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\hypersetup{pdftitle={Mollifying Quantum Field Theory},
  pdfsubject={DoE - Mollifying Quantum Field Theory},
  pdfauthor={Daniel Doro Ferrante <danieldf@het.brown.edu>},
  pdfkeywords={mollify, quantum, field, theory},
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\title{\textcolor{yellow}{Mollifying Quantum Field Theory}}
\subtitle{\textcolor{darkorange2}{\textsf{DoE 2004}}}
\author{\textcolor{lightyellow}{\href{http://www.het.brown.edu/}{Daniel Doro
      Ferrante and Gerald S. Guralnik}}}
\institution{%
  High Energy Theory \\
  \href{http://www.brown.edu/}{Brown University}}
\email{danieldf@het.brown.edu}
\DefaultTransition{Dissolve}
\Logo(-1.5,-1){\includegraphics[scale=0.07]{coco.eps}}
\slideCaption{\textsf{DoE '04}}
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\begin{document}
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\begin{slide}[Glitter]{\yellow\emph{\textsf{DoE '04}}}
  \vspace{-0.25cm}
  \begin{enumerate}
  \item Conventional Monte Carlo modified to examine phase structure. Mollification
    techniques. Ferrante will talk on this.
  \item Non-conventional numerical approach to numerical Quantum Field Theory:
    Source-Galerkin.
  \item General Relativity/Astrophysics: Unsuspected topological solutions to
    dimensionally reduced Hilbert-Einstein gravity.
  \end{enumerate}
\end{slide}
%%
\maketitle
%%
\begin{slide}[Glitter]{Outline}
  \begin{itemize}
  \item \hyperlink{lqft}{Lattice Quantum Field Theory}
  \bigskip
  \item \hyperlink{moll}{Mollifiers}
  \bigskip
  \item \hyperlink{piat}{Putting it all together\ldots}
    \begin{itemize}
    \item \hyperlink{mqft}{Mollifying QFT}
    \item \hyperlink{ga}{Genetic Algorithm}
    \item \hyperlink{ex}{Example}
    \end{itemize}
  \item \hyperlink{fw}{Future Work\ldots}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\hypertarget{lqft}{\yellow Lattice Quantum Field Theory}}
  \begin{description}
  \item[{\yellow Path Integral:}] Lorentzian QFT, (highly oscillatory),
    \smallskip
    \begin{equation*}
      \langle\mathcal{O}[\phi]\rangle = \varint\, \mathcal{O}[\phi]\, \exp\bigl\{i\,
      S[\phi,J]\bigr\}\, \mathcal{D}\phi
    \end{equation*}
  \medskip
  \item[\textcolor{yellow}{Wick rotated:}] Euclidean QFT, (exponentially decaying),
    \smallskip
    \begin{equation*}
      \langle\mathcal{O}[\phi]\rangle_E = \varint\, \mathcal{O}[\phi]\, \exp\bigl\{-
      S_E[\phi,J]\bigr\}\, \mathcal{D}\phi
    \end{equation*}
  \end{description}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Lattice Quantum Field Theory}
  \begin{description}
  \item[\yellow{Lattice QFT:}] Discrete Euclidean QFT, (Markov chains, Metropolis Monte Carlo),
    \begin{equation*}
      \langle\mathcal{O}[\varphi]\rangle_E = \frac{1}{||\Phi||}\, \sum_{\varphi\in\Phi}\,
      \Biggl\{\frac{1}{N^d}\, \sum_{i=1}^{N^d}\, \mathcal{O}[\varphi^{[i]}] \Biggr\} \; ,
    \end{equation*}
    \smallskip
    where $\varphi^{[i]}$ are chosen with a probability density of
    $\exp\bigl\{-S_E[\varphi]\bigr\}$.
