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\begin{document}
% slide 1
%slide 1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Multi-scale methods in quantum field theory \\
PRD 95,094501 (2017)
}
\end{center}

\vfill

\begin{center}
{\bf 
  W. N. Polyzou, Tracie Michlin and Fatih Bulut
}
\end{center}

\vfill

\begin{center}
{\bf
The University of Iowa (WP and TM),In\"on\"u University
(FB)
}
\end{center}



\begin{figure}
\begin{minipage}[t]{.3\linewidth}
\centering
\includegraphics[angle=000,scale=.3]{mumbai.pdf}
%\caption{Full matrix, $\lambda$=0, mass=1}
%\label{fig:8}
\end{minipage}
\begin{minipage}[t]{.4\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{doe.pdf}
%\caption{Full matrix, $\lambda$=0.2, mass=1}
%\label{fig:9}
\end{minipage}
\end{figure}


%\begin{figure}
%\centering
%\includegraphics[scale=.3]{mumbai.pdf}
% \caption{V(x)}
%\label{figure 1}
%\end{figure}  

\end{frame}

%slide 2 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Outline
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item[$\bullet$] Mutli-resolution (wavelet) basis.
\vfill
\item[$\bullet$] Exact multi-resolution decomposition of quantum fields. 
\vfill
\item[$\bullet$] Decoupling degrees of freedom by scale - 
construction of coarse-scale effective theories.
\vfill
\item[$\bullet$] Results, observations and concluding remarks.
}
\end{itemize}
\end{frame}

%slide 3 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Strategy
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item[$\bullet$] Construct multi-resolution basis.
\vfill
\item[$\bullet$] Exactly decompose fields into discrete 
degrees of freedom with different resolutions.  
\vfill
\item[$\bullet$] Perform resolution and volume truncations on fields
\vfill
\item[$\bullet$] Use truncated fields to construct a truncated Hamiltonian.
\vfill
\item[$\bullet$] Block diagonalize the truncated Hamiltonian by resolution. 
\vfill 
\item[$\bullet$] Evolve fields with truncated Hamiltonian.
\vfill
\item[$\bullet$] Construct correlation functions.
}
\end{itemize}
\end{frame}

%slide 4 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf Basis functions (Daubechies wavelets)}
\end{center}
\vfill  
\begin{center}
{\bf Generated by the fixed point, ${\color{red}s(x)}$, of a\\
renormalization group equation }
\end{center}
\[
{\color{red}s(x)} = \underbrace{D
\underbrace{(\sum_{l=0}^{2K-1} h_l T^l {\color{red}s(x)})}_{\mbox{block average}}
}_{\mbox{rescale}} .
\]
\vfill
\begin{center}
{\bf Unitary translations \qquad Unitary scale transformations}
\end{center}
\vfill
\[
Ts(x) = s(x-1) \qquad Ds(x) = \sqrt{2} s(2x) .
\]
\vfill
\begin{center}
{\bf Scale fixing}
\end{center}
\[
\int dx s(x) =1
\]
\end{frame}

%slide 5 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
{\footnotesize
\begin{center}
{\bf{\color{black}
Weights $h_l$ (constants) determine properties of basis
}
}
\end{center}    
\begin{table}[t]
{\color{black}  
\caption{\bf \color{black} Filter weights for Daubechies K=3 Wavelets}
\begin{tabular}{|l|l|}
\hline				      		      
$h_0$ & $(1+\sqrt{10}+\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}$ \\
$h_1$ & $(5+\sqrt{10}+3\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}$ \\
$h_2$ & $(10-2\sqrt{10}+2\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}$ \\
$h_3$ & $ (10-2\sqrt{10}-2\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2} $ \\
$h_4$ & $(5+\sqrt{10}-3\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}$ \\
$h_5$ & $(1+\sqrt{10}-\sqrt{5+2\sqrt{10}}\,)/16\sqrt{2}$ \\
\hline
\end{tabular}
\label{coef}
}
\end{table}
\vfill 
\begin{itemize}
{\color{black} \bf
\item[] Determined uniquely (up to reflection) by requiring 
orthonormal basis functions that can pointwise
represent polynomials of degree 2. 
}
\end{itemize}
}
\end{frame}

