Relativistic Particles and Fields in External Electromagnetic Potential

Home , Field equation, Relativistic particle, Thomas precession

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 Schrödinger 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 electromagnetic 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 substitution 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 interaction 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 substitution 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 Landéfactor. 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 magnetic 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. To lowest order in 1/c, ª we must add to ªLS the angular velocity T of the Thomas precession, such that the total angular velocity of precession becomes

g 1

ª ª ª = + − v v˙ . (6.42) LS T ≈ 2 × Since g is very close to 2, the Thomas precession explains why the spin-orbit splitting was initially found to be in agreement with a normal gyromagnetic ratio g = 1, the characteristic value for a rotating charged sphere. If there is also an external magnetic ﬁeld, this is transformed to the electron rest frame by a Lorentz transformation (4.283), where it leads to an approximate equation of motion for the spin

dS ′ v = B B E . (6.43) dt × ≈ × − c ×

Expressing via Eq. (6.31), this becomes dS eg v

S ªem S B E . (6.44) dt ≡ − × ≈ 2Mc × − c ×

This equation deﬁnes the frequency ªem of precession due to the magnetic and electric ﬁelds in the rest frame of the electron. Expressing E in terms of the acceleration via Eq. (6.37), this becomes dS g M S eB (v v˙ ) . (6.45) dt ≈ 2Mc × − c × The acceleration can be expressed in terms of the central Coulomb potential V (r) as x 1 dV v˙ = . (6.46) − r M dr The spin precession rate in the electron’s rest frame is

dS g x 1 dV g 1 dV = S eB + v = S eB L . (6.47) dt 2Mc × × r c dr ! 2Mc × − Mc dr !

There exists a simple Hamiltonian operator for the spin-orbit interaction HLS(t), from which this equation can be derived via Heisenberg’s equation (1.280): i S˙ (t)= [S(t),HLS(t)]. (6.48) h¯ 442 6 Relativistic Particles and Fields in External Electromagnetic Potential

The operator is

LS 1 dV

H (r) = B L − · − Mc e dr ! ge g 1 dV = S B + S L . (6.49) −2Mc · 2M 2c2 · r dr Indeed, using the commutation rules (6.32), we ﬁnd immediately (6.46). Historically, the interaction energy (6.49) was used to explain the experimental level splittings assuming a gyromagnetic ratio g 1 for the electron. ≈ Without the external magnetic ﬁeld, the angular velocity of precession caused by spin-orbit coupling is g 1 ∂V

ª = L . (6.50) LS 2M 2c2 r ∂r It was realized by Thomas in 1927 that the relativistic motion of the electron changes the factor g to g 1, as in (6.42), so that the true precession frequency is −

g 1 1 ∂V

ª ª ª = + = − L . (6.51) LS T 2M 2c2 r ∂r This implied that the experimental data should give g 1 1, so that g is really twice as large as expected for a rotating charged sphere.− Indeed≈ , the value g 2 was predicted by the Dirac theory of the electron. ≈ In Section 12.15 we shall ﬁnd that the magnetic moment of the electron has a g-factor slightly larger than the Dirac value 2, the relative deviations a (g 2)/2 being deﬁned as the anomalous magnetic moments. From measurements≡ − of the above precession rate, experimentalists have deduced the values

a(e−) = (115965.77 0.35) 10−8, (6.52) ± × a(e+) = (116030 120) 10−8, (6.53) ± × a(µ±) = (116616 31) 10−8. (6.54) ± × In quantum electrodynamics, the gyromagnetic ratio will receive further small corrections, as will be discussed in detail in Chapter 12.

6.1.3 Relativistic Equation of Motion for Spin Vector and Thomas Precession If an electron moves in an orbit under the influence of a torque-free central force, such as an electron in the Coulomb field of an atomic nucleus, the total angular momentum is conserved. The spin, however, performs a Thomas precession as discussed in the previous section. There exists a covariant equation of motion for the spin four-vector introduced in Eq. (4.767) which describes this precession. Along a particle orbit parametrized by a parameter τ, for instance the proper time, we form 6.1 Charged Point Particles 443 the derivative with respect to τ, assuming that the motion proceeds at a fixed total angular momentum:

dSˆ dpκ µ = ǫ Jˆνλ . (6.55) dτ µνλκ dτ The right-hand side can be simpliﬁed by multiplying it with the trivial expression 1 gστ p p =1. (6.56) M 2c2 σ τ Now we use the identity for the ǫ-tensor

ǫµνλκgστ = ǫµνλσgκτ + ǫµνσκgλτ + ǫµσλκgντ + ǫσνλκgµτ , (6.57) which can easily be verified using the antisymmetry of the ǫ-tensor and considering µνλκ = 0123. Then the right-hand side becomes a sum of the four terms 1 ǫ J νλpσpκp′ +ǫ J νλp pσp′κ +ǫ J νλp pσp′κ +ǫ J νλpσp p′κ , (6.58) M 2c2 µνλσ κ µνσκ λ µσλκ ν σνλκ µ ′µ µ κ ′ 2 where p dp /dτ. The first term vanishes, since p pκ = (1/2)dp /dτ = (1/2)dM 2c2≡/dτ = 0. The last term is equal to Sˆ p′κp /M 2c2. Inserting the iden- − κ µ tity (6.57) into the second and third terms, we obtain twice the left-hand side of (6.55). Taking this to the left-hand side, we find the equation of motion

dS 1 dpλ µ = S p . (6.59) dτ −M 2c2 λ dτ µ

Note that on account of this equation, the time derivative dSµ/dτ points in the direction of pµ. Let us verify that this equation yields indeed the Thomas precession. Denoting the derivatives with respect to the time t = γτ by a dot, we can rewrite (6.59) as

dS 1 dS 1 γ2 S˙ = = S0p˙0 + S p˙ p = (S v˙ ) v, (6.60) ≡ dt γ dτ −M 2c2 · c2 · dS 1 d γ2 S˙ 0 = (S v)= (S v˙ ) . (6.61) 0 ≡ dt c dt · c2 · We now diﬀerentiate Eq. (4.780) with respect to the time using the relationγ ˙ = γ3vv˙ /c2, and ﬁnd

γ 1 γ 1 γ3 1 S˙ = S˙ S˙ 0v S0v˙ (v v˙ ) S0 v. (6.62) R − γ +1 c2 − γ +1 c2 − (γ + 1)2 c4 · Inserting here Eqs. (6.60) and (6.61), we obtain

γ2 1 γ 1 γ3 S˙ = (S v˙ )v S0 v˙ (v v˙ ) S0v. (6.63) R γ +1 c2 · − γ +1 c2 − (γ + 1)2 · 444 6 Relativistic Particles and Fields in External Electromagnetic Potential

On the right-hand side we return to the spin vector SR using Eqs. (4.779) and (4.782), and ﬁnd

γ2 1

S˙ = [(S v˙ )v (S v)v˙ ]= ª S , (6.64) R γ +1 c2 R · − R · T × R with the Thomas precession frequency

γ2 1

ª = v v˙ , (6.65) T −(γ + 1) c2 × which agrees with the result (4B.26) derived from purely group-theoretic consider- ations. In an external electromagnetic ﬁeld, there is an additional precession. For slow particles, it is given by Eq. (6.45). If the electron moves fast, we transform the electromagnetic ﬁeld to the electron rest frame by a Lorentz transformation (4.283), and obtain an equation of motion for the spin:

v 2 v v

˙ ′ γ SR = B = γ B E B . (6.66) × ×" − c × − γ +1 c c · #

Expressing via Eq. (6.31), this becomes

eg v γ v v

S˙ R SR ªem = SR B E B , (6.67) ≡ − × 2Mc × " − c × − γ +1 c c · # which is the relativistic generalization of Eq. (6.44). It is easy to see that the associated fully covariant equation is

µ dS g µν 1 µ d λ eg µν 1 µ λκ = eF Sν + p Sλ p = F Sν + 2 2 p SλF pκ . (6.68) dτ 2Mc " Mc dτ # 2Mc M c On the right-hand side we have inserted the relativistic equation of motion of a point particle (6.21) in an external electromagnetic ﬁeld. If we add to this the relativistic Thomas precession rate (6.59), we obtain the covariant Bargmann-Michel-Telegdi equation1

µ dS 1 µν g 2 µ d λ e µν g 2 µ λκ = egF Sν + − p Sλ p = gF Sν + −2 2 p SλF pκ .(6.69) dτ 2Mc" Mc dτ # 2Mc M c

For the spin vector SR in the electron rest frame this implies a change in the electromagnetic precession rate in Eq. (6.67) to

ª ª S˙ = S ª S ( + ) (6.70) R − R × em ≡ − R × em T 1V. Bargmann, L. Michel, and V.L. Telegdi, Phys. Rev. Lett. 2 , 435 (1959). 6.2 Charged Particle in Schr¨odinger Theory 445 with a frequency given by the Thomas equation2

T e g 1 g γ v v g γ v ªem = 1+ B 1 B E .(6.71) −Mc " 2 − γ ! − 2 − γ+1 c · c − 2 − γ+1! c × # The contribution of the Thomas precession is the part without the gyromagnetic ratio g:

e 1 γ 1 γ 1

ªT = 1 B + (v B) v + v E . (6.72) −Mc "− − γ ! γ+1 c2 · γ+1 c × #

This agrees with the Thomas frequency (6.65), after inserting the acceleration (6.24). The Thomas equation (6.71) can be used to calculate the time dependence of the helicity h SR vˆ of an electron, i.e., its component of the spin in the direction of motion. Using≡ the· chain rule of diﬀerentiation, we can express the change of the helicity as dh d 1 d = (S vˆ)= S˙ vˆ + [S (vˆ S )vˆ] v, (6.73) dt dt R · R · v R − · R dt Inserting (6.70) as well as the equation for the acceleration (6.24), we obtain

dh e g gv c = SR⊥ 1 vˆ B + E . (6.74) dt −Mc · 2 − × 2c − v where SR⊥ is the component of the spin vector orthogonal to v. This equation shows that for a Dirac electron, which has the g-factor g = 2, the helicity remains constant in a purely magnetic field. Moreover, if the electron moves ultra-relativistically (v c), the value g = 2 makes the last term extremely small, (e/Mc)γ−2S E, ≈ ≈ R⊥ · so that the helicity is almost unaffected by an electric field. The anomalous magnetic moment of the electron, however, changes this to a finite value (e/Mc)aS ≈ − R⊥ · E. This drastic effect was exploited to measure the experimental values listed in Eqs. (6.52)–(6.54).

