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How to implement this simple product rule in mathematica




Firstly i have defined a simple function below.


   dotPro[a_, b_] := a[1]*b[1] + a[2]*b[2];

Then i create two terms using the above function.


  t1=dotPro[q,σ];
  t2=dotPro[ϵ,σ];

Then i have some rules regarding how these parameters multiply


  r1 = Rule[σ[1]^2, 1];
  r2 = Rule[σ[2]^2, 1];

  r3 = Rule[σ[1] σ[2], I*σ[3]]; 
  r4 =Rule[σ[2] σ[1], -I*σ[3]];

Now i expand the product


  res = Expand[t1*t2]

Finally i apply the said rules to my expanded terms


  res /. r1 /. r2 /. r3 /. r4

The answer i get is the following



 (q[1] ϵ[1] + q[2] ϵ[2] + I q[2] ϵ[1] σ[3] + I q[1] ϵ[2] σ[3])

What i want to get is this(minus sign)


 q[1] ϵ[1] + q[2] ϵ[2] -I q[2] ϵ[1] σ[3] + I q[1] ϵ[2] σ[3]

I know what the problem is,its related to mathematica assuming commutative product.so at the heart it deals with implementing non-commutative algebra.


So i did find a link to a package here but it was not working and i think this simple thing can be achieved without resort to any packages.


All i want is whenever there is a term with σ[1] σ[2] it should be replaced by I*σ[3] and for σ[2] σ[1] it should be replaced by -I*σ[3]


I tried to achieve it like this,but no success


list = {{1, 1}, {1, 2}, {2, 1}, {2, 2}};

t1[[#[[1]]]]*t2[[#[[2]]]] & /@ list /. r1 /. r2 /. r3 /. r4

Answer



The following code is a quick and dirty solution to your request. The basic idea is to avoid using the usual multiplication and use Dot instead.


dotPro[a_, b_] := a[1].b[1] + a[2].b[2];
dotPro[q, σ].dotPro[ε, σ]
Distribute[%]
% /. {Dot[x___, a_, y___, a_, z___] :> Dot[x, y, z]
  , Dot[x___, a_[1], y___, a_[2], z___] :> I Dot[x, y, a[3]]
  , Dot[x___, a_[2], y___, a_[1], z___] :> -I Dot[x, y, a[3]]}

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