en>Chetvorno: DAB oscillator

2013-06-25T14:57:05Z

DAB oscillator

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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}}

en>BattyBot: changed {{no footnotes}} to {{more footnotes}} & general fixes using AWB (8097)

2012-07-17T05:27:11Z

changed {{no footnotes}} to {{more footnotes}} & general fixes using AWB (8097)

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Backward wave oscillator - Revision history

en>Chetvorno: DAB oscillator

en>BattyBot: changed {{no footnotes}} to {{more footnotes}} & general fixes using AWB (8097)