14,804 research outputs found
Spacetime-Free Approach to Quantum Theory and Effective Spacetime Structure
Motivated by hints of the effective emergent nature of spacetime structure,
we formulate a spacetime-free algebraic framework for quantum theory, in which
no a priori background geometric structure is required. Such a framework is
necessary in order to study the emergence of effective spacetime structure in a
consistent manner, without assuming a background geometry from the outset.
Instead, the background geometry is conjectured to arise as an effective
structure of the algebraic and dynamical relations between observables that are
imposed by the background statistics of the system. Namely, we suggest that
quantum reference states on an extended observable algebra, the free algebra
generated by the observables, may give rise to effective spacetime structures.
Accordingly, perturbations of the reference state lead to perturbations of the
induced effective spacetime geometry. We initiate the study of these
perturbations, and their relation to gravitational phenomena
Spencer Operator and Applications: From Continuum Mechanics to Mathematical physics
The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely
used in mathematics today and, up to our knowledge, has never been used in
engineering applications or mathematical physics. The main purpose of this
paper, an extended version of a lecture at the second workshop on Differential
Equations by Algebraic Methods (DEAM2, february 9-11, 2011, Linz, Austria) is
to prove that the use of the Spencer operator constitutes the common secret of
the three following famous books published about at the same time in the
beginning of the last century, though they do not seem to have anything in
common at first sight as they are successively dealing with elasticity theory,
commutative algebra, electromagnetism and general relativity: (C) E. and F.
COSSERAT: "Th\'eorie des Corps D\'eformables", Hermann, Paris, 1909. (M) F.S.
MACAULAY: "The Algebraic Theory of Modular Systems", Cambridge University
Press, 1916. (W) H. WEYL: "Space, Time, Matter", Springer, Berlin, 1918 (1922,
1958; Dover, 1952). Meanwhile, we shall point out the importance of (M) for
studying control identifiability and of (C)+(W) for the group theoretical
unification of finite elements in engineering sciences, recovering in a purely
mathematical way well known field-matter coupling phenomena (piezzoelectricity,
photoelasticity, streaming birefringence, viscosity, ...). As a byproduct and
though disturbing it could be, we shall prove that these unavoidable new
diferential and homological methods contradict the mathematical foundations of
both engineering (continuum mechanics,electromagnetism) and mathematical (gauge
theory, general relativity) physics.Comment: Though a few of the results presented are proved in the recent
references provided, the way they are combined with others and patched
together around the three books quoted is new. In view of the importance of
the full paper, the present version is only a summary of the definitive
version to appear later on. Finally, the reader must not forget that "each
formula" appearing in this new general framework has been used explicitly or
implicitly in (C), (M) and (W) for a mechanical, mathematical or physical
purpos
Macaulay inverse systems revisited
Since its original publication in 1916 under the title "The Algebraic Theory
of Modular Systems", the book by F. S. Macaulay has attracted a lot of
scientists with a view towards pure mathematics (D. Eisenbud,...) or
applications to control theory (U. Oberst,...).However, a carefull examination
of the quotations clearly shows that people have hardly been looking at the
last chapter dealing with the so-called "inverse systems", unless in very
particular situations. The purpose of this paper is to provide for the first
time the full explanation of this chapter within the framework of the formal
theory of systems of partial differential equations (Spencer operator on
sections, involution,...) and its algebraic counterpart now called "algebraic
analysis" (commutative and homological algebra, differential modules,...). Many
explicit examples are fully treated and hints are given towards the way to work
out computer algebra packages.Comment: From a lecture at the International Conference : Application of
Computer Algebra (ACA 2008) july 2008, RISC, LINZ, AUSTRI
A discussion on the origin of quantum probabilities
We study the origin of quantum probabilities as arising from non-boolean
propositional-operational structures. We apply the method developed by Cox to
non distributive lattices and develop an alternative formulation of
non-Kolmogorvian probability measures for quantum mechanics. By generalizing
the method presented in previous works, we outline a general framework for the
deduction of probabilities in general propositional structures represented by
lattices (including the non-distributive case).Comment: Improved versio
Modular Invariance on the Torus and Abelian Chern-Simons Theory
The implementation of modular invariance on the torus as a phase space at the
quantum level is discussed in a group-theoretical framework. Unlike the
classical case, at the quantum level some restrictions on the parameters of the
theory should be imposed to ensure modular invariance. Two cases must be
considered, depending on the cohomology class of the symplectic form on the
torus. If it is of integer cohomology class , then full modular invariance
is achieved at the quantum level only for those wave functions on the torus
which are periodic if is even, or antiperiodic if is odd. If the
symplectic form is of rational cohomology class , a similar result
holds --the wave functions must be either periodic or antiperiodic on a torus
times larger in both direccions, depending on the parity of .
