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\title{Local Quantum Physics \& The Bisognano-Wichmann Theorem}

\author[C.\ Solveen]{Christoph Solveen}
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\begin{frame}{Introduction}
We discuss the modular data associated to local algebras in quantum field theory.\\\quad\\ \pause
This talk is a ``physical digression'': We consider nets of algebras of observables on $4d$ Minkowski spacetime.\\\quad\\ \pause
Modular objects of algebras associated to so-called wedge regions allow a geometric interpretation in terms of the symmetries of spacetime.\\\quad\\ \pause
This has interesting consequences for physics: Unruh effect, Hawking radiation \dots
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\section{Minkowski Spacetime and its Symmetries}

\begin{frame}{Spacetime}
$4d$-Minkowski spacetime $M^4$ is the real manifold $\mathbbm{R}^4$, equipped with the Minkowski pseudometric $\eta$ with signature $(+,-,-,-)$.\\\quad\\ \pause
The group of diffeomorphisms that leave $\eta$ invariant is the Poincar\'e group $\mathcal{P}=\mathcal{L}\ltimes\mathbbm{R}^4$, $\mathcal{L}$ being the the Lorentz group $O(1,3)$.\\\quad\\ \pause

We consider the subgroup $\mathcal{P}^{\uparrow}_{+}=\mathcal{L}^{\uparrow}_{+}\ltimes\mathbbm{R}^4$, where $\mathcal{L}^{\uparrow}_{+}$ is the proper orthochronous Lorentz group, i.e.\ the identity component of $\mathcal{L}$.
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\begin{frame}{Boosts}
Elements of the Lorentz group $\mathcal{L}$ can be represented as $4\times4$-matrices. A particular kind of Lorentz transformation are the ``boosts'', for example in the $x^1$-direction:
\begin{displaymath}
\begin{pmatrix}
	\cosh(s)&-sinh(s)&0&0\\
	-sinh(s)&cosh(s)&0&0\\
	0&0&1&0\\
	0&0&0&1
\end{pmatrix},
\end{displaymath}
relating two inertial frames in relative uniform motion.
\\\quad\\\pause
Here, $s\in\mathbb{R}$ is called the ``rapidity'' and relates to the relative velocity $v$ as $\cosh(s)=1/\sqrt{1+v^2/c^2}$.
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\begin{frame}{Causal structure}
The Lorentz group preserves the (indefinite) inner product $x\cdot y=x^0y^0-x^1y^2-x^2y^2-x^3y^3$ on Minkowksi spacetime (now seen as a vector space). With the help of this inner product one defines the light cone at a given point x, which encodes the causal relations of $x$ with all events throughout spacetime.\\\quad\\ \pause

Example: the closed \emph{forward light cone} at the origin $V^{+}=\{x\in M^4:x\cdot x\geq0~\text{and}~ x^0\geq 0\}$.\\\quad\\\pause

We also need the \emph{causal complement} $\mathcal{O}'$ of a given subregion $\mathcal{O}\subset M^4$:
\begin{displaymath}
\mathcal{O}':=\{x\in M^4: (x-y)\cdot(x-y)<0~\text{for all}~y\in\mathcal{O}\}.
\end{displaymath}\pause
A region $\mathcal{O}$ is called \emph{causally complete} if $\mathcal{O}=\mathcal{O}''$. Basic example: ``diamonds'' or ``double cones''.
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\section{Axioms for Quantum Field Theory}

