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## Hot answers tagged matrix-model

21

matrix string theory may be viewed just as a variation of BFSS Matrix Theory, although arguably an important one, and the original papers are the full introductions at the same moment. http://arxiv.org/abs/hep-th/9701025 http://arxiv.org/abs/hep-th/9702187 http://arxiv.org/abs/hep-th/9703030 Some of the few hundred followups deal with some more ...

10

Among normal books, Becker-Becker-Schwarz probably matches your summary most closely. However, you may want to look at a list of string theory books: http://motls.blogspot.com/2006/11/string-theory-textbooks.html Don't miss the "resource letter" linked at the bottom which is good for more specialized issues such as string field theory. An OK review of ...

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The BFSS matrix model is a quantum mechanical model – i.e. quantum field theory in 0+1 dimensions - that describes uncompactified M-theory in 11 dimensions assuming that we study the large $N$ limit of the model with the $U(N)$ symmetry. As myself and later Susskind determined, one may also directly interpret the finite $N$ BFSS matrix model as describing M-...

6

I can offer up something similar to this, which is an isomorphism between something called the Tsirelson bound and the spacetime metric. This is not exactly the emergence of spacetime from quantum mechanics, but it does illustrate how spacetime could be seen as quantum mechanics in diguise. Suppose we have four operators $A_1, A_2, B_1, B_2$  such that: $$... 6 I) More generally, Let V be a (say, finite dimensional) vector space over a field \mathbb{F}. Let (e_i)_{i\in I} be a basis for V. Let A\in {\rm End}(V) be an endomorphism in V, i.e. a \mathbb{F}-linear map A:V\to V. Let the matrix (M^i{}_j)_{i,j\in I} be the unique \mathbb{F}-valued matrix that represents the linear map A in the ... 5 Matrix string theory http://arxiv.org/abs/hep-th/9701025 http://arxiv.org/abs/hep-th/9702187 http://arxiv.org/abs/hep-th/9703030 is indeed an exact description of fundamental type IIA strings (and similarly E_8\times E_8 heterotic strings) at any (e.g. weak) coupling where you can explicitly see the off-diagonal degrees of freedom. You could say ... 4 First of all, regardless of speculative comments in the original BFSS paper, different superselection sectors – different parts of the Hilbert space specified by the "background" (at least at infinity) – require different matrix model descriptions. The matrix model is known for T^p with any radii; for p\leq 3, the theory is a super Yang-Mills theory on ... 3 In quantum mechanics (QM), normally we talk about the time evolution of some quantum state (QS), |QS(t)\rangle, of a particle. A QS is an entity that contains all the information we could possibly know about the particle and it can be treated as a vector in the Hilbert space. In the case of unitary evolution, mathematically, we have$$|QS(t_2)\rangle=U(...

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The Christoffel–Darboux formula is not an asymptotic (in the sense of $N$ going to infinity) result, whereas the semicircle is valid for random matrices of infinite size. For finite matrices you obtain the oscillations you've got. To see this check out and plot formula (97) in http://arxiv.org/abs/math-ph/0412017 As to the traceless GUE, I'm no expert but ...

2

I actually solved the problem. The key idea is to use the fact that the metric depends on the internal manifold (the flag) only through the Maurer-Cartan forms and hence the scalar curvature cannot depend on the position in the internal manifold. One can then expand the elements of $SU(N)$ near the origin. Keep the metric exact in terms of the lambdas and to ...

2

It is not just a distribution of eigenvalues. The underlying distribution for the Gaussian orthogonal ensemble GOE(n) of real symmetric $n\times n$ matrices is $\frac{n(n+1)}{2}$ dimensional, and given in terms of the $\frac{n(n+1)}{2}$ independent matrix elements, each with a Gaussian weight factor: $$\left[\prod_{i,j\in\{1,\ldots, n\}}^{i\leq j} \int_{\... 2 In the first action the A_{\mu} are Hermitian. In the second action the A_{\mu} are anti-Hermitian since we let A_{\mu}\to\frac{i}{g}A_{\mu}. The commutator of anti-Hermitian matrices are also anti-Hermitian. If we have \text{Tr}(M^{2}) , with M being anti-Hermitian, then we can write it as \text{Tr}(M^{2})=-\text{Tr}((iM)^{2}) , with iM ... 1 You can think of an MPS as being built up by objects with three indices. How to easily represent such an object? We can think of this as a matrix where each entry is a vector (in particular the vector will be a vector in the on-site Hilbert space, so e.g. for a spin 1/2 system it will be of the form \alpha |\uparrow \rangle + \beta |\downarrow \rangle). ... 1 I"m not sure that I understand what you mean by "random". Do you mean "arbitrary"? That question aside, the unitary matrix is used in physics to effect similarity transformations of matrices in physics.$$A'=U^{\dagger} A U$$where U is the unitary transformation matrix and U^{\dagger} is the Hermitian conjugate. With this transformation, the ... 1 The space of ADHM quadruples (B_1,B_2,I,J) \in \mathbb{C}^{k\times k}\times\mathbb{C}^{k\times k}\times\mathbb{C}^{k\times N}\times\mathbb{C}^{N\times k} carries a group action of \mathrm{GL}(k,\mathbb{C}) as$$ (B_1,B_2,I,J)\mapsto (UB_1 U^{-1},UB_2U^{-1},IU^{-1},UJ) for $U\in\mathrm{GL}(k,\mathbb{C})$. This descends to a $\mathrm{U}(k)$ action on the ...

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I would like to begin by your last question. I am of the opinion that this is correct. I am not versed in systems engineering but I see the equivalence in the mathematical treatments. And QM is very much about the mathematics, is more a descriptive body of knowledge than it is an explanatory. And proof of this is the many interpretations it has, all of ...

1

Let $M$ be the mass matrix for fermions $\psi_+$ and for $\psi_-$ (separately). It is obtained by $\not{D}\not{D^+}= -\partial_t^2+ M^2$ Then $M^2=r^2 \mathbb{Id_{16}}- \not{v}$, Now, the $16*16$ matrix $\not{v}$ has a zero trace, and it square is $\vec v^2 Id_{16}$, so the only possibility is that the matrix $\not{v}$ has 8 eigenvalues $v$, and 8 ...

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