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Electromagnetic tensor
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<h2 id="definition">Definition</h2> <p>The electromagnetic tensor, conventionally labelled <i>F</i>, is defined as the <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a> of the <a href="/facts/Electromagnetic_four-potential/S8gTFNAt">electromagnetic four-potential</a>, <i>A</i>, a differential 1-form:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> F = d e f d A . {\displaystyle F\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} A.} <p>Therefore, <i>F</i> is a <a href="/facts/Differential_form/T9gyHtMv">differential 2-form</a>— an antisymmetric rank-2 tensor field—on Minkowski space. In component form, </p> F μ ν = ∂ μ A ν − ∂ ν A μ . {\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }.} <p>where ∂ {\displaystyle \partial } is the <a href="/facts/Four-gradient/dkVV0lnY">four-gradient</a> and A {\displaystyle A} is the <a href="/facts/Electromagnetic_four-potential/S8gTFNAt">four-potential</a>. </p><p><a href="/facts/Maxwell%2527s_equations/LfJ1drHS">SI units for Maxwell's equations</a> and the <a href="/facts/Sign_convention/0yGl2bzD">particle physicist's sign convention</a> for the <a href="/facts/Metric_signature/w8d57bQG">signature</a> of <a href="/facts/Minkowski_space/vDkfkQJn">Minkowski space</a> (+ − − −), will be used throughout this article. </p> <h3>Relationship with the classical fields</h3> <p>The Faraday <a href="/facts/Differential_form/T9gyHtMv">differential 2-form</a> is given by </p> F = ( E x / c ) d x ∧ d t + ( E y / c ) d y ∧ d t + ( E z / c ) d z ∧ d t + B x d y ∧ d z + B y d z ∧ d x + B z d x ∧ d y , {\displaystyle 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,} <p>where d t {\displaystyle dt} is the time element times the speed of light c {\displaystyle c} . </p><p>This is the <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a> of its 1-form antiderivative </p> A = A x d x + A y d y + A z d z − ( ϕ / c ) d t {\displaystyle A=A_{x}\ dx+A_{y}\ dy+A_{z}\ dz-(\phi /c)\ dt} , <p>where ϕ ( x → , t ) {\displaystyle \phi ({\vec {x}},t)} has − ∇ → ϕ = E → {\displaystyle -{\vec {\nabla }}\phi ={\vec {E}}} ( ϕ {\displaystyle \phi } is a scalar potential for the <a href="/facts/Conservative_vector_field/MoS3V9aV">irrotational/conservative vector field</a> E → {\displaystyle {\vec {E}}} ) and A → ( x → , t ) {\displaystyle {\vec {A}}({\vec {x}},t)} has ∇ → × A → = B → {\displaystyle {\vec {\nabla }}\times {\vec {A}}={\vec {B}}} ( A → {\displaystyle {\vec {A}}} is a vector potential for the <a href="/facts/Solenoidal_vector_field/9ohQ6clr">solenoidal vector field</a> B → {\displaystyle {\vec {B}}} ). </p><p>Note that </p> { d F = 0 ⋆ d ⋆ F = J {\displaystyle {\begin{cases}dF=0\\{\star }d{\star }F=J\end{cases}}} <p>where d {\displaystyle d} is the exterior derivative, ⋆ {\displaystyle {\star }} is the <a href="/facts/Hodge_star_operator/AH6tmHhU">Hodge star</a>, J = − J x d x − J y d y − J z d z + ρ d t {\displaystyle J=-J_{x}\ dx-J_{y}\ dy-J_{z}\ dz+\rho \ dt} (where J → {\displaystyle {\vec {J}}} is the <a href="/facts/Current_density/IlUKSe2B">electric current density</a>, and ρ {\displaystyle \rho } is the <a href="/facts/Charge_density/xFDPXUDM">electric charge density</a>) is