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\hypersetup{pdftitle={Loop Quantum Cosmology},
  pdfsubject={ApJC - Loop Quantum Gravity and its Cosmological Effects},
  pdfauthor={Daniel Doro Ferrante <danieldf@brown.edu>},
  pdfkeywords={loop, quantum, gravity, cosmology},
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\rightheader{\href{http://olympus.het.brown.edu/~danieldf/}{Daniel Doro Ferrante}}
\leftheader{\href{http://www.brown.edu/}{Brown University}}
\rightfooter{\quad\textsf{\thepage}}
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\title{\textcolor{yellow}{Loop Quantum Cosmology}}
\author{\href{http://olympus.het.brown.edu/~danieldf/}{\uline{Daniel Doro Ferrante}}}
\maketitle
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\foilhead[-3cm]{}
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\begin{center}
  \scalebox{1.5}{\includegraphics{out.pdf}}

  by \href{http://www.tug.org/}{\uline{\TeX}}\/ and
  \href{http://www.gnu.org/software/emacs/emacs.html}{\uline{GNU Emacs}}
\end{center}
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\foilhead[-2cm]{\uwave{Outline}}
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\begin{itemize}
\item \hyperlink{qgi}{\uline{Quantum Gravity's importance}};
\item \hyperlink{paqg}{\uline{Present Approaches to Quantum Gravity}};
\item \hyperlink{blqg}{\uline{Basics of Loop Quantum Gravity}};
\item \hyperlink{lqgr}{\uline{LQG Results}};
\item \hyperlink{lqc}{\uline{Loop Quantum Cosmology}}; \&
\item \hyperlink{lqcra}{\uline{LQC Recent Applications}}.
\end{itemize}
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\foilhead[-1.5cm]{\hypertarget{qgi}{Quantum Gravity: \emph{Why is it important?!} \blacksmiley}}
\pauseDissolve
\begin{description}
\item[\textcolor{yellow}{Relation to other problems}] \pauseOBox
  \vspace{-1cm}
  \begin{itemize}
  \item Quantum Cosmology \& Origin of the Universe;
  \item Quantum Theory ``without time'', Unitarity;
  \item Structure and Interpretation of Quantum Mechanics,
    \href{http://math.ucr.edu/home/baez/topos.html}{\uline{Topos theory}};
  \item Wave Function Collapse;
  \item Unification of All Interactions, TOEs;
  \item Final State of a Black Hole; \&
  \item Ultraviolet Divergences.
  \end{itemize}
\end{description}
\pauseDissolve
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\foilhead{\hypertarget{paqg}{Present Approaches to Quantum Gravity}}
\pauseDissolve
\begin{description}
\item[\textcolor{yellow}{Main Directions}] \pauseOBox
  \vspace{1cm}
  \begin{itemize}
  \item \textbf{String Theory}; \&
  \vspace{3cm}
  \item \textbf{Loop Quantum Gravity}.
  \end{itemize}
\end{description}
\pauseDissolve
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\foilhead{}
\begin{description}
\item[\textcolor{yellow}{Traditional Approaches}] \pauseOBox
  \begin{itemize}
  \item \textbf{Discrete Approaches} (Regge calculus, dynamical triangulations);
  \vspace{1.3cm}
  \item \textbf{Old Hopes, Approximate Theories} (Euclidean QG, QFT on curved spacetime); \&
  \vspace{1.3cm}
  \item \textbf{``Unorthodox Approaches''} (causal sets, twistors, Finkelstein's ideas).
  \end{itemize}
\end{description}
\pauseDissolve
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\foilhead{}
\begin{description}
\item[\textcolor{yellow}{New Directions}] \pauseOBox
  \vspace{2cm}
  \begin{itemize}
  \item \textbf{Noncommutative Geometry};
  \vspace{1.3cm}
  \item \textbf{Null Surface Formulations}; \&
  \vspace{1.3cm}
  \item \textbf{Spin Foam Models} (TQFTs).
