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Dr. Rhett Herman, Dept. of Chemistry and Physics
Commutation Relations Involving Gauge Covariant Derivatives

I my work, I need to be able to handle a large number of the commutators of gauge covariant derivatives. With the program MathTensor being able to automatically handle the commutator of covariant derivatives, it is a relatively straightforward exercise to build a library of gauge covariant commutators for future use. All that is involved is the recognition of a few patterns, and then the use of an iterative procedure to build as many of these as needed.

In the work below, the covariant derivative will be denoted by the traditional semicolon ";" whole the gauge covariant derivative will be indicated by the vertical bar "|". I am assuming the reader is familiar with the notation and commands of MathTensor. I am assuming the reader is familiar with LaTeX, since it will be helpful for me to use some of the LaTeX style to emphasize subscripts, superscripts, etc.

The basic idea is the correct reordering of the gauge covariant derivatives of an object "a" which carries information about the electromagnetic field present in a spacetime:

a    = ad2   - I a e F
 |ab      ab          ab

                                    p
a     = a     - I e a   F   + a   R
 |acb    |abc        |a  bc    |p  a bc

                                          p             p
a      = a      - I e ad2   F   + ad2   R     + ad2   R
 |abdc    |abcd          ab  cd      pb  a cd      ap  b cd

                                                                p
a     = a     - I a e F     - I e a   F   - I e a   F   + a   R
 |cab    |abc          ac;b        |b  ac        |a  bc    |p  a bc
Here, I is the square root of (-1), while "e" is the coupling constant (charge) of the electromagnetic field, and MaxwellF represents the Maxwell Tensor. The object a_{|abdc} represents four gauge covariant derivatives of "a". The OrderCD[] command within MathTensor does this for covariant derivatives very rapidly, but my work needs the additional information carried by the MaxwellF tensor, as shown above.


We first need to define the generic tensors to serve as carriers of the derivative information. These are ad1_{a},ad2_{ab},...,ad6_{abcdef} below. They must be assigned no symmetries: These new tensors will represent 1, 2, 3, 4, 5, and 6 gauge covariant derivatives of the scalar "a", a scalar which depends on the electromagnetic (gauge) field present in the spacetime. 

DefineTensor[ad1,{{1},1}]
DefineTensor[ad2,{{1,2},1}]
DefineTensor[ad3,{{1,2,3},1}]
DefineTensor[ad4,{{1,2,3,4},1}]
DefineTensor[ad5,{{1,2,3,4,5},1}]
DefineTensor[ad6,{{1,2,3,4,5,6},1}]


PermWeight::"def": "Object \!\(\"ad1\"\) defined"
PermWeight::"def": "Object \!\(\"ad2\"\) defined"
PermWeight::"def": "Object \!\(\"ad3\"\) defined"
PermWeight::"def": "Object \!\(\"ad4\"\) defined"
PermWeight::"def": "Object \!\(\"ad5\"\) defined"
PermWeight::"def": "Object \!\(\"ad6\"\) defined"

Next, we need to define some rules that will convert one gauge covariant derivative of ad2_{ab} into ad3_{abc}: For example, a_{|a}->ad1[la], ad1_{a|b}->ad2[la,lb], ad3_{abc|d}->ad4[la,lb,lc,ld], etc.


RuleUnique[ad5d1rule,CD[ad5[la_,lb_,lc_,ld_,le_],lf_],ad6[la,lb,lc,ld,le,lf]]
RuleUnique[ad4d1rule,CD[ad4[la_,lb_,lc_,ld_],le_],ad5[la,lb,lc,ld,le]]
RuleUnique[ad3d1rule,CD[ad3[la_,lb_,lc_],ld_],ad4[la,lb,lc,ld]]
RuleUnique[ad2d1rule,CD[ad2[la_,lb_],lc_],ad3[la,lb,lc]]
RuleUnique[ad1d1rule,CD[ad1[la_],lb_],ad2[la,lb]]
RuleUnique[ad0d1rule,CD[a,la_],ad1[la]]


These first definitions are done by hand in that they simply commute the gauge covariant derivative on the final two indices of the tensors ad1, ad2, etc. These are printed out just to see the results. The main thing is that they are redefined afterwards for later use by the DefUnique[] command.

