Question

Why does the free hermitian scalar field Lagrangian need a factor of 1/2 even though the equation of motion is invariant if the Lagrangian is multiplied by any number, as in: \[L=\alpha \frac12(\partial_\mu\phi\partial^\mu\phi-m^2\phi^2)\]

Comments

This is for convenience as otherwise a factor of 2 would appear in the Euler-lagrange equations, though this doesn't change the final field equations. The factor can be changed to any positive number by rescaling the field; hence there is no loss of generality assuming it to be $1/2$.

Note also that the scale matters in the quantum case, as the path integral depends on $e^{i S/\hbar}$, where the action $S$ is obtained by integrating the Lagrangian density.

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I wasn't sure whether the factor of half had significance. Does such a scale, which shows up in the action, have any non-trivial consequences in QFT?

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@conservedcharge: As Arnold has explained, yes, it matters because we normalize fields for them to have some physical meaning and the normalization conditions are not contained in the field equations. When you calculate the transition probability amplitude, say from a free quantum motion to the same free quantum motion, you have to have this $1/2$ in $S$ in order to obtain unity.