Masujima M. Applied Mathematical Methods in Theoretical Physics
Подождите немного. Документ загружается.
518 10 Calculus of Variations: Applications
the axion is the good candidate for the dark matter of the universe.
If the axion indeed serve to close the universe, it is possible to search for
their traces experimentally. The axions in the dark matter of the universe can
be converted in an external electromagnetic field into photons. Since the axions
move with the nonrelativistic velocity, the converted photons will have a very sharp
frequency distribution centered around the axion mass, m
a
.Thuswecantryto
detect the signal of these axions by means of a variable frequency resonant cavity
placed in an external electromagnetic field. Exploring the mass range of the axions
of the order of
4.5 ×10
−4
< m
a
< 5 ×10
−4
eV,
preliminary experimental result provided the nontrivial upper bound on the axion
–photon–photon coupling constant as
K
aγγ
< 1.6.
There are compelling astrophysical and cosmological reasons for wanting some
dark matter in the universe. The ‘‘invisible’’ axions have never been in thermal
equilibrium and hence they are cold. The ‘‘invisible’’ axions are the most sensible
candidate for this dark matter of the universe.
Extensive worldwide attempts to detect the axion from the universe with the
variable frequency resonant cavity placed in an external electromagnetic field are
underway at the moment.
Peccei–Quinn axion hypotheses and the invisible axion scenario are the exten-
sions of the standard model.
10.17
Lattice Gauge Field Theory and Quark Confinement
Gauge Field Sector of Lattice Gauge Field Theory
In this section, we discuss the gauge sector of the lattice gauge field theory, its
classical continuum limit, the path integral quantization of the lattice gauge field
theory and the strong coupling limit of the lattice gauge field theory to realize the
linearly rising potential for the quark confinement in the form of Wilson’s area
law.
The lattice gauge field theory is defined on the discrete Euclidean space–time by
cutting the continuum Euclidean space–time into the hyper-square lattice of side
a. We associate the link variables U’s which connect the neighboring lattice sites
with
U
µ
(n) ≡ exp[iaA
µ
] ≡ exp[iaA
αµ
T
α
]. (10.17.1)
10.17 Lattice Gauge Field Theory and Quark Confinement 519
Letting ˆµ to be the unit vector in the µ-direction with µ = 1, 2, 3, 4, we define
U
−µ
(n + a ˆµ) ≡ U
µ
(n)
−1
. (10.17.2)
The gauge transformation is defined by
U
µ
(n) −→ U(ω(n))U
µ
(n)U(ω(n + ˆµ))
†
. (10.17.3)
The minimum lattice square, the plaquet P
µν
,inthe ˆµ–ˆν plane is defined by
n
U
µ
(n)
⇒ n +a ˆµ
U
ν
(n+a ˆµ)
⇒ n + a ˆµ + aˆν
U
−µ
(n+a ˆµ+aˆν)
⇒ n + aˆν
U
−ν
(n+aˆν)
⇒ n.
(10.17.4)
The Euclidean action functional of the lattice gauge field theory is defined by
I
gauge
E
≡
1
2g
2
P
µν
1
Tr
)
U
µ
(n)U
ν
(n + a ˆµ)U
−µ
(n + a ˆµ + aˆν)U
−ν
(n +aˆν)
*
+ h.c. −2
2
. (10.17.5)
The action functional for the gauge field, (10.17.5), is given by the sum over all
plaquets P
µν
of the products of the four link variables, U’s, of each plaquet P
µν
.
The action functional for the gauge field, (10.17.5), is clearly gauge invariant under
the discrete gauge transformation, (10.17.3).
We observe that
U
ν
(n + a ˆµ) = exp
)
iaA
ν
(n + a ˆµ)
*
≈ exp
)
iaA
ν
(n) + ia
2
∂
µ
A
ν
(n)
*
,
U
−µ
(n + a ˆµ + aˆν) = U
µ
(n + aˆν)
−1
≈ exp
)
−iaA
µ
(n) −ia
2
∂
ν
A
µ
(n)
*
,
U
−ν
(n +aˆν) = U
ν
(n)
−1
= exp
)
−iaA
ν
(n)
*
.
