arXiv subjects

Gabriele Carcassi

Publications and source records attributed to Gabriele Carcassi.

At least 19 recordsLinked to original sources

arXiv2026

Reverse Quantum Mechanics

Reverse Physics is a methodology that breaks physical theories into separate mathematical and physical conditions to establish their logical relationships. To showcase the power of the methodology, we present several results for quantum mechanics and their related insights. The standard Hilbert-space formulation conflicts with basic physical requirements, while a minimal topological modification can solve these problems. The ensemble space, rather than the pure-state space, distinguishes classical from quantum systems. The Born rule is an additional assumption linking orthogonality, mutual exclusivity and information entropy. Under explicit background conditions, unitary evolution is equivalent to deterministic and reversible evolution. Nonselective projective measurements can be characterized as Lindblad equilibration processes, while unitary evolution can be characterized as a limit of infinitesimal projective processes. Classical mechanics is recovered as the high-entropy limit of quantum mechanics, and every quantum state, pure or mixed, is a dynamical, spectral and thermodynamic equilibrium. These results are self-contained, use the standard vector-space representation and can thus be used as common tools and constraints for teaching, interpretations, reconstructions and future theories.

quant-ph↗

arXiv2026

What non-additive integral for ensemble spaces?

In a previous work we were able to define a non-additive measure that can be used to represent both classical and quantum states in physics. We further extended this idea to work on a generic space of statistical ensembles (i.e. an ensemble space) in a way that connects to Choquet theory. The question of which non-additive integral is suitable to generalize the notion of expectation value remains open. In this paper we show that the Sugeno and Choquet integrals are not suitable.

quant-ph↗

arXiv2025

Non-additive measures for quantum probability?

It is well-established that quantum probability does not follow classical Kolmogorov probability calculus. Various approaches have been developed to loosen the axioms, of which the use of signed measures is the most successful (e.g. the Wigner quasiprobability distribution). As part of our larger effort Assumptions of Physics, we have been considering the various roles of measures, which are used in physics not only for probability, but also to quantify the count of possible states and configurations. These measures play a crucial role in classical mechanics, as they effectively define its geometric structure. If one tries to construct a parallel in quantum mechanics, the measure to quantify the count of states turns out to be non-additive. The proper extension of probability calculus may require the use of non-additive measures, which is something that, to our knowledge, has not yet been explored. The purpose of this paper is to present the general idea and the open problems to an audience that is knowledgeable of the subject of non-additive set functions, though not necessarily in quantum physics, in the hope that it will spark helpful discussions. We go through the motivation in simple terms, which stems from the link between the entropy of a uniform distribution and the logarithm of the measure associated to its support. If one extends this notion to quantum mechanics, the associated measure is non-additive. We will explore some properties of this "quantum measure", its reasonableness in terms of the physics, but its peculiarity on the math side. We will explore the need for a set of properties that can properly characterize the measure and a generalization of the Radon-Nikodym derivative to define a properly extended probability calculus that reduces to the standard additive one on sets of physically distinguishable cases (i.e. orthogonal measurement outcomes).

quant-ph↗

arXiv2025

Response to "Are Hilbert Spaces Unphysical? Hardly, My Dear!''

A recent criticism of our paper ``The unphysicality of Hilbert spaces'' by Nivaldo Lemos refutes our central argument that a state with finite expectation value can be mapped to a state with infinite expectation value by a coordinate transformation. By conflating coordinate transformation with change of basis in quantum mechanics, Lemos argues that expectation values are invariant under change of variables. In the present work, we clarify the distinction between coordinate transformation and change of basis, and rebut Lemos' main argument.

quant-ph↗

arXiv2024

A No-Go Theorem for {\psi}-ontic Models? Yes! Response to Criticisms

This short note addresses the criticisms recently proposed by Shan Gao against our article "On the Reality of the Quantum State Once Again: A No-Go Theorem for {\psi}-Ontic Models" (Found. Phys. 54:14). The essay aims to respond to such objections and to show once again that the theorem proved in our paper is correct, and therefore true - contrary to Gao's claims. Philosophical consequences of this fact are briefly discussed.