%slide 6 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf Scaling functions, $s^k_n(x)$}
\end{center}


\begin{center}
{\bf Rescale and translate fixed point, $s(x)$}
\end{center}
\[
s^k_n (x) := D^k T^n s(x) = 2^{k/2} s\left (2^k(x-2^{-k}n)\right ) .
\]

\begin{center}
{\bf ${\cal S}_k :=$ resolution $2^{-k}$ subspace, \qquad $\{s^k_n(x)\}$ basis
}
\end{center}

\[
{\cal S}_k := \{ f(x) \vert f(x)= \sum_{n=-\infty}^\infty c_n s^k_n(x),
\quad \sum_{n=-\infty}^\infty \vert c_n \vert^2 < \infty \}.
\]

\[
{\cal S}_k := D^k {\cal S}_0\qquad
\]

\[
{\cal S}_{k} \subset {\cal S}_{k+n} \qquad n\geq 0 
\]

\[
{\cal S}_{k+1} = {\cal S}_{k} \oplus {\cal W}_{k} \qquad {\cal W}_k \not= \{\emptyset \}.  
\]

\end{frame}

%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf Multi-resolution decomposition of Hilbert space 
}
\end{center}

\[
{\cal S}_{k+1} = {\cal S}_{k} \oplus {\cal W}_{k} 
\]
 

\[
L^2 (\mathbb{R}) = {\cal S}_{k}\oplus {\cal W}_{k} \oplus {\cal W}_{k+1}
\oplus {\cal W}_{k+2} \oplus {\cal W}_{k+3} \oplus  \cdots  =
\]

\[
\cdots \oplus {\cal W}_{k-2}\oplus {\cal W}_{k-1} \oplus {\cal W}_{k}
\oplus {\cal W}_{k+1} \oplus {\cal W}_{k+2} \oplus  \cdots
\]

\begin{center}
{\bf Wavelets (basis for ${\cal W}_k$)}
\end{center}

\[
w(x) :=\sum_{l=0}^{2K-1} g_l T^l s(x)
\qquad g_l = (-)^l h_{2K-1-l}
\]

\[
w^k_n (x) := D^k T^n w(x) = 2^{k/2} w\left (2^k(x-2^{-k}n)\right ) .
\]

\end{frame}

%slide 9 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf Multi-resolution basis}
\end{center}
\[
\boxed{
\{\xi_m(x)\}= \{ s^k_n(x)\}_{n=-\infty}^\infty \cup
\{ w^l_n(x)\}_{n=-\infty}^\infty{}_{l=k}^\infty 
}
\]
\begin{itemize} 
{\bf 
\item[$\bullet$] Complete, orthonormal
\vfill
\item[$\bullet$] Limited smoothness (increases with $K$) 
\vfill
\item[$\bullet$] Compact support $(s^k_n(x),w^k_n(x))\subset {\color{red}
2^{-k}[n,(n+2K-1)]}$  
\vfill
\item[$\bullet$] Partition of unity $1 = 2^{k/2} \sum_m s^k_m(x)$ 
\vfill
\item[$\bullet$] $x^n =\sum_m c_m s^k_m(x)$ (pointwise, $n<K$)
\vfill
\item[$\bullet$] $s_n^k(x)$ resolution $2^{-k}$, $w_n^l(x)$ resolution $2^{-{l+1}}$ not in $2^{-l}$
\vfill
\item[$\bullet$] $\int dx x^m w_n^l(x) =0, \qquad  m<K$   
\vfill
\item[$\bullet$] Basis functions are fractal  
}
\end{itemize}

\end{frame}

%slide 10 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf Multi-resolution decomposition of canonical fields}
\end{center}
\[
\xi_n(x) \in \{ s_n^k(x), w^l_n(x) \}
\qquad
\xi_{\mathbf{n}}(\mathbf{x}):= \xi_{n_1}(x)\xi_{n_2}(y) \xi_{n_3}(z)
\]
\vfill
\[
\Phi (\mathbf{x},t) =
\sum_{\mathbf{n}} \Phi^k (\mathbf{n},t) \xi_{\mathbf{n}}(\mathbf{x})   
\qquad
\Phi^k (\mathbf{n},t) = \int d\mathbf{x}\xi_{\mathbf{n}}(\mathbf{x}) 
\Phi (\mathbf{x},t)
\]
\vfill
\[
\Pi (\mathbf{x},t) =
\sum_{\mathbf{n}} \Pi^k (\mathbf{n},t)\xi_{\mathbf{n}}(\mathbf{x})  
\qquad
\Pi^k (\mathbf{n},t) = \int d\mathbf{x}\xi_{\mathbf{n}}(\mathbf{x}) 
\Pi (\mathbf{x},t)
\]
\vfill
\[
[\Phi (\mathbf{n},t), \Pi (\mathbf{m},t)]=
i \delta_{\mathbf{n},\mathbf{m}}
\]
\vfill
\[
[\Phi (\mathbf{n},t), \Phi (\mathbf{m},t)]=
[\Pi (\mathbf{n},t), \Pi (\mathbf{m},t)]=0 , 
\]

