scholarly journals MACROSCOPIC OBSERVABLES AND THE BORN RULE, I: LONG RUN FREQUENCIES

2008 ◽  
Vol 20 (10) ◽  
pp. 1173-1190 ◽  
Author(s):  
N. P. LANDSMAN
Keyword(s):  
Quantum Mechanics ◽  
Quantum System ◽  
Single Case ◽  
Born Rule ◽  
C Algebra ◽  
C Algebras ◽  
Long Run ◽  
Frequency Interpretation ◽  
Continuous Fields ◽  
The One

We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr's doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C*-algebra of observables is empirically accessible only through associated commutative C*-algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of single-case probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein [17] and Hartle [21], intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach refines previous elaborations of this program — notably the one due to Farhi, Goldstone, and Gutmann [15] as completed by Van Wesep [50] — in replacing infinite tensor products of Hilbert spaces by continuous fields of C*-algebras. Furthermore, instead of relying on the controversial eigenstate-eigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables.

scholarly journals THE GELFAND SPECTRUM OF A NONCOMMUTATIVE C*-ALGEBRA: A TOPOS-THEORETIC APPROACH

2011 ◽  
Vol 90 (1) ◽  
pp. 39-52 ◽  
Author(s):  
CHRIS HEUNEN ◽  
NICOLAAS P. LANDSMAN ◽  
BAS SPITTERS ◽  
SANDER WOLTERS
Keyword(s):  
Quantum Mechanics ◽  
Usual Sense ◽  
Theoretic Approach ◽  
Stone Duality ◽  
Complete Lattices ◽  
C Algebra ◽  
C Algebras ◽  
Gelfand Transform ◽  
Noncommutative Spaces ◽  
Gelfand Duality

AbstractWe compare two influential ways of defining a generalized notion of space. The first, inspired by Gelfand duality, states that the category of ‘noncommutative spaces’ is the opposite of the category of C*-algebras. The second, loosely generalizing Stone duality, maintains that the category of ‘point-free spaces’ is the opposite of the category of frames (that is, complete lattices in which the meet distributes over arbitrary joins). Earlier work by the first three authors shows how a noncommutative C*-algebra gives rise to a commutative one internal to a certain sheaf topos. The latter, then, has a constructive Gelfand spectrum, also internal to the topos in question. After a brief review of this work, we compute the so-called external description of this internal spectrum, which in principle is a fibred point-free space in the familiar topos of sets and functions. However, we obtain the external spectrum as a fibred topological space in the usual sense. This leads to an explicit Gelfand transform, as well as to a topological reinterpretation of the Kochen–Specker theorem of quantum mechanics.

DYNAMICAL SYSTEMS IN GRAPH C*-ALGEBRAS

2005 ◽  
Vol 16 (09) ◽  
pp. 999-1015 ◽  
Author(s):  
JA A JEONG ◽  
GI HYUN PARK
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Commutative Algebra ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Locally Finite ◽  
C Algebra ◽  
C Algebras ◽  
Shift Space ◽  
The One

Let E be a row finite directed graph with no sinks and (XE, σE) the one-sided edge shift space. Then the graph C*-algebra C*(E) contains the commutative algebra C0(XE). Moreover if E is locally finite so that the canonical completely positive map ϕE on C*(E) is well-defined, ϕE|C0(XE) coincides with the *-homomorphism [Formula: see text]. In this paper we first show that if two edge shift spaces (XE, σE) and (XF, σF) are topologically conjugate, there is an isomorphism of C*(E) onto C*(F), and if the graphs are locally finite the isomorphism transforms ϕE|C0(XE) onto ϕF|C0(XF), which has been known for Cuntz–Krieger algebras. Let ht(ϕE) be Voiculescu–Brown topological entropy of ϕE. In case E is finite, it is well-known that the values ht(ϕE), [Formula: see text], hl(E) and hb(E) all coincide, where [Formula: see text] is the AF core of C*(E) and hl(E), hb(E) are the loop, block entropies of E respectively. If E is irreducible and infinite, [Formula: see text] has been known recently, and here we show that [Formula: see text], where Et is the transposed graph of E. Also some dynamical systems related with AF subalgebras [Formula: see text] of [Formula: see text] are examined to prove that [Formula: see text] for each vertex v.

