Hilbert Space Quantum Mechanics Research Articles

Trace class operators and states in p-adic quantum mechanics

Within the framework of quantum mechanics over a quadratic extension of the non-Archimedean field of p-adic numbers, we provide a definition of a quantum state relying on a general algebraic approach and on a p-adic model of probability theory. As in the standard complex case, a distinguished set of physical states are related to a notion of trace for a certain class of bounded operators, and in fact, we show that one can define a suitable space of trace class operators in the non-Archimedean setting, as well. The analogies—but also the several (highly non-trivial) differences—with respect to the case of standard quantum mechanics in a complex Hilbert space are analyzed.

Generalised probabilistic theories in a new light

In this paper, a modified formulation of generalized probabilistic theories that will always give rise to the structure of Hilbert space of quantum mechanics, in any finite outcome space, is presented and the guidelines to how to extend this work to infinite dimensional Hilbert spaces are given. Moreover, this new formulation which will be called as extended operational probabilistic theories, applies not only to quantum systems, but also equally well to classical systems, without violating Bell's theorem, and at the same time solves the measurement problem. A new answer to the question of why our universe is quantum mechanical rather than classical will be presented. Besides, this extended probability theory shows that it is non determinacy, or to be more precise, the non deterministic description of the universe, that makes the laws of physics the way they are. In addition, this paper shows that there is still a possibility that there might be a deterministic level from which our universe emerges, which if understood correctly, may open the door wide to applications in areas such as quantum computing. In addition, this paper explains the deep reason why complex Hilbert spaces in quantum mechanics are needed.

"Mysteries" of Modern Physics and the Fundamental Constants c, h, and G

We review how the kinematic structures of special relativity and quantum mechanics both stem from the relativity principle, i.e., "no preferred reference frame" (NPRF). Essentially, NPRF applied to the measurement of the speed of light c gives the light postulate and leads to the geometry of Minkowski space, while NPRF applied to the measurement of Planck's constant h gives "average-only" projection and leads to the denumerable-dimensional Hilbert space of quantum mechanics. These kinematic structures contain the counterintuitive aspects ("mysteries") of time dilation, length contraction, and quantum entanglement. In this essay, we extend the application of NPRF to the gravitational constant G and show that it leads to the "mystery" of the contextuality of mass in general relativity. Thus, we see an underlying coherence and integrity in modern physics via its "mysteries" and the fundamental constants c, h, and G. It is well known that Minkowski and Einstein were greatly influenced by David Hilbert in their development of special relativity and general relativity, respectively, but relating those theories to quantum mechanics via its non-Boolean Hilbert space kinematics is perhaps surprising.Quanta 2022; 11: 5–14.

Comment on “Nonlocality claims are inconsistent with Hilbert-space quantum mechanics”

The view exists that the Bell inequality is a mere inconsistent application of classical concepts to a well-established quantum world. In the article, ``Nonlocality claims are inconsistent with Hilbert-space quantum mechanics'' [Phys. Rev. A, 101, 022117, (2020)], Robert B. Griffiths advocates for quantum theory's locality. Although R. B. Griffiths presents valuable insights in favor of quantum mechanics' local character, he argues the Bell inequality is an inconsistent application of classical physics to quantum mechanics. We explain why a correct assessment of the Bell inequality does not in fact conflict with Griffiths's views of quantum locality and, on the contrary, it already contemplates them. Hence, Bell inequality inconsistency is not necessary for Griffiths's quantum locality to hold.

Reply to “Comment on ‘Nonlocality claims are inconsistent with Hilbert-space quantum mechanics’ ”

In Phys. Rev. A 101, 022117 (2020), it was argued that Bell inequalities are based on classical, not quantum, physics, and hence their violation in experiments provides no support for the claimed existence of peculiar nonlocal and superluminal influences in the real (quantum) world. This Reply to Lambare's Comment, Phys. Rev. A 104, 066201 (2021), on that paper seeks to clarify some issues related to the correct use of Hilbert space quantum mechanics for identifying the microscopic causes of later macroscopic measurement outcomes, a matter not properly addressed by Bell, who used classical hidden variables in place of the Hilbert subspaces (equivalently, their projectors) employed by von Neumann in his \it{Mathematical Foundations of Quantum Mechanics}.

Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits

One of the lowest-order corrections to Gaussian quantum mechanics in infinite-dimensional Hilbert spaces are Airy functions: a uniformization of the stationary phase method applied in the path integral perspective. We introduce a "periodized stationary phase method" to discrete Wigner functions of systems with odd prime dimension and show that the π8 gate is the discrete analog of the Airy function. We then establish a relationship between the stabilizer rank of states and the number of quadratic Gauss sums necessary in the periodized stationary phase method. This allows us to develop a classical strong simulation of a single qutrit marginal on t qutrit π8 gates that are followed by Clifford evolution, and show that this only requires 3t2+1 quadratic Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as ∼30.8t for 10−2 precision) for any number of π8 gates to full precision.