\end{frame}

%slide 11 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{itemize} 
{\bf 
\item[$\bullet$] Expansion is exact.
\vfill
\item[$\bullet$] Operator valued distributions replaced by infinite
  sums of well-defined discrete field operators.
\vfill
\item[$\bullet$] Products of discrete fields are well defined.
\vfill
\item[$\bullet$] Discrete fields satisfy canonical commutation relations.
\vfill
\item[$\bullet$] Decomposes the field into local observables by resolution.
\vfill
\item[$\bullet$] Natural resolution (limit $l$) and volume (limit $n$) truncations.
}
\end{itemize}
\end{frame}

%slide 12 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


\begin{frame}
\begin{center}
{\bf
Resolution and volume truncated fields
}
\end{center}
\[
\Phi_T (\mathbf{x},t) =
\sum_{\mathbf{n}\in {\cal I}} \Phi^k (\mathbf{n},t) \xi_{\mathbf{n}}(\mathbf{x})   
\]
\vfill
\[
\Pi_T (\mathbf{x},t) =
\sum_{\mathbf{n}\in {\cal I}} \Pi^k (\mathbf{n},t)\xi_{\mathbf{n}}(\mathbf{x})  
\]
\vfill
\begin{itemize} 
{\bf 
\item[$\bullet$] Restrict index set to finite subset, ${\cal I}$ (finite volume, finite resolution).
  \vfill
\item[$\bullet$] Truncated fields are still differentiable functions of $\mathbf{x}$
}
\end{itemize}
\[
{\cal I} = \{k \leq l \leq l_{max}, -n_{l,max} \leq n_l \leq n_{l,max} \}
\]

\end{frame}


%slide 13 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


\begin{frame}
\begin{center}
{\bf Hamiltonians (exact multi-scale representation)}
\end{center}

\begin{center}
{\bf Mass and kinetic terms}
\end{center}

\[
\mu^2 \int d\mathbf{x} \Phi^2 (\mathbf{x},t) =
\mu^2 \sum \Phi (\mathbf{m},t)^2 
\qquad
\int d\mathbf{x} \Pi^2 (\mathbf{x},t) =
\sum \Pi (\mathbf{m},t)^2 
\]

\begin{center}
{\bf Derivative terms}
\end{center}
\[
\int d\mathbf{x} \pmb{\nabla} \Phi (\mathbf{x},t) \cdot \pmb{\nabla} \Phi (\mathbf{x},t) =
\sum \Phi (\mathbf{m},t)D_{\mathbf{m}\mathbf{n}}\Phi (\mathbf{n},t) 
\]
\[
D_{\mathbf{m}\mathbf{n}}:=
\int d\mathbf{x} \pmb{\nabla} \xi_{\mathbf{m}} (\mathbf{x}) \cdot \pmb{\nabla} 
\xi_{\mathbf{n}} (\mathbf{x}) 
\]

\begin{center}
{\bf Local interactions}
\end{center}
\[
\int d\mathbf{x} \Phi^N (\mathbf{x},t) =
\sum \Gamma_{\mathbf{n}_1 \cdots \mathbf{n}_N}
\Phi (\mathbf{n}_1,t) \cdots \Phi (\mathbf{n}_N,t) 
\]
\[
\Gamma_{\mathbf{n}_1 \cdots \mathbf{n}_N} =
\int d\mathbf{x}
\xi_{\mathbf{n}_1} (\mathbf{x}) \cdots \xi_{\mathbf{n}_N} (\mathbf{x}) 
\]

