Potential field approach Mathematical descriptions of the electromagnetic field

1 potential field approach

1.1 maxwell s equations in potential formulation
1.2 gauge freedom

1.2.1 coulomb gauge
1.2.2 lorenz gauge


1.3 extension quantum electrodynamics

potential field approach

many times in use , calculation of electric , magnetic fields, approach used first computes associated potential: electric potential,



φ


{\displaystyle \varphi }

, electric field, , magnetic potential, a, magnetic field. electric potential scalar field, while magnetic potential vector field. why electric potential called scalar potential , magnetic potential called vector potential. these potentials can used find associated fields follows:

e

=
−

∇

φ
−



∂

a



∂
t





{\displaystyle \mathbf {e} =-\mathbf {\nabla } \varphi -{\frac {\partial \mathbf {a} }{\partial t}}}







b

=

∇

×

a



{\displaystyle \mathbf {b} =\mathbf {\nabla } \times \mathbf {a} }



maxwell s equations in potential formulation

these relations can substituted maxwell s equations express latter in terms of potentials. faraday s law , gauss s law magnetism reduce identities (e.g., in case of gauss s law magnetism, 0 = 0). other 2 of maxwell s equations turn out less simply.

these equations taken powerful , complete maxwell s equations. moreover, problem has been reduced somewhat, electric , magnetic fields had 6 components solve for. in potential formulation, there 4 components: electric potential , 3 components of vector potential. however, equations messier maxwell s equations using electric , magnetic fields.

gauge freedom

these equations can simplified taking advantage of fact electric , magnetic fields physically meaningful quantities can measured; potentials not. there freedom constrain form of potentials provided not affect resultant electric , magnetic fields, called gauge freedom. these equations, choice of twice-differentiable scalar function of position , time λ, if (φ, a) solution given system, potential (φ′, a′) given by:

φ
′

=
φ
−



∂
λ


∂
t





{\displaystyle \varphi =\varphi -{\frac {\partial \lambda }{\partial t}}}








a

′

=

a

+

∇

λ
.


{\displaystyle \mathbf {a} =\mathbf {a} +\mathbf {\nabla } \lambda .}

this freedom can used simplify potential formulation. either of 2 such scalar functions typically chosen: coulomb gauge , lorenz gauge.

coulomb gauge

the coulomb gauge chosen in such way




∇

⋅


a

′

=
0


{\displaystyle \mathbf {\nabla } \cdot \mathbf {a} =0}

, corresponds case of magnetostatics. in terms of λ, means must satisfy equation

∇

2


λ
=
−

∇

⋅

a



{\displaystyle \nabla ^{2}\lambda =-\mathbf {\nabla } \cdot \mathbf {a} }

.

this choice of function results in following formulation of maxwell s equations:

∇

2



φ
′

=
−


ρ

ε

0






{\displaystyle \nabla ^{2}\varphi =-{\frac {\rho }{\varepsilon _{0}}}}







∇

2




a

′

−

μ

0



ε

0






∂

2




a

′



∂

t

2





=
−

μ

0



j

+

μ

0



ε

0


∇

(



∂

φ
′



∂
t



)



{\displaystyle \nabla ^{2}\mathbf {a} -\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {a} }{\partial t^{2}}}=-\mu _{0}\mathbf {j} +\mu _{0}\varepsilon _{0}\nabla \left({\frac {\partial \varphi }{\partial t}}\right)}

several features maxwell s equations in coulomb gauge follows. firstly, solving electric potential easy, equation version of poisson s equation. secondly, solving magnetic vector potential particularly difficult. big disadvantage of gauge. third thing note, , not obvious, electric potential changes instantly everywhere in response change in conditions in 1 locality.

for instance, if charge moved in new york @ 1 pm local time, hypothetical observer in australia measure electric potential directly measure change in potential @ 1 pm new york time. seemingly goes violates causality in special relativity, i.e. impossibility of information, signals, or travelling faster speed of light. resolution apparent problem lies in fact that, stated, no observers can measure potentials; measure electric , magnetic fields. so, combination of ∇φ , ∂a/∂t used in determining electric field restores speed limit imposed special relativity electric field, making observable quantities consistent relativity.

lorenz gauge

a gauge used lorenz gauge. in this, scalar function λ chosen such that

∇

⋅


a

′

=
−

μ

0



ε

0





∂

φ
′



∂
t



,


{\displaystyle \mathbf {\nabla } \cdot \mathbf {a} =-\mu _{0}\varepsilon _{0}{\frac {\partial \varphi }{\partial t}},}

meaning λ must satisfy equation

∇

2


λ
−

μ

0



ε

0






∂

2


λ


∂

t

2





=
−

∇

⋅

a

−

μ

0



ε

0





∂
φ


∂
t



.


{\displaystyle \nabla ^{2}\lambda -\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\lambda }{\partial t^{2}}}=-\mathbf {\nabla } \cdot \mathbf {a} -\mu _{0}\varepsilon _{0}{\frac {\partial \varphi }{\partial t}}.}

the lorenz gauge results in following form of maxwell s equations:

∇

2



φ
′

−

μ

0



ε

0






∂

2



φ
′



∂

t

2





=

◻

2



φ
′

=
−


ρ

ε

0






{\displaystyle \nabla ^{2}\varphi -\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=\box ^{2}\varphi =-{\frac {\rho }{\varepsilon _{0}}}}







∇

2




a

′

−

μ

0



ε

0






∂

2




a

′



∂

t

2





=

◻

2




a

′

=
−

μ

0



j



{\displaystyle \nabla ^{2}\mathbf {a} -\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {a} }{\partial t^{2}}}=\box ^{2}\mathbf {a} =-\mu _{0}\mathbf {j} }

the operator




◻

2




{\displaystyle \box ^{2}}

called d alembertian (some authors denote square



◻


{\displaystyle \box }

). these equations inhomogeneous versions of wave equation, terms on right side of equation serving source functions wave. wave equation, these equations lead 2 types of solution: advanced potentials (which related configuration of sources @ future points in time), , retarded potentials (which related past configurations of sources); former disregarded field analyzed causality perspective.

as pointed out above, lorenz gauge no more valid other gauge since potentials cannot measured. despite this, there quantum mechanical phenomena in potentials appear affect particles in regions observable field vanishes throughout region, example in aharonov–bohm effect. however, these phenomena not provide means directly measure potentials nor detect difference between different mutually gauge equivalent potentials. lorenz gauge has further advantage of equations being lorentz invariant.

extension quantum electrodynamics

canonical quantization of electromagnetic fields proceeds elevating scalar , vector potentials; φ(x), a(x), fields field operators. substituting 1/c = ε0μ0 previous lorenz gauge equations gives:

∇

2



a

−


1

c

2








∂

2



a



∂

t

2





=
−

μ

0



j



{\displaystyle \nabla ^{2}\mathbf {a} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {a} }{\partial t^{2}}}=-\mu _{0}\mathbf {j} }







∇

2


φ
−


1

c

2








∂

2


φ


∂

t

2





=
−


ρ

ε

0






{\displaystyle \nabla ^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}

here, j , ρ current , charge density of matter field. if matter field taken describe interaction of electromagnetic fields dirac electron given four-component dirac spinor field ψ, current , charge densities have form:

j

=
−
e

ψ

†



α

ψ


ρ
=
−
e

ψ

†


ψ

,


{\displaystyle \mathbf {j} =-e\psi ^{\dagger }{\boldsymbol {\alpha }}\psi \,\quad \rho =-e\psi ^{\dagger }\psi \,,}

where α first 3 dirac matrices. using this, can re-write maxwell s equations as:

which form used in quantum electrodynamics.