Rel Particles in External Fields

The most tragic word in the English language is ’potential’

Arthur Lotti

6

Relativistic Particles and Fields inExternal Electromagnetic Potential

Having found classical fields to describe relativistic particles, we may ask what cor-rections the relativistic motion causes to the Schrodinger description of atoms. Weshall therefore study Klein-Gordon and the Dirac equation in an external field. LetAµ(x) be the associated four-vector potential, accounting for electric and magneticfield strengths via (4.227) and (4.228). For classical relativistic point particles, aninteraction with these external fields is introduced via the so-called minimal sub-

stitution rule, which we are now going to discuss. Throughout this Chapter, thepresence of an external electromagnetic field will be assumed, without asking for itsorigin.

An important property of the electromagnetic field is its description in terms of avector potential Aµ(x) and the gauge invariance of this description. In Eqs. (4.230)and (4.231) we have expressed electric and magnetic field strength as componentsof a four-curl Fµν = ∂µAν − ∂νAµ of a vector potential Aµ(x). This four-curl isinvariant under gauge transformations

Aµ(x) → Aµ(x) + ∂µΛ(x). (6.1)

The gauge invariance restricts strongly the possibilities of introducing electromag-netic interactions into particle dynamics and the Lagrange densities (6.94) and (6.95)of charged scalar and Dirac fields.

The prescription for coupling electromagnetism has been known for a long timein the classical electrodynamics of point particles.

6.1 Charged Point Particles

A free relativistic particle moving along an arbitrarily parametrized path xµ(τ) infour-space is described by an action

A = −Mc∫

dτ√

qµ(τ)qµ(τ). (6.2)

428

6.1 Charged Point Particles 429

The physical time along the path is given by q0(τ) = ct, and the physical velocityby v(t) ≡ dq(t)/dt. In terms of these, the action reads:

A =∫

dt L(t) ≡ −Mc2∫

dt

[

1− v2(t)

c2

]1/2

, (6.3)

6.1.1 Coupling to Electromagnetism

If the particle has a charge e (in our convention, e has a negative value for electrons)and lies at rest at 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 have agreementwith the sign in the historic form of the Maxwell equations

∇ ·E(x) = −∇2φ(x) = ρ(x),

∇×B(x)− E(x) = ∇×∇×A(x)− E(x)

= −[

∇2A(x)−∇ ·∇A(x)

]

− E(x) =1

cj(x). (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

Aintpot = −e

∫

dtA0 (q(t), t) . (6.9)

Since the time t coincides with q0(τ)/c of the trajectory, this can be expressed as

Aintpot = −e

c

∫

dq0A0. (6.10)

In this form it is now quite simple to write down the complete electromagnetic inter-action purely on the basis of relativistic invariance. The direct invariant extensionof (6.11) is obviously

Aint = −ec

∫

dqµAµ(q). (6.11)

430 6 Relativistic Particles and Fields in External Electromagnetic Potential

Thus, the full action of a point particle can be written in covariant form as

A=−Mc∫

dτ√

qµ(τ)qµ(τ)−e

c

∫

dqµAµ(q), (6.12)

or more explicitly as

A=∫

dt L(t) = −Mc2∫

dt

[

1− v2

c2

]1/2

− e∫

dt(

A0 − 1

cv ·A

)

. (6.13)

The canonical formalism supplies us with the canonically conjugate momenta

P =∂L

∂v=M

v√

1− v2/c2+e

cA ≡ p+

e

cA. (6.14)

The Euler-Lagrange equation obtained by extremizing this equation is

d

dt

∂L

∂v(t)=

∂L

∂q(t), (6.15)

ord

dtp(t) = −e

c

d

dtA(q(t), t)− e∇A0(q(t), t) +

e

cvi∇Ai(q(t), t). (6.16)

We now split

d

dtA(q(t), t) = (v(t) ·∇)A(q(t), t) +

∂

∂tA(q(t), t), (6.17)

and obtain

d

dtp(t) = −e

c(v(t) ·∇)A(q(t), t)− e

c

∂

∂tA(q(t), t)− e∇A0(q(t), t)+

e

cvi∇Ai(q(t), t).

(6.18)The right-hand side contains the electric and magnetic fields (4.232) and (4.233), interms of which it takes the well-known form

d

dtp = e

(

E+v

c×B

)

. (6.19)

This can be rewritten in terms of the proper time τ ≡ t/γ as

d

dτp =

e

Mc

(

E p0 + p×B)

, (6.20)

Recalling Eqs. (4.230) and (4.231), this is recognized as the spatial part of thecovariant equation

d

dτpµ =

e

McF µ

νpν . (6.21)

The temporal component of this equation

d

dτp0 =

e

McE · p (6.22)


gives the energy increase of a particle running through an electromagnetic field. Inreal time this is

d

dtp0 = eE · v

c. (6.23)

Combining this with (6.19), we find the acceleration

d

dtv(t) = c

d

dt

p

p0=

e

γM

[

E+v

c×B− v

c

(

v

c· E)]

(6.24)

The velocity is related to the canonical momenta and external field via

v

c=

P− e

cA

√

(

P− e

cA

)2

+m2c2. (6.25)

This can be used to calculate the Hamiltonian via the Legendre transform

H =∂L

∂vv− L = P · v − L (6.26)

giving

H = c

√

(

P− e

cA

)2

+m2c2 + eA0. (6.27)

In the non-relativistic limit this has the expansion

H = mc2 +1

2m

(

P− e

cA

)2

+ eA0 + . . . (6.28)

Thus, the free theory goes over into the interacting theory by the minimal substitu-

tion rule

p → p− e

cA, H → H − e

cA0. (6.29)

or, in relativistic notation:

pµ → pµ − e

cAµ. (6.30)

6.1.2 Spin Precession in Atom

In 1926, Uhlenbeck and Goudsmit noticed that the observed Zeeman splitting ofatomic levels could be explained by an electron of spin 1

2. Its magnetic moment is

usually expressed in terms of the combination of fundamental constants which havethe dimension of a magnetic moment, the Bohr magneton µB = eh/Mc as

� = gµBS

h, µB ≡ eh

2Mc, (6.31)


where S = �/2 is the spin matrix which has the commutation rules

[Si, Sj] = ihǫijkSk, (6.32)

and g is a dimensionless number called the gyromagnetic ratio or Lande 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 angularmomentum is conserved. The spin, however, shows a precession just like a spinningtop. This precession has two main contributions: one is due to the magnetic couplingof the magnetic moment of the spin to the magnetic field of the electron orbit, calledspin-orbit coupling . The other part is purely kinematical, it is the Thomas precessiondiscussed in Section (4.16), 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

HLS(r) =g

2M2c2S · L 1

r

dV (r)

dr, (6.33)

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

dt= �×B, (6.34)

where � is the magnetic moment of the electron. In an atom, the magnetic field inthe rest frame of the electron is entirely due to the electric field in the rest frame ofthe atom. A Lorentz transformation (4.276) to the electron velocity produces themagnetic field in the electron rest frame:

B = Bel = −γ v

c× E, γ =

1√

1− v2/c2. (6.35)

Since an atomic electron has a small velocity ratio v/c which is of the order of thefine-structure constant α ≈ 1/137, we can write approximately

Bel ≈ −v

c× E. (6.36)

The electric field gives the electron an acceleration

v =e

ME, (6.37)

so that we may also write

Bel ≈ −Mc

e

1

c2v × v, (6.38)


and the precession equation (6.34) as

dS

dt≈ g

2c2(v × v)× S. (6.39)

This can be written asdS

dt= LS × S, (6.40)

where LS is the angular velocity of spin precession caused by the orbital magneticfield at in the rest frame of the electron:

LS ≡ g

2c2v × v. (6.41)

In the rest frame of the atom where the electron is accelerated towards the centeralong its orbit, this result receives a relativistic correction. To lowest order in 1/c,we must add to LS the angular velocity T of Thomas precession, such that thetotal angular velocity of precession becomes

= LS +T ≈ g − 1

2v× v. (6.42)

Since g is very close to 2, the Thomas precession explains why the spin-orbit splittingwas initially found to be in agreement with a normal gyromagnetic ratio g = 1, thecharacteristic value for a rotating charged sphere.

If there is also an external magnetic field, this is transformed to the electronrest frame by a Lorentz transformation (4.273), where it leads to an approximateequation of motion for the spin

dS

dt= �×B′ ≈ �×

(

B− v

c× E

)

. (6.43)

Expressing � via Eq. (6.31), this becomes

dS

dt≡ −S×em ≈ eg

2McS×

(

B− v

c×E

)

. (6.44)

This equation defines the frequency em of precession due to the magnetic end elec-tric fields in the rest frame of the electron. Expressing E in terms of the accelerationvia Eq. (6.37), this becomes

dS

dt≈ g

2McS×

[

eB− M

c(v × v)

]

. (6.45)

The acceleration can be expressed in terms of the central Coulomb potential V (r)as

v = −x

r

1

M

dV

dr. (6.46)

the spin precession rate in the electron’s rest frame is

dS

dt=

g

2McS×

(

eB + v × x

r

1

c

dV

dr

)

=g

2McS×

(

eB − 1

McLdV

dr

)

. (6.47)


There exists a simple Hamiltonian operator for spin-orbit interaction HLS(t)], fromwhich this equation can be derived via Heisenberg’s equation (1.279)

S(t) =i

h[S(t), HLS(t)]. (6.48)

The operator is

HLS(r) = −� ·(

B − 1

Mc eLdV

dr

)

= − ge

2McS ·B +

g

2M2c2S · L1

r

dV

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 splittingsassuming a gyromagnetic ratio g ≈ 1 for the electron.

