The most tragic word in the English language is ’potential’. Arthur Lotti 6 Relativistic Particles and Fields in an External Electromagnetic Potential Given the classical field theory of relativistic particles, we may ask which quantum phenomena arise in a relativistic generalization of the 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 (4.229) and (4.230). 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. (4.232) and (4.233) we have expressed electric and magnetic field strength as components of a four-curl Fµν = ∂µ Aν − ∂ν Aµ of a vector potential Aµ (x). This four-curl is invariant under gauge transformations Aµ (x) → Aµ (x) + ∂µ Λ(x). (6.1) 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 oringin 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 A = −Mc Z q dτ q˙µ (τ )q˙µ (τ ). (6.2) The physical time along the path is given by q 0 (τ ) = ct, and the physical velocity by v(t) ≡ dq(t)/dt. In terms of these, the action reads: A= Z dt L(t) ≡ −Mc 2 Z 428 v2 (t) dt 1 − 2 c " #1/2 . (6.3) 429 6.1 Charged Point Particles 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) φ(x, t) = A0 (x, t). (6.5) where In our convention, the charge of the electron e has a negative value to have agreement with the sign in the historic form of the Maxwell equations: ∇ · E(x) = −∇2 φ(x) = ρ(x), ˙ ˙ ∇ × B(x) − E(x) = ∇ × ∇ × A(x) − E(x) h i 1 ˙ = − ∇2 A(x) − ∇ · ∇A(x) − E(x) = j(x). c (6.6) 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 Aint pot = −e Z dt A0 (q(t), t) . (6.9) Since the time t coincides with q 0 (τ )/c of the trajectory, this can be expressed as Aint pot e =− c Z dq 0 A0 . (6.10) 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 A int e =− c Z dq µ Aµ (q). (6.11) Thus, the full action of a point particle can be written in covariant form as A = −Mc Z q dτ q˙µ (τ )q˙µ (τ ) − e c Z dq µ Aµ (q), (6.12) or more explicitly as A= Z dt L(t) = −Mc 2 Z v2 dt 1 − 2 c " #1/2 −e Z 1 dt A0 − v · A . c (6.13) 430 6 Relativistic Particles and Fields in an External Electromagnetic Potential The canonical formalism supplies us with the canonically conjugate momenta P= v e e ∂L = Mq + A ≡ p + A. ∂v c 1 − v2 /c2 c (6.14) The Euler-Lagrange equation obtained by extremizing this equation is ∂L d ∂L = , dt ∂v(t) ∂q(t) (6.15) or d ed e p(t) = − A(q(t), t) − e∇A0 (q(t), t) + v i ∇Ai (q(t), t). dt c dt c We now split ∂ d A(q(t), t) = (v(t) · ∇)A(q(t), t) + A(q(t), t), dt ∂t (6.16) (6.17) and obtain e e∂ e d p(t) = − (v(t) · ∇)A(q(t), t) − A(q(t), t) − e∇A0 (q(t), t) + v i ∇Ai (q(t), t). dt c c ∂t c (6.18) The right-hand side contains the electric and magnetic fields (4.234) and (4.235), in terms of which it takes the well-known form v d p= e E+ ×B . dt c (6.19) This can be rewritten in terms of the proper time τ ≡ t/γ as d e 0 Ep +p×B , p= dτ Mc (6.20) Recalling Eqs. (4.232) and (4.233), this is recognized as the spatial part of the covariant equation e µ ν d µ p = F νp . (6.21) dτ Mc The temporal component of this equation e d 0 p = E·p dτ Mc (6.22) gives the energy increase of a particle running through an electromagnetic field. In real time this is v d 0 p = eE· . (6.23) dt c Combining this with (6.19), we find the acceleration d v d p e v v E+ ×B− v(t) = c = ·E 0 dt dt p γM c c c . (6.24) 431 6.1 Charged Point Particles The velocity is related to the canonical momenta and external field via v = s c e P− A c . 2 e P − A + m2 c2 c (6.25) This can be used to calculate the Hamiltonian via the Legendre transform H= ∂L v − L = P · v − L, ∂v (6.26) giving H =c s e P− A c 2 + m2 c2 + eA0 . (6.27) In the non-relativistic limit this has the expansion H = mc2 + 1 e P− A 2m c 2 + eA0 + . . . . (6.28) Thus, the free theory goes over into the interacting theory by the minimal substitution rule e e H → H − A0 , (6.29) p → p − A, c c or, in relativistic notation: e pµ → pµ − Aµ . c 6.1.2 (6.30) Spin Precession in an Atom In 1926, Uhlenbeck and Goudsmit noticed that the observed Zeeman splitting of atomic levels could be explained by an electron of spin 12 . 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 = e¯h/Mc. It reads = gµB Sh¯ , µB ≡ e¯h , 2Mc (6.31) where S = /2 is the spin matrix which has the commutation rules [Si , Sj ] = i¯hǫijk Sk , (6.32) and g is a dimensionless number called the gyromagnetic ratio or Land´e factor. 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 432 6 Relativistic Particles and Fields in an 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 kinematical, 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 H LS (r) = 1 dV (r) g S·L , 2 2 2M c r dr (6.33) where V (r) is the atomic potential depending only on r = |x|. To derive this H LS (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 = dt × B, (6.34) 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 (4.278) that boosts an electron at rest to a velocity v produces a magnetic field in the electron’s rest frame: B = Bel = −γ v × E, c 1 γ=q . 1 − v 2 /c2 (6.35) 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 Bel ≈ − v × E. c (6.36) The electric field gives the electron an acceleration v˙ = e E, M (6.37) so that we may also write Mc 1 ˙ v × v, e c2 and Heisenberg’s precession equation (6.34) as Bel ≈ − dS g ˙ × S. ≈ 2 (v × v) dt 2c (6.38) (6.39) This can be expressed in the form dS = dt LS × S, (6.40) 433 6.1 Charged Point Particles where LS is the angular velocity of the spin precession caused by the orbital magnetic field in the rest frame of the electron: g ˙ 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, we must add to LS the angular velocity T of the Thomas precession, such that the total angular velocity of precession becomes = LS + T ≈ g −2 1 v × v.