---------------------------------------------------
S5g. Why locality and causal commutation relations?
---------------------------------------------------
In measurement terms, locality is the idea that a measurement here
and a simultaneous measurement there can be performed independently,
and in particular don't limit each other in precision. This is encoded
in the requirement that 'local' quantities described by fields
Phi_a(\x,t) here (at \x) and fields Phi_b(\y,t) there (at \y)
commute if the positions \x and \y are distinct.
The covariant form of this locality requirement is that,
with x=(ct,\x) and the +--- norm defined by x^2=x_0^2-\x^2,
[Phi_a(x),Phi_b(y)]=0 if (x-y)^2<0 (*)
Indeed, if x_0=y_0=ct then (x-y)^2=(x_0-y_0)^2-(\x-\y)^2=-(\x-\y)^2<0,
so this commutation relation holds at equal time. But then Lorentz
covariance implies that it must hold whenever (x-y)^2<0, since any
pair (x,y) with (x-y)^2<0 can be transformed into an equal time pair.
Thus locality is a property of distinguished fields satisfying (*),
called local fields. This property is completely independent of states,
since it is understood that the property holds independent of the
coincidental properties of the state.
Quantum field theory is physics in the Heisenberg picture, with
states fixed once and for all, and all spacetime dependence in
the fields. The universe is in a definite though largely unknown state,
and apart from the Lagrangian of the standard model plus gravitation,
all the history, present and future of the universe is encoded in
this universal state.
Lacking knowledge of this state, physicists are usually
contend with describing tiny portions of this state, namely the
restriction of the state to a subalgebra of accessible quantities
within the lab (or at least close to the solar system).
Since there are many such subsystems of interest, and all these
are in different states even if described by the same algebra
(more precisely by isomorphic ones), all generic properties of
physical systems must be independent of the states.