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Relativistic Particles and Fields in External Electromagnetic Potential

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Relativistic Particles and Fields in External Electromagnetic Potential The most tragic word in the English language is ‘potential’. Arthur Lotti 6 Relativistic Particles and Fields in External Electromagnetic Potential Given the classical field theory of relativistic particles, we may ask which quantum phenomena arise in a relativistic generalization of the Schr¨odinger theory of atoms. In a first step we shall therefore study the behavior of the Klein-Gordon and Dirac equations in an external electromagnetic field. Let Aµ(x) be the four-vector potential that accounts for electric and magnetic field strengths via the equations (4.230) and (4.231). For classical relativistic point particles, an interaction with these external fields is introduced via the so-called minimal substitution rule, whose gauge origin and experimental consequences will now be discussed. An important property of the electromagnetic field is its description in terms of a vector potential Aµ(x) and the gauge invariance of this description. In Eqs. (4.233) and (4.234) we have expressed electric and magnetic field strength as components of a four-curl Fµν = ∂µAν ∂νAµ of a vector potential Aµ(x). This four-curl is invariant under gauge transformations− A (x) A (x)+ ∂ Λ(x). (6.1) µ → µ µ The gauge invariance restricts strongly the possibilities of introducing electromag- netic interactions into particle dynamics and the Lagrange densities (6.92) and (6.94) of charged scalar and Dirac fields. We shall see that the origin of the minimal substi- tution rule lies precisely in the gauge-invariance of the vector-potential description of electromagnetism. 6.1 Charged Point Particles A free relativistic particle moving along an arbitrarily parametrized path xµ(τ) in four-space is described by an action = Mc dτ q˙µ(τ)q ˙ (τ). (6.2) A − µ Z q The physical time along the path is given by q0(τ) = ct, and the physical velocity by v(t) dq(t)/dt. In terms of these, the action reads: ≡ 2 1/2 2 v (t) = dt L(t) Mc dt 1 2 . (6.3) A Z ≡ − Z " − c # 436 6.1 Charged Point Particles 437 6.1.1 Coupling to Electromagnetism If the particle has a charge e and lies at rest at some position x, its electric potential energy is V (x, t)= eφ(x, t) (6.4) where φ(x, t)= A0(x, t). (6.5) In our convention, the charge of the electron e has a negative value to be in agreement with the sign in the historic form of the Maxwell equations: ∇ E(x) = ∇2φ(x)= ρ(x), · − ∇ B(x) E˙ (x) = ∇ ∇ A(x) E˙ (x) × − × × − 1 = ∇2A(x) ∇ ∇A(x) E˙ (x)= j(x). (6.6) − − · − c h i If the electron moves along a trajectory q(t), its potential energy is V (t)= eφ (q(t), t) . (6.7) In the Lagrangian L = T V , this contributes with the opposite sign − Lint(t)= eA0 (q(t), t) , (6.8) − giving a potential part of the interaction int 0 pot = e dt A (q(t), t) . (6.9) A − Z Since the time t coincides with q0(τ)/c of the trajectory, this can be expressed as int e 0 0 pot = dq A . (6.10) A −c Z In this form it is now quite simple to write down the complete electromagnetic inter- action purely on the basis of relativistic invariance. The direct invariant extension of (6.11) is obviously int e µ = dq Aµ(q). (6.11) A −c Z Thus, the full action of a point particle can be written in covariant form as e = Mc dτ q˙µ(τ)q ˙ (τ) dqµA (q), (6.12) A − µ − c µ Z q Z or more explicitly as v2 1/2 1 2 0 v A = dt L(t)= Mc dt 1 2 e dt A . (6.13) A Z − Z " − c # − Z − c · 438 6 Relativistic Particles and Fields in External Electromagnetic Potential The canonical formalism supplies us with the canonically conjugate momenta ∂L v e e P = = M + A p + A. (6.14) ∂v 1 v2/c2 c ≡ c − q The Euler-Lagrange equation obtained by extremizing this equation is d ∂L ∂L = , (6.15) dt ∂v(t) ∂q(t) or d e d e p(t)= A(q(t), t) e∇A0(q(t), t)+ vi∇Ai(q(t), t). (6.16) dt −c dt − c We now split d ∂ A(q(t), t)=(v(t) ∇)A(q(t), t)+ A(q(t), t), (6.17) dt · ∂t and obtain d e e ∂ e p(t)= (v(t) ∇)A(q(t), t) A(q(t), t) e∇A0(q(t), t)+ vi∇Ai(q(t), t). dt −c · − c ∂t − c (6.18) The right-hand