Without the external magnetic field, the angular velocity of precession causedby spin-orbit coupling is is

LS =g

2M2c2L1

r

∂V

∂r. (6.50)

It was realized by Thomas in 1927 that the relativistic motion of the electron changethe factor g as in (6.42) to g − 1, so that the true precession frequency is

= LS +T =g − 1

2M2c2L1

r

∂V

∂r. (6.51)

This implied that the experimental data gave g − 1 ≈ 1, so that g is really twiceas large as expected for a rotating charged sphere. Indeed, the value g ≈ 2. wasindeed predicted by the Dirac theory of the electron.

In Section 12.15 we shall find that the magnetic moment of the electron has a gfactor slightly larger than the Dirac value 2, the relative deviations a ≡ (g − 2)/2being defined as the anomalous magnetic moments . From measurements of theabove precession rate, experimentalists have deduced the values

a(e−) = (115 965.77± 0.35)× 10−8, (6.52)

a(e+) = (116 030± 120)× 10−8, (6.53)

a(µ±) = (116 616± 31)× 10−8. (6.54)

In quantum electrodynamics, the gyromagnetic ratio will receive further smallcorrections, as will be discussed in detail in Chapter 12.

6.1.3 Relativistic Equation of Motion for Spin Vector andThomas 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 asdiscussed in the previous section. There exists a covariant equation of motion for thespin four-vector introduced in Eq. (4.761) which describes this precession. Along aparticle orbit parametrized by a parameter τ , for instance the proper time, we formthe derivative with respect to τ , assuming that the motion proceeds at fixed totalangular momentum:

dSµ

dτ= ǫµνλκJ

νλdpκ

dτ. (6.55)

The right-hand side can be simplified by multiplying it with the trivial expression

1

M2c2gστpσpτ = 1. (6.56)

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 consideringµνλκ = 0123. Then the right-hand side becomes a sum of the four terms

1

M2c2

(

ǫµνλσJνλpσpκp′κ+ǫµνσκJ

νλpλpσp′κ+ǫµσλκJ

νλpνpσp′κ+ǫσνλκJ

νλpσpµp′κ)

, (6.58)

where p′µ ≡ dpµ/dτ . The first term vanishes, since pκp′κ = (1/2)dp2/dτ =(1/2)dM2c2/dτ = 0. The last term is equal to −Sκp

′κpµ/M2c2. Inserting the iden-

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

dSµ

dτ= − 1

M2c2Sλdpλ

dτpµ. (6.59)

Note that on account of this equation, the time derivative dSµ/dτ points in thedirection of pµ.

Let us verify that this equation yields indeed the Thomas precession. Denotingthe derivatives with respect to the time t = γτ by a dot, we can rewrite (6.59) as

S ≡ dS

dt=

1

γ

dS

dτ= − 1

M2c2

(

S0p0 + S · p)

p =γ2

c2(S · v)v, (6.60)

S0 ≡ dS0

dt=

1

c

d

dt(S · v) = γ2

c2(S · v) . (6.61)

We now differentiate Eq. (4.774) with respect to the time using the relation γ =γ3vv/c2, and find

SR = S− γ

γ + 1

1

c2S0v − γ

γ + 1

1

c2S0v − γ3

(γ + 1)21

c4(v · v)S0 v. (6.62)


Inserting here Eqs. (6.60) and (6.61), we obtain

SR =γ2

γ + 1

1

c2(S · v)v− γ

γ + 1

1

c2S0 v − γ3

(γ + 1)2(v · v)S0v. (6.63)

On the right-hand side we return to the spin vector SR using Eqs. (4.773) and(4.776), and find

SR =γ2

γ + 1

1

c2[(SR · v)v− (SR · v)v] = T × SR, (6.64)

with the Thomas precession frequency

T = − γ2

(γ + 1)

1

c2v × v, (6.65)

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 slowparticles, it is given by Eq. (6.45). If the electron moves fast, we transform theelectromagnetic field to the electron rest frame by a Lorentz transformation (4.273),and obtain an equation of motion for the spin becomes

SR=�×B′ = �×[

γ(

B− v

c× E

)

− γ2

γ + 1

v

c

(

v

c·B)

]

. (6.66)

Expressing � via Eq. (6.31), this becomes

SR ≡ −SR ×em =eg

2McSR ×

[

(

B− v

c×E

)

− γ

γ + 1

v

c

(

v

c·B)

]

, (6.67)

which is the relativistic generalization of Eq. (6.44). It is easy to see that theassociated fully covariant equation is

dSµ

dτ=

g

2Mc

[

eF µνSν +1

McpµSλ

d

dτpλ]

=eg

2Mc

[

F µνSν +1

M2c2pµSλF

λκpκ

]

. (6.68)

On the right-hand side we have inserted the relativistic equation of motion (6.21)of a point particle in an external electromagnetic field.

If we add to this the relativistic Thomas precession rate (6.59), we obtain thecovariant Bargmann-Michel-Telegdi equation1

dSµ

dτ=

1

2Mc

[

egF µνSν +g − 2

McpµSλ

d

dτpλ]

=e

2Mc

[

gF µνSν +g − 2

M2c2pµSλF

λκpκ

]

.

(6.69)

1V. Bargmann, L. Michel, and V.L. Telegdi, Phys. Rev. Lett. 2 , 435 (1959).

6.2 Charged Particle in Schrodinger Theory 437

For the spin vector SR in the electron rest frame this implies a change in theelectromagnetic precession rate in Eq. (6.67) to2

dS

dt= emT × S ≡ (em +T)× S (6.70)

with a frequency given by the Thomas equation

emT = − e

Mc

[(

g

2−1 +

1

γ

)

B−(

g

2−1

)

γ

γ+1

(

v

c·B)

v

c−(

g

2− γ

γ+1

)

v

c×E

]

.

(6.71)The contribution of the Thomas precession is the part without the gyromagneticretio g:

T = − e

Mc

[

−(

1− 1

γ

)

B+γ

γ+1

1

c2(v ·B)v +

γ

γ+1

1

cv × E

]

. (6.72)

This is 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 ofthe helicity h ≡ SR · v of an electron, i.e., its component of the spin in the directionof motion. Using the chain rule of differentiation,

d

dt(SR · v) = SR · v +

1

v[SR − (v · SR)v]

d

dtv (6.73)

and inserting (6.70) as well as the equation for the acceleration (6.24), we obtain

dh

dt= − e

McSR⊥ ·

[(

g

2− 1

)

v ×B+(

gv

2c− c

v

)

E

]

. (6.74)

where SR⊥ is the component of the spin vector orthogonal to v. This equation showsthat for a Dirac electron which has g = 2 the helicity remains constant in a purelymagnetic field. Moreover, if the electron moves ultrarelativistically (v ≈ c), thevalue g = 2 makes the last term extremely small, ≈ (e/Mc)γ−2SR⊥ · E, so that thehelicity is almost unaffected by an electric field. The anomalous magnetic momentof the electron, however, changes this a finite value ≈ −(e/Mc)aSR⊥ ·E. This drasticeffect was used to measure the experimental values listed in Eqs. (6.52)–(6.54).

6.2 Charged Particle in Schrodinger Theory

When going over from quantum mechanics to second quantized field theories wefound the rule that a non-relativistic Hamiltonian

H =p2

2m+ V (x) (6.75)

2L.T. Thomas, Phil. Mag. 3 , 1 (1927).


became an operator

H =∫

d3xψ†(x, t)

[

−∇2

2m+ V (x)

]

ψ(x, t), (6.76)

where we have omitted the operator hats, for brevity. With the same rules we seethat the second quantized form of the interacting nonrelativistic Hamiltonian in astatic A(x) field,

H =(p− eA)2

2m+e

cA0, (6.77)

is given by

H =∫

d3xψ†(x, t)

[

− 1

2m

(

∇− ie

cA

)2

+ eA0(x)

]

ψ(x, t). (6.78)

When going to the action of this theory we find

A =∫

dtL =∫

dt∫

d3x[

ψ†(x, t)(

i∂t + eA0)

ψ(x, t)

+1

2mψ†(x, t)

(

∇− ie

cA

)2

ψ(x, t)]

. (6.79)

It is easy to verify that (6.78) reemerges from the Legendre transform

H =∂L

∂ψ(x, t)ψ(x, t)− L. (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

∇ → ∇− ie

cA(x, t),

∂t → ∂t + ieA0(x, t), (6.81)

or covariantly:

∂µ → ∂µ + ie

cAµ(x). (6.82)

This substitution rule has the important property that the gauge invariance ofthe free photon action is preserved by the interacting theory: If we perform thegauge transformation

Aµ(x) → Aµ(x) + ∂µΛ(x), (6.83)

i.e.,

A0(x, t) → A0(x, t) + ∂tΛ(x, t)

A(x, t) → A(x, t)−∇Λ(x, t), (6.84)

6.2 Charged Particle in Schrodinger Theory 439

the action remains invariant if we simultaneously change the fields ψ(x, t) of thecharged particles by a spacetime-dependent phase

