Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
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:
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
