The measure of distinguishability between two neighboring preparations of a physical system by a measurement apparatus naturally defines the line element of the preparation space of the system. We point out that quantum mechanics can be derived from the invariance of this line element in the canonical formulation. The canonical formulation of quantum statistical mechanics is also discussed.
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 concept of an injective affine embedding of the quantum states into a set of classical states, i.e., into the set of the probability measures on some measurable space, as well as its relation to statistically complete observables is revisited, and its limitation in view of a classical reformulation of the statistical scheme of quantum mechanics is discussed. In particular, on the basis of a theorem concerning a non-denseness property of a set of coexistent effects, it is shown that an injective classical embedding of the quantum states cannot be supplemented by an at least approximate clas…
Within the recent reformulation of quantum mechanics where a potential function is not required, we show how to reconstruct the potential so that a correspondence with the standard formulation could be established. However, severe restriction is placed by the correspondence on the kinematics of such problems.
In a comparison of the principles of special relativity and of quantum mechanics, the former theory is marked by its relative economy and apparent explanatory simplicity. A number of theorists have thus been led to search for a small number of postulates - essentially information theoretic in nature - that would play the role in quantum mechanics that the relativity principle and the light postulate jointly play in Einstein's 1905 special relativity theory. The purpose of the present paper is to resist this idea, at least in so far as it is supposed to reveal the fundamental form of the theory…
Bell's Theorem rules out many potential reformulations of quantum mechanics, but within a generalized framework, it does not exclude all "locally-mediated" models. Such models describe the correlations between entangled particles as mediated by intermediate parameters which track the particle world-lines and respect Lorentz covariance. These locally-mediated models require the relaxation of an arrow-of-time assumption which is typically taken for granted. Specifically, some of the mediating parameters in these models must functionally depend on measurement settings in their future, i.e., on in…
The widely known but also somewhat esoteric Mach principle envisages a fully relational formulation of physical theories without any reference to a concept of `absolute space'. When applied to classical mechanics, under the guise of an extended symmetry group, this procedure is known to lead to an equation of motion with inertial-like forces that are sourced by the mass distribution of the system itself. In this paper we follow a similar procedure and reformulate the Schrödinger equation of non-relativistic quantum mechanics in a fully Machian way. Just like its classical counterpart, the resu…
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…
Gao (2017) presents a new mentalistic reformulation of the well-known measurement problem affecting the standard formulation of quantum mechanics. According to this author, it is essentially a determinate-experience problem, namely a problem about the compatibility between the linearity of the Schroedinger's equation, the fundamental law of quantum theory, and definite experiences perceived by conscious observers. In this essay I aim to clarify (i) that the well-known measurement problem is a mathematical consequence of quantum theory's formalism, and that (ii) its mentalistic variant does not…
A structural similarity between Classical Mechanics (CM) and Quantum Mechanics (QM) was revealed by P.A.M. Dirac in terms of Lie Algebras: while in CM the dynamics is determined by the Lie algebra of Poisson brackets on the manifold of scalar fields for classical position/momentum observables q/p, d/dt q={q,H}, d/dt p={p,H}, QM evolves (in Heisenberg's picture) according to the formally similar Lie algebra of commutator brackets of the corresponding operators Q/P: d/dt Q=i/h [Q,H], d/dt P=i/h [P,H] where QP-PQ=ih. A further common framework for comparing CM and QM is the category of symplectic…
The non-Hermitian Schrödinger equation is re-expressed generally in the form of Hamilton's canonical equation without any approximation. Its quantization called non-Hermitian quantum field theory is discussed. By virtue of the canonical equation, the theory of non-Hermitian quantum mechanics is totally reformulated, including the probability amplitudes of states, the expectations of operators, as well as the expressions of interaction terms. The conventional difficulties in non-Hermitian quantum mechanics are totally overcome by the reformulation. Specifically, the imaginary parts the non-Herm…
Some of the important non-classical aspects of quantum mechanics can be described in more intuitive terms if they are reformulated in a geometrical picture based on an extension of the classical phase space. This contribution presents various phase-space properties of moments describing a quantum state and its dynamics. An example of a geometrical reformulation of a non-classical quantum effect is given by an equivalence between conditions imposed by uncertainty relations and centrifugal barriers, respectively.