\begin{center}
{\bf Terms almost local due to support conditions}
\end{center}

\end{frame}

%slide 14 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
\begin{frame}
\begin{center}
{\bf
All coefficients can be computed analytically using the
renormalization group equation, scale fixing condition,
and weight coefficients, $h_l$.  For example the non-zero 
matrix elements
}    
\end{center}
\[
D^0_{mn} := \int {d s^0_m(x) \over dx}{d s^0_n(x) \over dx} dx
\]
\begin{center}
{\bf
have the following exact rational values
}
\end{center}
\[
D^0_{s;40}=D^0_{s;-40}=- 3/560 
\]
%\label{a:15}
%\eeq
%\beq
\[
D^0_{s;30}=D^0_{s;-30}= - 4/35 
\]
%\label{a:16}
%\eeq
%\beq
\[
D^0_{s;20}= D^0_{s;-20} =  92/105 
\]
%\label{a:17}
%\eeq
%\beq
\[
D^0_{s;10}= D^0_{s;-10} = -356/105 
\]
%\label{a:18}
%\eeq
%\beq
\[
D^0_{s;00} = 295/56 .
\]
\end{frame}

\begin{frame}

\begin{center}
{\bf Truncated Hamiltonian \\
replace fields by truncated fields}
\end{center}

\begin{center}
{\bf Dynamics}
\end{center}
\[
\dot{\Phi}_{\mathbf{n}}(t) = i [H_T, {\Phi}_{\mathbf{n}}(t)]  
\qquad
\dot{\Pi}_{\mathbf{n}}(t) = i [H_T, {\Pi}_{\mathbf{n}}(t)]   
\]
%\eeq

\begin{center}
{\bf Initial conditions}
\end{center}
\[
[\Phi (\mathbf{n},0), \Pi (\mathbf{m},0)]=
i \delta_{\mathbf{n},\mathbf{m}}
\]
\[
[\Phi (\mathbf{n},0), \Phi (\mathbf{m},0)]=
[\Pi (\mathbf{n},0), \Pi (\mathbf{m},0)]=0 , 
\]

\begin{center}
{\bf Hilbert space - the truncated problem well-defined in the free field Fock space}
\end{center}


\end{frame}

%slide 15 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}{\bf 
Advantage of wavelet basis
}
\end{center}
  
\begin{center}{\bf 
Scaling properties of coefficients 
(dim $= 1+1$) 
}
\end{center}
\vfill
\[
{\cal S}_{k+m} = {\cal S}_{k} \oplus {\cal W}_{k} \oplus \cdots \oplus 
{\cal W}_{k+m}
\]

\[
\sum D^k_{mn} \Phi_m^k \Phi_n^k =
2^{2k} \sum D^0_{mn} \Phi_m^k \Phi_n^k 
\]
\vfill
\begin{center}
{\bf 
and}
\end{center}
\vfill
\[
\sum \Gamma^k_{n+1 \cdots n_N} \Phi_{n_1}^k \cdots \Phi_{n_N}^k=
2^{3k({n\over 2}-1)} \sum \Gamma^0_{n+1 \cdots n_N} \Phi_{n_1}^k \cdots \Phi_{n_N}^k
\]

\end{frame}

\begin{frame}
\[
\Downarrow
\]
  
\begin{center}{\bf 
Operator renormalization group equation relates
$H$ truncated at different scales (rescale $+$ canonical transformation)    
}
\end{center}

\vfill

\[
H^k(\Phi^k, \Pi^k, m^{2k} ,\gamma_N^k) =
2^k H^0(\Phi^0, \Pi^0, 2^{-2k}m^{0} , 2^{k(N-4)}\gamma_N^0) 
\]

\begin{center}{\bf 
Canonical transformation
}
\end{center}

\[
\Phi^k = \eta \Phi^0 \qquad \Pi^k = \eta^{-1} \Pi^0 \qquad \eta= 2^{-k/2}  
\]

\begin{center}{\bf 
can be realized as a unitary transformation
}
\end{center}


\end{frame}

%slide 16 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf The problem of constructing a coarse-scale effective
theory that includes the effects of eliminated fine-scale degrees of freedom
is studied first using a free field theory in 1+1 dimension
}
\end{center}

\[
H= 
{1 \over 2}\int ({\Pi}({x},0){\Pi}({x},0)
+\pmb{\nabla}\Phi ({x},0) \cdot \pmb{\nabla}\Phi ({x},0)
+ \mu^2 \Phi({x},0)\Phi({x},0))d{x} ,
\]
\begin{center}
{\bf Spatial derivatives give non-trivial scale coupling
}
\end{center}

\[
H_{Ts}^{k+n} = H^k_{Ts} + H_{Tw} +H_{Tsw}
\]

\begin{itemize}
{\bf
\item[$\bullet$] Resolution $1/2^{n+k}$ Hamiltonian $=$ resolution
$1/2^k$ Hamiltonian $+$ fine scale corrections $+$ scale coupling terms

