18 Gauge felds, coordinate systems, and geometric phases 18.1 Maxwell’s equations Maxwell said 1 ∂B! Faraday’s law E! = (18.1) ∇× − c ∂t 1 ∂D! Ampère’s law H! = !j + (18.2) ∇× c ! ∂t " Coulomb’s law D! = ρ (18.3) ∇ no magnetic monopoles B! =0 (18.4) ∇ and it was light [43, 44]. Together with Newton’s laws of classical mechanics and the Lorentz force !v F! = q E! + B! (18.5) c × # $ that was all what was known fundamentally about physics by the end of the 19th century. Maxwell originally didn’t write his equations in the modern notation using vectors we are familiar with. This was introduced later by Oliver Heaviside [45] and Josiah Willard Gibbs, making the equations much easier to read (and evaluate). Instead, Maxwell used 20 scalar felds, inspired by the then-popular fuid mechanics. Also his equations contain explicitly the vector potential A!, which can be eliminated from his equations. The equations for the 16 parameters above above include the relations B! = µ0 H! + λM! , (18.6) % & D! = %0E! + λP,! (18.7) introducing the macroscopic magnetization M! and electric polarization P! with the mag- netic permeability µ0 and the dielectric constant %0. The factor λ denotes a dimension- 281 18 Gauge felds, coordinate systems, and geometric phases Table 18.1: Defnitions of µ0, %0, and λ in different unit systems Unit system µ0 %0 λ 7 2 Système Internationale 4π 10− T /(A/m) 1/(µ0c )1 × Gauß 1 14π Heaviside-Lorentz 1 11 esu 1/c2 14π emu 1 1/c2 4π less constant depending on the unit system chosen.1 We are using Heaviside-Lorentz units (named after Oliver Heaviside and Hendrik Antoon Lorentz), equivalent to setting µ0 = %0 = λ =1. This makes the structure of Maxwell’s equations particularly easy to understand.2 18.1.1 The scalar and vector potentials In the following, we set the magnetization M! =0and the polarization P! =0. That is, we express felds E! and B! in terms of all charges and currents on a microscopic level. In this form, Maxwell’s equations are often named «in vacuum» (which is wrong or at least misleading) or «microscopic» in contrast to the «macroscopic» equations above involving M! and P! . 1Setting λ =4π (Gauß, esu, emu) scales electromagnetic quantities by an integration over the solid angle. The factor 4π then appears explicitly in Coulomb’s law (18.3), however drops out of calculations involving planar geometries, for example parallel charged plates. 2Maxwell’s equations are equations each establishing a certain proportionality. Historically, different pro- portionality constants have emerged, leading to different electromagnetic unit systems. Among these are the Système Internationale d’Unités, the Gaussian unit system, and the Heaviside-Lorentz system, see Table 18.1 for the corresponding values of the constants. Using these constants, we can express the Coulomb force between two point-like charged particles as λ q1q2 F = 2 (18.8) 4π#0 r and the force between two infnitely thin wires of length $ at distance r as λµ0 $ F = 2I1I2 . (18.9) 4π r Maxwell’s equations then have the form 1 1 ∂B% 1 1 ∂D% √#0 E% = , √µ0 H% = λ%j + , D% = λρ, B% =0. ∇× − √µ0 c ∂t ∇× √#0 c ! ∂t " ∇ ∇ Everything else one needs to know about electromagnetic units can be found in the appendix of John David Jackson’s book on electromagnetism. 282 18.1 Maxwell’s equations With Stokes’ theorem in differential form, ( A!)=0, (18.10) ∇· ∇× ( φ)=0 (18.11) ∇× ∇· for any suffciently well-behaved vector feld A! = A!(!x, t) and scalar feld φ = φ(!x, t),we can use the divergence-freeness of the magnetic feld (18.4) and Faraday’s law (18.1)to express E! and B! through B! = A,! (18.12) ∇× 1 ∂ E! + A! =0 (18.13) ∇× c ∂t # $ 1 ∂ E! = A! + φ . (18.14) ⇒ − c ∂t ∇ # $ We replace E! and B! accordingly in Ampère’s law (18.2) and obtain 1 ∂ 1 ∂ 1 A! φ A! = !j. (18.15) ∇× ∇× − c ∂t −∇ − c ∂t c % & # $ Using the identity A! = A! ( ) A! (18.16) ∇× ∇× ∇ ∇ − ∇·∇ % & % & ∆ and the d’Alembert operator ' () * 1 ∂2 := ∆ (18.17) ! c2 ∂t2 − we obtain 1 ∂ 1 A! + φ + A! = !j. (18.18) ! ∇ c ∂t ∇ c # $ Replacing E! from Eq. (18.14) in Coulomb’s law (18.3) yields 1 ∂ φ A! = ρ (18.19) ∇ −∇ − c ∂t # $ 1 ∂ ∆φ A! = ρ (18.20) ⇒− − c ∂t∇ 1 ∂ 1 ∂ φ φ + A! = ρ, (18.21) ⇒ ! − c ∂t c ∂t ∇ # $ where in the last line we just added and subtracted (1/c2)∂2φ/∂t2. We note that the equations (18.18) and (18.21) are exactly equivalent to Maxwell’s equations: Four frst- order differential equations are replaced by two second-order equations. Typeset June 8, 2021 283 18 Gauge