Zee A. Quantum Field Theory in a Nutshell
Подождите немного. Документ загружается.
III.4 Gauge Invariance: A Photon Can Find No Rest
When the central identity blows up
I explained in chapter I.7 that the path integral for a generic field theory can be formally
evaluated in what deserves to be called the Central Identity of Quantum Field Theory:
Dϕe
−
1
2
ϕ
.
K
.
ϕ−V(ϕ)+J
.
ϕ
= e
−V(δ/δJ)
e
1
2
J
.
K
−1
.
J
(1)
For any field theory we can always gather up all the fields, put them into one giant column
vector, and call the vector ϕ. We then single out the term quadratic in ϕ write it as
1
2
ϕ
.
K
.
ϕ,
and call the rest V(ϕ). I am using a compact notation in which spacetime coordinates and
any indices on the field, including Lorentz indices, are included in the indices of the formal
matrix K. We will often use (1) with V = 0:
Dϕe
−
1
2
ϕ
.
K
.
ϕ+J
.
ϕ
= e
1
2
J
.
K
−1
.
J
(2)
But what if K does not have an inverse?
This is not an esoteric phenomenon that occurs in some pathological field theory, but
in one of the most basic actions of physics, the Maxwell action
S(A) =
d
4
xL =
d
4
x
1
2
A
μ
(∂
2
g
μν
− ∂
μ
∂
ν
)A
ν
+ A
μ
J
μ
. (3)
The formal matrix K in (2) is proportional to the differential operator (∂
2
g
μν
− ∂
μ
∂
ν
) ≡
Q
μν
. A matrix does not have an inverse if some of its eigenvalues are zero, that is, if when
acting on some vector, the matrix annihilates that vector. Well, observe that Q
μν
annihilates
vectors of the form ∂
ν
(x) : Q
μν
∂
ν
(x) = 0. Thus Q
μν
has no inverse.
There is absolutely nothing mysterious about this phenomenon; we have already en-
countered it in classical physics. Indeed, when we first learned the concepts of electricity,
we were told that only the “voltage drop” between two points has physical meaning. At
a more sophisticated level, we learned that we can always add any constant (or indeed
any function of time) to the electrostatic potential (which is of course just “voltage”) since
III.4. Gauge Invariance | 183
by definition its gradient is the electric field. At an even more sophisticated level, we see
that solving Maxwell’s equation (which of course comes from just extremizing the action)
amounts to finding the inverse Q
−1
. [In the notation I am using here Maxwell’s equation
∂
μ
F
μν
= J
ν
is written as Q
μν
A
ν
= J
μ
, and the solution is A
ν
= (Q
−1
)
νμ
J
μ
.]
Well, Q
−1
does not exist! What do we do? We learned that we must impose an additional
constraint on the gauge potential A
μ
, known as “fixing a gauge.”
A mundane nonmystery
To emphasize the rather mundane nature of this gauge fixing problem (which some
older texts tend to make into something rather mysterious and almost hopelessly dif-
ficult to understand), consider just an ordinary integral
+∞
−∞
dAe
−A
.
K
.
A
, with A =
(a, b) a 2-component vector and K =
10
00
, a matrix without an inverse. Of course
you realize what the problem is: We have
+∞
−∞
+∞
−∞
da db e
−a
2
and the integral over
b does not exist. To define the integral we insert into it a delta function δ(b − ξ).
The integral becomes defined and actually does not depend on the arbitrary num-
ber ξ . More generally, we can insert δ[f(b)] with f some function of our choice. In
the context of an ordinary integral, this procedure is of course ludicrous overkill, but
we will use the analog of this procedure in what follows. In this baby problem, we
could regard the variable b, and thus the integral over it, as “redundant.” As we will
see, gauge invariance is also a redundancy in our description of massless spin 1 parti-
cles.
A massless spin 1 field is intrinsically different from a massive spin 1 field—that’s the
crux of the problem. The photon has only two polarization degrees of freedom. (You already
learned in classical electrodynamics that an electromagnetic wave has two transverse
degrees of freedom.) This is the true physical origin of gauge invariance.
In this sense, gauge invariance is, strictly speaking, not a “real” symmetry but merely a
reflection of the fact that we used a redundant description: a Lorentz vector field to describe
two physical degrees of freedom.