\item[$\bullet$] Goal - block diagonalize truncated Hamiltonian by resolution
}    
\end{itemize}  
%\[
%H^k_{Ts} :=  
%{1 \over 2} (\sum_{{n}} {\Pi}^k(s,{n},0){\Pi}^k(s,n,0)
%+\sum_{{m}{n}} \Phi^k (s,{m} ,0) D^k_{s;{m}{n}} \Phi^k (s,{n},0)
%\]
%\[+ \mu^2 \sum_{{n}} {\Phi}^k(s,{n},0){\Phi}^k(s,{n},0)),
%\]

\end{frame}  

%slide 17 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\[
H^k_T :=  
{1 \over 2} (\sum_{{n}} {\Pi}^k(s,{n},0){\Pi}^k(s,n,0)
+\sum_{{m}{n}} \Phi^k (s,{m} ,0) D^k_{s;{m}{n}} \Phi^k (s,{n},0)
\]
\[+ \mu^2 \sum_{{n}} {\Phi}^k(s,{n},0){\Phi}^k(s,{n},0)),
\]

\[
H_w :=  
{1 \over 2} (\sum_{{n},l} {\Pi}^{l}(w,{n},0)
{\Pi}^l(w,{n},0)
+ \sum_{{m},l,{n},j}\Phi^l (w,{m} ,0) 
D^{lj}_{w;{m}{n}}
\Phi^j (w,{n},0)
\]
%\beq
\[
+ \mu^2 \sum_{l,{n}} {\Phi}^l(w,{n},0)
{\Phi}^l(w,{n},0)),
\]

\[
H_{sw} :=  
{1 \over 2} 
\sum_{{m},l,{n}}\Phi^l (w,{m} ,0) 
{\color{red}D^{lk}_{sw;{m}{n}}}
\Phi^k (s,{n},0) .
\]
\end{frame}

%slide 18 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
  {\bf Use similarity renormalization group evolution to decouple scales
  (eliminate ${\color{red}D^{lk}_{sw;{m}{n}}}$)
}
\end{center}


\begin{center}
{\bf Generate unitarily equivalent Hamiltonians
}
\end{center}

\[
H(\lambda) = U(\lambda)H(0)U^{\dagger}(\lambda) 
\]

\[
{dU(\lambda) \over d\lambda} = {dU(\lambda) \over d\lambda}
U^{\dagger}(\lambda) U(\lambda) =
K(\lambda) U(\lambda)     
\]

\[
K(\lambda) = {dU(\lambda) \over d\lambda}U^{\dagger}(\lambda) = -
K^{\dagger}(\lambda) 
\]


\begin{center}
{\bf Use a generator $K(\lambda)$ of the form
}
\end{center}

\[
K(\lambda) =
[G(\lambda),H(\lambda)]
\]

\[
{dH(\lambda) \over d\lambda}= [K(\lambda),H(\lambda)] = [[G(\lambda),H(\lambda)],H(\lambda)] =
[H(\lambda),[H(\lambda),G(\lambda)]]
\]


\end{frame}

%slide 19 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Initial conditions
}
\end{center}
\[
H(0) = H_T^{k+n}
\qquad 
G(0) = H_{Tsw} 
\]
\[
G(\lambda) = \mbox{\bf part of } H(\lambda) \quad \mbox{\bf that couples scales}
\]
\[
H(\lambda) = H_{block}(\lambda) + G(\lambda)
\]

\begin{center}
{\bf 
Equations separate (free field case)
}
\end{center}

\[
{dH_{block}(\lambda)\over d\lambda} =
[G(\lambda),[G(\lambda),H_{block}(\lambda)]]
\]
\[
{dG(\lambda)\over d\lambda} =
-[H_{block}(\lambda),[H_{block}(\lambda),G(\lambda)]]
\]

