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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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