the 4-current density 1-form, is the differential forms version of Maxwell's equations. </p><p>The <a href="/facts/Electric_field/tt4MooBs">electric</a> and <a href="/facts/Magnetic_field/Ta8Ev588">magnetic fields</a> can be obtained from the components of the electromagnetic tensor. The relationship is simplest in <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">Cartesian coordinates</a>: </p> E i = c F 0 i , {\displaystyle E_{i}=cF_{0i},} <p>where <i>c</i> is the speed of light, and </p> B i = − 1 / 2 ϵ i j k F j k , {\displaystyle B_{i}=-1/2\epsilon _{ijk}F^{jk},} <p>where ϵ i j k {\displaystyle \epsilon _{ijk}} is the <a href="/facts/Levi-Civita_tensor/WBKUA3Tw">Levi-Civita tensor</a>. This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor will <a href="/facts/Covariant_transformation/aFKj2UaT">transform covariantly</a>, and the fields in the new frame will be given by the new components. </p><p>In contravariant <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> form with metric signature (+,-,-,-), </p> F μ ν = [ 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 ] . {\displaystyle F^{\mu \nu }={\begin{bmatrix}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{bmatrix}}.} <p>The covariant form is given by <a href="/facts/Raising_and_lowering_indices/cPsSVpgI">index lowering</a>, </p> F μ ν = η α ν F β α η μ β = [ 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 ] . {\displaystyle F_{\mu \nu }=\eta _{\alpha \nu }F^{\beta \alpha }\eta _{\mu \beta }={\begin{bmatrix}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{bmatrix}}.} <p>The Faraday tensor's <a href="/facts/Hodge_star_operator/AH6tmHhU">Hodge dual</a> is </p> G α β = 1 2 ϵ α β γ δ F γ δ = [ 0 − B x − B y − B z B x 0 E z / c − E y / c B y − E z / c 0 E x / c B z E y / c − E x / c 0 ] {\displaystyle {G^{\alpha \beta }={\frac {1}{2}}\epsilon ^{\alpha \beta \gamma \delta }F_{\gamma \delta }={\begin{bmatrix}0&-B_{x}&-B_{y}&-B_{z}\\B_{x}&0&E_{z}/c&-E_{y}/c\\B_{y}&-E_{z}/c&0&E_{x}/c\\B_{z}&E_{y}/c&-E_{x}/c&0\end{bmatrix}}}} <p>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. </p> <h3>Properties</h3> <p>The matrix form of the field tensor yields the following properties:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <ol><li><a href="/facts/Skew-symmetric_matrix/nw5CQeLN">Antisymmetry</a>: F μ ν = − F ν μ {\displaystyle F^{\mu \nu }=-F^{\nu \mu }} </li> <li>Six independent components: In Cartesian coordinates, these are simply the three spatial components of the electric field (<i>Ex, Ey, Ez</i>) and magnetic field (<i>Bx, By, Bz</i>).</li> <li>Inner product: If one forms an inner product of the field strength tensor a <a href="/facts/Lorentz_invariant/K5AEuGKg">Lorentz invariant</a> is formed F μ ν F μ ν = 2 ( B 2 − E 2 c 2 ) {\displaystyle F_{\mu \nu }F^{\mu \nu }=2\left(B^{2}-{\frac {E^{2}}{c^{2}}}\right)} meaning this number does not change from one <a href="/facts/Frame_of_reference/15yYfB9M">frame of reference</a> to another.