=0
d
F
=
0
β=β
d
β
F
=
β
J
where in fact the second equality is not needed because of the identity 2=0
d
2
=
0
. In abstract tensor notation, these equations expand to
=2β[]
F
a
b
=
2
β
[
a
A
b
]
β[]=0
β
[
a
F
b
c
]
=
0
β=
β
a
F
a
b
=
J
b
where in the above, additional compact notation has been used to simplify things: square brackets mean antisymmetrizing indices, and the covariant derivative β
β
a
is related to a flat derivative β
β
a
and a connection Ξ
Ξ
b
a
c
. Now, performing a 3+1 decomposition (here in flat space-time) Maxwell's equations break down as
ββ
β=0
β
β
B
β
=
0
ββ
β=4
β
β
E
β
=
4
Ο
Ο
βββ=ββΓβ
β
B
β
β
t
=
β
β
Γ
B
β
βββ=+βΓββ4β.
β
E
β
β
t
=
+
β
Γ
B
β
β
4
Ο
J
β
.
In the above, there is yet more compactifying notation: the divergence is defined as
ββ
β=ββ
β
β
v
β
=
β
v
i
β
x
i
and the curl is defined as
(βΓβ)=ββ.
(
β
Γ
v
β
)
i
=
Ο΅
i
j
k
β
v
k
β
x
j
.
In the above, still more compactifying notation has been used, namely the Einstein summation convention. Written out in components, the divergence is
ββ
β=ββ+ββ+ββ
β
β
v
β
=
β
v
x
β
x
+
β
v
y
β
y
+
β
v
z
β
z
and curl is defined as
(βΓβ)=βββββ
(
β
Γ
v
β
)
z
=
β
v
y
β
x
β
β
v
x
β
y
and similarly for the other two components.
As you can see, the extremely compact and powerful notation
=0
d
F
=
0
β=β
d
β
F
=
β
J
expands out to a large set of component equationsβbut 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:
=+Higgs+lep.+qrk.+Yuk.
L
=
L
Y
M
+
L
H
i
g
g
s
+
L
l
e
p
.
+
L
q
r
k
.
+
L
Y
u
k
.
where
=β14Tr
L
Y
M
=
β
1
4
T
r
F
a
b
F
a
b
Higgs=β(β )()+14(β β122)2
L
H
i
g
g
s
=
β
(
D
a
Ο
β
)
(
D
a
Ο
)
+
1
4
Ξ»
(
Ο
β
Ο
β
1
2
v
2
)
2
lep.=ββ β§Έβ+Β―β β§ΈΒ―
L
l
e
p
.
=
i
β
I
β
β§Έ
D
β
I
+
i
e
Β―
I
β
β§Έ
D
e
Β―
I
qrk.=β β§Έ+Β―β β§ΈΒ―+Β―β β§ΈΒ―
L
q
r
k
.
=
i
q
I
β
β§Έ
D
q
I
+
i
u
Β―
I
β
β§Έ
D
u
Β―
+
i
d
Β―
I
β
β§Έ
D
d
Β―
I
Yuk.=ββΒ―ββ²Β―
L
Y
u
k
.
=
β
y
I
J
Ο
β
I
e
Β―
J
β
y
I
J
β²
Ο
q
I
d
Β―
J
ββ³β Β―+β..
β
y
I
J
β³
Ο
β
q
I
u
Β―
J
+
h
.
c
.
and specifying that F is the curvature on the gauge group β(3)β(2)β(1)
G
β
S
U
(
3
)
β
S
U
(
2
)
β
U
(
1
)
,
D
a
is the gauge covariant derivative on G, β§Έ
β§Έ
D
is the gauge covariant Dirac operator on G, and specifying the representations of the fields: the scalar
Ο
in the rep (1,2,-1/2), and the fermions β
β
in the rep (1,2,-1/2), Β―
e
Β―
in (1,1,+1),
q
in (3,2,+1/6), Β―
u
Β―
in (3Β―,1,β2/3)
(
3
Β―
,
1
,
β
2
/
3
)
, and Β―
d
Β―
in (3Β―,1,+1/3)
(
3
Β―
,
1
,
+
1
/
3
)
; where I is a generation index running over 1,2,3, and where everything else is a constant.
42.7K viewsView 50 upvotes Β· Answer requested by Mario Ng
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