ψ (x, t) → e−i ecΛ(x,t)ψ(x, t). (6.85)

Under this transformation, the derivatives of the field change like

∇ψ(x, t) → e−i ecΛ(x,t)

[

∇− ie

c∇Λ(x, t)

]

ψ,

∂tψ → e−i ecΛ(x,t) (∂t − ie∂tΛ)ψ(x, t). (6.86)

The modified derivatives appearing in the action have therefore the following simpletransformation law:

(

∇− ie

cA

)

ψ(x, t) → e−i ecΛ(x,t)

(

∇− ie

cA

)

ψ(x, t),(

∂t + ie

cA0)

ψ(x, t) → e−i ecΛ(x,t)

(

∂t + ieA0)

ψ(x, t). (6.87)

These combinations of derivatives and gauge fields are called covariant derivatives

and written as

Dψ(x, t) ≡(

∇− ie

cA

)

ψ(x, t)

Dtψ(x, t) ≡(

∂t + ieA0)

ψ(x, t) (6.88)

or, in four-vector notation,

Dµψ(x) =(

∂µ + ie

cAµ

)

ψ(x). (6.89)

Here the adjective covariant does not refer to the Lorentz group but to the gaugegroup. It records the fact that Dµψ transforms under local gauge changes (6.81) ofψ in the same way as ψ itself in (6.85):

Dµψ(x) → e−i ecΛ(x)Dµψ(x). (6.90)

With the help of this covariant derivative any action, which is invariant under globalphase changes by a constant phase angle

ψ(x) → e−iφψ(x), (6.91)

can easily be made invariant under local gauge transformations (6.83). We merelyhave to replace all derivatives by covariant derivatives (6.89) and add the gaugeinvariant photon action (4.234).


6.3 Charged Relativistic Fields

6.3.1 Scalar Field

The Lagrangian density of a free charged scalar field is from Eqs (4.162)

L = ∂µφ∗(x)∂µφ(x)−M2φ∗(x)φ(x) (6.92)

For a relativistic scalar field of charge e, the interacting Lagrange density is astraightforward generalization of the Schrodinger case (6.79):

L = [Dµφ(x)]∗Dµφ(x)−M2φ∗(x)φ(x)

=[

∂µ − ie

cAµ(x)

]

φ(x)[

∂µ + ie

cAµ(x)

]

φ(x)−M2φ∗(x)φ(x) (6.93)

6.3.2 Dirac Field

The Lagrangian densities of a free charged spin-1/2 field is from Eq. (4.498):

L = ∂µφ∗(x)∂µφ(x)−M2φ∗(x)φ(x) (6.94)

and

L(x) = ψ(x) (i/∂ −m)ψ(x). (6.95)

If the particle carries a charge e, we replace in this Lagrangian:

/∂ = γµ∂µ → γµ(

∂µ + ie

cAµ

)

=(

/∂ + ie

c/A)

≡ /D . (6.96)

In this way we arrive at the Lagrangian of quantum electrodynamics (QED)

L(x) = ψ(x) (i/D −m)ψ(x)− 1

4F 2µν . (6.97)

The classical field equations can easily be found by extremizing the action andvariation with respect to all fields which gives

δAδψ(x)

= (i/D −M)ψ(x) = 0 (6.98)

δAδAµ(x)

= ∂νFνµ(x)− 1

cjµ(x) = 0. (6.99)

with the current density

jµ(x) ≡ e c ψ(x)γµψ(x). (6.100)

Equation (6.99) is the Maxwell equation for the electromagnetic field around a clas-sical four-dimensional vector current jµ(x):

∂νFνµ(x) =

1

cjµ(x). (6.101)

6.4 Pauli Equation from Dirac Theory 441

In the Lorentz gauge ∂µAµ(x) = 0, this equation reads simply

−∂2Aµ(x) =1

cjµ(x). (6.102)

The current jµ combines the charge density ρ(x) and the current density j ofparticles of charge e in a four-vector

jµ = (cρ, j) . (6.103)

In terms of electric and magnetic fields Ei = F i0, Bi = −F jk, the field equations(6.101) turn into the Maxwell equations

∇ · E = ρ = eψγ0ψ = eψ†ψ

∇×B− E =1

cj =

e

cψ ψ. (6.104)

The first is Coulomb’s law, the second Ampere’s law in the presence of charges andcurrents.

Note that the physical units employed here differ from those used in many booksof classical electrodynamics3 by the absence of a factor 1/4π on the right-hand side.The Lagrangian used in those books is

L(x) = − 1

8πF 2µν(x)−

1

cjµ(x)Aµ(x) (6.105)

=1

4π

[

E2 −B2]

(x)−[

ρφ− 1

cj ·A

]

(x),

which leads to Maxwell’s field equations

∇ · E = 4πρ

∇×B =4π

cj. (6.106)

The form employed conventionally in quantum field theory arises from this by re-placing A →

√4πA and e →

√4πe. The charge of the electron in our units has

therefore the numerical value

e = −√4πα ≈ −

√

4π/137 (6.107)

rather than e = −√α.

6.4 Pauli Equation from Dirac Theory

It is instructive to take the Dirac equation (6.98) to a two-component form corre-sponding to (4.561) and (4.573), and further to (4.579). Due to the fundamental

3See for example J.D. Jackson, Classical Electrodynamics, Wiley and Sons, New York, 1967.


nature of the equations to be derived we shall not work with natural units in this sec-tion but carry all fundamental constants along explicitly. As in (4.580), we multiply(6.98) by (ih/D −Mc) and work out the product

(ih/D −Mc) (ih/D +Mc) =[

iγµ(

h∂µ + ie

cAµ

)

+Mc] [

iγµ(

h∂µ + ie

cAµ

)

−Mc]

(6.108)

We now use the relation

γµγν =1

2(γµγν + γνγµ) +

1

2(γµγν − γνγµ) = gµν − iσµν , (6.109)

with σµν from (4.514), and find

γµγν(

h∂µ + ie

cAµ

)(

∂µ + ie

cAµ

)

= gµν(

h∂µ + ie

cAµ

)(

∂µ + ie

cAµ

)

− i

2σµν [h∂µ + i

e

cAµ, h∂ν + i

e

cAν ]

=(

h∂µ + ie

cAµ

)2

+1

2

eh

cσµνFµν . (6.110)

Thus we obtain as a generalization of Eqs. (4.579) the Pauli equation[

−(

h∂µ + ie

cAµ

)2

− 1

2

eh

cσµνFµν −M2c2

]

ψ(x) = 0, (6.111)

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 a term

1

2σµνFµν = −� ·H+ i� · E. (6.112)

where in the chiral and Dirac representations

� =

(

−� 00 �

)

, �D =

(

0 �

� 0

)

, (6.113)

respectively. Thus, in the chiral representation, Eq. (6.111) decomposes into twoseparate two-component equations for the upper and lower spinor components ξ(x)and η(x) in ψ(x):

[

−(

h∂µ + ie

cAµ

)2

+ � · (H± iE)−M2c2]{

ξ(x)η(x)

}

= 0. (6.114)

In the nonrelativistic limit where c → ∞, we remove the fast oscillations fromξ(x), setting ξ(x) ≡ e−iMc2t/hΨ(x, t)/

√2M as in (4.153), and find for Ψ(x, t) the

nonrelativistic equation[

i∂t +h2

2M

(

∇− ie

chA

)2

+e

2Mc� ·H− eA0(x)

]

Ψ(x, t) = 0. (6.115)

6.5 Relativistic Wave Equations in Coulomb Potential 443

This corresponds to a magnetic interaction energy

Hmag = − eh

2Mc� ·H. (6.116)

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

Hmag = −� ·H. (6.117)

A Dirac particle with has therefore a magnetic moment

� =e

Mc

h�

2= 2

e

2McS, (6.118)

where S = �/2 is the spin matrix. Experimentally, one parametrizes a magneticmoment of a fundamental particles as

� = ge

2McS (6.119)

with the gyromagnetic ratio g, which is equal to unity for a uniformly charged sphere.According to the Dirac theory, an electron has a gyromagnetic ratio

ge|Dirac = 2. (6.120)

The experimental value is very close to this. The small deviation called anomalous

magnetic moment will be explained in Chapter 12.The nonrelativistic Pauli equation (6.115) could also have been obtained by intro-

ducing a minimal coupling directly into the nonrelativistic two-component equation(4.576). This changes i∂t → i∂t − eA0 and (� ·∇)2 → [� · (∇− ieA)]2. The latteris worked out as in (6.121), yielding

[� · (∇− ieA)]2 =[

δij + iǫijkσk]

(∇i − ieAi)(∇j − ieAj)

= (∇− ieA)2 + iǫijkσk(∇i − ieAi)(∇j − ieAj) + e� ·H, (6.121)

so that Eq. (4.578) goes over into (6.115), after reinserting all fundamental constants,

6.5 Relativistic Wave Equations in Coulomb Potential

It is now easy to write down field equations for a Klein-Gordon and a Dirac fieldin the presence of an external Coulomb potential of charge Ze. In natural units wehave

VC(x) = −Zαr, r =

√x2, (6.122)

corresponding to a four-vector potential

eAµ(x) = (VC(x, 0), 0). (6.123)


Since this does not depend on time, we can consider the wave equations for wavefunctions φ(x) = e−iEtφE(x) and ψ(x) = e−iEtψE(x), and find the time-independentequations

(E2 +∇2 −M)φE(x) = 0 (6.124)

and

(γ0E + i ·∇−M)ψE(x) = 0. (6.125)

In these equations we simply perform the minimal substitution

E → E +Zα

r. (6.126)

The energy-eigenvalues obtained from the resulting equations can be compared withthose of hydrogen-like atoms. The velocity of an electron in the ground state is ofthe order αZc. Thus for rather high Z, the electron has a relativistic velocity andthere must be corrections significant deviations from Schrodinger theory. We shallsee that the Dirac equation in an external field reproduces quite well a number offeatures resulting from the relativistic motion.