˙ (6.42) 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 (4.275), where it leads to an approximate equation of motion for the spin dS = dt Expressing ×B ≈ ′ v × B− ×E . c via Eq. (6.31), this becomes (6.43) (6.44) em This equation defines the frequency em of precession due to the magnetic and elecdS ≡ −S × dt v eg S× B− ×E . ≈ 2Mc c tric fields in the rest frame of the electron. Expressing E in terms of the acceleration via Eq. (6.37), this becomes g M dS ˙ . ≈ S × eB − (v × v) dt 2Mc c (6.45) The acceleration can be expressed in terms of the central Coulomb potential V (r) as x 1 dV , (6.46) v˙ = − r M dr the spin precession rate in the electron’s rest frame is dS g x 1 dV = S× eB + v × dt 2Mc r c dr ! g 1 dV = S× eB − L 2Mc Mc dr ! . (6.47) There exists a simple Hamiltonian operator for the spin-orbit interaction H LS (t)], from which this equation can be derived via Heisenberg’s equation (1.280): i ˙ S(t) = [S(t), H LS (t)]. h ¯ (6.48) 434 6 Relativistic Particles and Fields in an External Electromagnetic Potential The operator is H LS ! dV 1 L (r) = − · B − Mc e dr ge g 1 dV = − S·B+ S·L . 2 2 2Mc 2M c r dr (6.49) 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 1 ∂V g L . (6.50) LS = 2 2 2M c r ∂r It was realized by Thomas in 1927 that the relativistic motion of the electron changes the factor g as in (6.42) to g − 1, so that the true precession frequency is g − 1 1 ∂V = LS + T = 2M L . 2 c2 r ∂r (6.51) 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− ) = (115 965.77 ± 0.35) × 10−8 , a(e+ ) = (116 030 ± 120) × 10−8 , a(µ± ) = (116 616 ± 31) × 10−8 . (6.52) (6.53) (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. (4.742) which describes this precession. Along a particle orbit parametrized by a parameter τ , for instance the proper time, we form 435 6.1 Charged Point Particles the derivative with respect to τ , assuming that the motion proceeds at a fixed total angular momentum: dpκ dSˆµ = ǫµνλκ Jˆνλ . dτ dτ (6.55) The right-hand side can be simplified by multiplying it with the trivial expression 1 g στ pσ pτ = 1. M 2 c2 (6.56) Now we use the identity for the ǫ-tensor ǫµνλκ gστ = ǫµνλσ gκτ + ǫµνσκ gλτ + ǫµσλκ gντ + ǫσνλκ gµτ , (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 νλ σ κ ′ νλ σ ′κ νλ σ ′κ νλ σ ′κ , (6.58) ǫ J p p p +ǫ J p p p +ǫ J p p p +ǫ J p p p µνλσ µνσκ λ µσλκ ν σνλκ µ κ M 2 c2 where p′µ ≡ dpµ /dτ . The first term vanishes, since pκ p′κ = (1/2)dp2/dτ = (1/2)dM 2 c2 /dτ = 0. The last term is equal to −Sˆκ p′κ pµ /M 2 c2 . Inserting the identity (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 1 dpλ dSµ = − 2 2 Sλ pµ . dτ M c dτ (6.59) 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 S˙ ≡ = dt dS0 = S˙ 0 ≡ dt γ2 1 dS 1 ˙ v, = − 2 2 S 0 p˙ 0 + S · p˙ p = 2 (S · v) γ dτ M c c 1d γ2 ˙ . (S · v) = 2 (S · v) c dt c (6.60) (6.61) We now differentiate Eq. (4.755) with respect to the time using the relation γ˙ = 2 ˙ γ 3 vv/c , and find S˙ R = S˙ − γ 1 ˙0 γ 1 0 γ3 1 ˙ S 0 v. ˙ S v − S (v · v) v − 2 2 2 γ +1c γ +1c (γ + 1) c4 (6.62) Inserting here Eqs. (6.60) and (6.61), we obtain S˙ R = γ2 1 γ 1 0 γ3 ˙ ˙ S 0 v. ˙ (S · v)v − S (v · v) v − γ + 1 c2 γ + 1 c2 (γ + 1)2 (6.63) 436 6 Relativistic Particles and Fields in an External Electromagnetic Potential On the right-hand side we return to the spin vector SR using Eqs. (4.754) and (4.757), and find S˙ R = γ2 1 ˙ − (SR · v)v] ˙ = [(SR · v)v γ + 1 c2 T × SR, (6.64) with the Thomas precession frequency T = − (γ γ+ 1) c12 v × v,˙ 2 (6.65) which agrees with the result (4B.26) derived from purely group-theoretic considerations. 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 (4.275), and obtain an equation of motion for the spin: γ2 v v v S˙ R = ×B = × γ B − × E − ·B c γ+1 c c Expressing ′ " # . (6.66) via Eq. (6.31), this becomes S˙ R ≡ −SR × em eg = SR × 2Mc " v γ v B− ×E − c γ+1 c v ·B c # , (6.67) which is the relativistic generalization of Eq. (6.44). It is easy to see that the associated fully covariant equation is 1 µ d λ 1 eg g dS µ eF µν Sν + F µν Sν + 2 2 pµ Sλ F λκ pκ . (6.68) = p Sλ p = 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 dS µ g−2 µ d λ g−2 e 1 egF µν Sν + gF µν Sν + 2 2 pµ Sλ F λκ pκ .(6.69) = p Sλ p = 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 S˙ R = −SR × 1 Tem ≡ −SR × ( em + T) V. Bargmann, L. Michel, and V.L. Telegdi, Phys. Rev. Lett. 2 , 435 (1959). (6.70) 437 6.2 Charged Particle in the Schr¨ odinger Theory with a frequency given by the Thomas equation2 T em = − e Mc " ! g 1 γ g v g γ v B− −1 −1 + ·B − − 2 γ 2 γ +1 c c 2 γ +1 ! # v × E .(6.71) c The contribution of the Thomas precession is the part without the gyromagnetic ratio g: T " ! # e 1 γ 1 γ 1 =− − 1− B+ (v · B) v + v×E . Mc γ γ +1 c2 γ +1 c (6.72) 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 ˆ of an electron, i.e., its component of the spin in the direction the helicity h ≡ SR · v of motion. Using the chain rule of differentiation, we can express the change of the helicity as d 1 d dh ˆ ) = S˙ R · v ˆ + [SR − (ˆ = (SR · v v · SR )ˆ v] v, (6.73) dt dt v dt Inserting (6.70) as well as the equation for the acceleration (6.24), we obtain e dh =− SR⊥ · dt Mc g gv c ˆ ×B+ E . −1 v − 2 2c v (6.74) 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 −2 SR⊥ · E, 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)aSR⊥ · E. This drastic effect was exploited to measure the experimental values listed in Eqs. (6.52)–(6.54). 6.2 Charged Particle in the Schr¨ odinger Theory When going over from quantum mechanics to second quantized field theories we found the rule that a non-relativistic Hamiltonian H= p2 + V (x) 2m (6.75) becomes an operator H= 2 Z ∇2 d xψ (x, t) − + V (x) ψ(x, t), 2m 3 † L.T. Thomas, Phil. Mag. 3 , 1 (1927). " # (6.76) 438 6 Relativistic Particles and Fields in an 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 0 H= + A, 2m c (6.77) is given by H= Z " e 1 ∇−i A d xψ (x, t) − 2m c 3 † 2 0 # + eA (x) ψ(x, t). (6.78) When going to the action of this theory we find A= Z dtL = Z dt Z 3 d x ψ † (x, t) i∂t + eA0 ψ(x, t) e 1 † ψ (x, t) ∇ − i A + 2m c 2 ψ(x, t) . (6.79) It is easy to verify that (6.78) reemerges from the Legendre transform H= ∂L ˙ ψ(x, t) − L. ˙ ∂ ψ(x, t) (6.80) 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 ∂t → ∂t + ieA0 (x, t), (6.81) or covariantly: e (6.82) ∂µ → ∂µ + i Aµ (x). c 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) + ∂ µ Λ(x), (6.83) A0 (x, t) → A0 (x, t) + ∂t Λ(x, t) A(x, t) → A(x, t) − ∇Λ(x, t), (6.84) i.e., the action remains invariant provided we simultaneously change the fields ψ(x, t) of