String Theory
All Roads Lead to String Theory (Polchinski)
Prior to the First Superstring Revolution
Early History S-Matrix Theory
Regge Trajectory
Bosonic String Theory Worldsheet
Bosonic String Theory
String Perturbation Theory
Tachyon Condensation
Supersymmetric Revolution Supersymmetry
RNS Formalism
GS Formalism
Superstring Revolutions
First Superstring Revolution GSO Projection
Type II String Theory
Type IIB String Theory
Type IIA String Theory
Type I String Theory
Type H String Theory
Type HO String Theory
Type HE String Theory
Second Superstring Revolution T-Duality
Horava-Witten String Theory
Holographic Principle
N=4 Super-Yang-Mills Theory
BFSS Matrix Theory
Matrix String Theory
(2,0) Theory
Twistor String Theory
String Field Theory
Pure Spinor Formalism
After the Revolutions
Phenomenology String Theory Landscape
Minimal Supersymmetric Standard Model
String Phenomenology

Matrix String Theory is a non-perturbative formulation of Type IIA String Theory and Type HE String Theory. It was discovered by Lubos Motl in 1997 [1] [2] [3] [4] [5] .

Type IIA Matrix String TheoryEdit this section

Type IIA Matrix String Theory is the Matrix Model formulation of Type IIA String Theory.

The momenta in BFSS Matrix Theory are quantised as $ p=\frac{N}{R} $/. In BFSS Matrix Theory, $ N\to\infty $. Therefore, the explanation given for this by Motl is that in uncompactified M-Theory, $ R $ is $ \infty $.

Since when M-Theory is compactified on a light-like circle, this $ R $ becomes finite, so should $ N $, to have a well-defined momenta.

M-Theory compactified on a light-circle is related by a Lorentz Transform to M-Theory compactified on a space-like circle. This is because a Lorentz Transform transforms between space-like coordinates and time-like ones.

The actual Lorentz Transform2 can be calculated as follows:

$ \begin{align} & \gamma \left[ \begin{matrix} 1 & -v \\ -v & 1 \\ \end{matrix} \right]\gamma \left[ \begin{matrix} 1 & -v \\ -v & 1 \\ \end{matrix} \right]\left[ \begin{matrix} {{x}_{0}}-2\pi R \\ {{x}_{1}}+2\pi R \\ \end{matrix} \right]=\left[ \begin{matrix} {{x}_{0}} \\ {{x}_{1}}+2\pi R \\ \end{matrix} \right] \\ & \gamma \left[ \begin{matrix} {{x}_{0}}-2\pi R-v{{x}_{1}}-2\pi Rv \\ {{x}_{1}}+2\pi R-v{{x}_{0}}+2\pi Rv \\ \end{matrix} \right]=\left[ \begin{matrix} {{x}_{0}} \\ {{x}_{1}}+2\pi R \\ \end{matrix} \right] \\ & \left. \begin{align} & \frac{1-\sqrt{1-{{v}^{2}}}}{\sqrt{1-{{v}^{2}}}}{{x}_{0}}-2\pi R\left( 1+v \right)-\frac{v}{\sqrt{1-{{v}^{2}}}}{{x}_{1}}=0 \\ & \frac{1-\sqrt{1-{{v}^{2}}}}{\sqrt{1-{{v}^{2}}}}{{x}_{1}}+2\pi Rv-\frac{v}{\sqrt{1-{{v}^{2}}}}{{x}_{0}}=0 \\ \end{align} \right\} \\ \end{align} $

Type IIA Matrix String Theory still has a gauge group of $ U(N) $, but with a finite $ N $, .

Type HE Matrix String TheoryEdit this section

Type HE Matrix String Theory is much more difficult than Type IIA Matrix String Theory, because to have an 11-Dimensional Interpretation (a consistent M-Theory which gives rise to this theory when the=re is a Horava-Witten Boundary), extra fields, called the $ \chi $-fields (also known as the "Chi-Fields"), need to be added. See [1]. Equation 3.1 of [1] is basically, as one may see from the Dirac Field Lagrangian Density, that:

$ \begin{align} & \mathsf{\mathcal{L}}=\chi \nabla \chi \\ & \text{ }=\chi \left( {{\partial }_{\mu }}-i{{A}_{\mu }} \right)\chi \\ & \text{ }=\chi \left( {{\partial }_{0}}+2\pi {{R}_{1}}{{\partial }_{1}}-i\left( {{A}_{0}}+i{{A}_{1}} \right) \right)\chi \\ \end{align} $

Note that the first $ \chi $ is not conjugated, because this is unnecessary (the $ \chi $-field is real).

ReferencesEdit this section