6.2 Charged Particle in Schr¨odinger Theory

When going over from quantum mechanics to second quantized ﬁeld theories we found the rule that a non-relativistic Hamiltonian p2 H = + V (x) (6.75) 2m becomes an operator

∇2 H = d3xψ†(x, t) + V (x) ψ(x, t), (6.76) Z "−2m # 2L.T. Thomas, Phil. Mag. 3 , 1 (1927). 446 6 Relativistic Particles and Fields in External Electromagnetic Potential where we have omitted the operator hats, for brevity. With the same rules we see that the second quantized form of the interacting nonrelativistic Hamiltonian in a static A(x) ﬁeld,

(p eA)2 e H = − + A0, (6.77) 2m c is given by 1 e 2 H = d3xψ†(x, t) ∇ i A + eA0(x) ψ(x, t). (6.78) Z "−2m − c # When going to the action of this theory we ﬁnd

3 † 0 = dtL = dt d x ψ (x, t) i∂t + eA ψ(x, t) A Z Z Z 1 e 2 + ψ†(x, t) ∇ i A ψ(x, t) . (6.79) 2m − c It is easy to verify that (6.78) reemerges from the Legendre transform ∂L H = ψ˙(x, t) L. (6.80) ∂ψ˙(x, t) − The action (6.79) holds also for time-dependent Aµ(x) ﬁelds. We can now deduce the second quantized form of the minimal substitution rule (6.29) which is e ∇ ∇ i A(x, t), → − c ∂ ∂ + ieA0(x, t), (6.81) t → t or covariantly: e ∂ ∂ + i A (x). (6.82) µ → µ c µ This substitution rule has the important property that the gauge invariance of the free photon action is preserved by the interacting theory: If we perform the gauge transformation

Aµ(x) Aµ(x)+ ∂µΛ(x), (6.83) → i.e.,

A0(x, t) A0(x, t)+ ∂ Λ(x, t) → t A(x, t) A(x, t) ∇Λ(x, t), (6.84) → − the action remains invariant provided we simultaneously change the ﬁelds ψ(x, t) of the charged particles by a spacetime-dependent phase

x ψ (x, t) e−i(e/c)Λ( ,t)ψ(x, t). (6.85) → 6.3 Charged Relativistic Fields 447

Under this transformation, the derivatives of the ﬁeld change like

−i(e/c)Λ(x,t) ∂tψ(x, t) e (∂t ie∂tΛ) ψ(x, t), → − e ∇ψ(x, t) e−i(e/c)Λ(x,t) ∇ i ∇Λ(x, t) ψ(x, t). (6.86) → − c The modiﬁed derivatives appearing in the action have therefore the following simple transformation law:

e 0 −i(e/c)Λ(x,t) 0 ∂t + i A ψ(x, t) e ∂t + ieA ψ(x, t), c → e e ∇ i A ψ(x, t) e−i(e/c)Λ(x,t) ∇ i A ψ(x, t). (6.87) − c → − c These combinations of derivatives and gauge ﬁelds are called covariant derivatives. They occur so frequently in gauge theories that they deserve their own symbols:

D ψ(x, t) ∂ + ieA0 ψ(x, t), t ≡ t e Dψ(x, t) ∇ i A ψ(x, t), (6.88) ≡ − c or, in four-vector notation, e Dµψ(x) = ∂µ + i Aµ ψ(x). (6.89) c Here the adjective of the covariant derivative does not refer to the Lorentz group but to the gauge group. It records the fact that Dµψ transforms under local gauge changes (6.81) of ψ in the same way as ψ itself in (6.85):

D ψ(x) e−i(e/c)Λ(x)D ψ(x). (6.90) µ → µ With the help of this covariant derivative, any action that is invariant under a global multiplication change of the ﬁeld by a constant phase factor e−iφ,

ψ(x) e−iφψ(x), (6.91) → can also be made invariant under a local version of this transformation, in which φ is an arbitrary function φ(x). For this, we merely have to replace all derivatives by covariant derivatives (6.89), and add to the ﬁeld action the gauge-invariant photon action (4.237).

6.3 Charged Relativistic Fields 6.3.1 Scalar Field The Lagrangian density of a free relativistic scalar ﬁeld was stated in Eq. (4.165):

= ∂ φ∗(x)∂µφ(x) M 2φ∗(x)φ(x). (6.92) L µ − 448 6 Relativistic Particles and Fields in External Electromagnetic Potential

If the ﬁeld carries a charge e, the derivatives are simply replaced by the covariant derivatives (6.89), thus leading to a straightforward generalization of the Schr¨odinger action in (6.79):

= [D φ(x)]∗Dµφ(x) M 2φ∗(x)φ(x) L µ − e µ e µ 2 ∗ = ∂µ i Aµ(x) φ(x) ∂ + i A (x) φ(x) M φ (x)φ(x). (6.93) − c c − The associated scalar ﬁeld action = d4x must be extended by the gauge- A L invariant photon action (4.237). R

6.3.2 Dirac Field The Lagrangian density of a free charged spin-1/2 ﬁeld was stated in Eq. (4.501):

(x) = ψ¯(x)(i/∂ M) ψ(x). (6.94) L − If the particle carries a charge e, we must replace the derivatives in this Lagrangian by their covariant versions (6.89):

µ µ e e /∂ = γ ∂µ γ ∂µ + i Aµ = /∂ + i /A /D. (6.95) → c c ≡ Adding again the gauge-invariant photon action (4.237), we arrive at the Lagrangian of quantum electrodynamics (QED) 1 (x)= ψ¯(x)(i/D M) ψ(x) F 2 . (6.96) L − − 4 µν The classical ﬁeld equations can easily be found by extremizing the action under variations of all ﬁelds, which gives δ A = (i/D M) ψ(x)=0, (6.97) δψ¯(x) −

δ νµ 1 µ A = ∂νF (x) j (x)=0, (6.98) δAµ(x) − c where jµ(x) is the current density

jµ(x) ec ψ¯(x)γµψ(x). (6.99) ≡ Equation (6.98) is the Maxwell equation for the electromagnetic ﬁeld around a classical four-dimensional vector current jµ(x): 1 ∂ F νµ(x)= jµ(x). (6.100) ν c µ In the Lorentz gauge ∂µA (x) = 0, this equation reads simply 1 ∂2Aµ(x)= jµ(x). (6.101) − c 6.4 Pauli Equation from Dirac Theory 449

The current jµ combines the charge density ρ(x) and the current density j of particles of charge e in a four-vector jµ =(cρ, j) . (6.102) In terms of electric and magnetic fields Ei = F i0, Bi = F jk, the field equations (6.100) turn into the Maxwell equations − ∇ E = ρ = eψγ¯ 0ψ = eψ†ψ · 1 e ¯ ∇ B E˙ = j = ψ ψ. (6.103) × − c c The first is Coulomb’s law, the second Ampère’s law in the presence of charges and currents. Note that the physical units employed here differ from those used in many books of classical electrodynamics3 by the absence of a factor 1/4π on the right-hand side. The Lagrangian used in those books is 1 1 (x) = F 2 (x) jµ(x)A (x) (6.104) L −8π µν − c µ 1 1 = E2 B2 (x) ρφ j A (x), 4π − − − c · h i which leads to Maxwell’s field equations ∇ E = 4πρ, · 4π ∇ B = j. (6.105) × c The form employed conventionally in quantum field theory arises from this by replacing A √4πA and e √4πe. The charge of the electron in our units has therefore the→ numerical value→ e = √4πα 4π/137 (6.106) − ≈ − q rather than e = √α. − 6.4 Pauli Equation from Dirac Theory

It is instructive to take the Dirac equation (6.97) to a two-component form corresponding to (4.567) and (4.569), and further to (4.585). Due to the fundamental nature of the equations to be derived we shall not work with natural units in this section but carry along explicitly all fundamental constants. As in (4.586), we multiply (6.97) by (ih/D¯ Mc) and work out the product − µ e µ e (ih/D¯ Mc)(ih/D¯ + Mc) = iγ h∂¯ µ + i Aµ + Mc iγ h∂¯ µ + i Aµ Mc . − c c − (6.107)

3See for example J.D. Jackson, Classical Electrodynamics, Wiley and Sons, New York, 1967. 450 6 Relativistic Particles and Fields in External Electromagnetic Potential