the charged particles by a spacetime-dependent phase ψ (x, t) → e−i(e/c)Λ(x,t) ψ(x, t). (6.85) 439 6.3 Charged Relativistic Fields Under this transformation, the derivatives of the field change like ∂t ψ(x, t) → e−i(e/c)Λ(x,t) (∂t − ie∂t Λ) ψ(x, t), e ∇ψ(x, t) → e−i(e/c)Λ(x,t) ∇ − i ∇Λ(x, t) ψ(x, t). c (6.86) The modified derivatives appearing in the action have therefore the following simple transformation law: e ∂t + i A0 ψ(x, t) → e−i(e/c)Λ(x,t) ∂t + ieA0 ψ(x, t), c e e −i(e/c)Λ(x,t) ∇ − i A ψ(x, t) → e ∇ − i A ψ(x, t). c c (6.87) These combinations of derivatives and gauge fields are called covariant derivatives. They occur so frequently in gauge theories that they deserve their own symbols: ∂t + ieA0 ψ(x, t), e Dψ(x, t) ≡ ∇ − i A ψ(x, t), c Dt ψ(x, t) ≡ (6.88) or, in four-vector notation, Dµ ψ(x) = e ∂µ + i Aµ ψ(x). c (6.89) Here the adjective of the covariant derivative does not refer to the Lorentz group but to the gauge group. It records the fact that Dµ ψ transforms under local gauge changes (6.81) of ψ in the same way as ψ itself in (6.85): Dµ ψ(x) → e−i(e/c)Λ(x) Dµ ψ(x). (6.90) With the help of this covariant derivative, any action that is invariant under a global multiplication change of the field ψ(x) → e−iφ ψ(x) (6.91) by a spacetime-independent phase factor e−iφ , can also be made invariant under a local version of this transformation, in which φ becomes 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 (4.236). 6.3 6.3.1 Charged Relativistic Fields Scalar Field The Lagrangian density of a free relativistic scalar field was stated in Eq. (4.164): L = ∂µ φ∗ (x)∂ µ φ(x) − M 2 φ∗ (x)φ(x). (6.92) 440 6 Relativistic Particles and Fields in an External Electromagnetic Potential If the field carries a of 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): L = [Dµ φ(x)]∗ D µ φ(x) − M 2 φ∗ (x)φ(x) e e = ∂µ − i Aµ (x) φ(x) ∂ µ + i Aµ (x) φ(x) − M 2 φ∗ (x)φ(x). c c The associated scalar field action A = invariant photon action (4.236). 6.3.2 R (6.93) d4 x L must be extended by the gauge- Dirac Field The Lagrangian density of a free charged spin-1/2 field was stated in Eq. (4.476): ¯ (i/ ∂ − M) ψ(x). L(x) = ψ(x) (6.94) If the particle carries a charge e, we must replace the derivatives in this Lagrangian by their covariant vesions (6.89): e e ∂/ = γ µ ∂µ → γ µ ∂µ + i Aµ = ∂/ + i A / c c ≡D /. (6.95) Adding again the gauge-invariant photon action (4.236), we we arrive at the Lagrangian of quantum electrodynamics (QED) 1 2 ¯ (i/ . L(x) = ψ(x) D − M) ψ(x) − Fµν 4 (6.96) The classical field equations can easily be found by extremizing the action and variation with respect to all fields, which gives δA = (i/ D − M) ψ(x) = 0, ¯ δ ψ(x) δA 1 = ∂ν F νµ (x) − j µ (x) = 0, δAµ (x) c (6.97) (6.98) where j µ (x) is the current density µ ¯ j µ (x) ≡ ec ψ(x)γ ψ(x). (6.99) Equation (6.98) is the Maxwell equation for the electromagnetic field around a classical four-dimensional vector current j µ (x): 1 ∂ν F νµ (x) = j µ (x). c (6.100) In the Lorentz gauge ∂µ Aµ (x) = 0, this equation reads simply 1 −∂ 2 Aµ (x) = j µ (x). c (6.101) 441 6.4 Pauli Equation from Dirac Theory The current j µ combines the charge density ρ(x) and the current density j of particles of charge e in a four-vector j µ = (cρ, j) . (6.102) In terms of electric and magnetic fields E i = F i0 , B i = −F jk , the field equations (6.100) turn into the Maxwell equations ¯ 0 ψ = eψ † ψ ∇ · E = ρ = eψγ ˙ = 1 j = e ψ¯ ψ. ∇×B−E c c (6.103) The first is Coulomb’s law, the second Amp`ere’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 2 Fµν (x) − j µ (x)Aµ (x) 8π c i 1 h 2 1 2 = E − B (x) − ρφ − j · A (x), 4π c (6.104) L(x) = − which leads to Maxwell’s field equations ∇ · E = 4πρ, 4π j. ∇×B = c (6.105) 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 q √ (6.106) e = − 4πα ≈ − 4π/137 √ 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 (4.542) and (4.544), and further to (4.560). 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 (4.561), we multiply (6.97) by (i¯hD / − Mc) and work out the product e e ¯ ∂µ + i Aµ + Mc iγ µ h (i¯hD / − Mc) (i¯hD / + Mc) = iγ µ h ¯ ∂µ + i Aµ − Mc . c c (6.107) 3 See for example J.D. Jackson, Classical Electrodynamics, Wiley and Sons, New York, 1967. 442 6 Relativistic Particles and Fields in an External Electromagnetic Potential We now use the relation 1 1 γ µ γ ν = (γ µ γ ν + γ ν γ µ ) + (γ µ γ ν − γ ν γ µ ) = g µν − iσ µν , 2 2 (6.108) with σ µν from (4.492), and find µ ν γ γ e h ¯ ∂µ + i Aµ c e ∂µ + i Aµ c e e i e e ¯ ∂µ + i Aµ ∂µ + i Aµ − σ µν [¯h∂µ + i Aµ , h = g µν h ¯ ∂ν + i Aν ] c c 2 c c e = h ¯ ∂µ + i Aµ c 2 + 1 e¯h µν σ Fµν . 2 c (6.109) Thus we obtain, as a generalization of Eqs. (4.560), the Pauli equation: " e − h ¯ ∂µ + i Aµ c 2 # 1 e¯h µν σ Fµν − M 2 c2 ψ(x) = 0, − 2 c (6.110) 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 · H + i · E, 1 µν σ Fµν = − 2 (6.111) where in the chiral and Dirac representations = − 0 0 ! D = 0 , 0 ! , (6.112) 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): " e − h ¯ ∂µ + i Aµ c 2 + · (H ± iE) − M c 2 2 #( ξ(x) η(x) ) = 0. (6.113) In the nonrelativistic limit where c√→ ∞, we remove the fast oscillations from 2 ξ(x), setting ξ(x) ≡ e−iM c t/¯h Ψ(x, t)/ 2M as in (4.155), and find for Ψ(x, t) the nonrelativistic Pauli equation h ¯2 e i∂t + ∇−i A 2M c¯h " 2 # e · H − eA0 (x) Ψ(x, t) = 0. + 2Mc (6.114) This corresponds to a magnetic interaction energy Hmag = − e¯h · H. 2Mc (6.115) 443 6.5 Relativistic Wave Equations in the Coulomb Potential For a small magnet with a magnetic moment , the magnetic interaction energy is Hmag = − · H. (6.116) A Dirac particle has therefore a magnetic moment ¯ e e h =2 S, = Mc 2 2Mc (6.117) where S = /2 is the spin matrix. Experimentally, one parametrizes a magnetic moment of a fundamental particles as e =g S, (6.118) 2Mc with the gyromagnetic ratio g, which is equal to unity for a uniformly charged sphere. Within the Dirac theory, an electron has a gyromagnetic ratio ge Dirac = 2. (6.119) 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 electromegnetic coupling directly into the nonrelativistic two-component