Questions tagged [matrix-model]
A matrix model is a non-peturbative formulation of a theory, such as string theory based on Matrix quantum mechanics
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Does Wick theorem hold for ensembles of random unitary matrices?
Consider ensembles of random unitary matrices with weight function $w(\phi)$, which have partition functions of the form
$$
Z= \int_{U(N)} w(U)dU = \int \prod_{i=1}^N \frac{d \phi_i}{2\pi } w(\phi_i)\...
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How to understand Heisenberg time in random matrix theory?
Recently, from few papers, I have encountered the word 'Heisenberg time' $t_{\text{H}}$ which is an inverse of a mean level spacing $\Delta(\hat{\mathcal{H}})$ of a finite system Hamiltonian $\hat{\...
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D0, D1, D2... DN branes consist of D0 branes?
In BFSS Matrix theory, D0-branes connect the ends of open strings. Do other D-branes (D1, D2... DN) consist of D0-branes?
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Trace identity for $SU(N)$ matrix integral
I would like to know if there's a nice way to compute the following:
$$ \int_{SU(N)} \underbrace{ dU}_{\text{Haar Measure}} \mathrm{tr} \left(U^n \right)~?$$
The following is necessary:
$U \in SU(N)$
$...
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Extremizing the action of matrix model
I know that in order to find the equations of motion for a certain functional, it is required to solve the Euler-Lagrange equations. But I'm wondering how to do this in the case of string matrix ...
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Is there a general equivalence of spectral correlations between hermitian and unitary random matrix ensembles?
It is common lore that the Gaussian Unitary Ensemble (GUE) and Circular Unitary Ensemble (CUE) have the same spectral correlations as the order $N$ of the matrix goes to infinity, see e.g. section 5.9 ...
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Integral relation on Lie (Super)algebra
I have the following integral relation where $X$ is an element of a Lie (super)algebra, $Y_i$ are functions on the respective Lie group and $M$ is an element of the Lie group:
$$Y_1(M)=\int{D}Xe^{-\...
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Feynman Rules for the BFSS Model. Large $N$ Super Matrix Quantum Mechanics
BFSS model is a theory of super-symmetric matrix quantum mechanics describing $N$ coincident $D0$-branes, defined by the action
$$S=\frac{1}{g^2}\int dt\ \text{Tr}\left\{ \frac{1}{2}(D_t X^I)^2 + \...
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Gaussian Matrix Integral
I couldn't quite understand how we calculate the Gaussian Matrix Integral
$$\mathcal{Z}=\int dM\ e^{-N\text{tr}\left(\frac{1}{2}M^2+JM\right)},$$
where the integration measure $dM$ over the $N$ by $N$ ...
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Difference between resolvent and degeneracy
I am studying https://arxiv.org/abs/1903.11115. In equation (62), the resolvent is defined as the integral transform of partition function as
$$
R(E) = -\int_0^\infty d\beta\ e^{\beta E} Z(\beta)
$$
...
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Analytical expression for density of random matrix level ratios
Consider a hermitian matrix $H$ with eigenvalues $E_{i-1}<E_i$. The level spacings are defined as $s_i=E_i-E_{i-1}$ and the level ratios as $r_i = s_i/s_{i-1}$. To make the support of an underlying ...
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Matrix Integrals, Riemann Surfaces & Black Holes. A question regarding one of J.M. Maldacena's talks
I was watching this presentation of Juan Martin Maldacena at Princeton: https://www.youtube.com/watch?v=OMb_P5qPpMc&ab_channel=GraduatePhysics. In one slide he shows an interesting integral. (I ...
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How can I show that $1/N$ expansion for large $N$ matrix models have a string theoretical perturbation expansion?
While surfing through some further reading suggestions on string theory, I stumbled upon this slide from a talk by Nathan Seiberg. I wanted to derive the main argument by applying a perturbation ...
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Why is a static potential between supergravitons problematic?
When deriving the one and two-loop result for the effective potential between two scattering supergravitons, for example from here, we see that it is always a velocity dependent potential. ...
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Validity of DLCQ Matrix Theory near the Big Bang
In this paper,section 3.2, Craps, Sethi and Verlinde claim that DLCQ matrix theory is valid near the big bang if
The open string oscillators decouple and
Gravity decouples from the matrix description
...
