Mathematical object that describes the electromagnetic field in spacetime
Inelectromagnetism , theelectromagnetic tensor orelectromagnetic field tensor (sometimes called thefield strength tensor ,Faraday tensor orMaxwell bivector ) is a mathematical object that describes theelectromagnetic field in spacetime. The field tensor was developed byArnold Sommerfeld after the four-dimensionaltensor formulation ofspecial relativity was introduced byHermann Minkowski .[ 1] : 22 quantization of the electromagnetic field by the Lagrangian formulation describedbelow .
The electromagnetic tensor, conventionally labelledF , is defined as theexterior derivative of theelectromagnetic four-potential ,A , a differential 1-form:[ 2] [ 3]
F = d e f d A . {\displaystyle F\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} A.} Therefore,F is adifferential 2-form — an antisymmetric rank-2 tensor field—on Minkowski space. In component form,
F μ ν = ∂ μ A ν − ∂ ν A μ . {\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }.} where∂ {\displaystyle \partial } four-gradient andA {\displaystyle A} four-potential .
SI units for Maxwell's equations and theparticle physicist's sign convention for thesignature ofMinkowski space (+ − − −) , will be used throughout this article.
Relationship with the classical fields [ edit ] The Faradaydifferential 2-form is given by
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,} whered t {\displaystyle dt} c {\displaystyle c}
This is theexterior derivative of its 1-form antiderivative, the covariant form of the four-potential, is[ 4] : 315
A = ( ϕ / c ) d t − A x d x − A y d y − A z d z , {\displaystyle A=(\phi /c)\ dt-A_{x}\ dx-A_{y}\ dy-A_{z}\ dz,} whereϕ ( x → , t ) {\displaystyle \phi ({\vec {x}},t)} − ∇ → ϕ = E → {\displaystyle -{\vec {\nabla }}\phi ={\vec {E}}} ϕ {\displaystyle \phi } irrotational/conservative vector field E → {\displaystyle {\vec {E}}} A → ( x → , t ) {\displaystyle {\vec {A}}({\vec {x}},t)} ∇ → × A → = B → {\displaystyle {\vec {\nabla }}\times {\vec {A}}={\vec {B}}} A → {\displaystyle {\vec {A}}} solenoidal vector field B → {\displaystyle {\vec {B}}}
Theelectric andmagnetic fields can be obtained from the components of the electromagnetic tensor. The relationship is simplest inCartesian coordinates :
E i = c F 0 i , {\displaystyle E_{i}=cF_{0i},} wherec is the speed of light, and
B i = − 1 / 2 ϵ i j k F j k , {\displaystyle B_{i}=-1/2\epsilon _{ijk}F^{jk},} whereϵ i j k {\displaystyle \epsilon _{ijk}} Levi-Civita tensor . This gives the fields in a particular reference frame; if the reference frame is changed, the components of the electromagnetic tensor willtransform covariantly , and the fields in the new frame will be given by the new components.
In contravariantmatrix form with metric signature (+,-,-,-),[ 4] : 313
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}}.} The covariant form is given byindex lowering ,
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}}.} The Faraday tensor'sHodge dual is
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}}}} 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.
The matrix form of the field tensor yields the following properties:[ 5]
Antisymmetry :F μ ν = − F ν μ {\displaystyle F^{\mu \nu }=-F^{\nu \mu }} 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 aLorentz invariant is formedF 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)} frame of reference to another.Pseudoscalar invariant:F μ ν {\displaystyle F^{\mu \nu }} Hodge dual G μ ν {\displaystyle G^{\mu \nu }} Lorentz invariant :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} \,} ϵ α β γ δ {\displaystyle \epsilon _{\alpha \beta \gamma \delta }} Levi-Civita symbol . The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here isϵ 0123 = − 1 {\displaystyle \epsilon _{0123}=-1} Determinant :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}} Trace :F = F μ μ = 0 {\displaystyle F={{F}^{\mu }}_{\mu }=0} This tensor simplifies and reducesMaxwell's equations as four vector calculus equations into two tensor field equations. Inelectrostatics andelectrodynamics ,Gauss's law andAmpère's circuital law are respectively:
∇ ⋅ 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} } and reduce to the inhomogeneous Maxwell equation:
∂ α F β α = − μ 0 J β {\displaystyle \partial _{\alpha }F^{\beta \alpha }=-\mu _{0}J^{\beta }} J α = ( c ρ , J ) {\displaystyle J^{\alpha }=(c\rho ,\mathbf {J} )} four-current .Inmagnetostatics and magnetodynamics,Gauss's law for magnetism andMaxwell–Faraday equation are respectively:
∇ ⋅ 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} } which reduce to theBianchi identity :
∂ 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} or using theindex notation with square brackets [note 1]
∂ [ α F β γ ] = 0 {\displaystyle \partial _{[\alpha }F_{\beta \gamma ]}=0} 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}
The field tensor derives its name from the fact that the electromagnetic field is found to obey thetensor transformation law , this general property of physical laws being recognised after the advent ofspecial relativity . This theory stipulated that all the laws of physics should take the same form in all coordinate systems – this led to the introduction oftensors . The tensor formalism also leads to a mathematically simpler presentation of physical laws.
