Geometric and algebraic approaches to quantum theory
- URL: http://arxiv.org/abs/2102.09176v4
- Date: Fri, 15 Oct 2021 06:03:52 GMT
- Title: Geometric and algebraic approaches to quantum theory
- Authors: Albert Schwarz
- Abstract summary: We show how to formulate physical theory taking as a starting point the set of states.
The equations of motion and the formulas for probabilities of physical quantities are analyzed.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We show how to formulate physical theory taking as a starting point the set
of states (geometric approach). We discuss the relation of this formulation to
the conventional approach to classical and quantum mechanics and the theory of
complex systems. The equations of motion and the formulas for probabilities of
physical quantities are analyzed. A heuristic proof of decoherence in our
setting is used to justify the formulas for probabilities. We show that any
physical theory theory can be obtained from classical theory if we restrict the
set of observables. This remark can be used to construct models with any
prescribed group of symmetries; one can hope that this construction leads to
new interesting models that cannot be build in the conventional framework.
The geometric approach can be used to formulate quantum theory in terms of
Jordan algebras, generalizing the algebraic approach to quantum theory. The
scattering theory can be formulated in geometric approach.
Related papers
- On the applicability of Kolmogorov's theory of probability to the description of quantum phenomena. Part I [0.0]
I show that it is possible to construct a mathematically rigorous theory based on Kolmogorov's axioms and physically natural random variables.
The approach can in principle be adapted to other classes of quantum-mechanical models.
arXiv Detail & Related papers (2024-05-09T12:11:28Z) - An algebraic formulation of nonassociative quantum mechanics [0.0]
We develop a version of quantum mechanics that can handle nonassociative algebras of observables.
Our approach is naturally probabilistic and is based on using the universal enveloping algebra of a general nonassociative algebra.
arXiv Detail & Related papers (2023-11-07T01:36:23Z) - Connecting classical finite exchangeability to quantum theory [69.62715388742298]
Exchangeability is a fundamental concept in probability theory and statistics.
We show how a de Finetti-like representation theorem for finitely exchangeable sequences requires a mathematical representation which is formally equivalent to quantum theory.
arXiv Detail & Related papers (2023-06-06T17:15:19Z) - Quantum mechanics and quantum field theory. Algebraic and geometric
approaches [0.0]
This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory.
It is based on the algebraic approach where the starting point is a star-algebra and on the geometric approach where the starting point is a convex set of states.
arXiv Detail & Related papers (2023-01-10T06:13:10Z) - Weyl Geometry and Quantum Corrections [0.0]
Weyl Geometry can be used to merge quantum theory and general relativity consistently as classical field theories.
In the Weyl Geometric framework, it seems that both quantum theory and gravity can merge consistently, once quantum theory is geometrized.
arXiv Detail & Related papers (2021-12-24T06:38:02Z) - Quantum Simulation of Conformal Field Theory [77.34726150561087]
We describe a quantum algorithm to simulate the dynamics of conformal field theories.
A full analysis of the approximation errors suggests near-term applicability.
arXiv Detail & Related papers (2021-09-29T06:44:33Z) - Quantum simulation of gauge theory via orbifold lattice [47.28069960496992]
We propose a new framework for simulating $textU(k)$ Yang-Mills theory on a universal quantum computer.
We discuss the application of our constructions to computing static properties and real-time dynamics of Yang-Mills theories.
arXiv Detail & Related papers (2020-11-12T18:49:11Z) - Topological Quantum Gravity of the Ricci Flow [62.997667081978825]
We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow.
First, we use BRST quantization to construct a "primitive" topological Lifshitz-type theory for only the spatial metric.
We extend the primitive theory by gauging foliation-preserving spacetime symmetries.
arXiv Detail & Related papers (2020-10-29T06:15:30Z) - Classicality without local discriminability: decoupling entanglement and
complementarity [0.0]
An operational probabilistic theory where all systems are classical, and all pure states of composite systems are entangled, is constructed.
We demonstrate that the presence of entanglement is independent of the existence of incompatible measurements.
We also prove the existence, in the theory, of a universal processor.
arXiv Detail & Related papers (2020-08-10T10:30:47Z) - Emergence of classical behavior in the early universe [68.8204255655161]
Three notions are often assumed to be essentially equivalent, representing different facets of the same phenomenon.
We analyze them in general Friedmann-Lemaitre- Robertson-Walker space-times through the lens of geometric structures on the classical phase space.
The analysis shows that: (i) inflation does not play an essential role; classical behavior can emerge much more generally; (ii) the three notions are conceptually distinct; classicality can emerge in one sense but not in another.
arXiv Detail & Related papers (2020-04-22T16:38:25Z) - From a quantum theory to a classical one [117.44028458220427]
We present and discuss a formal approach for describing the quantum to classical crossover.
The method was originally introduced by L. Yaffe in 1982 for tackling large-$N$ quantum field theories.
arXiv Detail & Related papers (2020-04-01T09:16:38Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.