</li> <li><a href="/facts/Pseudoscalar/3ahEmnIW">Pseudoscalar</a> invariant: The product of the tensor F μ ν {\displaystyle F^{\mu \nu }} with its <a href="/facts/Hodge_dual/AH6tmHhU">Hodge dual</a> G μ ν {\displaystyle G^{\mu \nu }} gives a <a href="/facts/Lorentz_invariant/K5AEuGKg">Lorentz invariant</a>: G γ δ F γ δ = 1 2 ϵ α β γ δ F α β F γ δ = − 4 c B ⋅ E {\displaystyle G_{\gamma \delta }F^{\gamma \delta }={\frac {1}{2}}\epsilon _{\alpha \beta \gamma \delta }F^{\alpha \beta }F^{\gamma \delta }=-{\frac {4}{c}}\mathbf {B} \cdot \mathbf {E} \,} where ϵ α β γ δ {\displaystyle \epsilon _{\alpha \beta \gamma \delta }} is the rank-4 <a href="/facts/Levi-Civita_symbol/WBKUA3Tw">Levi-Civita symbol</a>. The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here is ϵ 0123 = − 1 {\displaystyle \epsilon _{0123}=-1} .</li> <li><a href="/facts/Determinant/eGYqJ2TF">Determinant</a>: det ( F ) = 1 c 2 ( B ⋅ E ) 2 {\displaystyle \det \left(F\right)={\frac {1}{c^{2}}}\left(\mathbf {B} \cdot \mathbf {E} \right)^{2}} which is proportional to the square of the above invariant.</li> <li><a href="/facts/Trace_(linear_algebra)/dchFFvHt">Trace</a>: F = F μ μ = 0 {\displaystyle F={{F}^{\mu }}_{\mu }=0} which is equal to zero.</li></ol> <h3>Significance</h3> <p>This tensor simplifies and reduces <a href="/facts/Maxwell%2527s_equations/LfJ1drHS">Maxwell's equations</a> as four vector calculus equations into two tensor field equations. In <a href="/facts/Electrostatic/bxkuW6s4">electrostatics</a> and <a href="/facts/Electrodynamic/oAkmALHT">electrodynamics</a>, <a href="/facts/Gauss%2527s_law/Am2X21DU">Gauss's law</a> and <a href="/facts/Amp%25C3%25A8re%2527s_circuital_law/B6WVCpc6">Ampère's circuital law</a> are respectively: </p> ∇ ⋅ E = ρ ϵ 0 , ∇ × B − 1 c 2 ∂ E ∂ t = μ 0 J {\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}},\quad \nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}=\mu _{0}\mathbf {J} } <p>and reduce to the inhomogeneous Maxwell equation: </p> ∂ α F β α = − μ 0 J β {\displaystyle \partial _{\alpha }F^{\beta \alpha }=-\mu _{0}J^{\beta }} , where J α = ( c ρ , J ) {\displaystyle J^{\alpha }=(c\rho ,\mathbf {J} )} is the <a href="/facts/Four-current/C54racI8">four-current</a>. <p>In <a href="/facts/Magnetostatic/DFmGU4a7">magnetostatics</a> and magnetodynamics, <a href="/facts/Gauss%2527s_law_for_magnetism/MGaCT5GL">Gauss's law for magnetism</a> and <a href="/facts/Faraday%2527s_law_of_induction/PFBreB0F">Maxwell–Faraday equation</a> are respectively: </p> ∇ ⋅ B = 0 , ∂ B ∂ t + ∇ × E = 0 {\displaystyle \nabla \cdot \mathbf {B} =0,\quad {\frac {\partial \mathbf {B} }{\partial t}}+\nabla \times \mathbf {E} =\mathbf {0} } <p>which reduce to the <a href="/facts/Bianchi_identity/LWp7DMDM">Bianchi identity</a>: </p> ∂ γ F α β + ∂ α F β γ + ∂ β F γ α = 0 {\displaystyle \partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0} <p>or using the <a href="/facts/Ricci_calculus/We4yudeq">index notation with square brackets</a>[note 1] for the antisymmetric part of the tensor: </p> ∂ [ α F β γ ] = 0 {\displaystyle \partial _{[\alpha }F_{\beta \gamma ]}=0} <p>Using the expression relating the Faraday tensor to the four-potential, one can prove that the above antisymmetric quantity turns to zero identically ( ≡ 0 {\displaystyle \equiv 0} ). This tensor equation reproduces the homogeneous Maxwell's equations. </p> <h2 id="relativity">Relativity</h2> <p class="note">Main article: <a href="/facts/Maxwell%2527s_equations_in_curved_spacetime/ofggksaC">Maxwell's equations in curved spacetime</a></p> <p>The field tensor derives its name from the fact that the electromagnetic field