scholarly journals Nonflatness and totality

2018 ◽  
Vol 29 (2) ◽  
pp. 309-338
Author(s):  
BASIL A. KARÁDAIS
Keyword(s):  
Density Theorem ◽  
Compact Element ◽  
Separation Property ◽  
Strong Version ◽  
Finite Sets ◽  
Finite Density ◽  
Flat Base

We interpret finite types as domains over nonflat inductive base types in order to bring out the finitary core that seems to be inherent in the concept of totality. We prove a strong version of the Kreisel density theorem by providing a total compact element as a witness, a result that we cannot hope to have if we work with flat base types. To this end, we develop tools that deal adequately with possibly inconsistent finite sets of information. The classical density theorem is reestablished via a ‘finite density theorem,’ and corollaries are obtained, among them Berger's separation property.

Relationship Between Structure and Separation Property

2021 ◽  
pp. 881-922
Author(s):  
Zhanfeng Ju ◽  
El‐Sayed M. El‐Sayed ◽  
Daqiang Yuan
Keyword(s):  
Separation Property

scholarly journals Algebraic singularities of scattering amplitudes from tropical geometry

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
James Drummond ◽  
Jack Foster ◽  
Ömer Gürdoğan ◽  
Chrysostomos Kalousios
Keyword(s):  
Scattering Amplitudes ◽  
Natural Language ◽  
Tropical Geometry ◽  
Cluster Algebras ◽  
Finite Alphabet ◽  
Square Root ◽  
Finite Sets ◽  
Yang Mills ◽  
Square Roots ◽  
Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

scholarly journals Complex Langevin calculations in QCD at finite density

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Yuta Ito ◽  
Hideo Matsufuru ◽  
Yusuke Namekawa ◽  
Jun Nishimura ◽  
Shinji Shimasaki ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Chemical Potential ◽  
Expansion Method ◽  
Finite Density ◽  
Fermi Sphere ◽  
Expectation Value ◽  
Sign Problem ◽  
Zero Momentum ◽  
Effective Theories ◽  
Severe Sign

Abstract We demonstrate that the complex Langevin method (CLM) enables calculations in QCD at finite density in a parameter regime in which conventional methods, such as the density of states method and the Taylor expansion method, are not applicable due to the severe sign problem. Here we use the plaquette gauge action with β = 5.7 and four-flavor staggered fermions with degenerate quark mass ma = 0.01 and nonzero quark chemical potential μ. We confirm that a sufficient condition for correct convergence is satisfied for μ/T = 5.2 − 7.2 on a 83 × 16 lattice and μ/T = 1.6 − 9.6 on a 163 × 32 lattice. In particular, the expectation value of the quark number is found to have a plateau with respect to μ with the height of 24 for both lattices. This plateau can be understood from the Fermi distribution of quarks, and its height coincides with the degrees of freedom of a single quark with zero momentum, which is 3 (color) × 4 (flavor) × 2 (spin) = 24. Our results may be viewed as the first step towards the formation of the Fermi sphere, which plays a crucial role in color superconductivity conjectured from effective theories.

Representation and manipulation of finite sets

1980 ◽  
Vol 10 (4) ◽  
pp. 8-12 ◽  
Author(s):  
B. L. McAllister
Keyword(s):  
Finite Sets

Relationship Between Structure and Separation Property

2021 ◽  
pp. 922-952
Author(s):  
Zhanfeng Ju ◽  
El‐Sayed M. El‐Sayed ◽  
Daqiang Yuan
Keyword(s):  
Separation Property

scholarly journals Aspects of quantum information in finite density field theory

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Lucas Daguerre ◽  
Raimel Medina ◽  
Mario Solís ◽  
Gonzalo Torroba
Keyword(s):  
Field Theory ◽  
Quantum Information ◽  
Mutual Information ◽  
Fermi Surface ◽  
Relative Entropy ◽  
Chemical Potential ◽  
Operator Product ◽  
Dirac Fermions ◽  
Long Distance ◽  
Finite Density

Abstract We study different aspects of quantum field theory at finite density using methods from quantum information theory. For simplicity we focus on massive Dirac fermions with nonzero chemical potential, and work in 1 + 1 space-time dimensions. Using the entanglement entropy on an interval, we construct an entropic c-function that is finite. Unlike what happens in Lorentz-invariant theories, this c-function exhibits a strong violation of monotonicity; it also encodes the creation of long-range entanglement from the Fermi surface. Motivated by previous works on lattice models, we next calculate numerically the Renyi entropies and find Friedel-type oscillations; these are understood in terms of a defect operator product expansion. Furthermore, we consider the mutual information as a measure of correlation functions between different regions. Using a long-distance expansion previously developed by Cardy, we argue that the mutual information detects Fermi surface correlations already at leading order in the expansion. We also analyze the relative entropy and its Renyi generalizations in order to distinguish states with different charge and/or mass. In particular, we show that states in different superselection sectors give rise to a super-extensive behavior in the relative entropy. Finally, we discuss possible extensions to interacting theories, and argue for the relevance of some of these measures for probing non-Fermi liquids.

On Isbell's density theorem for bitopological pointfree spaces I

2020 ◽  
Vol 273 ◽  
pp. 106962
Author(s):  
M. Andrew Moshier ◽  
Imanol Mozo Carollo ◽  
Joanne Walters-Wayland
Keyword(s):  
Density Theorem
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