The trace term in the action functional, (10.17.5), in the classical continuum limit
a → 0isgivenby
Tr
)
U
µ
(n)U
ν
(n + a ˆµ)U
−µ
(n + a ˆµ + aˆν)U
−ν
(n + aˆν)
*
≈ Tr
%
e
iaA
µ
(n)
e
iaA
ν
(n)+ia
2
∂
µ
A
ν
(n)
e
−iaA
µ
(n)−ia
2
∂
ν
A
µ
(n)
e
−iaA
ν
(n)
'
≈ Tr exp
)
ia
2
{∂
µ
A
ν
− ∂
ν
A
µ
+i[A
µ
, A
ν
]}
*
, (10.17.6)
where the use has been made of the Baker–Campbell–Hausdorff formula whose
iterated commutator is truncated of the order a
2
,
exp[iaA
µ
(n)] exp
)
iaA
ν
(n)
*
≈ exp
ia(A
µ
(n) + A
ν
(n)) −
a
2
2
)
A
µ
(n), A
ν
(n)
*
.
(10.17.7)
520 10 Calculus of Variations: Applications
The action functional, I
gauge
E
, in the classical continuum limit a → 0, is given by
I
gauge
E
≈−
1
2g
2
P
µν
a
4
Tr
F
µν
(n)F
µν
(n)
≈−
1
2g
2
d
4
x Tr
F
µν
(x)F
µν
(x)
.
(10.17.8)
Hence, we recover the action functional of the continuum non-Abelian gauge field
theory. In (10.17.6), the terms of the order O(a
3
) and the higher order terms do not
contribute to the action functional, (10.17.8), due to the fact that the generators are
traceless,
TrT
α
= 0.
Path integral quantization of the lattice gauge field theory is carried out by
$
dU
µ
exp[I
gauge
E
],
with the invariant Haar measure,
dU = 1,
dUf (U
0
U) =
d(U
0
U)f (U
0
U) =
dUf (U). (10.17.9)
Wilson operator P
C
is defined as the operator which takes the products of the
link variables, Us, starting from some starting point and back to the starting point
along the closed contour C. Its vacuum expectation value is given by
P
C
UU ···U=
$
dU
µ
(P
C
UU ···U)exp
)
I
gauge
E
*
, (10.17.10)
wherewetakethecontourC to be the rectangular contour with the sides R a in
thespace-likedirectionandT a in the time-like direction. We define β ≡ 1g
2
in the action functional, (10.17.5), and consider the strong coupling limit g →∞.
We expand exp[I
gauge
E
(10.17.5)] in powers of β. The dominant contribution to (10)
comes from the plaquets, numbering RTa
2
, which are the contribution from the
action functional and densely cover the interior of the closed contour C. Then, in
the strong coupling limit g →∞,wehave
P
C
UU ···U≈
1
2g
2
RTa
2
= exp
−RT
1
a
2
ln
2g
2
. (10.17.11)
The term multiplying −T is the potential between the heavy quark and the heavy
antiquark in the static limit,
V(R) = αR with α =
1
a
2
ln
2g
2
. (10.17.12)
We have proven that QCD provides the quark confinement potential, if we can find
the linearly rising potential in an appropriate continuum limit at some finite g.
10.17 Lattice Gauge Field Theory and Quark Confinement 521
Fermion Field Sector of Lattice Gauge Field Theory
In this section, we discuss the fermion sector of the lattice gauge field theory.
The action functional, I
F
E
, for the fermion in the lattice gauge field theory is
defined by
I
F
E
≡ a
4
x
(
µ
i
2a
%
ψ(x)γ
µ
U
µ
(x)ψ (x + a
µ
) − ψ(x + a
µ
)γ
µ
U
µ
(x)
†
ψ(x)
'
−m
0
ψ(x)ψ(x) +
µ
r
2a
%
ψ(x)γ
µ
U
µ
(x)ψ (x + a
µ
)
+
ψ(x + a
µ
)γ
µ
U
µ
(x)
†
ψ(x)
'
− 2ψ(x)ψ(x)
, (10.17.13)
where the summation over x is taken over each lattice site and a
µ
≡ a ˆµ.This
fermion action functional, I
F
E
, is locally gauge invariant under
U
µ
(x) −→ U(ω(x))U
µ
(x)U(ω(x + a
µ
))
†
,
ψ(x) −→ U(ω(x))ψ(x),
ψ(x) −→ ψ(x)U(ω(x))
†
.
(10.17.14)
Observing that, up to the second order, we have the following expansions of U
µ
(x)
and ψ(x + a
µ
):
U
µ
(x) = exp
)
iaA
αµ
(x)T
α
*
≈ 1 +iaA
µ
(x) +
1
2
iaA
µ
(x)
2
,
ψ(x + a
µ
) ≈ ψ(x) + a∂
µ
ψ(x) +
1
2
a
2
∂
2
µ
ψ(x),
(10.17.15)
the limit a → 0 of the fermion action functional, I
F
E
,isgivenby
I
F
E
≈
d
4
x
i
2
)
ψγ
µ
D
µ
ψ − D
µ
ψγ
µ
ψ
*
− m
0
ψψ
+
ar
4
%
ψD
µ
D
µ
ψ + ψ
←−
D
µ
←−
D
µ
ψ
'
. (10.17.16)
The term proportional to r in the action functional, I
F
E
, for the fermion in the
lattice gauge field theory is called Wilson term. In order to understand its relevance,
we shall consider the model with
U
µ
= 1andr = 0. (10.17.17)
Regarding p
µ
as the momentum operator in the coordinate representation, we have
exp[iap
µ
]ψ(x) = ψ(x + a
µ
).
