By John S Shearer
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38) gives a sum of simple harmonic oscillator Lagrangians. After quantization, the energy eigenstates are labeled by the occupation numbers nj of the various oscillators, with their energies being Equad = nj + j 1 2 ωj . 42) There is an immediate problem, because the sum of the zero-point energies is inﬁnite. Putting aside for the moment the details of how to regulate this inﬁnity, we subtract from the Lagrangian an appropriate constant, δE, whose value is chosen to make the energy of the vacuum ﬁnite (and perhaps zero).
Instead, it seems more reasonable to simply say that all of the kink states have become doubly degenerate. 5 Kinks in more spacetime dimensions 27 momentum P and vanishing occupation numbers for all bosonic modes and all nonzero fermionic modes are related by a† |P − = |P + , a|P + = |P − . 92) One way to express this is to assign a fractional fermion number, ±1/2, to these two basic states, with integer increments then arising from the action of the b†q and d†q . One can also envision situations with more than one fermionic zero mode; this is often the case for solitons in higher dimensions.
Examination of the largetime behavior of Eq. 98) shows that there is a net time advance √ 2 1 − u2 ln |u|. 100) Δt = − mu Since one can show that the long-range force between kinks is repulsive, Eq. 100) seems inconsistent with the former interpretation, but it is quite reasonable if the solution is interpreted as two solitons that approach each other but reverse direction before they meet. 101) that describes two solitons that collide, pass through each other, and then emerge undeformed, as shown in Fig.