  \end{description}  
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Lattice Quantum Field Theory}
  \begin{description}
  \item[\yellow{Question:}] \emph{``Is it possible to do Lorentzian Lattice QFT ($\exp\bigl\{i\,
      S[\phi]\bigr\}$)?''}
  \bigskip
  \item[\yellow{Importance:}]\hfill
    \begin{itemize}
    \item Highly oscillatory integrands;
    \item Unified treatment for broken and symmetric phases;
    \item etc\ldots
    \end{itemize}
  \end{description}  
\end{slide}
%%
\begin{slide}[Dissolve]{\hypertarget{moll}{\yellow Mollifiers}}
  \vspace{-0.5cm}
  \begin{itemize}
  \item \textcolor{lightyellow}{Convolution} works as a \textcolor{lightyellow}{band-pass}
    filter;
  \item\textcolor{yellow}{Standard Mollifier}:
    \begin{equation*}
      \eta_{\epsilon}(x) = 
      \begin{cases}
        C\, \exp\Bigl\{\frac{1}{|x/\epsilon|^2 - 1}\Bigr\} &; |x| < 1 \\
        0 &; |x| \geqslant 1
      \end{cases}
    \end{equation*}
  \item\textcolor{yellow}{Gaussian Mollifier}:
    \begin{equation*}
      \eta_{\epsilon}(x) = \frac{1}{\sqrt{2\, \pi\, \epsilon^2\,{}}}\,
      \exp\biggl\{-\frac{1}{2}\, (x/\epsilon)^2\biggr\}
    \end{equation*}
  \item Normalization: $\int\eta_{\epsilon}(x) = 1$
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\hypertarget{piat}{\yellow Putting it all together\ldots}}
  \begin{itemize}
  \item Smooth out the highly oscillatory parts (via a convolution with a mollifier),
    leaving only the most important contributions.
  \bigskip
  \item Code: Genetic Algorithm (non-local correlations) + Metropolis Monte Carlo.
  \end{itemize}
\end{slide}
%%
\begin{slide}[Glitter]{\hypertarget{mqft}{\yellow Mollifying Quantum Field Theory}}
  \begin{itemize}
  \item Mollified Generating Functional
    \begin{equation*}
      \mathcal{Z}[J] \mapsto \mathcal{Z}_{\epsilon}[J] \equiv\varint\biggl\{\varint
        \eta_{\epsilon}[\phi - \varphi]\, e^{i\, S[\varphi,J]}\, \mathcal{D}\varphi
        \biggr\}\, \mathcal{D}\phi
    \end{equation*}
  \item property of convolutions (Fubini thm)
    \begin{equation*}
      \mathcal{Z}[J] = \mathcal{Z}_{\epsilon}[J] 
    \end{equation*}
  \item property of mollifiers (approx by smooth functs)
    \begin{equation*}
      \lim_{\epsilon\rightarrow 0}\mathcal{Z}_{\epsilon}[J] = \mathcal{Z}[J]
    \end{equation*}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Glitter]{\hypertarget{ga}{\yellow Genetic Algorithm}}
  \begin{algorithm}
    \caption{Genetic Algorithm}
    \label{alg:ga}
    \begin{algorithmic}[1]
      \STATE Choose the recombination rate such that: $0 \leqslant r\leqslant 1$ \COMMENT{usually
        between $0.5\%$ and $1.0\%$}
      \STATE Draw a random number $\beta\in [0,1]$ and compare with $r$
      \IF{$\beta \geqslant r$}
      \STATE Metropolis Monte Carlo loop \COMMENT{using the \emph{random walk} technique}
      \ELSE
      \STATE Genetic loop
      \ENDIF
    \end{algorithmic}
  \end{algorithm}
\end{slide}
%%
\begin{slide}[Glitter]{\yellow Genetic Algorithm}
  \begin{algorithm}
    \caption{Genetic Loop}
    \label{alg:gl}
    \begin{algorithmic}[1]
      \STATE Choose a random pair: $\{\phi_1, \phi_2\} = \{\phi(t_1,x_1,y_1,z_1),
      \phi(t_2,x_2,y_2,z_2)\}$
      \STATE Generate the pair $\{\phi'_1, \phi'_2\} = \{\phi(t'_1,x'_1,y'_1,z'_1),
      \phi(t'_2,x'_2,y'_2,z'_2)\} = \mathds{G}(\phi_1, \phi_2)$
      \STATE Draw a random number $c\in [0,1]$
      \IF{$c > P(\phi'_1, \phi'_2)/P(\phi_1, \phi_2)$}
      \STATE do \textbf{nothing}
      \ELSE
      \STATE perform the exchange $(\phi_1, \phi_2) \mapsto (\phi'_1, \phi'_2)$
      \ENDIF
    \end{algorithmic}
  \end{algorithm}
\end{slide}
%%
\begin{slide}[Glitter]{\hypertarget{ex}{\yellow Example}}
  \begin{description}
  \item[\yellow{Airy Function:}] 
