arXiv subjects

Charles B. Thorn

Publications and source records attributed to Charles B. Thorn.

At least 19 recordsLinked to original sources

arXiv2022

Four String Amplitudes for the Generalized Protostring

We specialize the $N$ string scattering amplitudes for the generalized protostring to $N=4$. This allows for a much more detailed and explicit study of their basic physical and mathematical properties, such as singularity structure and high energy behavior. Since this class of models does not enjoy full Poincaré invariance, the high energy behavior depends on the Lorentz frame. The relative simplicity of the four string amplitudes allows a direct understanding of the complications due to this non-covariance.

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arXiv2022

String Bit Description of Antiperiodic Fermion Worldsheet Fields

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.

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arXiv2021

Heisenberg spin chain as a worldsheet coordinate for lightcone quantized string

Although the energy spectrum of the Heisenberg spin chain on a circle defined by $H=\frac{1}{4}\sum_{k=1}^M(σ_k^xσ_{k+1}^x+σ_k^yσ_{k+1}^y +Δσ_k^zσ_{k+1}^z)$ is well known for any fixed $M$, the boundary conditions vary according to whether $M\in 4\mathbb{N}+r$, 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 $M_1<M$ and $M-M_1$ spins (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 $M\to\infty$ the spectrum is that of a free compactified worldsheet field. Such a field can be interpreted as a compact transverse string coordinate $x(σ)\equiv x(σ)+R_0$. 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 $Δ\neq0$ is a change in the compactification radius $R_0\to R_Δ$. As $Δ\to-1$ 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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arXiv2020

Internal and Super Symmetry in String Bit Models

We study in a general way the construction of string bit Hamiltonians which are supersymmetric, We construct several quadratic and quartic polynomials in string bit creation and annihilation operators ${\barϕ}^A_{a_1\cdots a_n}$, $ϕ^A_{a_1\cdots a_n}$,which commute with the supersymmetry generators $Q^a$. Among these operators are ones with the spinor tensor structure required to provide the lightcone worldsheet vertex insertion factors needed to give the correct interactions for the IIB superstring, whenever a closed string separates into two closed strings or two closed strings join into one.

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arXiv2019

Color Characters for White Hot String Bits

The state space of a generic string bit model is spanned by $N\times N$ matrix creation operators acting on a vacuum state. Such creation operators transform in the adjoint representation of the color group $U(N)$ (or $SU(N)$ if the matrices are traceless). We consider a system of $b$ species of bosonic bits and $f$ speciesof fermionic bits. The string, emerging in the $N\to\infty$ limit, identifies $P^+=mM\sqrt{2}$ with $M$ the bit number operator and $P^-=H\sqrt{2}$ with $H$ the system Hamiltonian. We study the thermal properties of this string bit system in the case $H=0$, which can be considered the tensionless string limit: the only dynamics is restricting physical states to color singlets. Then the thermal partition function ${\rm Tr} e^{-βmM}$ can be identified, putting $x=e^{-βm}$, with a generating function $χ_0^{bf}(x)$, for which the coefficient of $x^n$ in its expansion about $x=0$ is the number of color singlets with bit number $M=n$. This function is a purely group theoretic object, which is well-studied in the literature. We show that at $N=\infty$ this system displays a Hagedorn divergence at $x=1/(b+f)$ with ultimate temperature $T_H=m/\ln(b+f)$. The corresponding function for finite $N$ is perfectly finite for $0<x<1$, so the $N=\infty$ system exhibits a phase transition at temperature $T_H$ which is absent for any finite $N$. We demonstrate that the low temperature phase is unstable above $T_H$. The lowest-order $1/N$ asymptotic correction, for $x\to1$ in the high temperature phase, is computed for large $N$. Remarkably, this is related to the number of labeled Eulerian digraphs with $N$ nodes. Systematic methods to extend our results to higher orders in $1/N$ are described.

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arXiv2017

String Bits at Finite Temperature and the Hagedorn Phase

We study the behavior of a simple string bit model at finite temperature. We use thermal perturbation theory to analyze the high temperature regime. But at low temperatures we rely on the large $N$ limit of the dynamics, for which the exact energy spectrum is known. Since the lowest energy states at infinite $N$ are free closed strings, the $N=\infty$ partition function diverges above a finite temperature $β_H^{-1}$, the Hagedorn temperature. We argue that in these models at finite $N$, which then have a finite number of degrees of freedom, there can be neither an ultimate temperature nor any kind of phase transition. We discuss how the discontinuous behavior seen at infinite $N$ can be removed at finite $N$. In this resolution the fundamental string bit degrees of freedom become more active at temperatures near and above the Hagedorn temperature.

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arXiv2016

Scientific Biography of Stanley Mandelstam, Part I: 1955-1980

I review Stanley Mandelstam's many contributions to particle physics, quantum field theory and string theory covering the years 1955 through 1980. His more recent work will be reviewed by Nathan Berkovits. This is my contribution to the Memorial Volume for Stanley Mandelstam (World Scientific, 2017).