\end{frame}

%slide 20 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Expand first equation in basis of eigenstates of $H_C:=G(\lambda)$,
second in eigenstates of $H_B:=H_{block}(\lambda)$
}
\end{center}


\[
{dH_{Bmn} (\lambda) \over d\lambda} =
(e_{cm}(\lambda)-e_{cn}(\lambda))^2 H_{Bmn}(\lambda)
\]
{\bf and}
\[
{dH_{Cmn} (\lambda) \over d\lambda} =
-(e_{bm}(\lambda)-e_{bn}(\lambda))^2 H_{Cmn}(\lambda) .
\]

\begin{center}
{\bf
Integrating
}
\end{center}
\[
H_{Bmn}(\lambda) = e^{\int_0^\lambda (e_{cm}(\lambda')-e_{cn}(\lambda'))^2 d\lambda'}
H_{Bmn}(0)
\]
\[
H_{Cmn}(\lambda) = e^{-\int_0^\lambda (e_{bm}(\lambda')-e_{bn}(\lambda'))^2 d\lambda'}
H_{Cmn}(0).
\]
\end{frame}   

%slide 21 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] Formal solution shows coupling terms exponentially suppressed.
\vfill
\item [${\color{red}\bullet}$] Evolves to decoupled effective Hamiltonians
involving degrees of freedom on different scales.
\vfill
\item [${\color{red}\bullet}$] Can stall when eigenvalues get close or cross.
\vfill
\item [${\color{red}\bullet}$] The choice of generator is specific to the
free field case.
\vfill
\item [${\color{red}\bullet}$] Evolution easily computed - Hamiltonian
has discrete canonical fields with constant coefficients.
\vfill
\item  [${\color{red}\bullet}$] Free field case allows a detailed analysis
  of the evolution of scales.
}
\end{itemize}
\end{frame}

%slide 22 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Test case: 16 scaling functions, 16 wavelets, BC - vanish at boundary    
}
\end{center}    
\vfill
\begin{center}
{\bf
$H(\lambda)$ has 16 quadratic types obtained by taking
products of $\Phi_s (n,0)$, $\Pi_s (m,0)$, $\Phi_w (n,0)$,
and $\Pi_w (m,0)$.
}    
\end{center}
\vfill
\begin{center}
{\bf
The $\lambda$ evolution should drive the coefficients of the
$\Phi_s(n,0) \Phi_w(m,0), \cdots$ to $0$. 
}    
\end{center}

\begin{center}
{\bf
  Plots show the Hilbert-Schmidt norms of the coefficient matrices of each
  type of operator as a function of $\lambda$.
}    
\end{center}

\end{frame}

%slide 23 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{4.0in}{!}{
\includegraphics{{hs1x.pdf}}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

%slide 24 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{4.0in}{!}{
\includegraphics{{hs2x.pdf}}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

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\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{4.0in}{!}{
\includegraphics{{hs3x.pdf}}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

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\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{4.0in}{!}{
\includegraphics{{hs4x.pdf}}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}


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\begin{frame}
\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{mass_1_s_0.pdf}
\caption{Full matrix, $\lambda$=0, mass=1}
\label{fig:8}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{mass_1_s_p2.pdf}
\caption{Full matrix, $\lambda$=0.2, mass=1}
\label{fig:9}
\end{minipage}
\end{figure}
\end{frame}

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\begin{frame}
\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{mass_1_s_2.pdf}
\caption{Full matrix, $\lambda$=2.0, mass=1}
\label{fig:8}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{mass_1_s_20.pdf}
\caption{Full matrix, $\lambda$=20.0, mass=1}
\label{fig:9}
\end{minipage}
\end{figure}
\end{frame}

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\begin{frame}
\begin{center}
{\bf
Observations
}
\end{center}  
\vfill
\begin{itemize}
{\bf     
\item[$\bullet$] Coefficients exhibit expected behavior.
\vfill
\item[$\bullet$] The decay of the coefficients of the scale coupling
terms is initially fast, but slows significantly.
\vfill
\item[$\bullet$] Truncated free fields are equivalent to coupled oscillators;
how does the SRG evolution distribute the normal modes among the blocks?   
}
\end{itemize}  
\end{frame}  

%slide 36 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
To get a more detailed understanding of the effects of the
$\lambda$ flow note that the truncated Hamiltonian has the form
}
\end{center}  
\vfill
\[
H_T = {1 \over 2} 
[
(\Pi^s ,\Pi^w) 
\left (
\begin{array}{cc}
I_s & 0\\
0 & I_w 
\end{array}
\right ) 
\left (
\begin{array}{c}
\Pi^s \\
\Pi^w  
\end{array}
\right ) +
\]
\[
(\Phi^s ,\Phi^w) 
\left (
\begin{array}{cc}
\mu^2I+D_{s} & {\color{red}D_{sw}}\\
{\color{red}D_{ws}}  & \mu^2I + D_{w} 
\end{array}
\right ) 
\left (
\begin{array}{c}
\Phi^s \\
\Phi^w  
\end{array}
\right ) 
]
\]
\end{frame}