  \end{itemize}
\end{description}
\pauseDissolve
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\foilhead[-1cm]{\hypertarget{blqg}{Loop Quantum Gravity: The Basics}}
\pauseDissolve
\begin{itemize}
\item \textbf{Quantum Field Theory on a Differentiable Manifold} \pauseOBox
  \begin{itemize}
  \item background independence;
  \bigskip
  \item pseudo-1-form (densitized triad); \&
  \bigskip
  \item $SU(2)$-connection.
  \end{itemize}
\end{itemize}
  \vspace{-1cm}
  \pauseDissolve
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\foilhead[-1cm]{Loop Quantum Gravity: The Basics {\small(cont'd)}}
\pauseDissolve
\begin{itemize}
\item \textbf{Wilson Loop Operators} \pauseOBox
  \begin{itemize}
  \item \textbf{well defined} in the Hilbert space of the theory; \&
  \item non-perturbative.
  \end{itemize}
  \vspace{-1cm}
  \pauseDissolve
\item \textbf{Physical meaning of diffeomorphism invariance and its implementation in the
    quantum theory} \pauseOBox
  \begin{itemize}
  \item \textbf{all} variables are \textbf{dynamical}!
  \end{itemize}
\end{itemize}

\hyperlink{app}{\uline{Appendix}}
\pauseDissolve
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\foilhead{\hypertarget{lqgr}{LQG: Results}}
\begin{itemize}
\item \textbf{Technical} \pauseOBox
  \begin{itemize}
  \item Solution of the Hamiltonian constraint
  \bigskip
  \item Time Evolution, Topological Feynman rules
  \bigskip
  \item Fermions
  \bigskip
  \item Maxwell \& Yang-Mills
  \bigskip
  \item Lattice \& Simplicial models
  \end{itemize}
\end{itemize}
\pauseDissolve
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\foilhead[-2cm]{LQG: Results {\small(cont'd)}}
\begin{itemize}
\item \textbf{Physical} \pauseOBox
  \vspace{1cm}
  \begin{itemize}
  \item Planck Scale Discreteness of space (area or volume operators have a discrete spectrum)
  \vspace{.3cm}
    \begin{center}
      \includegraphics{img199.png}\scalebox{0.5}{\includegraphics{img168.png}}
    \end{center}
  \item Classical Limit, quantum states for flat spacetimes
  \vspace{1cm}
  \item Black Hole Entropy
  \end{itemize}
\end{itemize}
\pauseDissolve
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\foilhead[-2cm]{\hypertarget{lqc}{Loop Quantum Cosmology}}
\begin{itemize}
\item Hamiltonian constraint:
  \pauseOBox
  \vspace{1cm}
  \begin{boldequation*}
    \hspace{-1.1cm}
    \frac{12\, \sqrt{|p|}}{\gamma^2}\, \Biggl[c\, (c - k) + (1 + \gamma^2)\, \frac{k^2}{4} \Biggr] =
      8\, \pi\, G\, H_{m}(p,\phi,p_{\phi})
  \end{boldequation*}
  \vspace*{1cm}\\
  where $|p| = a^2$, $c = 1/2\, (k - \gamma\, \dot{a})$, $k = 0, 1$ for a flat or closed
  model, $H_m$ is the matter Hamiltonian. ($p$ and $c$ are part of the \textbf{isotropic}
  triad and connection, respectively.)
\end{itemize}
\pauseDissolve
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\foilhead{LQC {\small(cont'd)}}
\begin{itemize}
\item This is just Friedman's eq,
  \begin{boldequation*}
    3\, (\dot{a}^2 + k^2)\, a = 8\, \pi\, G\, a^3\rho_{m}(a,\phi,p_{\phi})
  \end{boldequation*}
\bigskip
\item LQC: \textbf{\textcolor{yellow}{symmetric}} LQG states.
\end{itemize}
\pauseDissolve
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\foilhead{LQC {\small(cont'd)}}
\begin{itemize}
\item Discrete Evolution
  \pauseDissolve
  \begin{itemize}
  \item \textcolor{yellow}{\textbf{Indirect}} quantization: via holonomies, i.e.,
    $\nexists\; c$ operator, just a holonomy related to it.
  \pauseDissolve
  \vspace{.7cm}
  \item Quantized Friedman eq: \textbf{difference} eq!
