Masujima M. Applied Mathematical Methods in Theoretical Physics
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528 10 Calculus of Variations: Applications
We then have
σµν
c
∗
σν
v
µσ ,µν
=
σµν
c
∗
σν
v
σµ,νµ
=
σµν
c
∗
σµ
v
σν,µν
, (10.19.10a)
σµν
v
µν,µσ
c
σν
=
σµν
v
νµ,σµ
c
σν
=
σµν
v
µν,σν
c
σµ
. (10.19.10b)
Now
φ
H
φ
−
φ
H
φ
=
σµ
c
∗
σµ
t
σµ
+
A
ν=1
v
(a)
σν,µν
+
σµ
t
µσ
+
A
ν=1
v
(a)
µν,σν
c
σµ
+ quadratic term in (c, c
∗
). (10.19.9)
The vanishing of the variational derivatives with respect to c and c
∗
,
δ
δc
σµ
φ
H
φ
−
φ
H
φ
=
δ
δc
∗
σµ
φ
H
φ
−
φ
H
φ
= 0,
gives
t
σµ
+ U
σµ
= 0, t
µσ
+U
µσ
= 0, (10.19.12)
or
h
σµ
= 0, h
µσ
= 0, (10.19.13)
where we define U
σµ
, U
µσ
, h
σµ
,andh
µσ
as
U
σµ
≡
A
ν=1
v
(a)
σν,µν
, U
µσ
≡
A
ν=1
v
(a)
µν,σν
, (10.19.14)
h
σµ
≡ t
σµ
+ U
σµ
, h
µσ
≡ t
µσ
+ U
µσ
. (10.19.15)
We define
h
αβ
≡ t
αβ
+ U
αβ
, (10.19.16)
U
αβ
≡
A
ν=1
v
(a)
αν,βν
=
A
ν=1
ϕ
∗
α
(1)ϕ
∗
ν
(2)v
12
ϕ
β
(1)ϕ
ν
(2), (10.19.17)
for all indices, α and β, irrespective of α ≤ A, α>A, β ≤ A or β>A.
Table of h
αβ
.
β = ν ≤ A β = τ>A
α = µ ≤ Ah
µν
, h
µτ
≡ 0,
α = σ>Ah
σν
≡ 0, h
στ
.
10.20 Problems for Chapter 10 529
The upper right entry and the lower left entry of the table are what the variational
principle tells us. This implementation of the variational principle can be sharpened
to h
αβ
diagonal. Diagonalization is carried out by
ϕ
µ
−→ ϕ
µ
=
ν
R
µν
ϕ
ν
, R
µν
:unitary. (10.19.18)
Then the Slater determinant gets changed into
φ → φ
= (const)φ,
and the result of the variation is unaltered. Then the orbital ϕ
α
is the eigenfunction
of the operator h with the eigenvalue ε
α
. Thus we obtain Hartree–Fock equation,
t
αβ
+ U
αβ
= ε
α
δ
αβ
. (10.19.19)
Hartree–Fock equation is a nonlinear equation. Its solution is effected by iteration,
starting from some initial guess ϕ
(0)
α
.
There remains two important questions. One question has to do with the stability
of the iterative solutions. Namely, do they provide the true minimum? The second
variation of
φ
|
H
|
φ
should be examined. Hence Legendre test and Jacobi test
are in order. Another question has to do with the degeneracy of the Hartree–Fock
solution.
10.20
Problems for Chapter 10
10.1. (due to H. C.). Find the solution or solutions q(t) which extremize
I ≡
T
0
m
2
˙
q
2
−
1
6
q
6
dt,
subject to
q(0) = q(T) = 0.
10.2. (due to H. C.). Find the solution q(t) which extremizes
I ≡
T
0
m
2
˙
q
2
−
λ
3
q
3
dt,
subject to
q(0) = q(T) = 0.