equation (4.557). The minimal substitution rule (6.87) changes i∂t → i∂t − eA0 and ( · ∇)2 → [ · (∇ − ieA)]2 . The latter is processed as Eq. (4.558), and leads to [ · (∇ − ieA)]2 h i = δ ij + iǫijk σ k (∇i − ieAi )(∇j − ieAj ) = (∇ − ieA)2 + iǫijk σ k (∇i − ieAi )(∇j − ieAj ) + e · H. (6.120) This brings the free-field equation (4.559) 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 VC (x) = − , r = x2 , (6.121) r corresponding to a four-vector potential eAµ (x) = (VC (x, 0), 0). (6.122) Since this does not depend on time, we can consider the wave equations for wave functions φ(x) = e−iEt φE (x) and ψ(x) = e−iEt ψE (x), and find the time-independent equations (E 2 + ∇2 − M)φE (x) = 0 (6.123) 444 6 Relativistic Particles and Fields in an External Electromagnetic Potential and (γ 0 E + i · ∇ − M)ψE (x) = 0. (6.124) In these equations we simply perform the minimal substitution E→E+ Zα . r (6.125) 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 Schr¨odinger 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 Zα − ∇2 − − E ψE (x) = 0. 2M r (6.126) The Laplacian may be decomposed into radial and angular parts by writing ∇2 = ˆ2 L 2 ∂ ∂2 + − , ∂r 2 r ∂r r2 (6.127) ˆ = x×p ˆ are the differential operators for the generators of angular momenwhere L tum [the spatial part of Li = L23 of (4.96)]. Then (6.126) reads ˆ 2 2ZαM L ∂2 2 ∂ + 2 − − 2ME ψE (x) = 0. − 2− ∂r r ∂r r r ! (6.128) ˆ 2 are the spherical harmonics Ylm (θ, φ), which diagonalize also The eigenstates of L ˆ with the eigenvalues the third component of L, ˆ 2 Ylm (θ, φ) = l(l + 1)Ylm (θ, φ), L ˆ 3 Ylm (θ, φ) = mYlm (θ, φ). L (6.129) 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) 445 6.5 Relativistic Wave Equations in the Coulomb Potential Explicitly, v u u (n + l)! 1 t Rnl (r) = 1/2 (2l + 1)! (n − l − 1)! aB n 1 (6.131) ×(2r/naB )l+1 e−r/naB M(−n + l + 1, 2l + 2, 2r/naB ) = 1 1/2 aB n v u u (n − l t − 1)! −r/naB 2l+1 e (2r/naB )l+1 Ln−l−1 (2r/naB ), (n + l)! where aB is the Bohr radius which, in natural units with h ¯ = c = 1, is equal to 1 aB = . (6.132) ZMα For a hydrogen atom with Z = 1, this is about 1/137 times the Compton wavelength of the electron λe ≡ h ¯ /Me c. The classical velocity of the electron on the lowest Bohr orbit is vB = α c. Thus it is almost nonrelativistic, which is the reason why the Schr¨odinger 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 + 1) z a + ... . M(a, b, z) ≡ F1,1 (a, b, z) = 1 + z + b b(b + 1) z! (6.133) For b = −n, they are polynomials related to the Laguerre polynomials4 Lµn (z) by (n + µ)! M(−n, µ + 1, z). n!µ! The radial wave functions are normalized to Lµn (z) ≡ Z 0 (6.134) ∞ drRnr l (r)Rn′r l (r) = δnr n′r . (6.135) They have an asymptotic behavior Pnl (r/n)e−r/naB , where Pnl (r/n) is a polynomial of degree nr = n − l − 1, which defines the radial quantum number. The energy eigenvalues depend on n in the well-known way: En = 2 2 Mα . −Z 2n2 (6.136) The number α2 M/2 is the Rydberg-constant: 27.21 α2 M ≈ eV ≈ 3.288 × 1015 Hz. (6.137) Ry = 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 1 |ψn00 (0)| = √ π 4 s 3 ZMα 1 =√ n π s 3 1 . naB (6.138) I.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): Our Lµn = (−)µ /(n + µ)!Ln+µ µ |L.L. ). 446 6 Relativistic Particles and Fields in an External Electromagnetic Potential 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): " Zα E+ r 2 2 +∇ −M 2 # φE (x) = 0. (6.139) With the angular decomposition (6.127), this becomes ˆ 2 − Z 2 α2 2ZαE ∂2 2 ∂ L − 2− + − − (E 2 − M 2 ) φE (x) = 0. ∂r r ∂r r2 r " # (6.140) The solutions of this equation can be obtained from those of the nonrelativistic Schr¨odinger equation (6.128) by replacing ˆ2 → L ˆ 2 − Z 2 α2 , L E α → α , M E2 − M 2 . E → 2M (6.141) (6.142) (6.143) The replacement (6.141) is done most efficiently if we define the eigenvalues l(l + ˆ 2 − Z 2 α2 by analogy with those of L ˆ 2 as 1) − Z 2 α2 of the operator L λ(λ + 1) ≡ l(l + 1) − Z 2 α2 . (6.144) Then the quantum number l of the Schr¨odinger wave functions is simply replaced by λl = l − δl , where δl " 2 1 1 = l+ − l+ 2 2 Z 2 α2 + O(α4 ). = 2l + 1 2 −Z α 2 #1/2 (6.145) 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 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 nr + λ + 1 = nr + l + 1 − δl = n − δl . (6.146) It leads to the equation for the energy eigenvalues Enl 2 − M 2 Z 2 Mα2 Enl 2 1 =− , 2M 2 M 2 (n − δl )2 (6.147) 447 6.5 Relativistic Wave Equations in the Coulomb Potential with the solution M Enl = ± q 1 + Z 2 α2 /(n − δl )2 (Zα)2 3 (Zα)4 (Zα)4 = ±M 1 − + − 3 + O(Z 6 α6 ) . 2 4 2n 8 n n (2l + 1) " # (6.148) The first two terms correspond to the Schr¨odinger 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 Schr¨odinger 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 γ + i ·∇ − M ψE (x) = 0. E+ r (6.149) In order to find the energy spectrum it is useful to establish contact with the Klein-Gordon case. Multiplying (6.149) by the operator E+ we obtain " Zα E+ r 2 Zα 0 γ + i · ∇ + M, r 2 + ∇ − iγ 0 # Zα − M 2 ψE (x) = 0. ·∇ r (6.150) (6.151) has a block-diagonal form In the chiral representation, the 4 × 4 -matrix γ 0 = (4.538). We therefore decompose ψE (x) = ξE (x) ηE (x) ! , (6.152) and find the equation for the upper two-component spinors " Zα E+ r 2 # Zα − M 2 ξE (x) = 0. +∇ +i ·∇ r 2 (6.153) 448 6 Relativistic Particles and Fields in an External Electromagnetic Potential The lower bispinor ηE (x) satisfies the same equation with i replaced by −i. Expressing ∇2 via (6.127) and writing ∇ 1/r = −ˆ x/r 2 , we obtain the differential equation ˆ 2 − Z 2 α2 + iZα x ˆ 2ZαE L + − − (E 2 − M 2 ) ξE (x) = 0, 2 r r (6.154) and a corresponding equation for ηE (x). ˆ , the total angular momentum Due to the rotation invariance of · x " ∂2 2 ∂ − + 2 ∂r r ∂r ! # ˆ=L ˆ +S=L ˆ+ J (6.155) 2 ˆ 2 and commutes with the differential operator in (6.154). Thus we can diagonalize J 3 1 ˆ 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 + 21 and l− = j − 1/2. The two states ˆ is a pseudoscalar, and must necessarily have opposite parities. The operator · x ˆ is the unit matrix, change the parity of the wave function. Since the square of · x ˆ changes l by one unit. its eigenvalues must be ±1. Moreover, the unit vector x 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 (6.156) ˆ |jm, +i = −1. hjm, −| · x (6.157) ˆ |jm, +i = 0, hjm, +| · x ˆ |jm, −i = 0. hjm, −| · x For the off-diagonal elements we easily calculate ˆ |jm, −i = 1, hjm, +| · x The central parentheses in (6.154) have therefore the matrix elements ! (j + 21 )(j + 32 ) − Z 2 α2 ±iZα . L −Z α ± iZαˆ x = 1 ±iZα (j − 2 )(j + 12 ) − Z 2 α2 2 2 2 (6.158) 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 λj + = " 1 j+ 2 2 2 −Z α 2 #1/2 , λj− = λj+ − 1. (6.159) These may be written as λj ± = j ± 1 − δj ≡ l± − δj , 2 (6.160) where l± ≡ ± 21 is the orbital angular momentum, and 1 δj ≡ j + − 2 " 1 j+ 2 2 2 −Z α 2 #1/2 = Z 2 α2 + O(Z 4 α4 ). 