# What is observable in a relativistic quantum field theory?

2019 Award
There are two distinct ways to look at relativistic quantum theory of systems with variable number of particles (aka QFT).
Since you are promoting here your deviating version of QED rather than discussing the topic of the thread, I replied in post #12 of your own thread https://www.physicsforums.com/showthread.php?p=3155870

2019 Award
the question to be discussed is; Which fields (or other observables) represent the operational content of relativistic quantum field theory?

It will take some time till I have prepared the proper background for a full discussion.
Ok, now I have made up my mind how to discuss this in a fruitful way. I think that one needs to begin with classical field theory, since there certain problems are still ablsent, and one can better see the impact of relativistic thinking.

So let me begin with answering the question: What is the operational content of classical electrodynamics?

In a region Omega without charges or currents, such as in a vacuum, Maxwell's equations read
$$\nabla \cdot E (t,x)= 0,~~~ \nabla \cdot B(t,x)= 0,$$
$$\nabla \times E (t,x)= - \partial_t B(t,x),~~~ \nabla \times B(t,x) = c^{-2} \partial_t E(t,x).$$
Experimentally, one can check the values of the field by measuring the Lorentz force
$$F(t,x)=q(E(t,x)+v \times B(t,x))$$
that the electromagnetic field exerts at time t on a particle with charge q and velocity v at position x. Assuming that the fields don't vary too much in space and time, such tests allow one to determined the field everywhere in Omega to a certain accuracy. High frequency behavior cannot be tested in that way, but needs the methods of quantum optics. In that case, averages of products of the fluctuating part of the fields, collected into coherence matrices (that may depend on one or two position arguments), can be measured using optical tools.

We conclude that both the components of the fields, their arguments of the fields have an immediate, observer-independent physical meaning.

The Maxwell equations in vacuum are Poincare covariant, having the standard transformation behavior under translations, rotations, and Lorentz boosts. But things may look different when considered by different observers in their particular decomposition into space and time. The Lorentz transformations that mediate between observers in different Lorentz frames cause the standard effects of relativity when different observers look at the same objective situation.

There is an observer-dependent decomposition of space-time position x into a time component t=c^{-1} u dot x -- where c is the speed of light, u=(u_0,\u) is the 4-velocity of the observer, a future-pointing unit vector in the Minkowski inner product, u^2 = u_0^2-\u^2 --, and a 3-dimensional space orthogonal to u and passing through the observer's 4-position. In particular, in coordinates where the observer is at rest, \u=0, u_0=1, the time coordinate is t=x_0/c, and x=(ct,\x) with \x labeling the space coordinates.

This decomposition is convenient for our non-relativistic intuition. But this decomposition has no operational meaning at all, not even for the observer itself.

Indeed, an observer at rest in the origin of its rest frame cannot know at time t anything about the world in the present, the instantaneous space slice consisting of all x=(x_0,\x) with fixed x_0=ct -- except what happens at the origin itself. The reason is that relativity forbids the communication of information at a speed >c, so that whatever information an observer may have recorded at time t in its memory must be due to events happening in the past causal cone, consisting of all x=(x_0,\x) with x_0<=ct-|\x|. (For example, we don't see what happens at the sun now, only what happened 8 minutes ago.)

Similarly, an observer at rest in the origin of its rest frame cannot influence at time t anything about the world in the present -- except what happens at the origin itself. The reason is that relativity forbids the propagation of influences at a speed >c, so that whatever an observer does at time t can affect only events in the future causal cone, consisting of all x=(x_0,\x) with x_0>=ct+|\x|.

Thus the present is completely inaccessible and completely uncapable of being influenced by the observer, except at the origin. In particular, measurements can be made only at the origin itself, and learning about measurements made by others now is possible only in the future. Moreover, since space is unbounded but the intersection of any space slice with the past causal cone, one never can learn _everything_ about any particular time in the past.

We conclude that the local observer frames, and the associated splitting of space-time into slices of 3-spaces at fixed time, have no observational relevance and are spurious remnants of a nonrelativistic view of a relativistic theory. They become relevant only in as far one wants to make a nonrelativistic approximation and a corresponding expansion in c^{-1}.

This completes my exposition of the meaning of classical fields and their arguments.

Consequences for quantum fields will be drawn in a later post.