We now use the relation 1 1 γµγν = (γµγν + γνγµ)+ (γµγν γνγµ)= gµν iσµν , (6.108) 2 2 − − with σµν from (4.517), and ﬁnd

µ ν e e γ γ h∂¯ µ + i Aµ ∂µ + i Aµ c c µν e e i µν e e = g h∂¯ µ + i Aµ ∂µ + i Aµ σ [¯h∂µ + i Aµ, h∂¯ ν + i Aν] c c − 2 c c 2 e 1 eh¯ µν = h∂¯ µ + i Aµ + σ Fµν . (6.109) c 2 c Thus we obtain, as a generalization of Eqs. (4.585), the Pauli equation:

2 e 1 eh¯ µν 2 2 h∂¯ µ + i Aµ σ Fµν M c ψ(x)=0, (6.110) "− c − 2 c − # and the same equation once more for the other two-component spinor ﬁeld η(x). Note that, in this equation, electromagnetism is not coupled minimally. In fact, there is a non-minimal coupling of the spin via the tensor term

1 µν « σ F = ¦ H + i E, (6.111) 2 µν − · ·

where in the chiral and Dirac representations, the matrix « is

0 0 «

« = − , D = , (6.112) 0 ! 0 ! respectively. Thus, in the chiral representation, Eq. (6.110) decomposes into two separate two-component equations for the upper and lower spinor components ξ(x) and η(x) in ψ(x):

2 e 2 2 ξ(x) h∂¯ µ + i Aµ + (H iE) M c =0. (6.113) "− c · ± − #( η(x) ) In the nonrelativistic limit where c , we remove the fast oscillations from 2 → ∞ ξ(x), setting ξ(x) e−iMc t/¯hΨ(x, t)/√2M as in (4.156), and ﬁnd for Ψ(x, t) the nonrelativistic Pauli≡ equation

2 2 h¯ e e 0 i∂t + ∇ i A + H eA (x) Ψ(x, t)=0. (6.114) " 2M − ch¯ 2Mc · − # This corresponds to a magnetic interaction energy eh¯

H = H. (6.115) mag −2Mc · 6.5 Relativistic Wave Equations in the Coulomb Potential 451

For a small magnet with a magnetic moment , the magnetic interaction energy is

H = H. (6.116) mag − · A Dirac particle has therefore a magnetic moment

e h¯ e

= =2 S, (6.117) Mc 2 2Mc

where S = /2 is the spin matrix. Experimentally, one parametrizes the magnetic moment of a fundamental particles as e

= g S, (6.118) 2Mc where g is the so-called gyromagnetic ratio. It is normalized to unity for a uniformly charged sphere. Within the Dirac theory, an electron has a gyromagnetic ratio

ge =2. (6.119) Dirac

The experimental value is very close to this. A small deviation from it is called anomalous magnetic moment. It is a consequence of the quantum nature of the electromagnetic ﬁeld and will be explained in Chapter 12. The nonrelativistic Pauli equation (6.114) could also have been obtained by introducing the electromagnetic coupling directly into the nonrelativistic two-component 0 equation (4.582). The minimal substitution rule (6.87) changes i∂t i∂t eA and

2 2 → − ( ∇) [ (∇ ieA)] . The latter is worked out in detail as in Chapter 4, Eq.· (4.583),→ and· leads− to

2 ij k

[ (∇ ieA)] = δ + iǫ σ ( ieA )( ieA ) · − ijk ∇i − i ∇j − j h 2 i k = (∇ ieA) + iǫ σ ( ieA )( ieA )+ e H. (6.120) − ijk ∇i − i ∇j − j · This brings the free-ﬁeld equation (4.584) to the nonrelativistic Pauli expression (6.114), after reinserting all fundamental constants.

6.5 Relativistic Wave Equations in the Coulomb Potential

It is now easy to write down ﬁeld equations for a Klein-Gordon and a Dirac ﬁeld in the presence of an external Coulomb potential of charge Ze. In natural units we have Zα V (x)= , r = √x2, (6.121) C − r corresponding to a four-vector potential

eAµ(x)=(VC(x, 0), 0). (6.122) 452 6 Relativistic Particles and Fields in External Electromagnetic Potential

Since this does not depend on time, we can consider the wave equations for wave −iEt −iEt functions φ(x)= e φE(x) and ψ(x)= e ψE(x), and ﬁnd the time-independent equations

(E2 + ∇2 M)φ (x) = 0 (6.123) − E and

(γ E + i ∇ M)ψ (x)=0. (6.124) · − E In these equations we simply perform the minimal substitution

Zα E E + . (6.125) → r The energy-eigenvalues obtained from the resulting equations can be compared with those of hydrogen-like atoms. The velocity of an electron in the ground state is of the order αZc. Thus for rather high Z, the electron has a relativistic velocity and there must be significant deviations from the Schrödinger theory. We shall see that the Dirac equation in an external field reproduces quite well a number of features resulting from the relativistic motion.

6.5.1 Reminder of the Schr¨odinger Equation in a Coulomb Potential The time-independent Schr¨odinger equation reads

1 2 Zα ∇ E ψE(x)=0. (6.126) −2M − r − The Laplacian may be decomposed into radial and angular parts by writing

∂2 2 ∂ Lˆ2 ∇2 = + , (6.127) ∂r2 r ∂r − r2 where Lˆ = x pˆ are the diﬀerential operators for the generators of angular momen- × 23 tum [the spatial part of Li = L of (4.97)]. Then (6.126) reads

∂2 2 ∂ Lˆ 2 2ZαM + 2ME ψE(x)=0. (6.128) −∂r2 − r ∂r r2 − r − !

2 The eigenstates of Lˆ are the spherical harmonics Ylm(θ,φ), which diagonalize also the third component of Lˆ, with the eigenvalues

2 Lˆ Ylm(θ,φ) = l(l + 1)Ylm(θ,φ),

Lˆ3Ylm(θ,φ) = mYlm(θ,φ). (6.129) 6.5 Relativistic Wave Equations in the Coulomb Potential 453

The wave functions may be factorized into a radial wave function Rnl(r) and a spherical harmonic: ψnlm(x)= Rnl(r)Ylm(θ,φ). (6.130) Explicitly,

1 1 (n + l)! R (r) = (6.131) nl 1/2 v a n (2l + 1)!u(n l 1)! B u − − t (2r/na )l+1e−r/naB M( n + l +1, 2l +2, 2r/na ) × B − B 1 (n l 1)! = e−r/naB (2r/na )l+1L2l+1 (2r/na ), 1/2 v − − B n−l−1 B a n u (n + l)! B u t where aB is the Bohr radius which, in natural units withh ¯ = c = 1, is equal to 1 a = . (6.132) B ZMα For a hydrogen atom with Z = 1, this is about 1/137 times the Compton wavelength of the electron λ h/M¯ c. The classical velocity of the electron on the lowest Bohr e ≡ e orbit is vB = αc. Thus it is almost nonrelativistic, which is the reason why the Schrödinger equation explains the hydrogen spectrum quite well. The functions M(a, b, z) are confluent hypergeometric functions or Kummer functions, defined by the power series

a a(a + 1) z M(a, b, z) F (a, b, z)=1+ z + + .... (6.133) ≡ 1,1 b b(b + 1) z!

For b = n, they are polynomials related to the Laguerre polynomials4 Lµ(z) by − n (n + µ)! Lµ(z) M( n, µ +1, z). (6.134) n ≡ n!µ! − The radial wave functions are normalized to ∞ ′ ′ drRnrl(r)Rnrl(r)= δnrnr . (6.135) Z0

−r/naB They have an asymptotic behavior Pnl(r/n)e , where Pnl(r/n) is a polynomial of degree n = n l 1, which deﬁnes the radial quantum number. The energy r − − eigenvalues depend on n in the well-known way:

Mα2 E = Z2 . (6.136) n − 2n2 4I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.970 (our deﬁnition diﬀers from that in L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon Press, New York, 1965, Eq. (d.13): µ µ µ Our L = ( ) /(n + µ)!L + L L ). n − n µ | . . 454 6 Relativistic Particles and Fields in External Electromagnetic Potential

The number α2M/2 is the Rydberg-constant: α2M 27.21 Ry = eV 3.288 1015Hz. (6.137) 2 ≈ 2 ≈ × Later in Section 12.21) we shall need the value of the wave function at the origin. It is non-zero only for s-waves where it is equal to

3 3 1 ZMα 1 1 ψn00(0) = = . (6.138) | | √π s n √π snaB

6.5.2 Klein-Gordon Field in a Coulomb Potential After the substitution of (6.125) into (6.123), we ﬁnd the Klein-Gordon equation in the Coulomb potential (6.121):

2 Zα 2 2 E + + ∇ M φE(x)=0. (6.139) " r − # With the angular decomposition (6.127), this becomes

2 ˆ 2 2 2 ∂ 2 ∂ L Z α 2ZαE 2 2 + − (E M ) φE(x)=0. (6.140) "−∂r2 − r ∂r r2 − r − − # The solutions of this equation can be obtained from those of the nonrelativistic Schr¨odinger equation (6.128) by replacing