522 10 Calculus of Variations: Applications
Letting ψ(x) ∼ exp[−ik
µ
x
µ
]andψ(x) ∼ exp[ik
µ
x
µ
], we obtain the kernel of the
quadratic part of the fermion action functional, I
F
E
,as
µ
i
2a
)
γ
µ
exp[−iak
µ
] −exp[iak
µ
]γ
µ
*
− m
0
. (10.17.18)
The two-point Green’s function is given by
µ
1
a
γ
µ
sin ak
µ
− m
0
−1
. (10.17.19)
The momentum is defined in the fundamental Brillouin zone and we choose
−
π
2a
≤ k
µ
<
3π
2a
, µ = 1, 2, 3, 4. (10.17.20)
While keeping the momentum in the region k
µ
≈ 0, we take the limit a → 0to
obtain the ‘‘free’’ two-point Green’s function for the fermion as
µ
γ
µ
k
µ
− m
0
−1
. (10.17.21)
However, if we let k
1
→ k
1
+ πa and take the limit a → 0, we obtain
4
µ=2
γ
µ
k
µ
− γ
1
k
1
−m
0
−1
, (10.17.22)
which is also the ‘‘free’’ two-point Green’s function for the fermion with the
momentum k
1
→−k
1
and provides the fermion of mass m
0
.Wehavetwofermion
poles in each direction, and end up with 2
4
= 16 fermion poles for the ‘‘free’’
two-point Green’s function.
This situation is called the species doubling. On the other hand, if we set r = 0,
we obtain the two-point Green’s function as
µ
1
a
γ
µ
sin ak
µ
− m
0
−
µ
r
a
1 −cos ak
µ
−1
, (10.17.23)
which removes the redundant fermion poles other than k
µ
≈ 0.
We consider the species doubling from the symmetry property. We define the
operators T
µ
by
T
µ
≡ γ
µ
γ
5
exp
iπ
x
µ
a
, (10.17.24)
10.18 WKB Approximation in Path Integral Formalism 523
with the property T
µ
T
ν
+T
ν
T
µ
= δ
µν
. We define the 16 independent operators by
1, T
1
T
2
, T
1
T
3
, T
1
T
4
, T
2
T
3
, T
2
T
4
, T
3
T
4
, T
1
T
2
T
3
T
4
,
T
1
, T
2
, T
3
, T
4
, T
1
T
2
T
3
, T
2
T
3
T
4
, T
3
T
4
T
1
, T
4
T
1
T
2
.
(10.17.25)
Designating one of these operators as T, the action functional, I
F
E
, for the fermion
in the lattice gauge field theory, (10.17.13), with r = 0, is invariant under the
transformation specified by
ψ(x) → Tψ(x),
ψ(x) → ψ (x)T
−1
.
(10.17.26)
Since T
µ
adds the momentum by πa in the µ direction, the 15 Ts other than
T = 1 produce the 15 poles in the fundamental Brillouin zone, (10.17.20), from the
pole with k
µ
≈ 0 for the ‘‘free’’ two-point Green’s function for the fermion.
The 8 Ts in the first line of (10.17.25) commute with γ
5
while the 8 Tsinthe
second line of (10.17.25) anticommute with γ
5
. Thus when we try to introduce
the left-handed Weyl-type fermion into the theory, we end up with the equal
number of the left-handed Weyl-type fermions and the right-handed Weyl-type
fermions. Hence we face the technical difficulty for the implementation of the
lattice regularization to the electro-weak gauge field theory. Success of the lattice
gauge field theory approach to the infrared regime is limited to the strong coupling
limit of QCD.
10.18
WKB Approximation in Path Integral Formalism
Customary WKB method in quantum mechanics is the short wavelength approx-
imation to wave mechanics. We leave the details of the standard approach as an
exercise to the reader. WKB method in quantum theory in path integral formal-
ism consists of the replacement of general Lagrangian (density) with a quadratic
Lagrangian (density). We begin with Euclidean quantum theory. As a preliminary,
we first discuss the method of steepest descent.