quant-ph↗

arXiv2024

Classical mechanics as the high-entropy limit of quantum mechanics

We show that classical mechanics can be recovered as the high-entropy limit of quantum mechanics. That is, the high entropy masks quantum effects, and mixed states of high enough entropy can be approximated with classical distributions. The mathematical limit $\hbar \to 0$ can be reinterpreted as setting the zero entropy of pure states to $-\infty$, in the same way that non-relativistic mechanics can be recovered mathematically with $c \to \infty$. Physically, these limits are more appropriately defined as $S \gg 0$ and $v \ll c$. Both limits can then be understood as approximations independently of what circumstances allow those approximations to be valid. Consequently, the limit presented is independent of possible underlying mechanisms and of what interpretation is chosen for both quantum states and entropy.

quant-ph↗

arXiv2023

How quantum mechanics requires non-additive measures

Measure theory is used in physics, not just to capture classical probability, but also to quantify the number of states. In previous works, we found that state quantification plays a foundational role in classical mechanics, and therefore, we set ourselves to construct the quantum equivalent of the Liouville measure. Unlike the classical counterpart, this quantized measure is non-additive and has a unitary lower bound (i.e. no set of states can have less than one state). Conversely, requiring that state quantification is finite for finite continuous regions and that each state counts as one already implies non-additivity, which in turn implies the failure of classical theory. In this article we show these preliminary results and outline a new line of inquiry that may provide a different insight into the foundations of quantum theory. Additionally, this new approach may prove to be useful to those interested in a quantized theory of space-time, as we believe this requires a quantized measure for the quantification of the independent degrees of freedom.

quant-ph↗

arXiv2023

The unphysicality of Hilbert spaces

We argue that Hilbert spaces are not suitable to represent quantum states mathematically, in the sense that they require properties that are untenable by physical entities. We first demonstrate that the requirements posited by complex inner product spaces are physically justified. We then show that completeness in the infinite-dimensional case requires the inclusion of states with infinite expectations, coordinate transformations that take finite expectations to infinite ones and vice versa, and time evolutions that transform finite expectations to infinite ones in finite time. This means we should be wary of using Hilbert spaces to represent quantum states as they turn a potential infinity into an actual infinity. The main point of the paper, then, is to raise awareness that results that rely on the completeness of Hilbert spaces may not be physically significant. While we do not claim to know what a physically more appropriate closure should be, we note that Schwartz spaces, among other things, guarantee that the expectations of all polynomials of position and momentum are finite, their elements are uniquely identified by these expectations, and they are closed under Fourier transform.

quant-ph↗

arXiv2022

Geometric and physical interpretation of the action principle

We give a geometrical interpretation for the principle of stationary action in classical Lagrangian particle mechanics. In a nutshell, the difference of the action along a path and its variation effectively ``counts'' the possible evolutions that ``go through'' the area enclosed. If the path corresponds to a possible evolution, all neighbouring evolutions will be parallel, making them tangent to the area enclosed by the path and its variation, thus yielding a stationary action. This treatment gives a full physical account of the geometry of both Hamiltonian and Lagrangian mechanics which is founded on three assumptions: determinism and reversible evolution, independence of the degrees of freedom and equivalence between kinematics and dynamics. The logical equivalence between the three assumptions and the principle of stationary action leads to a much cleaner conceptual understanding.

physics.class-ph↗

arXiv2022

On the Common Logical Structure of Classical and Quantum Mechanics

At the onset of quantum mechanics, it was argued that the new theory would entail a rejection of classical logic. The main arguments to support this claim come from the non-commutativity of quantum observables, which allegedly would generate a non-distributive lattice of propositions, and from quantum superpositions, which would entail new rules for quantum disjunctions. While the quantum logic program is not as popular as it once was, a crucial question remains unsettled: what is the relationship between the logical structures of classical and quantum mechanics? In this essay we answer this question by showing that the original arguments promoting quantum logic contain serious flaws, and that quantum theory does satisfy the classical distributivity law once the full meaning of quantum propositions is properly taken into account. Moreover, we show that quantum mechanics can generate a distributive lattice of propositions, which, unlike the one of quantum logic, includes statements about expectation values which are of undoubtable physical interest. Lastly, we show that the lattice of statistical propositions in classical mechanics follows the same structure, yielding an analogue non-commutative sublattice of classical propositions. This fact entails that the purported difference between classical and quantum logic stems from a misconstructed parallel between the two theories.

