scholarly journals Linear Transformations in Hilbert Space: III. Operational Methods and Group Theory

1930 ◽  
Vol 16 (2) ◽  
pp. 172-175 ◽  
Author(s):  
M. H. Stone
Keyword(s):  
Hilbert Space ◽  
Group Theory ◽  
Linear Transformations ◽  
Operational Methods

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.

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 Maximal nonnegative and $\theta$-accretive extensions of a positive definite linear relation

2020 ◽  
Vol 12 (2) ◽  
pp. 289-296
Author(s):  
O.G. Storozh
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Differential Operators ◽  
Boundary Value ◽  
Positive Definite ◽  
Complex Hilbert Space ◽  
Multivalued Operator ◽  
Linear Transformations ◽  
Accretive Extension

Let $L_{0}$ be a closed linear positive definite relation ("multivalued operator") in a complex Hilbert space. Using the methods of the extension theory of linear transformations in a Hilbert space, in the terms of so called boundary value spaces (boundary triplets), i.e. in the form that in the case of differential operators leads immediately to boundary conditions, the general forms of a maximal nonnegative, and of a proper maximal $\theta$-accretive extension of the initial relation $L_{0}$ are established.

scholarly journals A structure theorem for operators with closed range

1978 ◽  
Vol 18 (2) ◽  
pp. 169-186
Author(s):  
James Guyker
Keyword(s):  
Hilbert Space ◽  
Structure Theorem ◽  
Linear Transformations ◽  
Partial Isometries ◽  
Closed Range

A characterization has previously been given for linear transformations in Hilbert space whose first N + 1 powers are partial isometries. An analogous characterization is now obtained for transformations whose first N+ 1 powers have closed ranges. A hypothesis (that transformations have no isometric part) is found to be unnecessary in previous work.

Products of idempotent linear transformations

1985 ◽  
Vol 100 (1-2) ◽  
pp. 123-138 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bounded Operators ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

SynopsisIn 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

Sign in / Sign up

Export Citation Format

Share Document