poster23: emph eqs

poster/poster.tex

@@ -27,15 +27,22 @@
 \addbibresource{bib/main.bib}
 \addbibresource{bib/comp.bib}
 \addbibresource{bib/misc.bib}
+\AtEveryBibitem{
+ \clearfield{issn}
+ \clearfield{month}
+ \clearfield{doi}
+}
+
 
 \usepackage{multicol} % This is so we can have multiple columns of text side-by-side
 \columnsep=100pt % This is the amount of white space between the columns in the poster
 \columnseprule=3pt % This is the thickness of the black line between the columns in the poster
 
 \usepackage[svgnames]{xcolor} % Specify colors by their 'svgnames', for a full list of all colors available see here: https://www.latextemplates.com/svgnames-colors
 
-%\usepackage{newtxtext} % Use the times font for text, newer flavor
+% \usepackage{newtxtext} % Use the times font for text, newer flavor
 \usepackage{helvet}
+% \usepackage{mathpazo}
 \renewcommand{\familydefault}{\sfdefault} % sans serif font
 
 \usepackage{graphicx} % Required for including images
@@ -46,12 +53,14 @@
 \usepackage{wrapfig} % Allows wrapping text around tables and figures
 
 \usepackage{physics}
-\usepackage[T1]{fontenc}
+%\usepackage[T1]{fontenc}
 \usepackage{setspace}
 
+\setlength{\fboxsep}{5\fboxsep}
+
 \usepackage{bbm}
 
-\usepackage[top=2.8cm,right=2.8cm,bottom=2.8cm,left=2.6cm]{geometry}
+\usepackage[top=2.8cm,right=2.8cm,bottom=2.8cm,left=2.8cm]{geometry}
 
 \usepackage{lipsum}
 
@@ -96,6 +105,8 @@
 \newcommand{\dbradket}[2]{\left\langle\smallback\left\langle #1 \middle| #2 \right\rangle\smallback\right\rangle}
 \newcommand{\dketdbra}[2]{\left| #1 \left\rangle\smallback\left\rangle \smallback \right\langle\smallback\right\langle #2 \right|}
 
+\newcommand{\boxedcolor}{DarkSlateGrey}
+
 
 %\renewcommand{\abstractname}{\large Abstract} % abstract hacks
 
@@ -110,18 +121,15 @@
 % The second is 25% wide and houses a logo for your university/organization or a photo of you
 % The widths of these boxes can be easily edited to accommodate your content as you see fit
 \begin{minipage}[c]{0.70\linewidth}
-\VeryHuge \color{NavyBlue} \textbf{It's $\ket{1}$ o'clock} \color{Black}\\[0.5cm] % Title
+\VeryHuge\color{NavyBlue}\textbf{It's $\ket{1}$ o'clock} \color{Black}\\[0.5cm] % Title
 \huge\textit{Relational Time and Applications}\\[1cm] % Subtitle
 \huge University College Cork, School of Physics\\[0.66cm] % University/organization
-\large%
-Guido De Rosa (MSc.)
-|
-Andreas Ruschhaupt (Supervisor)%
+\Large{Guido De Rosa (MSc.) -- Andreas Ruschhaupt (Supervisor)}
 
 \end{minipage}%
 %
 \begin{minipage}[t]{0.30\linewidth}
-\includegraphics[width=19cm]{ucc_logo.pdf}\\ %% vector, from original SVG
+\includegraphics[width=18cm]{ucc_logo.pdf}\\ %% vector, from original SVG
 \end{minipage}
 
 \vspace{2cm} % A bit of extra whitespace between the header and poster content
@@ -149,31 +157,29 @@ \section*{\large Abstract}
 time is an observable with an associated self-adjoint operator.
 We also apply the Page-Wootters formalism
 to absorptive detector models with
-on-unitary evolution \cite{RuschhauptAbsorption}.
+non-unitary evolution \cite{RuschhauptAbsorption}.
 
 %\end{abstract}
 
-\setlength{\parindent}{1.5em} % Default is 15pt.
+\setlength{\parindent}{2em} % Default is 15pt.
 
 %----------------------------------------------------------------------------------------
 % INTRODUCTION
 %----------------------------------------------------------------------------------------
 
