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arXiv:2112.01588

String Bit Description of Antiperiodic Fermion Worldsheet Fields

Charles B. Thorn

Published Dec 2, 2021• 0 citations
arXivHTMLPDF

Abstract

We study a string bit Hamiltonian whose continuum limit describes anti-periodic (AP) anti-commuting worldsheet fields. We calculate the amplitude for transitions between an AP spin chain and a periodic (P) one in the continuum limit, M-> infinity where M is the bit number of either chain. We also numerically evaluate the corresponding amplitudes at increasing finite M to assess the convergence rate to the continuum. We then give the overlap equations for the transition AP+AP->AP, and numerically solve them for increasing $M$ values at a fixed value of x=K/M, where M is the bit number of the large chain and K is the bit number of one of the smaller chains. For this case, in contrast to the situation with an even number of AP chains, there is an obstacle to directly finding the continuum limit analytically. We suggest an indirect analytic approach to this problem: using the AP->P transition followed by a P->AP+AP transition, each of which has a relatively simple analytic continuum limit. We also show how bosonization of the fermion fields enables an analytic recursive evaluation of the AP$+$AP$\to$AP amplitudes.

DOI
10.48550/arxiv.2112.01588

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Although the energy spectrum of the Heisenberg spin chain on a circle defined by H=14Σk=1Mσkxσk+1x+σkyσk+1y+Δσkzσk+1z is well known for any fixed M, the boundary conditions vary according to whether Mϵ 4N + r, where r = -1,0,1,2, and also according to the parity of the number of overturned spins in the state, In string theory all these cases must be allowed because interactions involve a string with M spins breaking into strings with M1 < M and M - M1spins (or vice versa). We organize the energy spectrum and degeneracies of H in the case Δ = 0 where the system is equivalent to a system of free fermions. In spite of the multiplicity of special cases, in the limit M → ∞ the spectrum is that of a free compactified worldsheet field. Such a field can be interpreted as a compact transverse string coordinate x(σ) ≡ x(σ) + R0. We construct the bosonization formulas explicitly in all separate cases, and for each sector give the Virasoro conformal generators in both fermionic and bosonic formulations. Furthermore from calculations in the literature for selected classes of excited states, there is strong evidence that the only change for Δ ≠ 0 is a change in the compactification radius R0→ RΔ. As Δ→ -1 this radius goes to infinity, giving a concrete example of noncompact space emerging from a discrete dynamical system. Finally we apply our work to construct the three string vertex implied by a string whose bosonic coordinates emerge from this mechanism.
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