Causal fermion systems and the ETH approach to quantum theory

Abstract

After reviewing the theory of "causal fermion systems" (CFS theory) and the "Events, Trees, and Histories Approach" to quantum theory (ETH approach), we compare some of the mathematical structures underlying these two general frameworks and discuss similarities and differences. For causal fermion systems, we introduce future algebras based on causal relations inherent to a causal fermion system. These algebras are analogous to the algebras previously introduced in the ETH approach. We then show that the spacetime points of a causal fermion system have properties similar to those of "events", as defined in the ETH approach. Our discussion is underpinned by a survey of results on causal fermion systems describing Minkowski space that show that an operator representing a spacetime point commutes with the algebra in its causal future, up to tiny corrections that depend on a regularization length.

No Events on Closed Causal Curves

“…In the ETH approach to QT E x Ă E y is compatible with the statement that x is in the future of y. The second motivation comes from a recent paper [5] comparing the ETH approach to QT with the Causal Fermion Systems theory (see [1] for an introduction). In the CFS setting the relevant future algebras are constructed explicitly from the future light cone.…”

“…Here we only collect those structures and definitions of the ETH approach that are relevant to the argument in the later sections. For a reader unfamiliar with the approach the recent review by Fröhlich [6] or the discussion in [5] is a good place to start. For technical details on the non-relativistic formulation see [8].…”

“…For technical details on the non-relativistic formulation see [8]. Here, we limit the discussion to the relativistic formulation and closely follow [7] and [5] but restricting the exposition to a minimum of what is needed to understand the arguments in the present paper. We will use the occasion to point out similarities between the relativistic formulation of the ETH approach and the theory of "Causal Sets" [4] and use this to argue that the full QT prohibits the existence of closed causal chains.…”

The Time-Evolution of States in Quantum Mechanics according to the ETH-Approach

“…An important question to be answered is whether the space-time localization of projections describing actual events can be determined more precisely than has been possible, so far. Some ideas discussed in [47] may be relevant in this connection. 3.…”

Local Algebras for Causal Fermion Systems in Minkowski Space

“…This estimate is motivated by the notion of "events" in the ETH formulation of quantum theory [10,11]. This connection is explored further in [4].…”