III Classical and Quantum Solitons - ϕ<sup>4</sup> kinks

1ϕ⁴ kinks

III Classical and Quantum Solitons

1.4 Quantization of kink motion

We now briefly talk about how to quantize kinks. The most naive way of doing

so is pretty straightforward. We use the moduli space approximation, and then

we have a very simple kink Lagrangian.

L =

M ˙a

This is just a free particle moving in

with mass

. This

is known as the

collective coordinate of the kink. Quantizing a free particle is very straightforward.

It is just IB Quantum Mechanics. For completeness, we will briefly outline this

procedure.

We first put the system in Hamiltonian form. The conjugate momentum to

a is given by

P = M ˙a.

Then the Hamiltonian is given by

H = P ˙a − L =

Then to quantize, we replace

by the operator

−i~

∂

∂a

. In this case, the quantum

Hamiltonian is given by

H = −

∂

∂a

A wavefunction is a function of

and

, and this is just ordinary QM for a single

particle.

As usual, the stationary states are given by

ψ(a) = e

iκa

and the momentum and energy (eigenvalues) are

P = ~κ, H = E =

Is this actually “correct”? Morally speaking, we really should quantize the

complete 1 + 1 dimensional field theory. What would this look like?

In normal quantum field theory, we consider perturbations around a vacuum

solution, say

φ ≡

1, and we obtain mesons. Here if we want to quantize the

kink solution, we should consider field oscillations around the kink. Then the

solution contains both a kink and a meson. These mesons give rise to quantum

corrections to the kink mass M.

Should we be worried about these quantum corrections? Unsurprisingly, it

turns out these quantum corrections are of the order of the meson mass. So we

should not be worried when the meson mass is small.

Meson-kink scattering can also be studied in the full quantum theory. To

first approximation, since the kink is heavy, mesons are reflected or transmitted

with some probabilities, while the momentum of the kink is unchanged. But

when we work to higher orders, then of course the kink will move as a result.

This is all rather complicated.

For more details, see Rajaraman’s Solitons and Instantons, or Weinberg’s

Classical Solutions in Quantum Field Theory.

The thing that is really hard to understand in the quantum field theory

is kink-antikink pair production. This happens in meson collisions when the

mesons are very fast, and the theory is highly relativistic. What we have done

so far is perturbative and makes the non-relativistic approximation to get the

adiabatic picture. It is very difficult to understand the highly relativistic regime.