arXiv subjects

Gabriele Carcassi

Publications and source records attributed to Gabriele Carcassi.

30 records · Page 2Linked to original sources

arXiv2020

The fundamental connections between classical Hamiltonian mechanics, quantum mechanics and information entropy

We show that the main difference between classical and quantum systems can be understood in terms of information entropy. Classical systems can be considered the ones where the internal dynamics can be known with arbitrary precision while quantum systems can be considered the ones where the internal dynamics cannot be accessed at all. As information entropy can be used to characterize how much the state of the whole system identifies the state of its parts, classical systems can have arbitrarily small information entropy while quantum systems cannot. This provides insights that allow us to understand the analogies and differences between the two theories.

quant-ph↗

arXiv2019

Towards a general mathematical theory of experimental science

We lay the groundwork for a formal framework that studies scientific theories and can serve as a unified foundation for the different theories within physics. We define a scientific theory as a set of verifiable statements, assertions that can be shown to be true with an experimental test in finite time. By studying the algebra of such objects, we show that verifiability already provides severe constraints. In particular, it requires that a set of physically distinguishable cases is naturally equipped with the mathematical structures (i.e. second-countable Kolmogorov topologies and $σ$-algebras) that form the foundation of manifold theory, differential geometry, measure theory, probability theory and all the major branches of mathematics currently used in physics. This gives a clear physical meaning to those mathematical structures and provides a strong justification for their use in science. Most importantly it provides a formal framework to incorporate additional assumptions and constrain the search space for new physical theories.

cs.AI↗

arXiv2017

Topology and experimental distinguishability

In this work we introduce the idea that the primary application of topology in experimental sciences is to keep track of what can be distinguished through experimentation. This link provides understanding and justification as to why topological spaces and continuous functions are pervasive tools in science. We first define an experimental observation as a statement that can be verified using an experimental procedure and show that observations are closed under finite conjunction and countable disjunction. We then consider observations that identify elements within a set and show how they induce a Hausdorff and second-countable topology on that set, thus identifying an open set as one that can be associated with an experimental observation. We then show that experimental relationships are continuous functions, as they must preserve experimental distinguishability, and that they are themselves experimentally distinguishable by defining a Hausdorff and second-countable topology for this collection.

math.GN↗

arXiv2017

From physical assumptions to classical and quantum Hamiltonian and Lagrangian particle mechanics

The aim of this work is to show that particle mechanics, both classical and quantum, Hamiltonian and Lagrangian, can be derived from few simple physical assumptions. Assuming deterministic and reversible time evolution will give us a dynamical system whose set of states forms a topological space and whose law of evolution is a self-homeomorphism. Assuming the system is infinitesimally reducible---specifying the state and the dynamics of the whole system is equivalent to giving the state and the dynamics of its infinitesimal parts---will give us a classical Hamiltonian system. Assuming the system is irreducible---specifying the state and the dynamics of the whole system tells us nothing about the state and the dynamics of its substructure---will give us a quantum Hamiltonian system. Assuming kinematic equivalence, that studying trajectories is equivalent to studying state evolution, will give us Lagrangian mechanics and limit the form of the Hamiltonian/Lagrangian to the one with scalar and vector potential forces.

physics.class-ph↗

arXiv2015

From physical principles to relativistic classical Hamiltonian and Lagrangian particle mechanics

We show that classical particle mechanics (Hamiltonian and Lagrangian consistent with relativistic electromagnetism) can be derived from three fundamental assumptions: infinite reducibility, deterministic and reversible evolution, and kinematic equivalence. The core idea is that deterministic and reversible systems preserve the cardinality of a set of states, which puts considerable constraints on the equations of motion. This perspective links different concepts from different branches of math and physics (e.g. cardinality of a set, cotangent bundle for phase space, Hamiltonian flow, locally Minkowskian space-time manifold), providing new insights. The derivation strives to use definitions and mathematical concepts compatible with future extensions to field theories and quantum mechanics.

physics.class-ph↗

arXiv2014

From physical principles to classical Hamiltonian mechanics

We derive the Hamiltonian formulation of classical mechanics directly, without reference to Lagrangian mechanics. We start from the definition of states in terms of labels used to identify them, and show how, under a deterministic and reversible process, the conservation of the cardinality of the labels leads to Hamilton's equations.

physics.class-ph↗

arXiv2013

Derivation of Hamiltonian mechanics from determinism and reversibility

We put forth the idea that Hamilton's equations coincide with deterministic and reversible evolution. We explore the idea from five different perspectives (mathematics, measurements, thermodynamics, information theory and state mapping) and we show how they in the end coincide. We concentrate on a single degree of freedom at first, then generalize. We also discuss possible philosophical reasons why the laws of physics can only describe such processes, even if others must exist.

physics.class-ph↗

arXiv2012

Quantum mechanics: last stop for reductionism

The state space of a homogeneous body is derived under two different assumptions: infinitesimal reducibility and irreducibility. The first assumption leads to a real vector space, used in classical mechanics, while the second one leads to a complex vector space, used in quantum mechanics.

quant-ph↗

arXiv2009

How Einstein's quantum hypothesis requires a departure from classical mechanics

The aim of this work is to show how Einstein's quantum hypothesis leads immediately and necessarily to a departure from classical mechanics. First we note that the classical description and predictions are in terms of idealized measurements that are exact, instantaneous, non-perturbative, independent of each other and process agnostic. If we assume we cannot arbitrarily reduce the strength of a signal, measurements are ultimately perturbative to some degree. We show how a physical description in which the best measurement conceivable, i.e. the ideal measurement, perturbs the system leads to all the concepts present in quantum mechanics including conjugate variables, probabilistic predictions and measurements connected to symmetries.

quant-ph↗

arXiv2009

Classical mechanics as the many-particle limit of quantum mechanics

We derive the classical limit of quantum mechanics by describing the center of mass of a system constituted by a large number of particles. We will show that in that limit the commutator between the position and velocity of the center of mass is infinitesimal, which allows both to be known with great precision. We then show how the infinitesimal commutator allows for the definition of functions of position and velocity, and how the commutator reduces to a Poisson bracket.

quant-ph↗

arXiv2008

The Schroedinger equation as the time evolution of a deterministic and reversible process

In this paper we derive the Schroedinger equation by assuming it describes the time evolution of a deterministic and reversible process that leaves at each moment in time a different observable well defined; that is, it allows an accurate prediction for it. We also derive an expression for the well defined observable and apply this expression to the cases of spin precession and a free particle.

quant-ph↗

arXiv2008

Quantum mechanics as a consequence of discrete interactions

Quantum mechanics is usually presented starting from a series of postulates about the mathematical framework. In this work we show that those same postulates can be derived by assuming that measurements are discrete interactions: that is, that we measure at specific moments in time (as opposed to a continuous measurement that spans a long time interval) and that the system is in general affected by our measurement. We believe that this way of presenting quantum mechanics would make it easier to understand by laying out a more cohesive view of the theory and making it resonate more with our physics intuition.

quant-ph↗