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Question: Calculation of the trace of the SU(2) field strength tensor

Question:Calculation of the trace of the SU(2) field strength tensor

hendriksdf5 50 Maple 2024
tensor physics index
July 24 2024
1

I would like to calculate the following quantity: 

 

Where F is the SU(2) field strength tensor given by:

The gauge field V (in my code A) is defined as

 

where rj is the unit vector in spherical coordinates.

I tried to calculate it with maple, however, the result is not correct. I should get a scalar function, but my result still contains dependencies on x,y,z. And I really don't know why. I have defined the gauge field in (11) and the field strength tensor in (14). I could imagine that SumOverRepeatedIndices() in (16) does not work as I think (For each a = (1,2,3) I would like a summation over mu and nu). Greek letters are my spacetime indices and lowercase letters are my space indices. Do I perhaps have to use SU(2) indices instead of the space indices? But how exactly does a SU(2) index differ from a space index?    

restart

with(Physics)

__________________________________________________________________

 (1)

with(Vectors)NULL

Setup(spacetimeindices = greek, spaceindices = lowercaselatin, su2indices = uppercaselatin, signature = `+++-`, coordinates = spherical)

[coordinatesystems = {X}, signature = `+ + + -`, spaceindices = lowercaselatin, spacetimeindices = greek, su2indices = uppercaselatin]

 (2)

Setup(realobjects = {g, diff(x, x), diff(y(x), x), diff(z(x), x), f__A(X[1])})

[realobjects = {g, phi, r, rho, theta, x, `x'`, y, `y'`, z, `z'`, f__A(r)}]

 (3)

"x'(r,theta,phi) := r * sin(theta) * cos(phi)"

proc (r, theta, phi) options operator, arrow, function_assign; Physics:-`*`(r, sin(theta), cos(phi)) end proc

 (4)

"y'(r,theta,phi) := r * sin(theta) * sin(phi)"

proc (r, theta, phi) options operator, arrow, function_assign; Physics:-`*`(r, sin(theta), sin(phi)) end proc

 (5)

"z'(r,theta,phi) := r * cos(theta)"

proc (r, theta, phi) options operator, arrow, function_assign; Physics:-`*`(r, cos(theta)) end proc

 (6)

 

This ist my unit vector:

Define(R[a] = [(diff(x, x))/r, (diff(y(x), x))/r, (diff(z(x), x))/r]) 

{Physics:-Dgamma[mu], Physics:-Psigma[mu], R[a], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (7)

R[definition]

R[a] = [`x'`/r, `y'`/r, `z'`/r]

 (8)

Parse:-ConvertTo1D, "first argument to _Inert_ASSIGN must be assignable"

r^2

 (9)

"Define(A[mu,~a] =(1-`f__A`(X[1]) )/(g*X[1])*LeviCivita[a, mu,j,4]* R[j] ) "

{A[mu, `~a`], Physics:-Dgamma[mu], F[mu, nu, a], Physics:-Psigma[mu], R[a], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (10)

A[definition]

A[mu, `~a`] = (1-f__A(r))*Physics:-LeviCivita[4, a, j, mu]*R[j]/(g*r)

 (11)

A[]

A[mu, a] = Matrix(%id = 36893489989479580364)

 (12)

Define(F[mu, nu, a] = d_[nu](A[mu, a])-d_[mu](A[nu, a])+LeviCivita[a, b, c, 4]*A[mu, `~b`]*A[nu, `~c`])

{A[i, `~a`], Physics:-Dgamma[mu], F[mu, nu, a], Physics:-Psigma[mu], R[a], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

 (13)

F[definition]

F[mu, nu, a] = -Physics:-d_[nu](A[a, mu], [X])+Physics:-d_[mu](A[a, nu], [X])-Physics:-LeviCivita[4, a, b, c]*A[mu, `~b`]*A[nu, `~c`]

 (14)

simplify(F[])

F[mu, nu, a] = _rtable[36893489989585113204]

 (15)

"-1/(4)Simplify(SumOverRepeatedIndices(F[mu,nu,a]*F[~mu,~nu,a])); "

