\, \mathcal{D}\phi \mapsto
      \langle\mathcal{O}[\phi]\rangle_E = \varint \mathcal{O}[\phi]\,
      \exp\bigl\{- S[\phi]\bigr\}\, \mathcal{D}\phi \; ; \\
    \langle\mathcal{O}[\varphi]\rangle^{\text{latt}}_E = \frac{1}{||\Phi||}\,
      \sum_{\varphi\in\Phi}\, \Biggl\{\frac{1}{N^d}\, \sum_{i=1}^{N^d}\,
      \mathcal{O}[\varphi^{[i]}] \Biggr\} \; ;
  \end{gather*}
  where $\Phi$ is the set of all field configurations on the lattice and $\varphi^{[i]}$
  are chosen with a probability density of $\exp\bigl\{-S_E[\varphi]\bigr\}$ (Markov
  Chains [ergodicity], Metropolis Monte Carlo). \\

  The question of Lorentzian Lattice QFT is deeper than my appear at first sight. On the
  one hand, it would firstly enable the computation of highly oscillatory integrands
  including, but not limited to, phase transitions. Secondly, a single framework would be
  used for the treatment of all phases of a QFT. On the other hand, new techniques
  (analytic, numeric and algorithmic) have to be used in order to tackle this problem. Not
  to mention the fact that this \emph{exercise} sheds some new light and understanding on
  old friends of ours, e.g., Feynman's Path Integral, Schwinger-Dyson's equation, phase
  transitions, etc. \\

  \subsection{Mollifiers or Approximate Identities}
  This technique is vastly used in mathematics in order to build [smooth] functions out of
  distributions (a.k.a generalized functions), using convolutions as ``filters'' and
  making wildly-behaving objects more tractable. The idea is to convolve two objects so
  that the result is a smooth function:
  \begin{equation*}
    f_{\epsilon}(x) = \int_{\mathbb{R}} \eta_{\epsilon}(x-y)\, f(y)\, dy \; ;
  \end{equation*}
  where $\epsilon$ is a parameter to control the approximation and $f_{\epsilon}(x)$ is
  the mollification of $f(x)$. The more frequently used mollifiers are,
  \begin{align*}
    \text{Standard:}& & \eta_{\epsilon}(x) &= 
      \begin{cases}
        C\, \exp\Bigl\{\frac{1}{|x/\epsilon|^2 - 1}\Bigr\} &, |x| < 1 \; ; \\
        0 &, |x| \geqslant 1 \; ;
      \end{cases} \\
    \text{Gaussian:}& & \eta_{\epsilon}(x) &=  \frac{1}{\sqrt{2\, \pi\,
      \epsilon^2\,{}}}\, \exp\biggl\{-\frac{1}{2}\, (x/\epsilon)^2\biggr\} \; ;\\
    \text{Normalization:}& & \int_{\mathbb{R}}\eta_{\epsilon}(x) &= 1 \; .
  \end{align*}

  \subsection{Mollifying Quantum Field Theory}
  The idea is to use the mollification technology in order to smooth out (\textit{i.e.},
  filter) the highly oscillatory parts of the Path Integral in question, leaving only the
  most important contributions.
  \begin{align*}
    &\text{Mollified Generating Functional} & \mathcal{Z}[J] &\mapsto
      \mathcal{Z}_{\epsilon}[J] \equiv\varint_{\Gamma}\biggl\{\varint_{\mathbb{R}}
      \eta_{\epsilon}[\phi - \varphi]\, e^{i\, S[\varphi,J]}\, \mathcal{D}\varphi
      \biggr\}\, \mathcal{D}\phi \; ; \\
    &\text{Fubini's Theorem} & \mathcal{Z}[J] &= \mathcal{Z}_{\epsilon}[J] \; ; \\
    &\text{Property of Mollifiers} & \lim_{\epsilon\rightarrow 0}\mathcal{Z}_{\epsilon}[J]
      &= \mathcal{Z}[J] \; .
  \end{align*}
  Note that, by appropriately choosing $\Gamma$, you can pick out all the phase structure
  of the theory! Furthermore, this is a very nice way of introducing \emph{block
  variables} straight into the Path Integral formulation of a QFT.\\