6.5.1 Reminder of Schrodinger Equation with CoulombPotential

The time-independent Schrodinger equation reads

(

− 1

2M∇

2 − Zα

r− E

)

ψE(x) = 0. (6.127)

The Laplacian may be decomposed into radial and angular parts by writing

∇2 =

∂2

∂r2+

2

r

∂

∂r− L2

r2, (6.128)

where L = x× p are the differential operators for the generators of angular momen-tum [the spatial part of Li = L23 of (4.95)]. Then (6.127) reads

(

− ∂2

∂r2− 2

r

∂

∂r+

L2

r2− 2ZαM

r− 2ME

)

ψE(x) = 0. (6.129)

The eigenstates of L2 are the spherical harmonics Ylm(θ, φ), which diagonalize alsothe third component of L, with the eigenvalues

L2Ylm(θ, φ) = l(l + 1)Ylm(θ, φ),

L3Ylm(θ, φ) = mYlm(θ, φ). (6.130)


The wave functions may be factorized into a radial wave function Rnl(r) and aspherical harmonic:

ψnlm(x) = Rnl(r)Ylm(θ, φ). (6.131)

Explicitly,

Rnl(r) =1

a1/2B n

1

(2l + 1)!

√

√

√

√

(n + l)!

(n− l − 1)!(6.132)

×(2r/naB)l+1e−r/naBM(−n + l + 1, 2l + 2, 2r/naB)

=1

a1/2B n

√

√

√

√

(n− l − 1)!

(n+ l)!e−r/naB(2r/naB)

l+1L2l+1n−l−1(2r/naB),

where aB is the Bohr radius, which in natural units h = c = 1 is equal to

aB =1

ZMα, (6.133)

For a hydrogen atom with Z = 1, this is about 1/137 times the Compton wavelengthof the electron λe ≡ h/Mec. The classical velocity of the electron on the lowest Bohrorbit is vB = α c. Thus it is almost nonrelativistic, which is the reason what theSchrodinger equation explains the hydrogen spectrum quite well. The functionsM(a, b, z) are confluent hypergeometric functions or Kummer functions, defined bythe power series

M(a, b, z) ≡ F1,1(a, b, z) = 1 +a

bz +

a(a + 1)

b(b+ 1)

z

z!+ . . . . (6.134)

For b = −n they are polynomial related to the Laguerre polynomials4 Lµn(z) by

Lµn(z) ≡

(n + µ)!

n!µ!M(−n, µ + 1, z), (6.135)

The radial wave functions are normalized to∫ ∞

0drRnrl(r)Rn′

rl(r) = δnrn′r. (6.136)

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 energyeigenvalues depend on are n in the well-known way:

En = −Z2Mα2

2n2. (6.137)

4I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.970 (our definition differs from that inL.D. Landau and E.M. Lifshitz, Quantum Mechanics , Pergamon Press, New York, 1965, Eq. (d.13):Our Lµ

n = (−)µ/(n+ µ)!Ln+µµ|L.L.).


The number α2M/2 is the Rydberg-constant:

Ry =α2M

2≈ 27.21

2eV ≈ 3.288× 1015Hz. (6.138)

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

|ψn00(0)| =1√π

√

ZMα

n

3

=1√π

√

1

naB

3

. (6.139)

6.5.2 Klein-Gordon Field in Coulomb Potential

After the substitution (6.126) into (6.124), we find the Klein-Gordon equation inthe Coulomb potential (6.122):

[

(

E +Zα

r

)2

+∇2 −M2

]

φE(x) = 0. (6.140)

With the angular decomposition (6.128), this becomes

[

− ∂2

∂r2− 2

r

∂

∂r+

L2 − Z2α2

r2− 2ZαE

r− (E2 −M2)

]

φE(x) = 0. (6.141)

The solutions of this equation can be obtained from those of the nonrelativisticSchrodinger equation (6.129) by replacing

L2 → L2 − Z2α2, (6.142)

α → αE

M, (6.143)

E → E2 −M2

2M. (6.144)

The replacement (6.142) is done most efficiently by defining the eigenvalues l(l +1)− Z2α2 of the operator L2 − Z2α2 by analogy with those of L2 as λ(λ+ 1):

l(l + 1)− Z2α2 ≡ λ(λ+ 1). (6.145)

Then the quantum number l of the Schrodinger wave functions is simply replacedby λl = l − δl, where

δl = l +1

2−[

(

l +1

2

)2

− Z2α2

]1/2

=Z2α2

2l + 1+O(α4). (6.146)

The other solution of relation (6.145) with the opposite sign in front of the squareroot is unphysical since the associated wave functions are too singular at the origin


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

nr + λ+ 1 = nr + l + 1− δl = n− δl. (6.147)

This leads to the equation for the energy eigenvalues

Enl2 −M2

2M= −Z

2Mα2

2

Enl2

M2

1

(n− δl)2, (6.148)

with the solution

Enl = ± M√

1 + Z2α2/(n− δl)2

= ±M[

1− (Zα)2

2n2+

3

8

(Zα)4

n4− (Zα)4

n3(2l + 1)+O(Z6α6)

]

. (6.149)

The first two terms correspond to the Schrodinger energies (6.137) (plus rest energy),the next two are relativistic corrections. The first of these breaks the degeneracybetween the levels of the same n and different l, generated in the Schrodinger theoryby the Lentz-Runge vector [O(4)-invariance].

The correction terms become large for large central charge Z. In particular, thelowest energy and successively the higher ones become complex for central chargesZ > 137/2. The physical reason for this is that the large potential gradient near theorigin can create pairs of particles from the vacuum. This phenomenon can only beproperly 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 Coulomb Potential

After the substitution (6.126) into (6.125), we find the Dirac equation in theCoulomb potential (6.122):

[(

E +Zα

r

)

γ0 + i ·∇−M]

ψE(x) = 0. (6.150)

In order to find the energy spectrum it is useful to establish contact with theKlein-Gordon case. Multiplying (6.150) by the operator

(

E +Zα

r

)

γ0 + i ·∇+M, (6.151)

we obtain[

(

E +Zα

r

)2

+∇2 − iγ0 ·∇Zα

r−M2

]

ψE(x) = 0., (6.152)


In the chiral representation, 4×4 -matrix γ0 = � has a diagonal block form (4.557).We therefore decompose

ψE(x) =

(

ξE(x)ηE(x)

)

, (6.153)

and find the equation for the upper two-component spinors

[

(

E +Zα

r

)2

+∇2 + i� ·∇Zα

r−M2

]

ξE(x) = 0, (6.154)

and the same equation for ηE(x) with i replaced by −i. Expressing ∇2 via (6.128)

and writing ∇ 1/r = −x/r2, we obtain the differential equation

[

−(

∂2

∂r2+

2

r

∂

∂r

)

+L2 − Z2α2 + iZα�x

r2− 2ZαE

r− (E2 −M2)

]

ξE(x) = 0,

(6.155)and a corresponding equation for ηE(x).

Due to the rotation invariance of � · x, the total angular momentum

J = L+ S = L+�

2(6.156)

commutes with the differential operator in (6.155). Thus we can diagonalize J2 andJ3 with eigenvalues j(j + 1) and m. For a fixed value of j = 1

2, 1, 3

2, . . . , the orbital

angular momentum can have the value l+ = j + 12and l− = j− 1/2. The two states

have opposite parities. The operator � · x is a pseudoscalar and must necessarilychange the parities. Since the square of � · x is the unit matrix, its eigenvalues mustbe ±1. Moreover, the unit vector x changes necessarily the parity and thus l. Thus,in the two-component Hilbert space of fixed quantum numbers j and m with orbitalangular momenta l = l± = j ± 1/2, the diagonal matrix elements vanish

〈jm,+|� · x|jm,+〉 = 0, 〈jm,−|� · x|jm,−〉 = 0. (6.157)

For the off-diagonal elements we easily calculate

〈jm,+|� · x|jm,−〉 = 1, 〈jm,−|� · x|jm,+〉 = −1. (6.158)

The central parentheses in (6.155) have therefore the matrix elements

(

L2−Z2α2 ± iZαx)

=

(

(j + 12)(j + 3

2)− Z2α2 ±iZα

±iZα (j − 12)(j + 1

2)− Z2α2

)

.