-(1/4)*((2*(-1+f__A(r))*`z'`*r*g-(diff(f__A(r), r))*`z'`*r^2*g-(-1+f__A(r))^2*`z'`*`x'`)^2+4*(-1+f__A(r))^4*`y'`^2*`z'`^2+((diff(f__A(r), r))*`x'`*r^2*g+2*(1-f__A(r))*`x'`*r*g+(1-f__A(r))*(-1+f__A(r))*`z'`^2)^2+((diff(f__A(r), r))*`y'`*r^2*g+2*(1-f__A(r))*`y'`*r*g+(-1+f__A(r))^2*`y'`*`x'`)^2+(2*(-1+f__A(r))*`x'`*r*g-(diff(f__A(r), r))*`x'`*r^2*g-(-1+f__A(r))*(1-f__A(r))*`y'`^2)^2+(-2*(-1+f__A(r))*`z'`*r*g+(diff(f__A(r), r))*`z'`*r^2*g+(-1+f__A(r))^2*`z'`*`x'`)^2+(-(diff(f__A(r), r))*`x'`*r^2*g-2*(1-f__A(r))*`x'`*r*g-(1-f__A(r))*(-1+f__A(r))*`z'`^2)^2+2*(-1+f__A(r))^4*`x'`^4+2*(-1+f__A(r))^4*`y'`^2*`x'`^2+2*(-1+f__A(r))^4*`z'`^2*`x'`^2+(-(diff(f__A(r), r))*`y'`*r^2*g-2*(1-f__A(r))*`y'`*r*g-(-1+f__A(r))^2*`y'`*`x'`)^2+(-2*(-1+f__A(r))*`x'`*r*g+(diff(f__A(r), r))*`x'`*r^2*g+(-1+f__A(r))*(1-f__A(r))*`y'`^2)^2)/(r^8*g^4)

 (16)

L__FST := simplify(-(1/4)*((2*(-1+f__A(r))*`z'`*r*g-(diff(f__A(r), r))*`z'`*r^2*g-(-1+f__A(r))^2*`z'`*`x'`)^2+4*(-1+f__A(r))^4*`y'`^2*`z'`^2+((diff(f__A(r), r))*`x'`*r^2*g+2*(1-f__A(r))*`x'`*r*g+(1-f__A(r))*(-1+f__A(r))*`z'`^2)^2+((diff(f__A(r), r))*`y'`*r^2*g+2*(1-f__A(r))*`y'`*r*g+(-1+f__A(r))^2*`y'`*`x'`)^2+(2*(-1+f__A(r))*`x'`*r*g-(diff(f__A(r), r))*`x'`*r^2*g-(-1+f__A(r))*(1-f__A(r))*`y'`^2)^2+(-2*(-1+f__A(r))*`z'`*r*g+(diff(f__A(r), r))*`z'`*r^2*g+(-1+f__A(r))^2*`z'`*`x'`)^2+(-(diff(f__A(r), r))*`x'`*r^2*g-2*(1-f__A(r))*`x'`*r*g-(1-f__A(r))*(-1+f__A(r))*`z'`^2)^2+2*(-1+f__A(r))^4*`x'`^4+2*(-1+f__A(r))^4*`y'`^2*`x'`^2+2*(-1+f__A(r))^4*`z'`^2*`x'`^2+(-(diff(f__A(r), r))*`y'`*r^2*g-2*(1-f__A(r))*`y'`*r*g-(-1+f__A(r))^2*`y'`*`x'`)^2+(-2*(-1+f__A(r))*`x'`*r*g+(diff(f__A(r), r))*`x'`*r^2*g+(-1+f__A(r))*(1-f__A(r))*`y'`^2)^2)/(r^8*g^4))

(-g^2*r^4*(`x'`^2+(1/2)*`y'`^2+(1/2)*`z'`^2)*(diff(f__A(r), r))^2+4*g^2*r^3*(`x'`^2+(1/2)*`y'`^2+(1/2)*`z'`^2)*(-1+f__A(r))*(diff(f__A(r), r))-4*((1/8)*(`x'`^2+`y'`^2+`z'`^2)^2*f__A(r)^2-(1/4)*(`x'`^2+`y'`^2+`z'`^2)^2*f__A(r)+(1/8)*`x'`^4+(g^2*r^2+(1/4)*`y'`^2+(1/4)*`z'`^2)*`x'`^2+(1/2)*(g^2*r^2+(1/4)*`y'`^2+(1/4)*`z'`^2)*(`y'`^2+`z'`^2))*(-1+f__A(r))^2)/(r^8*g^4)

 (17)

 

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