2j + 1 (6.161) 449 6.5 Relativistic Wave Equations in the Coulomb Potential 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 nr + λj± + 1 = nr + l± + 1 − δl = n − δj . (6.162) In terms of δj , the energies obey the same equation as in (6.148), so that we obtain M Enj = ± q 1 + Z 2 α2 /(n − δj )2 (Zα)2 3 (Zα)4 (Zα)4 = ±M 1 − + − + O(Z 6 α6 ) . 2n2 8 n4 n3 (2j + 1) " # (6.163) The condition nr ≥ 0 implies that j ≤n− j ≤n− 3 2 1 2 for λj+ = j + 12 − δj , λj− = j − 21 − δj . (6.164) For n = 1, 2, 3, . . . , the total angular momentum runs through j = 12 , 23 , . . . , n − 12 . 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. 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 Schr¨odinger spectrum, there is now a twofold degeneracy of levels 450 6 Relativistic Particles and Fields in an External Electromagnetic Potential 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 occurs only once. The lowest degenerate pair consists of the levels 2S1/2 and 2P1/2 .5 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. 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) −MZ 4 α4 n3 /(2j+1). For the states 2P1/2 and 2P3/2 , the splitting is Z 4 α2 2 ∆fine E2P = α M. (6.165) 32 In a hydrogen atom, this is equal to ∆fine E2P = 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 2S 1/2 , 2P 1/2, and 2P 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 velocities 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 representation (4.525). Multiplying (6.149) by γ 0 and inserting the Dirac matrices (4.537) for γ 0 = , we obtain Zα i ·∇ E−M + r Zα E+M + i ·∇ r ξE (x) ηE (x) ! = 0. (6.167) This is of course just the time-independent version of (4.544) 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 ηE (x) ≈ −i 5 · ∇ξ 2M E (x). (6.168) Recall the notation in atomic physics for an electronic state: n2S+1 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. 6 See H.A. Bethe and E.E. Salpeter in Encyclopedia of Physics (Handbuch der Physik) 335 , Springer, Berlin, 1957, p. 196. 451 6.5 Relativistic Wave Equations in the Coulomb Potential We may take care of rotational symmetry of the system by splitting the spinor wave functions into radial and angular parts Gjl (r) l yj,m(θ, φ) i r , ψE (x) = F (r) jl l ˆ yj,m(θ, φ) ·x r (6.169) l where yj,m (θ, φ) denote the spinor spherical harmonics. They are composed from the ordinary spherical harmonics Ylm (θ, φ) and the basis spinors χ(s3 ) of (4.421) via Clebsch-Gordan coefficients (see Appendix 4C): l yj,m (θ, φ) = hj, m|l, m′ ; 12 , s3 iYlm′ (θ, φ)χ(s3 ). (6.170) The derivation is given in Appendix 6A. The explicit form of the spinor spherical harmonics (6.170) is for l = l± : l+ yj,m (θ, φ) l− yj,m (θ, φ) =√ 1 2l+ + 1 = √ 1 2l− + 1 √ ! l+ − m + 12 Yl+ ,m− 1 (θ, φ) 2 √ , − l+ + m + 12 Yl+ ,m+ 1 (θ, φ) (6.171) √ ! l− + m + 12 Yl− ,m− 1 (θ, φ) 2 √ . l− − m + 21 Yl− ,m+ 1 (θ, φ) (6.172) 2 2 On these eigenfunctions, the operator L · L· with has the eigenvalues l l yj,m (θ, φ) = −(1 + κ± )yj,m(θ, φ), ± ± 1 κ± = ∓(j + ), 2 (6.173) 1 j =l± . 2 (6.174) We can now go from Eqs. (6.167) to radial differential equations by using the trivial identity, i ·∇ f (r) l+ y ≡ r l,m · x ( · x) i ( · ∇) f (r) yl + r2 l,m , r (6.175) and the algebraic relation Eq. (4.439) in the form ( · a)( · b) = i · (a × b) + i(a · b), (6.176) to bring the right-hand side to · x (ir∂r − i · L) f (r) yl + r2 r l,m " f (r) f (r) = i∂r − i (1 + κ) 2 r r # l · xˆ yl,m . + (6.177) 452 6 Relativistic Particles and Fields in an External Electromagnetic Potential In this way we find the radial differential equations for the functions Fjl (r) and Gjl (r): d 1 Zα Gjl (r) = − Fjl (r) ∓ (j + 1/2) Fjl (r), (6.178) E−M + 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 − E 2 , writing F (r) = q 1 − E/Me−ρ/2 (F1 − F2 )(ρ), G(r) = q 1 + E/Me−ρ/2 (F1 + F2 )(ρ). (6.180) The functions F1,2 (ρ) satisfy a degenerate hypergeometric differential equation of the form # " d d2 (6.181) ρ 2 + (b − ρ) − a F (a, b; ρ) = 0, dρ dρ and the solutions are F2 (ρ) = ρl F (γ − ZαEλ, 2γ + 1; ρ), γ − ZαEλ F (γ + 1 − ZαEλ, 2γ + 1; ρ). F1 (ρ) = ρl −1/λ + ZαEλ (6.182) q The constant γ is Einstein’s gamma parameter γ = 1 − v 2 /c2 for the atomic unit velocity v = Zαc. It has the expansion γ = 1 − Z 2 α2 /2. As an example, we write down explicitly the ground state wave functions of the 1/2 1S state: 1 0 v u u (2MZα)3 0 1 1+γ e−mZαr t ψ1S 1/2 ,± 1 = 1−γ −iφ . i 1−γ cos θ 1−γ i sin θe 2 4π 2Γ(1 + 2γ) (2MZα) Zα Zα 1−γ 1−γ iφ i Zα sin θe −i Zα cos θ (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 − Z 2 α2 /2, and we see that for α → 0, the upper components of q the spinor wave functions tend to the nonrelativistic Schr¨odinger wave function 2 (ZαM)3 /4πe−ρ , multiplied by Pauli spinors (4.421). In general, l (6.184) ξj,m (x) = hj, m|l, m; 12 , s3 iψnlm (x)χ(s3 ). The lower (small) components vanish. 6.6 Green Function in 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. 