    \begin{align*}
      \mathcal{Z}[J] &= \frac{\displaystyle\int^{\infty}_{-\infty} \exp\bigl\{i\, x^3/3 +
        i\, J\, x\bigr\} \, dx}{\displaystyle\int^{\infty}_{-\infty} \exp\bigl\{i\,
        x^3/3\bigr\} \, dx} = \frac{\mathrm{Ai}(J)}{\mathrm{Ai}(0)} \\
      \mathcal{Z}_{\epsilon}[J] &= \frac{\displaystyle\int^{\infty}_{-\infty}
        \eta_{\epsilon}[y-x]\, \exp\bigl\{i\, x^3/3 + i\, J\, x\bigr\} \,
        dx\, dy}{\displaystyle\int^{\infty}_{-\infty} \eta_{\epsilon}[y-x]\,
        \exp\bigl\{i\, x^3/3\bigr\} \, dx\, dy}
    \end{align*}
  \end{description}
\end{slide}
%%
\begin{slide}[Glitter]{\yellow Example}
  \begin{description}
  \item[\yellow{Airy Function:}]\hfill\newline
    \vspace{3cm}
    \includegraphics[scale=0.2]{reeminuss2d.eps}
    \hspace{1cm}
    \includegraphics[scale=0.2]{reeminuss3d.eps}
  \end{description}
\end{slide}
%%
\begin{slide}[Dissolve]{\hypertarget{fw}{\yellow Future Work}}
  \vspace{-0.5cm}
  \begin{description}
  \item[\yellow{Topology Change in GR:}] It was found that, on the \emph{Gravitational
  Chern-Simons} theory, a broken symmetric phase emerges. The plan would be to study the
  \emph{Dirac Operator} defined in the symmetric and in the broken-symmetric phases, under
  the light of \emph{Spectral Geometry, the Atyiah-Singer Index Theorem and Morse Theory}
  in order to check for a \emph{topology change} between the two phases. This would be a
  new kind of \emph{topological defect} and, in turn, would have several implications to
  Cosmology.
  \end{description}
\end{slide}
%%
\begin{slide}[Glitter]{\yellow Source-Galerkin}
  \begin{itemize}
  \item Solve Schwinger-Dyson's eq using diff eq evaluation techniques.
  \item Obvious advantage: Do \textbf{not} have to deal with fermionic integrals (as
    opposed to Monte Carlo). Suggests that a symmetrical approach is possible.
  \item Furthermore, phases are easily handled.
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Example: $\phi^4$}
  \begin{align*}
    S = \int \frac{1}{2}\, \partial_{\mu}\phi\partial^{\mu}\phi &- \frac{m^2}{2}\, \phi^2
      - \frac{g}{4}\, \phi^4 + J\, \phi\, d^4x \\
    \Rightarrow\; \mathcal{Z} &= \varint e^{-S}\, [d\phi] \\
    \text{Monte Carlo} &\Longleftrightarrow\, [d\phi] = \int_{-\infty}^{\infty} d\phi_1 \dotsi
      \int_{-\infty}^{\infty} d\phi_n
  \end{align*}
  \textit{i.e.}, all integrals in the \textbf{real} line, $\mathbb{R}$.
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Example: $\phi^4$}
  \begin{itemize}
  \item Solutions are only for the phase that is \textbf{regular} as $g \rightarrow
    0$. Must integrate off of $\mathbb{R}$, (no direct MC approach), for other phases ---
    symmetry breaking.
  \item Other solutions through Schwinger-Dyson:
    \begin{align*}
      (-\partial^2 - m^2)\, \phi - g\, \phi^3 + J(x) &= 0 \\
      \biggl[(-\partial^2 - m^2)\, \frac{\delta}{\delta J(x)} - g\,
        \biggl(\frac{\delta}{\delta J(x)}\biggr)^3 + J(x)\biggr]\, \mathcal{Z} &= 0
    \end{align*}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Example: $\phi^4$}
  \begin{itemize}
  \item This is an infinite set of coupled linear diff eq: coupled through spacetime
    derivatives
  \item deconstruct to 0 dimensions: ordinary 3rd order diff eq
    \begin{equation*}
      \biggl[-m^2\, \frac{d}{d\, J} - g\, \biggl(\frac{d}{d\, J}\biggr)^3 + J\biggr]\, \mathcal{Z} = 0
    \end{equation*}
  \item Solution is still not trivial: three independent solutions
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Example: $\phi^4$}
  \begin{itemize}
  \item Can work them out by Taylor expansion:
    \begin{equation*}
      \mathcal{Z} = \sum_{n} \frac{a_n\, J^n}{n!}
    \end{equation*}
  \item However, we want to find a more rapidly converging approach
