# Causality in quantum field theory with particles going faster than light

So I have read Feynman's QED, and it seemed to me from this that, if we are to build any sort of semi-intuitive picture of what particles might be doing, we have to account for particles going faster than light in order to get the right results in experiments. Indeed, in the book we had to account for paths in which particles went back in time and appeared as their antiparticles (although it was never clear to me if this is the only way we observe 'antiparticles'- if these are necessarily particles going back in time.

But anyway, with the mention of particles going faster than light I started to think about what this theory had to say about causality. But looking up 'quantum field theory and causality' just yielded many results referring to the commutator. Having read posts like this, I understand how the vanishing commutator translates to a statement about causality, but this is just the mathematics! Usually I am all about the math and don't bother too much about physical pictures, but in this case the commutator 'explanation' really settles nothing in my mind. I am still utterly confused as to how this theory, in which we integrate over paths involving particles travelling faster than light and back in time, in order to get results that agree with experiment, doesn't suggest that causality would be violated.

Despite Quantum field theories(the relativistic kind) incorporating special relativity, the picture we have of it involves particles going faster than light? Is causality perhaps not violated because this faster-than-light travel does not carry information (as in the case of quantum entanglement)? But surely it must do if it affects the experimental results! Perhaps does the commutator argument somehow translate to a physical interpretation in our picture that would settle what seems to me to be a discrepancy?

"...Indeed, in the book we had to account for paths in which particles went back in time and appeared as their antiparticles (although it was never clear to me if this is the only way we observe 'antiparticles'- if these are necessarily particles going back in time..."

These are just pretty words about the interpretation of the antiparticle; these sentences are just a prose describing formal interpretation of the antiparticle (creation) operator as the one in the front of the "going backward in time" plane wave $e^{-ip^{\mu}x_{\mu}}$ in the expansion of the quantum field $\psi$ to which the particle and antiparticle correspond.

In reality, the antiparticle exists as well as its own particle, and no "travel backward in time" for its observation is needed. It only differs from the particle by the signs of all of the possible charges. And it definitely can be observed by various experiments; for example, in the proton beta decay we can observe the positron - the antiparticle to the electron. Moreover, the existence of antiparticle is in fact strictly required by the Lorentz invariance of the quantum theory.

"...Usually I am all about the math and don't bother too much about physical pictures, but in this case the commutator 'explanation' really settles nothing in my mind..."

The heart of this "commutator derivation" is the requirement that two measurements, which have been done at space-like separated points, doesn't affect each other; by the other words, two measurements at space-like separated points can be done simultaneously and independently on each other. This is just the precise statement of the causality principle, and this is very physical.

The "commutator derivation" and, in its turn, the causality principle, actually follows from the poincare invariance of the quantum theory. The statement follows from the definition of $S$-operator, which gives the amplitude of the transition $|\alpha\rangle \to |\beta\rangle$ between the states $\alpha$ and $\beta$: $$\hat{S} \equiv \sum_{i = 0}^{\infty}\frac{(-i)^{n}}{n!}\int d^{4}x_{1}...d^{4}x_{n}\hat{T}(\hat{H}_{\text{int}}(x_{1})...\hat{H}_{\text{int}}(x_{n}))$$ Here $\hat{T}$ is the chronological ordering and $\hat{H}_{\text{int}}(x)$ is the hamiltonian operator. It's not hard to realize that the poincare invariance of the $S$-operator leads to the fact that $$\tag 1 [\hat{H}_{\text{int}}(x),\hat{H}_{\text{int}}(y)] = 0 \quad \text{if} \quad (x-y)^{2}<0$$ The reason is just the fact that the chronological ordering for two points $x^{\mu}, y^{\mu}$ is poincare invariant only in the case of time-like or isotropic intervals. In the case of space-like intervals, it is not poincare invariant, and we must require $(1)$.

"...Despite Quantum field theories(the relativistic kind) incorporating special relativity, the picture we have of it involves particles going faster than light?.."

Definitely no. The formalism of the QFT is in particular based on the unitarity and existence of the stable vacuum. As long as tachyons are present in the theory, none of these principles is fulfilled.