\section{(11 February 2009)}
\subsection*{The construction of the algebra \( 
\mathcal{A}_{\mathcal{P}} \) (cont.)}
We have this algebra
\[ 
\begin{split}
    \mathcal{A}_{\mathcal{P}} & = C_{\mathcal{P}}(\mathbb{R}^{n}) \rtimes_{\alpha} 
    \mathbb{R}^{n} \cong C(\Omega_{\mathcal{P}}) \rtimes_{\alpha} 
    \mathbb{R}^{n}  \\
    \pi_{\mathcal{P}} & : \mathcal{A}_{\mathcal{P}} \rightarrow 
    B(L^{2}(\mathbb{R}^{n}))  \\
    f & : \mathbb{R}^{n} \rightarrow C_{\mathcal{P}}(\mathbb{R}^{n})  
    \\
    (\pi_{\mathcal{P}}(f)\psi)(x) & = \int f(x - y)(x)\psi(y)\mathrm{d}y
\end{split}
\]

\begin{theorem*}
    If \( H = \frac{\hat{p}^{2}}{2m} + V(\hat{q}) \), \( V \in 
    C_{\mathcal{P}}(\mathbb{R}^{n}) \), then \( \forall F \in 
    C_{0}(\mathbb{R}) \), \( F(H) \in 
    \pi_{\mathcal{P}}(\mathcal{A}_{\mathcal{P}}) \) \(( 
    \Leftrightarrow \exists A \in \mathcal{A}_{\mathcal{P}} \) such 
    that \( A \) is represented by \( F(H) )\).
\end{theorem*}

\begin{proof}
    \( \hat{p} = \frac{\hbar}{i}\Delta \), \( \hat{q} \) left 
    multiplication by \( x \mapsto x \).  Taking \( \hbar = 1,\,m = 1 
    \), \( H = -\Delta + V \), then \( H \) is bounded below (but not 
    above), and by a shift of \( V \), \( H \geq 0 \).
    
    Suppose \( F \) has a Laplace transform; \( F \) is regular 
    enough that
    \[ 
    \tilde{F}(t) = 2\pi 
    \int_{-\infty}^{\infty}\mathrm{e}^{tE}F(E)\mathrm{d}E
    \]
    \[ 
    \left(F(E) = 
    \int_{0}^{\infty}\mathrm{e}^{-tE}\tilde{F}(t)\mathrm{d}t \text{ 
    is the Laplace transform of } \tilde{F} : \mathbb{R} \rightarrow 
    \mathbb{R}\right)
    \]
    
    Then \( F(H) \) is defined by ``Laplace'' functional calculus
    \[ 
    F(H) = \int_{0}^{\infty}\mathrm{e}^{-tH}\tilde{F}(t)\mathrm{d}t 
    \text{; bounded because } H \text{ is bounded below}.
    \]
    
    To show that \( F \) belongs to a norm-closed subalgebra it 
    suffices to show that \( \mathrm{e}^{-tH} \) belongs; that is, 
    for these functions \( F \), it suffices to show that \( 
    \mathrm{e}^{-tH} \in \pi_{\mathcal{P}}(\mathcal{A}_{\mathcal{P}}) \).
    
    Consider first the case where 
    \[ 
    \mathrm{e}^{-t(-\Delta)} \text{ (the heat kernel)}\footnote{With 
    \( \hbar = 1 \), \( 
    \hat{p}^{2} = -\Delta \)}
    \]
    
    \begin{lemma*}
	\[ 
	\left(\mathrm{e}^{t \Delta \psi}\right)(x) = \int
	\underset{= \frac{c_{1}}{t^{n/2}}\mathrm{e}^{\frac{-(x - 
	y)^{2}}{4t}}}{\underbrace{\left(\mathrm{e}^{t \Delta}\right)_{xy}}} \psi(y)
	\mathrm{d}y
	\]
    \end{lemma*}
    
    \[ 
    \underset{f(x - y)(x)}{\pi_{\mathcal{P}}(f) = \mathrm{e}^{t
    \Delta} } \quad f : \underset{f(\xi)(x) =
    \frac{c_{1}}{t^{n/2}}\mathrm{e}^{\frac{-\xi^{2}}{4t}}}{\mathbb{R}^{n}
    \rightarrow C_{\mathcal{P}}(\mathbb{R}^{n})}
    \]
    