\begin{frame}{Observable Algebras and Haag-Kastler Axioms}
(Haag Kastler axioms for nets of algebras of observables) One considers a net of (say) von Neumann algebras on $M^4$ $\mathcal{O}\mapsto\mathcal{R}(\mathcal{O})$, where $\mathcal{O}$ are open bounded regions of $M^4$, which we take as causally complete.\pause
\begin{enumerate}
	\item $\mathcal{O}_1\subset \mathcal{O}_2\Rightarrow \mathcal{R}(\mathcal{O}_1)\subset\mathcal{R}(\mathcal{O}_2)$ (Isotony)\pause
	\item $\mathcal{O}_1\subset \mathcal{O}_2'\Rightarrow \mathcal{R}(\mathcal{O}_1)\subset\mathcal{R}(\mathcal{O}_2)'$ (Locality)\pause
	\item There is an automorphic action of $\mathcal{P}^{\uparrow}_{+}$ acting geometrically on the net: $\alpha_{g}(\mathcal{R}(\mathcal{O}))=\mathcal{R}(g\mathcal{O})$ for $g\in\mathcal{P}^{\uparrow}_{+}$. (Poincar\'e Symmetry)\pause
\end{enumerate}
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\begin{frame}{Vacuum Representation}
A representation $\pi_{0}$ of the Haag Kastler net $\mathcal{O}\mapsto\mathcal{R}(\mathcal{O})$ on some Hilbert space $H$ is called \emph{vacuum representation} if:\pause
\begin{enumerate}
	\item There exists a strongly continuous unitary representation $U$ of $\mathcal{P}^{\uparrow}_{+}$ that implements the Poincar\'e symmetry: $U(g)\pi_{0}(\mathcal{R}(\mathcal{O}))U(g)^*=\pi_{0}(\alpha_{g}(\mathcal{R}(\mathcal{O})))$. (Poincar\'e Covariance)\pause
	\item The joint spectrum of the generators of the translation subgroup lies in the closed forward light cone $V^{+}$. (Spectrum Condition)\pause
	\item There exists a unique translation invariant vector $\Omega\in H$ and the set $\{\pi_{0}(A)\Omega:A\in\mathcal{R}(\mathcal{O}), \mathcal{O} ~\text{double cone}\}$ is dense in $H$. (Vacuum)\pause
\end{enumerate}
Under these assumptions, $\Omega$ is Poincar\'e-invariant.
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\begin{frame}{Some Remarks conerning the Haag-Kastler Axioms}
Reeh-Schlieder Theorem (Borchers):
If the Haag-Kastler net is additive, i.e.\ $$\mathcal{O}=\cup_{i}\mathcal{O}_{i}~\Rightarrow~ \mathcal{R}(\mathcal{O})=\vee_{i}\mathcal{R}(\mathcal{O}_{i}),$$ then the vaccum vector $\Omega$ is cyclic and separating for any double cone.\\\quad\\\pause

Remark I: There are other representations relevant for QFT, for example those describing global thermal equilibrium situations (KMS property) or those describing charged particles (sector theory).\\\quad\\\pause

Remark II: The idea that physics is encoded into a net of observable algebras is very useful in QFT on curved spacetime.
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\begin{frame}{Garding-Wightman axioms for Quantum Field Theory (Bosonic)}
Let $H$ be a Hilbert space. Fields are operator valued tempered distributions $\mathcal{S}(M^4)\ni f\mapsto \phi(f)$ such that:\pause
\begin{enumerate}
\item There is a common dense invariant domain $\mathcal{D}$ for all fields $\phi(f)$. $\phi(\overline{f})\subset\phi(f)^*$ and for real valued $f$, the $\phi(f)$ are essentially self-adjoint.\pause
\item There is a strongly continuous, positive energy representation $U$ of $\mathcal{P}^{\uparrow}_{+}$ with $U(g)\phi(f)U(g)^*=\phi(f_{g})$.\pause
\item There is a unique (up to a phase) translation-invariant unit vector $\Omega\in\mathcal{D}$, the vacuum.\pause
\item If the supports of $f$ and $g$ are spacelike separated, then $[\phi(f),\phi(g)]=0$ on $\mathcal{D}$.
\end{enumerate}
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\begin{frame}{Nets from Fields}
Assuming the Wightman axioms on a Hilbert space $H$, one can obtain a Haag-Kastler net:\\\pause For any open region $\mathcal{O}$, denote by $\mathcal{R}(O)$ the von Neumann algebra generated by bounded functional calculi of the fields $\phi(f)$ with $\mathrm{supp}(f)\subset\mathcal{O}$. Then $\mathcal{O}\mapsto\mathcal{R}(\mathcal{O})$ obeys the Haag-Kastler axioms with vacuum representation on $H$.\\\quad\\ \pause
The other direction, fields from nets, is also discussed in the literature. There are conditions under which one can recover the field content of a given net of observables (Fredenhagen \& Hertel, Bostelmann and others).
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\begin{frame}{Reeh-Schlieder and PCT Theorem}
Reeh-Schlieder Theorem: Assume the the field is irreducible, i.e.\ there is no bounded operator that commutes with all fields apart from multiples of the identity. Then for any non-empty open region $\mathcal{O}$, the vacuum vector $\Omega$ is cyclic for the algebra $\mathcal{R}(\mathcal{O})$. In particular, if $\mathcal{O}'$ is non-empty, $\Omega$ is a standard vector for $\mathcal{R}(\mathcal{O})$.\\\quad\\\pause