THE DAUNS–HOFMANN THEOREM REVISITED

2011 ◽  
Vol 10 (01) ◽  
pp. 29-37 ◽  
Author(s):  
KARL HEINRICH HOFMANN
Keyword(s):  
New Orleans ◽  
Historical Record ◽  
Local Culture ◽  
Text Book ◽  
C Algebra ◽  
C Algebras ◽  
Long Time ◽  
Algebra Theory ◽  
Continuous Fields ◽  
Early Phases

John Dauns died on June 4, 2009 of cancer in New Orleans at the age of 73. His work on rings and modules is well-known in the algebra community. However, functional analysts working in the area of C*-algebras are likely to know his name from one theorem that is a corollary of results he and I had obtained in work we did in the mid-sixties of the last century ([17, 27, 18]) and which became known as the Dauns-Hofmann Theorem in C*-algebra theory. It has been known in the C*-text book and monograph literature up to the recent one under this name ([9, 20, 21, 39, 41]) since it provides apparently a useful tool that continues to be used in current research (see e.g. [7, 8, 25, 38, 40]). The problem with the historical record of the Dauns–Hofmann Theorem is that it used to be somewhat obscure how it originated and that the full weight of what was proved was not precisely understood for a long time. As John Dauns was deeply, if not subbornly involved in the development of the early phases of the representation of rings, algebras, C*-algebras (and other classes of algebras) by continuous sections in bundles (sometimes called continuous fields) [11–18], and since his work in this area was substantial and contributed much to a local culture of "sectional representation" at Tulane University ([19, 22, 27–34, 43, 44, 48]), I feel that it is justified to attempt a clarification. He can no longer participate himself in such an attempt; nor would he actually protest the occasional lack of acknowledgment were he alive, because that would be contrary to his ever gentle disposition. This small survey is devoted to shedding some additional light on this portion of John Dauns' work in mathematics; it is natural that it should have a personal tenor by a writer remembering his presence and his work as a collaborator.

scholarly journals KMS states on C*-algebras associated to local homeomorphisms

2014 ◽  
Vol 25 (08) ◽  
pp. 1450066 ◽  
Author(s):  
Zahra Afsar ◽  
Astrid an Huef ◽  
Iain Raeburn
Keyword(s):  
Finite Graph ◽  
Critical Value ◽  
Toeplitz Algebra ◽  
Graph Algebras ◽  
Kms States ◽  
C Algebra ◽  
C Algebras ◽  
Backward Shift ◽  
The One ◽  
Natural Dynamics

For every Hilbert bimodule over a C*-algebra, there are natural gauge actions of the circle on the associated Toeplitz algebra and Cuntz–Pimsner algebra, and hence natural dynamics obtained by lifting these gauge actions to actions of the real line. We study the KMS states of these dynamics for a family of bimodules associated to local homeomorphisms on compact spaces. For inverse temperatures larger than a certain critical value, we find a large simplex of KMS states on the Toeplitz algebra, and we show that all KMS states on the Cuntz–Pimsner algebra have inverse temperature at most this critical value. We illustrate our results by considering the backward shift on the one-sided path space of a finite graph, where we can use recent results about KMS states on graph algebras to see what happens below the critical value. Our results about KMS states on the Cuntz–Pimsner algebra of the shift show that recent constraints on the range of inverse temperatures obtained by Thomsen are sharp.

Migration and US agricultural competitiveness

2013 ◽  
Vol 10 (2) ◽  
pp. 159-179 ◽  
Author(s):  
Philip L. Martin
Keyword(s):  
Farm Workers ◽  
Easy Access ◽  
Foreign Born ◽  
Guest Workers ◽  
Long Run ◽  
Short Run ◽  
Lower Wage ◽  
The Government ◽  
The One

Agriculture has one of the highest shares of foreign-born and unauthorized workers among US industries; over three-fourths of hired farm workers were born abroad, usually in Mexico, and over half of all farm workers are unauthorized. Farm employers are among the few to openly acknowledge their dependence on migrant and unauthorized workers, and they oppose efforts to reduce unauthorized migration unless the government legalizes currently illegal farm workers or provides easy access to legal guest workers. The effects of migrants on agricultural competitiveness are mixed. On the one hand, wages held down by migrants keep labour-intensive commodities competitive in the short run, but the fact that most labour-intensive commodities are shipped long distances means that long-run US competitiveness may be eroded as US farmers have fewer incentives to develop labour-saving and productivity-improving methods of farming and production in lower-wage countries expands.