453 6.6 Green Function in External Electromagnetic Field 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: d4 p i ′ e−ip(x−x ) = 4 2 2 (2π) p − M + iη d4 p −ip(x−x′)+iτ (p2 −M 2 +iη) e . (2π)4 0 (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 complicated differential equation ′ G(x−x ) = Z n Z ∞ dτ Z o [i∂ − eA(x)]2 − M 2 G(x, x′ ) = iδ (4) (x − x′ ), (6.187) for which a Fourier decomposition is 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 G(x − x′ ) = Z 0 ∞ 2 dτ hx|eiτ [(i∂−eA) −M 2 +iη] |x′ i. (6.188) The integrand contains the time-evolution operator associated with the Hamiltonian operator ˆ H(x, i∂) ≡ − (i∂ − eA)2 + M 2 . (6.189) This is the Schroedinger representation of the operator ˆ = H(ˆ H x, pˆ) = −Pˆ 2 + M 2 , (6.190) where Pˆµ ≡ pˆµ − eAµ (ˆ x) is the canonical momentum in the presence of electromagnetism. We shall calculate the evolution operator in (6.188) by introducing timedependent Heisenberg operators for position and momentum. These obey the Heisenberg-Ehrenfest equations of motion [recall (1.277)]: h dˆ xµ (τ ) ˆ x = i H, ˆµ τ )] = 2Pˆ µ (τ ) dτ h i ˆ dP µ (τ ) ˆ Pˆ µ(τ ) = 2eF µ ν (ˆ = i H, x(τ ))Pˆ ν (τ ) + ie∂ ν Fµν (ˆ x(τ )). dτ (6.191) (6.192) In a constant field where F µ ν (ˆ x(τ )) is a constant matrix F µ ν , the last term in the second equation is absent, and we find directly the solution Pˆ µ (τ ) = e2eF τ µ ν Pˆ ν (0). (6.193) 454 6 Relativistic Particles and Fields in an External Electromagnetic Potential µ µ Here the matrix e2eF τ e2eF τ ν is defined by its formal power series expansion ν = δ µ ν + 2eF µ ν τ + 4e2 F µ λ F λ ν τ2 + ... . 2 (6.194) Inserting (6.193) into Eq. (6.191), we find the time-dependent operator xˆµ (τ ): µ µ xˆ (τ ) − xˆ (0) = e2eF τ − 1 eF !µ ν Pˆ ν (0), (6.195) where the matrix on the right-hand side is again defined by its formal power series e2eF τ − 1 eF !µ ν = 2τ + e2 F µ λ F λ ν (2τ )3 + ... . 3! (6.196) 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 P (0) = eF 2 sinh eF τ " ˆν #µ ν [ˆ x(τ ) − xˆ(0)]ν , (6.197) and, using (6.193), Pˆ ν (τ ) = Lµ ν (eF τ ) [ˆ x(τ ) − xˆ(0)]ν , with the matrix 1 eeF τ L ν (eF τ ) ≡ eF µ ν 2 sinh eF τ " µ #µ (6.198) . (6.199) By squaring (6.198) we obtain x(τ ) − xˆ(0)]ν , Pˆ 2 (τ ) = [ˆ x(τ ) − xˆ(0)]µ Kµ ν (eF τ ) [ˆ (6.200) where Kµ ν (eF τ ) = Lλ µ (eF τ )Lλ ν (eF τ ). (6.201) Using the antisymmetry of the matrix Fµν , we can rewrite this as 1 e2 F 2 Kµ (eF τ ) = Lµ (−eF τ )Lλ (eF τ ) = 4 sinh2 eF τ ν λ ν " # ν . (6.202) µ The commutator between two operators xˆ(τ ) at different times is e2eF τ − 1 [ˆ x (τ ), xˆν (0)] = i eF µ !µ ν , (6.203) 455 6.6 Green Function in External Electromagnetic Field and !µ e2eF τ − 1 xˆ (τ ), xˆν (0) + xˆν (τ ), xˆ (0) = i eF ! #µ " 2eF τ −2eF τ µ sinh 2eF τ e −e = 2i . = i ν ν eF eF h i µ h µ i T ν e2eF τ − 1 +i eF T !µ ν (6.204) With the help of this commutator, we can expand (6.200) in powers of operators xˆ(τ ) and xˆ(0). We must be sure to let the later operators xˆ(τ ) lie to the left of the earlier operators xˆ(0) as follows: H(ˆ x(τ ), xˆ(0); τ ) = −ˆ xµ (τ )Kµ ν (eF τ)ˆ xν (τ ) − xˆµ (0)Kµ ν (eF τ)ˆ xν (0) i + 2ˆ xµ (τ )Kµ ν (eF τ)ˆ xν (0) − tr [eF coth eF τ ] + M 2 . 2 (6.205) Given this form of the Hamiltonian operator, it is easy to calculate the time evolution amplitude in Eq. (6.188): ˆ hx, τ |x′ 0i ≡ hx|e−iHτ |x′ i. (6.206) It satisfies the differential equation ˆ ˆ h ˆ ˆ i ˆ e−iHτ |x′ i ˆ e−iHτ |x′ i = hx|e−iHτ eiHτ H i∂τ hx, τ |x′ 0i ≡ hx|H ˆ x(τ ), Pˆ (τ ))|x′ , 0i. = hx, τ |H(ˆ (6.207) Replacing the operator H(ˆ x(τ ), Pˆ (τ )) by H(ˆ x(τ ), xˆ(0); τ ) of Eq. (6.205), the matrix elements on the right-hand side can immediately be evaluated, using the property hx, τ |ˆ x(τ ) = xhx, τ |, xˆ(0)|x′ , 0i = x′ |x′ , 0i, (6.208) and the differential equation (6.209) becomes i∂τ hx, τ |x′ 0i ≡ H(x, x′ ; τ )hx, τ |x′ 0i, or hx, τ |x′ 0i = C(x, x′ )E(x, x′ ; τ ) ≡ C(x, x′ )e−i R dτ H(x,x′ ;τ ) (6.209) . (6.210) The prefactor C(x, x′ ) contains a possible constant of integration in the exponent which may have an arbitrary dependence on x and x′ . The following integrals are needed: Z dτ K(eF τ ) = 1Z 1 e2 F 2 = − eF coth eF τ, dτ 2 4 4 sinh eF τ (6.211) and Z dτ tr [eF coth eF τ ] = tr log sinh eF τ sinh eF τ = tr log + 4 log τ. eF eF τ (6.212) 456 6 Relativistic Particles and Fields in an External Electromagnetic Potential 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 ) ( i 1 sinh eF τ 1 . E(x, x ; τ ) = 2 exp − (x−x′ )µ [eF coth eF τ ]µ ν (x−x′ )ν −iM 2 τ − tr log τ 4 2 eF τ (6.213) The last term produces a prefactor ′ det sinh eF τ eF τ −1/2 ! . (6.214) The time-independent integration constant is fixed by the differential equation with respect to x: ˆ ˆ ˆ ˆ [i∂µ −eAµ (x)] hx, τ |x′ 0i = hx|Pˆµ e−iHτ |x′ i = hx|e−iHτ eiHτ Pˆµ e−iHτ |x′ i i h = hx, τ |Pˆµ (τ )|x′ 0i, (6.215) which becomes, after inserting (6.198): [i∂µ −eAµ (x)] hx, τ |x′ 0i = Lµ ν (eF τ )(x − x′ )ν hx, τ |x′ 0i. (6.216) Calculating the partial derivative we find i∂µ hx, τ |x′ 0i = [i∂µ C(x, x′ )]E(x, x′ ; τ ) + C(x, x′ )[i∂µ E(x, x′ ; τ )] 1 = [i∂µ C(x, x′ )]E(x, x′ ; τ ) + C(x, x′ ) [eF coth eF τ ]µ ν (x − x′ )ν E(x, x′ ; τ ). 2 Subtracting from this eAµ (x)hx, τ |x′ 0i, and inserting (6.210), the right-hand side of (6.216) is equal to [i∂µ C(x, x′ )]E(x, x′ ; τ ) plus 1 Lµ (eF τ )(x − x )ν − [eF coth eF τ ]µ ν (x − x′ )ν C(x, x′ )E(x, x′ ; τ ). (6.217) 2 ν ′ Inserting Eq. (6.199), this simplifies to e ν Fµ (x − x′ )ν C(x, x′ )E(x, x′ ; τ ), 2 (6.218) so that C(x, x′ ) satisfies the time-independent differential equation e i∂ − eA (x) − F µ ν (x − x′ )ν C(x, x′ ) = 0. 2 µ µ (6.219) This is solved by ′ C(x, x ) = C exp −ie Z x x′ dξ µ 1 Aµ (ξ) + Fµ ν (ξ − x′ )ν 2 . (6.220) 457 6.6 Green Function in External Electromagnetic Field The contour of integration is arbitrary since A′ (ξ) ≡ Aµ (ξ) + 12 Fµ ν (ξ − x′ )ν has a vanishing curl: ∂µ A′ν (x) − ∂ν A′µ (x) = 0. (6.221) 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 dξ µ points in the same direction of xµ − x′µ as ξ µ − x′µ and Fµν is antisymmetric. Hence we may write for a straight-line connection ′ Z C(x, x ) = C exp −ie x x′ µ dξ Aµ (ξ) . (6.222) The normalization constant C is finally fixed by the initial condition lim hx, τ |x′ 0i = δ (4) (x − x′ ), (6.223) τ →0 which requires C=− i . (4π)2 (6.224) Collecting all terms we obtain x i −1/2 sinh eF τ µ dξ A (ξ) det hx, τ |x 0i = − exp −ie µ (4πτ )2 eF τ x′ i × exp − (x−x′ )µ [eF coth eF τ ]µ ν (x−x′ )ν −iM 2 τ . 