We show that the principles of a ''complete physical theory'' and the conclusions of the standard quantum mechanics do not irreconcilably contradict each other as is commonly believed. In the algebraic approach, we formulate axioms that allow constructing a renewed mathematical scheme of quantum mechanics. This scheme involves the standard mathematical formalism of quantum mechanics. Simultaneously, it contains a mathematical object that adequately describes a single experiment. We give an example of the application of the proposed scheme.
We study the physical content of the PT-symmetric complex extension of quantum mechanics as proposed in Bender et al, Phys. Rev. Lett. 80, 5243 (1998) and 89, 270401 (2002), and show that as a fundamental probabilistic physical theory it is neither an alternative to nor an extension of ordinary quantum mechanics. We demonstrate that the definition of a physical observable given in the above papers is inconsistent with the dynamical aspect of the theory and offer a consistent notion of an observable.
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O of the quantum mechanics based on H. In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the under…
The paper provides an introduction into p-mechanics, which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics. p-Mechanics naturally provides a common ground for several different approaches to quantisation (geometric, Weyl, coherent states, Berezin, deformation, Moyal, etc.) and has a potential for expansions into field and string theories. The backbone of p-mechanics is solely the representation theory of the Heisenberg group. Keywords: Classical mechanics, quantum mechanics, Moyal brackets, Poisson brackets, commutator, Heisenberg grou…
It is proposed the scheme of quantum mechanics, in which a Hilbert space and the linear operators are not primary elements of the theory. Instead of it certain variant of the algebraic approach is considered. The elements of noncommutative algebra (observables) and the nonlinear functionals on this algebra (physical states) are used as the primary constituents. The functionals associate with results of a particular measurement. It is suggested to consider certain ensembles of the physical states as quantum states of the standart quantum mechanics. It is shown that in such scheme the mathematic…
We reconsider briefly the relation between "physical quantity" and "physical reality in the light of recent interpretations of Quantum Mechanics. We argue, that these interpretations are conditioned from the epistemological relation between these two fundamental concepts. In detail, the choice as ontic level of the concept affect, the relative interpretation. We note, for instance, that the informational view of quantum mechanics (primacy of the subjectivity) is due mainly to the evidence of the "random" physical quantities as ontic element. We will analyze four positions: Einstein, Rovelli, d…
We introduce the first complete and approximatively universal diagrammatic language for quantum mechanics. We make the ZX-Calculus, a diagrammatic language introduced by Coecke and Duncan, complete for the so-called Clifford+T quantum mechanics by adding four new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open questions in categorical quantum mechanics. We prove the completeness of the Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that t…
In their Erratum [Phys. Rev. Lett. {\bf 92}, 119902 (2004), quant-ph/0208076], written in reaction to [quant-ph/0310164], Bender, Brody and Jones propose a revised definition for a physical observable in PT-symmetric quantum mechanics. We show that although this definition avoids the dynamical inconsistency revealed in quant-ph/0310164, it is still not a physically viable definition. In particular, we point out that a general proof that this definition is consistent with the requirements of the quantum measurement theory is lacking, give such a proof for a class of PT-symmetric systems by esta…
We are focused on the idea that observables in quantum physics are a bit more than just hermitian operators and that this is, in general, a "tricky business". The origin of this idea comes from the fact that there is a subtle difference between symmetric, hermitian, and self-adjoint operators which are of immense importance in formulating Quantum Mechanics. The theory of self-adjoint extensions is presented through several physical examples and some emphasis is given on the physical implications and applications.
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…
The physical constructs underlying the properties of quantum mechanics are explored. Arguments are given that the particle wave function as well as photon and phonon quanta must derive from a more fundamental physical construct that has not yet been identified. An approach to identifying the construct is discussed and a specific construct is proposed and explained.
In the framework of the topos approach to quantum mechanics we give a representation of physical properties in terms of modal operators on Heyting algebras. It allows us to introduce a classical type study of the mentioned properties.