is found to obey the <a href="/facts/Tensor_transformation_law/pNjFo5tV">tensor transformation law</a>, this general property of physical laws being recognised after the advent of <a href="/facts/Special_relativity/vsFFc63E">special relativity</a>. This theory stipulated that all the laws of physics should take the same form in all coordinate systems – this led to the introduction of <a href="/facts/Tensor/pNjFo5tV">tensors</a>. The tensor formalism also leads to a mathematically simpler presentation of physical laws. </p><p>The inhomogeneous Maxwell equation leads to the <a href="/facts/Continuity_equation/GVav2Vet">continuity equation</a>: </p> ∂ α J α = J α , α = 0 {\displaystyle \partial _{\alpha }J^{\alpha }=J^{\alpha }{}_{,\alpha }=0} <p>implying <a href="/facts/Conservation_of_charge/WGQXmoE8">conservation of charge</a>. </p><p>Maxwell's laws above can be generalised to <a href="/facts/Curved_spacetime/93NBB86g">curved spacetime</a> by simply replacing <a href="/facts/Partial_derivative/JWHsBkAu">partial derivatives</a> with <a href="/facts/Covariant_derivative/sRGN0Xh6">covariant derivatives</a>: </p> F [ α β ; γ ] = 0 {\displaystyle F_{[\alpha \beta ;\gamma ]}=0} and F α β ; α = μ 0 J β {\displaystyle F^{\alpha \beta }{}_{;\alpha }=\mu _{0}J^{\beta }} <p>where the <a href="/facts/Covariant_derivative/sRGN0Xh6">semicolon notation</a> represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the <a href="/facts/Maxwell%2527s_equations_in_curved_spacetime/ofggksaC">curved space Maxwell equations</a>. Again, the second equation implies charge conservation (in curved spacetime): </p> J α ; α = 0 {\displaystyle J^{\alpha }{}_{;\alpha }\,=0} <p>The stress-energy tensor of electromagnetism </p> T μ ν = 1 μ 0 [ F μ α F ν α − 1 4 η μ ν F α β F α β ] , {\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right]\,,} <p>satisfies </p> T α β , β + F α β J β = 0 . {\displaystyle {T^{\alpha \beta }}_{,\beta }+F^{\alpha \beta }J_{\beta }=0\,.} <h2 id="lagrangian-formulation-of-classical-electromagnetism">Lagrangian formulation of classical electromagnetism</h2> <p class="note">See also: <a href="/facts/Classical_field_theory/7HuKY1qu">Classical field theory</a></p> <p><a href="/facts/Classical_electromagnetism/vK0YI6dj">Classical electromagnetism</a> and <a href="/facts/Maxwell%2527s_equations/LfJ1drHS">Maxwell's equations</a> can be derived from the <a href="/facts/Action_(physics)/NhWS2Mts">action</a>: S = ∫ ( − 1 4 μ 0 F μ ν F μ ν − J μ A μ ) d 4 x {\displaystyle {\mathcal {S}}=\int \left(-{\begin{matrix}{\frac {1}{4\mu _{0}}}\end{matrix}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }\right)\mathrm {d} ^{4}x\,} where d 4 x {\displaystyle \mathrm {d} ^{4}x} is over space and time. </p><p>This means the <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a> density is </p> L = − 1 4 μ 0 F μ ν F μ ν − J μ A μ = − 1 4 μ 0 ( ∂ μ A ν − ∂ ν A μ ) ( ∂ μ A ν − ∂ ν A μ ) − J μ A μ = − 1 4 μ 0 ( ∂ μ A ν ∂ μ A ν − ∂ ν A μ ∂ μ A ν − ∂ μ A ν ∂ ν A μ + ∂ ν A μ ∂ ν A μ ) − J μ A μ {\displaystyle {\begin{aligned}{\mathcal {L}}&=-{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }\\&=-{\frac {1}{4\mu _{0}}}\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 {1}{4\mu _{0}}}\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{aligned}}} <p>The two middle terms in the parentheses are the same, as are the two outer terms, so the Lagrangian density is </p> L = − 1 2 μ 0 ( ∂ μ A ν ∂ μ A ν − ∂ ν A μ ∂ μ A ν ) − J μ A μ . {\displaystyle {\mathcal {L}}=-{\frac {1}{2\mu _{0}}}\left(\partial _{\mu }A_{\nu }\partial ^{\mu }A^{\nu }-\partial _{\nu }A_{\mu }\partial ^{\mu }A^{\nu }\right)-J^{\mu }A_{\mu }.