The Causality in QFT is said not be violated by showing field operators in Klein Gordon field commute outside the light cone. This is necessary because If I here measure a observable A and my friend there at mars measure observable B that does not commute with A then he can influence my measurement by just measuring his observable. This is why operators defined at space separated points must commute. This is what has been described by Sidney Coleman in his QFT lectures. However, I am worried about the non-zero propagator i.e probability amplitude for particle to propagate from $y$ to $x$ and it does not vanish outside the light cone. This no zero probability amplitude means particle created at space point can be detected outside the light cone. Won't it violate causality. This is often stressed in QFT books that concept of antiparticle is required to preserve causality in QFT. I don't understand it. Moreover, It is also mention that the propagator $G(x1,x2)$ for time like separated events, give probability amplitude for antiparticle to travel from x1 to x2 if t2>t1 and probability of particle to travel from x2 to x1 if t1>t2. I want to know what actually propagates? Does the sign of the propagator changes particle or antiparticle to be created? I am asking this because let say I have a some kind of particle creator, will it create particle and antiparticle both, particle in future and antiparticle in past? Moreover it said that for space like separated points amplitude for particle to travel from x1 to x2 cancels probability for antiparticle to travel from x2 to x1 and they cancels each other and causality is preserved. No books seems to discuss this good way in intuitive way
1 Answers
Many of the quantities calculated in QFT are analogous to ones considered in ordinary QM, such as probability amplitudes, however, the interpretation of these quantities are notably different in the respective theories.
In ordinary QM one thinks about particles as being localized in some region of space, however, within the framework of QFT one should abandon this notion and think instead of "fields" distributed throughout space-time. For example, the excitations of a quantum field, i.e. $|\vec p\rangle$ yield the quantity $\langle 0|\phi(\vec x)|\vec p\rangle$, which can be understood as a wave function for the momentum eigenstate. Although one might try to imagine this as a particle at $\vec x$ (cf Peskin and Schroeder, 24) one must remember that no real particle has a definite momentum and position; such a notion violates the uncertainty principle. The only real physical notion we have is the projection of the field operator onto the Fock space: $$\phi(\vec x)|0\rangle=\int {d^3p\over (2\pi)^32E_p}e^{-i\vec p\cdot\vec x}|\vec p\rangle.$$ The QFT framework is concerned not with calculating wave functions, but with characterizing a quantum field by a set of quantum numbers and calculating scattering amplitudes for those fields as they interact with other fields such as the electromagnetic field, etc.
Now concerning the matter of causality, the real physical question is not whether propagation amplitudes vanish outside the light cone (which they do not) but whether or not commutators of the field operator ($\phi(x)$, $\phi(y)$ for $x$, $y$ space-like separated) vanish. Afterall, it is measurements that are important and not whether a particle exists in a region of space-time where one couldn't possibly probe.
The commutators do indeed vanish and is the reason why the Green's functions of QFT are built from the commutators as opposed to the propagation amplitudes as they are in ordinary QM. It is the Green's functions of QFT that by virtue of their containing two exactly cancelling terms: a propagation amplitude for a particle with a given mass and charge, and another for a particle with the same mass but opposite charge, that give rise to the concept that causality requires the real existence of matter and anti-matter. It is a mathematical necessity with real physical implications.
Issues of localization aside, whether or not a particle is propagating from $x$ to $y$ or is located in the past or future is no definite matter, as QFT is a full relativistic formulation and as such, it is one's frame of reference which determine whether a particle travels from $x$ to $ y$. For example, in one frame a particle propagates from $x$ to $y$ where $y$ is in the space-like future of $x$, however, since the interval is space-like one can always find a continuous Lorentz transformation (an element of $SO(3,1)$ as opposed to $SO(3,1)^{+}$) to a frame where the particle propagates from $y$ to $x$ and $x$ is in the space-like past of $y$.
In summary, Field excitations are states like ${\mathbf{a_p}}^\dagger|0\rangle=|\vec p\rangle$, we think of them as corresponding to the existence of a physical particle, however, the state itself should not be interpreted to mean that there is a physical particle located at $\vec x$ with momentum $\vec p$. Particle localization, a concept at home in QM must be abandoned in QFT. Further propagation should not be thought of as a particle travelling along a path in space but as transition amplitudes, i.e. the probability of a system undergoing a "quantum jump" as it were. Just as quantum mechanics forced physicists to give up classical determinism, quantum field theory demands that physicists give up quantum localization.
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