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I am trying to understand the meaning of the renormalization group equation and what i have understood is that, since observable (or physical?) quantities must not depend on arbitrary energy scales, the RG equation must hold. But why do then the coupling constants, for example depend on the energy scale? Are they not observable quantities?

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No, the coupling constants are not observable quantities. The only thing that we measure are correlation functions.

When correlation functions are computed naively, they apparently depend on the cut-off as well as the coupling constants. The couplings must depend on the cut-off just in the right way for the dependence on the cut-off to cancel out.

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  • $\begingroup$ Thank you very much, as i expected my confusion stems from not knowing exactly what an observable quantity is. Do you know of any books in which the definition of observable quantity is given clearly? $\endgroup$ – Egosphere Jan 30 '16 at 14:19
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    $\begingroup$ Sorry, I can't think of a book on this particular topic. The definition of observables depends on the theory that you consider and is usually clear. In quantum physics, they are computed by bracketing operators in the system state, $\langle \psi | O | \psi \rangle $. In statistical physics they are averages over microscopic fluctuations, $\int P(\{\mu \}) \, O(\{\mu \} )$. The $\{\mu \}$'s are the micro-states of the system and $P(\{\mu \})$ is the corresponding probability. $\endgroup$ – Steven Mathey Jan 30 '16 at 14:49

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