Scalar Field Theory - Classical Scalar Field Theory - Complex Scalar Field Theory

Complex Scalar Field Theory

In a complex scalar field theory, the scalar field takes values in the complex numbers, rather than the real numbers. The action considered normally takes the form

\mathcal{S}=\int \mathrm{d}^{D-1}x \, \mathrm{d}t \mathcal{L} = \int \mathrm{d}^{D-1}x \, \mathrm{d}t \left[\eta^{\mu\nu}\partial_\mu\phi^*\partial_\nu\phi -V(|\phi|^2)\right]

This has a U(1) symmetry, whose action on the space of fields rotates, for some real phase angle .

As for the real scalar field, spontaneous symmetry breaking is found if m2 is negative. This gives rise to a Mexican hat potential which is analogous to the double-well potential in real scalar field theory, but now the choice of vacuum breaks a continuous U(1) symmetry instead of a discrete one. This leads to a Goldstone boson.

Read more about this topic:  Scalar Field Theory, Classical Scalar Field Theory

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