530 10 Calculus of Variations: Applications
10.3. The action for a particle in a gravitational field is given by
I ≡−m
4
g
µν
dx
µ
dt
dx
ν
dt
dt,
where g
µν
is the metric tensor. Show that the motion of this particle is governed by
d
2
x
ρ
ds
2
=−
ρ
µν
dx
µ
ds
dx
ν
ds
,
with
ρ
µν
≡
1
2
g
ρσ
(∂
µ
g
σν
+ ∂
ν
g
σµ
− ∂
σ
g
µν
),
which is called the Christoffel symbol.
Hint: This problem corresponds to a free fall of the particle in the curved space–
time.
10.4. (due to H. C.). The space–time structure in the presence of a black hole is
given by
(ds)
2
=
1 −
1
r
(dt)
2
− (dr)
2
1 −
1
r
−r
2
)
sin
2
θ(dφ)
2
+ (dθ )
2
*
,
and the motion of a particle is such that
3
ds is minimized. Let the particle move
in the x–y plane and hence θ =
π
2
. Then the equations of motion of this particle
subject to the gravitational pull of the black hole are obtained by extremizing
s ≡
t
f
t
i
1 −
1
r
−
dr
dt
2
1 −
1
r
− r
2
dφ
dt
2
dt,
with initial and final coordinates fixed.
(a) Derive the equation of motion obtained by varying φ. Integrate this equation
once to obtain an equation with one integration constant.
(b) Derive the equation of motion obtained by varying r.Findawaytoobtainan
equation involving
dr
dt
and
dφ
dt
and a second integration constant.
(c) Let the particle be at r = 2, φ = 0, with
dr
dt
=
dφ
dt
= 0 at the initial time t = 0.
Determine the motion of this particle as best you can. How long does it take
for this particle to reach r = 1?
10.5. (due to H. C.). The invariant distance ds in the neigborhood of a black hole is
given by
(ds)
2
=
1 −
2M
r
(dt)
2
− (dr)
2
1 −
2M
r
,
10.20 Problems for Chapter 10 531
where r, θ , φ (we set θ = φ = constant) are the spherical polar coordinates and
M is the mass of the black hole. The motion of a particle extremizes the invariant
distance.
(a) Write down the integral which should be extremized. From the expression of
this integral, find a first-order equation satisfied by r(t).
(b) If (r − 2M) is small and positive, solve the first-order equation. How much time
does it take for the particle to fall to the critical distance r = 2M ?
10.6. (due to H. C.). Find the kink solution by extremizing
I ≡
+∞
−∞
dt
1
2
φ
2
t
−
m
2
2
φ
2
+
λ
4
φ
4
+
1
4
m
4
λ
.
10.7. Extremize
I ≡
x
0
0
dx
d
˜
f (x, θ )
cos θ
∂f (x, θ )
∂x
+ f (x, θ)
−
κ
4π
w(
n −
n
0
)f (x, θ
0
)d
0
,
treating f and
˜
f as independent. Here κ is a constant, the unit vectors,
n and
n
0
,
are pointing in the direction specified by spherical angles, (θ , ϕ)and(θ
0
, ϕ
0
), and
d
0
is the differential solid angle at
n
0
. Obtain the steady-state transport equation
for anisotropic scattering from the very heavy scatterers,
cos θ
∂f (x, θ )
∂x
=−f (x, θ) +
κ
4π
w(
n −
n
0
)f (x, θ
0
)d
0
,
−cos θ
∂
˜
f (x, θ )
∂x
=−
˜
f (x, θ ) +
κ
4π
w(
n
0
−
n)
˜
f (x, θ
0
)d
0
.
Interpret the result for
˜
f (x, θ).
10.8. Extremize
I ≡
dtd
3
x L
ψ,
∂
∂t
ψ,
∇ψ, ϕ,
∂
∂t
ϕ,
∇ϕ
,
where
L =−
∇ϕ
∇ψ −
a
2
2
ϕ
∂
∂t
ψ − ψ
∂
∂t
ϕ
,
treating ψ and ϕ as independent. Obtain the diffusion equation
∇
2
ψ(t,
x ) = a
2
∂ψ(t,
x)
∂t
,
∇
2
ϕ(t,
x) =−a
2
∂ϕ(t,
x)
∂t
.
