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Sagot :
Answer:
F=dA
dF=0
d⋆F=⋆J
where in fact the second equality is not needed because of the identity d2=0 . In abstract tensor notation, these equations expand to
Fab=2∇[aAb]
∇[aFbc]=0
∇aFab=Jb
where in the above, additional compact notation has been used to simplify things: square brackets mean antisymmetrizing indices, and the covariant derivative ∇a is related to a flat derivative ∂a and a connection Γbac . Now, performing a 3+1 decomposition (here in flat space-time) Maxwell's equations break down as
∇⋅B⃗ =0
∇⋅E⃗ =4πρ
∂B⃗ ∂t=−∇×B⃗
∂E⃗ ∂t=+∇×B⃗ −4πJ⃗ .
In the above, there is yet more compactifying notation: the divergence is defined as
∇⋅v⃗ =∂vi∂xi
and the curl is defined as
(∇×v⃗ )i=ϵijk∂vk∂xj.
In the above, still more compactifying notation has been used, namely the Einstein summation convention. Written out in components, the divergence is
∇⋅v⃗ =∂vx∂x+∂vy∂y+∂vz∂z
and curl is defined as
(∇×v⃗ )z=∂vy∂x−∂vx∂y
and similarly for the other two components.
As you can see, the extremely compact and powerful notation
dF=0
d⋆F=⋆J
expands out to a large set of component equations—but they have the same content, so it's not very meaningful that the equations in components are much longer than in terms of differential forms.
As another example, Elson Liu answered with an exploded-out form of the Standard Model Lagrangian. However, it can be written much more compactly:
L=LYM+LHiggs+Llep.+Lqrk.+LYuk.
where
LYM=−14TrFabFab
LHiggs=−(Daϕ†)(Daϕ)+14λ(ϕ†ϕ−12v2)2
Llep.=iℓ†I/DℓI+ie¯†I/De¯I
Lqrk.=iq†I/DqI+iu¯†I/Du¯+id¯†I/Dd¯I
LYuk.=−yIJϕℓIe¯J−y′IJϕqId¯J
−y′′IJϕ†qIu¯J+h.c.
and specifying that F is the curvature on the gauge group G≃SU(3)⊗SU(2)⊗U(1) , Da is the gauge covariant derivative on G, /D is the gauge covariant Dirac operator on G, and specifying the representations of the fields: the scalar ϕ in the rep (1,2,-1/2), and the fermions ℓ in the rep (1,2,-1/2), e¯ in (1,1,+1), q in (3,2,+1/6), u¯ in (3¯,1,−2/3) , and d¯ in (3¯,1,+1/3) ; where I is a generation index running over 1,2,3, and where everything else is a constant.
Step-by-step explanation:
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