A Modular Operator Approach to Entanglement of Causally Closed Regions

Abstract

Quantum entanglement is shown for causally separated regions along the radial direction by using a conformal quantum mechanical correspondence with conformal radial Killing fields of causal diamonds in Minkowski space. In particular, the theory of local von Neumann algebras and Tomita Takesaki modular operators is applied in the entanglement structure of causal diamonds in conformal quantum mechanics. The entanglement of local states in their respective causal regions is shown through the measures of concurrence and entanglement entropy using the Tomita Takesaki modular conjugation operator. A holographic entropy formula is derived for the conformal quantum mechanics causal diamond correspondence. A new connection is made between the thermal time flow defined by the modular group of automorphisms to the physical time flow in a causal diamond via the aforementioned correspondence. The thermal interpretation of these results via two-point thermal Green’s functions and modular group flow supports the idea of a possible emergent theory of spacetime.

Appendix: Von Neumann Algebras and Tomita-Takesaki Modular Operators

The general references for this section are [37,38,39].

Normed Algebra: Consider an algebra $A \in \mathcal {A}$ over $\mathbb {C}$. A normed algebra has a Norm Map: $A \rightarrow ||A|| \in \mathbb {R}^+$ such that

$$\begin{aligned}&||A|| > 0\\&||A|| = 0 \; \text {iff} \; A=0\\&\alpha \in \mathbb {C}, ||\alpha A||= |\alpha | ||A||\\&||A+B|| \le ||A|| + ||B||\\&||AB|| \le ||A|| \; ||B|| \end{aligned}$$

Banach Algebra: Let $\mathcal {A}$ be an algebra. A normed algebra has a norm map: $\mathcal {A} \rightarrow \mathbb {R}^+ , \,\,\, A \rightarrow ||A|| \in \mathbb {R}^+ , \forall A \in \mathcal {A}$ such that

$$\begin{aligned}&||A|| \ge 0,\\&||A|| = 0 \iff A=0, \\&\alpha \in \mathbb {C} , ||\alpha A || = |\alpha | \, ||A||, \\&|| A+B|| \le ||A|| + ||B||, \\&||AB|| \le ||A|| \, ||B|| . \end{aligned}$$

A Banach algebra is a complete normed algebra (complete in the norm map): Consider a Cauchy sequence $A_1, A_2, ... \in \mathcal {A}$. For any $\epsilon >0, \exists$ an integer N such that $\forall$ natural numbers $m,n > N$, $||A_m -A_n|| < \epsilon$ (a sequence whose elements become arbitrarily close to each other as the sequence progresses). By introducing a “distance” metric $d(A,B)=||A-B||$, once induces a topology on $\mathcal {A}$ where a neighborhood U in $\mathcal {A}$ is given by $U(A,\epsilon )=\{B; \; B \in \mathcal {A}, \; d(A,B) < \epsilon , \; \epsilon >0\}$. A metric space $(\mathcal {A}, d)$ in which every Cauchy sequence converges to an element in $\mathcal {A}$ is called complete in the standard norm. Counterexample: Rational numbers $(p/q) \in \mathbb {Q}$ are not complete. The Cauchy sequence $x+0=1, x_{n+1}=\frac{x_n +2/x_n}{2}$ converges to the irrational number $\sqrt{2}$.

Furthermore, for example, let $\mathcal {A} = C^0(X, \mathbb {C}) \equiv$ complex valued continuous functions f on a compact space X (a bounded region): $f: X \rightarrow \mathbb {C}$.

$$\begin{aligned}&\text {Let} x \in X, \;\;\ f,g \in \mathcal {A}\\&(f+g)(x) =f(x) +g(x) \ \text {(Addition)}\\&(\alpha f)(x) = \alpha f(x) \ \text {(scalar multiplication)}\\&\Big ( fg \Big ) (x) =f(x)g(x) \ \text {(Algebra product)}\\&||f|| = \underset{x\in X}{\sup } |f(x)| \ \text {(norm)} \end{aligned}$$

$\varvec{C}^{*}$ -Algebra: A $\varvec{C}^{*}$-algebra $\mathcal {A}$ is a Banach algebra with an involutive map $^*: \mathcal {A} \rightarrow \mathcal {A} , \,\,\, A \rightarrow A^* \,\,\, \forall A \in \mathcal {A} , \lambda \in \mathbb {C}$ such that

$$\begin{aligned}&(A^{*})^{*}=A\\&(AB)^{*}=B^{*} A^{*}\\&(\lambda A)^{*}= \bar{\lambda } A^{*}\\&(\lambda A + \alpha B )^{*} = \bar{\lambda } A^{*} +\bar{\alpha } B^{*} \text {(anti-linear) } \text { (so far, we have a *-algebra)}\\&||A^{*}|| = ||A|| \text { (norm condition for involutive Banach algebra),}\\&||A A^{*}|| = ||A|| \, ||A^{*}|| = ||A^{*} A|| = ||A^{*}|| \, ||A|| =||A||^{2} (C^{*} \text { condition for } \varvec{C}^{*} \text { -Algebra).}\\ \end{aligned}$$

If $A^{*} = A$, this element is called self-adjoint. If $A^{*}A = AA^{*} =I$ this element is called unitary.