} <p>Substituting this into the <a href="/facts/Euler%25E2%2580%2593Lagrange_equation/Ymq3r3Dh">Euler–Lagrange equation</a> of motion for a field: </p> ∂ μ ( ∂ L ∂ ( ∂ μ A ν ) ) − ∂ L ∂ A ν = 0 {\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }A_{\nu })}}\right)-{\frac {\partial {\mathcal {L}}}{\partial A_{\nu }}}=0} <p>So the Euler–Lagrange equation becomes: </p> − ∂ μ 1 μ 0 ( ∂ μ A ν − ∂ ν A μ ) + J ν = 0. {\displaystyle -\partial _{\mu }{\frac {1}{\mu _{0}}}\left(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }\right)+J^{\nu }=0.\,} <p>The quantity in parentheses above is just the field tensor, so this finally simplifies to </p> ∂ μ F μ ν = μ 0 J ν {\displaystyle \partial _{\mu }F^{\mu \nu }=\mu _{0}J^{\nu }} <p>That equation is another way of writing the two inhomogeneous <a href="/facts/Maxwell%2527s_equations/LfJ1drHS">Maxwell's equations</a> (namely, <a href="/facts/Gauss%2527s_law/Am2X21DU">Gauss's law</a> and <a href="/facts/Amp%25C3%25A8re%2527s_circuital_law/B6WVCpc6">Ampère's circuital law</a>) using the substitutions: </p> 1 c E i = − F 0 i ϵ i j k B k = − F i j {\displaystyle {\begin{aligned}{\frac {1}{c}}E^{i}&=-F^{0i}\\\epsilon ^{ijk}B_{k}&=-F^{ij}\end{aligned}}} <p>where <i>i, j, k</i> take the values 1, 2, and 3. </p> <h3>Hamiltonian form</h3> <p>The <a href="/facts/Hamiltonian_field_theory/aWRhRJF0">Hamiltonian</a> density can be obtained with the usual relation, </p> H ( ϕ i , π i ) = π i ϕ ˙ i ( ϕ i , π i ) − L . {\displaystyle {\mathcal {H}}(\phi ^{i},\pi _{i})=\pi _{i}{\dot {\phi }}^{i}(\phi ^{i},\pi _{i})-{\mathcal {L}}\,.} <p>Here ϕ i = A i {\displaystyle \phi ^{i}=A^{i}} are the fields and the momentum density of the EM field is </p> π i = T 0 i = 1 μ 0 F 0 α F i α = 1 μ 0 c E × B . {\displaystyle \pi _{i}=T_{0i}={\frac {1}{\mu _{0}}}F_{0}{}^{\alpha }F_{i\alpha }={\frac {1}{\mu _{0}c}}\mathbf {E} \times \mathbf {B} \,.} <p>such that the conserved quantity associated with translation from <a href="/facts/Noether%2527s_theorem/VFjqrEvz">Noether's theorem</a> is the total momentum </p> P = ∑ α m α x ˙ α + 1 μ 0 c ∫ V d 3 x E × B . {\displaystyle \mathbf {P} =\sum _{\alpha }m_{\alpha }{\dot {\mathbf {x} }}_{\alpha }+{\frac {1}{\mu _{0}c}}\int _{\mathcal {V}}\mathrm {d} ^{3}x\,\mathbf {E} \times \mathbf {B} \,.} <p>The Hamiltonian density for the electromagnetic field is related to the <a href="/facts/Electromagnetic_stress-energy_tensor/0JxwEylo">electromagnetic stress-energy tensor</a> </p> T μ ν = 1 μ 0 [ F μ α F ν α − 1 4 η μ ν F α β F α β ] . {\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right]\,.} <p>as </p> H = T 00 = 1 2 ( ϵ 0 E 2 + 1 μ 0 B 2 ) = 1 8 π ( E 2 + B 2 ) . {\displaystyle {\mathcal {H}}=T_{00}={\frac {1}{2}}\left(\epsilon _{0}\mathbf {E} ^{2}+{\frac {1}{\mu _{0}}}\mathbf {B} ^{2}\right)={\frac {1}{8\pi }}\left(\mathbf {E} ^{2}+\mathbf {B} ^{2}\right)\,.