Causal Processes in C*-Algebraic Setting

Abstract

In this paper, we attempt to explicate Salmon’s idea of a causal process, as defined in terms of the mark method, in the context of C*-dynamical systems. We prove two propositions, one establishing mark manifestation infinitely many times along a given interval of the process, and, a second one, which establishes continuous manifestation of mark with the exception of a countable number of isolated points. Furthermore, we discuss how these results can be implemented in the context of the Haag–Araki theories of relativistic quantum fields on Minkowski spacetime.

Appendix: On Analytic Elements for the One-Parameter Group of Time Translations in Local Quantum Physics

Given a $\sigma (X;F)$-continuous group of isometries of a complex Banach space X, an element $A\in X$ is said to be analytic for $\tau$ ( $\tau$-analytic) if there exist a strip

$$I_{\lambda }=\left\{ z\in {\mathbb {C}}:\mid Imz\mid <\lambda \right\}$$

in ${\mathbb {C}}$, and a function $f_{A}:I_{\lambda }\rightarrow X$ such that

(i) $f_{A}(t)=\tau _{t}(A)$ for $t\in {\mathbb {R}},$

(ii) $z\mapsto \eta (f_{A}(z))$ is analytic for all $\eta \in F$ and $z\in I_{\lambda }$.

On these conditions, $f_{A}(z)=\tau _{z}(A),\,z\in I_{\lambda }$ is an X-valued analytic function; if $\lambda =\infty$ then$f_{A}$ is entirely analytic [28, Def. 2.5.20, p. 99]. Moreover, condition (ii) is equivalent to the existence of the limit

$$\underset{h\rightarrow 0}{lim}\,\frac{f_{A}(z+h)-f_{A}(z)}{h}$$

in norm in X for $z\in I_{\lambda }$ as h tends to zero in ${\mathbb {C}}$, [28, Prop. 2.5.21, p. 99].

The groups of isometries we consider in this paper are $\sigma ({\mathcal {A}},{\mathcal {A}}^{*})$-continuous i.e. for all $A\in {\mathcal {A}},\,t\mapsto \eta (\tau _{t}(A))$ is continuous for every $\eta \in {\mathcal {A}}^{*},$which amounts to the requirement that $t\mapsto \tau _{t}(A)$ be continuous in norm for every $A\in \mathcal {A}$ where ${\mathcal {A}}$ is a von Neumann algebra.

Lemma

The sum and the product of two analytic elements for a group of automorphisms $\tau$ in a von Neumann algebra ${\mathcal {A}}$ is a $\tau$-analytic element.

Proof

Consider two $\tau$-analytic elements $A,B\in {\mathcal {A}}$ and let $f_{A}(z)=\tau _{z}(A),\,z\in I_{\lambda _{1}}$, $f_{B}(z)=\tau _{z}(B),\,z\in I_{\lambda _{2}}$ for $I_{\lambda _{i}}=\left\{ z\in {\mathbb {C}}:\mid Imz\mid <\lambda _{i}\right\} ,i=1,2$ be the two corresponding ${\mathcal {A}}$-valued functions that satisfy conditions (i) and (ii). To show that the product AB is $\tau$-analytic we should prove that the ${\mathcal {A}}$-valued function $f_{AB}:I_{\lambda }\rightarrow {\mathcal {A}},\;f_{AB}(z)=f_{A}(z)f_{B}(z)$ which extends the function $f_{AB}(t)=f_{A}(t)f_{B}(t)=\tau _{t}(A)\tau _{t}(B)=\tau _{t}(AB)$ for $t\in {\mathbb {R}},$satisfies the condition of analyticity:

$$\underset{h\rightarrow 0}{lim}\,\frac{f_{AB}(z+h)-f_{AB}(z)}{h}$$

in $I_{\lambda }=\left\{ z\in {\mathbb {C}}:\mid Imz\mid <\lambda \right\} ,\lambda =min\left\{ \lambda _{1},\lambda _{2}\right\}$. But

$$\begin{aligned}&\frac{f_{AB}(z+h)-f_{AB}(z)}{h}=\frac{f_{A}(z+h)f_{B}(z+h)-f_{A}(z)f_{B}(z)}{h}\\&\quad =\frac{f_{A}(z+h)-f_{A}(z)}{h}f_{B}(z+h)+f_{A}(z)\frac{f_{B}(z+h)-f_{B}(z)}{h}. \end{aligned}$$

Since $f_{A}$and $f_{B}$ satisfy the analyticity condition, which implies also the continuity of $f_{A}$and $f_{B}$, each factor in the last expression in the above equality has a well-defined limit for $h\rightarrow 0$ and the analyticity condition for $f_{AB}$ is satisfied, for every $z\in I_{\lambda }$. Hence, AB is a $\tau$-analytic element of ${\mathcal {A}}.$

The case of the sum of two $\tau$-analytic elements can be proven along the same line of reasoning by a simple direct calculation. $\square$

To obtain a family of analytic elements for the group of time translations one may begin with any element localized in some open bounded region O of Minkowski spacetime, $A\in {\mathcal {R}}(O)$, and smear its time translation over ${\mathbb {R}}$ with a Gaussian function depending on a parameter $n\in {\mathbb {N}}.$ Namely,

$$A_{n}=\sqrt{\frac{n}{\pi }}\int _{-\infty }^{+\infty }e^{-nt^{2}}\tau _{t}(A)dt,\quad n\in {\mathbb {N}}.$$

Each $A_{n}$ is an entire analytic element for $\tau$ and the family $\left\{ A_{n}\right\} _{n\in {\mathbb {N}}}$ converges to A in the $\sigma (X;F)$ topology. Moreover, in [28, Cor. 2.5.23, p. 101] it is shown that the set of entire analytic elements form $\sigma ({\mathcal {R}};{\mathcal {R}}^{*})$- continuous group of isometries form a norm-dense subset of ${\mathcal {R}}.$

Fact

(Fredenhagen, 2019)¹⁰ In a Haag–Araki theory of relativistic quantum fields on Minkowski spacetime, no non-trivial analytic elements for time translations can be localized in open bounded regions of Minkowski spacetime.

Proof

Let A be analytic for time translations element which is also localized in a bounded region O. Let B be localized in the spacelike complement of an $\varepsilon$-neighbourhood of O,  for some $\varepsilon >0$. Then the commutator,

$$\left[ \tau _{t}(A),B\right]$$

vanishes in an open interval of t. For $\left| t\right| <\varepsilon$ we have

$$\left\langle B^{*}\varOmega ,\tau _{t}(A)\varOmega \right\rangle =\left\langle \tau _{t}(A^{*})\varOmega ,B\varOmega \right\rangle ,$$

where $\varOmega \in {\mathcal {H}}$ is the vacuum vector.

By the spectrum condition, which demands the generator of the time translations to be positive, the term on the left hand side defines a function that is analytic in the upper halfplane while the term on the right hand side, an analytic function in the lower halfplane. By assumption, they are also analytic in a strip around the real axis. Since these analytic functions coincide in an open interval of the real axis, we obtain an entire bounded analytic function which, therefore, is constant.

By the Reeh–Schlieder Theorem, these observables B generate from the vacuum $\varOmega$ a dense set of vectors in the Hilbert space ${\mathcal {H}}$. Thus, $A\varOmega$ is invariant under time evolution and, due to the uniqueness of the vacuum,

$$A\varOmega =\left\langle \varOmega ,A\varOmega \right\rangle \varOmega .$$

Thereby, we get the result that an analytic local observable is a multiple of the identity of the algebra. $\square$