Editing C-symmetry (section)

==Scalar fields==
The Dirac field has a "hidden" <math>U(1)</math> gauge freedom, allowing it to couple directly to the electromagnetic field without any further modifications to the Dirac equation or the field itself.{{efn|This freedom is explicitly removed, constrained away in [[Majorana spinor]]s.}}  This is not the case for [[scalar field]]s, which must be explicitly "complexified" to couple to electromagnetism. This is done by "tensoring in" an additional factor of the [[complex plane]] <math>\mathbb{C}</math> into the field, or constructing a [[Cartesian product]] with <math>U(1)</math>.

One very conventional technique is simply to start with two real scalar fields, <math>\phi</math> and <math>\chi</math> and create a linear combination

:<math>\psi \mathrel\stackrel{\mathrm{def}}{=} {\phi + i\chi \over \sqrt{2}}</math>

The charge conjugation [[involution (mathematics)|involution]] is then the mapping <math>\mathsf{C}:i\mapsto -i</math> since this is sufficient to reverse the sign on the electromagnetic potential (since this complex number is being used to couple to it). For real scalar fields, charge conjugation is just the identity map: <math>\mathsf{C}:\phi\mapsto \phi</math> and <math>\mathsf{C}:\chi\mapsto \chi</math> and so, for the complexified field, charge conjugation is just <math>\mathsf{C}:\psi\mapsto \psi^*.</math> The "mapsto" arrow <math>\mapsto</math> is convenient for tracking "what goes where"; the equivalent older notation is simply to write <math>\mathsf{C}\phi=\phi</math> and <math>\mathsf{C}\chi = \chi</math> and <math>\mathsf{C}\psi = \psi^*.</math>

The above describes the conventional construction of a charged scalar field. It is also possible to introduce additional algebraic structure into the fields in other ways. In particular, one may define a "real" field behaving as <math>\mathsf{C}:\phi\mapsto -\phi</math>. As it is real, it cannot couple to electromagnetism by itself, but, when complexified, would result in a charged field that transforms as <math>\mathsf{C}:\psi\mapsto -\psi^*.</math>  Because C-symmetry is a [[discrete symmetry]], one has some freedom to play these kinds of algebraic games in the search for a theory that correctly models some given physical reality.

In physics literature, a transformation such as <math>\mathsf{C}:\phi \mapsto \phi^c = -\phi</math> might be written without any further explanation. The formal mathematical interpretation of this is that the field <math>\phi</math> is an element of <math>\mathbb{R}\times\mathbb{Z}_2</math> where <math>\mathbb{Z}_2 = \{+1, -1\}.</math> Thus, properly speaking, the field should be written as <math>\phi = (r, c)</math> which behaves under charge conjugation as <math>\mathsf{C}: (r, c) \mapsto (r, -c).</math> It is very tempting, but not quite formally correct to just multiply these out, to move around the location of this minus sign; this mostly "just works", but a failure to track it properly will lead to confusion.