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Consider a (classical) system of several interacting particles. Can it be shown that, if the Lagrangian of such a system is Lorenz invariant, there cannot be any space-like influences between the particles?

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Suggestion for the title(v2): Classically, does locality emerge from a Lorentz-invariant action? – Qmechanic Mar 31 '12 at 20:30
up vote 2 down vote accepted

A counterexample:

1) Take two particles in a laboratory S

2) Set the clocks attached to the particles so that at laboratory time $t=0$ both clocks show $\tau_1=0$, $\tau_2=0$

3) Leave the particles to evolve under the following lorentz-invariant lagrangian:

$$ \mathcal{L}=\mathcal{L}_1+\mathcal{L}_2+\mathcal{L}_{\mathrm{int}}\\ \mathcal{L}_{\mathrm{int}}=u^{~\mu}_1(\tau_1)u_{2\mu}(\tau_2), $$ where $\mathcal{L}_1,\mathcal{L}_2$ are free-particle lagrangians. The total lagrangian is lorenz-invariant, but the physical system is not due to its initial setup, which allows the lagrangian to transmit the interactions in a space-like manner.

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Very elegant, thank you. – Nathaniel Apr 1 '12 at 11:33

Lorentz covariance is not enough as there are superluminal representations of the Poincare group, featuring tachyons. (See the section ''What about particles faster than light (tachyons)?'' in Chapter A7: Time and space of my theoretical physics FAQ for some background on tachyons)

The condition required for causality beyond Lorentz invariance is that the resulting system of differential equations is symmetric hyperbolic.

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