Restricting the functional integral
I will now discuss the method for dealing with this redundancy invented by Faddeev and
Popov. As you will see presently, it is the analog of the method we used in our baby problem
above. Even in the context of electromagnetism this method is a bit of overkill, but it will
prove to be essential for nonabelian gauge theories (as we will see in chapter VII.1) and
for gravity. I will describe the method using a completely general and somewhat abstract
language. In the next section, I will then apply the discussion here to a specific example.
If you have some trouble with this section, you might find it helpful to go back and forth
between the two sections.
184 | III. Renormalization and Gauge Invariance
Suppose we have to do the integral I ≡
DAe
iS(A)
; this can be an ordinary integral
or a path integral. Suppose that under the transformation A → A
g
the integrand and
the measure do not change, that is, S(A) = S(A
g
) and DA = DA
g
. The transformations
obviously form a group, since if we transform again with g
, the integrand and the measure
do not change under the combined effect of g and g
and A
g
→ (A
g
)
g
= A
gg
. We would
like to write the integral I in the form I = (
Dg)J , with J independent of g. In other
words, we want to factor out the redundant integration over g. Note that Dg is the invariant
measure over the group of transformations and
Dg is the volume of the group. Be aware
of the compactness of the notation in the case of a path integral: A and g are both functions
of the spacetime coordinates x.
I want to emphasize that this hardly represents anything profound or mysterious. If
you have to do the integral I =
dx dy e
iS(x, y)
with S(x, y) some function of x
2
+y
2
, you
know perfectly well to go to polar coordinates I =(
dθ)J =(2π)J, where J =
dr re
iS(r)
is an integral over the radial coordinate r only. The factor 2π is precisely the volume of the
group of rotations in 2 dimensions.
Faddeev and Popov showed how to do this “going over to polar coordinates” in a
unified and elegant way. Following them, we first write the numeral “one” as
1 = (A)
Dgδ[f(A
g
)], an equality that merely defines (A). Here f is some function
of our choice and (A), known as the Faddeev-Popov determinant, of course depends on
f . Next, note that [(A
g
)]
−1
=
Dgδ[f(A
g
g
)] =
Dg
δ[f(A
g
)] = [(A)]
−1
, where the
second equality follows upon defining g
=g
g and noting that Dg
=Dg. In other words,
we showed that (A) = (A
g
) : the Faddeev-Popov determinant is gauge invariant. We
now insert 1 into the integral I we have to do:
I =
DAe
iS(A)
=
DAe
iS(A)
(A)
Dgδ[f(A
g
)]
=
Dg
DAe
iS(A)
(A)δ[f(A
g
)] (4)
As physicists and not mathematicians, we have merrily interchanged the order of
integration.
At the physicist’s level of rigor, we are always allowed to change integration variables
until proven guilty. So let us change A to A
g
−1
; then
I =
Dg
DAe
iS(A)
(A)δ[f (A)] (5)
where we have used the fact that DA, S(A), and (A) are all invariant under A → A
g
−1
.
That’s it. We’ve done it. The group integration (
Dg) has been factored out.
The volume of a compact group is finite, but in gauge theories there is a separate group
at every point in spacetime, and hence (
Dg) is an infinite factor. (This also explains
why there is no gauge fixing problem in theories with global symmetries introduced in
III.4. Gauge Invariance | 185
chapter I.10.) Fortunately, in the path integral Z for field theory we do not care about
overall factors in Z, as was explained in chapter I.3, and thus the factor (
Dg) can simply
be thrown away.
Fixing the electromagnetic gauge
Let us now apply the Faddeev-Popov method to electromagnetism. The transformation
leaving the action invariant is of course A
μ
→ A
μ
− ∂
μ
,sog in the present context is
denoted by and A
g
≡ A
μ
− ∂
μ
. Note also that since the integral I we started with is
independent of f it is still independent of f in spite of its appearance in (5). Choose
f (A) = ∂A − σ , where σ is a function of x . In particular, I is independent of σ and
so we can integrate I with an arbitrary functional of σ , in particular, the functional
e
−(i/2ξ)
d
4
xσ (x)
2
.
