What is the physical significance of an imaginary field?

Question

How should we understand a complex field? What is the physical significance of its imaginary part? For example, a complex scalar field is written as: \begin{equation} \phi = \phi_{1} + i \phi_{2} \end{equation} What is the physical significance of $\phi_{2}$? If it is the same as the significance of $\phi_{1}$, why is $\phi_{2}$ put in the imaginary part? What does imaginary mean? Does it mean unreal or virtual? If not, then what?

Answer 1

Complex fields appear in real-valued (Hermitian) Lagrangians. For example, your $\phi$ may appear in a kinetic term $\partial_\mu\phi^\ast \partial^\mu\phi=\partial_\mu\phi_1 \partial^\mu\phi_1 +\partial_\mu\phi_2 \partial^\mu\phi_2$. In theory, you could forget about $\phi$ altogether, thinking instead in terms of uncoupled real fields $\phi_1,\,\phi_2$; indeed, they remain uncoupled if the potential term is proportional to $\phi^\ast\phi=\phi_1^2+\phi_2^2$. But there is an advantage to writing everything instead in terms of one complex field, which is that the $U\left( 1\right)$ symmetry is obvious, whereas the biggest symmetry you might otherwise notice is $\mathbb{Z}_2^2$.

If $\phi$ denotes the Higgs field, the potential term also contains a contribution proportional to $\left( \phi^\ast\phi\right)^2=\phi_1^4+2\phi_1^2\phi_2^2+\phi_2^4$ which manifestly couples $\phi_1$ to $\phi_2$. However, it is only when we think in terms of the complex $\phi$ that the spontaneous symmetry breaking induced by a choice of a vacuum is evident.

Asking about the physical significance of $\phi_1,\,\phi_2$ separately is a bit like asking about the separate significance of Cartesian coordinates in a plane. If you've studied orbital mechanics for a 2-body solar system, you'll know that polar coordinates have a more obvious physical relevance and interpretation. And in the case of a Mexican hat potential, again a polar representation is more relevant. Let's consider the vacuum case. Writing $\phi=\rho e^{i\theta}$, $\rho$ is the vacuum amplitude while $\theta$ is an arbitrary phase that breaks the symmetry. This cleaving of a computable physical parameter from an arbitrary one implies a greater physical significance than either $\phi_1$ or $\phi_2$ has.