(6.159)In analogy with the Klein-Gordon case, we denote the eigenvalues of this matrix byλ(λ+ 1). The corresponding values of λ are found to be

λj+ =

[

(

j +1

2

)2

− Z2α2

]1/2

, λj− = λj+ − 1. (6.160)


These may be written as

λj± = j ± 1

2− δj ≡ l± − δj , (6.161)

where l± ≡ ±12is the orbital angular momentum, and

δj ≡ j +1

2−[

(

j +1

2

)2

− Z2α2

]1/2

=Z2α2

2j + 1+O(Z4α4). (6.162)

When solving Eq. (6.155), the solutions consist, as in the nonrelativistic hydrogenatom, of an exponential factor multiplied by a polynomial of degree nr which is theradial quantum number. It is related to the quantum numbers of spin and orbitalangular momentum, and to the principal quantum number n, by

nr + λj± + 1 = nr + l± + 1− δl = n− δj . (6.163)

In terms of δj , the energies obey the same equation as in (6.149), so that we obtain

Enj = ± M√

1 + Z2α2/(n− δj)2

= ±M[

1− (Zα)2

2n2+

3

8

(Zα)4

n4− (Zα)4

n3(2j + 1)+O(Z6α6)

]

. (6.164)

The condition nr ≥ 0 implies that

j ≤ n− 32

j ≤ n− 12

forλj+ = j + 1

2− δj ,

λj− = j − 12− δj .

(6.165)

For n = 1, 2, 3, . . . , the total angular momentum runs through j = 12, 32, . . . , n− 1

2.

The spectrum of the hydrogen atom according to the Dirac theory is shown inFig. 6.1. As a remnant of the O(4)-degeneracy of the levels with l = 0, 1, 2, . . . , n−1and fixed n in the Schrodinger spectrum, there is now a twofold degeneracy of levelsof equal n and j, with adjacent l-values, which are levels of opposite parity. Anexception is the highest total angular momentum j = n − 1/2 at each n whichoccurs 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 theLamb shift. Its explanation is one of the early triumphs of quantum electrodynamics,which will be discussed in detail in Chapter .

As in the Klein-Gordon case, there are complex energies, here for Z > 137, withS1/2 being the first level to become complex.

5Recall the notation in atomic physics for an electronic state: n2S+1LJ , where n is the principalquantum number, L the orbital angular momentum, J the total angular momentum, and S thetotal spin. In a one-electron system such as the hydrogen atom, the trivial superscript 2S + 1 = 2may be omitted.


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 struc-ture. States with the same n and l but with different j are split apart by the forthterm in Eq. (6.164)−MZ4α4n3/(2j+1). For the states 2P1/2 and 2P3/2, the splittingis

∆fineE2P =Z4α2

32α2M. (6.166)

In a hydrogen atom, this is equal to

∆fineE2P = 3.10.95 GHz. (6.167)

Thus it is roughly of the order of the splitting caused by the interaction of the mag-netic moment of the electron with that of the proton, the so-called hyperfine-splitting.This is for 2S1/2, 2P 1/2, and 2P 3/2 levels approximately equal to 1, 1/8, 1/24, 1/60times 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 notonly interested in the spectrum but also in the wave functions it is advantageousto solve directly the Dirac equation (6.150) 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.


representation (4.544). Multiplying (6.150) by γ0 and inserting for γ0 = � theDirac matrix (4.556), we obtain

E −M +Zα

ri� ·∇

i� ·∇ E +M +Zα

r

(

ξE(x)ηE(x)

)

= 0. (6.168)

This is of course just the time-independent version of (4.573) extended by theCoulomb potential according to the minimal substitution rule (6.126). To lowestorder in α, the lower spinor is related to the upper by

ηE(x) ≈ −i� ·∇2M

ξE(x). (6.169)

We may take care of rotational symmetry of the system by splitting the spinor wavefunctions into radial and angular parts

ψE(x) =

iGjl(r)

rylj,m(θ, φ)

Fjl(r)

r� · x ylj,m(θ, φ)

, (6.170)

where ylj,m(θ, φ) denote the spinor spherical harmonics. They are composed fromthe ordinary spherical harmonics Ylm(θ, φ) and the basis spinors χ(s3) of (4.443) viaClebsch-Gordan coefficients (see Appendix 4C):

ylj,m(θ, φ) = 〈j,m|l, m′; 1

2, s3〉Ylm′(θ, φ)χ(s3). (6.171)

The derivation is given in Appendix 6A.The explicit form of the spinor spherical harmonics (6.171) is for l = l±:

yl+j,m(θ, φ) =

1√2l+ + 1

( √l+ −m+ 1

2Yl+,m−

1

2

(θ, φ)

−√l+ +m+ 1

2Yl+,m+ 1

2

(θ, φ)

)

, (6.172)

yl−j,m(θ, φ) =

1√2l− + 1

( √l− +m+ 1

2Yl−,m−

1

2

(θ, φ)√l− −m+ 1

2Yl−,m+ 1

2

(θ, φ)

)

. (6.173)

On these eigenfunctions, the operator L · � has the eigenvalues

L · � yl±j,m(θ, φ) = −(1 + κ±)yl±j,m(θ, φ), (6.174)

with

κ± = ∓(j +1

2), j = l ± 1

2. (6.175)

We can now go from Eqs. (6.168) to radial differential equations by using thetrivial identity,

i� ·∇f(r)

ryl+l,m ≡ � · x

r2(� · x) i (� ·∇)

f(r)

ryl+l,m, (6.176)


and the algebraic relation Eq. (4.461) in the form

(� · a)(� · b) = i� · (a× b) + i(a · b), (6.177)

to bring the right-hand side to

� · xr2

(ir∂r − i� · L) f(r)ryl+l,m =

[

i∂rf(r)

r− i (1 + κ)

f(r)

r2

]

� · x yl+l,m. (6.178)

In this way we find the radial differential equations for the functions Fjl(r) andGjl(r):

(

E −M +Zα

r

)

Gjl(r) = − d

drFjl(r)∓ (j + 1/2)

1

rFjl(r), (6.179)

(

E +M +Zα

r

)

Fjl(r) =d

drGjl(r)∓ (j + 1/2)

1

rGjl(r). (6.180)

To solve these, dimensionless variables ρ ≡ 2r/λ are introduced, with λ =1/√M2 − E2, writing

F (r) =√

1−E/Me−ρ/2(F1 − F2)(ρ), G(r) =√

1 + E/Me−ρ/2(F1 + F2)(ρ). (6.181)

The functions F1,2(ρ) satisfy a degenerate hypergeometric differential equation ofthe form

[

ρd2

dρ2+ (b− ρ)

d

dρ− a

]

F (a, b; ρ) = 0, (6.182)

and the solutions are

F2(ρ) = ρlF (γ − ZαEλ, 2γ + 1; ρ),

F1(ρ) = ρlγ − ZαEλ

−1/λ+ ZαEλF (γ + 1− ZαEλ, 2γ + 1; ρ). (6.183)

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− Z2α2/2,As an example, we write down explicitly the ground state wave functions of the

1S1/2 state:

ψ1S1/2,± 1

2

=

√

√

√

√

(2MZα)3

4π

1 + γ

2Γ(1 + 2γ)

e−mZαr

(2MZα)1−γ

1 00 1

i1−γZα

cos θ i1−γZα

sin θe−iφ

i1−γZα

sin θeiφ −i1−γZα

cos θ

.

(6.184)

The first column is for m = 1/2, the second for m = −1/2. For small α, Einstein’sgamma parameter has the expansion γ = 1 − Z2α2/2, and we see that in theα → 0, the upper components of the spinor wave functions tend to the nonrelativistic

Schrodinger wave function 2√

(ZαM)3/4πe−ρ multiplied by Pauli spinors (4.443).In general,

ξlj,m(x) = 〈j,m|l, m; 1

2, s3〉ψnlm(x)χ(s3). (6.185)

The lower, small components vanish.

6.6 Green Function in External Electromagnetic Field 453

6.6 Green Function in External Electromagnetic Field

An important physical quantity of a propagating field is the Green function, definedas the solution of the equation of motion with a δ-function source term [recall (1.314)and (2.403)]. For external electromagnetic fields which are constant or plane waves,this Green function can be calculated exactly.

6.6.1 Scalar Field in Constant Electromagnetic Field

For a scalar field, the Green function G(x, x′) is defined by the inhomogeneousdifferential equation

(−∂2 −M2)G(x, x′) = iδ(4)(x− x′), (6.186)

whose solution can immediately be expressed as a Fourier integral:

G(x−x′) =∫ d4p

(2π)4i

p2 −M2 + iηe−ip(x−x′) =

∫ ∞

0dτ∫ d4p

(2π)4e−ip(x−x′)+iτ(p2−M2+iη).

(6.187)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 func-tion in the presence of a static electromagnetic field, which obeys the more compli-cated differential equation

{

[i∂ − eA(x)]2 −M2}

G(x, x′) = iδ(4)(x− x′), (6.188)

for which a Fourier decomposition is no longer helpful. For a constant and anoscillating electromagnetic field, however, this equation can be solved by an elegantmethod due to Fock and Schwinger [1].

Generalizing the right-hand side of (6.187), we find the representation

G(x− x′) =∫ ∞

0dτ 〈x|eiτ [(i∂−eA)2−M2+iη]|x′〉. (6.189)

The integrand contains the time-evolution operator associated with the Hamiltonianoperator

H(x, i∂) ≡ − (i∂ − eA)2 +M2. (6.190)

This is the Schroedinger representation of the operator

H = H(x, p) = −P 2 +M2, (6.191)

where Pµ ≡ pµ − eAµ(x) is the canonical momentum in the presence of electromag-netism.