Nonlinear description of quantum dynamics: Generalized coherent states

In this work, it is shown that there is an inherent nonlinear evolution in the dynamics of the so-called generalized coherent states. To show this, the immersion of a classical manifold into the Hilbert space of quantum mechanics is employed. Then, one may parameterize the time dependence of the wave function through the variation of parameters in the classical manifold. Therefore, the immersion allows us to consider the so-called principle of analogy, i.e., using the procedures and structures available from the classical setting to employ them in the quantum framework.

Quaternionic quantum harmonic oscillator

In this article we obtained the harmonic oscillator solution for quaternionic quantum mechanics ($\mathbbm{H}$QM) in the real Hilbert space, both in the analytic method and in the algebraic method. The quaternionic solutions have many additional possibilities if compared to complex quantum mechanics ($\mathbbm{C}$QM), and thus there are many possible applications to these results in future research.

Структура квантовой механики

Interest in quantum mechanics is growing in connection with new experiments with quantum particles and a deeper knowledge of nano- and subatomic principles. The practical use of quantum mechanic methods goes in parallel with the constant rethinking of its foundations, its ideological and epistemological role. A rigorous exposition of the foundations of quantum mechanics in abstract-Hilbert spaces was given in the book of J. Von Neumann in 1933. It is the first and only experiment that has been brought to the end, the presentation of the apparatus of quantum mechanics with the sequence and rigor which usually presented in a purely theory. Later in 1949 the group of French mathematicians under the pseudonym N. Bourbaki introduced the concept of mathematical into mathematics. To date, there is no narration of quantum mechanics, where rigor is combined with the concept of mathematical structure. In the framework of the approach formulated by the group of N. Burbaki, quantum mechanics is a composite structure consisting of three simple structures: the Hilbert space of complex-valued vectors, the space of linear self-adjoint operators and the structure of classical mechanics are showed in this article.

Fourier transform, quantum mechanics and quantum field theory on the manifold of general relativity

A proof is given for the Fourier transform for functions in a quantum mechanical Hilbert space on a non-compact manifold in general relativity. In the (configuration space) Newton–Wigner representation, we discuss the spectral decomposition of the canonical operators and give a proof of the Parseval–Plancherel relation and the Born rule for linear superposition. We then discuss the representations of pure quantum states and their dual vectors and construct the Fock space and the associated quantum field theory for Bose–Einstein and Fermi–Dirac statistics.

Deep Quantum Geometry of Matrices

We employ machine learning techniques to provide accurate variational wavefunctions for matrix quantum mechanics, with multiple bosonic and fermionic matrices. Variational quantum Monte Carlo is implemented with deep generative flows to search for gauge invariant low energy states. The ground state, and also long-lived metastable states, of an $\mathrm{SU}(N)$ matrix quantum mechanics with three bosonic matrices, as well as its supersymmetric `mini-BMN' extension, are studied as a function of coupling and $N$. Known semiclassical fuzzy sphere states are recovered, and the collapse of these geometries in more strongly quantum regimes is probed using the variational wavefunction. We then describe a factorization of the quantum mechanical Hilbert space that corresponds to a spatial partition of the emergent geometry. Under this partition, the fuzzy sphere states show a boundary-law entanglement entropy in the large $N$ limit.

Nonlocality claims are inconsistent with Hilbert-space quantum mechanics

It is shown that when properly analyzed using principles consistent with the use of a Hilbert space to describe microscopic properties, quantum mechanics is a local theory: one system cannot influence another system with which it does not interact. Claims to the contrary based on quantum violations of Bell inequalities are shown to be incorrect. A specific example traces a violation of the CHSH Bell inequality in the case of a spin-3/2 particle to the noncommutation of certain quantum operators in a situation where (non)locality is not an issue. A consistent histories analysis of what quantum measurements measure, in terms of quantum properties, is used to identify the basic problem with derivations of Bell inequalities: the use of classical concepts (hidden variables) rather than a probabilistic structure appropriate to the quantum domain. A difficulty with the original Einstein-Podolsky-Rosen (EPR) argument for the incompleteness of quantum mechanics is the use of a counterfactual argument which is not valid if one assumes that Hilbert-space quantum mechanics is complete; locality is not an issue. The quantum correlations that violate Bell inequalities can be understood using local quantum common causes. Wavefunction collapse and Schr\"odinger steering are calculational procedures, not physical processes. A general Principle of Einstein Locality rules out nonlocal influences between noninteracting quantum systems. Some suggestions are made for changes in terminology that could clarify discussions of quantum foundations and be less confusing to students.

