d
3
~xβ
(
~x
) (
h
Ω

φ
(
~x, t
)

Ω
i 
φ
0
(
~x
))
(5.23)
53
with respect to
α
,
β
(
~x
), and minimizing it with respect to unconstrained

Ω
i
.
We
obtain in particular
H

Ω
i
=
α

Ω
i
+
Z
d
3
~xβ
(
~x
)
φ
(
~x, t
)

Ω
i
.
(5.24)
This equation can be read as one that determines

Ω
i
given
α
and
β
(
~x
).
α
and
β
(
~x
)
are then adjusted so that (5.22) is satisfied (at a given time
t
). We can now interpret
β
(
~x
) as an external current, and
H

Z
d
3
~xβ
(
~x
)
φ
(
~x, t
)
(5.25)
as the Hamiltonian in the presence of the current
β
(
~x
), derived from the modified
action
Z
d
4
x
L
+
β
(
~x
)
φ
(
~x, t
)
.
(5.26)
α
is the ground state energy in the presence of the current
β
(
~x
). This follows from the
assumption that
h
Ω

H

Ω
i
is minimized by our

Ω
i
.
This ground state energy
α
can
also be computed from the transition amplitude
h
0
, out

0
, in
i
β
=
e

iαT
=
e
iW
[
β
]
,
(5.27)
where the current
β
(
~x
) is turned on adiabatically, and turned off adiabatically after
a long time
T
.
In this adiabatic process the vacuum changes into the new ground
state with respect to the current modified Hamiltonian. The result is nothing but our
generating function
e
iW
[
β
]
. Now we can derive the minimal energy
h
Ω

H

Ω
i
=
α
h
Ω

Ω
i
+
Z
d
3
~xβ
(
~x
)
h
Ω

φ
(
~x, t
)

Ω
i
=
α
+
Z
d
3
~xβ
(
~x
)
φ
0
(
~x
)
=
1
T

W
[
β
] +
Z
d
4
xβ
(
~x
)
φ
0
(
~x
)
=

1
T
Γ[
φ
0
] =
V
3
V
(
φ
0
)
.
(5.28)
This result shows that the effective potential
V
(
φ
0
) has the interpretation of the
min
imal
energy subject to the constraint
h
φ
(
x
)
i
=
φ
0
.
Note that the second order derivative of
V
(
φ
0
) in
φ
0
is the inverse of the twopoint
function of
φ
in the matrix sense, evaluated at zero momentum. The latter is positive
definite in the matrix sense. This implies that
V
(
φ
0
) is a convex function. Suppose
we have a scalar field theory with a quartic coupling, and negative
m
2
, in its classical
action. The potential is still bounded from below, but is not convex. What we just
54
saw is that the quantum effective potential must be convex, seemingly contradicting
(5.20).
In fact, in deriving (5.20) we have assumed a stable vacuum with vanishing
expectation value of
φ
(
x
), which is not the case when
m
2
<
0. In this case, there are
two vacuum states

Ω
1
i
and

Ω
2
i
, in which the expectation value of
φ
(
x
) (to leading
order in perturbation theory) is at the one of the two minima of the classical potential,
h
φ
(
x
)
i
=
φ
±
≡ ±
s
6

m
2

g
.
(5.29)
If we restrict
h
φ
(
x
)
i
to lie between
φ

and
φ
+
, we can minimize the energy with a state
of the form

Ω
i
=
c
1

Ω
1
i
+
c
2

Ω
2
i
,
h
φ
i
Ω
=

c
1

2
φ

+

c
2

2
φ
+
∈
(
φ

, φ
+
)
.
(5.30)
Thus the true effective potential
V
(
φ
0
) is a
constant
in between
φ

and
φ
+
! We see
that perturbation theory fails to capture this effective potential (in fact, the oneloop
calculation would give a complex effective potential in between
φ

and
φ
+
).
5.3
Renormalization of nonabelian gauge theory
We will now deal with the renormalization of nonabelian gauge theory by studying the
1PI effective action
Γ[
A
μ
, ψ, η,
¯
η
]
(5.31)
where
ψ
denotes matter fields (transforming in some representation
R
of the gauge
group
G
), and
η,
¯
η
are the FadeevPopov ghosts.
Γ is computed by shifting
A
μ
→
A
0
μ
+
A
μ
, etc., and summing up 1PI diagrams of the quantum fluctuations
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 Summer '19
 Physics, Quantum Field Theory, φ, Lorentz, soft photon