The inhomogeneous Maxwell equation leads to thecontinuity equation :
∂ α J α = J α , α = 0 {\displaystyle \partial _{\alpha }J^{\alpha }=J^{\alpha }{}_{,\alpha }=0} implyingconservation of charge .
Maxwell's laws above can be generalised tocurved spacetime by simply replacingpartial derivatives withcovariant derivatives :
F [ α β ; γ ] = 0 {\displaystyle F_{[\alpha \beta ;\gamma ]}=0} F α β ; α = μ 0 J β {\displaystyle F^{\alpha \beta }{}_{;\alpha }=\mu _{0}J^{\beta }} where thesemicolon notation represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as thecurved space Maxwell equations . Again, the second equation implies charge conservation (in curved spacetime):
J α ; α = 0 {\displaystyle J^{\alpha }{}_{;\alpha }\,=0} The stress-energy tensor of electromagnetism
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]\,,} satisfies
T α β , β + F α β J β = 0 . {\displaystyle {T^{\alpha \beta }}_{,\beta }+F^{\alpha \beta }J_{\beta }=0\,.} Lagrangian formulation of classical electromagnetism [ edit ] Classical electromagnetism andMaxwell's equations can be derived from theaction :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\,} d 4 x {\displaystyle \mathrm {d} ^{4}x}
This means theLagrangian density is
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}}} The two middle terms in the parentheses are the same, as are the two outer terms, so the Lagrangian density is
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 }.} Substituting this into theEuler–Lagrange equation of motion for a field:
∂ 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} So the Euler–Lagrange equation becomes:
− ∂ 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.\,} The quantity in parentheses above is just the field tensor, so this finally simplifies to
∂ μ F μ ν = μ 0 J ν {\displaystyle \partial _{\mu }F^{\mu \nu }=\mu _{0}J^{\nu }} That equation is another way of writing the two inhomogeneousMaxwell's equations (namely,Gauss's law andAmpère's circuital law ) using the substitutions:
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}}} wherei, j, k take the values 1, 2, and 3.
TheHamiltonian density can be obtained with the usual relation,
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}}\,.} Hereϕ i = A i {\displaystyle \phi ^{i}=A^{i}}
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} \,.} such that the conserved quantity associated with translation fromNoether's theorem is the total momentum
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} \,.} The Hamiltonian density for the electromagnetic field is related to theelectromagnetic stress-energy tensor
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]\,.} as
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)\,.} where we have neglected theenergy density of matter , assuming only the EM field, and the last equality assumes the CGS system. The momentum of nonrelativistic charges interacting with the EM field in theCoulomb gauge (∇ ⋅ A = ∇ i A i = 0 {\displaystyle \nabla \cdot \mathbf {A} =\nabla _{i}A^{i}=0}
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 })\,.} The total Hamiltonian of the matter + EM field system is
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}}\,.} where for nonrelativistic point particles in the Coulomb gauge
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 }|}}\,.} where the last term is identically1 8 π ∫ V d 3 x E ∥ 2 {\displaystyle {\frac {1}{8\pi }}\int _{\mathcal {V}}d^{3}x\mathbf {E} _{\parallel }^{2}} E ∥ i = ∇ i A 0 {\displaystyle {E}_{\parallel i}={\nabla _{i}}A_{0}}
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)\,.} where andE ⊥ i = − 1 c ∂ 0 A i {\displaystyle {E}_{\perp i}=-{\frac {1}{c}}\partial _{0}A_{i}}
Quantum electrodynamics and field theory [ edit ] TheLagrangian ofquantum electrodynamics extends beyond the classical Lagrangian established in relativity to incorporate the creation and annihilation of photons (and electrons):
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 },} where the first part in the right hand side, containing theDirac spinor ψ {\displaystyle \psi } Dirac field . Inquantum field theory 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.