We shall calculate the evolution operator in (6.189) by introducing time-dependent Heisenberg position and momentum operators obeying the Heisenberg-Ehrenfest equations of motion [recall (1.276)]:

dxµ(τ)

dτ= i

[

H, xµ(

τ)] = 2P µ(τ) (6.192)

dP µ(τ)

dτ= i

[

H, P µ(τ)]

= 2eF µν(x(τ))P

ν(τ) + ie∂νFµν(x(τ)). (6.193)


In a constant field where F µν(x(τ)) is a constant matrix F µ

ν , the last term inthe second equation is absent and we find directly the solution

P µ(τ) =(

e2eFτ)µ

νP ν(0). (6.194)

where the matrix(

e2eFτ)µ

νis defined by its formal power series expansion

(

e2eFτ)µ

ν= δµν + 2eF µ

ντ + 4e2F µλF

λντ 2

2+ . . . . (6.195)

Inserting (6.194) into Eq. (6.192), we find the time-dependent operator xµ(τ):

xµ(τ)− xµ(0) =

(

e2eFτ − 1

eF

)µ

νP ν(0). (6.196)

where the matrix on the right-hand side is again defined by its formal power series

(

e2eFτ − 1

eF

)µ

ν= 2τ + e2F µ

λFλν(2τ)3

3!+ . . . . (6.197)

Note that division by eF is not a matrix multiplication by the inverse of the matrixeF but indicates the reduction of the expansion powers of eF by one unit. This isdefined also if eF does not have an inverse.

We can invert Eq. (6.196) to find

P ν(0) =1

2

[

eFe−eFτ

sinh eFτ

]µ

ν[x(τ)− x(0)]ν , (6.198)

and, using (6.194),

P ν(τ) = Lµν(eFτ) [x(τ)− x(0)]ν , (6.199)

with the matrix

Lµν(eFτ) ≡

1

2

[

eF µν

eeFτ

sinh eFτ

]µ

. (6.200)

By squaring (6.199) we obtain

P 2(τ) = [x(τ)− x(0)]µKµν(eFτ) [x(τ)− x(0)]ν , (6.201)

where

Kµν(eFτ) = Lλ

µ(eFτ)Lλν(eFτ). (6.202)

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

Kµν(eFτ) = Lµ

λ(−eFτ)Lλν(eFτ) =

1

4

[

e2F 2

sinh2 eFτ

]

µ

ν

. (6.203)


The commutator between two operators x(τ) at different times is

[xµ(τ), xν(0)] = i

(

e2eFτ − 1

eF

)µ

ν, (6.204)

and

[

xµ(τ), xν(0)]

+[

xν(τ), xµ(0)

]

= i

(

e2eFτ − 1

eF

)µ

ν+ i

(

e2eFT τ − 1

eF T

)µ

ν

= i

(

e2eFτ − e−2eFτ

eF

)µ

ν= 2i

[

sinh 2eFτ

eF

]µ

ν, (6.205)

With the help of this commutator, we can expand (6.201) in such a way in powersof operators x(τ) and x(0), that the later operators x(τ) come to lie to the left ofthe earlier operators x(0) as follows:

H(x(τ), x(0); τ) = −xµ(τ)Kµν(eFτ)xν(τ)− xµ(0)Kµ

ν(eFτ)xν(0)

+ 2xµ(τ)Kµν(eFτ)xν(0)−

i

2tr [eF coth eFτ ] +M2. (6.206)

Given this form of the Hamiltonian operator it is easy to calculate the time evolutionamplitude in Eq. (6.189):

〈x, τ |x′ 0〉 ≡ 〈x|e−iHτ |x′〉. (6.207)

It satisfies the differential equation

i∂τ 〈x, τ |x′ 0〉 ≡ 〈x|H e−iHτ |x′〉 = 〈x|e−iHτ[

eiHτH e−iHτ]

|x′〉= 〈x, τ |H(x(τ), P (τ))|x′, 0〉. (6.208)

Replacing the operator H(x(τ), P (τ)) by H(x(τ), x(0); τ) of Eq. (6.206), the matrixelements on the right-hand side can immediately be evaluated using the property

〈x, τ |x(τ) = x〈x, τ |, x(0)|x′, 0〉 = x′|x′, 0〉, (6.209)

and the differential equation (6.210) becomes

i∂τ 〈x, τ |x′ 0〉 ≡ H(x, x′; τ)〈x, τ |x′ 0〉, (6.210)

or〈x, τ |x′ 0〉 = C(x, x′)E(x, x′; τ) ≡ C(x, x′)e−i

∫

dτ H(x,x′;τ). (6.211)

The prefactor C(x, x′) contains a possible constant of integration in the exponentwhich may have an arbitrary dependence on x and x′. The following integrals areneeded:

∫

dτ K(eFτ) =1

4

∫

dτe2F 2

sinh2 eFτ= −1

4eF coth eFτ, (6.212)


and∫

dτ tr [eF coth eFτ ] = tr logsinh eFτ

eF= tr log

sinh eFτ

eFτ+ 4 log τ. (6.213)

these results following again from a Taylor expansion of both sides. The exponentialfactor E(x, x′; τ) in (6.211) becomes, therefore,

E(x, x′; τ)=1

τ 2exp

{

− i

4(x−x′)µ [eF coth eFτ ]µ

ν(x−x′)ν−iM2τ−1

2tr log

sinh eFτ

eFτ

}

.

(6.214)The last term produces a prefactor

det −1/2

(

sinh eFτ

eFτ

)

. (6.215)

The time-independent integration constant is fixed by the differential equationwith respect to x:

[i∂µ−eAµ(x)] 〈x, τ |x′ 0〉 = 〈x|Pµe−iHτ |x′〉 = 〈x|e−iHτ

[

eiHτ Pµe−iHτ

]

|x′〉= 〈x, τ |Pµ(τ)|x′ 0〉, (6.216)

which becomes, after inserting (6.199):

[i∂µ−eAµ(x)] 〈x, τ |x′ 0〉 = Lµν(eFτ)(x− x′)ν〈x, τ |x′ 0〉, (6.217)

Calculating the partial derivative we find

i∂µ〈x, τ |x′ 0〉= [i∂µC(x, x′)]E(x, x′; τ) + C(x, x′)[i∂µE(x, x

′; τ)]

= [i∂µC(x, x′)]E(x, x′; τ) + C(x, x′)

1

2[eF coth eFτ ]µ

ν(x− x′)νE(x, x′; τ).

Subtracting from this eAµ(x)〈x, τ |x′ 0〉, and inserting (6.211), the right-hand side of(6.217) is equal to [i∂µC(x, x

′)]E(x, x′; τ) plus

{

Lµν(eFτ)(x− x′)ν −

1

2[eF coth eFτ ]µ

ν(x− x′)ν

}

C(x, x′)E(x, x′; τ). (6.218)

Inserting Eq. (6.200), this simplifies to

e

2Fµ

ν(x− x′)νC(x, x′)E(x, x′; τ), (6.219)

so that C(x, x′) satisfies the time-independent differential equation[

i∂µ − eAµ(x)− e

2F µ

ν(x− x′)ν]

C(x, x′) = 0. (6.220)

This is solved by

C(x, x′) = C exp{

−ie∫ x

x′

dξµ[

Aµ(ξ) +1

2Fµ

ν(ξ − x′)ν

]}

. (6.221)


The contour of integration is arbitrary since A′(ξ) ≡ Aµ(ξ) +12Fµ

ν(ξ − x′)ν has avanishing curl:

∂µA′ν(x)− ∂νA

′µ(x) = 0. (6.222)

We can therefore choose the contour to be a straight line connecting x′ and x, inwhich case the F -term does not contribute in (6.221), since dξµ points in the samedirection of xµ − x′µ as ξµ − x′µ and Fµν is antisymmetric. Hence we may write fora straight-line connection

C(x, x′) = C exp[

−ie∫ x

x′

dξµAµ(ξ)]

. (6.223)

The normalization constant C is finally fixed by the initial condition

limτ→0

〈x, τ |x′ 0〉 = δ(4)(x− x′), (6.224)

which requires

C = − i

(4π)2. (6.225)

Collecting all terms we obtain

〈x, τ |x′ 0〉 = − i

(4πτ)2exp

[

−ie∫ x

x′

dξµAµ(ξ)]

det −1/2

(

sinh eFτ

eFτ

)

× exp{

− i

4(x−x′)µ [eF coth eFτ ]µ

ν(x−x′)ν−iM2τ}

. (6.226)

For zero field, this reduces to the relativistic free-particle amplitude

〈x, τ |x′ 0〉 = − i

(4πτ)2exp

[

− i

2

(x− x′)2

2τ− iM2

]

. (6.227)

According to relation (6.189), the Green function of the scalar field is given bythe integral

G(x, x′) =∫ ∞

0dτ 〈x, τ |x′ 0〉. (6.228)

The functional trace of (6.226)

Tr〈x, τ |x 0〉 = V∆ti

(4πτ)2eEτ

sinh eEτ(6.229)

will be needed below. Due to tranlation invariance in spacetime, it carries a factortotal spatial volume V × total time ∆t of the universe.

