The Dirac field has a "hidden" U ( 1 ) gauge freedom, allowing it to couple directly to the electromagnetic field without any further modifications to the Dirac equation or the field itself. This is not the case for scalar fields, which must be explicitly "complexified" to couple to electromagnetism. This is done by "tensoring in" an additional factor of the complex plane C into the field, or constructing a Cartesian product with U ( 1 ) .