scholarly journals The Topological Origin of Quantum Randomness

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 581
Author(s):  
Stefan Heusler ◽  
Paul Schlummer ◽  
Malte S. Ubben
Keyword(s):  
High School ◽  
Hilbert Space ◽  
Group Theory ◽  
Lorentz Group ◽  
Quantum Physics ◽  
Time Development ◽  
Mathematical Structures ◽  
Stochastic Nature ◽  
Quantum Randomness ◽  
And Topology

What is the origin of quantum randomness? Why does the deterministic, unitary time development in Hilbert space (the ‘4π-realm’) lead to a probabilistic behaviour of observables in space-time (the ‘2π-realm’)? We propose a simple topological model for quantum randomness. Following Kauffmann, we elaborate the mathematical structures that follow from a distinction(A,B) using group theory and topology. Crucially, the 2:1-mapping from SL(2,C) to the Lorentz group SO(3,1) turns out to be responsible for the stochastic nature of observables in quantum physics, as this 2:1-mapping breaks down during interactions. Entanglement leads to a change of topology, such that a distinction between A and B becomes impossible. In this sense, entanglement is the counterpart of a distinction (A,B). While the mathematical formalism involved in our argument based on virtual Dehn twists and torus splitting is non-trivial, the resulting haptic model is so simple that we think it might be suitable for undergraduate courses and maybe even for High school classes.

The Essence of Quantum Computing

2018 ◽  
Author(s):  
Rajendra K. Bera
Keyword(s):  
Hilbert Space ◽  
Quantum Computing ◽  
Quantum Physics ◽  
Quantum Algorithms ◽  
Quantum Computers ◽  
Turing Machines ◽  
Classical Physics ◽  
Core Set ◽  
The World ◽  
Subordinate Role

It now appears that quantum computers are poised to enter the world of computing and establish its dominance, especially, in the cloud. Turing machines (classical computers) tied to the laws of classical physics will not vanish from our lives but begin to play a subordinate role to quantum computers tied to the enigmatic laws of quantum physics that deal with such non-intuitive phenomena as superposition, entanglement, collapse of the wave function, and teleportation, all occurring in Hilbert space. The aim of this 3-part paper is to introduce the readers to a core set of quantum algorithms based on the postulates of quantum mechanics, and reveal the amazing power of quantum computing.

Elements of Group Theory

2021 ◽  
pp. 51-110
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Group Theory ◽  
Lie Groups ◽  
Lorentz Group ◽  
Unitary Representations ◽  
Compact Groups ◽  
Mathematical Language ◽  
Group Representations ◽  
Infinite Groups ◽  
Infinite Dimensional ◽  
Symmetry Properties

The mathematical language which encodes the symmetry properties in physics is group theory. In this chapter we recall the main results. We introduce the concepts of finite and infinite groups, that of group representations and the Clebsch–Gordan decomposition. We study, in particular, Lie groups and Lie algebras and give the Cartan classification. Some simple examples include the groups U(1), SU(2) – and its connection to O(3) – and SU(3). We use the method of Young tableaux in order to find the properties of products of irreducible representations. Among the non-compact groups we focus on the Lorentz group, its relation with O(4) and SL(2,C), and its representations. We construct the space of physical states using the infinite-dimensional unitary representations of the Poincaré group.

Coupling coefficients and tensor operators for chains of groups

1975 ◽  
Vol 277 (1272) ◽  
pp. 545-585 ◽  
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Group Theory ◽  
Linear Operators ◽  
Coupling Coefficients ◽  
Vector Representation ◽  
General Statement ◽  
The Third ◽  
Representation Spaces ◽  
Definition Of

The action of an arbitrary (but finite or compact) group on an arbitrary Hilbert space is studied. The application of group theory to physical calculations is often based on the Wigner-Eckart theorem, and one of the aims is to lead up to a general proof of this theorem. The group’s action gives irreducible ket-vector representation spaces, products of which lead to a definition of coupling (Wigner, or Clebsch-Gordan) coefficients and jm and j symbols. The properties of these objects are studied in detail, beginning with properties that are independent of the basis chosen for the representation spaces. We then explore some of the consequences of choosing bases by using the action of a subgroup. This leads to the Racah factorization lemma and the definition of jm factors, also a general statement of Racah’s reciprocity. In the third part, we add to these ideas, some properties of the space of all linear operators taking the Hilbert space to itself. This leads to a proof of the Wigner—Eckart theorem which is both succinct and in the language of quantum mechanics.

scholarly journals FERMIONIC SUBSPACES OF THE BOSONIC STRING

2004 ◽  
Vol 19 (supp02) ◽  
pp. 117-125
Author(s):  
A. CHATTARAPUTI ◽  
F. ENGLERT ◽  
L. HOUART ◽  
A. TAORMINA
Keyword(s):  
Field Theory ◽  
Hilbert Space ◽  
Group Theory ◽  
Conformal Field Theory ◽  
Crucial Role ◽  
Predictive Power ◽  
Bosonic String ◽  
Two Dimensional ◽  
Conformal Field ◽  
Fermionic String

A universal symmetric truncation of the bosonic string Hilbert space yields all known closed fermionic string theories in ten dimensions, their D-branes and their open descendants. We highlight the crucial role played by group theory and two-dimensional conformal field theory in the construction and emphasize the predictive power of the truncation. Such circumstantial evidence points towards the existence of a mechanism which generates space-time fermions out of bosons dynamically within the framework of bosonic string theory.

scholarly journals Group Theory in Classical and Quantum Physics

Nature ◽  
1966 ◽  
Vol 211 (5047) ◽  
pp. 346-347
Author(s):  
B. L. MOISEIWITSCH
Keyword(s):  
Group Theory ◽  
Quantum Physics
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