# Three Levels Of Locality In Quantum Physics

Locality seems to be a deep principle of physics. It’s the idea that what happens at any given
place in the universe doesn’t (directly) depend on things happening in other places far away, but only on
what’s happening nearby. For instance, the lightspeed limit in relativity guarantees that if
you want to know will happen *here* in the next nanosecond, all you have to care about is
stuff happening within a light-nanosecond of here. Events farther than that are irrelevant. So
cause and effect operate locally—events can’t have an effect too far away too quickly.

Quantum physics has thrown a bit of a wrench in conventional notions of locality. Experiments on entanglement have confirmed violations of Bell inequalities, establishing that correlations exist between events happening well outside each other’s lightspeed horizons—correlations that cannot be explained on the basis of “local realism”, the idea that the outcome of a measurement depends only on what’s going on in or near the measuring device, and not on what’s going on in a widely separated location.

But this apparent failure of the principle of locality in quantum physics is very slippery. Both the general public and physicists themselves have often fallen into misunderstandings of it. For instance, it’s often thought that entanglement implies the ability to send signals faster than light, which it does not. Even people who know this fact often still think there must be some sort of “influence” (e.g. wavefunction collapse) propagating faster than light, even though things conspire to prevent us from using that influence to send signals; this is a misconception as well.

Moreover, there are other ways locality shows up, beyond the usual locality of cause and effect in ordinary space and time. In fact, there are three ways that I know of. Maybe what we need is not to throw out the principle of locality in quantum physics, but just tweak it a bit.

## Locality in Hilbert space

The first way locality shows up in quantum physics is kind of trivial. I think it’s worth mentioning for completeness, though.

The most general mathematical setting for quantum physics is a
Hilbert space—a vector space with an
inner product (a canonical notion of lengths and angles).
Each point of a Hilbert space represents a state that a quantum
system could be in at a moment in time. If we have a quantum system—anything from an electron spin
up to the entire universe—at any given moment in time it occupies some *single point* in its Hilbert
space. The Schrödinger equation tells us how that point moves around over time, i.e. how the
system evolves:
$$
\frac{d\Psi}{dt} = -\frac{i}{\hbar} H \Psi(t)
$$
Here, $\Psi(t)$ is the state—the moving point in Hilbert space—and $H$ is the Hamiltonian,
a linear operator that maps the current point to its current velocity vector (give or take some
constant factors).

So quantum physics is *local in Hilbert space*. Where the system goes next depends only on
where it is now, not where it was in the past, or what’s going on in some other part of Hilbert
space. (The physics is also completely deterministic.)

This mirrors the situation in classical mechanics, where dynamics is similarly local (and deterministic) in phase space.

But this is trivial because we’ve defined the state space to
*make* the dynamics local. If we had non-local dynamics—e.g. if a state was *several* points in
Hilbert space with each point’s motion controlled by the locations of other points, or some such—we
could just form a new, bigger state space defined so that every point consists of *all* the
information needed to define the state at a moment of time, then write a local equation of motion
for this new space.

## Locality in configuration space

In the Hilbert-space dynamics, any hermitian operator can serve as a Hamiltonian. Being hermitian
isn’t a very strong restriction—it’s like requiring a matrix to be symmetric—so there are
*lots* of possible Hamiltonians in any decent-sized Hilbert space, especially
infinite-dimensional ones. In practice, though, we don’t see arbitrary hermitian operators acting
as Hamiltonians in real-world quantum systems—we see a very specific kind of operator: ones that
are local in configuration space.

Any Hilbert space can be viewed as a space of functions from some domain to the complex numbers.
The domain over which these functions are defined is the
configuration space of a quantum system. There will be one *dimension* in the Hilbert space
for each *point* in the configuration space. So Hilbert spaces in quantum physics are usually
pretty enormous.

The configuration space as defined here isn’t unique. In fact there is at least one configuration space for
each orthonormal basis of Hilbert space. Each configuration space is what you might think of as a
“classicalized” state space of a quantum system. For instance, in the case of an electron, one
possible configuration space is the 3D space of its position in Cartesian coordinates. Another
possibility is the 3D space of its momentum; another is its position in spherical coordinates. In the
case of *two* electrons, a configuration space could be the 6D space of both their positions,
or both their momenta, or the position of one and the momentum of the other. Or the position of one
in Cartesian coordinates and of the other in spherical coordinates. You get the idea.

(More formally, a configuration space as I’ve defined it here is a set of possible outcomes of measurements of a CSCO. But we don’t need to be so technical here.)

If we pick a specific configuration space and its accompanying orthonormal basis of Hilbert space,
then the Hilbert-space dynamics—namely, the Schrödinger equation—can be “pushed down” to the configuration space, and we can
express the dynamics there as well. First, define
$$
\Psi(x, t) = \langle x | \Psi(t) \rangle
$$
On the right is the state vector in Hilbert space that we met before. We’re taking its inner
product with one of the basis vectors, labeled *x*. On the left is a new function $\Psi(x, t)$
we’re defining, which gives a complex number for each point *x* in the configuration space,
given by the inner product with the basis vector corresponding to that point. This is the wavefunction,
whose value at a point *x* is the quantum amplitude for the system to be at *x*.

Then, the Schrödinger equation can be written as $$ \frac{\partial}{\partial t} \Psi(x, t) = -\frac{i}{\hbar} (H \Psi)(x, t) $$ Before, we had the time derivative of the entire state vector in Hilbert space on the left; now we have the time derivative of the wavefunction at just one point, i.e. the component of the state along just one basis vector in Hilbert space. On the right, we have the same Hamiltonian operator, only now thought of as a linear map from the current wavefunction to the function that gives its current velocity at every point.