Lˆ 2 Lˆ 2 Z2α2, (6.141) → − E α α , (6.142) → M E2 M 2 E − . (6.143) → 2M The replacement (6.141) is done most efficiently if we define the eigenvalues l(l + 1) Z2α2 of the operator Lˆ 2 Z2α2 by analogy with those of Lˆ2 as − − λ(λ + 1) l(l + 1) Z2α2. (6.144) ≡ − Then the quantum number l of the Schrödinger wave functions is simply replaced by λ = l δ , where l − l 2 1/2 1 1 2 2 δl = l + l + Z α 2 − " 2 − # Z2α2 = + (α4). (6.145) 2l +1 O The other solution of relation (6.144) with the opposite sign in front of the square root is unphysical since the associated wave functions are too singular at the origin to 6.5 Relativistic Wave Equations in the Coulomb Potential 455

be normalizable. As before, the radial quantum number nr determining the degree of the polynomial Pnl(r/n) in the wave functions must be an integer. This is no longer true for the combination of quantum numbers which determines the energy. This is now given by

n + λ +1= n + l +1 δ = n δ . (6.146) r r − l − l It leads to the equation for the energy eigenvalues

E 2 M 2 Z2Mα2 E 2 1 nl − = nl , (6.147) 2M − 2 M 2 (n δ )2 − l with the solution M Enl = ± 1+ Z2α2/(n δ )2 − l q (Zα)2 3 (Zα)4 (Zα)4 = M 1 + + (Z6α6) . (6.148) ± " − 2n2 8 n4 − n3(2l + 1) O #

The first two terms correspond to the Schrödinger energies (6.136) including the rest energies of the atom. The next two are relativistic corrections. The first of these breaks the degeneracy between the levels of the same n and different l. This is caused in the Schrödinger theory by the famous Lentz-Runge vector [O(4)-invariance] [1]. The correction terms become large for large central charge Z. In particular, the lowest energy and successively the higher ones become complex for central charges Z > 137/2. The physical reason for this is that the large potential gradient near the origin can create pairs of particles from the vacuum. This phenomenon can only be properly understood after quantizing the field theory. As for the free Klein-Gordon field, the energy appears with both signs.

6.5.3 Dirac Field in a Coulomb Potential After the substitution (6.125) into (6.124), we ﬁnd the Dirac equation in the Coulomb potential (6.121):

Zα 0 E + γ + i ∇ M ψE(x)=0. (6.149) r · − In order to ﬁnd the energy spectrum it is useful to establish contact with the Klein-Gordon case. Multiplying (6.149) by the operator

Zα 0

E + γ + i ∇ + M, (6.150) r · we obtain 2 Zα 2 0 Zα 2 E + + ∇ iγ ∇ M ψE (x)=0. (6.151) " r − · r − # 456 6 Relativistic Particles and Fields in External Electromagnetic Potential

0 « In the chiral representation, the 4 4 -matrix γ = has a block-diagonal form (4.563). We therefore decompose ×

ξE(x) ψE(x)= , (6.152) ηE(x) ! and ﬁnd the equation for the upper two-component spinors

2 Zα 2 Zα 2 E + + ∇ + i ∇ M ξE(x)=0. (6.153) " r · r − #

The lower bispinor ηE(x) satisﬁes the same equation with i replaced by i. Express- 2 − ing ∇ via (6.127) and writing ∇ 1/r = xˆ/r2, we obtain the diﬀerential equation −

2 2 2 2 ˆ ∂ 2 ∂ L Z α + iZα xˆ 2ZαE 2 2 + + − (E M ) ξE(x)=0, "− ∂r2 r ∂r ! r2 − r − − # (6.154) and a corresponding equation for ηE(x).

Due to the rotation invariance of xˆ, the total angular momentum · Jˆ = Lˆ + S = Lˆ + (6.155) 2 commutes with the diﬀerential operator in (6.154). Thus we can diagonalize Jˆ2 and ˆ 1 3 J3 with eigenvalues j(j + 1) and m. For a ﬁxed value of j = 2 , 1, 2 ,..., the orbital angular momentum can have the value l = j + 1 and l = j 1/2. The two states + 2 − − have opposite parities. The operator xˆ is a pseudoscalar, so that multiplication · by it will necessarily change the parity of the wave function. Since the square of

xˆ is the unit matrix, its eigenvalues must be 1. Moreover, the unit vector xˆ changes· l by one unit. Thus, in the two-component± Hilbert space of ﬁxed quantum numbers j and m, with orbital angular momenta l = l± = j 1/2, the diagonal

matrix elements vanish ± jm, + xˆ jm, + =0, jm, xˆ jm, =0. (6.156) h | · | i h −| · | −i

For the oﬀ-diagonal elements we easily calculate jm, + xˆ jm, =1, jm, xˆ jm, + = 1. (6.157) h | · | −i h −| · | i − The central parentheses in (6.154) have therefore the matrix elements

1 3 2 2 2 2 2 (j + 2 )(j + 2 ) Z α iZα L Z α iZαxˆ = − 1 ± 1 2 2 . (6.158) − ± iZα (j )(j + ) Z α ! ± − 2 2 − By analogy with the Klein-Gordon case, we denote the eigenvalues of this matrix by λ(λ + 1). The corresponding values of λ are found to be

2 1/2 1 2 2 λj+ = j + Z α , λj− = λj+ 1. (6.159) " 2 − # − 6.5 Relativistic Wave Equations in the Coulomb Potential 457

These may be written as 1 λ = j δ l δ , (6.160) j± ± 2 − j ≡ ± − j where l 1 is the orbital angular momentum, and ± ≡ ± 2 2 1/2 2 2 1 1 2 2 Z α 4 4 δj j + j + Z α = + (Z α ). (6.161) ≡ 2 − " 2 − # 2j +1 O When solving Eq. (6.154), the solutions consist, as in the nonrelativistic hydrogen atom, of an exponential factor multiplied by a polynomial of degree nr which is the radial quantum number. It is related to the quantum numbers of spin and orbital angular momentum, and to the principal quantum number n, by

n + λ +1= n + l +1 δ = n δ . (6.162) r j± r ± − l − j

In terms of δj, the energies obey the same equation as in (6.148), so that we obtain M Enj = ± 1+ Z2α2/(n δ )2 − j q (Zα)2 3 (Zα)4 (Zα)4 = M 1 + + (Z6α6) . (6.163) ± " − 2n2 8 n4 − n3(2j + 1) O # The condition n 0 implies that r ≥ 3 1 j n λj+ = j + δj, ≤ − 2 for 2 − (6.164) j n 1 λ = j 1 δ . ≤ − 2 j− − 2 − j For n =1, 2, 3,..., the total angular momentum runs through j = 1 , 3 ,...,n 1 . 2 2 − 2 The spectrum of the hydrogen atom, according to the Dirac theory, is shown in Fig. 6.1. As a remnant of the O(4)-degeneracy of the levels with l =0, 1, 2,...,n 1 and fixed n in the Schrödinger spectrum, there is now a twofold degeneracy of levels− of equal n and j, with adjacent l-values, which are levels of opposite parity. An exception is the highest total angular momentum j = n 1/2 at each n, which − 5 occurs only once. The lowest degenerate pair consists of the levels 2S1/2 and 2P1/2. It was an important experimental discovery to find that this prediction is wrong. There is a splitting of about 10% of the fine-structure splitting. This is called the Lamb shift. Its explanation is one of the early triumphs of quantum electrodynamics, which will be discussed in detail in Section 12.21. As in the Klein-Gordon case, there are complex energies, here for Z > 137, with S1/2 being the first level to become complex.

5 2S+1 Recall the notation in atomic physics for an electronic state: n LJ , where n is the principal quantum number, L the orbital angular momentum, J the total angular momentum, and S the total spin. In a one-electron system such as the hydrogen atom, the trivial superscript 2S +1=2 may be omitted. 458 6 Relativistic Particles and Fields in External Electromagnetic Potential

Figure 6.1 Hydrogen spectrum according to Dirac’s theory. The splittings are shown only schematically. The ﬁne-structure splitting of the 2P -levels is about 10 times as big as the hyperﬁne splitting and Lamb shift.

An important correct prediction of the Dirac theory is the presence of fine structure. States with the same n and l but with different j are split apart by the forth term in Eq. (6.163) MZ4α4n3/(2j+1). For the states 2P and 2P , the splitting − 1/2 3/2 is Z4α2 ∆ E = α2M. (6.165) fine 2P 32

In a hydrogen atom, this is equal to

∆ﬁneE2P =3.10.95 GHz. (6.166)

Thus it is roughly of the order of the splitting caused by the interaction of the magnetic moment of the electron with that of the proton, the so-called hyperﬁne-splitting. For 2S1/2, 2P 1/2, and2P 3/2 levels, this is approximately equal to 1, 1/8, 1/24, 1/60 times 1 420 MHz.6 In a hydrogen atom, the electronic motion is only slightly relativistic, the veloc- ities being of the order αc, i.e., only about 1% of the light velocity. If one is not only interested in the spectrum but also in the wave functions it is advantageous to solve directly the Dirac equation (6.149) with the gamma matrices in the Dirac

6See H.A. Bethe and E.E. Salpeter in Encyclopedia of Physics (Handbuch der Physik) 335 , Springer, Berlin, 1957, p. 196. 6.5 Relativistic Wave Equations in the Coulomb Potential 459 representation (4.550). Multiplying (6.149) by γ0 and inserting the Dirac matrices

0 « (4.562) for γ = , we obtain Zα

E M + i ∇ ξE(x)  − r ·  =0. (6.167) Zα ηE(x) !