    If we have a constant potential (zero potential)
    \[ 
    \mathrm{e}^{-t H} \in \underset {\underset{ = \text{ functions of 
    }\hat{p}}{\cong C_{0}(\hat{\mathbb{R}}^{n})}}{\mathbb{C} \rtimes \mathbb{R}^{n}} \subset 
    \mathcal{A}_{\{x\}}
    \]    
    So [Dyson-Phillips expansion]
    \[ 
    \mathrm{e}^{-t H} = \mathrm{e}^{-t \Delta - t V} = \mathrm{e}^{t 
    \Delta} + \sum_{\nu = 0}^{\infty} \underset{s_{i} \geq 0 \,\, 
    \sum_{0}^{\nu} s_{i} = t}{\int \mathrm{d}s_{0} \cdots 
    \int\mathrm{d}s_{\nu}\mathrm{e}^{s_{1}\Delta} V 
    \mathrm{e}^{s_{2}\Delta} V \cdots V \mathrm{e}^{s_{\nu} 
    \Delta}}
    \]
    If \( V \) is bounded, the series converges in norm.  \( -\Delta 
    \geq 0 \Rightarrow \|\mathrm{e}^{s \Delta}\| \leq 1 \)
    \[ 
    \|\int \mathrm{d}s_{0} \cdots \int
    \mathrm{d}s_{\nu}\mathrm{e}^{s_{1} \Delta} \cdots \| \leq
    \underset{\text{volume of a } \nu \text{-simplex}}{\underset{s_{i} \geq 0 \,\, \sum_{0}^{\nu} s_{i} = t}{\int
    \mathrm{d}s_{1} \cdots \int \mathrm{d}s_{\nu}}}
    \|V\|^{\nu} \leq
    \frac{t^{\nu}}{\nu !} \|V\|^{\nu}
    \]
    So the series converges absolutely like \( \sum 
    \frac{t^{\nu}}{\nu !} \|V\|^{\nu} \leq \mathrm{e}^{t\|V\|} \)
    
    Now we are almost done.  Need to show \( \mathrm{e}^{s_{i}\Delta}
    \) is in the algebra, so that \( \mathrm{e}^{s_{1}\Delta}V
    \mathrm{e}^{s_{2}\Delta}V \cdots \mathrm{e}^{s_{\nu}\Delta} \) is
    in the algebra.
    
    It suffices to show that \( V\mathrm{e}^{t \Delta} \in
    \pi_{\mathcal{P}}(\mathcal{A}_{\mathcal{P}}) \):
    \[ 
    (V\mathrm{e}^{t \Delta}) \psi(x) = \int \underset{ \text{integral
    kernel } f(x - y)(x) }{\underbrace{V(x) \frac{c_{1}}{t^{n/2}}
    \mathrm{e}^{\frac{-|x - y|^{2}}{4t}}}} \psi (y) \mathrm{d}y
    \]
    \[ 
    \begin{split}
	f & : \mathbb{R}^{n} \rightarrow C_{\mathcal{P}}(\mathbb{R}^{n})  \\
	f(\xi)(x) & = V(x) \frac{c_{1}}{t^{n/2}} 
	\mathrm{e}^{\frac{-\xi^{2}}{4t}}  \\
	\text{then, }\pi_{\mathcal{P}} (f) & = V \mathrm{e}^{t \Delta}
    \end{split}
    \]
    
    
    
    \hfill \hfill \hfill \hfill \hfill \hfill QED
\end{proof}
    
    \textbf{Q}: This is a good representation (Schr\"{o}dinger); what 
    about the others? (Other representations from other characters.)
    
    \subsubsection*{Covariant families of operators}
    Comes from the physics of condensed matter.  Random potentials are
    described by potentials which are random variables over some
    probability space \( (\Omega, \mathbb{P}) \).  So
    \[ 
    \Omega \ni \omega \mapsto V_{\omega} : \mathbb{R}^{n} \rightarrow 
    \mathbb{R}
    \]
    
    Ideally, solve everything for \( H_{\omega} = H_{0} + V_{\omega} 
    \) (\( H_{0} = - \Delta \) for example).
    \[ 
    \leadsto \leadsto \int_{\Omega} 
    \mathrm{Tr}(\rho_{\omega}A_{\omega}) \mathrm{d}\mathbb{P}
    \]
    The density matrix might depend on \( \omega \).
    