For example, let $\mathcal {A} = C^0(X, \mathbb {C}) \equiv$ complex valued continuous functions f on a compact space X (a bounded region). From above, we know that this is a Banach algebra. We know introduce an involution map through complex conjugation:

$$\begin{aligned}&f^{*}: X \rightarrow \mathbb {C}, \;\;\; f^{*} \in \mathcal {A}\\&f^{*}(x) := \overline{f(x)}, \; \forall x \in X \text { (Involution)}\\&||f^{*}|| = \underset{x\in X}{\sup } |f^{*}(x)|=\underset{x\in X}{\sup } |\overline{f(x)}| =\underset{x\in X}{\sup } |f(x)|= ||f|| \text { (norm condition)}\\ \end{aligned}$$

Furthermore, consider complex $n \times n$ matrices A. Here, we have

$$\begin{aligned}&A^{*} := A^{\dagger } \ \text {(involution is the adjoint operation)}\\&||A|| := \sqrt{Tr(AA^{\dagger })} \ \text {(norm is the square root of trace)} \end{aligned}$$

The algebra is given by commutative matrix addition $A+B$ and non-commutative matrix multiplication $AB \ne BA$.

Bounded Linear Operators $\mathcal {B}(\mathcal {H})$ on a Hilbert Space: Let $\mathcal {H}$ be a Hilbert space. A bounded linear operator $A \in \mathcal {B} (\mathcal {H})$ acts on the Hilbert space $A: \mathcal {H} \rightarrow \mathcal {H}$ such that

$$\begin{aligned}&A(\psi + \phi )=A(\psi )+A(\phi ), \;\; \forall \psi ,\phi \in \mathcal {H} \text { (linearity)}\\&\exists \ \text { a positive real number } c < \infty \text { such that } ||A(\psi )|| \le c ||\psi ||, \;\; \forall \psi \in \mathcal {H} \ \text {(bounded)}\\&||A|| = \underset{\psi \in \mathcal {H}, ||\psi || =1}{\sup } \{||A(\psi )|| \} \text { (norm of an operator)} \end{aligned}$$

For all $A \in \mathcal {B}(\mathcal {H}), \;\exists$ a unique element called the adjoint operator $A^{\dagger } \in \mathcal {B}(\mathcal {H})$ such that (using the Hilbert space inner product $\langle \cdot , \cdot \rangle$

$$\begin{aligned}&\langle A^{\dagger }(\psi ), \phi \rangle = \langle \psi , A(\phi ) \rangle \\&(\alpha A + \lambda B)^{\dagger } = \overline{\alpha } A^{\dagger } + \overline{\lambda } B^{\dagger }\\&(A B)^{\dagger } = B^{\dagger } A^{\dagger } \\&(A^{\dagger })^{\dagger } = A\\&||A^{\dagger }A|| = ||A A^{\dagger }|| =||A||^2\\&||A^{\dagger }|| = ||A|| \end{aligned}$$

The $\varvec{C}^{*}$-Algebra features can be given to $\mathcal {B}(\mathcal {H})$ via the following

Banach Norm $\longrightarrow$ Operator Norm

Involution $*$-operator $\longrightarrow$ Adjoint operation $^\dagger$

$||A^{*}||=||A^{\dagger }|| = ||A||$ (Norm condition)

$||A A^*|| = ||A A^{\dagger }|| =||A|| \; ||A^{\dagger }|| = ||A||^2$ ($C^*$ condition for $\varvec{C}^{\varvec{*}}$-Algebra).