(part of) of the PCT-theorem: There exists an anti-unitary operator $\Theta$ uniquely defined by (unsmeared notation) $$\Theta\phi(x)\Theta^{-1}=\phi^{*}(-x)~~;~~\Theta\Omega=\Omega.$$
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\section{Bisognano-Wichmann Theorem, Borcher's Theorem}

\begin{frame}{Wedge Regions}
Let us consider the \emph{right wedge region} in Minkowski spacetime $M^4$:
$$W=\{x\in M^4: x^1>\left|x^0\right|\}.$$\pause

This region is invariant under the following one parameter subgroup of the Lorentz group:
\begin{displaymath}
\Lambda(s)=\begin{pmatrix}
	\cosh(s)&-sinh(s)&0&0\\
	-sinh(s)&cosh(s)&0&0\\
	0&0&1&0\\
	0&0&0&1
\end{pmatrix}~,~s\in\mathbb{R}.
\end{displaymath}
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\begin{frame}{Bisognano Wichmann}
Let $\mathcal{R}(W)$ denote the von Neumann algebra generated by the bounded functional calculi of $\phi(f)$, $\mathrm{supp}(f)\subset W$. By the Reeh-Schlieder Theorem, the vacuum $\Omega$ is standard for $\mathcal{R}(W)$, hence we find the modular conjugation $J_{W}$ and modular operator $\Delta_{W}$ for the pair $(\mathcal{R}(W),\Omega)$.\\\quad\\\pause

Bisognano-Wichmann Theorem: $$J_{W}=\Theta\cdot U(R_{23}(\pi))~\text{and}~\Delta^{it}=U(\Lambda(-2\pi t)).$$
(Here $U(R_{23}(s))$ and $U(\Lambda(t))$ denote the unitary implementations of rotations in the $(x^2,x^3)$-plane and $\Lambda(t)$.)\\\quad\\\pause

Hence the modular objects act geometrically on the algebra corresponding the the right wedge. This holds true for all regions that are images of the right wedge under the Poincar\'e group.
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\begin{frame}{A Variant of BW: Conformal symmetry and the Theorem of Hislop \& Longo}
In $4d$, the conformal group is a finite-dimensional Lie group, for $M^4$ its identity component is generated by the proper Poincar\'e group and the relativistic ray transformation $$x\mapsto\frac{x}{x\cdot x}$$ (going over to a suitable compactification of spacetime to avoid singular transformations).\\\quad\\\pause

Theorem (Hislop \& Longo): In the case of an irreducible neutral scalar field that satisfies the Wightman axioms where the confromal group is taken instead of the Poincar\'e group, the modular conjugations and groups of the von Neumann algebras associated with wedges, double cones and forward and backward light cones have a geometric meaning. 
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\begin{frame}{Borcher's Theorem}
For a long time, the BW-Theorem was a kind of paradoxical result: its formulation is natural in the Haag-Kastler formalism, but its proof was given in the Garding-Wightman setting (using analyticity properties of so-called $n$-point functions).\\\quad\\\pause

However, there is Theorem (Borchers): Let $\mathcal{R}$ be a von Neumann algebra with standard vector $\Omega$ with modular data $(J,\Delta)$ and $U(a)$ a one-parameter group of unitaries leaving $\Omega$ fixed such that for $a\geq0$, $U(a)\mathcal{R}U(a)^{*}\subset\mathcal{R}$. Then $$\Delta^{it}U(a)\Delta^{-it}=U(e^{-2\pi t}a)~\text{and}~JU(a)J=U(-a)~,~a,t\in\mathbb{R}.$$
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\begin{frame}{Conformal symmetry and AQFT}
The results of Hislop and Longo and the theorem of Borchers have been used to prove the following:\\\quad\\\pause

Theorem: Let $\mathcal{O}\mapsto\mathcal{R}(\mathcal{O})$ be a conformally covariant net of von Neumann Algebras. Then the modular conjugations and groups of the von Neumann algebras associated with wedges, double cones and forward and backward light cones have a geometric meaning.\\\quad\\\pause