Surfing the Quantum World

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Musical Instrument ◽  
Popular Science ◽  
Quantum Spin ◽  
Exclusion Principle ◽  
Maximum Height ◽  
Core Concepts ◽  
The Many ◽  
The One

Surfing the Quantum World bridges the gap between in-depth textbooks and typical popular science books on quantum ideas and phenomena. Among its significant features is the description of a host of mind-bending phenomena, such as a quantum object being in two places at once or a certain minus sign being the most consequential in the universe. Much of its first part is historical, starting with the ancient Greeks and their concepts of light, and ending with the creation of quantum mechanics. The second part begins by applying quantum mechanics and its probability nature to a pedagogical system, the one-dimensional box, an analog of which is a musical-instrument string. This is followed by a gentle introduction to the fundamental principles of quantum theory, whose core concepts and symbolic representations are the foundation for most of the subsequent chapters. For instance, it is shown how quantum theory explains the properties of the hydrogen atom and, via quantum spin and Pauli’s Exclusion Principle, how it accounts for the structure of the periodic table. White dwarf and neutron stars are seen to be gigantic quantum objects, while the maximum height of mountains is shown to have a quantum basis. Among the many other topics considered are a variety of interference phenomena, those that display the wave properties of particles like electrons and photons, and even of large molecules. The book concludes with a wide-ranging discussion of interpretational and philosophic issues, introduced in Chapters 14 by entanglement and 15 by Schrödinger’s cat.

scholarly journals How to Erase Quantum Monogamy?

2021 ◽  
Vol 3 (1) ◽  
pp. 53-67
Author(s):  
Ghenadie Mardari
Keyword(s):  
Quantum System ◽  
Quantum Entanglement ◽  
The Other ◽  
Arrow Of Time ◽  
Quantum Level ◽  
Delayed Choice ◽  
Other Hand ◽  
The One ◽  
Quantum Erasure ◽  
Non Locality

The phenomenon of quantum erasure exposed a remarkable ambiguity in the interpretation of quantum entanglement. On the one hand, the data is compatible with the possibility of arrow-of-time violations. On the other hand, it is also possible that temporal non-locality is an artifact of post-selection. Twenty years later, this problem can be solved with a quantum monogamy experiment, in which four entangled quanta are measured in a delayed-choice arrangement. If Bell violations can be recovered from a “monogamous” quantum system, then the arrow of time is obeyed at the quantum level.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

On C*-Algebras Associated with Subshifts

1997 ◽  
Vol 08 (03) ◽  
pp. 357-374 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Symbolic Dynamics ◽  
Markov Shift ◽  
C Algebra ◽  
C Algebras ◽  
Markov Shifts ◽  
Universal Properties

We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.

Subaltern mediators in the digital landscape

Target ◽  
2017 ◽  
Vol 29 (2) ◽  
pp. 319-338 ◽  
Author(s):  
Teresa Iribarren
Keyword(s):  
Cultural Capital ◽  
Web 2.0 ◽  
The Other ◽  
The Internet ◽  
Lingua Franca ◽  
Content Type ◽  
For Profit ◽  
Long Run ◽  
The One ◽  
Digital Landscape

This article explores translational literary Web 2.0 practices and user-generated cultural creations on the Internet, focusing on video poetry that re-creates canonical poets’ bodies of work. It will be argued that the use of for-profit platforms like YouTube and Vimeo by indie creators and translators of video poetry favours the emergence of new translational attitudes, practices and objects that have positive but also contentious effects. One the one hand, these online mediators explore new poetic expressions and tend to make the most of the potential for dissemination of poetic heritage, providing visibility to non-hegemonic literatures. On the other hand, however, these translational digitally-born practices and creations by voluntary and subaltern mediators might reinforce the hegemonic position of large American Internet corporations at the risk of commodifying cultural capital, consolidating English as a lingua franca and perhaps, in the long run, even fostering a potentially monocultural and internationally homogeneous aesthetics.

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