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…
We investigate two methods of constructing a solution of the Schrödinger equation from the canonical transformation in classical mechanics. One method shows that we can formulate the solution of the Schrödinger equation from linear canonical transformations, the other focuses on the generating function which satisfies the Hamilton-Jacobi equation in classical mechanics. We also show that these two methods lead to the same solution of the Schrödinger equation.
The representation of a Schrodinger equations as a classic Hamiltonian system allows to construct a unified perturbation theory both in classic, and in a quantum mechanics grounded on the theory of canonical transformations, and also to receive asymptotic estimations of affinity of the precisian approximated solutions of Schrodinger equations
$p$-Mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how non-linear canonical transformations affect $p$-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of $p$-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state…
The concept of number is fundamental to the formulation of any physical theory. We give a heuristic motivation for the reformulation of Quantum Mechanics in terms of non-standard real numbers called Quantum Real Numbers. The standard axioms of quantum mechanics are re-interpreted. Our aim is to show that, when formulated in the language of quantum real numbers, the laws of quantum mechanics appear more natural, less counterintuitive than when they are presented in terms of standard numbers.
Relational mechanics is a reformulation of mechanics (classical or quantum) for which space is relational. This means that the configuration of an $N$-particle system is a shape, which is what remains when the effects of rotations, translations, and dilations are quotiented out. This reformulation of mechanics naturally leads to a relational notion of time as well, in which a history of the universe is just a curve in shape space without any reference to a special parametrization of the curve given by an absolute Newtonian time. When relational mechanics (classical or quantum) is regarded as f…
We present a study on linear canonical transformation in the framework of a phase space representation of quantum mechanics that we have introduced in our previous work [1]. We begin with a brief recall about the so called phase space representation. We give the definition of linear canonical transformation with the transformation law of coordinate and momentum operators. We establish successively the transformation laws of mean values, dispersions, basis state and wave functions.Then we introduce the concept of isodispersion linear canonical transformation.
Canonical transformations using the idea of quantum generating functions are applied to construct a quantum Hamilton-Jacobi theory, based on the analogy with the classical case. An operator and a c-number forms of the time-dependent quantum Hamilton-Jacobi equation are derived and used to find dynamical solutions of quantum problems. The phase-space picture of quantum mechanics is discussed in connection with the present theory.
Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the ac…
The linear canonical transforms of position and momentum are used to construct the tomographic probability representation of quantum states where the fair probability distribution determines the quantum state instead of the wave function or density matrix. The example of Moshinsky shutter problem is considered.
Relational formulation of quantum mechanics is based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. In a recent paper (J. M. Yang, Sci. Rep. 8:13305, 2018), basic relational quantum mechanics framework is formulated to derive quantum probability, Born's Rule, Schrödinger Equations, and measurement theory. This paper further extends the reformulation effort in three aspects. First, it gives a clearer explanation of the key concepts behind the framework to c…
The quantum weak value draws many attentions recently from theoretical curiosity to experimental applications. Now we design an unusual weak measuring procedure as the pre-selection, mid-selection and post-selection to study the correlation function of two weak values, which we called the weak correlation function. In this paper, we proposed an weak measurement experiment to measure the canonical commutator $[\hat{x},\hat{p}]=i\hbar$ in quantum mechanics. Furthurmore, we found the intriguing equivalence between the canonical commutation relation and Riemann hypothesis, and then obtained the we…
In this Paper we present an approach to Quantum Mechanical Canonical Transformations. Our main result is that Time Dependent Quantum Canonical Transformations can always be cast in the form of Squeezing Operators. We revise the main properties of these operators in regard to its Lie group properties, how two of them can be combined to yield another operator of the same class and how can also be decomposed and fragmented. In the second part of the paper we show how this procedure works extremely well for the Time Dependent Quantum Harmonic Oscillator. The issue of the systematic construction of…
In order to establish a correspondence between the reformulation of quantum mechanics without potential function and the conventional quantum mechanics; we obtained the potential function of the New Wilson - Racah quantum system in [3] using any of the proposed formulas in [4]. To achieve this, we used the matrix elements of the potential function and the basis element of the configuration space.