The result (6.229) can be checked by a more elementary derivation [96]. Welet the constant electric field point in the z-direction, and represent it by a vectorpotential to have only a zeroth component

A3(x) = −Ex0. (6.230)


Then the Hamiltonian (6.191) becomes

H = −p20 + p2⊥ + (p3 + eEx0)

2 +M2, (6.231)

where p⊥ are the two-dimensional momenta in the xy-plane. Using the commutationrule [p0, x0] = i, this can be rewritten as

H = e−ip0p3/eEH ′eip0p3/eE (6.232)

where H ′ is the sum of two commuting Hamiltonians:

H ′ = −(p20 − e2E2x20) + p2⊥ +M2 ≡ HωE

+ H⊥. (6.233)

The first is a harmonic Hamiltonian with imaginary frequency ωE = ieE and anenergy 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 comlete setof momentum states on either side of (6.207), so that the functional trace becomes

Tr〈x, τ |x 0〉 =∫

d4x∫

d4p

(2π)4

∫

d4p′

(2π)4e−i(p−p′)x〈p|e−iτ(HωE

+H⊥)|p′〉 (6.234)

The matrix elements are

〈p|e−iτH |p′〉 = e−ip0(x0+p3/eE)〈p0|e−isHωE |p′0〉e−iτ(p2⊥+M2−iη)eip

′0(x0+p′3/eE)

× (2π)2δ(2)(p⊥ − p′⊥)(2π)δ(p

3 − p′3). (6.235)

Inserting this into (6.234) and performing the integrals over the spatial parts of p′

appearing in the δ-functions of (6.235) yields

Tr〈x, τ |x 0〉 = V∫

dx0

∫

d2p⊥(2π)2

e−iτ(p2⊥+M2−iη)

×∫

dp0dp3dp′0

(2π)3e−i(p0−p′

0)(x0+p3/eE)〈p0|e−isHωE |p′0〉, (6.236)

which can be reduced to

Tr〈x, τ |x 0〉= V∆t−i4πτ

e−iτ(M2−iη) eE

2π

[

∫

dp02π

〈p0|e−iτHωE |p0〉]

. (6.237)

The expression in brackets is the trace of e−iτHωE , which is conveniently calculatedin the eigenstates |n〉 of the harmonic oscillator with eigenvalues −2(n + 1/2)ωE:

Tre−iτHωE =∞∑

n=0

eiτ 2(n+1/2)eE =i

2 sinωE=

1

2 sinh τeE. (6.238)

Thus we obtain

Tr〈x, τ |x 0〉= V∆t−i

4(2π)2τ 2eEτ

sinh τeE, (6.239)


6.6.2 Dirac Field in Constant Electromagnetic Field

For a Dirac field we have to solve the inhomogeneous differential equation

{iγµ[∂µ − eAµ(x)]−M}S(x, x′) = iδ(4)(x− x′), (6.240)

rather than (6.188). The solution can formally be written as

S(x, x′) = {iγµ[∂µ − eAµ(x)] +M} G(x, x′) = iδ(4)(x− x′), (6.241)

where G(x, x′) solves a slight generalization of Eq. (6.188):{

[i∂ − eA(x)]2 − e

2σµ

νFµν −M2

}

G(x, x′) = iδ(4)(x− x′), (6.242)

which is the Green function of the Pauli equation (6.111), in natural units. Fora constant field, the extra term enters the final result (6.241) in a trivial way ifwe recall the relations to the Green function (6.189) and (6.228), which imply thatG(x, x′) contains the fields as follows:

G(x, x′) =∫ ∞

0dτ exp

(

−ie2σµ

νFµντ)

〈x, τ |x′ 0〉. (6.243)

Constant Electric Background Field

For a constant electric field in the z-direction, we choose the vector potential to haveonly a zeroth component

A3(x) = −Ex0. (6.244)

Then, since F 30 = E, we have F30 = −E and F0

3 = −E, the field tensor Fµν is

given by the matrix

F = −E

0 0 0 10 0 0 00 0 0 01 0 0 0

= iE M3, (6.245)

where M3 is the generator (4.60) of pure Lorentz transformations in the z-direction.The exponential eeFτ is therefore equal to the boost transformation (4.59) B3(ζ) =e−iM3ζ with a rapidity ζ = −Eτ . Then we find from (4.14) the explicit matrices

eeFτ =

cosh eEτ 0 0 − sinh eEτ0 1 0 00 0 1 0

− sinh eEτ 0 0 cosh eEτ

. (6.246)

and hence

sinh eFτ =

0 0 0 − sinh eEτ0 0 0 00 0 0 0

− sinh eEτ 0 0 0

, (6.247)


sinh eFτ

eFτ=

sinh eEτ

eE0 0 0

0 1 0 00 0 1 0

0 0 0sinh eEτ

eE

, (6.248)

and

eF coth eFτ = eE

coth eEτ 0 0 00 1 0 00 0 1 00 0 0 coth eEτ

. (6.249)

Thus we obtain

〈x, τ |x′ 0〉= i

(4πτ)2eEτ

sinh eEτexp

[

−ie∫ x

x′

dξµAµ(ξ)]

(6.250)

× ei4 [−(x−x′)0eE coth eEτ(x−x′)0+(x−x′)T 1

τ(x−x′)T+(x−x′)3eE coth eEτ(x−x′)3]−iM2τ ,

where the superscript T indicates transverse directions to E. The prefactorexp [−ie ∫ xx′ dξµAµ(ξ)] is found by inserting (6.244) and integrating along the straightline

ξ = x′ + s(x− x′), s ∈ [0, 1], (6.251)

to be

exp[

−ie∫ x

x′

dξµAµ(ξ)]

= e−ieE(x0−x′0)∫

1

0ds[z′+s(z−z′)] = e−ieE(x0−x′

0)(z+z′). (6.252)

The exponential prefactor in the fermionic Green function (6.243) is calculatedin the chiral representation of the Dirac algebra where, due to (6.112) and (6.113)

exp(

−ie2σµ

νFµντ)

= exp (e�Eτ) =

(

e−�eEτ 00 e�eEτ

)

, (6.253)

which is equal to

exp(

−ie2σµ

νFµντ)

=

(

cosh eEτ−sinh eEτ �E 0

0 cosh eEτ+sinh eEτ �E

)

.

(6.254)

Comparison with (4.503) shows that this is the Dirac representation of a Lorentzboost with in the direction of E with rapidity ζ = 2e|E|τ . The Dirac trace of theevolution amplitude for Dirac fields is then simply

tr〈x, τ |x 0〉=− i

(4πτ)2eEτ

sinh eEτ× 4 cosh eEτ, (6.255)

and the functional trace of this carries simply a total spacetime volume factor V∆tas in the scalar expression (6.229).


Note that the Lorentz-transformation (6.254) has twice the rapidity of the trans-formation (6.246) in the defining representation, this being a manifestation of the gy-romagnetic ratio of the electron in Dirac’s theory being equal to two [recall (6.120)].

The process of pair creation in a space- and time-dependent electromagetic fieldis discussed in Ref. [3].

The above discussion becomes especially simple in 1+1 spacetime dimensions,the so-called massive Schwinger model [4].

6.6.3 Dirac Field in Electromagnetic Plane-Wave Field

The constant-background field results in the last subsection simplify drastically ifelectric and magnetic fields have the same size and are orthogonal to each other.This is the case for a traveling plane wave of arbitrary shape [5] running along somedirection nµ with n2 = 0. If ξ denotes the spatial coordinate along n, we may writethe vector potential as

Aµ(x) = ǫµf(ξ), ξ ≡ nx. (6.256)

where ǫµ is some polarization vector with the normalization ǫ2 = −1 in the gaugeǫn = 0. The field tensor is

Fµν = ǫµνf′(ξ), ǫµν ≡ nµǫν − nνǫµ, (6.257)

where the constant tensor ǫµν satisfies

ǫµνnµ = 0, ǫµνǫ

µ = 0, ǫµνǫνλ = nµnλ. (6.258)

The Heisenberg equations of motion (6.192), (6.194) take the form

dxµ(τ)

dτ= i

[

H, xµ(

τ)] = 2P µ(τ) (6.259)

dP µ(τ)

dτ= i

[

H, P µ(τ)]

= 2eǫµνPν(τ)f ′(ξ(τ)) +

e

2nµǫλκσ

λκf ′′(ξ(τ)). (6.260)

Note that the last term in (6.194) vanishes for a sourceless plane wave: ∂νFµν = 0.Multiplying these equations by nµ we see that

nµdξµ(τ)

dτ= 2nµP

µ(τ), nµdP µ(τ)

dτ= 0. (6.261)

Hence

nP (τ) = nP (0) = const, ξ(τ)− ξ(0) = nx(τ)− nx(0) = 2τnP (τ). (6.262)

Whereas the components of P (τ) parallel to n are time independent, those orthog-onal to n have a nontrivial time dependence. To find it we multiply (6.260) by ǫµνand find

d

dτǫνµP

µ(τ) = 2eǫνµǫµρPρf

′(ξ) = enνf′(ξ)(2nP ) = enνf

′(ξ)dξ

dτ= enν

df(ξ)

dτ, (6.263)


which is integrated toǫνµP

µ(τ) = enνf(ξ) + Cν , (6.264)

with an operator integration constant Cν which commutes with the constant nPand satisfies nνCν = 0 and and

ǫµνCν = nµ(nP ) = nµ ξ(τ)− ξ(0)

2τ. (6.265)

Inserting this into (6.260) and integrating that equation yields

Pµ(τ) =1

2πn

[

2eCµf(ξ) + e2nµf2(ξ) +

e

2nµǫµνσ

µνf ′(ξ)]

+ Dµ, (6.266)

where Dµ is again an interaction constant commuting with nP . With this, we can

integrate the equation of motion (6.259) over dτ = dξ/2nP and find

1

2[x(τ)− x(0)] =

1

(2nP )2

∫ ξ(τ)

ξ(0)dξ[


e

2nµǫµνσ

µνf ′(ξ)]

+ Dµτ.