 \color{SaddleBrown} % SaddleBrown color for the introduction
 
-\large
+%\large
 
 \section*{Historical background: The Pauli Objection}
 The idea of time as an observable in quantum mechanics,
 with an associated self-adjoint operator in a Hilbert space of quantum states,
-was first excluded by W. Pauli,
-and not given particular attention by the Author himself,
-if only a footnote was dedicated to dismiss the whole subject. \cite{PauliFootnote}
-
+was first excluded by W. Pauli \cite{PauliFootnote}.
+%
 Quoting \cite{Maccone:Pauli},
 ``Schr\"odinger's equation says that the Hamiltonian is the generator of time translations. This seems
 to imply that any reasonable definition of time operator must be conjugate to the Hamiltonian''.
-
+%
 Pauli's argument
 is based on showing that
 the existence of
@@ -185,38 +191,8 @@ \section*{Historical background: The Pauli Objection}
 $
 implies that the spectrum of \emph{any} Hamiltonian must be the whole real line
 (contradicting the property of being bounded from below).
-A detailed proof is expanded in the paper by E. Galapon \cite{Galapon2002}.
-
-% Since T is self-adjoint, then for all
-% $\beta\in\mathbb{R}$, $U_{\beta} = \exp(- i \beta T / \hbar)$
-% is unitary. A formal
-% expansion of the exponential yields the commutator
-
-% \begin{multline}
-% [U_{\beta}, H] =
-% \left[
-% \sum_{k=0}^{\infty} \frac{1}{k!} \left(- \frac{i\beta T}{\hbar} \right)^k, H
-% \right] =
-% \sum_{k=0}^{\infty} \frac{1}{k!} \left(- \frac{i\beta}{\hbar} \right)^k [T^k, H]
-% =
-% \sum_{k=0}^{\infty} \frac{1}{k!} \left(- \frac{i\beta}{\hbar} \right)^k kT^{k-1}[T, H] \\ =
-% \beta\sum_{k=1}^{\infty} \frac{1}{(k-1)!} \left(- \frac{i\beta}{\hbar} \right)^{k-1} T^{k-1} =
-% \beta\sum_{\kappa=0}^{\infty} \frac{1}{\kappa!} \left(- \frac{i\beta T}{\hbar} \right)^{\kappa} =
-% \beta U_{\beta}
-% \end{multline}
-% where the term for $k=0$ in the first sum vanishes, hence we can start the sum from
-% $k=1$ then set $\kappa=k-1$.
-
-% Now, given an eigenvector $\varphi_{E}$ so that $H\varphi_{E}=E\varphi_{E}$, there has:
-
-% $$
-% HU_{\beta}\varphi_{E} = (U_{\beta}H - [U_{\beta}, H])\varphi_{E} =
-% EU_{\beta}\varphi_{E} - \beta U_{\beta}\varphi_{E} = (E-\beta)U_{\beta}\varphi_{E}
-% $$
-% showing that $U_{\beta}\varphi_{E}$ is another eigenvector of $H$ with eigenvalue
-% $E-\beta$. But $\beta$ is an arbitrary real number and $H$ a \emph{generic} Hamiltonian,
-% hence the spectrum of a generic Hamiltonian $H$ should
-% be the whole real line, which contradicts the semi-bounded nature of its spectrum.
+A detailed proof is expanded in~\cite{Galapon2002}.
+
 
 %----------------------------------------------------------------------------------------
 % THE MODEL
@@ -248,11 +224,13 @@ \section*{Evolution without evolution: the Page and Wootters model}
 As explained in \cite{Lloyd:Time, Maccone:Pauli}, the overall Hamiltonian,
 encompassing both position and time as observables, is given by
 \begin{equation}\label{eq:pwHamiltonian}
- \hat{\mathbb{J}} = \hbar\hat{\Omega}\ox\idop_S + \idop_T\ox\hat{H}_S \,\text{,}
+ \color{\boxedcolor}\boxed{\hat{\mathbb{J}} = \hbar\hat{\Omega}\ox\idop_S + \idop_T\ox\hat{H}_S}
+ \,\text{,}
 \end{equation}
 while the \term{Wheeler-DeWitt equation} holds:
 \begin{equation}\label{eq:Wheeler-DeWitt}
- \hat{\mathbb{J}}\dket{\Psi} = 0 \,\text{,}
+ \color{\boxedcolor}\boxed{\hat{\mathbb{J}}\dket{\Psi} = 0}
+ \,\text{,}
 \end{equation}
 describing a \emph{static} universe, where evolution is only
 in terms of relations between parts of a multipartite system
@@ -279,30 +257,30 @@ \section*{Evolution without evolution: the Page and Wootters model}
 
 The Pauli objection no longer holds as time as a quantum observable operates in a separate Hilbert space.
 