Application of these results to the Abelian Chern-Simons is discussed.Comment: 24 pages, latex, no figures; title changed; last version published in
JM
Vacuum Fluctuations, Geometric Modular Action and Relativistic Quantum Information Theory
A summary of some lines of ideas leading to model-independent frameworks of
relativistic quantum field theory is given. It is followed by a discussion of
the Reeh-Schlieder theorem and geometric modular action of Tomita-Takesaki
modular objects associated with the quantum field vacuum state and certain
algebras of observables. The distillability concept, which is significant in
specifying useful entanglement in quantum information theory, is discussed
within the setting of general relativistic quantum field theory.Comment: 26 pages. Contribution for the Proceedings of a Conference on Special
Relativity held at Potsdam, 200
Macaulay inverse systems and Cartan-Kahler theorem
During the last months or so we had the opportunity to read two papers trying
to relate the study of Macaulay (1916) inverse systems with the so-called
Riquier (1910)-Janet (1920) initial conditions for the integration of linear
analytic systems of partial differential equations. One paper has been written
by F. Piras (1998) and the other by U. Oberst (2013), both papers being written
in a rather algebraic style though using quite different techniques. It is
however evident that the respective authors, though knowing the computational
works of C. done during the first half of the last century in a way not
intrinsic at all, are not familiar with the formal theory of systems of
ordinary or partial differential equations developped by D.C. Spencer
(1912-2001) and coworkers around 1965 in an intrinsic way, in particular with
its application to the study of differential modules in the framework of
algebraic analysis. As a byproduct, the first purpose of this paper is to
establish a close link between the work done by F. S. Macaulay (1862-1937) on
inverse systems in 1916 and the well-known Cartan-K{\"a}hler theorem (1934).
The second purpose is also to extend the work of Macaulay to the study of
arbitrary linear systems with variable coefficients. The reader will notice how
powerful and elegant is the use of the Spencer operator acting on sections in
this general framework. However, we point out the fact that the literature on
differential modules mostly only refers to a complex analytic structure on
manifolds while the Spencer sequences have been created in order to study any
kind of structure on manifolds defined by a Lie pseudogroup of transformations,
not just only complex analytic ones. Many tricky explicit examples illustrate
the paper, including the ones provided by the two authors quoted but in a quite
different framework
Globally nilpotent differential operators and the square Ising model
We recall various multiple integrals related to the isotropic square Ising
model, and corresponding, respectively, to the n-particle contributions of the
magnetic susceptibility, to the (lattice) form factors, to the two-point
correlation functions and to their lambda-extensions. These integrals are
holonomic and even G-functions: they satisfy Fuchsian linear differential
equations with polynomial coefficients and have some arithmetic properties. We
recall the explicit forms, found in previous work, of these Fuchsian equations.
These differential operators are very selected Fuchsian linear differential
operators, and their remarkable properties have a deep geometrical origin: they
are all globally nilpotent, or, sometimes, even have zero p-curvature. Focusing
on the factorised parts of all these operators, we find out that the global
nilpotence of the factors corresponds to a set of selected structures of
algebraic geometry: elliptic curves, modular curves, and even a remarkable
weight-1 modular form emerging in the three-particle contribution
of the magnetic susceptibility of the square Ising model. In the case where we
do not have G-functions, but Hamburger functions (one irregular singularity at
0 or ) that correspond to the confluence of singularities in the
scaling limit, the p-curvature is also found to verify new structures
associated with simple deformations of the nilpotent property.Comment: 55 page
- …