(6.267)This determines Dµ, which is reinserted into (6.266) to yield

Pµ(τ) =1

2τ[xµ(τ)− xµ(0)]

− τ[

ξ(τ)− ξ(0)]2

∫ ξ(τ)

ξ(0)dξ[


e

2nµǫρνσ

ρνf ′(ξ)]

+τ

ξ(τ)− ξ(0)

[

2eCνf(ξ(τ)) + e2nµf2(ξ(τ)) +

e

2nµǫρνσ

ρνf ′(ξ(τ))]

. (6.268)

Multiplying this by ǫνµ, and recalling (6.258) and (6.265), we find

ǫνµPµ(τ) =1

2τǫνµ [xµ(τ)− xµ(0)] +

− enν

ξ(τ)− ξ(0)

∫ ξ(τ)

ξ(0)dξ f(ξ) + enνf(ξ(τ)). (6.269)

Inserting this into (6.264) determines the integration constant Cν :

Cν =1

2τǫνµ [xµ(τ)− xµ(0)]−

enν

ξ(τ)− ξ(0)

∫ ξ(τ)

ξ(0)dξ f(ξ). (6.270)

It is useful to introduce the notation

〈 f 〉 ≡ 1

ξ(τ)− ξ(0)

∫ ξ(τ)

ξ(0)dξ f(ξ). (6.271)

and〈 (δf)2 〉 ≡ 〈 (f − 〈f〉)2〉 = 〈 f 2 〉 − 〈 f 〉2 . (6.272)


In order to calculate the matrix elements

〈x τ |H|x 0〉 =⟨

x τ

∣

∣

∣

∣

−P 2 +e

2σµ

νFµν +M2

∣

∣

∣

∣

x 0⟩

, (6.273)

we must time-order the operators x(τ), x(0). For this we need the commutator

[xµ(τ), xν(0)] = 2iτgµν . (6.274)

This is deduced from Eq. (6.268) by commuting it with x(τ) and using the triviallyvanishing equal-time commutator [ξ(τ), xν(τ)] = 0 as well as the nonequal-timecommutator [ξ(0), xν(τ)] = 2inντ . From the latter follows [ξ(τ), xν(0)] = 0, whichis also needed for time-ordering. The result is

〈x τ |H|x 0〉 = − 1

4τ 2(x− x′)2 − 2

i

τ+(

M2 + e2〈(δφ)2〉)2

+ eǫµνσµν f(ξ)− f(ξ′)

ξ − ξ′.

(6.275)

Integrating this in τ we obtain the exponential factor of the time-evolution amplitude(6.211):

E(x, x′; τ)=1

τ 2exp

{

− i

4τ(x−x′)2+

(

M2+e2〈(δf)2〉)2−iτeǫµνσµν f(ξ)− f(ξ′)

ξ − ξ′

}

.

(6.276)

The time-independent prefactor C(x, x′) is again determined by the differential equa-tion Eq. (6.216), which reduces here to

[i∂µ−eAµ(x)] 〈x, τ |x′ 0〉 =[

ǫµν(x− x′)νξ − ξ′

〈 f 〉 − f(ξ)

]

〈x, τ |x′ 0〉, (6.277)

and is solved by

C(x, x′) =−i

(4π)2exp

(

ie∫ x

x′

dyµ

{

Aµ(y)− ǫµν(x−x′)νξ − ξ′

[

∫ ′ny

ξdy′

f(y′)

ny − ξ′−f(ny)

]})

.

(6.278)

For a straight-line integration contour, the second term does not contribute, asbefore.

Observe that in Eq. (6.276), the mass term M2 is replaced by

M2eff =M2 + e2〈(δf)2〉, (6.279)

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-handside becomes M2 + e2〈 f 2〉. If the photon number density is ρ, their energy densityis ρω (in units with h = 1), and we can calculate

e2〈 f 2〉 = 4πα〈E2〉ω2

= 4παρ

ω. (6.280)


Hence we find a relative mass shift:

∆M2

M2= 4παλ2eλ ρ, (6.281)

where λe ≡ h/Mec = 3.861592642(28)× 10−3A is the Compton wavelength of theelectron. For visible light, the right-hand side is of the order of A3ρ/100. Presentlasers achieve energy densities of 109W/sec corresponding to a photon density

ρ =1

hω× 109

W

sec≡ 2.082× 10−7 1

A3

eV

hω. (6.282)

which is too small to make ∆M2/M2 observable.

Appendix 6A Spinor Spherical Harmonics

Equation (6.171) defines spinor spherical harmonics. In these, an orbital wave func-tion of angular momentum l± is coupled with spin 1/2 to a total angular momentumj = l∓ ± 1/2. For the configurations j = l− + 1/2 with m2 = −1/2 the recursionrelation (4C.20) for the Clebsch-Gordan coefficients 〈s1m1; s2m2|sm〉, simplifies byhaving no second term. Inserting s1 = l−, s2 = 1/2 and s = j = l− + 1/2, we find

〈l−, m+ 1

2; 1

2,− 1

2|l−+ 1

2, m〉 =

√

√

√

√

l− −m+ 1/2

l− −m+ 3/2〈l−, m− 1

2; 1

2,− 1

2|l−+ 1

2, m−1〉. (6A.1)

This has to be iterated with the initial condition

〈l−,−l−; 1

2,− 1

2|l− + 1

2, −l− − 1

2〉 = 1, (6A.2)

which follows from the fact that the state 〈l−,−l−; 1

2,− 1

2〉 carries a unique magnetic

quantum number m = −l− − 1/2 of the irreducible representation of total angularmomentum s = j = l− + 1/2. The result of the iteration is

〈l+, m− 1

2; 1

2, 1

2|l+− 1

2, m〉 =

√

√

√

√

l+ −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

〈l−, m− 1

2; 1

2, 1

2|l−+ 1

2, m〉 =

√

√

√

√

l− +m+ 1/2

l− +m+ 3/2〈l−, m+ 1

2; 1

2, 1

2|l−+ 1

2, m+1〉, (6A.4)

and iterating this with the initial condition

〈l−, l−; 1

2n 1

2|l− + 1

2, l− + 1

2〉 = 1, (6A.5)

Notes and References 465

which expresses the fact that the state 〈l− l−; 1

2

1

2〉 is the state of maximal magnetic

quantum number m = l− + 1/2 of the irreducible representation of total angularmomentum s = j = l− + 1/2. The result of the iteration is

〈l−, m− 1

2; 1

2, 1

2|l++ 1

2, m〉 =

√

√

√

√

l+ +m+ 1/2

2l− + 1. (6A.6)

Inserting (6A.3) and (6A.6) the expression (6.171) for the spinor spherical harmonicof total angular momentum j = l− + 1/2, which now reads

yl−j,m(θ, φ) = 〈l−, m− 1

2; 1

2, 1

2|l−+ 1

2, m〉 Yl m−1/2(θ, φ)χ( 1

2)

+ 〈l−m+ 1

2; 1

2− 1

2|l−+ 1

2, m〉 Yl m+1/2(θ, φ)χ(− 1

2), (6A.7)

and separating the spin-up and spin-down components, we obtain precisely (6.173).In order to find 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

2, m|l − 1

2, m〉 = 0. (6A.8)

Inserting a complete set of states in the direct product space yields

〈l + 1

2, m|l, m− 1

2; 1

2, 1

2〉〈lm− 1

2; 1

2

1

2|l − 1

2, m〉

+〈l + 1

2, m|l, m+ 1

2; 1

2,− 1

2〉〈l, m+ 1

2; 1

2,− 1

2|l − 1

2, m〉 = 0. (6A.9)

Together with (6A.3) and (6A.6) we find

〈l+.m− 1

2; 1

2, 1

2|l+− 1

2, m〉 =

√

√

√

√

l+ +m+ 1/2

2l+ + 1,

〈l+, m+ 1

2; 1

2,− 1

2|l+− 1

2, m〉 = −

√

√

√

√

l+ −m+ 1/2

2l+ + 1,

(6A.10)

With this, the expression (6.171) for the spinor spherical harmonics written as

yl+j,m(θ, φ) = 〈l+, m− 1

2; 1

2, 1

2|l+− 1

2, m〉 Yl,m−1/2(θ, φ)χ( 1

2)

+ 〈l+, m+ 1

2; 1

2,− 1

2|l+− 1

2, m〉 Yl,m+1/2(θ, φ)χ(− 1

2) (6A.11)

has the components given in (6.172).

Notes and References

[1] J. Schwinger, Phys. Rev. 82, 664 (1951); 93, 615 (1954); 94, 1362 (1954).

[2] C. Itzykson and J.B. Zuber, Quantum Field Theory, McGraw-Hill (1985).

[3] H. Kleinert, R. Ruffini, and X. SheShengPhys. Rev. D 78, 025011 (2008);A. Chervyakov and H. Kleinert, Phys. Rev. D 80, 065010 (2009).

[4] M.P. Fry, Phys. Rev. D 45, 682 (1992).

[5] C. Itzykson and E. Brezin, Phys. Rev. D 2, 1191 (1970).

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