532 10 Calculus of Variations: Applications
Interpret the result for ϕ.
10.9. Extremize
I ≡
dtd
3
x
−
1
2m
i
∇−
e
c
A
ψ
∗
i
∇−
e
c
A
ψ
+
1
2
ψ
∗
i
∂
∂t
− eφ
ψ +
i
∂
∂t
−eφ
ψ
∗
ψ
− ψ
∗
Vψ
,
∇
A +
1
c
∂φ
∂t
= 0,
treating ψ and ψ
∗
as independent. Obtain the Schr
¨
odinger equation
i
∂
∂t
−eφ
ψ =
1
2m
i
∇−
e
c
A
2
ψ + Vψ,
−
i
∂
∂t
+eφ
ψ
∗
=
1
2m
i
∇+
e
c
A
2
ψ
∗
+ Vψ
∗
.
Demonstrate that the Schr
¨
odinger equation is invariant under the gauge transfor-
mation
A →
A
=
A +
∇,
φ → φ
= φ − (1c)(∂∂t),
ψ → ψ
= exp[(iec)]ψ,
where
∇
2
−
1
c
2
∂
2
∂t
2
= 0.
10.10. Extremize
I ≡
dtd
3
x
(
−
1
i
∇−e
A
ψ
2
+
i
∂
∂t
− eφ
ψ
2
− m
2
|
ψ
|
2
+
,
∇
A +
∂φ
∂t
= 0,
treating ψ and ψ
∗
as independent. Obtain the Klein–Gordon equation
i
∂
∂t
− eφ
2
ψ −
1
i
∇−e
A
2
ψ = m
2
ψ,
i
∂
∂t
+ eφ
2
ψ
∗
−
1
i
∇+e
A
2
ψ
∗
= m
2
ψ
∗
.
10.11. Extremize
I ≡
d
4
x L
tot
,
where L
tot
is given by
L
tot
=
1
4
)
ψ
α
(x), D
αβ
(x)ψ
β
(x)
*
+
1
4
)
D
T
βα
(−x)ψ
α
(x), ψ
β
(x)
*
+
1
2
φ(x)K(x )φ(x) +L
int
(φ(x), ψ(x), ψ(x)),
10.20 Problems for Chapter 10 533
with D
αβ
(x), D
T
βα
(−x)andK(x)givenby
D
αβ
(x) = (iγ
µ
∂
µ
− m + iε)
αβ
,
D
T
βα
(−x) = (−iγ
T
µ
∂
µ
− m + iε)
βα
,
K(x) =−∂
2
− κ
2
+ iε,
and L
int
is given by the Yukawa coupling specified by
L
int
(φ(x), ψ(x), ψ(x)) =−G
0
ψ
α
(x)γ
αβ
(x)ψ
β
(x)φ(x).
The γ
µ
saretheDiracγ matrices with the property specified by
{γ
µ
, γ
ν
}=2η
µν
,
(γ
µ
)
†
= γ
0
γ
µ
γ
0
.
The ψ (x) is the four-component Dirac spinor and the ψ (x) is the Dirac adjoint of
ψ(x)definedby
ψ(x) ≡ ψ
†
(x)γ
0
.
(a) Obtain the Euler–Lagrange equations of motion for the ψ field, the ψ field
and the φ field.
(b) Causal Green’s function
F
(x − x
) for Klein–Gordon field φ(x)isdefinedby
i
F
(x − x
) =
d
3
k
(2π)
3
2
√
k
2
+ m
2
1
θ(t − t
)exp
)
−ik(x − x
)
*
+ θ(t
− t)exp
)
ik(x − x
)
*2
= i
d
4
k
(2π)
4
exp
)
−ik(x − x
)
*
1
k
2
− m
2
+ iε
.
