Why is it required that interactions between fields must occur at single spacetime points in order for them to be local? For example, why must an interaction Lagrangian be of the form [tex]\mathcal{L}_{int}\sim (\phi(x))^{2}[/tex] why can't one have a case where [tex]\mathcal{L}_{int}\sim\phi(x)\phi(y)[/tex] where ##x^{\mu}## and ##y^{\mu}## are time-like separated?(adsbygoogle = window.adsbygoogle || []).push({});

Is it simply because the fields themselves are localised, i.e. they are described in terms of their values at each spacetime point, and as such to avoid action-at-a-distance they can only interact with one another when they are located at the same spacetime point (as otherwise two fields ##\phi(x)## and ##\phi(y)## at distinct spacetime points ##x^{\mu}## and ##y^{\mu}## could spontaneously interact with one another which would constitute action-at-a-distance)? Hence if they were located at two distinct spacetime points that are time-like separated, then any direct interaction between them would still constitute action-at-a-distance, as they would be able to spontaneously influence one another despite not being at the same spacetime point?

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# Interactions between field operators & locality in QFT

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