-\subsection*{Non-zero eigenvalues}
+% \subsection*{Non-zero eigenvalues}
 
-Up to an irrelevant global
-phase, the physical vectors $\dket{\Psi}$ can be identified also by
-imposing the constraint
-\[
- \hat{\mathbb{J}}\dket{\Psi} = \epsilon \dket{\Psi}
-\]
-with real $\epsilon$. The corresponding evolution in $\hilb{H}_S$ will then need
-a rigid energy shift of its spectrum by replacing $\ket{\psi(t)}_S$
-with $e^{-i \epsilon t / \hbar} \ket{\psi(t)}_S$
+% Up to an irrelevant global
+% phase, the physical vectors $\dket{\Psi}$ can be identified also by
+% imposing the constraint
+% \[
+%  \hat{\mathbb{J}}\dket{\Psi} = \epsilon \dket{\Psi}
+% \]
+% with real $\epsilon$. The corresponding evolution in $\hilb{H}_S$ will then need
+% a rigid energy shift of its spectrum by replacing $\ket{\psi(t)}_S$
+% with $e^{-i \epsilon t / \hbar} \ket{\psi(t)}_S$
 
-\section*{Example: One qubit ``universe'' and a $N=32$-level clock}
+\section*{Example: One qubit and a $N=32$-level clock ``universe''}
 
 The clock is built as having a time operator which is diagonal in the
 chosen computational basis:
 \[
- \hat{T} \repr \frac{2\pi}{N}
- \begin{pmatrix}
- 0 & & & \\
- &1 & & \\
- & &\ddots & \\
- & & &N-1
- \end{pmatrix} \,\text{.}
+ \hat{T} \repr \frac{2\pi}{N} \,\;\! \mathrm{diag}\qty(0, \dots, N-1)
+ % \begin{pmatrix}
+ %  0 & & & \\
+ %  &1 & & \\
+ %  & &\ddots & \\
+ %  & & &N-1
+ % \end{pmatrix} \,\text{.}
 \]
 With this choice, the clock spans the characteristic period of $\Delta T = 2\pi$.
 
@@ -320,12 +298,15 @@ \section*{Example: One qubit ``universe'' and a $N=32$-level clock}
 The frequency operator in $\hilb{H}_T$, as the canonically conjugate operator
 of $\hat{T}$ in a finite-dimensional Hilbert space, is derived via
 \emph{discrete Fourier transformation} \cite{FiniteHilb} (with $F_N$ unitary):
-\[
- \hat{\Omega} = \frac{N}{2\pi} F^{}_{N} \hat{T} F^{\dagger}_{N} \, \text{.}
-\]
+\begin{equation}
+ \color{\boxedcolor}\boxed{
+ \hat{\Omega} = \frac{N}{2\pi} F^{}_{N} \hat{T} F^{\dagger}_{N}
+ }
+ \, \text{.}
+\end{equation}
 
-We therefore need to find the eigenvectors of $\hat{\mathbb{J}}$ as in \eqref{eq:pwHamiltonian}
-(a $2N \times 2N$ matrix obtained as a Kronecker product). Such eigensolutions
+We therefore need to find the eigenvectors of $\hat{\mathbb{J}}$ as in \eqref{eq:pwHamiltonian}.
+Such eigensolutions
 will encode the whole (periodic) evolution of the qubit (in $\hilb{H}_S$), but do
 not ``evolve'' themselves (as there's no external ``time'' outside $\hilb{H}_T \ox \hilb{H}_S$).
 
@@ -336,22 +317,22 @@ \section*{Example: One qubit ``universe'' and a $N=32$-level clock}
 i.e. solutions that do not correspond to eigenstates of
 the ``ordinary'' Hamiltonian $\hat{H}_S$ of the qubit.
 
-We pick $\dket{\Phi_{40}}$ and $\dket{\Phi_{41}}$ corresponding to eigenvalues
-$\epsilon_{40} = 12$ and $\epsilon_{41} = 11$.
+We pick $\dket{\Phi_{41}}$ corresponding to eigenvalue
+$\epsilon_{41} = 11$.
 
-The first two components of each eigenvector in the computational basis
-are interpreted as the components of the qubit in $\hilb{H}_S$ at $t=0$. The components
+The first two components
+are interpreted as the components of the qubit in $\hilb{H}_S$ at $t=0$. 
+In general, the components
 $2k$\nobreakdash-th and $2k+1$\nobreakdash-th
-as the components of the qubit at $k$-th discrete temporal step ($t = k \frac{2\pi}{N}$).
-Comparison with ordinary quantum mechanics solution (Schr\"odinger equation) as shown in picture.
+are compared to the components of the qubit at $k$-th discrete temporal step ($t = k \frac{2\pi}{N}$).
 
 \begin{center}\vspace{1cm}
  \includegraphics[width=0.3\linewidth]{PWfit32.png}
  \includegraphics[width=0.4\linewidth]{PWfit32top.png}
  \captionof{figure}{
  \color{Green}
- Comparison of discrete P-W model prediction (points) with the ordinary QM Schr\"odinger solution (continuous line)
- for eigensolutions of $\hat{\mathbb{J}}$ labeled as $\epsilon_{40}$ and $\epsilon_{41}$. Complex values.
+ Comparison of discrete P-W model prediction (points) with the ordinary QM Schr\"odinger solution (continuous line).
+ Complex values.
  }
 \end{center}\vspace{1cm}