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Why is there an additional $NI$ term in this ${\rm SU}(N)$ generator, from Matrix Quantum Mechanics?
This question refers to equation (11) in the latest preprint of the following paper:
X. Han, S. A. Hartnoll and J. Kruthoff, "Bootstrapping Matrix Quantum Mechanics", Phys. Rev. Lett. 125, ...
user244484
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Applications of the Gaussian unitary ensemble
It is well known that the pair correlation function of the zeros of the Riemann zeta function reproduces the correlation function of the random matrices from the Gaussian unitary ensemble (GUE). ...
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How are the saddle-point equations derived in the single random matrix model?
In this question, I'm referring to a specific step in https://arxiv.org/abs/hep-th/9306153. I want to reproduce equation (2.4) on page 15. I think I lack the experience required for dealing with ...
user195631
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Equivalence of 2d quantum and topological gravity
I have heard of the statement that 2d quantum gravity (defined as minimal models of CFT coupled to Liouville theory) and 2d topological gravity are equivalent. The former is described by the continuum ...
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Good starting point for quantum Hall matrix models
I am a recent Masters in theoretical condensed matter physics and have experience in working on topological insulators and Weyl semimetals. I have also dabbled a bit in the fractional quantum Hall ...
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Can we generalize matrix model theory?
As in the question, can matrix model theory be generalized to a tensor model theory? Will the results be different or useful in describing real world phenomena?
Details: in matrix model theory we ...
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$\delta^{(2)}$ convention
In this note: https://arxiv.org/abs/hep-th/0410165 at page 12 there is a delta-function constraint written as:
\begin{align}
\delta \left( ^ { U } M \right) = \prod _ { i < j } \delta ^ { ( 2 ) } \...
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Simultaneous shifted diagonalization of bunch of operators
I have the following Lie algebra which is generated by $\{L_n|n\geq 0\}.$ It satisfies the following commutation rule
$$ \Big[ L_i ,L_j \Big]= (i-j)(L_{i+j}-\frac14 L_{i+j-1}).....*$$
My question is ...
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Singular behavior of pure gravity
Can anyone plese explain what means singular part of partition function for pure gravity?
Let me specify my question. I am dealing with 2D quantum gravity and starts from path integral formulation of ...
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How to relate random matrix theory with Quantum mechanics approach
In Quantum mechanics we first built Hamiltonian and then find its eigenvalues and eigenvectors. How I can relate this with random matrix theory?
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How to calculate correlators in a 1D Conformally invariant Matrix Model?
I am working on a 1D Conformally invariant Matrix Model with the following Partition function:
$$
Z(g) = \int \mathcal{D}M(t) \exp \left[ -\text{tr}\int dt \left( \frac{1}{2} \dot{M}^2(t)+V(M) \...
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How to get the eigenvalue density contribution $\rho_1(x)$?
I'm studying the $1/N$ expansion beyond the planar limit in matrix models. Currently I'm trying to understand and reproduce the results of:
Antisymmetric Wilson loops in $\mathcal N \geq 4$ SYM ...
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Basis of eigenvectors common to H and B
Considering a three-dimensional state space spanned by the orthonormal basis formed by the three kets $|u_1\rangle,|u_2\rangle,|u_3\rangle $. In the basis of these three vectors, taken in order, are ...
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Hamiltonian matrix for a delta potential with periodic boundary condition
I'm trying to find the energy eigenvalues of a Dirac delta potential:
$$V(x)=-\alpha\delta(x)$$
with periodic boundary condition over some length $L$:
$$\psi(x+L)=\psi(x)$$
and only even ...
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Massless limit of Matrix Quantum Mechanics
I am working on a Matrix Quantum Mechanics model that is related to 2d string theory as defined here: http://arxiv.org/abs/hep-th/0311273 §Chapter III
The action is defined as $$
S = \text{Tr} \int ...
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Lumped mechanical system - Base Change and asymmetric matrices
Let's assume I have a system which can be wrote as:
$$ \underbrace{\begin{bmatrix}
m_{1} & 0 & \dots & 0 \\
0 & m_{2} & \...
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Gaussian beam propagation with ABCD matrix through a grin lens
I am currently trying to simulate a Gaussian beam that has a transverse offset of around 20um from the optic axis where the Gaussian beam travels through a grin lens using the ABCD matrix method. I ...
