Supersymmetric Yang-Mills Quantum Mechanics

Abstract

Abstract. The new approach to quantum mechanical problems is proposed. Quantum states are represented in an algebraic program, by lists of variable length, while operators are well defined functions on these lists. Complete numerical solution of a given system can then be automatically obtained. The method is applied to Wess-Zumino quantum mechanics and D=2 and D=4 supersymmetric Yang-Mills quantum mechanics with the SU(2) gauge group. Convergence with increasing size of the basis was observed in various cases. Many old results were confirmed and some new ones, especially for the D=4 system, are derived. Preliminary results in higher dimensions are also presented. In particular the spectrum of the zero-volume glueballs in 4 < D < 11 is obtained for the first time.

Towards the ground state of the supermembrane

“…Performing the Taylor expansion of the ground state about X = 0, the 0th order term (i.e. the coordinate independent one) for the SU(2) model has been constructed explicitly [17] and proven to be unique [18,19] which confirmed earlier symbolic results using Mathematica [20]. The 1st order term is now also available and turns out to be unique as well [21].…”

“…The unique candidate for |φ which we denote here by |S , has been constructed in [17,20], and the unique candidate for |φ a i which satisfies (2.4) has also been constructed in [21]. Thus we have the starting points for the sequences m = 0 and 1.…”

On gauge transformation property of coordinate independent SO(9) vector states in SU(2) Matrix Theory

“…The first term of the expansion, which is independent of X a i , must be gauge and SO(9) invariant [1]. Construction of such states is not an easy mathematical problem, and in the case of SU(2) gauge group, it has been constructed in [2,3], and shown that it is unique [3,4].…”

“…where φ a 1 ...an i 1 ...in are states constructed by acting creation operators made of θ a α on the vacuum for those operators. The leading term |φ has been constructed in [2,3].…”

Optimized Fock space in the large N limit of quartic interactions in matrix models

“…Despite a number of profound results concerning zero energy eigenfunctions, e.g. an explicit construction in the pure fermionic sector for the n = 2 model [22], large x behaviour [23], or existence results on compact regions [24], the question posed in the full generality remains open. We believe that the approach discussed in this paper can shed new light on this problem since our results correspond to the purely bosonic sector of the supermembrane and our construction can be possibly extended to the fermionic sectors allowing to study the embedded part of the spectrum of the supermembrane.…”