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In
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was developed by Arnold Sommerfeld after the four-dimensional
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
formulation of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
was introduced by Hermann Minkowski. The tensor allows related physical laws to be written concisely, and allows for the quantization of the electromagnetic field by the Lagrangian formulation described below.


Definition

The electromagnetic tensor, conventionally labelled ''F'', is defined as the exterior derivative of the electromagnetic four-potential, ''A'', a differential 1-form: :F \ \stackrel\ \mathrmA. Therefore, ''F'' is a differential 2-form— an antisymmetric rank-2 tensor field—on Minkowski space. In component form, :F_ = \partial_\mu A_\nu - \partial_\nu A_\mu. where \partial is the four-gradient and A is the four-potential. SI units for Maxwell's equations and the particle physicist's sign convention for the
signature A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
of Minkowski space , will be used throughout this article.


Relationship with the classical fields

The Faraday differential 2-form is given by : F = (E_x/c)\ dx \wedge dt + (E_y/c)\ dy \wedge dt + (E_z/c)\ dz \wedge dt + B_x\ dy \wedge dz + B_y\ dz \wedge dx + B_z\ dx \wedge dy, where dt is the time element times the speed of light c . This is the exterior derivative of its 1-form antiderivative : A = A_x\ dx + A_y\ dy + A_z\ dz - (\phi/c)\ dt , where \phi(\vec,t) has -\vec\phi = \vec ( \phi is a scalar potential for the irrotational/conservative vector field \vec ) and \vec(\vec,t) has \vec \times \vec = \vec ( \vec is a vector potential for the solenoidal vector field \vec ). Note that : \begin dF = 0 \\ dF = J \end where d is the exterior derivative, is the Hodge star, J = -J_x\ dx - J_y\ dy - J_z\ dz + \rho\ dt (where \vec is the electric current density, and \rho is the electric charge density) is the 4-current density 1-form, is the differential forms version of Maxwell's equations. The
electric Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
and
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
s can be obtained from the components of the electromagnetic tensor. The relationship is simplest in
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
: :E_i = c F_, where ''c'' is the speed of light, and :B_i = -1/2\epsilon_ F^, where \epsilon_ is the Levi-Civita tensor. This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor will transform covariantly, and the fields in the new frame will be given by the new components. In contravariant matrix form with metric signature (+,-,-,-), : F^ = \begin 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end. The covariant form is given by index lowering, : F_ = \eta_F^\eta_ = \begin 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end. The Faraday tensor's Hodge dual is : From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is assumed, and the electric and magnetic fields are with respect to the coordinate system's reference frame, as in the equations above.


Properties

The matrix form of the field tensor yields the following properties: # Antisymmetry: F^ = - F^ #Six independent components: In Cartesian coordinates, these are simply the three spatial components of the electric field (''Ex, Ey, Ez'') and magnetic field (''Bx, By, Bz''). #Inner product: If one forms an inner product of the field strength tensor a Lorentz invariant is formed F_ F^ = 2 \left( B^2-\frac \right) meaning this number does not change from one
frame of reference In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system, whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale have been specified in physical space. It ...
to another. # Pseudoscalar invariant: The product of the tensor F^ with its Hodge dual G^ gives a Lorentz invariant: G_F^ = \frac\epsilon_F^ F^ = -\frac \mathbf \cdot \mathbf \, where \epsilon_ is the rank-4 Levi-Civita symbol. The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here is \epsilon_ = -1 . #
Determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
: \det \left( F \right) = \frac \left( \mathbf \cdot \mathbf \right)^2 which is proportional to the square of the above invariant. # Trace: F=_=0 which is equal to zero.


Significance

This tensor simplifies and reduces
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
as four vector calculus equations into two tensor field equations. In electrostatics and electrodynamics, Gauss's law and Ampère's circuital law are respectively: :\nabla \cdot \mathbf = \frac,\quad \nabla \times \mathbf - \frac \frac = \mu_0 \mathbf and reduce to the inhomogeneous Maxwell equation: :\partial_ F^ = - \mu_0 J^, where J^ = ( c\rho, \mathbf ) is the four-current. In magnetostatics and magnetodynamics, Gauss's law for magnetism and Maxwell–Faraday equation are respectively: :\nabla \cdot \mathbf = 0,\quad \frac + \nabla \times \mathbf = \mathbf which reduce to the Bianchi identity: : \partial_\gamma F_ + \partial_\alpha F_ + \partial_\beta F_ = 0 or using the index notation with square brackets for the antisymmetric part of the tensor: : \partial_ F_ = 0 Using the expression relating the Faraday tensor to the four-potential, one can prove that the above antisymmetric quantity turns to zero identically (\equiv 0). This tensor equation reproduces the homogeneous Maxwell's equations.