In[12]:=
ad2[la,lb]+CommuteCD[CD[a,lb,la],lb,la]-
 CD[a,la,lb]-I*e*MaxwellF[la,lb]*a

ad3[la,lb,lc]+CommuteCD[CD[ad1[la],lc,lb],lc,lb]-
 CD[ad1[la],lb,lc]-I*e*MaxwellF[lb,lc]*ad1[la]

ad4[la,lb,lc,ld]+CommuteCD[CD[ad2[la,lb],ld,lc],ld,lc]-
 CD[ad2[la,lb],lc,ld]-I*e*MaxwellF[lc,ld]*ad2[la,lb]

ad5[la,lb,lc,ld,le]+CommuteCD[CD[ad3[la,lb,lc],le,ld],le,ld]-
 CD[ad3[la,lb,lc],ld,le]-I*e*MaxwellF[ld,le]*ad3[la,lb,lc]

ad6[la,lb,lc,ld,le,lf]+CommuteCD[CD[ad4[la,lb,lc,ld],lf,le],lf,le]-
 CD[ad4[la,lb,lc,ld],le,lf]-I*e*MaxwellF[le,lf]*ad4[la,lb,lc,ld]


Out[12]=
ad2   - I a e F
   ab          ab
Out[13]=
                               p
ad3    - I e ad1  F   + ad1  R
   abc          a  bc      p  a bc
Out[14]=
                                  p             p
ad4     - I e ad2   F   + ad2   R     + ad2   R
   abcd          ab  cd      pb  a cd      ap  b cd
Out[15]=
                                     p              p              p
ad5      - I e ad3    F   + ad3    R     + ad3    R     + ad3    R
   abcde          abc  de      pbc  a de      apc  b de      abp  c de
Out[16]=
                                        p               p
ad6       - I e ad4     F   + ad4     R     + ad4     R     + 
   abcdef          abcd  ef      pbcd  a ef      apcd  b ef
 
            p               p
  ad4     R     + ad4     R
     abpd  c ef      abcp  d ef
In the notebook interface, I have found it easiest to take the "//InputForm" of expressions and directly paste them into input cells for later use. For extremely long expressions, this is absolutely crucial in order to avoid transcription errors with the large number of indices.

In[17]:=
ad3[la,lb,lc]+CommuteCD[CD[ad1[la],lc,lb],lc,lb]-
 CD[ad1[la],lb,lc]-I*e*MaxwellF[lb,lc]*ad1[la]//InputForm 


Out[17]//InputForm=
ad3[la, lb, lc] - I*e*ad1[la]*MaxwellF[lb, lc] + 
  ad1[l1]*RiemannR[la, u1, lb, lc]

In[18]:=
ad4[la,lb,lc,ld]+CommuteCD[CD[ad2[la,lb],ld,lc],ld,lc]-
 CD[ad2[la,lb],lc,ld]-I*e*MaxwellF[lc,ld]*ad2[la,lb]//InputForm


Out[18]//InputForm=
ad4[la, lb, lc, ld] - I*e*ad2[la, lb]*MaxwellF[lc, ld] + 
  ad2[l1, lb]*RiemannR[la, u1, lc, ld] + 
  ad2[la, l1]*RiemannR[lb, u1, lc, ld]
...and so forth. Now, to redefine these for later use...