∇  i E + M +   · r    This is of course just the time-independent version of (4.569) extended by the Coulomb potential according to the minimal substitution rule (6.125). To lowest order in α, the lower spinor is related to the upper by

∇ η (x) i · ξ (x). (6.168) E ≈ − 2M E We may take care of rotational symmetry of the system by splitting the spinor wave functions into radial and angular parts G (r) i jl yl (θ,φ) r j,m ψE(x)=   , (6.169) Fjl(r) l  xˆ yj,m(θ,φ)   r ·    l where yj,m(θ,φ) denotes the spinor spherical harmonics. They are composed from the ordinary spherical harmonics Ylm(θ,φ) and the basis spinors χ(s3) of (4.446) via Clebsch-Gordan coeﬃcients (see Appendix 4E):

l ′ y (θ,φ)= j, m l, m ; 1 ,s Y ′ (θ,φ)χ(s ). (6.170) j,m h | 2 3i lm 3 The derivation is given in Appendix 6A. The explicit form of the spinor spherical harmonics (6.170) is for l = l±:

1 1 1 √l+ m + 2 Y (θ,φ) l+ l+,m− 2 yj,m(θ,φ)= − 1 , (6.171) √2l+ +1 √l+ + m + 2 Y 1 (θ,φ) ! − l+,m+ 2

1 1 √l− + m + 2 Y 1 (θ,φ) l− l−,m− 2 y (θ,φ) = 1 . (6.172) j,m 1 √2l− +1 √l− m + 2 Y − (θ,φ) ! − l ,m+ 2

On these eigenfunctions, the operator L has the eigenvalues · l± l±

L y (θ,φ)= (1 + κ )y (θ,φ), (6.173) · j,m − ± j,m with 1 1 κ = (j + ), j = l . (6.174) ± ∓ 2 ± 2 We can now go from Eqs. (6.167) to radial diﬀerential equations by using the trivial identity,

f(r) l+ x f(r) l+

i ∇ y · ( x) i ( ∇) y , (6.175) · r l,m ≡ r2 · · r l,m 460 6 Relativistic Particles and Fields in External Electromagnetic Potential

and the algebraic relation Eq. (4.464) in the form

( a)( b)= i (a b)+ i(a b), (6.176) · · · × · to bring the right-hand side to

x f(r) l+ f(r) f(r) l+ · (ir∂r i L) yl,m = i∂r i (1 + κ) xˆ yl,m. (6.177) r2 − · r " r − r2 # ·

In this way we find the radial differential equations for the functions Fjl(r) and Gjl(r): Zα d 1 E M + Gjl(r) = Fjl(r) (j +1/2) Fjl(r), (6.178) − r −dr ∓ r Zα d 1 E + M + Fjl(r) = Gjl(r) (j +1/2) Gjl(r). (6.179) r dr ∓ r To solve these, dimensionless variables ρ 2r/λ are introduced, with λ = 1/√M 2 E2, writing ≡ − F (r)= 1 E/Me−ρ/2(F F )(ρ), G(r)= 1+ E/Me−ρ/2(F + F )(ρ). (6.180) − 1 − 2 1 2 q q The functions F1,2(ρ) satisfy a degenerate hypergeometric differential equation of the form d2 d ρ +(b ρ) a F (a, b; ρ)=0, (6.181) " dρ2 − dρ − # and the solutions are F (ρ) = ρlF (γ ZαEλ, 2γ + 1; ρ), 2 − γ ZαEλ F (ρ) = ρl − F (γ +1 ZαEλ, 2γ + 1; ρ). (6.182) 1 1/λ + ZαEλ − − The constant γ is Einstein’s gamma parameter γ = 1 v2/c2 for the atomic unit − velocity v = Zαc. It has the expansion γ =1 Z2αq2/2. − As an example, we write down explicitly the ground state wave functions of the 1S1/2 state: 1 0 3 −mZαr (2MZα) 1+ γ e  0 1  ψ 1/2 ± 1 = 1−γ 1−γ −iφ . 1S , 2 v 1−γ i cos θ i sin θe u 4π 2Γ(1+ 2γ) (2MZα)  Zα Zα  u  1−γ 1−γ  t  i sin θeiφ i cos θ   Zα − Zα   (6.183) The first column is for m =1/2, the second for m = 1/2. For small α, Einstein’s − gamma parameter has the expansion γ = 1 Z2α2/2, and we see that for α 0, the upper components of the spinor wave− functions tend to the nonrelativistic→ Schrödinger wave function 2 (ZαM)3/4πe−ρ, multiplied by Pauli spinors (4.446). In general, q ξl (x)= j, m l, m; 1 ,s ψ (x)χ(s ). (6.184) j,m h | 2 3i nlm 3 The lower (small) components vanish. 6.6 Green Function in an External Electromagnetic Field 461

6.6 Green Function in an External Electromagnetic Field

An important physical object of a field theory is the Green function, defined as the solution of the equation of motion having a δ-function source term [recall (1.315) and (2.402)]. For external electromagnetic fields which are constant or plane waves, this Green function can be calculated exactly.

6.6.1 Scalar Field in a Constant Electromagnetic Field For a scalar field, the Green function G(x, x′) is defined by the inhomogeneous differential equation ( ∂2 M 2)G(x, x′)= iδ(4)(x x′), (6.185) − − − whose solution can immediately be expressed as a Fourier integral: 4 ∞ 4 ′ d p i −ip(x−x′) d p −ip(x−x′)+iτ(p2−M 2+iη) G(x x )= 4 2 2 e = dτ 4 e . − Z (2π) p M + iη Z0 Z (2π) − (6.186) A detailed discussion of this function will be given in Subsection 7.2.2. Here we shall address the problem of calculating the corresponding Green function in the presence of a static electromagnetic field, which obeys the more compli- cated differential equation [i∂ eA(x)]2 M 2 G(x, x′)= iδ(4)(x x′), (6.187) − − − for which a Fouriern decomposition iso no longer helpful. For either a constant or an oscillating electromagnetic field, however, this equation can be solved by an elegant method due to Fock and Schwinger [2]. Generalizing the right-hand side of (6.186), we find the representation

∞ 2 2 G(x x′)= dτ x eiτ[(i∂−eA) −M +iη] x′ . (6.188) − Z0 h | | i The integrand contains the time-evolution operator associated with the Hamiltonian operator Hˆ (x, i∂) (i∂ eA)2 + M 2. (6.189) ≡ − − This is the Schr¨odinger representation of the operator Hˆ = H(ˆx, pˆ)= Pˆ2 + M 2, (6.190) − where Pˆ pˆ eA (ˆx) is the canonical momentum in the presence of electromag- µ ≡ µ − µ netism. We shall calculate the evolution operator in (6.188) by introducing time- dependent Heisenberg operators for position and momentum. These obey the Heisenberg-Ehrenfest equations of motion [recall (1.277)]: dxˆµ(τ) = i H,ˆ xˆµ τ)]=2Pˆµ(τ) (6.191) dτ h dPˆµ(τ) = i H,ˆ Pˆµ(τ) =2eF µ (ˆx(τ))Pˆν(τ)+ ie∂ν F (ˆx(τ)). (6.192) dτ ν µν h i 462 6 Relativistic Particles and Fields in External Electromagnetic Potential

µ µ In a constant ﬁeld where F ν(ˆx(τ)) is a constant matrix F ν, the last term in the second equation is absent, and we ﬁnd directly the solution

ˆµ 2eF τ µ ˆν P (τ)= e ν P (0). (6.193) µ 2eF τ Here the matrix e ν is deﬁned by its formal power series expansion µ τ 2 e2eF τ = δµ +2eF µ τ +4e2F µ F λ + .... (6.194) ν ν ν λ ν 2 Inserting (6.193) into Eq. (6.191), we ﬁnd the time-dependent operatorx ˆµ(τ):

e2eF τ 1 µ xˆµ(τ) xˆµ(0) = − Pˆν(0), (6.195) − eF ! ν where the matrix on the right-hand side is again deﬁned by its formal power series

2eF τ µ 3 e 1 2 µ λ (2τ) − =2τ + e F λF ν + .... (6.196) eF ! ν 3!

Note that division by eF is not a matrix multiplication by the inverse of the matrix eF but indicates the reduction of the expansion powers of eF by one unit. This is deﬁned also if eF does not have an inverse. We can invert Eq. (6.195) to ﬁnd

1 e−eF τ µ Pˆν(0) = eF [ˆx(τ) xˆ(0)]ν , (6.197) 2 " sinh eF τ # ν − and, using (6.193), Pˆν(τ)= Lµ (eF τ)[ˆx(τ) xˆ(0)]ν , (6.198) ν − with the matrix eF τ µ µ 1 µ e L ν (eF τ) eF ν . (6.199) ≡ 2 " sinh eF τ # By squaring (6.198) we obtain

Pˆ2(τ)=[ˆx(τ) xˆ(0)]µ K ν(eF τ)[ˆx(τ) xˆ(0)] , (6.200) − µ − ν where ν λ ν Kµ (eF τ)= L µ(eF τ)Lλ (eF τ). (6.201)

Using the antisymmetry of the matrix Fµν , we can rewrite this as

2 2 ν ν λ ν 1 e F Kµ (eF τ)= Lµ ( eF τ)Lλ (eF τ)= 2 . (6.202) − 4 " sinh eF τ #µ 6.6 Green Function in an External Electromagnetic Field 463

