6.9. Derivatives on inner vbundles

It is possible to define a derivative which acts on an inner vector bundle (as used in Yang-Mills gauge theories).

Define a connection on a complex inner vbundle:

In[513]:=

DefCovD[ICD[-a], InnerC, {"*", "D"}]

** DefCovD: Defining covariant derivative ICD[-a] .

** DefTensor: Defining vanishing torsion tensor TorsionICD[a, -b, -c] .

** DefTensor: Defining symmetric Christoffel tensor ChristoffelICD[a, -b, -c] .

** DefTensor: Defining Riemann tensor RiemannICD[-a, -b, -c, d] . Antisymmetric only in the first pair.

** DefTensor: Defining non-symmetric Ricci tensor RicciICD[-a, -b] .

** DefCovD:  Contractions of Riemann automatically replaced by Ricci.

** DefTensor: Defining nonsymmetric AChristoffel tensor  AChristoffelICD[, -b, -ℭ] .

** DefTensor: Defining nonsymmetric AChristoffel tensor  AChristoffelICD†[†, -b, -ℭ†] .

** DefTensor: Defining FRiemann tensor FRiemannICD[-a, -b, -ℭ, ] . Antisymmetric only in the first pair.

** DefTensor: Defining FRiemann tensor FRiemannICD†[-a, -b, -ℭ†, †] . Antisymmetric only in the first pair.

We see that a single torsion tensor has been defined, but two Christoffel and two Riemann tensors. The inner Christoffel is called AChristoffelICD and the inner Riemann tensor is called FRiemannICD. They are both complex tensors and therefore have their complex conjugates. Note the different vbundles of their indices.

In[514]:=

? AChristoffelICD

Global`AChristoffelICD

Dagger[AChristoffelICD]^=AChristoffelICD†
DependenciesOfTensor[AChristoffelICD]^={M3}
Info[AChristoffelICD]^={nonsymmetric AChristoffel tensor ,}
MasterOf[AChristoffelICD]^=ICD
PrintAs[AChristoffelICD]^=A[D]
ServantsOf[AChristoffelICD]^={AChristoffelICD†}
SlotsOfTensor[AChristoffelICD]^={InnerC,-TangentM3,-InnerC}
SymmetryGroupOfTensor[AChristoffelICD]^=StrongGenSet[{},GenSet[]]
TensorID[AChristoffelICD]^={AChristoffel,ICD,PD}
xTensorQ[AChristoffelICD]^=True

In[515]:=

? FRiemannICD

Global`FRiemannICD

Dagger[FRiemannICD]^=FRiemannICD†
DependenciesOfTensor[FRiemannICD]^={M3}
Info[FRiemannICD]^={FRiemann tensor,Antisymmetric only in the first pair.}
MasterOf[FRiemannICD]^=ICD
PrintAs[FRiemannICD]^=FICD
ServantsOf[FRiemannICD]^={FRiemannICD†}
SlotsOfTensor[FRiemannICD]^={-TangentM3,-TangentM3,-InnerC,InnerC}
SymmetryGroupOfTensor[FRiemannICD]^=StrongGenSet[{1},GenSet[-Cycles[{1,2}]]]
TensorID[FRiemannICD]^={FRiemann,ICD}
xTensorQ[FRiemannICD]^=True

In[516]:=

? AChristoffelICD†

Global`AChristoffelICD†

Dagger[AChristoffelICD†]^=AChristoffelICD
DependenciesOfTensor[AChristoffelICD†]^={M3}
Info[AChristoffelICD†]^={nonsymmetric AChristoffel tensor ,}
MasterOf[AChristoffelICD†]^=AChristoffelICD
PrintAs[AChristoffelICD†]^=A[D]†
SlotsOfTensor[AChristoffelICD†]^={InnerC†,-TangentM3,-InnerC†}
SymmetryGroupOfTensor[AChristoffelICD†]^=StrongGenSet[{},GenSet[]]
TensorID[AChristoffelICD†]^={AChristoffel,ICD,PD}
xTensorQ[AChristoffelICD†]^=True

In[517]:=

? FRiemannICD†

Global`FRiemannICD†

Dagger[FRiemannICD†]^=FRiemannICD
DependenciesOfTensor[FRiemannICD†]^={M3}
Info[FRiemannICD†]^={FRiemann tensor,Antisymmetric only in the first pair.}
MasterOf[FRiemannICD†]^=FRiemannICD
PrintAs[FRiemannICD†]^=FICD†
SlotsOfTensor[FRiemannICD†]^={-TangentM3,-TangentM3,-InnerC†,InnerC†}
SymmetryGroupOfTensor[FRiemannICD†]^=StrongGenSet[{1},GenSet[-Cycles[{1,2}]]]
TensorID[FRiemannICD†]^={FRiemann,ICD}
xTensorQ[FRiemannICD†]^=True