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Example: $\phi^4$}
  \begin{itemize}
  \item The idea is to make a guess as to what a solution to this diff eq looks like in
    terms of basis of a complete set of functions
  \item A very nice set of orthogonal functions are the Hermite polynomials because they
    are orthogonal with an exponentially decreasing weight
  \end{itemize}
\end{slide}
%%
%%
\begin{slide}[Dissolve]{\yellow Example: $\phi^4$}
  \vspace{-0.7cm}
  \begin{itemize}
  \item The modified Hermite polys are given as
    \vspace{-0.3cm}
    \begin{align*}
      H_n^{\epsilon} &= (-1)^n\, e^{(\xi/\epsilon)^2}\, \frac{d^n}{d\xi^n}
        e^{-(\xi/\epsilon)^2} \\
      H_n = (-1)^n\, e^{x^2}&\, \frac{d^n}{d\xi^n} e^{-x^2} \Rightarrow
        H_n^{\epsilon}(\xi) = \epsilon^n\, H_n(\xi/\epsilon) \\
      \hspace{-0.3cm}
      \delta_{m\, n}\, 2^n\, n!\, \sqrt{\pi\,{}}\, \epsilon^{2n-1} =& \int_{-\infty}^{\infty} 
        H_m^{\epsilon}(x/\epsilon)\, H_n^{\epsilon}(x/\epsilon)\, e^{-(x/\epsilon)^2}\, dx \\
      H_0^{\epsilon} &= 1 \; , \; H_1 = 2\,\frac{\xi}{\epsilon^2} = 2\,
        \,\frac{\xi}{\epsilon^2}\, H_0 \\
      \therefore\; H_{n+1}^{\epsilon}(\xi) &= 2\,\frac{\xi}{\epsilon^2}
        H_n^{\epsilon}(\xi) - H_n^{\prime\, \epsilon}(\xi)
    \end{align*}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Example: $\phi^4$}
  \begin{itemize}
  \item $\boldsymbol\epsilon$: control of the computation (dilation parameter)
  \item Guess truncated solution:
    \begin{equation*}
      \mathcal{Z}^*(J) = \sum_{i=0}^{N} a_i\, H_i^{\epsilon}(J)
    \end{equation*}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Example: $\phi^4$}
  \begin{itemize}
  \item Substitute $\mathcal{Z}^*(J)$ into Schwinger-Dyson to get
    \begin{equation*}
      \biggl[-m^2\, \frac{d}{d\, J} - g\,\biggl(\frac{d}{d\, J}\biggr)^3 + J\biggr]\,
        \mathcal{Z}^* = \mathrm{Res}(J)
    \end{equation*}
     when $N\rightarrow\infty$ exact solution $\Rightarrow\mathrm{Res}(J) = 0$
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Source-Galerkin}
  \begin{itemize}
  \item Idea: Set the $a_i$ by requiring that the eq vanishes on the average
  \item Note: 3 of the $a_i$ are arbitrary because this is a 3rd order diff eq
  \end{itemize}
  {\small
  \begin{equation*}
    \hspace{-0.3cm}
    \int_{-\infty}^{\infty} \sum a_i\, \biggl(m\, \frac{d}{d\, x}H_i^{\epsilon}(x) - g\,
      \frac{d^3}{d\, x^3}H_i^{\epsilon}(x) + J\, H_i^{\epsilon}(x)\biggr)\,
      H_k^{\epsilon}(x)\, e^{-(x/\epsilon)^2}\, dx
  \end{equation*}
  }
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Source-Galerkin}
  \begin{itemize}
  \item \textcolor{yellow}{Breakthrough in our thinking! \blacksmiley}
  \item Because of (we thought) constraints imposed by higher dimensions, we used to only
    look at Taylor expansions ($J^k$ complete but \textbf{not} orthogonal)
    \begin{equation*}
      \mathcal{Z}^*(J) = \sum_{k=0}^{N} \frac{a_k\, J^k}{k!}
    \end{equation*}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Source-Galerkin}
  \begin{itemize}
  \item We found $\mathrm{Res}(J)$ and determined $a_k$ (particular choice of BC) from the
  condition
    \begin{equation*}
      \int_{-\infty}^{\infty} J^n\, \mathrm{Res}(J)\, e^{-(J/\xi)^2}\, dJ = 0
    \end{equation*}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Arbitrary Dimension}
  \begin{itemize}
  \item Rules are the same but dealing with a guess is much harder
  \item Arbitrary interaction
    \begin{align*}
      &\mathcal{L}(\phi) = \frac{1}{2}\, \partial^{\mu}\phi\partial_{\mu}\phi -
        \frac{m^2}{2}\, \phi^2 - V(\phi) + J\,\phi \\
      \hspace{-0.3cm}
      0 &= \bigl(\partial^2 + m^2\bigr)\frac{\delta \mathcal{Z}[J]}{\delta J(x)} +