Method of Steepest Descent
Consider the integral,
Z ≡
∞
−∞
dx
9
2πg
exp
−
I(x)
g
. (10.18.1)
524 10 Calculus of Variations: Applications
In the neighborhood of the stationary point of I(x), dI(x)/dx
x=x
0
= 0, we Taylor-
expand I(x)aroundx = x
0
,
I(x) = I
0
+
I
(2)
0
2
(x − x
0
)
2
+
I
(3)
0
3!
(x − x
0
)
3
+
I
(4)
0
4!
(x − x
0
)
4
+···.
Substituting this into Z,andreplacingx − x
0
with x,wehave
Z exp
−
I
0
g
∞
−∞
dx
9
2πg
exp
−
I
(2)
0
2g
x
2
−
I
(3)
0
3!g
x
3
−
I
(4)
0
4!g
x
4
−···
=
x→
√
gx
exp
−
I
0
g
∞
−∞
dx
9
2πg
exp
−
I
(2)
0
2
x
2
−
√
g
I
(3)
0
3!
x
3
−g
I
(4)
0
4!
x
4
−····
exp
−
I
0
g
∞
−∞
dx
9
2πg
exp
−
I
(2)
0
2
x
2
1 −g
I
(4)
0
4!
x
4
+
g
2
(
I
(3)
0
3!
)
2
x
6
+ O(g
2
)
.
We apply the Gaussian integral formula to obtain
Z = exp
−
I
0
g
1
B
I
(2)
0
1 −
g
8
I
(4)
0
(I
(2)
0
)
2
+
5g
24
(I
(3)
0
)
2
(I
(2)
0
)
3
+ O(g
2
)
. (10.18.2)
WKB Approximation in Quantum Mechanics
Consider the Euclidean generating functional of (the connected parts of) Green’s
functions,
Z [
J] = exp
%
W[
J]/
'
=
D
q exp
%
1
I[
q] −(
q
J)
2
/
'
. (10.18.3)
In the neighborhood of the stationary point of I[
q],
δI[
q]
δ
q(t)
q=
q
0
−
J(t) = 0,
we Taylor-expand I[
q] − (
q
J)around
q =
q
0
:
I[
q] −(
q
J) = I
0
− (
q
0
J) +
1
2
I
(2)
0
(
q −
q
0
)
2
+
1
3!
I
(3)
0
(
q −
q
0
)
3
+···,
(I
(n)
0
q
n
) ≡
dt
1
···dt
n
δ
n
I
δq
r
1
(t
1
) ···δq
r
n
(t
n
)
q=
q
0
q
r
1
(t
1
) ···q
r
n
(t
n
).
10.18 WKB Approximation in Path Integral Formalism 525
Substituting this into Z[
J]andreplacing
q −
q
0
with
q,and
q with
√
q,
Z [
J] exp
%
1
I
0
− (
q
0
J)
2
/
'
D
q exp
+
1
2
I
(2)
0
q
2
+
1
3!
I
(3)
0
q
3
−···
= exp
%
1
I
0
− (
q
0
J)
2
/
'
Det
− I
(2)
0
−1/2
1 +O()
= exp
I
0
− (
q
0
J)
−
1
2
Tr ln
−I
(2)
0
+ O()
.
From this, it follows that
W[
J] = I
0
− (
q
0
J) −
2
Tr ln Det
−1
[
q
0
] +O(
2
),
I
0
≡ I[
q
0
],
−1
r
1
,r
2
[
q
0
](t
1
, t
2
) ≡−
δ
2
I
δq
r
1
(t
1
)δq
r
2
(t
2
)
q
=
q
0
.
This is the path integral version of WKB approximation .
WKB Approximation in Quantum Field Theory
Consider now the Euclidean generating functional of (the connected parts of)
Green’sfunctions,
Z [J] = exp
[
W[J]/
]
=
Dφ exp
)1
I[φ] − (φJ)
2
/
*
. (10.18.4)
In the neighborhood of the stationary point of I[φ],
δI[φ]
δφ(x)
φ=φ
0
− J(x) = 0,
we functional Taylor-expand I[φ] −(φJ)aroundφ = φ
0
,
I[φ] − (φJ) = I
0
− (φ
0
J) +
1
2
I
(2)
0
(φ − φ
0
2
) +
1
3!
I
(3)
0
(φ − φ
0
)
3
+···,
I
(n)
0
φ
n
≡
d
4
x
1
···d
4
x
n
δ
n
I
δφ(x
1
) ···δφ(x
n
)
φ=φ
0
φ(x
1
) ···φ(x
n
).
