Contents

# Contents

## Idea

A basic characteristic of physics in the context of special relativity and general relativity is that causal influences on a Lorentzian manifold spacetime propagate in timelike or lightlike directions but not spacelike.

The fact that any two spacelike-separated regions of spacetime thus behave like independent subsystems is called causal locality or, with a slightly stronger technical definition, Einstein causality.

(One sometimes sees a further criterion to causality, that the causal influences in timelike and lightlike directions only propagate into the future, but this is not so simply dealt with; it probably only makes sense as a statement about coarse-grained entropy in statistical physics.)

From (Grigor’ev 197x):

Microcausality condition

a requirement that the causality condition (which states that cause must precede effect) be satisfied down to an arbitrarily small distance and time interval. The microcausality condition usually refers to distances ≲ $10^{-16}$ cm and to times ≲ $10^{-24}$ sec.

It is shown in the theory of relativity that the assumption of the existence of physical signals that propagate with a velocity greater than the velocity of light leads to violation of the causality requirement. Thus, the microcausality condition prohibits the propagation of signals at a velocity greater than the velocity of light “in the small”.

In quantum theory, where operators correspond to physical quantities, the microcausality condition requires the interchangeability of any operators that pertain to two points of space-time if these points cannot be linked by a light signal. This interchangeability means that the physical quantities to which these operators correspond can be precisely determined independently and simultaneously. The microcausality condition is important in quantum field theory, especially in the dispersion and axiomatic approaches; these approaches are not based on specific model concepts of interaction and therefore can be used for direct verification of the microcausality condition. In the most highly developed branch of quantum field theoryquantum electrodynamics — the microcausality condition has been experimentally verified for distances ≲ $10^{-15}$ cm (and, correspondingly, for times ≲ $10^{-25}$ sec).

The violation of the microcausality condition would make it necessary to radically alter the method of describing physical processes and to reject the dynamic description used in modern theories, in which the state of a physical system at a given moment of time (the effect) is determined by the states of the system at preceding times (the cause).

Notice that $10^{-15}$cm $= 10^{-17}m = 10^{-2}$fm and that the (charge) radius of the proton is about 0.8 fm. So the bound cited by (Grigor’ev 197x) in the above quote is about 1/100 the diameter of a proton.

It seems that Grigor’ev 197x just cited the length scale resolution of particle accelerators at that time. More recently, the LHC (see there) probes scales $\simeq 10^{-20}m$.

## In algebraic quantum field theory

In algebraic quantum field theory causal locality is formalized as follows. This is a key statement in the Haag-Kastler axioms on causally local nets of quantum observables:

###### Definition

(causal locality of algebras of observables)

A co-presheaf of algebras of quantum observables $\mathcal{A}$ on some spacetime is causally local if the algebras $\mathcal{A}(\mathcal{O})$ localized in spacelike separated spacetime regions $\mathcal{O}$ commute with each other (inside any of the algebras of observables localized in the causal closure $\mathcal{O}$ of the union of the two spacetime regions).

$\left( \mathcal{O}_1 \;\text{spacelike separated from} \; \mathcal{O}2 \right) \;\Rightarrow\; \left( [\mathcal{A}(\mathcal{O}_1), \mathcal{A}(\mathcal{O}_2)] \;=\; 0 \;\;\;\; \in \mathcal{A}( \mathcal{O} ) \right) \,.$

Under Wick rotation, this causal locality becomes “statistical locality” (see at Osterwalder-Schrader theorem).

In perturbative algebraic quantum field theory this condition follows from the causal additivity of the S-matrix (see there the section Causal locality and Quantum obsrvables).

There are variants that one may consider:

###### Definition

(strong causal locality of algebras of observables)

A local net of quantum observables is strongly causally local if it is causally local in that algebras $A_1 = A(O_1)$ and $A_2 = A(O_2)$ associated with spacelike separated regions commute with each other, and in addition for all commutative subalgebras $C_1 \subset A_1$ and $C_2 \subset A_2$ the algebra $C_1 \vee C_2 \subset A(O_1 \vee O_2)$ satisfies

1. $(C_1 \vee C_2) \cap A_1 = C_1$

2. $(C_1 \vee C_2) \cap A_2 = C_2$.

This is (Nuiten 11, def. 14).

There have been various proposals to understand these conditions from other principles:

1. In (Schreiber 09) the condition is related to n-functoriality of a corresponding (Schrödinger picture) functorial quantum field theory.

2. In (Nuiten 11, theorem 4.2) the condition us shown to be implied by the associated pre-sheaf of Bohr toposes satisfying spatial descent by local geometric morphisms.

3. In (Brunetti-Fredenhagen-Imani-Rejzner 12) condition def. is shown to be equivalent to the co-presheaf of observables being a monoidal functor is a suitable way.

## In S-matrix theories and string theory

In S-matrix theories causal additivity is meant to also be incarnated in terms of analyticity properties of the scattering amplitudes (for this reason one often speaks of “the analytic S-matrix”).

One S-matrix theory is perturbative string theory. Discussion of causality in string theory includes Martinec 95 and (Erler-Gross 04). The latter write in their introduction:

Perhaps then it comes as a surprise that critical string theory produces an analytic S-matrix consistent with macroscopic causality. In absence of any other known theoretical mechanism which might explain this, despite appearances one is lead to believe that string interactions must be, in some sense, local.

and

We find that string theory avoids problems with nonlocality in a surprising way. In particular, we find that the Witten vertex is “local enough” to allow for a nonsingular description of the theory which is completely local along a single null direction.

and

unlike lightcone string field theory, it is clear that cubic string field theory at least has a local limit where all spacetime coordinates are taken to the midpoint. We investigate this limit with a careful choice of regulator and show that at any stage the theory is nonsingular but arbitrarily close to being local and manifestly causal. We believe that the existence of this limit, though singular, must account for the macroscopic causality of the string S-matrix. Thus, string theory is local enough to avoid the inconsistencies of a theory which is acausal and nonlocal in time, but is nonlocal enough to make string theory different from quantum field theory

Then they comment on Martinec’s account above, and other’s, by saying:

To motivate our particular perspective, it seems appropriate to discuss earlier attempts to understand the role of locality, causality and time in string theory, and explain why we feel these approaches do not adequately address the problems just raised.

## References

### In local quantum field theory

• V. I. Grigor’ev, Microcausality condition, The Great Soviet Encyclopedia, 3rd Edition (1970-1979) (web)

• Jessey Wright, Quantum field theory: Motivating the Axiom of Micorcausality, PhD thesis 2012 (pdf)

• Anthony Duncan, The Conceptual Framework of Quantum Field Theory – Dynamics IV: Aspects of locality: clustering, microcausality, and analyticity, Oxford Scholarship Online (web)

• John Bell, The theory of local beables (1975) (pdf)

• pdf

### In algebraic quantum field theory

For references on the traditional discussion in AQFT see at Haag-Kastler axioms and at causally local net of observables. Proposals to understand the causal locality axiom from other principles include

### In S-matrix theories and string theory

Discussion of causality in string theory includes the following:

A brief look at the causality property of the string 2-point function is in

• Emil Martinec, Strings and Causality (pdf) in L. Baulieu, V. Dotsenko, V. Kazakov, P. Windey (eds.) Quantum Field Theory and String Theory , NATO ASI Series B: Physics Vol. 328 (1995)

An in-depth discussion of causality of the string scattering S-matrix via open string field theory is in

This rediscovered some facts that had earlier been noticed in

• M. Maeno, Canonical quantization of Witten’s string field theory using midpoint light-cone time, Phys. Rev. D43 no. 12 (1991).