symbolic Pauli matrix expansions

The tellsimpafter code method for
post expansion and post pulling out
of scalars, etc., can be greatly simplified
and is now in file pauli1.mac.

-----------------------------------------------
/* file pauli1.mac
   tellsimpafter methods used
   after commutator has been expanded and
   scalars, numbers, and %i's have been
   factored out by commutators.mac.
   
   sig[1], etc, with lower case 's' for symbolic sigma matrix
   
   Sig[1], etc, with upper case 'S' for explicit matrix.
   
   See definitions of explicit matrices S[k], k = 1,2,3,
   
    as well as mcomm, Z2, stoex, and xcomm (A,B)
    with examples of use to compare symbolic calculation
    with explicit matrix calculation.   
   
   
   see examples of use at end of file.
   
   loads commutators.mac (code written by Robert Dodier).
   
    */
   
 /* copyright 2010 by Edwin Woollett
 * I release this work under terms of the GNU General Public License.
 */   
   
   
   
 load (commutators)$

 remove (comm, antisymmetric)$

  declare ([s1, s2, s3, s4], scalar)$
 
 /* note: nonscalarp (sig[1]) --->  true  */
 
 
   declare (eps3,antisymmetric)$
  
  tellsimpafter (eps3[1,2,3], 1)$
  
  matchdeclare ([ii,jj,kk],integerp, bb, scalarp )$    
  
  tellsimpafter (comm (sig[ii],bb), 0)$
  
  tellsimpafter (comm (bb, sig[ii]), 0)$
 
 tellsimpafter (comm (sig[ii], sig[jj]),
        sum (2*%i*eps3 [ii,jj,kk]*sig[kk],kk,1,3))$
 
 matchdeclare (aa, nonscalarp, nn, evenp)$
 
 tellsimpafter ( aa^^nn, 1)$
 
 odd1p(x) := (oddp (x) and x > 1)$
 
 matchdeclare (aa, nonscalarp, nn, odd1p)$
 
 tellsimpafter ( aa^^nn, aa)$
 
 pauli_simp(x) := (comm_simp(x),
                                 ev (%%,infeval))$
 
 /* ------------------------------------------*/
  
 /* explicit Pauli matrices */
 
 I2 : ident(2)$
 
 Z2 : zeromatrix (2,2)$
 
 Sig[1] : matrix ([0,1],[1,0])$
 
 Sig[2] : matrix ([0,-%i],[%i,0])$
 
 Sig[3] : matrix ([1,0],[0,-1])$
 
 /***********  commutator of two matrices  *************/

  mcomm (_M1%,_M2%) :=
              _M1% . _M2% - _M2% . _M1% $
  
  
  /* convert symbolic to explicit matrix */
  
  stoex (_e%) := (subst (sig = Sig, _e%), ev (%%))$
  
  /* compare symbolic to explicit matrix caculation */
  
  xcomm (A, B) := 
   block ([scomm,ecomm,Margs,Mcomm ],
   
      /* symbolic calculation */
      
      scomm : comm_simp (comm (A,B)),
      display (scomm),
      
      /*  convert symbolic to matrix */
      
      ecomm : stoex (scomm),
      display (ecomm),
      
      /* ab initio matrix calculation */
      
      Margs : subst (sig = Sig,[A,B]),
      /* display (Margs), */
      
      Mcomm : apply ('mcomm, Margs),
      Mcomm : ev (Mcomm),
      display (Mcomm),
      
      /* compare symbolic with matrix calc */
      
      if (equal (ecomm - Mcomm,Z2)) then true else false)$                 
  
 
 /*  examples of use:
 
 (%i1) load(pauli1);
(%o1) "c:/work5/pauli1.mac"
(%i2) comm (sig[2],1);
(%o2) 0
(%i3) comm (1,sig[2]);
(%o3) 0
(%i4) comm_simp(comm(sig[1],sig[3]+sig[2]));
(%o4) 2*%i*sig[3]-2*%i*sig[2]
(%i5) comm_simp(comm(sig[2]+sig[1],sig[3]));
(%o5) 2*%i*sig[1]-2*%i*sig[2]
(%i6) comm_simp(comm(sig[1],sig[3]^^2+sig[2]));
(%o6) 2*%i*sig[3]
(%i7) comm_simp(comm(sig[1],sig[3]^^3+sig[2]));
(%o7) 2*%i*sig[3]-2*%i*sig[2]
 

(%i8) comm (sig[2],%i);
(%o8) 0
(%i9) comm (%i,sig[2]);
(%o9) 0
(%i10) comm (sig[2],s1);
(%o10) 0
(%i11) comm (s2,sig[2]);
(%o11) 0
(%i12) comm (a2,sig[2]);
(%o12) comm(a2,sig[2])

(%i13) comm_simp(comm(sig[1],sig[3] + sig[2]^^2));
(%o13) -2*%i*sig[2]
(%i14) comm_simp(comm(sig[1],sig[3] + sig[2]^^3));
(%o14) 2*%i*sig[3]-2*%i*sig[2]
(%i15) comm_simp(comm(sig[1],sig[3]^^2 + sig[2]^^3));
(%o15) 2*%i*sig[3]
(%i16) comm_simp(comm(sig[1],sig[3]^^3 + sig[2]^^3));
(%o16) 2*%i*sig[3]-2*%i*sig[2]
(%i17) comm_simp(comm(sig[1]^^2,sig[3]^^3 + sig[2]^^3));
(%o17) 0
(%i18) comm_simp(comm(sig[1]^^3,sig[3]^^3 + sig[2]^^3));
(%o18) 2*%i*sig[3]-2*%i*sig[2]

(%i1) load(pauli1)$
(%i2) xcomm (sig[1]^^3,sig[2]^^5);
scomm = 2*%i*sig[3]

ecomm = matrix([2*%i,0],[0,-2*%i])

Mcomm = matrix([2*%i,0],[0,-2*%i])

(%o2) true
(%i3) xcomm (sig[1]^^3,sig[2]^^3 + sig[3]^^5);
scomm = 2*%i*sig[3]-2*%i*sig[2]

ecomm = matrix([2*%i,-2],[2,-2*%i])

Mcomm = matrix([2*%i,-2],[2,-2*%i])

(%o3) true


*/
-------------------------------------

The files commutators.mac, pauli1.mac,
and pauli2.mac (defrule methods for
post expansion processing) will be
posted on my webpage.

Ted Woollett
http://www.csulb.edu/~woollett