4 ′ Z ! (6.225) For a vanishing field Fµ ν , this reduces to the relativistic free-particle amplitude i i (x − x′ )2 hx, τ |x′ 0i = − exp − − iM 2 . (4πτ )2 2 2τ " # (6.226) According to relation (6.188), the Green function of the scalar field is given by the integral Z ∞ ′ G(x, x ) = dτ hx, τ |x′ 0i. (6.227) 0 The functional trace of (6.225) Trhx, τ |x 0i = V ∆t eEτ i 2 (4πτ ) sinh eEτ (6.228) 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) 458 6 Relativistic Particles and Fields in an External Electromagnetic Potential Then the Hamiltonian (6.190) becomes ˆ = −ˆ ˆ 2⊥ + (ˆ H p20 + p p3 + eEx0 )2 + M 2 , (6.230) where p⊥ are the two-dimensional momenta in the xy-plane. Using the commutation rule [p0 , x0 ] = i, this can be rewritten as ˆ = e−iˆp0 p3 /eE H ˆ ′eiˆp0 p3 /eE , H (6.231) ˆ ′ is the sum of two commuting Hamiltonians: where H ˆ ′ = −(ˆ ˆω + H ˆ ⊥. H p20 − e2 E 2 x20 ) + p2⊥ + M 2 ≡ H E (6.232) 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 xyplane. 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 Trhx, τ |x 0i = Z 4 dx d4 p (2π)4 Z Z d4 p′ −i(p−p′)x ˆ e hp|e−iτ (HωE +H⊥ ) |p′ i. 4 (2π) (6.233) The matrix elements are ˆ ˆ 3 2 hp|e−iτ H |p′ i = e−ip0 (x0 +p /eE) hp0 |e−isHωE |p′0 ie−iτ (p⊥ +M × (2π)2 δ (2) (p⊥ − p′⊥ )(2π)δ(p3 − p′3 ). 2 −iη) ′ eip0 (x0 +p ′3 /eE) (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 d2 p⊥ −iτ (p2 +M 2 −iη) ⊥ e (2π)2 Z dp0 dp3 dp′0 −i(p0 −p′0 )(x0 +p3 /eE) ˆ e hp0 |e−isHωE |p′0 i, × 3 (2π) Trhx, τ |x 0i = V Z dx0 Z (6.235) which can be reduced to −i −iτ (M 2 −iη) eE e Trhx, τ |x 0i = V ∆t 4πτ 2π "Z # dp0 ˆ hp0 |e−iτ HωE |p0 i . 2π (6.236) ˆ The expression in brackets is the trace of e−iτ HωE , which is conveniently calculated in the eigenstates |ni of the harmonic oscillator with eigenvalues −2(n + 1/2)ωE : ˆω −iτ H E Tre = ∞ X eiτ 2(n+1/2)eE = n=0 i 1 = . 2 sin ωE 2 sinh τ eE (6.237) Thus we obtain Trhx, τ |x 0i = V ∆t −i eEτ . 4(2π)2 τ 2 sinh τ eE (6.238) 459 6.6 Green Function in External Electromagnetic Field 6.6.2 Dirac Field in a Constant Electromagnetic Field For a Dirac field we have to solve the inhomogeneous differential equation {iγ µ [∂µ − eAµ (x)] − M} S(x, x′ ) = iδ (4) (x − x′ ), (6.239) rather than (6.187). The solution can formally be written as ¯ x′ ) = iδ (4) (x − x′ ), S(x, x′ ) = {iγ µ [∂µ − eAµ (x)] + M} G(x, (6.240) ¯ x′ ) solves a slight generalization of Eq. (6.187): where G(x, e ¯ x′ ) = iδ (4) (x − x′ ), [i∂ − eA(x)]2 − σ µ ν Fµ ν − M 2 G(x, 2 (6.241) which 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, if we recall the relations to the Green function (6.188) and (6.227), which imply that ¯ x′ ) contains the fields as follows: G(x, ¯ x′ ) = G(x, Z 0 ∞ e dτ exp −i σ µ ν Fµ ν τ hx, τ |x′ 0i. 2 (6.242) 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 F3 0 = −E and F0 3 = −E. The field tensor Fµ ν is given by the matrix F = −E 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 = iE M3 , (6.244) where M3 is the generator (4.59) of pure Lorentz transformations in the z-direction. The exponential eeF τ is therefore equal to the boost transformation (4.58) B3 (ζ) = e−iM3 ζ with a rapidity ζ = −Eτ . From (4.14) we find the explicit matrices cosh eEτ 0 0 − sinh eEτ 0 1 0 0 0 − sinh eEτ 0 0 1 0 0 cosh eEτ (6.245) 0 0 0 − sinh eEτ 0 0 0 0 0 − sinh eEτ 0 0 0 0 0 0 (6.246) eeF τ = and hence sinh eF τ = , , 460 6 Relativistic Particles and Fields in an External Electromagnetic Potential sinh eF τ = eF τ and sinh eEτ eE 0 0 eF coth eF τ = eE Thus we obtain 0 0 0 1 0 0 1 0 0 0 0 sinh eEτ eE 0 1 0 0 0 0 0 0 1 0 0 coth eEτ 0 coth eEτ 0 0 0 , (6.247) . (6.248) x i eEτ hx, τ |x 0i = dξ µ Aµ (ξ) (6.249) exp −ie 2 ′ (4πτ ) sinh eEτ x i 2 ′ 0 ′ 0 ′ T 1 ′ T ′ 3 ′ 3 × e 4 [−(x−x ) eE coth eEτ (x−x ) +(x−x ) τ (x−x ) +(x−x ) eE coth eEτ (x−x ) ]−iM τ , ′ Z where the superscript T indicates transverse directions to E. The prefactor Rx exp [−ie x′ dξ µ Aµ (ξ)] is found by inserting (6.243) and integrating along the straight line ξ = x′ + s(x − x′ ), s ∈ [0, 1], (6.250) to be exp −ie Z x x′ ′ dξ µ Aµ (ξ) = e−ieE(x0 −x0 ) R1 0 ds[z ′ +s(z−z ′ )] ′ ′ = e−ieE(x0 −x0 )(z+z ) . (6.251) 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 exp −i σ µ ν Fµ ν τ 2 0 e−eEτ 0 eeEτ = exp (e Eτ ) = ! , (6.252) which is equal to e exp −i σ µ ν Fµ ν τ = 2 ˆ cosh eEτ −sinh eEτ E 0 ! 0 ˆ . cosh eEτ +sinh eEτ E (6.253) Comparison with (4.481) 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 trhx, τ |x 0i = − i eEτ × 4 cosh eEτ, 2 (4πτ ) sinh eEτ (6.254) and the functional trace of this carries simply a total spacetime volume factor V ∆t that appeared before in Eq. (6.228). 461 6.6 Green Function in External Electromagnetic Field 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 being equal to two [recall (6.119)]. The process of pair creation in a space- and time-dependent electromagetic 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 [4] running along some direction nµ with n2 = 0. If ξ denotes the spatial coordinate along n, we may write the vector potential as Aµ (x) = ǫµ f (ξ), ξ ≡ nx, (6.255) where ǫµ is some polarization vector with the normalization ǫ2 = −1 in the gauge ǫn = 0. The field tensor is Fµν = ǫµν f ′ (ξ), ǫµν ≡ nµ ǫν − nν ǫµ , (6.256) where the constant tensor ǫµν satisfies ǫµν nµ = 0, ǫµν ǫµ = 0, ǫµν ǫνλ = nµ nλ . (6.257) The Heisenberg equations of motion (6.191) and (6.192) take the form h dˆ xµ (τ ) ˆ xˆµ τ )] = 2Pˆ µ(τ ) = i H, (6.258) dτ h i dPˆ µ (τ ) ˆ )). (6.259) ˆ )) + e nµ ǫλκ σ λκ f ′′ (ξ(τ ˆ Pˆ µ (τ ) = 2eǫµ ν Pˆ ν (τ )f ′ (ξ(τ = i H, dτ 2 Note that the last term in (6.259) vanishes for a sourceless plane wave: ∂ ν Fµν = 0. Multiplying these equations by nµ we see that nµ dξˆµ (τ ) = 2nµ Pˆ µ (τ ), dτ nµ dPˆ µ (τ ) = 0. dτ (6.260) Hence nPˆ (τ ) = nPˆ (0) = const, ˆ ) − ξ(0) ˆ = nˆ ξ(τ x(τ ) − nˆ x(0) = 2τ nPˆ (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 ˆ ˆ d ˆ = enν f ′ (ξ)(2n ˆ ˆ dξ = enν df (ξ) , (6.262) ǫνµ Pˆ µ (τ ) = 2eǫνµ ǫµρ Pˆρ f ′ (ξ) Pˆ ) = enν f ′ (ξ) dτ dτ dτ 462 6 Relativistic Particles and Fields in an External Electromagnetic Potential which is integrated to ˆ + Cˆν , ǫνµ Pˆ µ (τ ) = enν f (ξ) (6.263) with an operator integration constant Cˆν which commutes with the constant nPˆ and satisfies nν Cˆ ν = 0 and ˆ ) − ξ(0) ˆ ξ(τ ǫµν Cˆν = nµ (nPˆ ) = nµ . 