Locality

Core Idea

Locality is simultaneously:

1. A physical principle — the rejection of "spooky action at a distance." It is the principle that physical interactions occur only between entities that are sufficiently close to each other.
2. A mathematical structure — evolution equations utilize spatial derivatives rather than arbitrary couplings between distant points.

Physical Motivation

In classical particle mechanics, the degrees of freedom are finite and independent; there is no notion of "closeness" between them. To introduce locality and causality, we need fields (Classical Field Theory).

Coulomb's and Newton's laws imply action at a distance: the force on a particle changes instantaneously if another particle moves far away. For some people, this is philosophically unsatisfactory and experimentally incorrect. Field theory (Maxwell, Einstein) replaces action at a distance with interactions that are locally mediated by the field.

Mathematical Implementation

When we transition from a discrete system (index $i=1,\dots ,n$) to a continuous one (index $x\in \mathbb{R}$), a problem arises: how do we write the relationships between the infinite "components" $u(x)$ without losing our minds? Imposing locality provides the answer:

The evolution of $u$ at $x$ depends only on $u$ and its derivatives at that very same $x$.

This turns an otherwise intractable problem (infinite arbitrary relationships) into a structured pattern: spatial derivatives couple each point with its infinitesimal neighbors, and this pattern remains the same for all $x$.

See evolution equation#Infinite dimensional case for the transition from a discrete to a continuous index, and the discussion on "special patterns" (derivatives) as the only coherent way to couple infinite components.

Locality vs. Non-locality in Quantum Mechanics

Bell's theorem proves that no local hidden-variable theory can reproduce the predictions of quantum mechanics. Experiments force a choice:

• Non-locality (Bohm): the quantum potential connects everything instantaneously.
• Non-realism (Copenhagen): particles do not possess definite properties until they are measured.
• Many-Worlds: retains both locality and realism, but rejects uniqueness (every measurement branches the universe).

The paradox: according to Bell himself, even if we reject realism, perfect correlation demands instantaneous communication, so locality is lost anyway.

See Bell's experiment and On Many Worlds, entanglement and hidden variables.

Vault Connections

• Classical Field Theory — physical motivation: fields exist to render physical laws local.
• Classical Mechanics — why particle mechanics fails to capture locality.
• evolution equation — implementation: the continuous index and derivative-based patterns.
• evolutionary vector field — geometry: vector fields that preserve the Cartan distribution.
• Bell's experiment — quantum consequences: locality vs. realism.
• on theories, symmetries and gauge — local gauge invariance as a distinct type of locality (local symmetries).