The commutator between two operatorsx ˆ(τ) at diﬀerent times is

2eF τ µ µ e 1 [ˆx (τ), xˆν (0)] = i − , (6.203) eF ! ν and

2eF τ µ 2eF T τ µ µ µ e 1 e 1 xˆ (τ), xˆν (0) + xˆν (τ), xˆ (0) = i − + i − eF ! ν eF T ! ν h i h i e2eF τ e−2eF τ µ sinh 2eF τ µ = i − =2i . (6.204) eF ! ν " eF # ν

With the help of this commutator, we can expand (6.200) in powers of operators xˆ(τ) andx ˆ(0). We must be sure to let the later operatorsx ˆ(τ) lie to the left of the earlier operatorsx ˆ(0) as follows:

H(ˆx(τ), xˆ(0); τ) = xˆµ(τ)K ν(eF τ)ˆx (τ) xˆµ(0)K ν(eF τ)ˆx (0) − µ ν − µ ν i + 2ˆxµ(τ)K ν(eF τ)ˆx (0) tr [eF coth eF τ]+ M 2. (6.205) µ ν − 2 Given this form of the Hamiltonian operator, it is easy to calculate the time evolution amplitude in Eq. (6.188): x, τ x′ 0 x e−iHτˆ x′ . (6.206) h | i≡h | | i It satisﬁes the diﬀerential equation

i∂ x, τ x′ 0 x Hˆ e−iHτˆ x′ = x e−iHτˆ eiHτˆ Hˆ e−iHτˆ x′ τ h | i ≡ h | | i h | | i = x, τ Hˆ (ˆx(τ), Pˆ(τ)) x′, 0 . h i (6.207) h | | i Replacing the operator H(ˆx(τ), Pˆ(τ)) by H(ˆx(τ), xˆ(0); τ) of Eq. (6.205), the matrix elements on the right-hand side can immediately be evaluated, using the property

x, τ xˆ(τ)= x x, τ , xˆ(0) x′, 0 = x′ x′, 0 , (6.208) h | h | | i | i and the diﬀerential equation (6.209) becomes

i∂ x, τ x′ 0 H(x, x′; τ) x, τ x′ 0 , (6.209) τ h | i ≡ h | i or ′ x, τ x′ 0 = C(x, x′)E(x, x′; τ) C(x, x′)e−i dτ H(x,x ;τ). (6.210) h | i ≡ The prefactor C(x, x′) contains a possible constant ofR integration in the exponent which may have an arbitrary dependence on x and x′. The following integrals are needed: 1 e2F 2 1 dτ K(eF τ)= dτ 2 = eF coth eF τ, (6.211) Z 4 Z sinh eF τ −4 464 6 Relativistic Particles and Fields in External Electromagnetic Potential and sinh eF τ sinh eF τ dτ tr [eF coth eF τ] = tr log = tr log + 4 log τ. (6.212) Z eF eF τ These results follow again from a Taylor expansion of both sides. As a consequence, the exponential factor E(x, x′; τ) in (6.210) becomes

′ 1 i ′ µ ν ′ 2 1 sinh eF τ E(x, x ; τ)= exp (x x ) [eF coth eF τ]µ (x x )ν iM τ tr log . τ 2 (−4 − − − − 2 eF τ ) (6.213) The last term produces a prefactor sinh eF τ det −1/2 . (6.214) eF τ ! The time-independent integration constant is ﬁxed by the diﬀerential equation with respect to x:

[i∂ eA (x)] x, τ x′ 0 = x Pˆ e−iHτˆ x′ = x e−iHτˆ eiHτˆ Pˆ e−iHτˆ x′ µ − µ h | i h | µ | i h | µ | i = x, τ Pˆ (τ) x′ 0 , h i (6.215) h | µ | i which becomes, after inserting (6.198):

[i∂ eA (x)] x, τ x′ 0 = L ν (eF τ)(x x′) x, τ x′ 0 . (6.216) µ − µ h | i µ − νh | i Calculating the partial derivative we ﬁnd i∂ x, τ x′ 0 = [i∂ C(x, x′)]E(x, x′; τ)+ C(x, x′)[i∂ E(x, x′; τ)] µh | i µ µ 1 = [i∂ C(x, x′)]E(x, x′; τ)+ C(x, x′) [eF coth eF τ] ν(x x′) E(x, x′; τ). µ 2 µ − ν ′ Subtracting from this eAµ(x) x, τ x 0 , and inserting (6.210), the right-hand side of ′h | ′i (6.216) is equal to [i∂µC(x, x )]E(x, x ; τ) plus

ν ′ 1 ν ′ ′ ′ Lµ (eF τ)(x x )ν [eF coth eF τ]µ (x x )ν C(x, x )E(x, x ; τ). (6.217) − − 2 − Inserting Eq. (6.199), this simplifies to e F ν (x x′) C(x, x′)E(x, x′; τ), (6.218) 2 µ − ν so that C(x, x′) satisfies the time-independent differential equation

µ µ e µ ′ ν ′ i∂ eA (x) F ν(x x ) C(x, x )=0. (6.219) − − 2 − This is solved by x 1 C(x, x′)= C exp ie dξµ A (ξ)+ F ν(ξ x′) . (6.220) ′ µ µ ν − Zx 2 − 6.6 Green Function in an External Electromagnetic Field 465

′ 1 ν ′ The contour of integration is arbitrary since A (ξ) Aµ(ξ)+ 2 Fµ (ξ x )ν has a vanishing curl: ≡ −

∂ A′ (x) ∂ A′ (x)=0. (6.221) µ ν − ν µ We can therefore choose the contour to be a straight line connecting x′ and x, in which case the F -term does not contribute in (6.220), since dξµ points in the same µ ′µ µ ′µ direction of x x as ξ x and Fµν is antisymmetric. Hence we may write for a straight-line connection− −

x C(x, x′)= C exp ie dξµ A (ξ) . (6.222) ′ µ − Zx The normalization constant C is ﬁnally ﬁxed by the initial condition

lim x, τ x′ 0 = δ(4)(x x′), (6.223) τ→0h | i − which requires i C = . (6.224) −(4π)2 Collecting all terms we obtain

i x sinh eF τ x, τ x′ 0 = exp ie dξµ A (ξ) det −1/2 2 ′ µ h | i −(4πτ) − Zx eF τ ! i ′ µ ν ′ 2 exp (x x ) [eF coth eF τ]µ (x x )ν iM τ . (6.225) × −4 − − − ν For a vanishing ﬁeld Fµ , this reduces to the relativistic free-particle amplitude

i i (x x′)2 x, τ x′ 0 = exp − iM 2 . (6.226) h | i −(4πτ)2 "−2 2τ − # According to relation (6.188), the Green function of the scalar ﬁeld is given by the integral ∞ G(x, x′)= dτ x, τ x′ 0 . (6.227) Z0 h | i The functional trace of (6.225), i eEτ Tr x, τ x 0 = V ∆t , (6.228) h | i (4πτ)2 sinh eEτ will be needed below. Due to translation invariance in spacetime, it carries a factor equal to the total spacetime volume V ∆t of the universe. The result (6.228) can be checked by× a more elementary derivation [3]. We let the constant electric ﬁeld point in the z-direction, and represent it by a vector potential to have only a zeroth component

A3(x)= Ex0. (6.229) − 466 6 Relativistic Particles and Fields in External Electromagnetic Potential

Then the Hamiltonian (6.190) becomes

Hˆ = pˆ2 + pˆ2 +(ˆp3 + eEx )2 + M 2, (6.230) − 0 ⊥ 0 where p⊥ are the two-dimensional momenta in the xy-plane. Using the commutation rule [p0, x0]= i, this can be rewritten as

3 3 Hˆ = e−ipˆ0p /eEHˆ ′eipˆ0p /eE, (6.231) where Hˆ ′ is the sum of two commuting Hamiltonians:

Hˆ ′ = (ˆp2 e2E2x2)+ p2 + M 2 Hˆ + Hˆ . (6.232) − 0 − 0 ⊥ ≡ ωE ⊥

The ﬁrst is a harmonic Hamiltonian with imaginary frequency ωE = ieE and an energy spectrum 2(n +1/2)ieE. The second describes a free particle in the xy- plane. This makes− it easy to calculate the functional trace. We insert a complete set of momentum states on either side of (6.206), so that the functional trace becomes

4 4 ′ d p d p ′ ˆ 4 −i(p−p )x −iτ(HωE +H⊥) ′ Tr x, τ x 0 = d x 4 4 e p e p . (6.233) h | i Z Z (2π) Z (2π) h | | i The matrix elements are

3 2 2 ′ ′3 −iτHˆ ′ −ip0(x0+p /eE) −isHˆ ′ −iτ(p +M −iη) ip (x0+p /eE) p e p = e p e ωE p e ⊥ e 0 h | | i h 0| | 0i (2π)2δ(2)(p p′ )(2π)δ(p3 p′3). (6.234) × ⊥ − ⊥ − Inserting this into (6.233) and performing the integrals over the spatial parts of p′ appearing in the δ-functions of (6.234) yields

2 d p 2 2 ⊥ −iτ(p⊥+M −iη) Tr x, τ x 0 = V dx0 2 e h | i Z Z (2π) 3 ′ dp0dp dp ′ 3 ˆ 0 −i(p0−p0)(x0+p /eE) −isHωE ′ 3 e p0 e p0 , (6.235) × Z (2π) h | | i which can be reduced to

2 i −iτ(M −iη) eE dp0 −iτHˆω Tr x, τ x 0 = V ∆t − e p0 e E p0 . (6.236) h | i 4πτ 2π "Z 2π h | | i#