The derivative ICD has curvature in both the tangent bundle TangentM3 and in the inner bundle InnerC. That becomes most apparent when changing to a different derivative:

In[518]:=

DefTensor[X[a, ], M3, Dagger→Complex]

** DefTensor: Defining tensor X[a, ] .

** DefTensor: Defining tensor X†[a, †] .

In[519]:=

ICD[-c][X[a, ]]

Out[519]=

D_c^ X_  ^a

In[520]:=

ChangeCovD[%, ICD]

Out[520]=

A[D] _ ( c)^   X_  ^a + Γ[D] _ ( cb)^a   X_  ^b + ∂_c^ X_  ^a

In[521]:=

ChangeCovD[%, PD, ICD]

Out[521]=

D_c^ X_  ^a

or when commuting two derivatives:

In[522]:=

ICD[-c] @ ICD[-d] @ X[a, ]

Out[522]=

D_c^ D_d^ X_  ^a

In[523]:=

SortCovDs[%]

Out[523]=

FICD_dc ^(   ) X_  ^a + R[D] _dcb ^(   a) X_  ^b + D_d^ D_c^ X_  ^a

For their complex conjugates:

In[524]:=

ICD[-c][X[a, ]]//Dagger

Out[524]=

D_c^ X†_   ^a†

In[525]:=

ChangeCovD[%, ICD]

Out[525]=

In[526]:=

ICD[-c] @ ICD[-d] @ X[a, ]//Dagger

Out[526]=

D_c^ D_d^ X†_   ^a†

In[527]:=

SortCovDs[%]

Out[527]=

In[528]:=

UndefTensor[X]

** UndefTensor: Undefined tensor X†

** UndefTensor: Undefined tensor X

An important concept, introduced in version 0.9.1 of xTensor`, is that of extension of a derivative.  Given a derivative Cd it is possible to define a second one CD in such a way that the associated tensors of CD on its tangent bundle will be those of Cd. We say then that CD extends Cd, or that CD has been constructed by extension of Cd.

Let us take the Cd derivative as base covd:

In[529]:=

$CovDs

Out[529]=

{PD, Cd, ICD}

We now extend that derivative to CD:

In[530]:=

DefCovD[CD[-a], InnerC, {"#", "D"}, ExtendedFrom→Cd]

** DefCovD: Defining covariant derivative CD[-a] .

** DefTensor: Defining nonsymmetric AChristoffel tensor  AChristoffelCD[, -b, -ℭ] .

** DefTensor: Defining nonsymmetric AChristoffel tensor  AChristoffelCD†[†, -b, -ℭ†] .

** DefTensor: Defining FRiemann tensor FRiemannCD[-a, -b, -ℭ, ] . Antisymmetric only in the first pair.

** DefTensor: Defining FRiemann tensor FRiemannCD†[-a, -b, -ℭ†, †] . Antisymmetric only in the first pair.

Now the tensors associated to the extended derivative will be those of the base derivative:

In[531]:=

{ChristoffelCD[a, -b, -c], RiemannCD[-a, -b, -c, d], TorsionCD[a, -b, -c]}

Out[531]=

{Γ[▽] _ ( bc)^a  , R[▽] _abc ^(   d), 0}

And then we have the inner curvature of the extended derivative:

In[532]:=

{AChristoffelCD[, -b, -ℭ], FRiemannCD[-a, -b, -ℭ, ]}

Out[532]=

{A[D] _ ( bℭ)^  , FCD_abℭ ^(   )}

We can undefine the extended derivative, without removing any of the base derivative objects:

In[533]:=

UndefCovD[CD]

** UndefTensor: Undefined nonsymmetric AChristoffel tensor AChristoffelCD†

** UndefTensor: Undefined nonsymmetric AChristoffel tensor AChristoffelCD

** UndefTensor: Undefined FRiemann tensor FRiemannCD†

** UndefTensor: Undefined FRiemann tensor FRiemannCD

** UndefCovD: Undefined covariant derivative CD

In[534]:=

{ChristoffelCd[a, -b, -c], TorsionCd[a, -b, -c], RiemannCd[-a, -b, -c, d], RicciCd[-a, -b]}

Out[534]=

{Γ[▽] _ ( bc)^a  , 0, R[▽] _abc ^(   d), R[▽] _ab^  }

In[535]:=

{ChristoffelCD[a, -b, -c], TorsionCD[a, -b, -c], RiemannCD[-a, -b, -c, d], RicciCD[-a, -b]}

Out[535]=

{ChristoffelCD[a, -b, -c], TorsionCD[a, -b, -c], RiemannCD[-a, -b, -c, d], RicciCD[-a, -b]}

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