  \pauseDissolve
  \vspace{.7cm}
  \item $\Rightarrow$ \textcolor{yellow}{\textbf{\uwave{no}}} cosmological singularities
    (homogeneous models)!
  \end{itemize}
\end{itemize}
\pauseDissolve
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\foilhead{LQC {\small(cont'd)}}
\begin{itemize}
\item Finite Inverse Scale Factor Operator
  \pauseOBox
  \begin{itemize}
  \item Matter Hamiltonian: need to quantize $a^{-3}$
    \begin{itemize}
    \item Inverse power of $\hat{p}$: discrete spectrum with $0$
    \item Alternative way yields 3 phases:
      \pauseOBox
      \begin{enumerate}
      \item \textcolor{yellow}{$\hat{H}_{m}(0) = 0$}
      \item \textcolor{yellow}{Inflation}
      \item \textcolor{yellow}{Transition to the classical $a^{-3}$}
      \end{enumerate}
    \end{itemize}
  \end{itemize}
\end{itemize}
\pauseDissolve
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\foilhead{\hypertarget{lqcra}{Recent Applications}}
\begin{itemize}
\item Quantum Structure of Classical Singularities
  \pauseOBox
  \vspace{2cm}
  \begin{itemize}
  \item All homogeneous models: no cosmological singularities!
  \vspace{1cm}
  \item Inhomogeneous models: can be treated in the BKL scenario.
  \end{itemize}
\end{itemize}
\pauseDissolve
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\foilhead[-2.3cm]{Recent Applications {\small(cont'd)}}
\begin{itemize}
\item Phenomenology
  \pauseOBox
  \begin{itemize}
  \item Inflation:
    \begin{itemize}
    \item \textcolor{yellow}{\textbf{No}} graceful exit problem (phase transition on the
      scale factor $\Rightarrow$ classical regime);
    \item Exists for \textbf{any} matter content, even without an inflaton (spacetime
      property).
    \end{itemize}
  \pauseDissolve
  \vspace{-.8cm}
  \item Quantized spacetime: photons with different energies should have slightly
    different speeds (modified dispersion relations; gamma \& cosmic rays).
  \end{itemize}
\end{itemize}
\pauseDissolve
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\foilhead{Recent Applications {\small(cont'd)}}
\begin{itemize}
\item Perturbative Corrections
  \pauseOBox
  \begin{itemize}
  \item Leading order: Friedmann eq (as seen); \&
  \vspace{1cm}
  \item Two types of corrections:
    \pauseOBox
    \vspace{.5cm}
    \begin{enumerate}
    \item Higher curvature corrections ( $\simeq$ effective action); \&
    \vspace{.5cm}
    \item Other terms: reflect that $\nexists$ unique coherent state such as the
      Minkowski vacuum for an effective action.
    \end{enumerate}
  \end{itemize}
\end{itemize}
\pauseDissolve
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\foilhead[-1cm]{Bibliography}
{\tiny
\begin{itemize}
\item
  \href{http://www.sciam.com/article.cfm?articleID=0007E95C-9597-1DC9-AF71809EC588EEDF}
  {\uline{\emph{``Throwing Einstein for a Loop''}}};
\item \href{http://www.sciamdigital.com/browse.cfm?sequencenameCHAR=item2&methodnameCHAR=resource_getitembrowse&interfacenameCHAR=browse.cfm&ISSUEID_CHAR=66641B02-2B35-221B-6390A3FA94605D1A&ARTICLEID_CHAR=66752753-2B35-221B-619EB81F85536798}{\uline{\emph{``Atoms of Space and Time''}}};
\item
  \href{http://www.sciam.com/article.cfm?articleID=00096B95-41B1-1C6F-84A9809EC588EF21&pageNumber=1}
  {\uline{\emph{``A Spin on Spin Foam''}}};
\item
  \href{http://www.sciam.com/article.cfm?articleID=000073A5-C100-1F80-B57583414B7F0103&chanID=sa009}
  {\uline{\emph{``The Future of String Theory -- A Conversation with Brian Greene''}}};
\item \href{http://arxiv.org/abs/hep-th/0303185}{\uline{\emph{``How far are we from the quantum
      theory of gravity?''}}};
\item \href{http://arxiv.org/abs/gr-qc/9803024/}{\uline{\emph{``Strings, loops and others: a