Five physical assumptions are proposed that together entail the general qualitative results, including the Born rule, of non-relativistic quantum mechanics by physical and information-theoretic reasoning alone. Two of these assumptions concern fundamental symmetries of physical interactions. The third concerns the Hilbert-space dimensions and the fourth and fifth the self-interaction Hamiltonians of the systems that function as "observers" within the theory. These assumptions are shown to provide a sufficient motivation for the usual Hilbert-space formalism, and to obviate the observation-rela…
Quantum and classical mechanics are derived using four natural physical principles: (1) the laws of nature are invariant under time evolution, (2) the laws of nature are invariant under tensor composition, (3) the laws of nature are relational, and (4) positivity (the ability to define a physical state). Quantum mechanics is singled out by a fifth experimentally justified postulate: nature violates Bell's inequalities.
On the base of years of experience of working on the problem of the physical foundation of quantum mechanics the author offers principles of solving it. Under certain pressure of mathematical formalism there has raised a hypothesis of complexity of space and time by Minkovsky, being significant mainly for quantum objects. In this eight-dimensional space and time with six space and two time dimensions all the problems and peculiarities of quantum mechanical formalism disappear, the reasons of their appearance become clear, and there comes a clear and physically transparent picture of the founda…
To begin with, some of the conundrums concerning Quantum Mechanics and its interpretation(s) are recalled. Subsequently, a sketch of the "ETH-Approach to Quantum Mechanics" is presented. This approach yields a logically coherent quantum theory of "events" featured by physical systems and of direct or projective measurements of physical quantities, without the need to invoke "observers". It enables one to determine the stochastic time evolution of states of physical systems. We also briefly comment on the quantum theory of indirect or weak measurements, which is much easier to understand and mo…
The standard presentation of the principles of quantum mechanics is critically reviewed both from the experimental/operational point and with respect to the request of mathematical consistency and logical economy. A simpler and more physically motivated formulation is discussed. The existence of non commuting observables, which characterizes quantum mechanics with respect to classical mechanics, is related to operationally testable complementarity relations, rather than to uncertainty relations. The drawbacks of Dirac argument for canonical quantization are avoided by a more geometrical approa…
If quantum states exhibit small nonlinearities during time evolution, then quantum computers can be used to solve NP-complete problems in polynomial time. We provide algorithms that solve NP-complete and #P oracle problems by exploiting nonlinear quantum logic gates. It is argued that virtually any deterministic nonlinear quantum theory will include such gates, and the method is explicitly demonstrated using the Weinberg model of nonlinear quantum mechanics.
Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and non-linear theory that is defined on a symplectic manifold. However, after invention of general relativity, we are convinced that geometry is physical and effect us in all scale. Hence the geometric formulation of quantum mechanics sought to give a unified picture of physical systems based on its underling geometrical structures, e.g., now, the states ar…
The problem of reconstructing information on a physical system from data acquired in long sequences of direct (projective) measurements of some simple physical quantities - histories - is analyzed within quantum mechanics; that is, the quantum theory of indirect measurements, and, in particular, of non-demolition measurements is studied. It is shown that indirect measurements of time-independent features of physical systems can be described in terms of quantum-mechanical operators belonging to an algebra of asymptotic observables. Our proof involves associating a natural measure space with cer…
If the statement by Einstein, Podolsky and Rosen on incompleteness of Quantum-Mechanical description of nature is correct, then we can regard Quantum Mechanics as a Method of Indirect Computation. The problem is, whether the theory is incomplete or the nature itself does not allow complete description? And if the first option is correct, how is it possible to complete the Quantum-Mechanical description? Here we try to complement de-Broglie's idea on wave-pilot the stochastic gravitation gives origin to. We assume that de-Broglie's wave-pilots are gravitational stochastic ones, and we shall reg…
An ordinary unambiguous integral representation for the finite propagator of a quantum system is found by starting of a privileged skeletonization of the functional action in phase space, provided by the complete solution of the Hamilton-Jacobi equation. This representation allows to regard the propagator as the sum of the contributions coming from paths where the momenta generated by the complete solution of the Hamilton-Jacobi equation are conserved -as it does happen on the classical trajectory-, but are not restricted to having the classical values associated with the boundary conditions f…