The Relationship of Arithmetic As Two Twin Peano Arithmetic(s) and Set Theory: A New Glance From the Theory of Information

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincare’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested.

Quantum-Information Conservation. The Problem About ‘Hidden Variables’, or the ‘Conservation of Energy Conservation’ in Quantum Mechanics: A Historical Lesson for Future Discoveries

The explicit history of the “hidden variables” problem is well-known and established. The main events of its chronology are traced. An implicit context of that history is suggested. It links the problem with the “conservation of energy conservation” in quantum mechanics. Bohr, Kramers, and Slaters (1924) admitted its violation being due to the “fourth Heisenberg uncertainty”, that of energy in relation to time. Wolfgang Pauli rejected the conjecture and even forecast the existence of a new and unknown then elementary particle, neutrino, on the ground of energy conservation in quantum mechanics, afterwards confirmed experimentally. Bohr recognized his defeat and Pauli’s truth: the paradigm of elementary particles (furthermore underlying the Standard model) dominates nowadays. However, the reason of energy conservation in quantum mechanics is quite different from that in classical mechanics (the Lie group of all translations in time). Even more, if the reason was the latter, Bohr, Cramers, and Slatters’s argument would be valid. The link between the “conservation of energy conservation” and the problem of hidden variables is the following: the former is equivalent to their absence. The same can be verified historically by the unification of Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics in the contemporary quantum mechanics by means of the separable complex Hilbert space. The Heisenberg version relies on the vector interpretation of Hilbert space, and the Schrodinger one, on the wave-function interpretation. However the both are equivalent to each other only under the additional condition that a certain well-ordering is equivalent to the corresponding ordinal number (as in Neumann’s definition of “ordinal number”). The same condition interpreted in the proper terms of quantum mechanics means its “unitarity”, therefore the “conservation of energy conservation”. In other words, the “conservation of energy conservation” is postulated in the foundations of quantum mechanics by means of the concept of the separable complex Hilbert space, which furthermore is equivalent to postulating the absence of hidden variables in quantum mechanics (directly deducible from the properties of that Hilbert space). Further, the lesson of that unification (of Heisenberg’s approach and Schrodinger’s version) can be directly interpreted in terms of the unification of general relativity and quantum mechanics in the cherished “quantum gravity” as well as a “manual” of how one can do this considering them as isomorphic to each other in a new mathematical structure corresponding to quantum information. Even more, the condition of the unification is analogical to that in the historical precedent of the unifying mathematical structure (namely the separable complex Hilbert space of quantum mechanics) and consists in the class of equivalence of any smooth deformations of the pseudo-Riemannian space of general relativity: each element of that class is a wave function and vice versa as well. Thus, quantum mechanics can be considered as a “thermodynamic version” of general relativity, after which the universe is observed as if “outside” (similarly to a phenomenological thermodynamic system observable only “outside” as a whole). The statistical approach to that “phenomenological thermodynamics” of quantum mechanics implies Gibbs classes of equivalence of all states of the universe, furthermore re-presentable in Boltzmann’s manner implying general relativity properly … The meta-lesson is that the historical lesson can serve for future discoveries.

The Isomorphism of Minkowski Space and the Separable Complex Hilbert Space and Its Physical Interpretation

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincare’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”.

An elementary canonical classical and quantum dynamics for general relativity

A consistent canonical classical and quantum dynamics in the framework of special relativity was formulated by Stueckelberg in 1941, and generalized to many body theory by Horwitz and Piron in 1973 (SHP). In this paper, this theory is embedded into the framework of general relativity (GR), here denoted by SHPGR. The canonical Poisson brackets of the SHP theory remain valid (invariant under local coordinate transformations) on the manifold of GR , and provide the basis for formulating a canonical quantum theory; the result (here defined as SHPGR) is generalized to many-body theory. A scalar product is defined for constructing the Hilbert space and a Hermitian momentum operator defined. The Fourier transform is defined, connecting momentum and coordinate representations. The potential which may occur in the SHP theory emerges as a spacetime scalar mass distribution in GR, and electromagnetism corresponds to a gauge field on the quantum mechanical GR Hilbert space in both the single particle and many-body theory. A diffeomorphism covariant form of Newton's law is found as an immediate consequence of the canonical formulation of SHPGR. We compute the classical evolution of the off shell mass on the orbit of a particle and the force on a particle and its energy at the Schwarzschild horizon. The propagator for evolution of the one body state is studied and a scattering theory on the manifold is worked out.

