In this paper we analyze two different functional formulations of classical mechanics. In the first one the Jacobi fields are represented by bosonic variables and belong to the vector (or its dual) representation of the symplectic group. In the second formulation the Jacobi fields are given as condensates of Grassmannian variables belonging to the spinor representation of the metaplectic group. For both formulations we shall show that, differently from what happens in the case presented in paper no. (I), it is possible to endow the associated Hilbert space with a positive definite scalar product and to describe the dynamics via a Hermitian Hamiltonian. The drawback of this formulation is that higher forms do not appear automatically and that the description of chaotic systems may need a further extension of the Hilbert space.
The complex Hilbert space of standard quantum mechanics may be treated as a real Hilbert space. The pure states of the complex theory become mixed states in the real formulation. It is then possible to generalize standard quantum mechanics, keeping the same set of physical states, but admitting more general observables. The standard time reversal operator involves complex conjugation, in this sense it goes beyond the complex theory and may serve as an example to motivate the generalization. Another example is unconventional canonical quantization such that the harmonic oscillator of angular fr…
In this paper we study the Hilbert space structure underlying the Koopman-von Neumann (KvN) operatorial formulation of classical mechanics. KvN limited themselves to study the Hilbert space of zero-forms that are the square integrable functions on phase space. They proved that in this Hilbert space the evolution is unitary for every system. In this paper we extend the KvN Hilbert space to higher forms which are basically functions of the phase space points and the differentials on phase space. We prove that if we equip this space with a positive definite scalar product the evolution can turn o…
Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket, and a quasidensity operator. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Classical mechanics can now be viewed as a deformation of quantum mechanics. The forms of semiquantum approximations to classical mechanics are indicated.
There is compelling evidence that, when continuous spectrum is present, the natural mathematical setting for Quantum Mechanics is the rigged Hilbert space rather than just the Hilbert space. In particular, Dirac's bra-ket formalism is fully implemented by the rigged Hilbert space rather than just by the Hilbert space. In this paper, we provide a pedestrian introduction to the role the rigged Hilbert space plays in Quantum Mechanics, by way of a simple, exactly solvable example. The procedure will be constructive and based on a recent publication. We also provide a thorough discussion on the ph…
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into either a problem involving a set of infinitely coupled differential equations or a problem involving a Shrödinger propagator with some appropriate kernel. Either way, Mathematics and Physics could be combined for Hilbert's tenth problem and for the notion of effective computability.
The quantum mechanical formalism for position and momentum of a particle in a one dimensional cyclic lattice is constructively developed. Some mathematical features characteristic of the finite dimensional Hilbert space are compared with the infinite dimensional case. The construction of an unbiased basis for state determination is discussed.
Quantum mechanics in the Rigged Hilbert Space formulation describes quasistationary phenomena mathematically rigorously in terms of Gamow vectors. We show that these vectors exhibit microphysical irreversibility, related to an intrinsic quantum mechanical arrow of time, which states that preparation of a state has to precede the registration of an observable in this state. Moreover, the Rigged Hilbert Space formalism allows the derivation of an exact golden rule describing the transition of a pure Gamow state into a mixture of interaction-free decay products.
Arno Bohm and Ilya Prigogine's Brussels-Austin Group have been working on the quantum mechanical arrow of time and irreversibility in rigged Hilbert space quantum mechanics. A crucial notion in Bohm's approach is the so-called preparation/registration arrow. An analysis of this arrow and its role in Bohm's theory of scattering is given. Similarly, the Brussels-Austin Group uses an excitation/de-excitation arrow for ordering events, which is also analyzed. The relationship between the two approaches is initially discussed focusing on their semi-group operators and time arrows. Finally a possibl…
It is shown that quantum mechanics is noncontextual if quantum properties are represented by subspaces of the quantum Hilbert space (as proposed by von Neumann) rather than by hidden variables. In particular, a measurement using an appropriately constructed apparatus can be shown to reveal the value of an observable A possessed by the measured system before the measurement took place, whatever other compatible ([B,A]=0) observable B may be measured at the same time.
In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics. The kinematical cornerstone of our framework is the so called polymer representation of the Heisenberg-Weyl (H-W) algebra, which is the starting point of the construction. The dynamics is constructed as a continuum limit of effective theories characterized by a scale, and requires a renormalization of the inner product. The result is a physical Hilbert space in whic…
We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert space as a linear real manifold equipped with its canonical symplectic form and restricting only to the expectation-value functions of Hermitian operators. Then in this framework, we reformulate the structure of quantum mechanics in the language of symplectic manifolds and avoid linear structure of Hilbert space in such a way that the results can be stated f…
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 hid…
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…
We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. It's evolution is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The ment…
A resolution of the quantum measurement problem(s) using the consistent histories interpretation yields in a rather natural way a restriction on what an observer can know about a quantum system, one that is also consistent with some results in quantum information theory. This analysis provides a quantum mechanical understanding of some recent work that shows that certain kinds of quantum behavior are exhibited by a fully classical model if by hypothesis an observer's knowledge of its state is appropriately limited.