      critical survey of the present approaches to quantum gravity''}}};
\item \href{http://arxiv.org/abs/gr-qc/9710008/}{\uline{\emph{``Loop Quantum Gravity''}}};
\item \href{http://arxiv.org/abs/gr-qc/9806079/}{\uline{\emph{``Loop quantum gravity and quanta
      of spacetime: a primer''}}};
\item \href{http://arxiv.org/abs/gr-qc/0210094/}{\uline{\emph{``Lectures on Loop Quantum
      Gravity''}}};
\item \href{http://arxiv.org/abs/gr-qc/0402062/}{\uline{\emph{``Canonical quantum gravity and
      consistent discretizations''}}};
\item \href{http://arxiv.org/abs/gr-qc/0102069/}{\uline{\emph{``Absence of Singularity in Loop
      Quantum Cosmology''}}};
\item \href{http://arxiv.org/abs/math-ph/0202008/}{\uline{\emph{``Quantum Geometry in Action:
      Big-Bang and Black Holes''}}};
\item \href{http://arxiv.org/abs/gr-qc/0306008/}{\uline{\emph{``Cosmological applications of loop
      quantum gravity''}}}; \&
\item \href{http://arxiv.org/abs/gr-qc/0402053/}{\uline{\emph{``Loop Quantum Cosmology: Recent
      Progress''}}}.
\end{itemize}
}
\pause
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\foilhead{Just in case\ldots}
Oh, you thought I wouldn't be prepared for some questions, huh?!
\textcolor{yellow}{\blacksmiley}

Not at all: In what follows there are some slides to entertain the more curious
characters\ldots
\pauseDissolve
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\foilhead[-2cm]{\hypertarget{app}{Appendix: Loop Quantization}}
\begin{itemize}
\item Quantum variables:
  \begin{itemize}
  \item Holonomies: $h_e(A) = \mathcal{P} \exp\{\int_e A^i_a\, \dot{e}^a\, \tau_i\, dt\}\;
    \in SU(2)$
  \item Fluxes: $F_S(E) = \int_S E^a_i\, n_a\, \tau^i\, d^2y$
  \end{itemize}
  \item $\tau_i$: Pauli matrices; $\dot{e}^a$: tangent vector to the edge $e$; $n_a$:
    co-normal to the surface $S$.
  \item If all curves $e$ and all surfaces $\Sigma$ are allowed, holonomies and fluxes
    contain the same information as the original fields.
\end{itemize}
\pause
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\foilhead[-2cm]{Appendix: Loop Quantization {\small (cont'd)}}
\begin{itemize}
\item Holonomies and fluxes are better for quantization: smeared versions of the fields
  obtained by natural integration along curves and surfaces
\item One- and two-dimensional smearings can be done without introducing a
  background metric! Background independent quantization! \textcolor{yellow}{\blacksmiley}
\item Quantum theory: defined on a representation of the background independent
  holonomy-flux Poisson $\star$-algebra
\end{itemize}
\pause
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\foilhead{Appendix: Loop Quantization {\small (cont'd)}}
\begin{itemize}
\item The basis for the states is given by \textbf{spin network} states which are
  associated to graphs in $\Sigma$ whose edges are labeled by irreducible $SU(2)$
  representations
\item Diffeomorphism-invariant inner product: Ashtekar-\\-Lewandowski
\end{itemize}
\pause
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\foilhead[-2cm]{Appendix: Loop Quantization {\small (cont'd)}}
\begin{itemize}
\item Characteristics of the Loop Quantization:
  \begin{itemize}
  \item Hilbert space before imposing the [diffeomorphism invariance and Hamiltonian]
    constraints is non-separable: all spin network states in different graphs are
    orthogonal to each other!
  \item Holonomies are well defined operators by definition!
  \item Flux operators have discrete spectra $\Rightarrow$ discrete spatial geometry!
  \end{itemize}
\end{itemize}
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