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The Causal Axioms of Algebraic Quantum Field Theory: A Diagnostic

Francisco Calderón

Abstract

Algebraic quantum field theory (AQFT) puts forward three "causal axioms" that aim to characterize the theory as one that implements relativistic causation: the spectrum condition, microcausality, and primitive causality. In this paper, I aim to show, in a minimally technical way, that none of them fully explains the notion of causation appropriate for AQFT because they only capture some of the desiderata for relativistic causation I state or because it is often unclear how each axiom implements its respective desideratum. After this diagnostic, I will show that a fourth condition, local primitive causality (LPC), fully characterizes relativistic causation in the sense of fulfilling all the relevant desiderata. However, it only encompasses the virtues of the other axioms because it is implied by them, as I will show from a construction by Haag and Schroer (1962). Since the conjunction of the three causal axioms implies LPC and other important results in QFT that LPC does not imply, and since LPC helps clarify some of the shortcomings of the three axioms, I advocate for a holistic interpretation of how the axioms characterize the causal structure of AQFT against the strategy in the literature to rivalize the axioms and privilege one among them.

The Causal Axioms of Algebraic Quantum Field Theory: A Diagnostic

Abstract

Algebraic quantum field theory (AQFT) puts forward three "causal axioms" that aim to characterize the theory as one that implements relativistic causation: the spectrum condition, microcausality, and primitive causality. In this paper, I aim to show, in a minimally technical way, that none of them fully explains the notion of causation appropriate for AQFT because they only capture some of the desiderata for relativistic causation I state or because it is often unclear how each axiom implements its respective desideratum. After this diagnostic, I will show that a fourth condition, local primitive causality (LPC), fully characterizes relativistic causation in the sense of fulfilling all the relevant desiderata. However, it only encompasses the virtues of the other axioms because it is implied by them, as I will show from a construction by Haag and Schroer (1962). Since the conjunction of the three causal axioms implies LPC and other important results in QFT that LPC does not imply, and since LPC helps clarify some of the shortcomings of the three axioms, I advocate for a holistic interpretation of how the axioms characterize the causal structure of AQFT against the strategy in the literature to rivalize the axioms and privilege one among them.
Paper Structure (15 sections, 8 equations, 6 figures)

This paper contains 15 sections, 8 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Four desiderata for relativistic causation
  3. Symptomatology
  4. The Spectrum Condition
  5. Microcausality
  6. Separability/Independence conditions
  7. MC and causation
  8. Primitive Causality
  9. Taking stock: against Atomistic
  10. LPC as a defense of Holistic
  11. LPC and determinism
  12. LPC and superluminal signaling: The Fermi's Two-Atom System
  13. Further advantages of LPC
  14. Holistic redux: LPC is implied by the causal axioms
  15. Outlook

Figures (6)

  • Figure 1: ${\mathcal{O}}_1$ and ${\mathcal{O}}_3$ are spacelike separated.
  • Figure 2: ${\mathcal{O}}_2$ depends causally on ${\mathcal{O}}_1$.
  • Figure 3: Fermi's two-atom system.
  • Figure 4: The region shaded with slanted lines is $\mathcal{C}$. The region shaded with vertical lines is $\mathcal{C}'$, the causal complement of $\mathcal{C}$. The diamond $\mathcal{D}$, containing $\mathcal{C}$, is the causal complement of $\mathcal{C}'$. The red dotted lines extend spatially to form a time slice of the temporal size of $\mathcal{C}$. Call $\mathcal{C}_r$ the two caps outside of $\mathcal{C}$ that extend in the spatial direction.
  • Figure 5: Causal diamond.
  • ...and 1 more figures

Theorems & Definitions (3)

  • proof
  • proof
  • proof