P-form electrodynamics
Encyclopedia
In theoretical physics
, p-form electrodynamics is a generalization of Maxwell's theory of electromagnetism
.
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
.
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 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.
s.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, p-form electrodynamics is a generalization of Maxwell's theory of electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
.
Ordinary (viz. one-form) Abelian electrodynamics
We have a one-form A, a gauge symmetrywhere α is any arbitrary fixed 0-form and d is the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
, and a gauge-invariant vector current J with density
Tensor density
In differential geometry, a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system to another , except that it is additionally multiplied or weighted by a power of the Jacobian determinant of the...
1 satisfying the continuity equation
Continuity equation
A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
where * is the Hodge dual
Hodge dual
In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...
.
Alternatively, we may express J as a (d − 1)-closed form
Closed form
-Maths:* Closed-form expression, a finitary expression* Closed differential form, a differential form \alpha with the property that d\alpha = 0-Poetry:* In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern;...
.
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
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
where M is the spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
.
p-form Abelian electrodynamics
We have a p-form B, a gauge symmetrywhere α is any arbitrary fixed (p-1)-form and d is the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
,
and a gauge-invariant p-vector J with density
Tensor density
In differential geometry, a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system to another , except that it is additionally multiplied or weighted by a power of the Jacobian determinant of the...
1 satisfying the continuity equation
Continuity equation
A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
where * is the Hodge dual
Hodge dual
In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...
.
Alternatively, we may express J as a (d-p)-closed form
Closed form
-Maths:* Closed-form expression, a finitary expression* Closed differential form, a differential form \alpha with the property that d\alpha = 0-Poetry:* In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern;...
.
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
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
where M is the spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
.
Other sign convention
Sign convention
In physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of...
s do exist.
The Kalb-Ramond field
Kalb-Ramond field
In theoretical physics in general and string theory in particular, the Kalb–Ramond field, also known as the NS-NS B-field, is a quantum field that transforms as a two-form i.e. an antisymmetric tensor field with two indices....
is an example with p=2 in string theory; the Ramond-Ramond field
Ramond-Ramond field
In theoretical physics, Ramond–Ramond fields are differential form fields in the 10-dimensional spacetime of type II supergravity theories, which are the classical limits of type II string theory. The ranks of the fields depend on which type II theory is considered...
s whose charged sources are D-brane
D-brane
In string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Hořava in 1989...
s are examples for all values of p. In 11d supergravity
Supergravity
In theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry...
or M-theory
M-theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
, 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 gerbeGerbe
In mathematics, a gerbe is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as a generalization of principal bundles to the setting of 2-categories...
s.