    Homogeneous media = microscopically translation invariant.
    
    Idea: \( (\Omega, \mathbb{P}) \) should carry an action of 
    \( \mathbb{R}^{n} \): \( \omega \overset{\xi}{\mapsto} \omega - \xi \) such that
    \[      
    \forall \omega \in \Omega\,\, x \in \mathbb{R}^{n} : V_{\omega -
    x} = U_{x} V_{\omega} U_{-x}\,,\,\,((U_{x}\psi)(y) = \psi(y - x))
    \]
    
    Covariant system.
    Bellisard, Johnson and Moser: for aperiodic media take \( \Omega  
    \) = hull.
    \( \mathbb{P} \) should be translation invariant.
    Pure phases = uniquely ergodic \( \mathbb{P} \) with respect to 
    itself.
    
%     \subsubsection*{Appendix. The Dyson expansion}
%     Notes from \S 6.1 of Gerald Folland's book.
%     
%     The Hamiltonian is the sum of a Hamiltonian which is understood, 
%     and a perturbation: \( H = H_{0} + H_{I} \).  The time evolution 
%     for these is
%     \begin{equation}
% 	U_{0}(t) = \mathrm{e}^{-itH_{0}} \text{ and }
% 	    U(t) = \mathrm{e}^{-itH}
% 	\label{exp}
%     \end{equation}
%     Considerations of domains and questions of boundedness are 
%     temporarily set aside.
%     
%     The \emph{observables} evolve in the Heisenberg picture 
%     (\emph{interaction picture}) according 
%     to the unperturbed Hamiltonian:
%     \begin{equation}
% 	A(t) = U_{0}(-t)AU_{0}(t)
% 	\label{inter}
%     \end{equation}
%     and (p. 124)
%     \begin{quote}
% 	the states evolve in such a way as to correct \( U_{0}(t) 
% 	    \) to \( U(t) \):
% 	    \begin{equation}
% 		\psi(t) = V(t)\psi \,, \text{ where } \, V(t) = 
% 		U_{0}(-t)U(t).
% 		\label{schr}
% 	    \end{equation}
%     \end{quote}
%     
%     What Folland calls the matrix element of the observable \( A \) 
%     between states \( \psi_{1} \) and \( \psi_{2} \) at time \( t \) 
%     is 
%     \[ 
%     \begin{split}
% 	\underset{\text{Schr\"{o}dinger picture}}{\langle \psi_{2}(t)
% 	| A(t) | \psi_{1}(t) \rangle} & = \langle V(t)\psi_{2} |
% 	\underset{\text{using \eqref{inter}}}{U_{0}(-t) A U_{0}(t)} | 
% 	\underset{\text{using \eqref{schr}}}{V(t)\psi_{1}} \rangle \\
% 	& = \langle \psi_{2} | V(t)^{\dag} U_{0}(-t) A U_{0}(t) V(t) |
% 	\psi_{1} \rangle \\
% 	& = \underset{\text{Heisenberg picture}}{\langle \psi_{2} |
% 	U(-t) A U(t) | \psi_{1} \rangle}        
%     \end{split}
%     \]
%     
%     If we can find \( V(t) \), then we can find \( U(t) = 
%     U_{0}(t)V(t) \) using \eqref{schr}.  The differential equation 
%     for \( V(t) \) is 
%     \begin{equation}
%       \begin{split}
% 	\frac{\mathrm{d}}{\mathrm{d}t} V(t) 
% 	& \overset{\text{\eqref{schr}}}{=} 
% 	\frac{\mathrm{d}U_{0}(-t)}{\mathrm{d}t} U(t) + U_{0}(-t) 
% 	\frac{\mathrm{d}U(t)}{\mathrm{d}t}  \\
% 	& \overset{\text{\eqref{exp}}}{=} (-1)U_{0}(-t) (-iH_{0}) 
% 	U(t) + U_{0}(-t)U(t)(-iH)  \\
% 	& = iU_{0}(-t)H_{0}U(t) - iU_{0}(-t)U(t)H  \\
% 	& = \underset{U(t) \text{ commutes with } H}{iU_{0}(-t)H_{0}U(t) - iU_{0}(-t)HU(t)}   \\
% 	& = iU_{0}(-t)(H_{0} - H)U(t) = -iU_{0}(-t)H_{I}U(t)
% 	\label{de}
%       \end{split}
%     \end{equation}