−iτHˆ The expression in brackets is the trace of e ωE , which is conveniently calculated in the eigenstates n of the harmonic oscillator with eigenvalues 2(n +1/2)ω : | i − E ∞ −iτHˆ iτ 2(n+1/2)eE i 1 Tre ωE = e = = . (6.237) 2 sin ωE 2 sinh τeE nX=0 Thus we obtain i eEτ Tr x, τ x 0 = V ∆t − . (6.238) h | i 4(2π)2τ 2 sinh τeE 6.6 Green Function in an External Electromagnetic Field 467

6.6.2 Dirac Field in a Constant Electromagnetic Field For a Dirac ﬁeld we have to solve the inhomogeneous diﬀerential equation iγµ[∂ eA (x)] M S(x, x′)= iδ(4)(x x′), (6.239) { µ − µ − } − rather than (6.187). The solution can formally be written as S(x, x′)= iγµ[∂ eA (x)] + M G¯(x, x′)= iδ(4)(x x′), (6.240) { µ − µ } − where G¯(x, x′) solves a slight generalization of Eq. (6.187):

2 e µ ν 2 ′ (4) ′ [i∂ eA(x)] σ νFµ M G¯(x, x )= iδ (x x ). (6.241) − − 2 − − This is the Green function of the Pauli equation (6.110), in natural units. For a constant field, the extra term enters the final result (6.240) in a trivial way. We recall the relations (6.188) and (6.227) to the Green function, and see that G¯(x, x′) contains the fields as follows: ∞ ′ e µ ν ′ G¯(x, x )= dτ exp i σ νFµ τ x, τ x 0 . (6.242) Z0 − 2 h | i Constant Electric Background Field For a constant electric field in the z-direction, we choose the vector potential to have only a zeroth component A3(x)= Ex0. (6.243) − Then, since F 30 = E, we have F 0 = E and F 3 = E. The field tensor F ν is 3 − 0 − µ given by the matrix 0001 0000 F = E   = iE M , (6.244) − 0000 3      1000    where M3 is the generator (4.60) of pure Lorentz transformations in the z-direction. eF τ The exponential e is therefore equal to the boost transformation (4.59) B3(ζ)= e−iM3ζ with a rapidity ζ = Eτ. From (4.14) we find the explicit matrices − cosh eEτ 0 0 sinh eEτ − 0 10 0 eeF τ =   , (6.245) 0 01 0    sinh eEτ 0 0 cosh eEτ     −  and hence 0 00 sinh eEτ − 0 00 0 sinh eF τ =   , (6.246) 0 00 0      sinh eEτ 00 0   −  468 6 Relativistic Particles and Fields in External Electromagnetic Potential

sinh eEτ 00 0  eE  sinh eF τ 0 10 0 =   , (6.247) eF τ  0 01 0     sinh eEτ     0 00   eE    and coth eEτ 00 0 0 10 0 eF coth eF τ = eE  . (6.248) 0 01 0      0 0 0 coth eEτ    Thus we obtain i eEτ x x, τ x′ 0 = exp ie dξµ A (ξ) (6.249) 2 ′ µ h | i (4πτ) sinh eEτ − Zx i −(x−x′)0eE coth eEτ(x−x′)0+(x−x′)T 1 (x−x′)T +(x−x′)3eE coth eEτ(x−x′)3 −iM 2τ e 4 [ τ ] , × where the superscript T indicates transverse directions to E. The prefactor x µ exp [ ie ′ dξ A (ξ)] is found by inserting (6.243) and integrating along the straight − x µ line R ξ = x′ + s(x x′), s [0, 1], (6.250) − ∈ to be

x ′ 1 ′ ′ ′ ′ µ −ieE(x0−x0) ds[z +s(z−z )] −ieE(x0−x0)(z+z ) exp ie dξ Aµ(ξ) = e 0 = e . (6.251) − x′ Z R The exponential prefactor in the fermionic Green function (6.242) is calculated in the chiral representation of the Dirac algebra where, due to (6.111) and (6.112),

E −e τ e µ ν e 0

«E E exp i σ νFµ τ = exp (e τ)= e τ , (6.252) − 2 0 e !

which is equal to ˆ e µ ν cosh eEτ sinh eEτ E 0

exp i σ νFµ τ = − . ˆ − 2 0 cosh eEτ +sinh eEτ E ! (6.253)

Comparison with (4.506) shows that this is the Dirac representation of a Lorentz boost into the direction of E with rapidity ζ = 2e E τ. The Dirac trace of the evolution amplitude for Dirac ﬁelds is then simply | | i eEτ tr x, τ x 0 = 4 cosh eEτ, (6.254) h | i −(4πτ)2 sinh eEτ × and the functional trace of this carries simply a total spacetime volume factor V ∆t that appeared before in Eq. (6.228). 6.6 Green Function in an External Electromagnetic Field 469

Note that the Lorentz-transformation (6.253) has twice the rapidity of the transformation (6.245) in the deﬁning representation, this being a manifestation of the gyromagnetic ratio of the electron in Dirac’s theory which is equal to two [recall (6.119)]. The process of pair creation in a space- and time-dependent electromagnetic ﬁeld is discussed in Ref. [4]. The above discussion becomes especially simple in 1+1 spacetime dimensions, the so-called massive Schwinger model [5].

6.6.3 Dirac Field in an Electromagnetic Plane-Wave Field The results for constant-background fields in the last subsection simplify drastically if electric and magnetic fields have the same size and are orthogonal to each other. This is the case for a travelling plane wave of arbitrary shape [10] running along some direction nµ with n2 =0. If ξ denotes the spatial coordinate along n, we may write the vector potential as A (x)= ǫ f(ξ), ξ nx, (6.255) µ µ ≡ where ǫµ is some polarization vector with the normalization ǫ2 = 1 in the gauge ǫn = 0. The field tensor is − F = ǫ f ′(ξ), ǫ n ǫ n ǫ , (6.256) µν µν µν ≡ µ ν − ν µ where the constant tensor ǫµν satisfies µ µ νλ ǫµν n =0, ǫµν ǫ =0, ǫµν ǫ = nµnλ. (6.257) The Heisenberg equations of motion (6.191) and (6.192) take the form dxˆµ(τ) = i H,ˆ xˆµ τ)]=2Pˆµ(τ) (6.258) dτ h dPˆµ(τ) e = i H,ˆ Pˆµ(τ) =2eǫµ Pˆν(τ)f ′(ξˆ(τ)) + nµǫ σλκf ′′(ξˆ(τ)). (6.259) dτ ν 2 λκ h i ν Note that the last term in (6.259) vanishes for a sourceless plane wave: ∂ Fµν = 0. Multiplying these equations by nµ we see that dξˆµ(τ) dPˆµ(τ) n =2n Pˆµ(τ), n =0. (6.260) µ dτ µ µ dτ Hence nPˆ(τ)= nPˆ(0) = const, ξˆ(τ) ξˆ(0) = nxˆ(τ) nxˆ(0) = 2τnPˆ(τ). (6.261) − − Whereas the components of Pˆ(τ) parallel to n are time independent, those orthogonal to n have a nontrivial time dependence. To find it we multiply (6.259) by ǫµν and find d dξˆ df(ξˆ) ǫ Pˆµ(τ)=2eǫ ǫµρPˆ f ′(ξˆ)= en f ′(ξˆ)(2nPˆ)= en f ′(ξˆ) = en , (6.262) dτ νµ νµ ρ ν ν dτ ν dτ 470 6 Relativistic Particles and Fields in External Electromagnetic Potential which is integrated to µ ǫνµPˆ (τ)= enνf(ξˆ)+ Cˆν, (6.263) with an operator integration constant Cˆν, that commutes with the constant nPˆ, and satisfies the relations nνCˆν = 0 and ξˆ(τ) ξˆ(0) ǫµν Cˆ = nµ(nPˆ)= nµ − . (6.264) ν 2τ Inserting this into (6.259), and integrating the resulting equation yields

1 2 2 e µν ′ Pˆµ(τ)= 2eCµf(ξˆ)+ e nµf (ξ)+ nµǫµν σ f (ξˆ) + Dˆ µ, (6.265) 2πn 2 where Dˆ µ is again an interaction constant commuting with nPˆ. Now we can integrate the equation of motion (6.258) over dτ = dξ/ˆ 2nPˆ, and ﬁnd

ξˆ(τ) 1 1 2 2 e µν ′ [ˆx(τ) xˆ(0)] = dξˆ 2eCµf(ξˆ)+ e nµf (ξ)+ nµǫµν σ f (ξˆ) + Dˆ µτ. ˆ 2 ˆ 2 − (2nP ) Zξ(0) 2 (6.266) We determine Dˆ µ, and insert it into (6.265) to ﬁnd 1 Pˆ (τ) = [ˆx (τ) xˆ (0)] µ 2τ µ − µ ξˆ(τ) τ 2 2 e ρν ′ dξˆ 2eCˆµf(ξˆ)+ e nµf (ξˆ)+ nµǫρν σ f (ξˆ) 2 ˆ − ξˆ(τ) ξˆ(0) Zξ(0) 2 − h i τ ˆ 2 2 ˆ e ρν ′ ˆ + 2eCˆνf(ξ(τ)) + e nµf (ξ(τ)) + nµǫρνσ f (ξ(τ)) . (6.267) ξˆ(τ) ξˆ(0) 2 − After multiplication by ǫνµ, and recalling (6.257) and (6.264), we obtain 1 ǫνµPˆ (τ) = ǫνµ [ˆx (τ) xˆ (0)] + µ 2τ µ − µ enν ξˆ(τ) dξˆ f(ξˆ)+ enνf(ξˆ(τ)). (6.268) − ξ(τ) ξˆ(0) ξˆ(0) − Z Inserting this into (6.263) determines the integration constant Cˆν :