} <p>where we have neglected the <a href="/facts/Perfect_fluid/wWpblQcg">energy density of matter</a>, 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 <a href="/facts/Coulomb_gauge/eJW4uffl">Coulomb gauge</a> ( ∇ ⋅ A = ∇ i A i = 0 {\displaystyle \nabla \cdot \mathbf {A} =\nabla _{i}A^{i}=0} ) is </p> p α = m α x ˙ α + q α c A ( x α ) . {\displaystyle \mathbf {p} _{\alpha }=m_{\alpha }{\dot {\mathbf {x} }}_{\alpha }+{\frac {q_{\alpha }}{c}}\mathbf {A} (\mathbf {x} _{\alpha })\,.} <p>The total Hamiltonian of the matter + EM field system is </p> H = ∫ V d 3 x T 00 = H m a t + H e m . {\displaystyle H=\int _{\mathcal {V}}d^{3}x\,T_{00}=H_{\rm {mat}}+H_{\rm {em}}\,.} <p>where for nonrelativistic point particles in the Coulomb gauge </p> H m a t = ∑ α m α | x ˙ α | 2 + ∑ α < β q α q β | x α − x β | = ∑ α 1 2 m α [ p α − q α c A ( x α ) ] 2 + ∑ α < β q α q β | x α − x β | . {\displaystyle H_{\rm {mat}}=\sum _{\alpha }m_{\alpha }|{\dot {\mathbf {x} }}_{\alpha }|^{2}+\sum _{\alpha <\beta }{\frac {q_{\alpha }q_{\beta }}{|\mathbf {x} _{\alpha }-\mathbf {x} _{\beta }|}}=\sum _{\alpha }{\frac {1}{2m_{\alpha }}}\left[\mathbf {p} _{\alpha }-{\frac {q_{\alpha }}{c}}\mathbf {A} (\mathbf {x} _{\alpha })\right]^{2}+\sum _{\alpha <\beta }{\frac {q_{\alpha }q_{\beta }}{|\mathbf {x} _{\alpha }-\mathbf {x} _{\beta }|}}\,.} <p>where the last term is identically 1 8 π ∫ V d 3 x E ∥ 2 {\displaystyle {\frac {1}{8\pi }}\int _{\mathcal {V}}d^{3}x\mathbf {E} _{\parallel }^{2}} where E ∥ i = ∇ i A 0 {\displaystyle {E}_{\parallel i}={\nabla _{i}}A_{0}} and </p> H e m = 1 8 π ∫ V d 3 x ( E ⊥ 2 + B 2 ) . {\displaystyle H_{\rm {em}}={\frac {1}{8\pi }}\int _{\mathcal {V}}d^{3}x\left(\mathbf {E} _{\perp }^{2}+\mathbf {B} ^{2}\right)\,.} <p>where and E ⊥ i = − 1 c ∂ 0 A i {\displaystyle {E}_{\perp i}=-{\frac {1}{c}}\partial _{0}A_{i}} . </p> <h3>Quantum electrodynamics and field theory</h3> <p class="note">Main articles: <a href="/facts/Quantum_electrodynamics/zPteAr6G">Quantum electrodynamics</a> and <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a></p> <p>The <a href="/facts/Lagrangian_(field_theory)/p9BCUz57">Lagrangian</a> of <a href="/facts/Quantum_electrodynamics/zPteAr6G">quantum electrodynamics</a> extends beyond the classical Lagrangian established in relativity to incorporate the creation and annihilation of photons (and electrons): </p> L = ψ ¯ ( i ℏ c γ α D α − m c 2 ) ψ − 1 4 μ 0 F α β F α β , {\displaystyle {\mathcal {L}}={\bar {\psi }}\left(i\hbar c\,\gamma ^{\alpha }D_{\alpha }-mc^{2}\right)\psi -{\frac {1}{4\mu _{0}}}F_{\alpha \beta }F^{\alpha \beta },} <p>where the first part in the right hand side, containing the <a href="/facts/Dirac_spinor/IDhhu8Eu">Dirac spinor</a> ψ {\displaystyle \psi } , represents the <a href="/facts/Dirac_field/KKaGVgX6">Dirac field</a>. In <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> 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. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Classification_of_electromagnetic_fields/w0zVfyWR">Classification of electromagnetic fields</a></li> <li><a href="/facts/Covariant_formulation_of_classical_electromagnetism/pM1kDz2n">Covariant formulation of classical electromagnetism</a></li> <li><a href="/facts/Electromagnetic_stress%25E2%2580%2593energy_tensor/0JxwEylo">Electromagnetic stress–energy tensor</a></li> <li><a href="/facts/Gluon_field_strength_tensor/6QIxCTDx">Gluon field strength tensor</a></li> <li><a href="/facts/Ricci_calculus/We4yudeq">Ricci calculus</a></li> <li><a href="/facts/Riemann%25E2%2580%2593Silberstein_vector/Xg8lYYTN">Riemann–Silberstein vector</a></li></ul> <h2 id="notes">Notes</h2> <ol><li>^ By definition, T [ a b c ] = 1 3 ! ( T a b c + T b c a + T c a b − T a c b − T b a c − T c b a ) {\displaystyle T_{[abc]}={\frac {1}{3!