Show that
F
(x − x
) satisfies the following differential equation:
(−∂
2
− m
2
)
F
(x − x
) = δ
4
(x − x
).
(c) Causal Green’s function S
F
(x − x
)forDiracfieldψ(x)isdefinedby
iS
F
(x − x
) =
d
3
p
(2π)
3
m
E
1
θ(t − t
)
+
(p)exp
)
−ip(x − x
)
*
+ θ(t
− t)
−
(p)exp
)
ip(x − x
)
*2
= i
d
4
p
(2π)
4
exp
)
−ip(x − x
)
*
γ
µ
p
µ
+m
p
2
− m
2
+ iε
,
534 10 Calculus of Variations: Applications
with the positive (negative) energy projection operator,
±
(p), defined by
+
(p) =
γ
µ
p
µ
+ m
2m
and
−
(p) =
−γ
µ
p
µ
+m
2m
.
Show that S
F
(x − x
) satisfies the following differential equation:
(iγ
µ
∂
µ
− m)S
F
(x − x
) = δ
4
(x − x
).
(d) Show that
F
(x − x
)andS
F
(x − x
) are related by
S
F
(x − x
) = (iγ
µ
∂
µ
+ m)
F
(x − x
).
10.12. Derive Lagrange equations of motion for quantum mechanics,
d
dt
∂L
ˆ
q
s
(t),
·
ˆ
q
s
(t)
∂
·
ˆ
q
r
(t)
−
∂L(
ˆ
q
s
(t),
·
ˆ
q
s
(t))
∂
ˆ
q
r
(t)
= 0,
from the Heisenberg equations of motion, the definition of the Hamiltonian, and
the equal time canonical commutators,
d
ˆ
q
r
(t)
dt
= i
)
H (
ˆ
q
s
(t),
ˆ
p
s
(t)),
ˆ
q
r
(t)
*
,
d
ˆ
p
r
(t)
dt
= i
)
H (
ˆ
q
s
(t),
ˆ
p
s
(t)),
ˆ
p
r
(t)
*
,
H (
ˆ
q
s
(t),
ˆ
p
s
(t)) =
r
ˆ
p
r
(t)
·
ˆ
q
r
(t) −L(
ˆ
q
s
(t),
·
ˆ
q
s
(t)),
[
ˆ
q
r
(t),
ˆ
p
s
(t)] = iδ
rs
,[
ˆ
q
r
(t),
ˆ
q
s
(t)] = [
ˆ
p
r
(t),
ˆ
p
s
(t)] = 0,
where the canonical momentum is defined by
ˆ
p
r
(t) =
∂L(
ˆ
q
s
(t),
·
ˆ
q
s
(t))
∂
·
ˆ
q
r
(t)
.
Hint for Problem 10.12:
Nishijima, K.: Theory of Fields, Kinokuniya-shoten, 1986, Tokyo. Chapter 1. (In
Japanese).
10.13. Derive Euler–Lagrange equations of motion for quantum field theory,
∂
µ
∂L(
ˆ
φ(x), ∂
µ
ˆ
φ(x))
∂(∂
µ
ˆ
φ(x))
−
∂L(
ˆ
φ(x), ∂
µ
ˆ
φ(x))
∂
ˆ
φ(x)
= 0,
10.20 Problems for Chapter 10 535
from the Heisenberg equations of motion, the definition of the Hamiltonian, the
equal time canonical commutators, and the definition of the canonical momentum.
Hint for Problem 10.13:
Nishijima, K.: Fields and Particles: Field Theory and Dispersion Relations,Benjamin
Cummings, Massachusetts, 1969 and 1974.
10.14. Consider the Stueckelberg Lagrangian density for vector meson theory,
L(A
ρ
) =−
1
4
F
ρσ
F
ρσ
+
1
2
µ
2
A
ρ
A
ρ
−
1
2
ξ(∂
ρ
A
ρ
)
2
−J
ρ
A
ρ
; F
ρσ
≡ ∂
ρ
A
σ
− ∂
σ
A
ρ
,
with J
ρ
the conserved current, ∂
ρ
J
ρ
= 0.