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Generalization of Itzykson-Zuber Formula to Path Integrals?
Some Background: In mathematical physics (matrix quantum mechanics in particular), one often runs into path integrals of the form:
$$ Z = \int \prod_{ab} DM^1_{ab}(\tau) DM^2_{ab}(\tau) e^{-NS(M^1,M^2,...
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References to Random Matrix Theory
I am looking for some good references - books/lecture notes/articles which contains Random Matrix Theory for Physicists. I am not particularly looking for mathematical rigor in derivations. I am more ...
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Relation between $1/N$ and perturbative expansions in QFT
I heard many times phrases like "large $N$ is a clever way to organize the diagrammatic expansion" or "each diagram in the large $N$ expansion contains an infinite number of usual perturbative ...
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Matching measured eigenvalue of a matrix [closed]
Given a Matrix of the form below, we have 8 variables, $\omega_{i}$ and $J_{ij}$. we want to diagonalise the Matrix to obtain values to match the observed Eigenvalue in an Experiment. i.e. $\bar{\...
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Emergence of space from quantum mechanics
Once talking to a visiting professor at my institute, I heard about some simple model that captures the emergence of space coordinates as the eigenvalues of some infinite-dimensional quantum ...
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How are unitary matrices and unitary random matrices associated with physics or quantum mechanics? [closed]
Please forgive my ignorance, my back ground is not physics. I am looking for distance measure between two unitary matrix (for my work). So my starting point is where else unitary matrix is applicable?...
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ADHM construction and Momentum Map
while I was reading about ADHM construction I had some troubles with precise geometrical identification of the various quantities. My doubts is well manifest in these two Wikipedia pages
1) ADHM ...
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Chern-Simons Theory over $S^3$ as integral - what is domain of integration?
I found these nice lecture notes by Marcos Marino, Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories, arXiv:1104.0783, so I am hoping to understand some parts ...
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Bloch's theorem for Semi-Infinite Lattice
If we have a lattice Hamiltonian $$ \sum_{n'\in\mathbb{Z}}H_{n,n'}\psi_{n'} = E\psi_{n} \,\forall n\in\mathbb{Z} $$
such that $ H_{n,n'} = H_{n+q,n'+q}$ for some $q\in\mathbb{N}$ and for all $\left(n,...
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On the Equivalence of Schrodinger and Heisenberg Descriptions of Quantum Mechanics and Observability
I'm not a physicist, but rather a control (feedback) systems engineer eager to understand more than just a cursory explanation of quantum mechanics. The StackExchange has been an excellent forum for ...
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What information does the trace of a matrix give?
I was recently thinking what information do we get from a matrix. So if we say the columns (or rows) of a matrix define the basis of a system, say vectors of 3 dimensional space. Then the determinant ...
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What is the current state of research about the Hayden-Preskill circuit? [duplicate]
Can someone summarize as to what are the problems and/or the open questions with the Hayden-Preskill circuit? (in the context of understanding black-holes or as a computer science question)It gives a ...
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Is there a physical interpretation to invariant random matrix ensembles?
Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
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The curvature of the space of commuting hermitian matrices
This is a question that I asked in the mathematics section, but I believe it may get more attention here. I am working on a project dedicated to the quantisation of commuting matrix models. In the ...
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What do matrices in the Gaussian orthogonal ensemble look like?
I've been reading a fair amount about quantum chaos, and random matrix theory comes up a lot. I get that they're looking at the distribution of eigenvalues from an ensemble of random matrices, but I ...
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Relation between holography and matrix models
Let's consider a 0-dimensional $N \times N$ Hermitean one matrix model.
It is defined by a potential V(M). Its partition function
is
$$Z = \int_{H_{N}} dM e^{-\frac{1}{g}V(M)}$$
where $H_{N}$ is the ...
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Normalizability of the Hartle-Hawking state in Liouville theory
I'm confused about how to normalize the Hartle-Hawking state in 2D quantum gravity. We can compute the HH state for two circles of length $\ell_1$ and $\ell_2$ in the matrix model as $\langle W(\ell_1)...
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Matrix integral in multi-matrix model
Though it is a mathematical problem, maybe more physicists know it well.
In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral
$$Z \sim \int \prod_{i=1}^r ...
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