        V'\bigl[\delta\mathcal{Z}[J]/\delta J(x)\bigr] - J(x)\, \mathcal{Z}[J]
    \end{align*}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Arbitrary Dimension}
  \begin{itemize}
  \item Most general solution
    \begin{equation*}
      \mathcal{Z}[J] = \sum_{m=0}^{\infty} \frac{1}{m!}\, \int G_m(x_1,\dotsc, x_m)\,
        J(x_1)\dotsb J(x_m)
    \end{equation*}
  \item In this form
    \begin{align*}
      \mathcal{Z}^*[J] &= c + \int G^*(x_1)\, J(x_1) d^nx_1 + \\
      &\; + \int G^*(x_1,x_2)\,\frac{J(x_1)\, J(x_2)}{2}\, d^nx_1\, d^nx_2 + \dotsb
    \end{align*}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Arbitrary Dimension}
  \begin{itemize}
  \item Inserting into Schwinger-Dyson's eq we obtain $\mathrm{Res}(J)$
  \item We characterize $G^*(x_1,\dotsc,x_n)$ by numerical parameters and set their values
    such that $\langle \mathrm{Res}(J) \rangle = 0$
  \item In analogy with the 0-dim case, we define a [non-unique] measure of source space
    {\tiny
    \begin{equation*}
      \hspace{-1cm}
      \int J(x_1)\dotsb J(x_n)\, \exp\biggl\{-\int \frac{J^2(x)}{2\, \epsilon^2}\,
        dx\biggr\}\, [dJ] = 
        \begin{cases}
          \epsilon^{n+1}\, \delta^+(x_1,\dotsc,x_n) & n \text{ even} \\
          0 & n \text{ odd}
        \end{cases}
    \end{equation*}
    }
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Arbitrary Dimension}
  \begin{itemize}
  \item This makes sure integrals converge
  \item $\epsilon \rightarrow 0$ emphasizes low order Green's functions strongly and
    $\epsilon \rightarrow \infty$ emphasizes high order
  \item With this measure we require
    \begin{align*}
      \bigl(\mathrm{Res}(J), J(x_1)\bigr) &= 0 \\
      & \vdots \\
      \bigl(\mathrm{Res}(J), J(x_1)\dotsb J(x_n)\bigr) &= 0
    \end{align*}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Arbitrary Dimension}
  \begin{itemize}
  \item Very important: to do the integrals we must put in forms of $G_1, G_2, \dotsc,
    G_n$.
  \item Parametrize with Feynman graphs with arbitrary internal topology, weight and mass
  \item This spans the Poincaré invariant space of this theory
  \item If there are other symmetries, one can easily build them into this guess
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Arbitrary Dimension}
  \begin{itemize}
  \item Note there also is a weighted integral wrt spatial variable to produce a number
  \item This works but does not converge well because the $J$'s while complete are not
    orthogonal
  \item \textcolor{yellow}{Breakthrough! \blacksmiley} Complete orthogonal function of
    arbitrary spacetime dim
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Arbitrary Dimension}
  \begin{itemize}
  \item There is a ``simple'' generalization of 1-dim Hermite polys
    \begin{equation*}
      H_n(\xi_1,\dotsc,\xi_n) = e^{\frac{1}{2}\, \xi^T\xi}\, (-1)^n\,
        \frac{\partial^n}{\partial\xi_1 \dotsb \partial\xi_n} e^{-\frac{1}{2}\xi^T\xi}
    \end{equation*}
  \item Weighted average
    \begin{equation*}
      \langle f(\xi)\, g(\xi) \rangle \equiv \int f(\xi) \, g(\xi) \frac{e^{-\frac{1}{2}\,
        \xi^T\xi}}{(2\pi)^{n/2}}\, d\xi
    \end{equation*}
  \end{itemize}
\end{slide}
%%
\begin{slide}[Dissolve]{\yellow Arbitrary Dimension}
  \begin{itemize}
  \item We deal with these just as we dealt with $J(x_1)\dotsb J(x_n)$ with
    \begin{equation*}
      \mathcal{Z}^* = G_0^{H} + G_1^H\, H_1 + G_2^H\, H_2 + \dotsb + G_m^H\, H_m
    \end{equation*}
  \item Represent the $G_m$ as sums of Feynman's graphs up to order $m$
  \end{itemize}
\end{slide}
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\end{document}
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% EnD
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