}}(T_{abc}+T_{bca}+T_{cab}-T_{acb}-T_{bac}-T_{cba})} <p>So if </p> ∂ γ F α β + ∂ α F β γ + ∂ β F γ α = 0 {\displaystyle \partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0} <p>then </p> 0 = 2 6 ( ∂ γ F α β + ∂ α F β γ + ∂ β F γ α ) = 1 6 { ∂ γ ( 2 F α β ) + ∂ α ( 2 F β γ ) + ∂ β ( 2 F γ α ) } = 1 6 { ∂ γ ( F α β − F β α ) + ∂ α ( F β γ − F γ β ) + ∂ β ( F γ α − F α γ ) } = 1 6 ( ∂ γ F α β + ∂ α F β γ + ∂ β F γ α − ∂ γ F β α − ∂ α F γ β − ∂ β F α γ ) = ∂ [ γ F α β ] {\displaystyle {\begin{aligned}0&={\begin{matrix}{\frac {2}{6}}\end{matrix}}(\partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha })\\&={\begin{matrix}{\frac {1}{6}}\end{matrix}}\{\partial _{\gamma }(2F_{\alpha \beta })+\partial _{\alpha }(2F_{\beta \gamma })+\partial _{\beta }(2F_{\gamma \alpha })\}\\&={\begin{matrix}{\frac {1}{6}}\end{matrix}}\{\partial _{\gamma }(F_{\alpha \beta }-F_{\beta \alpha })+\partial _{\alpha }(F_{\beta \gamma }-F_{\gamma \beta })+\partial _{\beta }(F_{\gamma \alpha }-F_{\alpha \gamma })\}\\&={\begin{matrix}{\frac {1}{6}}\end{matrix}}(\partial _{\gamma }F_{\alpha \beta }+\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }-\partial _{\gamma }F_{\beta \alpha }-\partial _{\alpha }F_{\gamma \beta }-\partial _{\beta }F_{\alpha \gamma })\\&=\partial _{[\gamma }F_{\alpha \beta ]}\end{aligned}}} </li></ol> <ul><li>Brau, Charles A. (2004). <i>Modern Problems in Classical Electrodynamics</i>. <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-514665-4.</li> <li>Jackson, John D. (1999). <a href="https://archive.org/details/classicalelectro0000jack_e8g9"><i>Classical Electrodynamics</i></a>. <a href="/facts/John_Wiley_%2526_Sons%2c_Inc./53g3LqJc">John Wiley & Sons, Inc.</a> <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-30932-X.</li> <li>Peskin, Michael E.; Schroeder, Daniel V. (1995). <a href="https://archive.org/details/introductiontoqu0000pesk"><i>An Introduction to Quantum Field Theory</i></a>. Perseus Publishing. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-50397-2.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Darrigol, O. (2005). The genesis of the theory of relativity. In Einstein, 1905–2005: Poincaré Seminar 2005 (pp. 1-31). Basel: Birkhäuser Basel <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>J. A. Wheeler; C. Misner; K. S. Thorne (1973). Gravitation. W.H. Freeman & Co. ISBN 0-7167-0344-0. <a href="0-7167-0344-0" target="_blank">0-7167-0344-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>D. J. Griffiths (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education, Dorling Kindersley. ISBN 978-81-7758-293-2. <a href="978-81-7758-293-2" target="_blank">978-81-7758-293-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>J. A. Wheeler; C. Misner; K. S. Thorne (1973). Gravitation. W.H. Freeman & Co. ISBN 0-7167-0344-0. <a href="0-7167-0344-0" target="_blank">0-7167-0344-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>