scholarly journals The noncommutative geometry of the Landau Hamiltonian: Differential aspects

2021 ◽  
Author(s):  
Giuseppe De Nittis ◽  
Maximiliano Sandoval
Keyword(s):  
Dirac Operator ◽  
Noncommutative Geometry ◽  
Differential Calculus ◽  
Quantum Hall ◽  
Geometric Interpretation ◽  
Quantum Harmonic Oscillator ◽  
C Algebra ◽  
Dense Subalgebra ◽  
Dixmier Trace ◽  
Do So

Abstract In this work we study the differential aspects of the noncommutative geometry for the magnetic C*-algebra which is a 2-cocycle deformation of the group C*-algebra of R2. This algebra is intimately related to the study of the Quantum Hall Effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo's formula. Taking inspiration from the ideas developed by Bellissard during the 80's, we build an appropriate Fredholm module for the magnetic C*-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard's theory, the so-called second Connes' formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic C*-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes' quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.

scholarly journals The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

2020 ◽  
Author(s):  
Giuseppe De Nittis ◽  
◽  
Maximiliano Sandoval ◽  
◽  
◽  
...  
Keyword(s):  
Dirac Operator ◽  
Hall Effect ◽  
Noncommutative Geometry ◽  
Quantum Hall Effect ◽  
Quantum Hall ◽  
Spectral Triple ◽  
C Algebra ◽  
Compact Resolvent ◽  
Landau Hamiltonian ◽  
The Continuum

This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the C∗-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.

scholarly journals Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

2017 ◽  
Vol 69 (3) ◽  
pp. 548-578 ◽  
Author(s):  
Michael Hartglass
Keyword(s):  
Differential Calculus ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Weighted Graph ◽  
Positive Cone ◽  
Von Neumann ◽  
Planar Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

scholarly journals STRINGS, NONCOMMUTATIVE GEOMETRY AND THE SIZE OF THE TARGET SPACE

1999 ◽  
Vol 14 (28) ◽  
pp. 4501-4517 ◽  
Author(s):  
FEDELE LIZZI
Keyword(s):  
String Theory ◽  
Dirac Operator ◽  
Noncommutative Geometry ◽  
Low Energy ◽  
Target Space ◽  
World Sheet ◽  
Crucial Point ◽  
Spectral Action ◽  
Energy Eigenvalues ◽  
The World

We describe how the presence of the antisymmetric tensor (torsion) on the world sheet action of string theory renders the size of the target space a gauge noninvariant quantity. This generalizes the R ↔ 1/R symmetry in which momenta and windings are exchanged, to the whole O(d,d,ℤ). The crucial point is that, with a transformation, it is possible always to have all of the lowest eigenvalues of the Hamiltonian to be momentum modes. We interpret this in the framework of noncommutative geometry, in which algebras take the place of point spaces, and of the spectral action principle for which the eigenvalues of the Dirac operator are the fundamental objects, out of which the theory is constructed. A quantum observer, in the presence of many low energy eigenvalues of the Dirac operator (and hence of the Hamiltonian) will always interpreted the target space of the string theory as effectively uncompactified.

scholarly journals DIFFERENTIAL CALCULUS ON FUZZY SPHERE AND SCALAR FIELD

1998 ◽  
Vol 13 (19) ◽  
pp. 3235-3243 ◽  
Author(s):  
URSULA CAROW-WATAMURA ◽  
SATOSHI WATAMURA
Keyword(s):  
Scalar Field ◽  
Dirac Operator ◽  
Differential Calculus ◽  
Spectral Triple ◽  
Fuzzy Sphere ◽  
Alternative Possibility ◽  
Operator Ordering

We find that there is an alternative possibility to define the chirality operator on the fuzzy sphere, due to the ambiguity of the operator ordering. Adopting this new chirality operator and the corresponding Dirac operator, we define Connes' spectral triple on the fuzzy sphere and the differential calculus. The differential calculus based on this new spectral triple is simplified considerably. Using this formulation the action of the scalar field is derived.