A formulation of quaternionic quantum mechanics ($\mathbb{H}$QM) is presented in terms of a real Hilbert space. Using a physically motivated scalar product, we prove the spectral theorem and obtain a novel quaternionic Fourier series. After a brief discussion on unitary operators in this formalism, we conclude that this quantum theory is indeed consistent, and can be a valuable tool in the search for new physics.
W consider the problem of testing if a given matrix in the Hilbert space formulation of quantum mechanics or a function in the phase space formulation of quantum theory represent a quantum state. We propose several practical criteria to recognise states in these both versions of quantum physics. After minor modifications they can be applied to check positivity of any operators acting in a Hilbert space or positivity of any functions from an algebra with a Weyl type * -- product.
It is standardly assumed in discussions of quantum theory that physical systems can be regarded as having well-defined Hilbert spaces. It is shown here that a Hilbert space can be consistently partitioned only if its components are assumed not to interact. The assumption that physical systems have well-defined Hilbert spaces is, therefore, physically unwarranted.
We investigate consequences of allowing the Hilbert space of a quantum system to have a time-dependent metric. For a given possibly nonstationary quantum system, we show that the requirement of having a unitary Schreodinger time-evolution identifies the metric with a positive-definite (Ermakov-Lewis) dynamical invariant of the system. Therefore the geometric phases are determined by the metric. We construct a unitary map relating a given time-independent Hilbert space to the time-dependent Hilbert space defined by a positive-definite dynamical invariant. This map defines a transformation that …
Smooth composite bundles provide the adequate geometric description of classical mechanics with time-dependent parameters. We show that the Berry's phase phenomenon is described in terms of connections on composite Hilbert space bundles.
Classically, electromagnetic pulses are described by real fields that couple to charged matter and propagate causally. We will show here that real fields of the form used in standard classical electromagnetic theory have a quantum mechanical interpretation in which the probability density for a photon to be at x is positive definite and operators representing all of the standard physical observables exist. A covariant alternative to the ω_{k}^{1/2} dependence that appears in most (but not all) presentations of quantum optics and quantum field theory is presented and real scalar one photon adva…
Any Hilbert space with composite dimension can be factorized into a tensor product of smaller Hilbert spaces. This allows to decompose a quantum system into subsystems. We propose a simple tractable model for a constructive study of decompositions of quantum systems.
Using the bicomplex numbers $\mathbb{T}$ which is a commutative ring with zero divisors defined by $\mathbb{T}=\{w_0 + w_1 i_1 + w_2 i_2 + w_3 j | w_0, w_1, w_2, w_3 \in \mathbb{R}\}$ where $i_{1}^{2} = -1, i_{2}^{2} = -1, j^2 = 1, i_1 i_2 = j = i_2 i_1$, we construct hyperbolic and bicomplex Hilbert spaces. Linear functionals and dual spaces are considered and properties of linear operators are obtained; in particular it is established that the eigenvalues of a bicomplex self-adjoint operator are in the set of hyperbolic numbers.
The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditionally been used. This paper reviews some work which has been devoted to generalizing the target space of this imbedding to Hilbert modules of a more general type. In particular, detailed discussion is give…
An adapted representation of quantum mechanics sheds new light on the relationship between quantum states and classical states. In this approach the space of quantum states splits into a product of the state space of classical mechanics and a Hilbert space, and expectation values of observables decompose into their classical value plus a quantum correction. The splitting is preserved under time evolution of the Schrödinger equation under certain assumptions, and the time evolution of the classical part of a quantum state is governed by Hamilton's equation. The new representation is obtained fr…
We consider the probabilistic description of nonrelativistic, spinless one-particle classical mechanics, and immerse the particle in a deformed noncommutative phase space in which position coordinates do not commute among themselves and also with canonically conjugate momenta. With a postulated normalized distribution function in the quantum domain, the square of the Dirac delta density distribution in the classical case is properly realised in noncommutative phase space and it serves as the quantum condition. With only these inputs, we pull out the entire formalisms of noncommutative quantum …
The upside-down simple harmonic oscillator system is studied in the contexts of quantum mechanics and classical statistical mechanics. It is shown that in order to study in a simple manner the creation and decay of a physical system by ways of Gamow vectors we must formulate the theory in a time-asymmetric fashion, namely using two different rigged Hilbert spaces to describe states evolving towards the past and the future. The spaces defined in the contexts of quantum and classical statistical mechanics are shown to be directly related by the Wigner function.