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arXiv2016

Protostring Scattering Amplitudes

We calculate some tree level scattering amplitudes for a generalization of the protostring, which is a novel string model implied by the simplest string bit models. These bit models produce a lightcone worldsheet which supports $s$ integer moded Grassmann fields. In the generalization we supplement this Grassmann worldsheet system with $d=24-s$ transverse coordinate worldsheet fields. The protostring corresponds to $s=24$ and the bosonic string to $s=0$. The interaction vertex is a simple overlap with no operator insertions at the break/join point. Assuming that $s$ is even we calculate the multi-string scattering amplitudes by bosonizing the Grassmann fields, each pair equivalent to one compactified bosonic field, and applying Mandelstam's interacting string formalism to a system of $s/2$ compactified and $d$ uncompactified bosonic worldsheet fields. We obtain all amplitudes for open strings with no oscillator excitations and for closed strings with no oscillator excitations and zero winding number. We then study in detail some simple special cases. Multi-string processes with maximal helicity violation have much simplified amplitudes. We also specialize to general four string amplitudes and discuss their high energy behavior. Most of these models are not covariant under the full Lorentz group $O(d+1,1)$. The exceptions are the bosonic string whose Lorentz group is $O(25,1)$ and the protostring whose Lorentz group is $O(1,1)$. The models in between only enjoy an $O(1,1)\times O(d)$ spacetime symmetry.

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arXiv2015

1/N Perturbations in Superstring Bit Models

We develop the 1/N expansion for stable string bit models, focusing on a model with bit creation operators carrying only transverse spinor indices a=1,...,s. At leading order (1/N=0), this model produces a (discretized) lightcone string with a "transverse space' of $s$ Grassmann worldsheet fields. Higher orders in the 1/N expansion are shown to be determined by the overlap of a single large closed chain (discretized string) with two smaller closed chains. In the models studied here, the overlap is not accompanied with operator insertions at the break/join point. Then the requirement that the discretized overlap have a smooth continuum limit leads to the critical Grassmann "dimension" of s=24. This "protostring", a Grassmann analog of the bosonic string, is unusual, because it has no large transverse dimensions. It is a string moving in one space dimension and there are neither tachyons nor massless particles. The protostring, derived from our pure spinor string bit model, has 24 Grassmann dimensions, 16 of which could be bosonized to form 8 compactified bosonic dimensions, leaving 8 Grassmann dimensions--the worldsheet content of the superstring. If the transverse space of the protostring could be "decompactified", string bit models might provide an appealing and solid foundation for superstring theory.

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arXiv2014

Space from String Bits

We develop superstring bit models, in which the lightcone transverse coordinates in D spacetime dimensions are replaced with d=D-2 double-valued "flavor" indices $x^k-> f_k=1,2$; $k=2,...,d+1$. In such models the string bits have no space to move. Letting each string bit be an adjoint of a "color" group U(N), we then analyze the physics of 't Hooft's limit $N->\infty$, in which closed chains of many string bits behave like free lightcone IIB superstrings with d compact coordinate bosonic worldsheet fields $x^k$, and s pairs of Grassmann fermionic fields $θ_{L,R}^a$, a=1,..., s. The coordinates $x^k$ emerge because, on the long chains, flavor fluctuations enjoy the dynamics of d anisotropic Heisenberg spin chains. It is well-known that the low energy excitations of a many-spin Heisenberg chain are identical to those of a string worldsheet coordinate compactified on a circle of radius $R_k$, which is related to the anisotropy parameter $-1<Δ_k<1$ of the corresponding Heisenberg system. Furthermore there is a limit of this parameter, $Δ_k->\pm 1$, in which $R_k->\infty$. As noted in earlier work [Phys.Rev.D{\bf 89}(2014)105002], these multi-string-bit chains are strictly stable at $N=\infty$ when d<s and only marginally stable when d=s. (Poincare supersymmetry requires d=s=8, which is on the boundary between stability and instability.)

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arXiv2014

Stable String Bit Models

In string bit models, the superstring emerges as a very long chain of "bits", in which s fermionic degrees of freedom contribute positively to the ground state energy in a way to exactly cancel the destabilizing negative contributions of d=s bosonic degrees of freedom. We propose that the physics of string formation be studied nonperturbatively in the class of string bit models in which s>d, so that a long chain is stable, in contrast to the marginally stable (s=d=8) superstring chain. We focus on the simplest of these models with s=1 and d=0, in which the string bits live in zero space dimensions. The string bit creation operators are N X N matrices. We choose a Hamiltonian such that the large N limit produces string moving in one space dimension, with excitations corresponding to one Grassmann lightcone worldsheet field (s=1) and no bosonic worldsheet field (d=0). We study this model at finite N to assess the role of the large N limit in the emergence of the spatial dimension. Our results suggest that string-like states with large bit number M may not exist for N<(M-1)/2. If this is correct, one can have finite chains of string bits, but not continuous string, at finite N. Only for extremely large N can such chains behave approximately like continuous string, in which case there will also be the (approximate) emergence of a new spatial dimension. In string bit models designed to produce critical superstring at N=infinity, we can then expect only approximate Lorentz invariance at finite N, with violations of order 1/N^2.

