Natural Science Forum / Physics / Research / January 2005

When I think of locality, what comes to mind is this thing is here and
that thing is there, etc. . This is my naive expectation of locality.
If a photon passed through a slit at time t, it means the photon was
localized at the slit up to the radius of the slit. However, a
relativistic particle can only be localized in a Lorentz covariant
manner in a certain region at a particular time up to its Compton
wavelength. For a photon, this means there is no Lorentz covariant
localization. There is no contradiction with the example of the slit
because the slit only localizes in two dimensions and if the photon is
a wave packet in the third dimension, the resolution is the same as the
pole-in-the-barn "paradox".

But let's take another example. Suppose now that the photon gets
absorbed by a photographic plate or some other detector with a fine
resolution. Because the Compton wavelength of the molecules/atoms
making up the detector is much smaller, it might appear as if the
photon has been localized in a Lorentz covariant manner. However, this
overlooks the entanglement of the photon, or more precisely, the
electromagnetic field with the detector (and also us, since we notice
where the spot on the photographic plate is).

There is also the Reeh-Schlieder theorem which states that any state in
the vacuum superselection sector can be arbitrarily approximated by
fields localized in ANY region. But that's really not all that strange.
We have the same thing with any fully entangled state. Suppose a system
decomposes into subsystem A and subsystem B and we have a state which
is fully entangled between both of them. This is only possible if the
dimension of the Hilbert space describing A is the same as the one
describing B. Then, by applying an operator solely on A, it is possible
to obtain any other state of the system. But this is nothing other than
a consequence of quantum entanglement, not nonlocality. Since the
vacuum, which is by definition the lowest energy state is fully
entangled, this explains the Reeh-Schlieder property. Even for
classical systems, minimizing the energy often leads to long distance
correlations. And as cosmologists often point out, the horizon problem
is "unnatural" and needs to be explained by some mechanism like
inflation or something else. To put it more plainly, we do not live in
the same superselection sector as the vacuum and long distance
correlations only extend up to the inflated radius and more
importantly, the kind of quantum entanglement needed for the
Reeh-Schlieder theorem is far weaker than the large scale temperature
and density correlations and can be explained by a prior causal
contact.

But this brings me to my question. What exactly do we mean by locality?
In quantum field theory, locality doesn't apply to states, but instead,
to the operators (fields) acting upon the states. This most definitely
does not agree with what we expect of locality, of what we think
locality is. If we interprete the formalism of quantum field theory
literally, it means we can't say "this thing is here", but only "the
operator measuring the existence of this thing here is localized here".