This is the first $4d$ example of a Bisognano-Wichmann type result in a purely algebraic setting. (It needs the full conformal symmetry, however.)
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\section{Bisognano-Wichmann-Theorem and Physics}

\begin{frame}{States in Physics}
States of a $*$-algebra $\mathcal{A}$ with unit are linear functionals $\omega$ on $\mathcal{A}$ such that for all $a\in\mathcal{A}$\pause
\begin{enumerate}
	\item $\omega(a^{*})=\overline{\omega(a)}$\pause
	\item $\omega(a^{*}a)\geq 0$\pause
	\item $\omega(1)=1$.\pause
\end{enumerate}
States give rise to representations of $\mathcal{A}$ via the famous GNS-construction (example: vacuum states give rise to vacuum representations).\\\quad\\\pause

In physics, states of an algebra $\mathcal{A}$ of observables are interpreted as expectation value functionals for measurements of observables (selfadjoint elements of $\mathcal{A}$) in a given physical configuration of the system.
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\begin{frame}{KMS-Property and Global Thermal Equilibrium}
In quantum statistical mechanics, global thermal equilibrium situations are described via the KMS condition:
For an observer whose time-evolution (dynamics) is described by an automorphism $\alpha_{t}$, a state $\omega$ is a global thermal equilbrium state at ``inverse temperature`` $\beta\in\mathbb{R}$ if $\omega$ is an $(\alpha_{t},\beta)$-KMS state.\\\quad\\\pause

Definition: $\omega$ is a $(\alpha_{t},\beta)$-KMS state if, for each pair of operators $a,b$ there is a function $h:\mathbb{C}\rightarrow\mathbb{C}$, analytic in the strip $\{z\in\mathbb{C}:0<\mathrm{Im}(z)<\beta\}$ and continuous at the boundaries such that $$h(t)=\omega(a\,\alpha_{t}(b))~~\text{and}~~h(t+i\beta)=\omega(\alpha_{t}(b)\,a)$$
for $t\in\mathbb{R}$.
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\begin{frame}{Modular Theory and KMS}
Let $\omega$ be a state over a (say) von Neumann algebra $\mathcal{R}$ such that for the corresponding GNS-triple $(H_{\omega},\pi_{\omega},\Omega_{\omega})$, the pair $(\pi_{\omega}(\mathcal{R})'',\Omega_{\omega})$ is standard.\\\quad\\\pause

Proposition: Denote by $\sigma_{\tau}(\cdot)=\Delta^{it}(\cdot)\Delta^{-it}$ the modular group corresponding to $(\pi_{\omega}(\mathcal{R})'',\Omega_{\omega})$. Then the state $\omega$ is $(\sigma_{\tau},-1)$-KMS.\\\quad\\\pause

Consequence for physics: An equilibrium state with inverse temperature $\beta$ may be characterized as a state over the observable algebra whose modular automorphism group $\sigma_{\tau}$ is the time translation group (that generates the dynamics), the modular parameter $\tau$ being related to physical time $t$ by $t=-\beta\tau$.
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\begin{frame}{Bisognano-Wichmann Theorem and Unruh Temperature}
\small Let the assumptions of the Bisognano-Wichmann theorem be fulfilled for a Haag-Kastler net of observables $\mathcal{O}\mapsto\mathcal{R}(\mathcal{O})$ in the vacuum representation (say, in case it is generated by Wightman fields). Denote by $\mathcal{R}(W)$ the algebra corresponding to the right wedge $W$.\\\quad\\\pause

Important physical fact: $\{\Lambda(t)x\,:\,t\in\mathbb{R}\}$ with $x=(0,a^{-1},0,0)\in W$ can be interpreted as trajectory of a uniformly accelerated observer with acceleration $a$. His time evolution on the net is given by $\alpha_t(\cdot):=U(\Lambda(t))(\cdot)U(\Lambda(t))^{*}$.\\\quad\\\pause

The Bisognano-Wichmann Theorem tells us that the modular conjugation of $\mathcal{R}(W)$ is given by $\sigma_{\tau}:=U(\Lambda(2\pi\tau))(\cdot)U(\Lambda(2\pi\tau))^{*}$. Consequence: For an observer with acceleration $a$ the vacuum is a $(\alpha_t,2\pi/a)$-KMS state, i.e.\ a he feels immersed in a heat bath of Temperature $T=a/(2\pi)$ (Boltzmann's constant set to $1$)! This is related to the famous Hawking radiation.

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