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arXiv2013

6-Vertex Model on an Open String Worldsheet

We propose boundary conditions on a two dimensional 6-vertex model, which is defined on the lightcone lattice for an open string worldsheet. We show that, in the continuum limit, the degrees of freedom of this 6-vertex model describe a target space coordinate compactified on a circle of radius R, which is related to the vertex weights. This conclusion had already been established for the case of a 6-vertex model on the worldsheet lattice for the propagator of a closed string. This exercise illustrates how the Bethe ansatz works in the presence of boundaries, at least of this particular type.

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arXiv2013

Open String Self-energy on the Lightcone Worldsheet Lattice

We continue our study of open string perturbation theory on the lightcone worldsheet lattice, which is an $M\times N$ rectangular grid. Here $M$ is the number of $P^+$ units and $N$ is the number of $ix^+$ units. We extend our previous analysis to the bosonic open string one planar loop self-energy. We find that, when all open string coordinates satisfy Neumann conditions, the ultraviolet worldsheet divergences associated with the closed string tachyon and boundary effects can be cancelled by renormalization of bulk ($AM^1$) and boundary ($BM^0$) worldsheet "cosmological constants". The bulk divergence for the open string matches that for the closed string. The open string tachyon mass shift displays the dilaton logarithmic divergence with the correct coefficient for its consistent absorption by renormalization of the string tension. The ultraviolet contribution to the open string gluon mass shift vanishes, in accord with its interpretation as a gauge particle. We also find that when the bosonic string ends on a D-brane additional negative powers of $\ln M$ multiply the bulk and boundary divergences. These can no longer be cancelled by the "cosmological constants", perhaps pointing to the need, in the presence of D-branes, for the cancellations of divergences provided by supersymmetry.

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arXiv2013

Worldsheet Propagator on the Lightcone Worldsheet Lattice

We develop new more powerful techniques, based on an almost closed form for the lattice worldsheet propagator, for analyzing planar open string worldsheets defined on a lightcone lattice. We show that results obtained in earlier work are easily reproduced with far more precision. In particular, consistency checks which required numerical analysis in the earlier work can now be confirmed exactly.

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arXiv2012

Null Physical States in String Models

This note is a brief addendum to my article Nucl. Phys. B 864 (2012) 285, [arXiv: 1110.5510], which discusses the noghost theorem in Ramond sectors of string models. In this addendum we derive additional information about the structure of null physical states in the Ramond-Neveu-Schwarz model.

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arXiv2012

Closed String Self-energy on the Lightcone Worldsheet Lattice

We study the one loop correction to the closed bosonic string propagator, including the possibile presence of D-branes, by discretizing the light cone worldsheet on an M times N rectangular lattice, with M proportional to P^+ and N+1 proportional to ix^+. The integrals over the moduli then become sums which we evaluate numerically. The main purpose of this study is to assess the reliability of the worldsheet lattice as a regulator of the divergences in string perturbation theory. There are two natural geometrical counterterms for the lightcone worldsheet, one proportional to the area of the worldsheet and the other proportional to the length of worldsheet boundaries, tracing the ends of open strings. We show that the divergences in the closed string self-energy can be cancelled by the area counterterm and a renormalization of the Regge slope parameter. The residual finite part is compatible with Lorentz invariance, provided a novel regularization, natural to the lightcone worldsheet lattice and described in this article, is employed.

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arXiv2012

Improved Proof of the No-ghost Theorem for Fermion States of the Superstring

The purpose of this note is to extend the improved proof of the no-ghost theorem for the bosonic and Neveu-Schwarz dual resonance models, presented in my article Nuclear Physics B286 (1987) 61, to cover the Ramond fermion string. As in that paper, the improvement involves the identification of an efficient basis for string state space and a self-contained proof, based on the super-Virasoro algebra, of the linear independence of the basis elements. We use our results to calculate the BRST cohomology for this system.

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arXiv2012

Determinants for the Lightcone Worldsheet

The evaluation of the determinant of the Laplacian defined on two dimensional regions of various shapes is an essential ingredient in calculating the scattering amplitudes of strings. In lightcone parameterization the regions are rectangular in shape with several slits of different length and location cut parallel to the $τ$ axis of the rectangle. This paper offers a compendium of applications of the methods of Kac and McKean and Singer to the calculation of such worldsheet determinants. Particular attention is paid to the effect of corners on the determinants. The effect of corners joining edges with like boundary conditions is implicit in Kac's results. We discuss the generalization to a corner joining a Dirichlet edge to a Neumann edge, and apply it to a scattering amplitude involving D-branes.

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