quant-ph↗

arXiv2022

On the reality of the quantum state once again: A no-go theorem for $\psi$-ontic models

In this paper we show that $\psi$-ontic models, as defined by Harrigan and Spekkens (HS), cannot reproduce quantum theory. Instead of focusing on probability, we use information theoretic considerations to show that all pure states of $\psi$-ontic models must be orthogonal to each other, in clear violation of quantum mechanics. Given that (i) Pusey, Barrett and Rudolph (PBR) previously showed that $\psi$-epistemic models, as defined by HS, also contradict quantum mechanics, and (ii) the HS categorization is exhausted by these two types of models, we conclude that the HS categorization itself is problematic as it leaves no space for models that can reproduce quantum theory.

quant-ph↗

arXiv2021

Reverse Physics: From Laws to Physical Assumptions

We introduce an approach to the foundations of physics that is more in line with the foundations of mathematics. The idea is to examine current theories and find a set of starting physical assumptions that are sufficient to rederive them, which is the physical equivalent of the reverse mathematics program. This allows us to identify physical starting points (as opposed to mathematical ones), compare their deductive power, and find those that can better serve as a foundation for our theories.

physics.gen-ph↗

arXiv2021

Experimental verifiability and topology

We briefly show how the use of topological spaces and $\sigma$-algebras in physics can be rederived and understood as the fundamental requirement of experimental verifiability. We will see that a set of experimentally distinguishable objects will necessarily be endowed with a topology that is Kolmogorov (i.e. $T_0$) and second countable, which both puts constraints on well-formed scientific theories and allows us to give concrete physical meaning to the mathematical constructs. These insights can be taken as a first step in a general mathematical theory for experimental science.

physics.gen-ph↗

arXiv2021

Classical mechanics and infinitesimal reducibility

We briefly show how classical mechanics can be rederived and better understood as a consequence of three assumptions: infinitesimal reducibility, deterministic and reversible evolution, and kinematic equivalence.

physics.class-ph↗

arXiv2020

Space-time structure may be topological and not geometrical

In a previous effort [arXiv:1708.05492] we have created a framework that explains why topological structures naturally arise within a scientific theory; namely, they capture the requirements of experimental verification. This is particularly interesting because topological structures are at the foundation of geometrical structures, which play a fundamental role within modern mathematical physics. In this paper we will show a set of necessary and sufficient conditions under which those topological structures lead to real quantities and manifolds, which are a typical requirement for geometry. These conditions will provide a physically meaningful procedure that is the physical counter-part of the use of Dedekind cuts in mathematics. We then show that those conditions are unlikely to be met at Planck scale, leading to a breakdown of the concept of ordering. This would indicate that the mathematical structures required to describe space-time at that scale, while still topological, may not be geometrical.

physics.gen-ph↗

arXiv2020

Hamiltonian mechanics is conservation of information entropy

In this work we show the equivalence between Hamiltonian mechanics and conservation of information entropy. We will show that distributions with coordinate independent values for information entropy require that the manifold on which the distribution is defined is charted by conjugate pairs (i.e. it is a symplectic manifold). We will also show that further requiring that the information entropy is conserved during the evolution yields Hamilton's equations.

physics.hist-ph↗

arXiv2020

The four postulates of quantum mechanics are three

The tensor product postulate of quantum mechanics states that the Hilbert space of a composite system is the tensor product of the components' Hilbert spaces. All current formalizations of quantum mechanics that do not contain this postulate contain some equivalent postulate or assumption (sometimes hidden). Here we give a natural definition of composite system as a set containing the component systems and show how one can logically derive the tensor product rule from the state postulate and from the measurement postulate. In other words, our paper reduces by one the number of postulates necessary to quantum mechanics.

quant-ph↗

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↗