felds, coordinate systems, and geometric phases 18.2 Gauge invariance The felds A! and φ are not uniquely determined by these equations: Shifting with Λ= Λ(!x, t) according to A! A! + Λ=: A!", (18.22) → ∇ 1 ∂ φ φ Λ=: φ", (18.23) → − c ∂t doesn’t change the physical felds B! and E! : B! = A! A!" = A! + Λ ∇× →∇× ∇× ∇ (18.24) = A! = B,! % & ∇× 1 ∂ 1 ∂ 1 ∂ 1 ∂ E! = φ A! φ" A!" = φ Λ A! + Λ −∇ − c ∂t → −∇ − c ∂t −∇ − c ∂t − c ∂t ∇ # $ (18.25) 1 ∂ % & = φ A! = E.! −∇ − c ∂t This is called a gauge degree of freedom: The felds A! and φ are defned only up to a gradient of a (possibly space- and time-dependent) scalar feld Λ, and any particular choice of Λ (called «fxing the gauge») leads to exactly the same physics. Therefore any theory involving electromagnetic felds must be formulated in a form which respects this gauge invariance. Including the Lorentz force into Lagrange’s equations ∂ d ∂ L L =0 (18.26) ∂xi − dt ∂x˙ i where ∂ /∂xi are the components of the canonical force and pi = ∂ /∂x˙ i are the com- L L ponents of the canonical momentum leads to e p! = m!x˙ + A! (18.27) c for the canonical momentum of a particle with charge e. We stress that the canonical momentum p! , being just a variable conjugate to the position !x explicitly depends on the gauge chosen. Thus in order to write the kinetic momentum in the Lagrange function in a gauge invariant form, we need to substitute it according to e m!x˙ p! A,! (18.28) → − c a substitution we will meet again in quantum mechanics, where in real-space represen- tation p! =(h¯/i) is the operator entering the canonical commutation relation [!x,p ! ]= ∇ i¯h. 284 18.2 Gauge invariance 18.2.1 Common gauges The Lorenz gauge This is named after the danish physicist Ludvig Valentin Lorenz, not to be confused with Hendrik Lorentz mentioned above. It reads 1 ∂ φ + A! =0, (18.29) c ∂t ∇ turning Maxwell’s equations into a set of inhomogeneous wave equations, particularly popular in the calculation of time dependent electromagnetic felds. This does not fx the gauge completely, as any scalar ψ = ψ(!x, t) satisfying a homogeneous wave equation ! ψ =0 (18.30) can still be used to shift A! A! + ψ and φ φ (1/c)∂ψ/∂t without changing the → ∇ → − physics. We note that the Lorenz gauge is Lorentz invariant. The Coulomb gauge Here we set A! =0. (18.31) ∇ Eq. (18.21) then reduces to Poisson’s equation, ∆φ = ρ, (18.32) − the solution of which is the instantaneous Coulomb potential.3 This also allows us to write the vector potential for a static uniform magnetic feld (a very common experi- mental setup in solid-state physics) as 1 A! = !x B.! (18.33) −2 × Two remarks: (a) The Coulomb gauge is not Lorentz invariant. (b) A! diverges. The temporal gauge In this gauge, the scalar feld itself φ =0. (18.34) Used by Hermann Weyl who discovered the principle of gauge invariance in quantum mechanics in 1918, quite some years before Schrödinger’s equation was established. 3A contradiction to Einstein’s postulate of a fnite speed of light c, refecting the fact that the scalar po- tential itself is not a physical object. Typeset June 8, 2021 285 18 Gauge felds, coordinate systems, and geometric phases The axial gauge A very special choice, A3 =0. (18.35) Used in particle physics to avoid «ghost felds». 18.3 Nonrelativistic quantum mechanics In this section we discuss the Hamiltonian for a single nonrelativistic scalar particle of mass m, charge e. For vanishing electromagnetic felds, its Hamiltonian and the Schrödinger equation of motion simply reads 2 1 h¯ H = , (18.36) 2m i ∇ # $ ∂ HΨ=i¯h Ψ. (18.37) ∂t A fundamental principle of quantum mechanics states that the properties of a physi- cal object cannot be determined by measuring its wavefunction (which is impossible), but rather only by determining expectation values of operators expressed in a certain representation.4 That is equivalent to the freedom to choose an arbitrary phase for the wavefunction: This phase cannot be expressed as the expectation value of an operator, thus cannot be measured, hence its choice is arbitrary and doesn’t affect the physics of the system. So if we set ie/(hc¯ )Λ Ψ ΨΛ := e Ψ, (18.38) → Schrödinger’s equation should not be changed. However taking the gradient of the shifted wavefunction shifts the momentum operator and the time derivative h¯ h¯ e + Λ, (18.39) i ∇→ i ∇ c∇ ∂ ∂ e ∂ i¯h i¯h Λ (18.40) ∂t → ∂t − c ∂t and therefore does change the form of Eq. (18.37) except for the trivial case of a constant, single-valued global phase.