The Paradigms introduced in philosophy of science one century ago are shown to be quite more satisfactory of that introduced by Galileo. This is particularly evident in the physics based on Hilbert Spaces and related mathematical structures that we apply in this paper to Quantum Mechanics and to Theory of Images. An exhaustive discussion, that include the algebraic analysis of the operators acting on them, exhibits that the Hilbert Spaces -- that have fixed dimension -- must be generalized to the Rigged Hilbert Spaces that contains right inside spaces with continuous and discrete dimensions. T…
We treat space and time as bona fide quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations in quantum gravity, we focus on a paradigm dealing with linear, first-order Hamiltonian and momentum constraints that lead to emergent features of temporal and spatial translations. Unlike the conventional treatment, we show that Klein-Gordon and Dirac equations in relativistic quantum mechanics can be unified in our paradigm by applying relativistic dispersion relations to eigenvalues rather than treating them as operator-valued equations. With time and space being…
The problem of the determination of the Hilbert-space metric which renders a given Hamiltonian $H$ self-adjoint is addressed from the point of view of applicability of computer-assisted algebraic manipulations. An exactly solvable example of the so called Gegenbauerian quantum-lattice oscillator is recalled for the purpose. Both the construction of suitable metric (basically, the solution of the Dieudonne's operator equation) and the determination of its domain of positivity are shown facilitated by the symbolic algebraic manipulations and by MAPLE-supported numerics and graphics.
The list of basic axioms of quantum mechanics as it was formulated by von Neumann includes only the mathematical formalism of the Hilbert space and its statistical interpretation. We point out that such an approach is too general to be considered as the foundation of quantum mechanics. In particular in this approach any finite-dimensional Hilbert space describes a quantum system. Though such a treatment might be a convenient approximation it can not be considered as a fundamental description of a quantum system and moreover it leads to some paradoxes like Bell's theorem. I present a list from …
Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functional relativity is introduced which states that quantum theory is indeed a functional tensor theory, i.e., it can be described by functional tensor equations. The main equations of quantum theory are show…
In this paper we find a simple rule to reproduce the algebra of quantum observables using only the commutators and operators which appear in the Koopman-von Neumann (KvN) formulation of classical mechanics. The usual Hilbert space of quantum mechanics becomes embedded in the KvN Hilbert space: in particular it turns out to be the subspace on which the quantum positions Q and momenta P act irreducibly.
In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of "physical experiment" and assuming "experimental accessibility and simplicity" as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper quant-ph/0506034. Pivotal roles are played by the "local observability principle", which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of "informationally complete observables" and of a …
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 …
The approximations of classical mechanics resulting from quantum mechanics are richer than a correspondence of classical dynamical variables with self-adjoint Hilbert space operators. Assertion that classical dynamic variables correspond to self-adjoint Hilbert space operators is disputable and sets unnatural limits on quantum mechanics. Well known examples of classical dynamical variables not associated with self-adjoint Hilbert space operators are discussed as a motivation for the realizations of quantum field theory that lack Hermitian field operators but exhibit interaction.
We study the question of how to decompose Hilbert space into a preferred tensor-product factorization without any pre-existing structure other than a Hamiltonian operator, in particular the case of a bipartite decomposition into "system" and "environment." Such a decomposition can be defined by looking for subsystems that exhibit quasi-classical behavior. The correct decomposition is one in which pointer states of the system are relatively robust against environmental monitoring (their entanglement with the environment does not continually and dramatically increase) and remain localized around…
Quantum theory's Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The reconstruction process is not complete, since it excludes the two-dimensional Hilbert space and still includes the exceptional Jordan algebras, which are not part of the Hilbert space apparatus. Options for physically meaningful potential generalizations of the apparatus are discussed.
A pseudo-Hermitian coupled-channel square-well model is proposed, solved and discussed. The domain of parameters is determined where all the bound-state energies (twice degenerate with respect to the second observable which we call "spin") remain real. The standard probabilistic interpretation of the wave functions is achieved by the transition from the original elementary non-physical indefinite pseudo-metric $θ$ to another, more involved but correct positive-definite physical metric $Θ$. In our model the latter step remains comparatively easily feasible. Another fortunate circumstance emerge…
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 …
Invariant Subspace Problem in Hilbert Spaces: Exploring Applications in Quantum Mechanics, Control Theory, Operator Algebras, Functional Analysis and Accelerator Physics
It is shown that a Hilbert space can be constructed for a quantum system starting from a framework in which histories are fundamental. The Decoherence Functional provides the inner product on this "History Hilbert space". It is also shown that the History Hilbert space is the standard Hilbert space in the case of non-relativistic quantum mechanics.