DefUnique[ad21[lb_,la_],ad2[la,lb]-I*a*e*MaxwellF[la,lb] ]

DefUnique[ad132[la_,lc_,lb_],ad3[la,lb,lc]-I*e*ad1[la]*MaxwellF[lb,lc]+
 ad1[l1]*RiemannR[la,u1,lb,lc] ]

DefUnique[ad1243[la_,lb_,ld_,lc_],
 ad4[la,lb,lc,ld]-I*e*ad2[la,lb]*MaxwellF[lc,ld]+
 ad2[l1,lb]*RiemannR[la,u1,lc,ld]+ad2[la,l1]*RiemannR[lb,u1,lc,ld] ]

DefUnique[ad12354[la_,lb_,lc_,le_,ld_],
 ad5[la,lb,lc,ld,le] - I*e*ad3[la,lb,lc]*MaxwellF[ld,le] + 
 ad3[l1,lb,lc]*RiemannR[la,u1,ld,le] + 
 ad3[la,l1,lc]*RiemannR[lb,u1,ld,le] +
 ad3[la,lb,l1]*RiemannR[lc,u1,ld,le] ]

DefUnique[ad123465[la_,lb_,lc_,ld_,lf_,le_],
 ad6[la,lb,lc,ld,le,lf] - I*e*ad4[la,lb,lc,ld]*MaxwellF[le,lf] + 
 ad4[l1,lb,lc,ld]*RiemannR[la,u1,le,lf] + 
 ad4[la,l1,lc,ld]*RiemannR[lb,u1,le,lf] + 
 ad4[la,lb,l1,ld]*RiemannR[lc,u1,le,lf] + 
 ad4[la,lb,lc,l1]*RiemannR[ld,u1,le,lf]  ]

The above definitions will now come in handy in the iterative procedure for constructing the many gauge covariant commutators. a_{|acb} has already been done above and DefUnique'd as ad132. (My notation is simplistic but helps me keep track of which derivatives have been reordered.)
Below, we reorder the three derivatives of a_{|bac} and a_{|cab}. These will be DefUnique'd as ad213 and ad312, respectively.

To reorder a_|{bac}, start by using the definition for ad21 above to reorder two derivatives of a: ad21[lb,la]. Then, take the covariant derivative with respect to the index "c". Finally, replace the covariant derivatives of the tensor ad2_{ab|c} and the scalar a{|c} with the generic tensors ad3_{abc} and ad1{c}, respectively. 

In[53]:=
temp1=ad21[lb,la]
temp2=CD[temp1,lc] /. { CD[e,lc] -> 0 }
temp3=Expand[
 temp2/.{CD[ad2[la,lb],lc]->ad3[la,lb,lc],CD[a,lc]->ad1[lc]}]
temp3//InputForm


Out[53]=
ad2   - I a e F
   ab          ab
Out[54]=
ad2     - I (a e F     + e a   F  )
   ab;c           ab;c      ;c  ab
Out[55]=
ad3    - I a e F     - I e ad1  F
   abc          ab;c          c  ab
Out[56]//InputForm=
ad3[la, lb, lc] - I*a*e*CD[MaxwellF[la, lb], lc] - 
  I*e*ad1[lc]*MaxwellF[la, lb]
The last line is now pasted directly into a DefUnique[] cell for later use.

Reordering a_|{cab} is a bit more involved. Start by commuting the "c" and "a" indices of a_{|ca} using ad21[lc,la]. Then, take the covariant derivative with respect to "b" ("temp3" below). "temp4" converts derivatives of the tensors into other tensors using the rules {ad2d1rule,ad1d1rule,ad0d1rule}. However, note that "temp4" contains ad3_{acb}, which has the indices out of order. This must be replaced with the appropriate expression which has the indices in order. This replacement is in "temp5" and uses the entity ad132_{acb} to replace ad3_{acb}. Finally, temp5 is the desired result, with a_|{cab} rewritten in terms of a_|{abc} plus terms involving both the Riemann and MaxwellF tensors. 

