Advanced Search

Question: Physics Commutator does not sum over repeated indices

Question:Physics Commutator does not sum over repeated indices

digerdiga 390 Maple 2019
commutator physics
March 22 2020
1

Hello,

 

 

 

> 

restart; with(Physics); Setup(mathematicalnotation = true); Setup(dimension = 3, metric = Euclidean, spacetimeindices = lowercaselatin); g_[]; Define(X[i], P[k]); Setup(quantumop = {P, X}, algebrarules = {%Commutator(P[i], P[j]) = 0, %Commutator(X[i], X[j]) = 0, %Commutator(X[j], P[k]) = Physics:-`*`(I, g_[j, k])}); L2 := Simplify(Physics:-`^`(Physics:-`*`(Physics:-`*`(LeviCivita[i, j, k], X[j]), P[k]), 2)); L2X := Simplify(Commutator(L2, X[k])); L2L2X__1 := Simplify(Commutator(L2, L2X)); L2L2X__2 := Simplify(Expand(AntiCommutator(Physics:-`*`(2, L2), X[k]))); Simplify(L2L2X__1-L2L2X__2)

[mathematicalnotation = true]

  

`The dimension and signature of the tensor space are set to `[3, `- - +`]

  

`The Euclidean metric in cartesian coordinates`

  

`Changing the signature of the tensor spacetime to: `*`+ + +`

  

[dimension = 3, metric = {(1, 1) = 1, (2, 2) = 1, (3, 3) = 1}, spacetimeindices = lowercaselatin]

  

Physics:-g_[a, b] = Matrix(%id = 18446746397905475934)

  

`Defined objects with tensor properties`

  

{Physics:-Dgamma[a], P[k], Physics:-Psigma[a], X[i], Physics:-d_[a], Physics:-g_[a, b], Physics:-LeviCivita[a, b, c]}

  

`* Partial match of '`*quantumop*`' against keyword '`*quantumoperators*`' `

  

_______________________________________________________

  

[algebrarules = {%Commutator(P[i], P[j]) = 0, %Commutator(X[i], X[j]) = 0, %Commutator(X[j], P[k]) = I*Physics:-g_[j, k]}, quantumoperators = {P, X}]

  

(2*I)*Physics:-`*`(X[j], P[j])+Physics:-`*`(Physics:-`^`(X[j], 2), Physics:-`^`(P[k], 2))-Physics:-`*`(X[j], X[k], P[j], P[k])

  

2*X[k]-(2*I)*Physics:-`*`(Physics:-`^`(X[b], 2), P[k])+(2*I)*Physics:-`*`(X[c], X[k], P[c])

  

(12*I)*Physics:-`*`(X[c], X[k], P[c])-(4*I)*Physics:-`*`(Physics:-`^`(X[b], 2), P[k])+4*X[k]+4*Physics:-`*`(Physics:-`^`(X[b], 2), X[k], Physics:-`^`(P[h], 2))-4*Physics:-`*`(X[a], Physics:-`^`(X[b], 2), P[a], P[k])+4*Physics:-`*`(Physics:-`^`(X[b], 2), X[f], P[f], P[k])-4*Physics:-`*`(X[f], X[i], X[k], P[f], P[i])

  

4*X[k]+(12*I)*Physics:-`*`(X[c], X[k], P[c])+4*Physics:-`*`(Physics:-`^`(X[b], 2), X[k], Physics:-`^`(P[d], 2))-(4*I)*Physics:-`*`(Physics:-`^`(X[b], 2), P[k])-4*Physics:-`*`(X[e], X[f], X[k], P[e], P[f])

  

0

 (1)
> 

NULL

 

 

 

 

Download Commutator_Algebra_with_L2.mw

 

 

 

I have the following issue.

The way it is set at the moment, works kinda in that the final result is 0, which I tried to check.

However, the expression L2L2X[1] is not fully Simplified, since the term

-(4*X[a])*X[b]^2*P[a]*P[k] + (4*X[b]^2)*X[f]*P[f]*P[k]

is actually zero. I suppose that he doesn't manage because of the different indices "a" and "f" here. How can this be fixed?

 

When I comment out "Define" and use "Coordinates(X,P)" instead, it doesn't work. Seems to give the wrong result (Difference is not zero).

 
Please Wait...