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\begin{frame}
\[
M:= 
\left (
\begin{array}{cc}
\mu^2I+D_{s} & {\color{red}D_{sw}}\\
{\color{red}D_{ws}}  & \mu^2I + D_{w} 
\end{array}
\right ) 
\]
\[
O^t M O = 
%\left (
%\begin{array}{cc}
%Im^2 + D^{ss} & D^{sw}0\\
%D^{ws} & Im^2 + D^{ww}  
%\end{array}
%\right ) 
%O
\left (
\begin{array}{cc}
m^s & 0\\
0 & m^w  
\end{array}
\right )
%\label{an.1}
\]
\begin{center}
{\bf   
where $m^s$ and $m^w$ are diagonal matrices consisting 
of eigenvalues of the
matrix $M$.
}
\end{center}
\begin{center}
{\bf   
Transformed discrete fields are related by the canonical
transformation
}
\end{center}
\[
\left (
\begin{array}{c}
\Phi^{\prime s} \\
\Phi^{\prime w}  
\end{array}
\right )  := 
O^t
\left (
\begin{array}{c}
\Phi^s \\
\Phi^w  
\end{array}
\right ) 
\qquad \mbox{and} \qquad
\left (
\begin{array}{c}
\Pi^{\prime s} \\
\Pi^{\prime w}  
\end{array}
\right )  := 
O^t \left (
\begin{array}{c}
\Pi^s \\
\Pi^w  
\end{array}
\right ) .
\]
\end{frame}
\begin{frame}
\[
H'= UHU^{\dagger} = {1 \over 2} 
[
(\Pi^{\prime s} ,\Pi^{\prime w})
\left (
\begin{array}{cc}
I & 0\\
0 & I 
\end{array}
\right ) 
\left (
\begin{array}{c}
\Pi^{\prime s} \\
\Pi^{\prime w}  
\end{array}
\right ) +
\]
\[
(\Phi^{\prime s} ,\Phi^{\prime w}) 
\left (
\begin{array}{cc}
m^{s} & 0\\
0  & m^{w} 
\end{array}
\right ) 
\left (
\begin{array}{c}
\Phi^{\prime s} \\
\Phi^{\prime w}  
\end{array}
\right ) 
]
\]

\begin{itemize}
{\bf
\item[$\bullet$] Transformed Hamiltonian is the Hamiltonian for uncoupled
harmonic oscillators
with frequencies $\sqrt{m_i}$

\item[$\bullet$] $\Pi \Pi$ coefficients in flow-evolved Hamiltonian are
approximately $(1/2)(I_s \oplus I_w)$ so $\sqrt{}$ of eigenvalues of
$M$ correspond to normal mode frequencies. 

\item[$\bullet$] A general $O$ will separate the eigenvalues of $M$ into two
distinct groups - there is no general relation between normal mode
frequencies and scale.
}
\end{itemize}
\end{frame}


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\begin{frame}
{\footnotesize
\color{black}    
\begin{table}[t]
\caption{\color{black}Normal mode frequencies}
\begin{tabular}{|l|l|l|l|}  
\hline		
$\lambda=20, \mu=1$& truncated & exact 1:16 & exact 17:32 \\
\hline
1.037e+00 &  1.037e+00 &  1.041e+00 &  1.665e+01 \\   
1.145e+00 &  1.146e+00 &  1.153e+00 &  1.925e+01 \\
1.326e+00 &  1.333e+00 &  1.340e+00 &  2.208e+01 \\
1.583e+00 &  1.609e+00 &  1.604e+00 &  2.512e+01 \\
1.919e+00 &  1.995e+00 &  1.947e+00 &  2.834e+01 \\
2.341e+00 &  2.525e+00 &  2.373e+00 &  3.167e+01 \\
2.861e+00 &  3.236e+00 &  2.890e+00 &  3.507e+01 \\
3.493e+00 &  4.161e+00 &  3.508e+00 &  3.846e+01 \\
4.263e+00 &  5.317e+00 &  4.243e+00 &  4.178e+01 \\
5.201e+00 &  6.689e+00 &  5.112e+00 &  4.495e+01 \\
6.346e+00 &  8.232e+00 &  6.134e+00 &  4.789e+01 \\
7.722e+00 &  9.859e+00 &  7.332e+00 &  5.053e+01 \\
9.309e+00 &  1.145e+01 &  8.729e+00 &  5.279e+01 \\
1.102e+01 &  1.289e+01 &  1.034e+01 &  5.462e+01 \\
1.274e+01 &  1.403e+01 &  1.219e+01 &  5.597e+01 \\
1.435e+01 &  1.476e+01 &  1.429e+01 &  5.679e+01 \\
\hline
\end{tabular}
\label{omega}
\end{table}
}
\end{frame}