In[62]:=
temp1=ad2[lc,la]
temp2=ad21[lc,la]
temp3=CD[temp2,lb] /. { CD[e,lb] -> 0 }
temp4=ApplyRules[temp3,{ad2d1rule,ad1d1rule,ad0d1rule}]
temp5=temp4 /. { ad3[la,lc,lb] -> Dum[ad132[la,lc,lb]] }
temp5//InputForm

Out[62]=
ad2
   ca
Out[63]=
ad2   - I a e F
   ac          ac
Out[64]=
ad2     - I (a e F     + e a   F  )
   ac;b           ac;b      ;b  ac
Out[65]=
ad3    - I a e F     - I e ad1  F
   acb          ac;b          b  ac
Out[66]=
                                                            p
ad3    - I a e F     - I e ad1  F   - I e ad1  F   + ad1  R
   abc          ac;b          b  ac          a  bc      p  a bc
Out[67]//InputForm=
ad3[la, lb, lc] - I*a*e*CD[MaxwellF[la, lc], lb] - 
  I*e*ad1[lb]*MaxwellF[la, lc] - I*e*ad1[la]*MaxwellF[lb, lc] + 
  ad1[l1]*RiemannR[la, u1, lb, lc]

As always, I DefUnique things that I will be using again later. 
In[68]:=
DefUnique[ad213[lb_,la_,lc_],
 ad3[la,lb,lc]-
 I*a*e*CD[MaxwellF[la,lb],lc]-I*e*ad1[lc]*MaxwellF[la,lb] ]

DefUnique[ad312[lc_,la_,lb_],
 ad3[la,lb,lc]-
 I*a*e*CD[MaxwellF[la,lc],lb]-I*e*ad1[lb]*MaxwellF[la,lc] - 
 I*e*ad1[la]*MaxwellF[lb,lc]+ad1[l1]*RiemannR[la,u1,lb,lc] ]

Finally, let's reorder a_{|bacd}, a_{|cabd}, and a_{|dabc}. Recall that a_{|abdc} was already done at the very beginning, and is used below in reordering a_{|dabc}. Note also how things are starting to get quite long and the idea of copying long expressions straight from "//InputForm" is now becoming more and more attractive!! 
In[20]:=
temp1=ad213[lb,la,lc]
temp2=CD[temp1,ld] /. { CD[e,ld] -> 0 }
temp3=ApplyRules[temp2,{ad3d1rule,ad1d1rule,ad0d1rule}]
temp3//InputForm

Out[20]=
ad3    - I a e F     - I e ad1  F
   abc          ab;c          c  ab
Out[21]=
ad3      - I (e a   F     + a e F     ) - I e (ad1  F     + ad1    F  )
   abc;d         ;d  ab;c        ab;cd            c  ab;d      c;d  ab
Out[22]=
ad4     - I e ad1  F     - I e ad1  F     - I a e F      - I e ad2   F
   abcd          d  ab;c          c  ab;d          ab;cd          cd  ab
Out[23]//InputForm=
ad4[la, lb, lc, ld] - I*e*ad1[ld]*CD[MaxwellF[la, lb], lc] - 
  I*e*ad1[lc]*CD[MaxwellF[la, lb], ld] - 
  I*a*e*CD[MaxwellF[la, lb], lc, ld] - I*e*ad2[lc, ld]*MaxwellF[la, lb]

In[24]:=
temp1=Dum[ad312[lc,la,lb]]
temp2=CD[temp1,ld] /. { CD[e,ld] -> 0 }
temp3=ApplyRules[temp2,{ad3d1rule,ad1d1rule,ad0d1rule}]
temp3//InputForm

Out[24]=
                                                            p
ad3    - I a e F     - I e ad1  F   - I e ad1  F   + ad1  R
   abc          ac;b          b  ac          a  bc      p  a bc
Out[25]=
                  p
ad3      + ad1  R       - I (e a   F     + a e F     ) - 
   abc;d      p  a bc;d         ;d  ac;b        ac;bd
 
