Tensor[FormInnerProduct] - compute the inner product of two forms with respect to a given metric tensor
Calling Sequences
FormInnerProduct(g, α, β,keyword)
FormInnerProduct(g, g1, α1, β1, keyword)
Parameters
g - a covariant metric tensor on a manifold or on a Lie algebra with frame name, e.g., M
α, β - two forms (of the same degree) on M, or lists of such
α1, β1 - two forms (of the same degree) on M, or lists of such, where M is a Lie algebra with coefficients in a representation space V
g1 - a covariant metric tensor on the representation space V
keyword - the keyword argument inversemetric = h, where h is the inverse of the metric g.
Description
Examples
Let g = gij dxi dxj and let h = hij∂∂xi∂∂xj be the inverse metric. If α= ai dxi and β= bj dxj are 1-forms, then their inner product is α, β = hijai bj. For monomial p-forms α1∧α2 ∧ ... ∧ αp and β1∧β2 ∧ ... ∧ βp , the inner product is given by
α1∧α2 ∧ ... ∧ αp , β1∧β2 ∧ ... ∧ βp = detαr βs.
This formula is extended by bi-linearity to give the general formula for the inner product of a pair of p−forms.
In the special case of forms defined on a Lie algebra with coefficients x and y in a representation, the inner product formula for monomials becomes
x α1∧α2 ∧ ... ∧ αp , y β1∧β2 ∧ ... ∧ βp= gVx,y det αr βs
where x, y ϵ V and gV is the inner product on V.
with⁡DifferentialGeometry:with⁡Tensor:with⁡LieAlgebras:
First define a manifold M with local coordinates x,y,z and define a metric on M.
DGsetup⁡x,y,z,M:
g ≔ evalDG⁡a⁢dx &t dx+b⁢dy &t dy+c⁢dz &t dz
g:=_DG⁡tensor,M,cov_bas,cov_bas,,1,1,a,2,2,b,3,3,c,_DG⁡tensor,M,cov_bas,cov_bas,,1,1,a,2,2,b,3,3,c
Example 1.
Compute the inner product of two 1-forms
α1 ≔ evalDG⁡a1⁢dx+a2⁢dy+a3⁢dz
α1:=_DG⁡form,M,1,1,a1,2,a2,3,a3,_DG⁡form,M,1,1,a1,2,a2,3,a3
β1 ≔ evalDG⁡b1⁢dx+b2⁢dy+b3⁢dz
β1:=_DG⁡form,M,1,1,b1,2,b2,3,b3,_DG⁡form,M,1,1,b1,2,b2,3,b3
FormInnerProduct⁡g,α1,β1
a1⁢b1a+a2⁢b2b+a3⁢b3c
Example 2.
Compute the inner products of a list of monomial 2-forms.
g2 ≔ evalDG⁡a⁢dx &t dx+b⁢dy &t dy+c⁢dz &t dz
g2:=_DG⁡tensor,M,cov_bas,cov_bas,,1,1,a,2,2,b,3,3,c,_DG⁡tensor,M,cov_bas,cov_bas,,1,1,a,2,2,b,3,3,c
Ω ≔ evalDG⁡dx &w dy,dx &w dz,dy &w dz
Ω:=_DG⁡form,M,2,1,2,1,_DG⁡form,M,2,1,2,1,_DG⁡form,M,2,1,3,1,_DG⁡form,M,2,1,3,1,_DG⁡form,M,2,2,3,1,_DG⁡form,M,2,2,3,1
FormInnerProduct⁡g2,Ω,Ω
Compute the inner product of a pair of 2-forms.
α2 ≔ evalDG⁡2⁢dx &w dy+dy &w dz
α2:=_DG⁡form,M,2,1,2,2,2,3,1,_DG⁡form,M,2,1,2,2,2,3,1
β2 ≔ evalDG⁡3⁢dx &w dz+4⁢dy &w dz
β2:=_DG⁡form,M,2,1,3,3,2,3,4,_DG⁡form,M,2,1,3,3,2,3,4
FormInnerProduct⁡g2,α2,α2
4a⁢b+1b⁢c
Example 3.
In this example we compute the inner products of forms defined on a Lie algebra with coefficients in a representation.
LD ≔ SimpleLieAlgebraData⁡so(4),so4
LD:=e1,e2=e4,e1,e3=e5,e1,e4=−e2,e1,e5=−e3,e2,e3=e6,e2,e4=e1,e2,e6=−e3,e3,e5=e1,e3,e6=e2,e4,e5=e6,e4,e6=−e5,e5,e6=e4
DGsetup⁡LD
Lie algebra: so4
DGsetup⁡x1,x2,x3,x4,V
frame name: V
ρ ≔ StandardRepresentation⁡so4,representationspace=V
ρ:=_DG⁡Representation,so4,0,V,0,,0−100100000000000,00−10000010000000,000−1000000001000,000000−1001000000,0000000−100000100,00000000000−10010,_DG⁡Representation,so4,0,V,0,,0−100100000000000,00−10000010000000,000−1000000001000,000000−1001000000,0000000−100000100,00000000000−10010
DGsetup⁡ρ,so4V,O,o
Lie algebra with coefficients: so4V
g ≔ KillingForm⁡so4V
g:=_DG⁡tensor,so4V,cov_bas,cov_bas,,1,1,−4,2,2,−4,3,3,−4,4,4,−4,5,5,−4,6,6,−4,_DG⁡tensor,so4V,cov_bas,cov_bas,,1,1,−4,2,2,−4,3,3,−4,4,4,−4,5,5,−4,6,6,−4
h ≔ InverseMetric⁡g
h:=_DG⁡tensor,so4V,con_bas,con_bas,,1,1,−14,2,2,−14,3,3,−14,4,4,−14,5,5,−14,6,6,−14,_DG⁡tensor,so4V,con_bas,con_bas,,1,1,−14,2,2,−14,3,3,−14,4,4,−14,5,5,−14,6,6,−14
gV ≔ evalDG⁡dx1 &t dx1+dx2 &t dx2+dx3 &t dx3+dx4 &t dx4
gV:=_DG⁡tensor,V,cov_bas,cov_bas,,1,1,1,2,2,1,3,3,1,4,4,1,_DG⁡tensor,V,cov_bas,cov_bas,,1,1,1,2,2,1,3,3,1,4,4,1
Compute the inner product of a pair of zero forms.
FormInnerProduct⁡g,gV,a⁢x1+b⁢x2,c⁢x1+d⁢x2
a⁢c+b⁢d
Compute the inner product of a pair of 1-forms.
FormInnerProduct⁡g,gV,x1⁢o1,x1⁢o3
0
FormInnerProduct⁡g,gV,x2⁢o1,x1⁢o1
FormInnerProduct⁡g,gV,x2⁢o2,x2⁢o2
−14
FormInnerProduct⁡g,gV,x2⁢o1 &w o2,x2⁢o1 &w o2
116
Compute the length of a 2-form.
α3 ≔ evalDG⁡a⁢x2⁢o1 &w o2+b⁢x4⁢o1 &w o3+c⁢x⁢o2 &w o5
α3:=_DG⁡form,so4V,2,1,2,a⁢x2,1,3,b⁢x4,2,5,c⁢x,_DG⁡form,so4V,2,1,2,a⁢x2,1,3,b⁢x4,2,5,c⁢x
FormInnerProduct⁡g,gV,α3,α3
14⁢a2+b2
See Also
DifferentialGeometry
Tensor
ContractIndices
InverseMetric
RaiseLowerIndices
SpinorInnerProduct
TensorInnerProduct
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