Relativity

The field tensor derives its name from the fact that the electromagnetic field is found to obey the tensor transformation law, this general property of physical laws being recognised after the advent of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
. This theory stipulated that all the laws of physics should take the same form in all coordinate systems – this led to the introduction of
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
s. The tensor formalism also leads to a mathematically simpler presentation of physical laws. The inhomogeneous Maxwell equation leads to the continuity equation: :\partial_\alpha J^\alpha = J^\alpha_ = 0 implying conservation of charge. Maxwell's laws above can be generalised to curved spacetime by simply replacing
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
s with
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
s: :F_ = 0 and F^_ = \mu_0 J^ where the semicolon notation represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the curved space Maxwell equations. Again, the second equation implies charge conservation (in curved spacetime): :J^\alpha_ \, = 0 The stress-energy tensor of electromagnetism :T^ = \frac \left F^F^\nu_ - \frac \eta^F_ F^\right\,, satisfies :_ + F^ J_\beta = 0\,.


Lagrangian formulation of classical electromagnetism

Classical electromagnetism and
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
can be derived from the action: \mathcal = \int \left( -\begin \frac \end F_ F^ - J^\mu A_\mu \right) \mathrm^4 x \, where \mathrm^4 x is over space and time. This means the Lagrangian density is :\begin \mathcal &= -\frac F_ F^ - J^\mu A_\mu \\ &= -\frac \left( \partial_\mu A_\nu - \partial_\nu A_\mu \right) \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) - J^\mu A_\mu \\ &= -\frac \left( \partial_\mu A_\nu \partial^\mu A^\nu - \partial_\nu A_\mu \partial^\mu A^\nu - \partial_\mu A_\nu \partial^\nu A^\mu + \partial_\nu A_\mu \partial^\nu A^\mu \right) - J^\mu A_\mu \\ \end The two middle terms in the parentheses are the same, as are the two outer terms, so the Lagrangian density is :\mathcal = - \frac \left( \partial_\mu A_\nu \partial^\mu A^\nu - \partial_\nu A_\mu \partial^\mu A^\nu \right) - J^\mu A_\mu. Substituting this into the Euler–Lagrange equation of motion for a field: : \partial_\mu \left( \frac \right) - \frac = 0 So the Euler–Lagrange equation becomes: : - \partial_\mu \frac \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) + J^\nu = 0. \, The quantity in parentheses above is just the field tensor, so this finally simplifies to : \partial_\mu F^ = \mu_0 J^\nu That equation is another way of writing the two inhomogeneous
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
(namely, Gauss's law and Ampère's circuital law) using the substitutions: :\begin \fracE^i &= -F^ \\ \epsilon^ B_k &= -F^ \end where ''i, j, k'' take the values 1, 2, and 3.


Hamiltonian form

The
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
density can be obtained with the usual relation, :\mathcal(\phi^i,\pi_i) = \pi_i \dot^i(\phi^i,\pi_i) - \mathcal \,. Here \phi^i=A^ are the fields and the momentum density of the EM field is :\pi_i= T_=\frac F_0^F_=\frac \mathbf\times\mathbf \,. such that the conserved quantity associated with translation from Noether's theorem is the total momentum :\mathbf= \sum_ m_\alpha \dot_ + \frac\int_ \mathrm^3 x\, \mathbf\times\mathbf \,. The Hamiltonian density for the electromagnetic field is related to the electromagnetic stress-energy tensor :T^ = \frac \left F^F^\nu_ - \frac \eta^F_ F^\right \,. as :\mathcal = T_ = \frac\left(\epsilon_0 \mathbf^2+\frac\mathbf^2\right) = \frac\left(\mathbf^2+\mathbf^2\right)\,. where we have neglected the energy density of matter, assuming only the EM field, and the last equality assumes the CGS system. The momentum of nonrelativistic charges interarcting with the EM field in the Coulomb gauge (\nabla\cdot \mathbf=\nabla_i A^i = 0) is :\mathbf_\alpha = m_\alpha \dot_ + \frac \mathbf(\mathbf_\alpha) \,. The total Hamiltonian of the matter + EM field system is :H = \int_\mathcal d^3 x \,T_ = H_ + H_ \,. where for nonrelativistic point particles in the Coulomb gauge :H_ = \sum_\alpha m_ , \dot_, ^2+ \sum_ \frac = \sum_\alpha \frac \left mathbf_ - \frac \mathbf(\mathbf_\alpha)\right2 + \sum_ \frac \,. where the last term is identically \frac \int_\mathcal d^3 x \mathbf_^2 where _ = A_0 and :H_ = \frac \int_\mathcal d^3 x \left(\mathbf_^2+\mathbf^2\right) \,. where and _ = -\frac\partial_0 A_i.


Quantum electrodynamics and field theory

The Lagrangian of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
extends beyond the classical Lagrangian established in relativity to incorporate the creation and annihilation of photons (and electrons): :\mathcal = \bar\psi \left(i\hbar c \, \gamma^\alpha D_\alpha - mc^2\right) \psi - \frac F_ F^, where the first part in the right hand side, containing the Dirac spinor \psi, represents the Dirac field. In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
it is used as the template for the gauge field strength tensor. By being employed in addition to the local interaction Lagrangian it reprises its usual role in QED.


See also

* Classification of electromagnetic fields * Covariant formulation of classical electromagnetism * Electromagnetic stress–energy tensor *
Gluon field strength tensor In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks. The strong interaction is one of the fundamental interactions of nature, and the quantum fiel ...
* Ricci calculus * Riemann–Silberstein vector


Notes


References

* * * {{tensors Electromagnetism Minkowski spacetime Theory of relativity Tensor physical quantities Tensors in general relativity