In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages, arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian) formulation of Quantum Mechanics has been revisited. In the present continuation of this study (with the spaces in question denoted as ${\cal H}^{\rm (auxiliary)}$ and ${\cal H}^{\rm (standard)}$) we spot a weak point of the 2HS formalism which lies in the double role played by ${\cal H}^{\rm (auxiliary)}$. As long as this confluence of roles may (and did!) lead to confusion in the literature, we propose an amended, three-Hilbert-space (3HS) reformulation of the sa…
The overall principles of what is now widely known as PT-symmetric quantum mechanics are listed, explained and illustrated via a few examples. In particular, models based on an elementary local interaction V(x) are discussed as motivated by the naturally emergent possibility of an efficient regularization of an otherwise unacceptable presence of a strongly singular repulsive core in the origin. The emphasis is put on the constructive aspects of the models. Besides the overall outline of the formalism we show how the low-lying energies of bound states may be found in closed form in certain dyna…
We investigate modifications of quantum mechanics (QM) that replace the unitary group in a finite dimensional Hilbert space with a finite group and determine the minimal sequence of subgroups necessary to approximate QM arbitrarily closely for general choices of Hamiltonian. This mathematical study reveals novel insights about 't Hooft's Ontological Quantum Mechanics, and the derivation of statistical mechanics from quantum mechanics. We show that Kornyak's proposal to understand QM as classical dynamics on a Hilbert space of one dimension higher than that describing the universe, supplemented…
In conventional Schrödinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h} \to H=Ω\,\mathfrak{h}\,Ω\neq H^\dagger$ induces the change ${\cal L} \to {\cal K}$ of the Hilbert space of states, reflected by the loss of the Hermiticity of $H\neq H^\dagger$. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts ${\cal K}$ into the third, correct physical Hilbert space ${\cal H}$, unitarily equivalent to ${\c…
This series of papers is devoted to an open-ended project aimed at the solution of Hilbert's sixth problem (concerning joint axiomatization of physics and probability theory) proposed to be constructed in the framework of an all-embracing mechanics. In this first paper, the bare skeleton of such a mechanics is constructed in the form of noncommutative Hamiltonian mechanics (NHM) which combines elements of noncommutative symplectic geometry and noncommutative probability in the framework of topological superalgebras; it includes, besides NHM basics, a treatment of Lie group actions in NHM and n…
We discuss, within the simplified context provided by the polymeric harmonic oscillator, a construction leading to a separable Hilbert space that preserves some of the most important features of the spectrum of the Hamiltonian operator. This construction may be applied to other polymer quantum mechanical systems, including those of loop quantum cosmology, and is likely generalizable to certain formulations of full loop quantum gravity. It is helpful to sidestep some of the physically relevant issues that appear in that context, in particular those related to superselection and the definition o…
The purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the mathematics of set partitions (which specify indefiniteness and definiteness) linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to more definite states…
A model for the information transfer from DNA to protein using quantum information and computation techniques is presented. DNA is modeled as the sender and proteins are modeled as the receiver of this information. On the DNA side, a 64-dimensional Hilbert space is used to describe the information stored in DNA triplets (codons). A Hamiltonian matrix is constructed for this space, using the 64 possible codons as base states. The eigenvalues of this matrix are not degenerate. The genetic code is degenerate and proteins comprise only 20 different amino acids. Since information is conserved, the …
Using the notions of frame transform and of square integrable projective representation of a locally compact group $G$, we introduce a class of isometries (tight frame transforms) from the space of Hilbert-Schmidt operators in the carrier Hilbert space of the representation into the space of square integrable functions on the direct product group $G\times G$. These transforms have remarkable properties. In particular, their ranges are reproducing kernel Hilbert spaces endowed with a suitable 'star product' which mimics, at the level of functions, the original product of operators. A 'phase spa…
From the standpoint of Hilbert's Sixth Problem, which is the axiomatisation of Physics, the famous paper of Lucien Hardy's, Quantum Theory from Five Reasonable Axioms, is not relevant. The present paper argues that Hardy does not give a physical definition of `limit', and if we assume the usual mathematical definition of limit of a sequence, he fails to define a sequence in physical terms to which the usual definition is applicable. We argue that one should not, in fact, try to define probability in terms of the usual notion of limit of a sequence of results of a measurement because of seeming…
In a recent paper Griffiths [38] has argued, based on the consistent histories interpretation, that Hilbert space quantum mechanics (QM) is noncontextual. According to Griffiths the problem of contextuality disappears if the apparatus is "designed and operated by a competent experimentalist" and we accept the Single Framework Rule (SFR). We will argue from a representational realist stance that the conclusion is incorrect due to the misleading understanding provided by Griffiths to the meaning of quantum contextuality and its relation to physical reality and measurements. We will discuss how t…
