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Register a new userNumerical Study of the Simplest String Bit Model
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String bit models provide a possible method to formulate a string as a discrete chain of pointlike string bits. When the bit number $M$ is large, a chain behaves as a continuous string. We study the simplest case that has only one bosonic bit and one fermionic bit. The creation and annihilation operators are adjoint representations of the $Uleft(Nright)$ color group. We show that the supersymmetry reduces the parameter number of a Hamiltonian from 7 to 3 and, at $N=infty$, ensures a continuous energy spectrum, which implies the emergence of one spatial dimension. The Hamiltonian $H_{0}$ is constructed so that in the large $N$ limit it produces a world sheet spectrum with one Grassmann world sheet field. We concentrate on numerical study of the model in finite $N$. For the Hamiltonian $H_{0}$, we find that the would-be ground energy states disappear at $N=left(M-1right)/2$ for odd $Mleq11$. Such a simple pattern is spoiled if $H$ has an additional term $xiDelta H$ which does not affect the result of $N=infty$. The disappearance point moves to higher (lower) $N$ when $xi$ increases (decreases). Particularly, the $pmleft(H_{0}-Delta Hright)$ cases suggest a possibility that the ground state could survive at large $M$ and $Mgg N$. Our study reveals that the model has stringy behavior: when $N$ is fixed and large enough, the ground energy decreases linearly with respect to $M$, and the excitation energy is roughly of order $M^{-1}$. We also verify that a stable system of Hamiltonian $pm H_{0}+xiDelta H$ requires $xigeqmp1$.
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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.
We provide a formalism to calculate the cubic interaction vertices of the stable string bit model, in which string bits have $s$ spin degrees of freedom but no space to move. With the vertices, we obtain a formula for one-loop self-energy, i.e., the $mathcal{O}left(1/N^{2}right)$ correction to the energy spectrum. A rough analysis shows that, when the bit number $M$ is large, the ground state one-loop self-energy $Delta E_{G}$ scale as $M^{5-s/4}$ for even $s$ and $M^{4-s/4}$ for odd $s$. Particularly, in $s=24$, we have $Delta E_{G}sim 1/M$, which resembles the Poincare invariant relation $P^{-}sim 1/P^{+}$ in $(1+1)$ dimensions. We calculate analytically the one-loop correction for the ground energies with $M=3$ and $s=1,,2$. We then numerically confirm that the large $M$ behavior holds for $sleq4$ cases.
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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 ${barphi}^A_{a_1cdots a_n}$, ${phi}^A_{a_1cdots a_n}$,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.
We show that planar cal N=4 Yang-Mills theory at zero t Hooft coupling can be efficiently described in terms of 8 bosonic and 8 fermionic oscillators. We show that these oscillators can serve as world-sheet variables, the string bits, of a discretized string. There is a one to one correspondence between the on shell gauge invariant words of the free Y-M theory and the states in the oscillators Hilbert space, obeying a local gauge and cyclicity constraints. The planar two-point functions and the three-point functions of all gauge invariant words are obtained by the simple delta-function overlap of the corresponding discrete string world sheet. At first order in the t Hooft coupling, i.e. at one-loop in the Y-M theory, the logarithmic corrections of the planar two-point and the three-point functions can be incorporated by nearest neighbour interactions among the discretized string bits. In the SU(2) sub-sector we show that the one-loop corrections to the structure constants can be uniquely determined by the symmetries of the bit picture. For the SU(2) sub-sector we construct a gauged, linear, discrete world-sheet model for the oscillators, with only nearest neighbour couplings, which reproduces the anomalous dimension Hamiltonian up to two loops. This model also obeys BMN scaling to all loops.
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