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O r d i n a r y ( v i a . o n e - f o r m ) A b e l i a n e l e c t r o d y n a m i c s
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p - f o r m A b e l i a n e l e c t r o d y n a m i c s
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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Ordinary (via. one-form) Abelian electrodynamics [ edit ]
We have a one-form
A
{\displaystyle \mathbf {A} }
, a gauge symmetry
A
→
A
+
d
α
,
{\displaystyle \mathbf {A} \rightarrow \mathbf {A} +d\alpha ,}
where
α
{\displaystyle \alpha }
is any arbitrary fixed 0-form and
d
{\displaystyle d}
is the exterior derivative , and a gauge-invariant vector current
J
{\displaystyle \mathbf {J} }
with density 1 satisfying the continuity equation
d
⋆
J
=
0
,
{\displaystyle d{\star }\mathbf {J} =0,}
where
⋆
{\displaystyle {\star }}
is the Hodge star operator .
Alternatively, we may express
J
{\displaystyle \mathbf {J} }
as a closed (n − 1) -form, but we do not consider that case here.
F
{\displaystyle \mathbf {F} }
is a gauge-invariant 2-form defined as the exterior derivative
F
=
d
A
{\displaystyle \mathbf {F} =d\mathbf {A} }
.
F
{\displaystyle \mathbf {F} }
satisfies the equation of motion
d
⋆
F
=
⋆
J
{\displaystyle d{\star }\mathbf {F} ={\star }\mathbf {J} }
(this equation obviously implies the continuity equation).
This can be derived from the action
S
=
∫
M
[
1
2
F
∧
⋆
F
−
A
∧
⋆
J
]
,
{\displaystyle S=\int _{M}\left[{\frac {1}{2}}\mathbf {F} \wedge {\star }\mathbf {F} -\mathbf {A} \wedge {\star }\mathbf {J} \right],}
where
M
{\displaystyle M}
is the spacetime manifold .
p -form Abelian electrodynamics[ edit ]
We have a p -form
B
{\displaystyle \mathbf {B} }
, a gauge symmetry
B
→
B
+
d
α
,
{\displaystyle \mathbf {B} \rightarrow \mathbf {B} +d\mathbf {\alpha } ,}
where
α
{\displaystyle \alpha }
is any arbitrary fixed (p − 1) -form and
d
{\displaystyle d}
is the exterior derivative , and a gauge-invariant p -vector
J
{\displaystyle \mathbf {J} }
with density 1 satisfying the continuity equation
d
⋆
J
=
0
,
{\displaystyle d{\star }\mathbf {J} =0,}
where
⋆
{\displaystyle {\star }}
is the Hodge star operator .
Alternatively, we may express
J
{\displaystyle \mathbf {J} }
as a closed (n − p ) -form.
C
{\displaystyle \mathbf {C} }
is a gauge-invariant (p + 1) -form defined as the exterior derivative
C
=
d
B
{\displaystyle \mathbf {C} =d\mathbf {B} }
.
B
{\displaystyle \mathbf {B} }
satisfies the equation of motion
d
⋆
C
=
⋆
J
{\displaystyle d{\star }\mathbf {C} ={\star }\mathbf {J} }
(this equation obviously implies the continuity equation).
This can be derived from the action
S
=
∫
M
[
1
2
C
∧
⋆
C
+
(
−
1
)
p
B
∧
⋆
J
]
{\displaystyle S=\int _{M}\left[{\frac {1}{2}}\mathbf {C} \wedge {\star }\mathbf {C} +(-1)^{p}\mathbf {B} \wedge {\star }\mathbf {J} \right]}
where M is the spacetime manifold .
Other sign conventions do exist.
The Kalb–Ramond field is an example with p = 2 in string theory; the Ramond–Ramond fields whose charged sources are D-branes are examples for all values of p . In eleven-dimensional supergravity or M-theory , we have a 3-form electrodynamics.
Non-abelian generalization [ edit ]
Just as we have non-abelian generalizations of electrodynamics, leading to Yang–Mills theories , we also have nonabelian generalizations of p -form electrodynamics. They typically require the use of gerbes .
References [ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=P-form_electrodynamics&oldid=1228735478 "
C a t e g o r i e s :
● E l e c t r o d y n a m i c s
● S t r i n g t h e o r y
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n i s d i f f e r e n t f r o m W i k i d a t a
● T h i s p a g e w a s l a s t e d i t e d o n 1 2 J u n e 2 0 2 4 , a t 2 1 : 5 7 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
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