side contains the electric and magnetic fields (4.235) and (4.236), in terms of which it takes the well-known form d v p = e E + B . (6.19) dt c × This can be rewritten in terms of the proper time τ t/γ as ≡ d e p = E p0 + p B , (6.20) dτ Mc × Recalling Eqs. (4.233) and (4.234), this is recognized as the spatial part of the covariant equation d e pµ = F µ pν. (6.21) dτ Mc ν The temporal component of this equation d e p0 = E p (6.22) dτ Mc · gives the energy increase of a particle running through an electromagnetic field. In real time this is d v p0 = e E . (6.23) dt · c Combining this with (6.19), we find the acceleration d d p e v v v v E B E (t)= c 0 = + . (6.24) dt dt p γM c × − c c · 6.1 Charged Point Particles 439 The velocity is related to the canonical momenta and external field via e v P A = − c . (6.25) c e 2 P A + m2c2 s − c This can be used to calculate the Hamiltonian via the Legendre transform ∂L H = v L = P v L, (6.26) ∂v − · − giving e 2 H = c P A + m2c2 + eA0. (6.27) s − c In the non-relativistic limit this has the expansion 1 e 2 H = mc2 + P A + eA0 + .... (6.28) 2m − c Thus, the free theory goes over into the interacting theory by the minimal substitu- tion rule e e p p A, H H A0, (6.29) → − c → − c or, in relativistic notation: e pµ pµ Aµ. (6.30) → − c 6.1.2 Spin Precession in an Atom In 1926, Uhlenbeck and Goudsmit noticed that the observed Zeeman splitting of 1 atomic levels could be explained by an electron of spin 2 . Its magnetic moment is usually expressed in terms of the combination of fundamental constants which have the dimension of a magnetic moment, the Bohr magneton µB = eh/Mc¯ . It reads S eh¯ = gµ , µ , (6.31) B h¯ B ≡ 2Mc where S = /2 is the spin matrix which has the commutation rules [Si,Sj]= ihǫ¯ ijkSk, (6.32) and g is a dimensionless number called the gyromagnetic ratio or Land´efactor. If an electron moves in an orbit under the influence of a torque-free central force, as an electron does in the Coulomb field of an atomic nucleus, the total angular momentum is conserved. The spin, however, shows a precession just like a spinning 440 6 Relativistic Particles and Fields in External Electromagnetic Potential top. This precession has two main contributions: one is due to the magnetic coupling of the magnetic moment of the spin to the magnetic field of the electron orbit, called spin-orbit coupling. The other part is purely kinematic, it is the Thomas precession discussed in Section (4.15), caused by the slightly relativistic nature of the electron orbit. The spin-orbit splitting of the atomic energy levels (to be pictured and discussed further in Fig. 6.1) is caused by a magnetic interaction energy g 1 dV (r) HLS(r)= S L , (6.33) 2M 2c2 · r dr where V (r) is the atomic potential depending only on r = x . To derive this HLS(r), | | we note that the spin precession of the electron at rest in a given magnetic field B is given by the Heisenberg equation dS = B, (6.34) dt × where is the magnetic moment of the electron. In an atom, the magnetic field in the rest frame of the electron is entirely due to the electric field in the rest frame of the atom. A Lorentz transformation (4.286) that boosts an electron at rest to a velocity v produces a magnetic field in the electron’s rest frame: v 1 B = Bel = γ E, γ = . (6.35) − c × 1 v2/c2 − q Since an atomic electron has a small velocity ratio v/c which is of the order of the fine-structure constant α 1/137, the field has the approximate size ≈ v B E. (6.36) el ≈ − c × The electric field gives the electron an acceleration e v˙ = E, (6.37) M so that we may also write Mc 1 B v v˙ , (6.38) el ≈ − e c2 × and Heisenberg’s precession equation (6.34) as dS g (v v˙ ) S. (6.39) dt ≈ 2c2 × × This can be expressed in the form dS = ª S, (6.40) dt LS × 6.1 Charged Point Particles 441 where ªLS is the angular velocity of the spin precession caused by the orbital mag- netic field in the rest frame of the electron: g ª v v˙ . (6.41) LS ≡ 2c2 × In the rest frame of the atom where the electron is accelerated towards the center along its orbit, this result receives a relativistic correction.
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