(a) Extremize the action functional, I[A
ρ
] =
3
d
4
xL(A
ρ
). Obtain the field
equation,
(∂
2
+ µ
2
)A
ρ
−(1 −ξ )∂
ρ
(∂
σ
A
σ
) = J
ρ
.
Takingthedivergenceofbothsidesoftheabovefieldequation,weobtain
ξ
∂
2
+
µ
2
ξ
(∂
ρ
A
ρ
) = 0.
We let m
2
≡ µ
2
/ξ with ξ>0. Show that the ‘‘transverse’’ field defined by
A
T
ρ
= A
ρ
+
1
m
2
∂
ρ
(∂
σ
A
σ
) = A
ρ
+
ξ
µ
2
∂
ρ
(∂
σ
A
σ
)
is divergenceless.
(b) By canonical quantization with operators satisfying
%
a
(λ)
(k), a
(λ
)†
(k
)
'
= δ
λλ
(2π)
3
2
B
k
2
+µ
2
δ
3
(
k −
k
), 1 ≤ λ, λ
≤ 3,
)
a
(0)
(k), a
(0)†
(k
)
*
=−(2π )
3
2
B
k
2
+ m
2
δ
3
(
k −
k
),
and all other commutators vanishing, show that the field has the Fourier
decomposition on two hyperboloids,
A
ρ
(x) =
d
3
k
2(2π)
3
B
k
2
+ µ
2
×
3
λ=1
%
a
(λ)
(k)ε
(λ)
ρ
(k)exp[−ikx] +a
(λ)†
(k)ε
(λ)∗
ρ
(k)exp[ikx]
'
+
d
3
k
2(2π)
3
9
k
2
+ m
2
k
ρ
µ
)
a
(0)
(k)exp[−ikx] +a
(0)†
(k)exp[ikx]
*
.
536 10 Calculus of Variations: Applications
The polarizations, ε
(λ)
ρ
(k), for λ = 1, 2, 3 are three orthonormal space-like
four-vectors orthogonal to k
ρ
(k
2
= µ
2
). Show that the two-point ‘‘free’’
Green’s function is given by
1
i
0
|
T(A
ρ
(x)A
σ
(y))
|
0
=−
d
4
k
(2π)
4
exp
)
−ik(x − y)
*
η
ρσ
−k
ρ
k
σ
/µ
2
k
2
− µ
2
+ iε
+
k
ρ
k
σ
/µ
2
k
2
− µ
2
/ξ + iε
.
(c) When ξ → 0andµ = 0, show that we recover Proca formalism of massive
vector meson theory.
(d) Prove the identity,
η
ρσ
− k
ρ
k
σ
/µ
2
k
2
− µ
2
+ iε
+
k
ρ
k
σ
/µ
2
k
2
−µ
2
/ξ + iε
=
η
ρσ
k
2
− µ
2
+ iε
+
1 −ξ
ξ
k
ρ
k
σ
(k
2
− µ
2
/ξ +iε)(k
2
− µ
2
+ iε)
.
In the limit µ → 0andξ = 0, (m goes to zero), we obtain the two-point
‘‘free’’ Green’s function as
1
i
0
|
T(A
ρ
(x)A
σ
(y))
|
0
=−
d
4
k
(2π)
4
exp
)
−ik(x − y)
*
η
ρσ
k
2
+iε
+
1 −ξ
ξ
k
ρ
k
σ
(k
2
+iε)
2
,
which is the covariant gauge ‘‘free’’ Green’s function for the electromagnetic
field.
Hint for Problem 10.14:
Stueckelberg, E.C.G.; Helv. Phys. Acta. 11, (1938), 225.
Nishijima, K.; Fields and Particles: Field Theory and Dispersion Relations,Benjamin
Cummings, Massachusetts, 1969 and 1974, Chapter 2.
Itzykson, C., and Zuber, J.B. ; Quantum Field Theory, McGraw-Hill, New York,
1985, Chapter 3.
