scholarly journals Holographic map for cosmological horizons

2020 ◽  
Vol 35 (26) ◽  
pp. 2050158
Author(s):  
Chang Liu ◽  
David A. Lowe
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Chaotic Dynamics ◽  
Einstein Gravity ◽  
Spin Model ◽  
De Sitter ◽  
Finite Dimensional ◽  
Large N ◽  
Sitter Background ◽  
Finite Dimensional Hilbert Space

We propose a holographic map between Einstein gravity coupled to matter in a de Sitter background and large N quantum mechanics of a system of spins. Holography maps a spin model with a finite-dimensional Hilbert space defined on a version of the stretched horizon into bulk gravitational dynamics. The full Hamiltonian of the spin model contains a nonlocal piece which generates chaotic dynamics, widely conjectured to be a necessary part of quantum gravity, and a local piece which recovers the perturbative spectrum in the bulk.

scholarly journals Quantum Mechanics in Finite Dimensions

1978 ◽  
Vol 31 (4) ◽  
pp. 233 ◽  
Author(s):  
TS Santhanam ◽  
KB Sinha
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Commutation Relation ◽  
Canonical Commutation Relation ◽  
Unitary Operators ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Finite Dimensional Hilbert Space

This paper contributes to a recent series discussing quantum mechanics defined on a finite-dimensional Hilbert space in which Weyl's commutation relation for unitary operators holds. In an earlier paper, Santhanam and Tekumalla (1976) showed that the commutation relation for hermitian operators with a bounded spectrum tends to Hdsenberg's standard canonical commutation relation as the spectrum becomes continuous and the dimension n -> 00. The present paper offers a formulation which is coordinate free in the limit n -> 00 and makes the limiting procedure especially ransparent.

scholarly journals On the Hydrogen Atom in the Holographic Universe

2021 ◽  
Author(s):  
S. Jalalzadeh ◽  
S. Abarghouei Nejad ◽  
P. V. R L Moniz
Keyword(s):  
Hilbert Space ◽  
Hydrogen Atom ◽  
Neutral Hydrogen ◽  
Hydrogen Gas ◽  
Quantum Deformation ◽  
De Sitter ◽  
Hydrogen Atoms ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Finite Dimensional Hilbert Space

Abstract We investigate the holographic bound utilizing a homogeneous, isotropic, and non-relativistic neutral hydrogen gas present in the de Sitter space. Concretely, we propose to employ de Sitter holography intertwined with quantum deformation of the hydrogen atom using the framework of quantum groups. Particularly, the $\mathcal U_q(so(4))$ quantum algebra is used to construct a finite-dimensional Hilbert space of the hydrogen atom. As a consequence of the quantum deformation of the hydrogen atom, we demonstrate that the Rydberg constant is dependent on the de Sitter radius, $L_\Lambda$. This feature is then extended to obtain a finite-dimensional Hilbert space for the full set of all hydrogen atoms in the de Sitter universe. We then show that the dimension of the latter Hilbert space satisfies the holographic bound. We further show that the mass of a hydrogen atom $m_\text{atom}$, the total number of hydrogen atoms at the universe, $N$, and the retrieved dimension of the Hilbert space of neutral hydrogen gas, $\text{Dim}{\mathcal H}_\text{bulk}$, are related to the de Sitter entropy, $S_\text{dS}$, the Planck mass, $m_\text{Planck}$, the electron mass, $m_\text{e}$, and the proton mass $m_\text{p}$, by $m_\text{atom}\simeq m_\text{Planck}S_\text{dS}^{-\frac{1}{6}}$, $N\simeq S_\text{dS}^\frac{2}{3}$ and $\text{Dim}{\mathcal H}_\text{bulk}=2^{\frac{m_\text{e}}{m_\text{p}}\alpha^2S_\text{dS}}$, respectively.

Linear Algebra and the Dirac Notation

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Vector Spaces ◽  
Complex Vector ◽  
Dual Vector ◽  
Finite Dimensional ◽  
Basic Facts

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.

Quantum mechanics on (anti)-de Sitter background II: Ramsauer–Townsend effect and WKB method

2018 ◽  
Vol 33 (26) ◽  
pp. 1850150 ◽  
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Energy Level ◽  
Harmonic Oscillator ◽  
Wkb Method ◽  
De Sitter ◽  
Quantum Harmonic Oscillator ◽  
One Dimensional ◽  
Sitter Background ◽  
The One

Based on the one-dimensional quantum mechanics on (anti)-de Sitter background [W. S. Chung and H. Hassanabadi, Mod. Phys. Lett. A 32, 26 (2107)], we discuss the Ramsauer–Townsend effect. We also formulate the WKB method for the quantum mechanics on (anti)-de Sitter background to discuss the energy level of the quantum harmonic oscillator and quantum bouncer.

scholarly journals THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED

2019 ◽  
Vol 7 ◽  
Author(s):  
WILLIAM SLOFSTRA
Keyword(s):  
Hilbert Space ◽  
Linear System ◽  
Tensor Product ◽  
Quantum Correlations ◽  
Embedding Theorem ◽  
Product Strategy ◽  
Finite Dimensional ◽  
Universal Embedding ◽  
Quantum Strategies ◽  
Finite Dimensional Hilbert Space

We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.

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