We now turn the crank. First, we calculate
[(A)]
−1
≡
Dgδ[f(A
g
)] =
Dδ(∂A − ∂
2
− σ) (6)
Next we note that in (5) (A) appears multiplied by δ[f (A)] and so in evaluating [(A)]
−1
in (6) we can effectively set f (A) = ∂A − σ to zero. Thus from (6) we have (A) “=”
[
Dδ(∂
2
)]
−1
. But this object does not even depend on A, so we can throw it away. Thus,
up to irrelevant overall factors that could be thrown away I is just
DAe
iS(A)
δ(∂A − σ).
Integrating over σ(x)as we said we were going to do, we finally obtain
Z =
Dσe
−(i/2ξ)
d
4
xσ (x)
2
DAe
iS(A)
δ(∂A − σ)
=
DAe
iS(A)−(i/2ξ)
d
4
x(∂A)
2
(7)
Nifty trick by Faddeev and Popov, eh?
Thus, S(A) in (3) is effectively replaced by
S
eff
(A) = S(A) −
1
2ξ
d
4
x(∂A)
2
=
d
4
x
1
2
A
μ
∂
2
g
μν
−
1 −
1
ξ
∂
μ
∂
ν
A
ν
+ A
μ
J
μ
(8)
and Q
μν
by Q
μν
eff
= ∂
2
g
μν
− (1 − 1/ξ)∂
μ
∂
ν
or in momentum space Q
μν
eff
=−k
2
g
μν
+ (1 −
1/ξ )k
μ
k
ν
, which does have an inverse. Indeed, you can check that
Q
μν
eff
−g
νλ
+ (1 − ξ)
k
ν
k
λ
k
2
1
k
2
= δ
μ
λ
Thus, the photon propagator can be chosen to be
(−i)
k
2
g
νλ
− (1 − ξ)
k
ν
k
λ
k
2
(9)
in agreement with the conclusion in chapter II.7.
186 | III. Renormalization and Gauge Invariance
While the Faddeev-Popov argument is a lot slicker, many physicists still prefer the explicit
Feynman argument given in chapter II.7. I do. When we deal with the Yang-Mills theory
and the Einstein theory, however, the Faddeev-Popov method is indispensable, as I have
already noted.
A photon can find no rest
Let us understand the physics behind the necessity for imposing by hand a (gauge fixing)
constraint in gauge theories. In chapter I.5 we sidestepped this whole issue of fixing the
gauge by treating the massive vector meson instead of the photon. In effect, we changed
Q
μν
to (∂
2
+m
2
)g
μν
−∂
μ
∂
ν
, which does have an inverse (in fact we even found the inverse
explicitly). We then showed that we could set the mass m to 0 in physical calculations.
There is, however, a huge and intrinsic difference between massive and massless
particles. Consider a massive particle moving along. We can always boost it to its rest
frame, or in more mathematical terms, we can always Lorentz transform the momentum
of a massive particle to the reference momentum q
μ
= m(1, 0, 0, 0). (As is the case
elsewhere in this book, if there is no risk of confusion, we write column vectors as row
vectors for typographical convenience.) To study the spin degrees of freedom, we should
evidently sit in the rest frame of the particle and study how its states respond to rotation.
The fancy pants way of saying this is we should study how the states of the particle
transform under that particular subgroup of the Lorentz group (known as the little group)
consisting of those Lorentz transformations that leave q
μ
invariant, namely
μ
ν
q
ν
=q
μ
.
For q
μ
=m(1, 0, 0, 0), the little group is obviously the rotation group SO(3). We then apply
what we learned in nonrelativistic quantum mechanics and conclude that a spin j particle
has (2j + 1) spin states (or polarizations in classical physics), as already noted back in
chapter I.5.
But if the particle is massless, we can no longer find a Lorentz boost that would bring
us to its rest frame. A photon can find no rest!
For a massless particle, the best we can do is to transform the particle’s momentum to the
reference momentum q
μ
= ω(1, 0, 0, 1) for some arbitrarily chosen ω. Again, this is just
a fancy way of saying that we can always call the direction of motion the third axis. What is
the little group that leaves q
μ
invariant? Obviously, rotations around the third axis, forming
the group O(2), leave q
μ
invariant. The spin states of a massless particle of any spin around
its direction of motion are known as helicity states, as was already mentioned in chapter
II.1. For a particle of spin j , the helicities ±j are transformed into each other by parity
and time reversal, and thus both helicities must be present if the interactions the particle
participates in respect these discrete symmetries, as is the case with the photon and the
graviton.
