Backward wave oscillator: Difference between revisions
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In [[theoretical physics]], '''p-form electrodynamics''' is a generalization of Maxwell's theory of [[electromagnetism]]. | |||
==Ordinary (viz. one-form) Abelian electrodynamics== | |||
We have a one-form '''A''', a [[gauge symmetry]] | |||
:<math>\mathbf{A} \rightarrow \mathbf{A} + d\alpha</math> | |||
where α is any arbitrary fixed [[0-form]] and d is the [[exterior derivative]], and a gauge-invariant [[vector current]] '''J''' with [[tensor density|density]] 1 satisfying the [[continuity equation]] | |||
:<math>d*\mathbf{J}=0</math> | |||
where * is the [[Hodge dual]]. | |||
Alternatively, we may express '''J''' as a (''d'' − 1)-[[closed form]]. | |||
'''F''' is a [[gauge invariant]] [[2-form]] defined as the exterior derivative <math>\mathbf{F}=d\mathbf{A}</math>. | |||
'''A''' satisfies the equation of motion | |||
:<math>d*\mathbf{F}=*\mathbf{J}</math> | |||
(this equation obviously implies the continuity equation). | |||
This can be derived from the [[action (physics)|action]] | |||
:<math>S=\int_M \left[\frac{1}{2}\mathbf{F}\wedge *\mathbf{F} - \mathbf{A} \wedge *\mathbf{J}\right]</math> | |||
where M is the [[spacetime]] [[manifold]]. | |||
==p-form Abelian electrodynamics== | |||
We have a [[p-form]] '''B''', a [[gauge symmetry]] | |||
:<math>\mathbf{B} \rightarrow \mathbf{B} + d\mathbf{\alpha}</math> | |||
where '''α''' is any arbitrary fixed (p-1)-form and d is the [[exterior derivative]], | |||
and a gauge-invariant [[p-vector]] '''J''' with [[tensor density|density]] 1 satisfying the [[continuity equation]] | |||
:<math>d*\mathbf{J}=0</math> | |||
where * is the [[Hodge dual]]. | |||
Alternatively, we may express '''J''' as a (d-p)-[[closed form]]. | |||
'''C''' is a [[gauge invariant]] (p+1)-form defined as the exterior derivative <math>\mathbf{C}=d\mathbf{B}</math>. | |||
'''B''' satisfies the equation of motion | |||
:<math>d*\mathbf{C}=*\mathbf{J}</math> | |||
(this equation obviously implies the continuity equation). | |||
This can be derived from the [[action (physics)|action]] | |||
:<math>S=\int_M \left[\frac{1}{2}\mathbf{C}\wedge *\mathbf{C} +(-1)^p \mathbf{B} \wedge *\mathbf{J}\right]</math> | |||
where M is the [[spacetime]] [[manifold]]. | |||
Other [[sign convention]]s do exist. | |||
The [[Kalb-Ramond field]] is an example with ''p=2'' in string theory; the [[Ramond-Ramond field]]s whose charged sources are [[D-brane]]s are examples for all values of ''p''. In 11d [[supergravity]] or [[M-theory]], we have a 3-form electrodynamics. | |||
==Non-abelian generalization== | |||
Just as we have non-abelian generalizations of electrodynamics, leading to [[Yang-Mills theory|Yang-Mills theories]], we also have nonabelian generalizations of p-form electrodynamics. They typically require the use of [[gerbe]]s. | |||
==References== | |||
* Henneaux; Teitelboim (1986), "p-Form electrodynamics", ''Foundations of Physics'' '''16''' (7): 593-617, [[Digital object identifier|doi]]:[http://dx.doi.org/10.1007/BF01889624 10.1007/BF01889624] | |||
*{{cite doi|10.1103/PhysRevD.83.125015|noedit}} | |||
* Navarro; Sancho (2012), "Energy and electromagnetism of a differential k-form ", ''J. Math. Phys.'' '''53''', 102501 (2012) [[Digital object identifier|doi]]:[http://dx.doi.org/10.1063/1.4754817 10.1063/1.4754817] | |||
{{DEFAULTSORT:P-Form Electrodynamics}} | |||
[[Category:Quantum field theory]] | |||
{{quantum-stub}} |
Latest revision as of 16:57, 25 June 2013
In theoretical physics, p-form electrodynamics is a generalization of Maxwell's theory of electromagnetism.
Ordinary (viz. one-form) Abelian electrodynamics
We have a one-form A, a gauge symmetry
where α is any arbitrary fixed 0-form and d is the exterior derivative, and a gauge-invariant vector current J with density 1 satisfying the continuity equation
where * is the Hodge dual.
Alternatively, we may express J as a (d − 1)-closed form.
F is a gauge invariant 2-form defined as the exterior derivative .
A satisfies the equation of motion
(this equation obviously implies the continuity equation).
This can be derived from the action
where M is the spacetime manifold.
p-form Abelian electrodynamics
We have a p-form B, a gauge symmetry
where α is any arbitrary fixed (p-1)-form and d is the exterior derivative,
and a gauge-invariant p-vector J with density 1 satisfying the continuity equation
where * is the Hodge dual.
Alternatively, we may express J as a (d-p)-closed form.
C is a gauge invariant (p+1)-form defined as the exterior derivative .
B satisfies the equation of motion
(this equation obviously implies the continuity equation).
This can be derived from the action
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 11d supergravity or M-theory, we have a 3-form electrodynamics.
Non-abelian generalization
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
- Henneaux; Teitelboim (1986), "p-Form electrodynamics", Foundations of Physics 16 (7): 593-617, doi:10.1007/BF01889624
- Template:Cite doi
- Navarro; Sancho (2012), "Energy and electromagnetism of a differential k-form ", J. Math. Phys. 53, 102501 (2012) doi:10.1063/1.4754817