${}^{\varvec{*}}$ -Homomorphism $\varvec{\xi }$: Let $\mathcal {A}$ and $\mathcal {B}$ be $\varvec{C}^{*}$-algebras. A ${}^{\varvec{*}}$-homomorphism $\xi$ is a mapping $\xi :\mathcal {A} \rightarrow \mathcal {B}$ that preserves the algebraic and ${}^{\varvec{*}}$ structures of $\mathcal {A}$. That is, $\forall A,A_1,A_2 \in \mathcal {A}$

$$\begin{aligned}&\xi ( A_1 + \ A_2)= \xi (A_1) + \xi (A_2) \;\text {(linearity)}\\&\xi (A_1 A_2)=\xi (A_1)\xi (A_2) \ \text {(homomorphism)}\\&\xi (A^*)=\xi (A)^{*}\ (^{*}\text{-preserving)} \end{aligned}$$

In general, $\varvec{\xi }$ is norm decreasing, i.e. $||\xi (A)|| \le ||A||$. If $\varvec{\xi }$ is an ${}^{\varvec{*}}$-isomorphism (one-to-one, onto), then it is norm preserving, $||\xi (A)|| = ||A||$. A ${}^{\varvec{*}}$-automorphism is a ${}^{\varvec{*}}$-isomorphism from a ${C}^{\varvec{*}}$-algebra to itself, i.e. $\xi :\mathcal {A} \rightarrow \mathcal {A}$.

Representation $\varvec{\pi }$ of a $\varvec{C}^{\varvec{*}}$ -Algebra: A representation of a $\varvec{C}^{\varvec{*}}$-algebra $\mathcal {A}$ is a pair $(\mathcal {H}, \pi )$ where $\mathcal {H}$ is a Hilbert space and $\pi$ is a ${}^{\varvec{*}}$-Homomorphism from $\mathcal {A}$ to $\mathcal {B}(\mathcal {H})$, $\pi :\mathcal {A} \rightarrow \mathcal {B}(\mathcal {H})$. If $\pi$ is an ${}^{\varvec{*}}$-isomorphism, it is called faithful such that

$$\begin{aligned}&x \text {ker} \; \pi = \{0\} \ \text {(faithful)}\\&||\pi (A)|| = ||A||, \forall A \in \mathcal {B}(\mathcal {H}) \ \text {(norm preserving)}\\&||\pi (A)||> 0, \forall A > 0 \ \text {(positive)} \end{aligned}$$

A subset $S \subset \mathcal {H}$ is called invariant under $\mathcal {A}$ if $\pi (\mathcal {A})S := \{\mathcal {A}\psi | A\in \mathcal {A}, \psi \in \mathcal {H}\} \subset S$

A representation $\pi$ is called irreducible if the only subspace of $\mathcal {H}$ invariant under $\pi (\mathcal {A})$ are $\{0\}$ and $\mathcal {H}$.

An operator ${A \in \mathcal {B}(\mathcal {H})}$ is called dense if its range exists only in the dense subspace of the Hilbert space $\mathcal {D} \in \mathcal {H}$.

Cyclic Vector $\; \Omega \in \mathcal {H}$ : A vector $\; \Omega \in \mathcal {H}$ is called a cyclic vector for a set of bounded operators $\mathcal {B}(\mathcal {H})$ if $\{A \; \Omega | A \in \mathcal {B}(\mathcal {H})\}$ is dense in the whole $\mathcal {H}$.

Cyclic Representation $\varvec{\pi }$ of a $\varvec{C}^{\varvec{*}}$ -Algebra: A cyclic representation of a $\varvec{C}^{\varvec{*}}$-algebra $\mathcal {A}$ is a triple $(\mathcal {H}, \pi , \Omega )$ where $(\mathcal {H}, \pi )$ is a representation of $\mathcal {A}$ and $\Omega \in \mathcal {H}$ is cyclic in the representation $\pi$.

Linear Functionals (and States): Given an algebra $\mathcal {A}$, a linear functional (or state) on $\mathcal {A}$ (a scalar valued ’function’ on $\mathcal {A}$) is a map $\omega :\mathcal {A} \rightarrow \mathbb {C}$ such that $\omega (\lambda A +\beta B)=\lambda \omega (A) +\beta \omega (B)$. Furthermore,

If $\mathcal {A}$ is a $^*$-algebra, $\omega$ is called a positive functional if $\omega (A^* A) \ge 0, \forall A \in \mathcal {A}$

A state $\omega$ is a positive functional with $\omega (A^*)=\overline{\omega (A)}$.

If $\mathcal {A}$ is a $^*$-algebra, and $\omega _i$ are positive functionals then for $\lambda _i \in \mathbb {R}^+, \sum _i \lambda _i = 1$, one may construct a positive convex linear functional as $\omega = \sum _i \lambda _i \omega _i$.

If $\mathcal {A}^{sa}$ is self-adjoint subspace, i.e. $A^* = A$ and $\omega$ is a positive functional, then $\omega :\mathcal {A}^{sa} \rightarrow \mathbb {R}$ (real scalars)

If $\mathcal {A}$ is a $\varvec{C}^{\varvec{*}}$-algebra, every positive functional $\omega$ is continuous.