One of the basic assumptions underlying Bell's theorem is the causal arrow of time, having to do with temporal order rather than spatial separation. Nonetheless, the physical assumptions regarding causality are seldom studied in this context, and often even go unmentioned, in stark contrast with the many different possible locality conditions which have been studied and elaborated. In the present work, some retrocausal toy-models which reproduce the predictions of Quantum Mechanics for Bell-type correlations are reviewed. It is pointed out that a certain toy-model which is ostensibly superdete…
A new formulation of relativistic quantum mechanics is presented and applied to a free, massive, and spin zero elementary particle in the Minkowski spacetime. The reformulation requires that time and space, as well as the timelike and spacelike intervals, are treated equally, which makes the new theory fully symmetric and consistent with the Special Theory of Relativity. The theory correctly reproduces the classical action of a relativistic particle in the path integral formalism, and allows for the introduction of a new quantity called vector-mass, whose physical implications for nonlocality,…
Interpretational problems with quantum mechanics can be phrased precisely by only talking about empirically accessible information. This prompts a mathematical reformulation of quantum mechanics in terms of classical mechanics. We survey this programme in terms of algebraic quantum theory.
We propose a mechanism for system-induced spontaneous time reparametrization symmetry breaking in canonical gravity. We consider a model of extended canonical gravity, based on Sen's spinorial expression of the spatial-diffeomorphism-gauge independent part of the shift vector, with one additional long-range self-interacting massive spin-$1/2$ particle whose spin is coupled to Sen's spinor in the Hamiltonian. This additional particle plays the role of creating the system-induced time reparametrization invariant potential. We use this potential to break time reparametrization symmetry spontaneou…
There are four reasons why our present knowledge and understanding of quantum mechanics could be regarded as incomplete. Firstly, the principle of linear superposition has not been experimentally tested for position eigenstates of objects having more than about a thousand atoms. Secondly, there is no universally agreed upon explanation for the process of quantum measurement. Thirdly, there is no universally agreed upon explanation for the observed fact that macroscopic objects are not found in superposition of position eigenstates. Fourthly, and perhaps most importantly, the concept of time is…
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations $F=\frac{dP}{dt}$, $P=m\frac{dV}{dt}$, and $\left[X,P\right]=i$. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mec…
Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) is shown to provide a unified origin for: i) the complex vector space formulation of Quantum Mechanics (QM); ii) the momentum and space commutation relations and the corresponding representations; iii) the Dirac Hamiltonian in the formulation of Relativistic Quantum Mechanics (RQM); iv) the existence of a self adjoint Time Operator that circumvents Pauli's objection.
