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 11-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
Henneaux; Teitelboim (1986), "p -Form electrodynamics", Foundations of Physics 16 (7): 593-617, doi :10.1007/BF01889624
Bunster, C.; Henneaux, M. (2011). "Action for twisted self-duality". Physical Review D . 83 (12): 125015. arXiv :1103.3621 . Bibcode :2011PhRvD..83l5015B. doi :10.1103/PhysRevD.83.125015. S2CID 119268081.
Navarro; Sancho (2012), "Energy and electromagnetism of a differential k -form ", J. Math. Phys. 53 , 102501 (2012) doi :10.1063/1.4754817