Substituting this into Z[J]andreplacingφ − φ
0
with φ,andφ with
√
φ,weget
Z [J] exp
)1
I
0
− (φ
0
J)
2
/
*
Dφ exp
+
1
2
I
(2)
0
φ
2
+
1
3!
I
(3)
0
φ
3
−···
= exp
)1
I
0
−(φ
0
J)
2
/
*
Det(−I
(2)
0
)
−1/2
1 +O()
= exp
I
0
− (φ
0
J)
−
1
2
Tr ln
−I
(2)
0
+ O()
.
526 10 Calculus of Variations: Applications
From this, it follows that
W[J] = I
0
−(φ
0
J) −
2
Tr ln Det
−1
[φ
0
] +O(
2
),
I
0
≡ I[φ
0
],
−1
[φ
0
](x
1
, x
2
) ≡−
δ
2
I
δφ(x
1
)δφ(x
2
)
φ=φ
0
.
This is the path integral version of WKB approximation .
Establishing the bounds on the error in the WKB approximation in Euclidean
quantum theory is tedious but possible. When it comes to the Minkowskian
quantum theory, the approximation method is the method of a stationary phase
and then establishing the bounds on the error is almost impossible.
10.19
Hartree–Fock Equation
We consider the computation of the ground state energy E of the system of A identical
fermions (electrons or nucleons).
Density Matrix Approach: By defining the density matrices for the one-body
operators (for the kinetic energy term) and the two-body operators (for the potential
energy term), it appears that the variational program based on the density matrices
is straightforward. But it is not easy.
Hartree–Fock Program: Hartree introduced the product wavefunction of the one
particle wavefunctions (orbitals) as the trial ground state wavefunction. Fock
antisymmetrized the product wavefunction introduced by Hartree.
Slater employed the Slater determinant φ,
φ(1, 2, ..., A) ≡
1
√
A!
ϕ
1
(1) ϕ
2
(1) · ϕ
A
(1)
ϕ
1
(2) ϕ
2
(2) · ϕ
A
(2)
····
ϕ
1
(A) ϕ
2
(A) · ϕ
A
(A)
, (10.19.1)
as the trial ground state wavefunction. In the Slater determinant, each orbital ϕ
α
has
thesingleparticleenergyε
α
. Hartree–Fock Program consists of choosing the Slater
determinant φ(1, 2, ..., A) as the ground state wavefunction
1, 2, 3, ..., A
|
and
finding the best determinantal wavefunction by demanding
δ
φ
|
H
|
φ
= 0, (10.19.2)
where the Hamiltonian operator is known,
H =
A
i=1
t
i
+
A
i<j
v
ij
.
10.19 Hartree–Fock Equation 527
If the Slater determinant φ is chosen, we have
E =
A
µ=1
t
µµ
+
1
2
A
µ,ν=1
(v
µν,µν
− v
µν,νµ
). (10.19.3)
Here the matrix elements are defined by
t
αβ
≡
α
|
t
|
β
=
dτϕ
∗
α
tϕ
β
, (10.19.4a)
v
αβ,γδ
≡
αβ
|
v
|
γδ
=
dτ
1
· dτ
2
ϕ
∗
α
(1)ϕ
∗
β
(2)v
12
ϕ
γ
(1
)ϕ
δ
(2
). (10.19.4b)
Note that v
12
is the matrix in the spin-space and isospin-space.
In implementing the variational program,
δ
φ
|
H
|
φ
= 0,
we consider the variation of the orbitals specified as
ϕ
µ
→ ϕ
µ
+ δϕ
µ
for all µ with δϕ
µ
=
σ>A
c
σµ
ϕ
σ
, (10.19.5)
to induce φ
= φ + δφ.Inδϕ
µ
, the inclusion of the term,
#
ν≤A
c
νµ
ϕ
ν
,isexcluded
because such term does not contribute to the Slater determinant (recall the property
of the determinant).
Now we have
φ
H
φ
=
φ
H
φ
+Series in (c
σµ
, c
∗
σµ
). (10.19.6)
Defining v
(a)
µν,µν
as
v
(a)
µν,µν
≡ v
µν,µν
− v
µν,νµ
, (10.19.7)
we explicitly have
φ
H
φ
−
φ
H
φ
=
σµ
c
∗
σµ
t
σµ
+t
µσ
c
σµ
+
1
2
σµν
c
∗
σµ
v
(a)
σν,µν
+ c
∗
σν
v
(a)
µσ ,µν
+ v
(a)
µν,σν
c
σµ
+ v
(a)
µν,µσ
c
σν
. (10.19.8)
Recall symmetry properties,
v
αβ,γδ
= v
βα,δγ
,
µν
v
µν,µν
=
µν
v
µν,νµ
. (10.19.9)