Let $\mathcal {A}$ be a $\varvec{C}^{\varvec{*}}$-algebra. $\forall A \in \mathcal {A}, \exists$ a positive functional $\omega _A$ such that, $||\omega _A||=1$ and $\omega _A(A^* A) = ||A||^2$.

GNS Construction-Representation from a State: Let $\mathcal {A}$ be a $\varvec{C}^{\varvec{*}}$-algebra (of observables). Let $\omega$ be a functional (state) on $\mathcal {A}$. A representation of $\mathcal {A}$ constructed from $\omega$ is a pair $(\mathcal {H}, \pi _{\omega })$ where given the inner product on $\mathcal {H}$, $\left\langle \;\; | \;\; \right\rangle : \mathcal {H} \times \mathcal {H} \rightarrow \mathbb {C}$ and a cyclic vector $\Omega _\omega$ with respect to $\pi _{\omega }$ $\bigl ( \pi _{\omega }(\mathcal {A}) |\left. \Omega _\omega \right\rangle$ is dense in $\mathcal {H}\bigr )$, one has the action of operators on $\mathcal {H}$ such that

$$\begin{aligned} \omega (A) := \frac{\left\langle \Omega _\omega |\pi _{\omega }(A) \Omega _\omega \right\rangle }{\left\langle \Omega _\omega | \Omega _\omega \right\rangle } \end{aligned}$$

Remark

This method requires one to choose a preferred representation $\pi _{\omega }$. In quantum mechanics, the Stone von Neumann theorem states that all representations of the Weyl algebra are unitarily equivalent to the Schrödinger representation so this is not a problem. In Minkowski QFT, Poincaré invariance chooses an appropriate cyclic vector (vaccuum state). The choice of a representation becomes an issue in QFT on curved spacetime manifolds.

Von Neumann Algebra: Consider a $C^*$-algebra $\mathcal {B(H)} =\{A\}$ of bounded linear operators on a Hilbert space, $A: \mathcal {H} \rightarrow \mathcal {H}$. Let $\mathcal {C}$ be a subset of $\mathcal {B(H)}$. An operator $A \in \mathcal {B(H)}$ belongs to the commutant $\mathcal {C}'$ of the set $\mathcal {C}$ $\iff$ $AC =CA, \,\,\, \forall C \in \mathcal {C}$. A von Neumann algebra $\mathcal {A}$ is a unital $C^*$-subalgebra of $\mathcal {B(H)}$ such that $\mathcal {A}'' = \mathcal {A}$. Consider a von Neumann algebra $\mathcal {A} \subset \mathcal {B(H)}$. A von Neumann algebra in standard form is one where there exists an element $| \Omega \rangle \in \mathcal {H}$ which is both cyclic (operating on $| \Omega \rangle$ with elements in $\mathcal {A}$ can generate a space dense in $\mathcal {H}$) and separating (if $A | \Omega \rangle = 0$, then $A=0$).

Tomita Takesaki Modular Operators: Consider a von Neumann algebra $\mathcal {A} \subset \mathcal {B(H)}$ in standard form with a cyclic and separating vector $| \Omega \rangle \in \mathcal {H}$. Let $S:\mathcal {H} \rightarrow \mathcal {H}$ be a anti-unitary operator defined by $S A | \Omega \rangle = A^* | \Omega \rangle$. Let the closure of S have a polar decomposition given by $S=J \Delta ^{\frac{1}{2}} =\Delta ^{-\frac{1}{2}} J$, where J is called the modular conjugation operator and $\Delta$ is called the modular operator. J is anti-linear and anti-unitary whereas $\Delta$ is self-adjoint and positive. Furthermore, the following relations hold:

• 1. $J \Delta ^{\frac{1}{2}} J = \Delta ^{-\frac{1}{2}}$

• 2. $J^2 =I, \,\,\, J^{*} = J$

• 3. $J |\Omega \rangle = |\Omega \rangle$

• 4. $J \mathcal {A} J = \mathcal {A}^{\prime }$

• 5. $\Delta = S^{*} S$

• 6. $\Delta |\Omega \rangle = |\Omega \rangle$

• 7. $\Delta ^{it} \mathcal {A} \Delta ^{-it} = \mathcal {A}$ (one parameter-t group of automorphisms of $\mathcal {A})$

• 8. If $\omega (A) = \langle \Omega | A \Omega \rangle$, $\forall A \in \mathcal {A}$, then $\omega$ is a KMS (Kubo-Martin-Schwinger) functional (state) on $\mathcal {A}$ with respect to the automorphism of 7.

• 9. $|\Omega \rangle$ is cyclic for $\mathcal {A}$ if and only if $|\Omega \rangle$ is separating for $\mathcal {A}'$