An Elementary Canonical Classical and Quantum Dynamics for General Relativity

A consistent canonical classical and quantum dynamics in the framework of special relativity was formulated by Stueckelberg in 1941, and generalized to many body theory by Horwitz and Piron in 1973 (SHP). In this paper, using local coordinate transformations, following the original procedure of Einstein, this theory is embedded into the framework of general relativity (GR) both for potential models (where the potential appears as a spacetime mass distribution with dimension of mass) and for electromagnetism (emerging as a gauge field on the quantum mechanical Hilbert space). The canonical Poisson brackets of the SHP theory remain valid (invariant under local coordinate transformations) on the manifold of GR, and provide the basis, following Dirac’s quantization procedure, for formulating a quantum theory. The theory is developed both for one and many particles.

On the Energy-Momentum Spectrum and One-Meson Dispersion Curves in (3 + 1)-Dimensional Strongly Coupled Lattice QCD with three Flavors

We use an imaginary-time formulation of quantum field theory to analyze a lattice QCD model with the improved Wilson action on a Euclidean (3 + 1)-dimensional unit hypercubic lattice, with three quark flavors, local SU(3)c gauge symmetry and global SU(3)f flavor symmetry. We work in the strong coupling regime, with a small hopping parameter κ > 0 and a much smaller plaquette coupling β > 0. This work extends our previous results on the pointwisely determined one-meson dispersion curves in the low-lying one-particle energy-momentum spectrum of strongly coupled lattice QCD models. Here, we give a smoothness characterization of the dispersion curves and provide bounds on the more than one-particle contributions to the spectral measure for the two-meson correlations. Denoting by H the underlying quantum mechanical physical Hilbert space, constructed according to the Osterwalder-Seiler prescription, our spectral results are derived from spectral representations for the positive self-adjoint energy and self-adjoint momentum operators in H. After establishing a lattice Feynman-Kac formula, these representations are used to rigorously identify complex momentum singularities of the lattice Fourier transform of special two-particle correlations with energy-momentum spectral points and identify the meson particles with isolated dispersion curves. To show isolated dispersion curves we must show both lower and upper spectral gaps. To determine the eightfold-way mesons, the usual spacetime and internal symmetries are heavily used to block decompose the lattice Fourier transform of the two-meson correlation matrix. Then, to solve the implicit equations determining the dispersion curves, we apply the classical analytic implicit function theorem to the blocks associated with total spin zero mesons and the Weierstrass preparation theorem to the blocks corresponding to total spin one. The Weierstrass preparation theorem is the natural tool to deal with the determination of the spectrum when degeneracies are present and the hypotheses for the analytic implicit function theorem are not fulfilled. All the 36 meson dispersion curves are of the form, for β = 0, w(p→)=−2lnκ+r(κ,p→), with r(κ,p→) of order κ2, giving an asymptotic mass -2 ln κ. A mass splitting of 2κ4 between the pseudoscalar and vector mesons occurs at order κ4. We determine smoothness properties of the functions r(κ,p→). The results for β = 0 are extended to small β > 0 using β-analyticity properties of correlations in the infinite-volume limit.

Infinite-dimensional Categorical Quantum Mechanics

We use non-standard analysis to define a category $^\star\!\operatorname{Hilb}$ suitable for categorical quantum mechanics in arbitrary separable Hilbert spaces, and we show that standard bounded operators can be suitably embedded in it. We show the existence of unital special commutative $\dagger$-Frobenius algebras, and we conclude $^\star\!\operatorname{Hilb}$ to be compact closed, with partial traces and a Hilbert-Schmidt inner product on morphisms. We exemplify our techniques on the textbook case of 1-dimensional wavefunctions with periodic boundary conditions: we show the momentum and position observables to be well defined, and to give rise to a strongly complementary pair of unital commutative $\dagger$-Frobenius algebras.

Irreversible evolution of a wave packet in the rigged-Hilbert-space quantum mechanics

It is well known that a state with complex energy cannot be the eigenstate of a self-adjoint operator, like the Hamiltonian. Resonances, i.e. states with exponentially decaying observables, are not vectors belonging to the conventional Hilbert space. One can describe these resonances in an unusual mathematical formalism, based on the so-called Rigged Hilbert Space (RHS). In the RHS, the states with complex energy are denoted as Gamow Vectors (GV), and they model decay processes. We study GV of the Reversed Harmonic Oscillator (RHO), and we analytically and numerically investigate the unstable evolution of wave packets. We introduce the background function to study initial data not composed only by a summation of GV and we analyse different wave packets belonging to specific function spaces. Our work furnishes support to the idea that irreversible wave propagations can be investigated by means of Rigged Hilbert Space Quantum Mechanics and provides insights for the experimental investigation of irreversible dynamics.