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I define:

Unprotect[NonCommutativeMultiply];
NonCommutativeMultiply[left___, HoldPattern[Times[u[x_], p : (a[k_] | SuperDagger[a][k_])]], 
right___] := u[x] NonCommutativeMultiply[left, p, right]
Protect[NonCommutativeMultiply];

since I want the function u[x_] to commute with a[k_], SuperDagger[a][k_]

Now if i try

In:= a[k] ** u[k]*a[q]
Out= a[q] a[k] ** u[k]    (*  which is not the desired output*)

if i try

In:= a[k] ** (u[k] a[q])
Out= a[k] ** a[q] u[k]

which is better but still it's not u[k] a[k] ** a[q]

I can't figure out why my definition isn't working. What am I missing?

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    $\begingroup$ The problem is that ** distributes over *. That means: a[k] ** u[k]*a[q] is interpreted as: (a[k] ** u[k]) * ( a[k] **a[q]). If you write: a[k] ** (u[k]*a[q]) you get: a[k] ** a[q] u[k] $\endgroup$
    – Daniel Huber
    Dec 20 '20 at 12:27
  • $\begingroup$ @DanielHuber I thought my rule would take care of that? Even with a[k] ** (u[k] a[q]) the output places u[k] after a[k] ** a[q] not before. How should I change it to have the desired output? $\endgroup$
    – geom
    Dec 20 '20 at 12:32
  • $\begingroup$ "Times" has the attribute "Orderless". Therefore MMA will rearrange the expression according to its own rules (I think, lexicographically, but I am not sure) $\endgroup$
    – Daniel Huber
    Dec 20 '20 at 12:47
  • $\begingroup$ @DanielHuber Geez, you are right. Even if I add orderless to NonCommutativeMultiply nothing changes.The order is decided by Sort which orders symbols by their names, and in the event of a tie, by their contexts. Please post your comments as an answer. They are helpful. $\endgroup$
    – geom
    Dec 20 '20 at 13:08
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The problem is that NonCommutativeMultiply (**) distributes over Times.

That means: a[k] ** u[k]*a[q] is interpreted as: (a[k] ** u[k]) * (a[k] ** a[q]). If you write: a[k] ** (u[k]*a[q]) you get: a[k] ** a[q] u[k]

Furthermore, Times has the attribute Orderless. Therefore MMA will rearrange the expression according to its own rules (I think lexicographically, but I am not sure).

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