In 2008 Coecke and Duncan proposed the graphical ZX-calculus rewrite system which came to formalize reasoning with quantum circuits, measurements and quantum states. The ZX-calculus is sound for qubit quantum mechanics. Hence, equality of diagrams under ZX-equivalent transformations lifts to an equality of corresponding equations over matrices. Conversely, in 2014 Backens proved completeness, establishing that any derivation done in stabilizer quantum mechanics with matrices can be derived graphically using the ZX-calculus. A graphical rewrite system that is both confluent and also terminates …
A proof is given that an invertible and a unitary operator can be used to reproduce the effect of a q-deformed commutator of annihilation and creation operators. In other words, the original annihilation and creation operators are mapped into new operators, not conjugate to each other, whose standard commutator equals the identity plus a correction proportional to the original number operator. The consistency condition for the existence of this new set of operators is derived, by exploiting the Stone theorem on 1-parameter unitary groups. The above scheme leads to modified equations of motion …
The relational interpretation of quantum mechanics (RQM) has received a growing interest since its first formulation in 1996. Usually presented as an interpretational layer over the usual quantum mechanics formalism, it appears as a philosophical perspective without proper mathematical counterparts. This state of affairs has direct consequences on the scientific debate on RQM which still suffers from misunderstandings and imprecise statements. In an attempt to clarify those debates, the present paper proposes a radical reformulation of the mathematical framework of quantum mechanics which is r…
Discrete quantum mechanics is here defined to be a quantum theory of wave functions defined on integers P_i and Q_i, while canonical quantum mechanics is assumed to be based on wave functions on the real numbers, R^n. We study reversible mappings from the position operators q_i and their quantum canonical operators p_i of a canonical theory, onto the discrete, commuting operators Q_i and P_i. In this paper we are particularly interested in harmonic oscillators. In the discrete system, these turn into deterministic models, which is our motivation for this study. We regard the procedure worked o…
This paper presents the general theory of canonical transformations of coordinates in quantum mechanics. First, the theory is developed in the formalism of phase space quantum mechanics. It is shown that by transforming a star-product, when passing to a new coordinate system, observables and states transform as in classical mechanics, i.e., by composing them with a transformation of coordinates. Then the developed formalism of coordinate transformations is transferred to a standard formulation of quantum mechanics. In addition, the developed theory is illustrated on examples of particular clas…
We show that the supersymmetric algebra of Witten's quantum mechanics is invariant under a given point canonical transformation. It is shown that Witten's supersymmetric quantum mechanics can be isospectral or not to the seed Hamiltonian depending on the space coordinate you work on. We illustrate our results by generating a new class of exactly solvable supersymmetric partner Hamiltonians which are not isospectral to the seed Hamiltonian.
We present a reformulation of quantum mechanics in terms of probability measures and functions on a general classical sample space and in particular in terms of probability densities and functions on phase space. The basis of our proceeding is the existence of so-called statistically complete observables and the duality between the state spaces and the spaces of the observables, the latter holding in the quantum as well as in the classical case. In the phase-space context, we further discuss joint position-momentum observables, Hilbert spaces of infinitely differentiable functions on phase spa…
We show that a continuous quantum non-demolition measurement of the energy of a nanomechanical resonator can be achieved by monitoring the resonator with a quantum point contact via a Cooper-pair box. This technique can further be used to prepare highly non-classical states of two resonators, such as canonical phase-reference states, and so-called "noon" states.
A seldom recognized fundamental difficulty undermines the concept of individual ``state'' in the present formulations of quantum statistical mechanics (and in its quantum information theory interpretation as well). The difficulty is an unavoidable consequence of an almost forgotten corollary proved by E. Schroedinger in 1936 and perused by J.L. Park, Am. J. Phys., Vol. 36, 211 (1968). To resolve it, we must either reject as unsound the concept of state, or else undertake a serious reformulation of quantum theory and the role of statistics. We restate the difficulty and discuss a possible resol…
A reformulation of quantum mechanics called ``hybrid interaction picture'' (HIP) is proposed. In contrast to its current non-Hermitian-interaction-picture (NIP) predecessor the new formalism combines a simplification of the NIP Hilbert-space metric with an innovative preconditioning of the observables of interest. Via an illustrative example it is shown that the idea might open new ways towards optimal model-building strategies in quantum theory of unitary systems.