  As a final point, note that you need to worry about non-local correlations when talking
  about Lattice QFT. A very novel approach is to use \emph{Genetic Algorithms} in order to
  deal with this problem. (There are works that show that this is a better choice than the
  Swendsen-Wang and/or the Wolff algorithm.) This is a pioneering idea pursued by us!
  %
  \columnbreak
  %%
  \section{Topology Change in General Relativity}
  %
  \subsection{Chern-Simons Theory}
  Chern-Simons (CS) theory has a curious history: It was discovered in the context of
  anomalies in the 70's. It was only by the mid-80's that it was realized that ordinary
  Einstein gravity in $(2+1)$-dimensions is a natural example of a CS system. There is an
  intrinsic connection between CS and the [mathematical] theory of knots and link
  invariants. (Established by Witten more than 10 years ago: summer of 1988.) The table
  below shows some of the analogies: \\

  \begin{center}
    \begin{tabular}{|c|c|} \hline\hline
      \textbf{Knot Theory}  & \textbf{Chern-Simons Theory} \\ \hline
      knots and links       & Wilson loops \\
      polynomial invariants & vev's of products of Wilson loops \\
      singular knots        & operators of singular knots \\
      Vassiliev invariants  & coefficients of the perturbative series \\
      configuration space integral & Landau gauge \\ \hline\hline
    \end{tabular}
  \end{center}
  \medskip

  The key to construct the CS form in 3-dim is as follows: the Pontryagin form,
  $\mathcal{P} = \tr(\mathcal{F}\wedge \mathcal{F})$, is closed, $\dop\mathcal{P} =
  0$. ($\mathcal{F} = \dop \mathcal{A} + \mathcal{A}\wedge \mathcal{A}$ is the curvature
  of the Lie algebra-valued connection 1-form $\mathcal{A}$, taken in the adjoint
  representation. Upon a gauge transformation, $\mathcal{F} \mapsto \mathcal{F}' =
  g^{-1}\, \mathcal{F}\, g$, where $g\in \mathfrak{g}$, the Lie algebra of the gauge group
  $\mathfrak{G}$. So using the cyclic property of the trace we see that the Pontryagin
  form, $\mathcal{P}$, remains invariant under gauge transformations.) By Poincaré's
  Lemma, $\mathcal{P}$ is locally exact, \textit{i.e.}, $\mathcal{P} =
  \dop\mathcal{Q}$. Thus, $\mathcal{Q} = L_{\text{SC}}$ is the CS Lagrangian, found to be
  $\mathcal{Q} = L_{\text{SC}} = \tr(\mathcal{A}\wedge \dop \mathcal{A} + \tfrac{2}{3}\,
  \mathcal{A}\wedge \mathcal{A}\wedge \mathcal{A})$. \\

  The essential ingredient in going to higher dimensions, $n$, is the existence of a
  closed $2\, n$-form, invariant under gauge transformations. It is straightforward to see
  that $\mathcal{Q}_{2\, n} = \tr(\mathcal{F}\wedge\dotsb\wedge \mathcal{F}) =
  \tr(\mathcal{F}^n)$ is what we are looking for. \\

%   More generally speaking, these invariants belong to the family of characteristic classes
%   known as Chern-Weil invariants. Chern-Weil Theory revealed a major connection between
%   algebraic topology and global differential geometry (a.k.a differential topology). For a
%   principal bundle, $E$, over a $n$-dim manifold, $M$, its $k$-th Pontryagin class,
%   $p_k(E)$, can be realized by the $4\, k$-form $\tr(\Omega^k)$, constructed with $k$
%   copies of the curvature $2$-form $\Omega$. In particular, $p_k(E) = \tr(\Omega^k) \in
%   H^{4\, k}_{\text{dR}}(M)$, where $H^{4\, k}_{\text{dR}}(M)$ is the $4\, k$-cohomology
%   group of $M$, does not depend on the choice of connection (a.k.a gauge invariant). If
%   two principal bundles are homeomorphic then their Pontryagin classes are the same
%   (Novikov's theorem). \\

