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These to me sound like they're two sides of the same coin. If you lose locality of interaction, then you lose locality of energy conservation, and you therefore have, among other things, combinations of energy transfer which simply push the energy out to $\infty$ instantaneously, creating a pathological global violation. I am not sure that I buy your ...

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I agree with your definition of locality (probably not surprising :)). Causality I would say is the statement that an event in the future should not affect an event in the past. We can formulate this in classical physics terms. Causality is necessary in order for there to be a well defined initial value problem: I should be able to choose an initial time ...

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Causality means that if something happens before in one reference frame of your choice, it happens before in any other existing reference frame in the universe. Locality means that if two events are space-like separated then it exists at least one reference frame where they happen at the same time; if two events are time-like separated, then it exists at ...

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The use of the Lagrangian density is a convenience, and it is not directly related to causality or relativity, and neither strictly to quantum theories. What I mean is that it is possible to formulate non-relativistic (quantum or classical) field theories using exactly the same language. The difference between "mechanics" and "field theory" is that, instead ...

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Locality is a physical requirement we impose (for good reasons). Locality is implemented in the theory by using fields, with local interactions in a Lagrangian density (ie, the Lagrangian only depends on products of fields and derivatives at a single point). I would definitely not say that locality occurs because Lagrangian densities show up in field theory, ...

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The difference is that in classical mechanics positions are exactly the fields you are looking at, whereas in general field theories they are the variables the actual fields depend on. In classical mechanics the solution of the dynamics is given by the knowledge of the position and the velocity $(q(t),\dot{q}(t))$ at any time $t$. Time plays the role of the ...

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The reason why the commutation relations between a field and its conjugate at equal times are of the form $$\left[\phi(t,\textbf{x}),\pi(t,\textbf{y})\right]=i\hbar\,\delta^{(3)}(\textbf{x}-\textbf{y})$$ is only to mirror and copy the canonical hamiltonian commutation relations $[q_i,p_j]=i\hbar\,\delta_{ij}$. No causality is involved, rather it is somehow ...

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One way to define spacelike separation in special relativity is that any two events are spacelike separated if and only if there exists a reference frame in which the two events have the same time coordinate. So yes, if $x^0 = y^0$ the separation is spacelike. Alternatively you can work from the definition where two events are spacelike separated if (and ...

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To reach the Lienard-Wiechert potentials or to prove Feynman's equation (exposed in his lectures without proof), it's necessary to begin with the so-called retarded potentials expressed here conveniently by the following. \phi\left(\mathbf{r},t\right)=\dfrac{1}{4\pi\varepsilon_{o}}\iiint ...

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To add to ACuriousMind's answer on the Liénard-Weichert potentials, you can put these formulas into an even more wonderfully descriptive form since you can derive Feynman's formula from them for the radiation from a moving charge: \vec{E} = ...

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The force does not change instantaneously, the correct way the electromagnetic field of (and thus the force exerted by) a moving electric charge is given by the Liénard-Wiechert potential, where one can see that the effect of the charge does not travel faster than light.

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The force is not propagated instantly. It takes time for the information to get from one point to another. You can treat that as an instant if you are working with small enough distances and velocities, but it's not. If you'll ever study field theory you'll meet retarded potentials that are just this: the field propagates at the speed of light and it's no ...

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