scholarly journals Spectral action and the electroweak θ-terms for the Standard Model without fermion doubling

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
A. Bochniak ◽  
A. Sitarz ◽  
P. Zalecki
Keyword(s):  
Standard Model ◽  
Symmetry Breaking ◽  
Dirac Operator ◽  
Noncommutative Geometry ◽  
Gauge Fields ◽  
Wick Rotation ◽  
Spectral Action ◽  
Geometry Model ◽  
The Standard Model ◽  
Cp Symmetry

Abstract We compute the leading terms of the spectral action for a noncommutative geometry model that has no fermion doubling. The spectral triple describing it, which is chiral and allows for CP-symmetry breaking, has the Dirac operator that is not of the product type. Using Wick rotation we derive explicitly the Lagrangian of the model from the spectral action for a flat metric, demonstrating the appearance of the topological θ-terms for the electroweak gauge fields.

scholarly journals DENSE m-CONVEX FRÉCHET SUBALGEBRAS OF OPERATOR ALGEBRA CROSSED PRODUCTS BY LIE GROUPS

1993 ◽  
Vol 04 (04) ◽  
pp. 601-673 ◽  
Author(s):  
LARRY B. SCHWEITZER
Keyword(s):  
Continuous Functions ◽  
Crossed Product ◽  
C Algebra ◽  
Schwartz Functions ◽  
Differentiable Functions ◽  
Dense Subalgebra ◽  
Fréchet Algebra ◽  
Locally Convex ◽  
Locally Convex Algebra ◽  
Projective Completion

Let A be a dense Fréchet *-subalgebra of a C*-algebra B. (We do not require Fréchet algebras to be m-convex.) Let G be a Lie group, not necessarily connected, which acts on both A and B by *-automorphisms, and let σ be a sub-polynomial function from G to the nonnegative real numbers. If σ and the action of G on A satisfy certain simple properties, we define a dense Fréchet *-subalgebra G ⋊σ A of the crossed product L1 (G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in σ. We give conditions on σ and the action of G on A which imply the m-convexity of the dense subalgebra G ⋊σ A. A locally convex algebra is said to be m-convex if there is a family of submultiplicative seminorms for the topology of the algebra. The property of m-convexity is important for a Fréchet algebra, and is useful in modern operator theory. If G acts as a transformation group on a locally compact space M, we develop a class of dense subalgebras for the crossed product L1 (G, C0 (M)), where C0 (M) denotes the continuous functions on M vanishing at infinity with the sup norm topology. We define Schwartz functions S (M) on M, which are differentiable with respect to some group action on M, and are rapidly vanishing with respect to some scale on M. We then form a dense Fréchet *-subalgebra G ⋊σ S (M) of rapidly vanishing, G-differentiable functions from G to S (M). If the reciprocal of σ is in Lp (G) for some p, we prove that our group algebras Sσ (G) are nuclear Fréchet spaces, and that G ⋊σ A is the projective completion [Formula: see text].

scholarly journals Generating Functionals for Locally Compact Quantum Groups

2020 ◽  
Author(s):  
Adam Skalski ◽  
Ami Viselter
Keyword(s):  
Quantum Group ◽  
Compact Quantum Group ◽  
Locally Compact ◽  
Generating Functional ◽  
Positive Functional ◽  
C Algebra ◽  
Locally Compact Quantum Groups ◽  
Dense Subalgebra ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

Abstract Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital *-subalgebra with core-like properties in its domain. On the other hand we prove that every normalised, symmetric, hermitian conditionally positive functional on a dense *-subalgebra of the unitisation of the universal C$^*$-algebra of a locally compact quantum group, satisfying certain technical conditions, extends in a canonical way to a generating functional. Some consequences of these results are outlined, notably those related to constructing cocycles out of convolution semigroups.

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