So far we haven’t said anything new, just re-expressed the same dynamics in a different form. The
surprise comes in what kind of operator $H$ looks like in this form. Of course it takes
different forms in different configuration spaces. But it turns out that for many quantum systems
of interest, there is a special subset of configuration spaces (position and momentum spaces), in which $H$ is a
*differential* operator—one consisting of spatial derivatives of $\Psi(x, t)$, plus maybe
some scalar factors. For instance, the Hamiltonian for a free particle in position space is
$$
H = -\frac{\hbar^2}{2m} \nabla^2
$$
Differential Hamiltonians mean the dynamics are *local in configuration space*: what happens
to the wavefunction at a particular point depends only on its current value in an infinitesimal
region around that point, but not its value in the past, or on the value of the wavefunction
anywhere else in configuration space. Nothing in the Hilbert-space dynamics requires this: it
would be perfectly possible for the Hamiltonian to turn out to be an
*integral* operator, for instance, or for it to have some other form that would make the
wavefunction’s behavior at one point depend on its values at distant points. The fact that this
*doesn’t* happen—that real Hamiltonians seem to always be local in some configuration
space—suggests to me that a deeper principle is at work.

There is a question here, though, about how special this fact really is. There are an infinity of possible configuration spaces, and in most of them the Hamiltonian won’t be local—indeed, in most of them it won’t have any simple form at all. Consider taking 3D position space and remapping it to 1D using a space-filling curve, for instance. This is another perfectly valid configuration space, but one in which the simple free-particle Hamiltonian would look totally pathological. Turning this around, is it possible that almost any arbitrary hermitian operator could be made to look local by picking the right configuration space? If so, local Hamiltonians aren’t so special after all. But I don’t know enough about functional analysis to even begin to answer this question.

## Locality in ordinary space

So far, we’ve been talking about configuration spaces. Let’s get even more specific and talk about
position spaces—the subset of configuration spaces in which the coordinates directly represent the
positions of particles in ordinary 3D space. In a one-particle system, position space *is*
just ordinary 3D space. But for a multi-particle system, position space has three dimensions *for
each particle*, since a single point in the configuration space represents a complete “classical
state” of the whole system. So locality in multi-particle position space doesn’t necessarily imply
locality in ordinary space.

This is exactly where entanglement comes into play. When two particles aren’t entangled, their wavefunction in their mutual configuration space has a distribution that can (at least approximately) be factored into two independent distributions, one for each particle. But this is a very special and unstable state, and if the particles interact, in general their wavefunction in their mutual configuration space will evolve into a shape that is not even approximately factorable. That’s entanglement.

To be a bit more concrete, consider the position space of a system containing two particles moving in 1D space. The position space is 2D, with one dimension for each particle. If the system evolves to a state where, say, there are two “blobs” of amplitude, centered around some two points $(x_1, y_1)$ and $(x_2, y_2)$, then the particles’ positions are entangled: if the first particle is found near $x_1$ then the second particle is very likely to be found near $y_1$, but not near $y_2$. This could be the case even though the first and second particles are light-years apart. The point is that although $x_1$ and $y_1$ are far apart in ordinary space, there is still a point $(x_1, y_1)$ in the configuration space, and if the wavefunction has some amplitude around there, then events around those two distant points will be correlated with each other.

If this were the whole story, we could wave goodbye to any remaining form of locality in ordinary space. Any collection of particles anywhere in the universe could be linked to each other, in the sense of being part of “the same blob” (granting that a “blob” can only be defined fuzzily at best) of amplitude in configuration space. Despite the well-behaved local dynamics in configuration space, there would be no reason to expect anything better than chaos in ordinary space.

But in fact, real-world systems don’t have such pathological behavior. We don’t
see collections of widely separated particles spontaneously entangling or disentangling themselves.
Entanglement, although it leads to non-local correlations in the future, always has a local origin:
particles have to get reasonably close to each other—close in our regular old 3D space—to actually
interact and become entangled. That, plus the speed-of-light limit that limits how fast particles
can move around in ordinary space, ensures that we *do* still have local dynamics in
ordinary space.

From the perspective of configuration space, this condition seems rather bizarre. For instance, back in our two-particle 1D world, if we require that the particles can only interact with each other when they’re close by, the interaction region forms a thin diagonal “pencil” in 2D configuration space. We’re saying that the two-particle dynamics has to factor into independent one-particle dynamics everywhere except along this pencil. Only in this one special region of configuration space are the particles actually allowed to affect each other.

Up in Hilbert space, quantum physics is local by definition; Hilbert space is the space of all possible quantum states, so only one point—the current state—matters. But we get a surprise when we find that we can come down to a configuration space and have local physics there as well. We get surprised again when we find that we can come from configuration space down to ordinary 3D space and still have a pretty strong form of locality—locality of interaction—despite the fact that long-distance entanglement violates the intuitive, classical form of locality. Each of these deeper forms of locality seems to place strong restrictions going back up to the bigger, more abstract spaces. Locality in ordinary space implies that particles can only interact in a tiny, special region of configuration space. Locality in configuration space implies that the Hamiltonian operator acting on Hilbert space has a very special form.

As far as I know, there being no *a priori* logical reason, on the basis of general quantum
theory, why these restrictions should exist. The origin of locality is still mysterious, but it
seems to be a deep principle of reality.