1ϕ4 kinks
III Classical and Quantum Solitons
1.2 Dynamic kink
We now want to look at kinks that move. Given what we have done so far, this
is trivial. Our theory is Lorentz invariant, so we simply apply a Lorentz boost.
Then we obtain a field
φ(x, t) = tanh γ(x − vt),
where, as usual
γ = (1 − v
2
)
−1/2
.
But this isn’t all. Notice that for small
v
, we can approximate the solution
simply by
φ(x, t) = tanh(x − vt).
This looks like a kink solution with a modulus that varies with time slowly. This
is known as the adiabatic point of view.
More generally, let’s consider a “moving kink” field
φ(x, t) = tanh(x − a(t))
for some function
a
(
t
). In general, this is not a solution to the field equation,
but if ˙a is small, then it is “approximately a solution”.
We can now explicitly compute that
˙
φ = −
da
dt
φ
0
.
Let’s consider fields of this type, and look at the Lagrangian of the field theory.
The kinetic term is given by
T =
Z
1
2
˙
φ
2
dx =
1
2
da
dt
2
Z
φ
02
dx =
1
2
M
da
dt
2
.
To derive this result, we had to perform the integral
R
φ
02
d
x
, and if we do
that horrible integral, we will find a value that happens to be equal to
M
=
4
3
.
Of course, this is not a coincidence. We can derive this result from Lorentz
invariance to see that the result of integration is manifestly M .
The remaining part of the Lagrangian is less interesting. Since it does not
involve taking time derivatives, the time variation of
a
is not seen by it, and we
simply have a constant
V =
4
3
.
Then the original field Lagrangian becomes a particle Lagrangian
L =
1
2
M ˙a
2
−
4
3
.
Note that when we first formulated the field theory, the action principle
required us to find a field that extremizes the action among all fields. However,
what we are doing now is to restrict to the set of kink solutions only, and then
when we solve the variational problem arising from this Lagrangian, we are
extremizing the action among fields of the form
tanh
(
x −a
(
t
)). We can think of
this as motion in a “valley” in the field configuration space. In general, these
solutions will not also extremize the action among all fields. However, as we
said, it will do so “approximately” if ˙a is small.
We can obtain an effective equation of motion
M¨a = 0,
which is an equation of motion for the variable a(t) in the moduli space.
Of course, the solution is just given by
a(t) = vt + const,
where
v
is an arbitrary constant, which we interpret as the velocity. In this
formulation, we do not have any restrictions on
v
, because we took the “non-
relativistic approximation”. This approximation breaks down when v is large.
There is a geometric interpretation to this. We can view the equation of
motion
M¨a
= 0 as the geodesic equation in the moduli space
R
, and we can think
of the coefficient
M
as specifying a Riemannian metric on the moduli space. In
this case, the metric is (a scalar multiple of) the usual Euclidean metric (da)
2
.
This seems like a complicated way of describing such a simple system, but
this picture generalizes to higher-dimensional systems and allows us to analyze
multi-soliton dynamics, in particular, the dynamics of vortices and monopoles.
We might ask ourselves if there are multi-kinks in our theory. There aren’t in
the
φ
4
theory, because we saw that the solutions are classified by the boundary
conditions, and we have already enumerated all the possible boundary conditions.
In more complicated theories like sine-Gordon theory, multiple kinks are possible.
However, while we cannot have two kinks in
φ
4
theory, we can have a kink
followed by an anti-kink, or more of these pairs. This actually lies in the “vacuum
sector” of the theory, but it still looks like it’s made up of kinks and anti-kinks,
and it is interesting to study these.