1
In particular, the photon, as we have seen repeatedly, has only two polarization
degrees of freedom, instead of three, since we no longer have the full rotation group SO(3).
1
But not with the neutrino.
III.4. Gauge Invariance | 187
(You already learned in classical electrodynamics that an electromagnetic wave has two
transverse degrees of freedom.) For more on this, see appendix B.
In this sense, gauge invariance is strictly speaking not a “real” symmetry but merely a
reflection of the fact that we used a redundant description: we used a vector field A
μ
with
its four degrees of freedom to describe two physical degrees of freedom. This is the true
physical origin of gauge invariance.
The condition
μ
ν
q
ν
=q
μ
should leave us with a 3-parameter subgroup. To find the other
transformations, it suffices to look in the neighborhood of the identity, that is, at Lorentz
transformations of the form (α, β) =I + αA + βB +
...
. By inspection, we see that
A =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
010 0
100−1
000 0
010 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
= i(K
1
+ J
2
), B =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
001 0
000 0
100−1
001 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
= i(K
2
− J
1
) (10)
where we used the notation for the generators of the Lorentz group from chapter II.3. Note
that A and B are to a large extent determined by the fact that J and K are symmetric and
antisymmetric, respectively.
By direct computation or by invoking the celebrated minus sign in (II.3.9), we find
that [A, B] = 0. Also, [J
3
, A] = B and [J
3
, B] =−A so that, as expected, (A, B) form a
2-component vector under O(2) rotations around the third axis. (For those who must
know, the generators A, B, and J
3
generate the group ISO(2), the invariance group of
the Euclidean 2-plane, consisting of two translations and one rotation.)
The preceding paragraph establishing the little group for massless particles applies
for any spin, including zero. Now specialize to a spin 1 massless particle with the two
polarization vectors
±
(q) = (1/
√
2)(0, 1, ±i,0). The polarization vectors are defined by
how they transform under rotation e
iφJ
3
. So it is natural to ask how
±
(q) transform under
(α, β). Inspecting (10), we see that
±
(q) →
±
(q) +
1
√
2
(α ± iβ)q (11)
We recognize (11) as a gauge transformation (as was explained in chapter II.7). For a mass-
less spin 1 particle, the gauge transformation is contained in the Lorentz transformations!
Suppose we construct the corresponding spin 1 field as in chapter II.5 [and in analogy
to (I.8.11) and (II.2.10)]:
A
μ
(x) =
d
3
k
(2π)
3
2ω
k
α=1,2
[a
(α)
(
k)ε
(α)
μ
(k)e
−i(ω
k
t−
k
.
x)
+ a
†(α)
(
k)ε
∗(α)
μ
(k)e
i(ω
k
t−
k
.
x))
] (12)
with ω
k
=|
k|. The polarization vectors ε
(α)
μ
(k) are of coursed determined by the condition
k
μ
ε
(α)
μ
(k) = 0, which we could easily satisfy by defining ε
(α)
(k) = (q → k)ε
(α)
(q), where
(q → k) denotes a Lorentz transformation that brings the reference momentum q to k.
Note that the ε
(α)
(k) thus constructed has a vanishing time component. (To see this,
first boost ε
(α)
μ
(q) along the third axis and then rotate, for example.) Hence, k
μ
ε
(α)
μ
(k) =
−
k
.
ε
(α)
(k) = 0. These properties of ε
(α)
μ
(k) translate into A
0
(x) = 0 and
−→
∇
.
A(x) = 0.
188 | III. Renormalization and Gauge Invariance
These two constraints cut the four degrees of freedom contained in A
μ
(x) down to two
and fix what is known as the Coulomb or radiation gauge.
2
Given the enormous importance of gauge invariance, it might be instructive to review
the logic underlying the “poor man’s approach” to gauge invariance (which, as I mentioned
in chapter I.5, I learned from Coleman) adopted in this book for pedagogical reasons. You
could have fun faking Feynman’s mannerism and accent, saying, “Aw shucks, all that fancy
talk about little groups! Who needs it? Those experimentalists won’t ever be able to prove
that the photon mass is mathematically zero anyway.”