The new relativistic equations of motion for the particles with spin s=1, s=3/2, s=2 and nonzero mass have been introduced. The description of the relativistic canonical quantum mechanics of the arbitrary mass and spin has been given. The link between the relativistic canonical quantum mechanics of the arbitrary spin and the covariant local field theory has been found. The manifestly covariant field equations that follow from the quantum mechanical equations, have been considered. The covariant local field theory equations for spin s=(1,1) particle-antiparticle doublet, spin s=(1,0,1,0) partic…
For classical canonical transformations, one can, using the Wigner transformation, pass from their representation in Hilbert space to a kernel in phase space. In this paper it will be discussed how the time-dependence of the uncertainties of the corresponding time-dependent quantum problems can be incorporated into this formalism.
The covariant canonical method of quantization based on the De Donder-Weyl covariant canonical formalism is used to formulate a world-sheet covariant quantization of bosonic strings. To provide the consistency with the standard non-covariant canonical quantization, it is necessary to adopt a Bohmian deterministic hidden-variable equation of motion. In this way, string theory suggests a solution to the problem of measurement in quantum mechanics.
In the paper is presented an invariant quantization procedure of classical mechanics on the phase space over flat configuration space. Then, the passage to an operator representation of quantum mechanics in a Hilbert space over configuration space is derived. An explicit form of position and momentum operators as well as their appropriate ordering in arbitrary curvilinear coordinates is demonstrated. Finally, the extension of presented formalism onto non-flat case and related ambiguities of the process of quantization are discussed.
The investigation of arXiv 1409.2766v2 [quant-ph] has been continued by the general form of the numerous equations with partial values of arbitrary spin, which were considered in above mentioned preprint. The general forms of quantum-mechanical and covariant equations for arbitrary spin together with the general description of the arbitrary spin field formalism are presented. The corresponding relativistic quantum mechanics of arbitrary spin is given as the system of axioms. Previously ignored partial example of the spin s=(0,0) particle-antiparticle doublet is considered. The partial example …
We propose a manifestly covariant canonical method of field quantization based on the classical De Donder-Weyl covariant canonical formulation of field theory. Owing to covariance, the space and time arguments of fields are treated on an equal footing. To achieve both covariance and consistency with standard noncovariant canonical quantization of fields in Minkowski spacetime, it is necessary to adopt a covariant Bohmian formulation of quantum field theory. A preferred foliation of spacetime emerges dynamically owing to a purely quantum effect. The application to a simple time-reparametrizatio…
We explain the approximate nature of particle trajectories in Bohm's quantum mechanics. They are streamlines of a superfluid in Madelung's reformulation of the Schrödinger wave function, around which the proper particle trajectories perform their quantum mechanical fluctuations to ensure Heisenberg's uncertainty relation between position and momentum.
We generalize classical statistical mechanics to describe the kinematics and the dynamics of systems whose variables are constrained by a single quantum postulate (discreteness of the spectrum of values of at least one variable of the theory). This is possible provided we adopt Feynman's suggestion of dropping the assumption that the probability for an event must always be a positive number. This approach has the advantage of allowing a reformulation of quantum theory in phase space without introducing the unphysical concept of probability amplitudes, together with all the problems concerning …
Classical nonlinear canonical (Poisson) maps have a distinguished role in quantum mechanics. They act unitarily on the quantum phase space and generate $\hbar$-independent quantum canonical maps. It is shown that such maps act in the noncommutative phase space as dictated by the classical covariance. A crucial observation made is that under the classical covariance the local quantum mechanical picture can become nonlocal in the Hilbert space. This nonlocal picture is made equivalent by the Weyl map to a noncommutative picture in the phase space formulation of the theory. The connection between…
Symmetries are widely used in modeling quantum systems but they do not contribute in postulates of quantum mechanics. Here we argue that logical, mathematical, and observational evidence require that symmetry should be considered as a fundamental concept in the construction of physical systems. Based on this idea, we propose a series of postulates for describing quantum systems, and establish their relation and correspondence with axioms of standard quantum mechanics. Through some examples we show that this reformulation helps better understand some of ambiguities of standard description. None…