The analysis in this problem is the intuitive motivation for the Nakanishi–Lautrup
B field.
10.15. Extremize the action functional for the electromagnetic field A
µ
,
I =
d
4
x
−
1
4
F
µν
F
µν
+ B∂
µ
A
µ
+
1
2
αB
2
,
F
µν
≡ ∂
µ
A
ν
− ∂
ν
A
µ
.
Obtain the Euler–Lagrange equations of motion for A
µ
field and B field. Can you
perform the q-number gauge transformation after canonical quantization?
10.20 Problems for Chapter 10 537
10.16. Extremize the action functional for the neutral massive vector field U
µ
,
I =
d
4
x
−
1
4
F
µν
F
µν
+
1
2
m
2
0
U
µ
U
µ
,
F
µν
≡ ∂
µ
U
ν
− ∂
ν
U
µ
.
Obtain the Euler–Lagrange equation of motion for U
µ
field. Examine the massless
limit m
0
→ 0 after canonical quantization.
10.17. Extremize the action functional for the neutral massive vector field A
µ
,
I =
d
4
x
−
1
4
F
µν
F
µν
+
1
2
m
2
0
A
µ
A
µ
+ B∂
µ
A
µ
+
1
2
αB
2
,
F
µν
≡ ∂
µ
A
ν
− ∂
ν
A
µ
.
Obtain the Euler–Lagrange equations of motion for A
µ
field and B field. Examine
the massless limit m
0
→ 0 after canonical quantization.
Hint for Problems 10.15, 10.16, and 10.17:
Lautrup,B.:Mat.Fys.Medd.Dan.Vid.Selsk.35(11). 29. (1967).
Nakanishi, N.: Prog. Theor. Phys. Suppl. 51. 1. (1972).
Yokoyama,K.:Prog.Theor.Phys.51. 1956. (1974), 52. 1669. (1974).
10.18. Derive the Schwinger–Dyson equation for the self-interacting scalar field
ˆ
φ(x), whose Lagrangian density is given by
L
ˆ
φ(x), ∂
µ
ˆ
φ(x)
=
1
2
∂
µ
ˆ
φ(x)∂
µ
ˆ
φ(x) −
1
2
m
2
ˆ
φ
2
(x) −
λ
4
4!
ˆ
φ
4
(x).
Hint: Introduce the proper self-energy part
∗
(x, y) and the vertex operator
4
(x, y, z, w), and mimic the discussion in Section 10.4.
10.19. Derive the Schwinger–Dyson equation for the self-interacting scalar field
ˆ
φ(x) whose Lagrangian density is given by
L
ˆ
φ(x), ∂
µ
ˆ
φ(x)
=
1
2
∂
µ
ˆ
φ(x)∂
µ
ˆ
φ(x) −
1
2
m
2
ˆ
φ
2
(x) −
λ
3
3!
ˆ
φ
3
(x) −
λ
4
4!
ˆ
φ
4
(x).
Hint: Introduce the proper self-energy part
∗
(x, y) and the vertex operators
3
(x, y, z)and
4
(x, y, z, w), and mimic the discussion in Section 10.4.
10.20. Derive the Schwinger–Dyson equation for the ps–ps meson theory whose
Lagrangian density is given by
L =
1
4
%
&
ψ
α
(x), D
αβ
(x)
ˆ
ψ
β
(x)
'
+
1
4
%
D
T
βα
(−x)
&
ψ
α
(x),
ˆ
ψ
β
(x)
'
+
1
2
ˆ
φ(x)K(x )
ˆ
φ(x) −ig
0
&
ψ
α
(x)(γ
5
)
αβ
ˆ
ψ
β
(x)
ˆ
φ(x),
where the kernels of the quadratic part of the Lagrangian density are given by
D
αβ
(x) = (iγ
µ
∂
µ
− m + iε)
αβ
, D
T
βα
(−x) = (−iγ
T
µ
∂
µ
− m + iε)
βα
,