  I e (ad1  F     + ad1    F  ) - I e (ad1  F     + ad1    F  ) + 
          b  ac;d      b;d  ac            a  bc;d      a;d  bc
 
           p
  ad1    R
     p;d  a bc
Out[26]=
ad4     - I e ad1  F     - I e ad1  F     - I e ad1  F     + 
   abcd          d  ac;b          b  ac;d          a  bc;d
 
         p
  ad1  R       - I a e F      - I e ad2   F   - I e ad2   F   + 
     p  a bc;d          ac;bd          bd  ac          ad  bc
 
          p
  ad2   R
     pd  a bc
Out[27]//InputForm=
ad4[la, lb, lc, ld] - I*e*ad1[ld]*CD[MaxwellF[la, lc], lb] - 
  I*e*ad1[lb]*CD[MaxwellF[la, lc], ld] - 
  I*e*ad1[la]*CD[MaxwellF[lb, lc], ld] + 
  ad1[l1]*CD[RiemannR[la, u1, lb, lc], ld] - 
  I*a*e*CD[MaxwellF[la, lc], lb, ld] - 
  I*e*ad2[lb, ld]*MaxwellF[la, lc] - I*e*ad2[la, ld]*MaxwellF[lb, lc] + 
  ad2[l1, ld]*RiemannR[la, u1, lb, lc]

In[45]:=
temp1=ad3[ld,la,lb]
temp2=Dum[ad312[ld,la,lb]]
temp3=CD[temp2,lc] /. { CD[e,lc] -> 0 }
temp4=ApplyRules[temp3,{ad3d1rule,ad1d1rule,ad0d1rule}]
temp5=temp4 /. { ad4[la,lb,ld,lc] -> Dum[ad1243[la,lb,ld,lc]] }
temp5//InputForm

Out[45]=
ad3
   dab
Out[46]=
                                                            p
ad3    - I a e F     - I e ad1  F   - I e ad1  F   + ad1  R
   abd          ad;b          b  ad          a  bd      p  a bd
Out[47]=
                  p
ad3      + ad1  R       - I (e a   F     + a e F     ) - 
   abd;c      p  a bd;c         ;c  ad;b        ad;bc
 
  I e (ad1  F     + ad1    F  ) - I e (ad1  F     + ad1    F  ) + 
          b  ad;c      b;c  ad            a  bd;c      a;c  bd
 
           p
  ad1    R
     p;c  a bd
Out[48]=
ad4     - I e ad1  F     - I e ad1  F     - I e ad1  F     + 
   abdc          c  ad;b          b  ad;c          a  bd;c
 
         p
  ad1  R       - I a e F      - I e ad2   F   - I e ad2   F   + 
     p  a bd;c          ad;bc          bc  ad          ac  bd
 
          p
  ad2   R
     pc  a bd
Out[49]=
ad4     - I e ad1  F     - I e ad1  F     - I e ad1  F     + 
   abcd          c  ad;b          b  ad;c          a  bd;c
 
         p
  ad1  R       - I a e F      - I e ad2   F   - I e ad2   F   - 
     p  a bd;c          ad;bc          bc  ad          ac  bd
 
                          p             p             p
  I e ad2   F   + ad2   R     + ad2   R     + ad2   R
         ab  cd      pc  a bd      pb  a cd      ap  b cd
Out[50]//InputForm=
ad4[la, lb, lc, ld] - I*e*ad1[lc]*CD[MaxwellF[la, ld], lb] - 
  I*e*ad1[lb]*CD[MaxwellF[la, ld], lc] - 
  I*e*ad1[la]*CD[MaxwellF[lb, ld], lc] + 
  ad1[l1]*CD[RiemannR[la, u1, lb, ld], lc] - 
  I*a*e*CD[MaxwellF[la, ld], lb, lc] - 
  I*e*ad2[lb, lc]*MaxwellF[la, ld] - I*e*ad2[la, lc]*MaxwellF[lb, ld] - 
  I*e*ad2[la, lb]*MaxwellF[lc, ld] + 
  ad2[l1, lc]*RiemannR[la, u1, lb, ld] + 
  ad2[l1, lb]*RiemannR[la, u1, lc, ld] + 
  ad2[la, l1]*RiemannR[lb, u1, lc, ld]
As always, the next step is to DefUnique these results for use in the next iteration, which would reorder five gauge covariant derivatives of "a".