Locality

Prerequisites: Special Relativity. Full path (6 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

Locality is a fundamental concept in modern physics and means that object can only be influenced by their immediate surroundings, i.e. locally.

For example, a billiard ball only starts rolling when it gets hit with something. It doesn't start rolling around magically.

This concept is made even more precise through special relativity. The fundamental fact at the heart of special relativity is that nothing can move with a velocity faster than the speed of light. This means tells us immediately how far an object can be away to influence another one since the effect always needs time to travel to the object in question.

Concrete

Local field theories. In a classical field theory, locality means that the action is an integral over spacetime of a Lagrangian density that depends on the fields and finitely many of their derivatives at the same point:

$$S[\phi] = \int d^4x\; \mathcal L\big(\phi(x), \partial_\mu\phi(x)\big) .$$

The Euler-Lagrange equations are then partial differential equations. They relate the field at a point only to the field and its derivatives in an infinitesimal neighbourhood. A term like

$$\int d^4x\, d^4y\; \phi(x)\, K(x - y)\, \phi(y)$$

with a kernel $K$ that does not vanish for $x \neq y$ couples distant points directly and is nonlocal.

Action at a distance versus propagation. In Newtonian gravity the potential obeys the Poisson equation $\nabla^2\Phi = 4\pi G\rho$. Its solution

$$\Phi(\vec x, t) = -G\int d^3y\, \frac{\rho(\vec y, t)}{|\vec x - \vec y|}$$

responds at the same instant $t$ to a change of $\rho$ anywhere in space. In electrodynamics in the Lorenz gauge, the potentials obey wave equations, $\Box A^\mu = \mu_0 J^\mu$ with $\Box = \frac{1}{c^2}\partial_t^2 - \nabla^2$, and the physical (retarded) solution is

$$A^\mu(\vec x, t) = \frac{\mu_0}{4\pi} \int d^3y\, \frac{J^\mu\big(\vec y,\, t - |\vec x - \vec y|/c\big)}{|\vec x - \vec y|} .$$

A change of the source at $\vec y$ reaches $\vec x$ only after the time $|\vec x - \vec y|/c$. More generally, for hyperbolic equations whose characteristic cones are the light cones, the solution at a point depends only on data in its past light cone. This is the precise form of relativistic causality for classical fields.

Locality in quantum field theory. In quantum field theory locality is expressed by microcausality: observables at spacelike separated points commute,

$$[\mathcal O_1(x), \mathcal O_2(y)] = 0 \qquad \text{for } (x - y)^2 < 0 \quad \text{(signature } (+,-,-,-)\text{)} .$$

For a free real scalar field one can check this directly: $[\phi(x), \phi(y)]$ is a c-number that vanishes outside the light cone. Fermionic fields anticommute at spacelike separation instead; observables are built from even products of them and do commute. Microcausality implies that no operation in one region can change the measurement statistics in a spacelike separated region, so no signal travels faster than light.

Locality and Bell's theorem. The word is also used for a different property. In a local hidden variable model, the probability of each outcome depends only on the setting of the local apparatus and on variables shared by the particles at their common source. Quantum mechanics violates the Bell inequalities that such models obey, as confirmed in experiments with entangled particles. This does not conflict with microcausality: the correlations between distant outcomes cannot be used to send signals.

Abstract

Q:Locality principle - What does it mean physically?

A: Global field configurations have to be obtained by summing up local data. That means by gluing local field configurations together.

Q: What does it mean mathematically?

A: This is captured by the theory of sheaves.

https://ncatlab.org/schreiber/files/EggertsonGaugeAndGroupoids14.pdf

Path to this page

1. Calculus
2. Linear Algebra
3. Newton's Second Law
4. Vector Calculus
5. Newtonian Mechanics
6. Special Relativity