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In QFT, the time-ordering causality is generally used.

There are 4 ways to bypassing the pole called time-ordering, anti-time-ordering, retarded and advanced.

But in many case only time-ordered causality is used.

Could I ask the reason why? (Source in the future couldn't affect something in the past? or just convention?)

In Srednicki QFT book, and also in Bjorken RQM book, deriving the Relativistic version of the Schrödinger equation, from the Einstein energy momentum reaction, one could get $H \psi = \sqrt{ p^{2} +m^{2}} \psi $. By Taylor expanding the sqrt part, there is infinite-order of Laplacian, so the theory is non-local. why is the theory non-local with the infinite-order of laplacian?

What is the local theory? and Why should the locality be kept? (is it related the Dirac light cone?)

Does one can tell that one theory is local or non-local in classical theories?

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  • $\begingroup$ Locality is manifested in the action itself. When we write action in terms of fields, fields are at the same position. Examples are like scalar fields. But if we require our theory should respect relativity, then we need causality. It means signal can't go past outside the lightcone. So that is the reason we use Feynman propgator in which particles and anti particles move forward in time. $\endgroup$ – Hare Jun 27 at 23:58
  • $\begingroup$ We use time-ordered propagators for some calculations and retarded/advanced for others. The question you should be asking is what these different propagators compute. The reason time-ordered propagator is commonly used is that the perturbation theory is most conveniently formulated using it. $\endgroup$ – Peter Kravchuk Jun 28 at 4:06
  • $\begingroup$ Thank you for your coments! $\endgroup$ – Summal Jun 28 at 5:25
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Casuality: two quantum systems at spacelike distances are not correlated, they cannot communicate, or influence each other. This involves the assumption, that there was no correlation (entanglement) before (in time).

In general, a process has many causes,[3] which are also said to be causal factors for it, and all lie in its past. An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future. Causality is one of the most fundamental and essential notions of physics.[48] Causal efficacy cannot 'propagate' faster than light. Otherwise, reference coordinate systems could be constructed (using the Lorentz transform of special relativity) in which an observer would see an effect precede its cause (i.e. the postulate of causality would be violated).

https://en.wikipedia.org/wiki/Causality

Locality: two quantum systems can only interact if they are close locally in space. An object is influenced directly only by its surroundings.

The concept is that for an action at one point to have an influence at another point, something in the space between those points such as a field must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence. The special theory of relativity limits the speed at which all such influences can travel to the speed of light, {\displaystyle c} c. Therefore, the principle of locality implies that an event at one point cannot cause a simultaneous result at another point. An event at point {\displaystyle A} A cannot cause a result at point {\displaystyle B} B in a time less than {\displaystyle T=D/c} {\displaystyle T=D/c}, where {\displaystyle D} D is the distance between the points.

https://en.wikipedia.org/wiki/Principle_of_locality

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