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Why is it that sometimes I see kinetic term of scalar Lagrangians written like this $$\mathcal{L}=\partial_\mu\phi^\dagger\partial^\mu\phi+\dots$$ like for example in scalar electrodynamics, while some other times (in this question for example) there is an explicit $\frac{1}{2}$ factor, making the Lagrangian appear like $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi^\dagger\partial^\mu\phi+\dots$$ Is it a different normalization of the fields? Maybe it's different for complex and real fields, or maybe for the representation of the fields if the theory is a gauge theory?

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It's more convenient to have the factor of $1/2$ for the real scalar field, while it's more convenient to have the factor of $1$ for the complex scalar field. Roughly, the factor of $1/2$ eliminates the "symmetry factor" associated with the two real $\phi$ fields being equivalent. The most direct way to see this from elementary QFT is to simply diagonalize the Hamiltonians for the real and complex scalar field theories using the mode expansion in terms of creation and annihilation operators.

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