%slide 38 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf   
The table shows that the SRG flow equation puts the lowest normal modes in coarse scale
Hamiltonian and the highest normal mode frequencies in the fine scale Hamiltonian
}
\end{center}
\end{frame}




%slide 40 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Conclusions
}  
\end{center}
  
\begin{itemize}
{\bf 

\item[1.] Flow equation methods with a suitable
generator can be used to construct an effective field theory
with coarse scale degrees of freedom.

\item[2.] The generator used separates both energy and distance
scales.

\item[3.] Increasing the truncated volume generated new low frequency
modes, while increasing the resolution increased the separation
between modes.  The mass sets a lower bound on the normal mode frequencies.

\item[4.] The flow equation also exhibited convergence for mass 0.

}
\end{itemize}  
\end{frame}

\begin{frame}
\begin{center}
{\bf
Summary
}  
\end{center}
  
\begin{itemize}
{\bf 

\item[5.] For this problem, the flow equation was successfully applied
directly to the Hamiltonian, without projecting on a subspace.

\item[6.] We found the that flow equation could be integrated using the
Euler method, but perturbation theory failed to converge.

\item[7.] The final evolved Hamiltonian was approximately local.   

\item[8.] The spectral properties suggest the in order to approach
the continuous spectrum of the exact theory, the volume and resolution
truncations need to be removed together.  

\item[9.] In our test the coefficients of the coupling terms initially
fell off quickly, but the rate of fall off slowed down significantly
as the flow parameter increased.  The method reduced the coupling
coefficients by a factors of about 100 for a modest value of the
flow parameter.

\item[10.] The convergence of the flow equation will slow down as
the separation between normal modes decreases.
}
\end{itemize}
\end{frame}

%slide 42 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Outlook
}  
\end{center}

\begin{itemize}
{\bf 
\item[1] It has been shown that the truncated correlation functions
converge to the exact free field Wightman functions as the resolution
is increased (Singh-Brennan Arxiv 1606:050686).

\item[2] Basis functions have only a finite number of derivatives - but
the Fourier transform of the test functions are entire.

\item[3] Solution of Heisenberg equations of motion + ground state
can be used to construct space-real-time correlation functions
as functions of continuous variables

\item[4] In interacting theories integrating the flow equation generates
infinite numbers of operators.

\item[5] Fields differentiable functions of $x$ - mathematics discrete.
}
\end{itemize}
\end{frame}

%slide 27 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{4.0in}{!}{
\includegraphics{{hs5x.pdf}}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

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\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{4.0in}{!}{
\includegraphics{{hs6x.pdf}}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

%slide 29 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{4.0in}{!}{
\includegraphics{{hs7x.pdf}}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

%slide 30 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{4.0in}{!}{
\includegraphics{{hs8x.pdf}}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

%slide 33 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{mass_0_s_p2.pdf}
\caption{Full matrix, $\lambda$=0.2, mass=0}
\label{fig:8}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{mass_0_s_2.pdf}
\caption{Full matrix, $\lambda$=2.0, mass=0}
\label{fig:9}
\end{minipage}
\end{figure}
\end{frame}

%slide 34 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{figure}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{mass_4_s_p2.pdf}
\caption{Full matrix, $\lambda$=0.2, mass=4}
\label{fig:8}
\end{minipage}
\begin{minipage}[t]{.45\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{mass_4_s_2.pdf}
\caption{Full matrix, $\lambda$=2.0, mass=4}
\label{fig:9}
\end{minipage}
\end{figure}
\end{frame}

\end{document}
%slide 43 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
