What is the reason that $zz^*$ is not written as $ |z|^2$ sometimes?

Question

For a general (complex) number $z$ I sometimes get terms that explicitly writes (for instance)

$$2zz^*+|z|^2. $$ I should add that these expressions come from long calculations involving packages (FeynCalc). How come FullSimplify[] does not simply put this expression to $3|z|^2$ instead?

Answer 1

Running FullSimplify I get

FullSimplify[z Conjugate[z] + 2 Abs[z]^2]

3 z Conjugate[z]

This is actually more simple than 3 Abs[z]^2 in the eyes of FullSimplify. See here for more info.

Simplify`SimplifyCount[3 z Conjugate[z]]

5

Simplify`SimplifyCount[3 Abs[z]^2]

6

Luckily there are many ways to guide FullSimplify. Here is one such way.

FullSimplify[z Conjugate[z] + 2 Abs[z]^2, ExcludedForms -> {_Abs}]

3 Abs[z]^2

Edit

A more general way is to use a combination of ComplexityFunction and TransformationFunctions. Basically we'll assign a penalty whenever the unwanted pattern is present and teach FullSimplify the rule we want to use.

ToAbs[a_. z_ Conjugate[z_]] := a Abs[z]^2
ToAbs[e_] := e

FullSimplify[a1 Conjugate[a1] + a1 Conjugate[a1] + b1 Conjugate[b1], 
  ComplexityFunction -> (LeafCount[#] + 100 Count[#, _Conjugate, ∞] &), 
  TransformationFunctions -> {ToAbs, Automatic}
]

2 Abs[a1]^2 + Abs[b1]^2