Abstract
We attempt to systematically develop the matrix-model/quantum-theory
correspondence by working out explicitly various non-trivial examples.
Semester
Fall 2019-2020
Level
Master II, Theoretical Physics, Badji-Mokhtar Annaba University.
Phi-four in two dimensions, the disordered/uniform 2nd order phase transition, Metropolis algorithm, critical behavior in 2 and 4 dimensions, continuum limit and renormalization group equation.
-Model II: Matrix Models (Real Quartic Matrix Model)
The quartic matrix model, the disordered/non-uniform 3rd order phase transition, Metropolis algorithm, multitrace matrix models and noncommutative phi-four, Wigner semi-circle law and emergent geometry.
-Model III: Gauge Models (M-(atrix) Theory)
M-(atrix) theory in large dimension limit, Metropolis algorithm for the coordinates and the holonomy angles, Hagedorn/deconfinement 2nd order phase transition, Gross-Witten-Wadia 3rd order phase transition and the emergence of a gap, relation to the black-string/black-hole first order phase transition via gauge/gravity duality.
Contents
-Introductory Remarks:
Path integrals and partition functions, thermal field theory, phase transitions, Monte Carlo methods (only Metropolis algorithm).
-Model I: Vector Models (Ising Model)Phi-four in two dimensions, the disordered/uniform 2nd order phase transition, Metropolis algorithm, critical behavior in 2 and 4 dimensions, continuum limit and renormalization group equation.
-Model II: Matrix Models (Real Quartic Matrix Model)
The quartic matrix model, the disordered/non-uniform 3rd order phase transition, Metropolis algorithm, multitrace matrix models and noncommutative phi-four, Wigner semi-circle law and emergent geometry.
-Model III: Gauge Models (M-(atrix) Theory)
M-(atrix) theory in large dimension limit, Metropolis algorithm for the coordinates and the holonomy angles, Hagedorn/deconfinement 2nd order phase transition, Gross-Witten-Wadia 3rd order phase transition and the emergence of a gap, relation to the black-string/black-hole first order phase transition via gauge/gravity duality.
Simulations
Simulation I- The Lattice $\Phi_2^4$.
Monte Carlo results: tests_Ising.pdf
statistical errors: error.f
random number generators: random.f
gnuplot script: script
main code: ising.f
Newton-Raphson method: newton-raphson-phi**4.f
Monte Carlo results: tests_Ising.pdf
statistical errors: error.f
random number generators: random.f
gnuplot script: script
main code: ising.f
Newton-Raphson method: newton-raphson-phi**4.f
Simulation II- Quartic Matrix Model:
Monte Carlo results: tests_multitrace.pdf
main code: matrix_four.f
eigenvalues sorting: eigen-sorting.f
eigenvalues histogram: eigen-histogram.f
Note: Results are obtained by one of our students.
Monte Carlo results: tests_multitrace.pdf
main code: matrix_four.f
eigenvalues sorting: eigen-sorting.f
eigenvalues histogram: eigen-histogram.f
Note: Results are obtained by one of our students.
Simulation III- Large number of dimensions of the BFSS Matrix Model:
Monte Carlo results: tests_BFSS.pdfcode without gauge field: matrixQM_v1.f
code with gauge field: matrixQM_v2.f
Background Material
Ising Model
Lattice QFT (of Matrix Models)
Phases of gauge theory in lower dimensions and the black-hole/black-string transition
The multitrace matrix models of noncommutative geometry and quantum gravity
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