2τ (6.264) Inserting this into (6.259), and integrating the resulting equation yields 1 ˆ + e2 nµ f 2 (ξ) + e nµ ǫµν σ µν f ′ (ξ) ˆ +D ˆ µ, 2eCµ f (ξ) Pˆµ (τ ) = 2πn 2 (6.265) ˆ µ is again an interaction constant commuting with nPˆ . Now we can integrate where D ˆ Pˆ , and find the equation of motion (6.258) over dτ = dξ/2n 1 1 [ˆ x(τ ) − xˆ(0)] = 2 (2nPˆ )2 ˆ ) ξ(τ ˆ +D ˆ + e2 nµ f 2 (ξ) + e nµ ǫµν σ µν f ′ (ξ) ˆ µ τ. dξˆ 2eCµ f (ξ) ˆ 2 ξ(0) (6.266) ˆ µ , and insert it into (6.265) to find We determine D Z 1 Pˆµ (τ ) = [ˆ xµ (τ ) − xˆµ (0)] 2τ Z ξ(τ ˆ ) τ e 2 2 ˆ ρν ′ ˆ ˆ ˆ ˆ − h dξ 2eCµ f (ξ) + e nµ f (ξ) + nµ ǫρν σ f (ξ) i2 ˆ 2 ξ(0) ˆ ) − ξ(0) ˆ ξ(τ τ ˆ )) . (6.267) ˆ )) + e2 nµ f 2 (ξ(τ ˆ )) + e nµ ǫρν σ ρν f ′ (ξ(τ + 2eCˆν f (ξ(τ ˆ ˆ 2 ξ(τ ) − ξ(0) After multiplication by ǫνµ , and recalling (6.257) and (6.264), we obtain 1 νµ ǫ [ˆ xµ (τ ) − xˆµ (0)] + 2τ Z ξ(τ ˆ ) enν ˆ + enν f (ξ(τ ˆ )). − dξˆ f (ξ) ˆ ˆ ξ(0) ξ(τ ) − ξ(0) ǫνµ Pˆµ (τ ) = (6.268) Inserting this into (6.263) determines the integration constant Cˆ ν : Z ξ(τ ˆ ) enν 1 νµ ˆ dξˆ f (ξ). ǫ [ˆ xµ (τ ) − xˆµ (0)] − Cˆ ν = ˆ ˆ 2τ ξ(0) ξ(τ ) − ξ(0) (6.269) It is useful to introduce the notations 1 hf i ≡ ˆ ξ(τ ) − ξ(0) and Z ˆ ) ξ(τ ˆ ξ(0) ˆ dξˆ f (ξ) h (δf )2 i ≡ h (f − hf i)2 i = h f 2 i − h f i2 . (6.270) (6.271) 463 6.6 Green Function in External Electromagnetic Field In order to calculate the matrix elements ˆ 0i = x τ −Pˆ 2 + e σ µ ν Fµ ν + M 2 x 0 , hx τ |H|x 2 (6.272) we must time-order the operators xˆ(τ ), xˆ(0). For this we need the commutator [ˆ xµ (τ ), xˆν (0)] = 2iτ gµν . (6.273) This is deduced from Eq. (6.267) by commuting it with xˆ(τ ) and using the trivially ˆ ), xˆν (τ )] = 0, as well as the nonequal-time vanishing equal-time commutator [ξ(τ ˆ ˆ ), xˆν (0)] = 0, which commutator [ξ(0), xˆν (τ )] = 2inν τ . The latter implies that [ξ(τ is also needed for time-ordering. The result is ′ ˆ 0i = − 1 (x − x′ )2 − 2 i + M 2 + e2 h(δφ)2 i 2 + eǫµν σ µν f (ξ) − f (ξ ) . hx τ |H|x 4τ 2 τ ξ − ξ′ (6.274) Integrating this over τ we obtain the exponential factor of the time-evolution amplitude (6.210): ( ) 2 1 i f (ξ) − f (ξ ′ ) E(x, x′ ; τ ) = 2 exp − (x−x′ )2 + M 2 +e2 h(δf )2 i −iτ eǫµν σ µν . τ 4τ 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 ǫµν (x − x′ )ν h f i − f (ξ) hx, τ |x′ 0i, [i∂µ −eAµ (x)] hx, τ |x 0i = ξ − ξ′ # " ′ (6.276) and is solved by −i C(x, x ) = exp ie (4π)2 ′ Z x x′ dyµ ( ǫµν (x−x′ )ν A (y)− ξ − ξ′ µ "Z ξ ′ny #)! f (y ′) −f (ny) . dy ny − ξ ′ (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 2 Meff = M 2 + e2 h(δf )2 i, (6.278) 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 h f 2 i. If the photon number density is ρ, their energy density is ρω (in units with h ¯ = 1), and we can calculate e2 h f 2i = 4πα h E 2i ρ = 4πα . 2 ω ω (6.279) 464 6 Relativistic Particles and Fields in an External Electromagnetic Potential Hence we find a relative mass shift: ∆M 2 = 4πα¯ λ2e λ ρ, M2 (6.280) ˚ is the Compton wavelength of the where λ¯e ≡ h ¯ /Me c = 3.861592642(28) × 10−3 A electron. For visible light, the right-hand side is of the order of ˚ A3 ρ/100. Present lasers achieve energy densities of 109 W/sec corresponding to a photon density ρ= W 1 eV 1 × 109 ≡ 2.082 × 10−7 , ˚ h ¯ω sec ¯ω A3 h (6.281) 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 (4C.20) for the Clebsch-Gordan coefficients hs1 m1 ; s2 m2 |smi, simplifies by having no second term. Inserting s1 = l− , s2 = 1/2, and s = j = l− + 1/2, we find hl− , m + 21 ; 21 , − 12 |l− + 21 , mi = v u u l− t − m + 1/2 hl− , m − 21 ; 21 , − 12 |l− + 21 , m−1i. l− − m + 3/2 (6A.1) This has to be iterated with the initial condition hl− , −l− ; 21 , − 12 |l− + 21 , −l− − 12 i = 1, (6A.2) which follows from the fact that the state hl− , −l− ; 21 , − 12 i carries a unique magnetic quantum number m = −l− − 1/2 of the irreducible representation of total angular momentum s = j = l− + 1/2. The result of the iteration is hl+ , m − 21 ; 12 , 21 |l+ − 12 , mi = v u u l+ t − m + 1/2 . 2l+ + 1 (6A.3) Similarly we may simplify the recursion relation (4C.21) for the configurations j = l+ − 1/2 with m2 = 1/2 to hl− , m − 12 ; 21 , 21 |l− + 12 , mi = v u u l− t + m + 1/2 hl− , m + 12 ; 21 , 12 |l− + 21 , m+1i, l− + m + 3/2 (6A.4) and iterate this with the initial condition hl− , l− ; 21 n 21 |l− + 12 , l− + 21 i = 1, (6A.5) 465 Notes and References which expresses the fact that the state hl− l− ; 21 12 i is the state of the maximal magnetic quantum number m = l− + 1/2 in the irreducible representation of total angular momentum s = j = l− + 1/2. The result of the iteration is hl− , m − 12 ; 21 , 12 |l+ + 12 , mi = v u u l+ t + m + 1/2 . 2l− + 1 (6A.6) 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 − yj,m (θ, φ) = hl− , m − 12 ; 21 , 21 |l− + 12 , mi Yl m−1/2 (θ, φ)χ( 21 ) + hl− m + 21 ; 12 − 21 |l− + 21 , mi Yl m+1/2 (θ, φ)χ(− 12 ). (6A.7) 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: hl + 21 , m|l − 21 , mi = 0. (6A.8) Inserting a complete set of states in the direct product space yields hl + 12 , m|l, m − 21 ; 21 , 21 ihlm − 12 ; 21 12 |l − 12 , mi +hl + 21 , m|l, m + 21 ; 12 , − 21 ihl, m + 12 ; 21 , − 12 |l − 21 , mi = 0. (6A.9) Together with (6A.3) and (6A.6) we find hl+ .m − 21 ; 21 , 12 |l+ − 12 , mi = hl+ , m + 21 ; 21 , − 12 |l+ − 12 , mi = v u u l+ t + m + 1/2 , 2l+ + 1 v u u l+ −t − m + 1/2 . 2l+ + 1 (6A.10) With this, the expression (6.170) for the spinor spherical harmonics written as l + yj,m (θ, φ) = hl+ , m − 12 ; 21 , 21 |l+ − 12 , mi Yl,m−1/2 (θ, φ)χ( 12 ) + hl+ , m + 21 ; 21 , − 12 |l+ − 12 , mi Yl,m+1/2 (θ, φ)χ(− 12 ) (6A.11) 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). 466 6 Relativistic Particles and Fields in an External Electromagnetic Potential [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 X. SheSheng, Phys. 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. Brezin, Phys. Rev. D 2, 1191 (1970).

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