ν ξˆ(τ) ν 1 νµ en Cˆ = ǫ [ˆxµ(τ) xˆµ(0)] dξˆ f(ξˆ). (6.269) 2τ − − ξ(τ) ξˆ(0) ξˆ(0) − Z It is useful to introduce the notations 1 ξˆ(τ) f dξˆ f(ξˆ) (6.270) h i ≡ ξ(τ) ξˆ(0) ξˆ(0) − Z and (δf)2 (f f )2 = f 2 f 2 . (6.271) h i≡h −h i i h i−h i 6.6 Green Function in an External Electromagnetic Field 471

In order to calculate the matrix elements

2 e µ ν 2 x τ Hˆ x 0 = x τ Pˆ + σ νFµ + M x 0 , (6.272) h | | i − 2

we must time-order the operatorsx ˆ( τ), xˆ(0). For this we need the commutator

[ˆxµ(τ), xˆν (0)] = 2iτgµν . (6.273) This is deduced from Eq. (6.267) by commuting it withx ˆ(τ) and using the trivially vanishing equal-time commutator [ξˆ(τ), xˆν (τ)] = 0, as well as the nonequal-time commutator [ξˆ(0), xˆν(τ)] = 2inντ. The latter implies that [ξˆ(τ), xˆν (0)] = 0, which is also needed for time-ordering. The result is ′ 1 i 2 f(ξ) f(ξ ) x τ Hˆ x 0 = (x x′)2 2 + M 2 + e2 (δφ)2 + eǫ σµν − . h | | i −4τ 2 − − τ h i µν ξ ξ′ − (6.274) Integrating this over τ we obtain the exponential factor of the time-evolution amplitude (6.210):

′ ′ 1 i ′ 2 2 2 2 2 µν f(ξ) f(ξ ) E(x, x ; τ)= exp (x x ) + M +e (δf) iτeǫµν σ − . τ 2 (−4τ − h i − ξ ξ′ ) − (6.275) The time-independent prefactor C(x, x′) is again determined by the diﬀerential equation Eq. (6.215), which reduces here to

µν ′ ′ ǫ (x x )ν ′ [i∂µ eAµ(x)] x, τ x 0 = − f f(ξ) x, τ x 0 , (6.276) − h | i " ξ ξ′ h i − # h | i − and is solved by

x µν ′ ′ny ′ ′ i µ ǫ (x x )ν ′ f(y ) C(x, x )= − exp ie dyµ A (y) − ′ dy ′ f(ny) . (4π)2 x′ ( − ξ ξ " ξ ny ξ − #)! Z − Z − (6.277) For a straight-line integration contour, the second term does not contribute, as before. Observe that in Eq. (6.275), the mass term M 2 is replaced by M 2 = M 2 + e2 (δf)2 , (6.278) eﬀ h i implying that, in an electromagnetic wave, a particle acquires a larger eﬀective mass. If the wave is periodic with frequency ω and wavelength λ =2πc/ω, the right-hand side becomes M 2 + e2 f 2 . If the photon number density is ρ, their energy density h i is ρω (in units withh ¯ = 1), and we can calculate E2 ρ e2 f 2 =4παh i =4πα . (6.279) h i ω2 ω 472 6 Relativistic Particles and Fields in External Electromagnetic Potential

Hence we ﬁnd a relative mass shift: ∆M 2 =4παλ¯2λ ρ, (6.280) M 2 e whereλ ¯ h/M¯ c = 3.861592642(28) 10−3A˚ is the Compton wavelength of the e ≡ e × electron. For visible light, the right-hand side is of the order of A˚3ρ/100. Present lasers achieve energy densities of 109 W/sec corresponding to a photon density

1 W 1 eV ρ = 109 2.082 10−7 , (6.281) hω¯ × sec ≡ × A˚3 hω¯ which is too small to make ∆M 2/M 2 observable.

Appendix 6A Spinor Spherical Harmonics

Equation (6.170) defines spinor spherical harmonics. In these, an orbital wave function of angular momentum l± is coupled with spin 1/2 to a total angular momentum j = l∓ 1/2. For the ± configurations j = l− + 1/2 with m2 = 1/2 the recursion relation (4E.20) for the Clebsch- − Gordan coefficients s1m1; s2m2 sm becomes simple by having no second term. Inserting s1 = l−, h | i s2 =1/2, and s = j = l− +1/2, we find

1 1 1 1 l− m +1/2 1 1 1 1 l−,m + 2 ; 2 , 2 l− + 2 ,m = − l−,m 2 ; 2 , 2 l− + 2 ,m 1 . (6A.1) h − | i l− m +3/2h − − | − i s − This has to be iterated with the initial condition

1 1 1 1 l−, l−; , l− + , l− =1, (6A.2) h − 2 − 2 | 2 − − 2 i 1 1 which follows from the fact that the state l−, l−; , carries a unique magnetic quantum h − 2 − 2 i number m = l− 1/2 of the irreducible representation of total angular momentum s = j = − − l− +1/2. The result of the iteration is

1 1 1 1 l+ m +1/2 l+,m 2 ; 2 , 2 l+ 2 ,m = − . (6A.3) h − | − i s 2l+ +1

Similarly we may simplify the recursion relation (4E.21) for the conﬁgurations j = l+ 1/2 with − m2 =1/2 to

1 1 1 1 l− + m +1/2 1 1 1 1 l−,m 2 ; 2 , 2 l− + 2 ,m = l−,m + 2 ; 2 , 2 l− + 2 ,m+1 , (6A.4) h − | i sl− + m +3/2h | i and iterate this with the initial condition

1 1 1 1 l−, l−; n l− + , l− + =1, (6A.5) h 2 2 | 2 2 i 1 1 which expresses the fact that the state l− l−; is the state of the maximal magnetic quantum h 2 2 i number m = l− +1/2 in the irreducible representation of total angular momentum s = j = l− +1/2. The result of the iteration is

1 1 1 1 l+ + m +1/2 l−,m 2 ; 2 , 2 l+ + 2 ,m = . (6A.6) h − | i s 2l− +1 Notes and References 473

Using (6A.3) and (6A.6), the expression (6.170) for the spinor spherical harmonic of total angular momentum j = l− +1/2 reads

l− 1 1 1 1 1 yj,m(θ, φ) = l−,m 2 ; 2 , 2 l− + 2 ,m Ylm−1/2(θ, φ)χ( 2 ) h −1 1 |1 1 i 1 + l− m + ; l− + ,m Y +1 2(θ, φ)χ( ). (6A.7) h 2 2 − 2 | 2 i lm / − 2 Separating the spin-up and spin-down components, we obtain precisely (6.172). In order to ﬁnd the corresponding result for j = l+ 1/2, we use the orthogonality relation for states with the same l but diﬀerent j = l 1/2: − ± l + 1 ,m l 1 ,m =0. (6A.8) h 2 | − 2 i Inserting a complete set of states in the direct product space yields

l + 1 ,m l,m 1 ; 1 , 1 lm 1 ; 1 1 l 1 ,m h 2 | − 2 2 2 ih − 2 2 2 | − 2 i + l + 1 ,m l,m + 1 ; 1 , 1 l,m + 1 ; 1 , 1 l 1 ,m =0. (6A.9) h 2 | 2 2 − 2 ih 2 2 − 2 | − 2 i Together with (6A.3) and (6A.6) we ﬁnd

1 1 1 1 l+ + m +1/2 l+.m 2 ; 2 , 2 l+ 2 ,m = , h − | − i s 2l+ +1

1 1 1 1 l+ m +1/2 l+,m + 2 ; 2 , 2 l+ 2 ,m = − . (6A.10) h − | − i −s 2l+ +1

With this, the expression (6.170) for the spinor spherical harmonics written as

l+ 1 1 1 1 1 yj,m(θ, φ) = l+,m 2 ; 2 , 2 l+ 2 ,m Yl,m−1/2(θ, φ)χ( 2 ) h − 1 1 |1 − 1 i 1 + l+,m + ; , l+ ,m Y +1 2(θ, φ)χ( ) (6A.11) h 2 2 − 2 | − 2 i l,m / − 2 has the components given in (6.171).

Notes and References [1] W. Lenz, Zeitschr. Phys. A 24, 197 (1924); P.J. Redmond, Phys. Rev. 133, B 1352 (1964); See also H. Kleinert, Group Dynamics of the Hydrogen Atom, Boulder Summer School Lectures in Theoretical Physics, ed. by W.E. Brittin and A.O. Barut, Gordon and Breach, N.Y. 1968, p. 427 (http://klnrt.de/4). [2] J. Schwinger, Phys. Rev. 82, 664 (1951); 93, 615 (1954); 94, 1362 (1954). [3] C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill (1985). [4] H. Kleinert, R. Ruﬃni, and S.S. XuePhys. Rev. D 78, 025011 (2008); A. Chervyakov and H. Kleinert, Phys. Rev. D 80, 065010 (2009). [5] M.P. Fry, Phys. Rev. D 45, 682 (1992). [6] C. Itzykson and E. Br´ezin, Phys. Rev. D 2, 1191 (1970).