So start, as in chapter I.5, with the two equations needed for describing a spin 1 massive
particle:
(∂
2
+ m
2
)A
μ
= 0 (13)
and
∂
μ
A
μ
= 0 (14)
Equation (14) is needed to cut the number of degrees of freedom contained in A
μ
down
from four to three.
Lo and behold, (13) and (14) are equivalent to the single equation
∂
μ
(∂
μ
A
ν
− ∂
ν
A
μ
) + m
2
A
ν
= 0 (15)
Obviously, (13) and (14) together imply (15). To verify that (15) implies (13) and (14), we
act with ∂
ν
on (15) and obtain
m
2
∂A =0 (16)
which for m = 0 requires ∂A = 0, namely (14). Plugging this into (15) we obtain (13).
Having packaged two equations into one, we note that we can derive this single equation
(15) by varying the Lagrangian
L =−
1
4
F
μν
F
μν
+
1
2
m
2
A
2
(17)
with F
μν
≡ ∂
μ
A
ν
− ∂
ν
A
μ
.
Next, suppose we include a source J
μ
for this particle by changing the Lagrangian to
L =−
1
4
F
μν
F
μν
+
1
2
m
2
A
2
+ A
μ
J
μ
(18)
with the resulting equation of motion
∂
μ
(∂
μ
A
ν
− ∂
ν
A
μ
) + m
2
A
ν
=−J
ν
(19)
But now observe that when we act with ∂
ν
on (19) we obtain
m
2
∂A =−∂J (20)
2
For a much more detailed and leisurely discussion, see S. Weinberg, Quantum Theory of Fields, pp. 69–74
and 246–255.
III.4. Gauge Invariance | 189
We recover (14) only if ∂
μ
J
μ
= 0, that is, if the source producing the particle, commonly
know as the current, is conserved.
Put more vividly, suppose the experimentalists who constructed the accelerator (or
whatever) to produce the spin 1 particle messed up and failed to insure that ∂
μ
J
μ
= 0;
then ∂A = 0 and a spin 0 excitation would also be produced. To make sure that the beam
of spin 1 particles is not contaminated with spin 0 particles, the accelerator builders must
assure us that the source J
μ
in the Lagrangian (18) is indeed conserved.
Now, if we want to study massless spin 1 particles, we simply set m = 0 in (18). The
“poor man” ends up (just like the “rich man”) using the Lagrangian
L =−
1
4
F
μν
F
μν
+ A
μ
J
μ
(21)
to describe the photon. Lo and behold (as we exclaimed in chapter II.7), L is left invariant
by the gauge transformation A
μ
→A
μ
−∂
μ
for any (x). (As was also explained in that
chapter, the third polarization decouples in the limit m → 0.) The “poor man” has thus
discovered gauge invariance!
However, as I warned in chapter I.5, depending on his or her personality, the poor man
could also wake up in the middle of the night worrying that physics might be discontinuous
in the limit m → 0. Thus the little group discussion is needed to remove that nightmare.
But then a “real” physicist in the Feynman mode could always counter that for any physical
measurement everything must be okay as long as the duration of the experiment is short
compared to the characteristic time 1/m. More on this issue in chapter VIII.1.
A reflection on gauge symmetry
As we will see later and as you might have heard, much of the world beyond electro-
magnetism is also described by gauge theories. But as we saw here, gauge theories are
also deeply disturbing and unsatisfying in some sense: They are built on a redundancy
of description. The electromagnetic gauge transformation A
μ
→ A
μ
− ∂
μ
is not truly a
symmetry stating that two physical states have the same properties. Rather, it tells us that
the two gauge potentials A
μ
and A
μ
− ∂
μ
describe the same physical state. In your or-
derly study of physics, the first place where A
μ
becomes indispensable is the Schr
¨
odinger
equation, as I will explain in chapter IV.4. Within classical physics, you got along perfectly
well with just
E and
B. Some physicists are looking for a formulation of quantum elec-
trodynamics without using A
μ
, but so far have failed to turn up an attractive alternative
to what we have. It is conceivable that a truly deep advance in theoretical physics would
involve writing down quantum electrodynamics without writing A
μ
.
III.5 Field Theory without Relativity
Slower in its maturity
Quantum field theory at its birth was relativistic. Later in its maturity, it found applications
in condensed matter physics. We will have a lot more to say about the role of quantum field
theory in condensed matter, but for now, we have the more modest goal of learning how
to take the nonrelativistic limit of a quantum field theory.