%   The question \emph{``What is the 3-form whose exterior derivative is the 4-dim Euler
%   density?''} is answered by the Einstein-Hilbert action with nonzero cosmological
%   constant, ($S = \int_M (R - 2\, \Lambda)\, \sqrt{-g\,{}}\, d^4x$, where $R$ is the Ricci
%   scalar and $\Lambda$ is the cosmological constant): this is the relation between CS and
%   Einstein's gravity in $(2+1)$-dim. However, in this gravitational setting (tangent
%   bundle), the connection (\emph{resp.}\/ curvature) used is the one induced by the metric
%   (equivalently, one can think of tangent bundles as principal bundles whose structure
%   group is given by $O(n)$ or $SO(n)$ --- thus, the connection/curvature would be those
%   induced by these groups). The Euler density, a.k.a Euler characteristic, is given by the
%   generalized Gauss-Bonet Theorem, which states that: the integral of the Pfaffian of the
%   curvature is equal to $(2\, \pi)^n\, \chi(M)$, where $\chi(M)$ denotes the Euler
%   characteristic of the manifold $M$; $\int_M \text{Pf}(\Omega) = (2\, \pi)^n\,
%   \chi(M)$. \\

  More than sixty years of frustrated efforts to quantize this theory can explain the
  immediate attention drawn by Witten's observation that gravity in $(2+1)$-dim is an
  exactly solvable model! This means that the quantum theory can be completely and
  explicitly spelled out. This is due to the fact that $(2+1)$-dim gravity has no
  propagating degrees of freedom and, therefore, its quantum description is like that of a
  system of point particles. It is a particular case of a Topological [Quantum] Field
  Theory!

  \subsection{The Dirac Operator: Spectral Geometry, the Atyiah-Singer
    Index Theorem and [Non-]Commutative Geometry}
  With the help of the Dirac Operator --- \textit{i.e.}, a derivation of the spinor fields
  (defined on the Clifford [Geometric] Algebra of the bundle considered) --- an analysis
  of the wave equation can be made, based on its eigenvalues and eigenvectors. That is,
  the idea is to tackle the question \emph{Can one hear the shape of the drum?}, first
  posed by M. Kac (1966). This means to use the eigenvalues found above in order to
  reconstruct the manifold/bundle in which the wave equation is defined (boundary
  conditions are an essential ingredient of this potion). This subject is known as
  Spectral Geometry and, after the work of A. Connes, it has an intrinsic relation to
  [non-]commutative geometry, where a triple is given: $(\mathcal{A}, \mathcal{H}, D)$,
  where $\mathcal{A}$ is a $C^*$-algebra, $\mathcal{H}$ is the Hilbert space of
  square-integrable spinors and $D$ is the Dirac Operator. \\

  The Atyiah-Singer Index Theorem is essential in all of this construction, because it
  gives a tool that enables us to distinguish between different Dirac Operators and the
  topologies of the manifolds/bundles in which they are defined.

  \subsection{Morse Theory, Cobordisms, Topology Change and Phase Transitions}
  Given that the Chern-Simons form is a topological invariant, one can use it in order to
  measure whether a \emph{topological phase transition} has occurred. The mathematical
  tools available for the analysis of topology change are, essentially, Morse Theory
  (generalization of the calculus of variations) and Cobordism Theory (relation between
  manifolds based on their boundaries). Furthermore, the study of Instantons/Solitons is
  highly based on the topological properties of the manifold/bundle where the particular
  QFT is defined.
\end{multicols}
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