The Lorentz invariant scalar field theory
L =(∂
†
)(∂) − m
2
†
− λ(
†
)
2
(1)
(with λ>0 as always) describes a bunch of interacting bosons. It should certainly contain
the physics of slowly moving bosons. For clarity consider first the relativistic Klein-Gordon
equation
(∂
2
+ m
2
) = 0 (2)
for a free scalar field. A mode with energy E = m + ε would oscillate in time as ∝e
−iEt
.
In the nonrelativistic limit, the kinetic energy ε is much smaller than the rest mass
m. It makes sense to write (x , t) = e
−imt
ϕ(x, t), with the field ϕ oscillating in time
much more slowly than e
−imt
. Plugging into (2) and using the identity (∂/∂t)e
−imt
(
...
) =
e
−imt
(−im +∂/∂t)(
...
) twice, we obtain (−im +∂/∂t)
2
ϕ −
∇
2
ϕ + m
2
ϕ = 0. Dropping the
term (∂
2
/∂t
2
)ϕ as small compared to −2im(∂/∂t)ϕ, we find Schr
¨
odinger’s equation, as we
had better:
i
∂
∂t
ϕ =−
∇
2
2m
ϕ (3)
By the way, the Klein-Gordon equation was actually discovered before Schr
¨
odinger’s
equation.
III.5. Nonrelativistic Field Theory | 191
Having absorbed this, you can now easily take the nonrelativistic limit of a quantum
field theory. Simply plug
(x , t) =
1
√
2m
e
−imt
ϕ(x, t) (4)
into (1) . (The factor 1/
√
2m is for later convenience.) For example,
∂
†
∂t
∂
∂t
− m
2
†
→
1
2m
im +
∂
∂t
ϕ
†
−im +
∂
∂t
ϕ
− m
2
ϕ
†
ϕ
1
2
i
ϕ
†
∂ϕ
∂t
−
∂ϕ
†
∂t
ϕ
(5)
After an integration by parts we arrive at
L = iϕ
†
∂
0
ϕ −
1
2m
∂
i
ϕ
†
∂
i
ϕ − g
2
(ϕ
†
ϕ)
2
(6)
where g
2
= λ/4m
2
.
As we saw in chapter I.10 the theory (1) enjoys a conserved Noether current J
μ
=
i(
†
∂
μ
− ∂
μ
†
). The density J
0
reduces to ϕ
†
ϕ, precisely as you would expect, while
J
i
reduces to (i/2m)(ϕ
†
∂
i
ϕ −∂
i
ϕ
†
ϕ). When you first took a course in quantum mechanics,
didn’t you wonder why the density ρ ≡ ϕ
†
ϕ and the current J
i
= (i/2m)(ϕ
†
∂
i
ϕ − ∂
i
ϕ
†
ϕ)
look so different? As to be expected, various expressions inevitably become uglier when
reduced from a more symmetric to a less symmetric theory.
Number is conjugate to phase angle
Let me point out some differences between the relativistic and nonrelativistic case.
The most striking is that the relativistic theory is quadratic in time derivative, while
the nonrelativistic theory is linear in time derivative. Thus, in the nonrelativistic theory
the momentum density conjugate to the field ϕ, namely δL/δ∂
0
ϕ, is just iϕ
†
, so that
[ϕ
†
(x , t), ϕ(x
, t)] =−δ
(D)
(x −x
). In condensed matter physics it is often illuminating to
write ϕ =
√
ρe
iθ
so that
L =
i
2
∂
0
ρ − ρ∂
0
θ −
1
2m
ρ(∂
i
θ)
2
+
1
4ρ
(∂
i
ρ)
2
− g
2
ρ
2
(7)
The first term is a total divergence. The second term tells us something of great impor-
tance
1
in condensed matter physics: in the canonical formalism (chapter I.8), the momen-
tum density conjugate to the phase field θ(x) is δL/δ∂
0
θ =−ρ and thus Heisenberg tells
us that
[ρ(x, t), θ(x
, t)] =iδ
(D)
(x −x
) (8)
1
See P. Anderson, Basic Notions of Condensed Matter Physics, p. 235.