In this paper we are going to introduce a new dynamical postulate in Quantum Mechanics. This new principle is defined using path integrals over the set of normalized wave functions. We will show in a qualitative way that this postulate is equivalent to the Schrodinger equation and to the measurement postulates. Then we propose a new set of fundamental postulates for Quantum Mechanics. In this approach to Quantum Mechanics we separate the fundamental postulates and the physical interpretation. The fundamental postulates are abstract mathematical principles. Their physical interpretation depend …
The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics, meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound, i.e. any equality that can be derived graphically can also be derived using matrix mechanics. Here, we show that the ZX-calculus is complete for pure qubit stabilizer quantum mechanics, meaning any equality that can be derived using matrices can also be derived pictorially. The proof rel…
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…
Symmetric informationally complete positive operator valued measures (SIC-POVMs) are studied within the framework of the probability representation of quantum mechanics. A SIC-POVM is shown to be a special case of the probability representation. The problem of SIC-POVM existence is formulated in terms of symbols of operators associated with a star-product quantization scheme. We show that SIC-POVMs (if they do exist) must obey general rules of the star product, and, starting from this fact, we derive new relations on SIC-projectors. The case of qubits is considered in detail, in particular, th…
The article is dedicated to discussion of irreversibility and foundation of statistical mechanics "from the first principles". Taking into account infinitesimal and, as it seems, neglectful for classical mechanics fluctuations of the physical vacuum, makes a deterministic motion of unstable dynamic systems is broken ("spontaneous determinism breaking", "spontaneous stochastization"). Vacuum fluctuations play part of the trigger, starting the powerful mechanism of exponent instability. The motion of the dynamic systems becomes irreversible and stochastic. Classical mechanics turns out to be app…
Recently, we gave a complete axiomatisation of the ZX-calculus for the overall pure qubit quantum mechanics. Based on this result, here we also obtain a complete axiomatisation of the ZX-calculus for the Clifford+T quantum mechanics by restricting the ring of complex numbers to its subring corresponding to the Clifford+T fragment resting on the completeness theorem of the ZW-calculus for arbitrary commutative ring. In contrast to the first complete axiomatisation of the ZX-calculus for the Clifford+T fragment, we have two new generators as features rather than novelties: the triangle can be em…
The physical world is quantum. However, our description of the quantum physics still relies much on concepts in classical physics and in some cases with `quantized' interpretations. The most important case example is that of spacetime. We examine the picture of the physical space as described by simple, so-called non-relativisitic, quantum mechanics instead of assuming the Newtonian model. The key perspective is that of (relativity) symmetry representation, and the idea that the physical space is to be identified as the configuration space for a free particle. Parallel to the case of the phase…
The purpose of this short article is to build on the work of Ghirardi, Marinatto and Weber (Ghirardi, Marinatto & Weber 2002; Ghirardi & Marinatto 2003, 2004, 2005) and Ladyman, Linnebo and Bigaj (2013), in supporting a redefinition of entanglement for "indistinguishable" systems, particularly fermions. According to the proposal, non-separability of the joint state is insufficient for entanglement. The re-definition is justified by its physical significance, as enshrined in three biconditionals whose analogues hold of "distinguishable" systems.
We find a sufficient set of equations between quantum circuits from which we can derive any other equation between stabilizer quantum circuits. To establish this result, we rely upon existing work on the completeness of the graphical ZX language for quantum processes. The complexity of the circuit equations, as opposed to the very intuitive reading of the much smaller number of ZX-equations, advocates the latter for performing computations with quantum circuits.
We consider the so called Legget inequalities which are deduced from the assumption of general (local or non-local) realism plus the arrow of time preservation. Then, instead of assuming cryto-nonlocal hidden variables, we assume any (local or non-local) realism compatible with the joint and non-joint expected values dictated by quantum mwechaanics. Hence, we prove that this double assumption is not consistent, since the corresponding general Leggett inequalities are violated by quantum mechanics. Thus, realism plus arrow of time preservation and quntum mechanics are not compatible. In other w…
One of the postulates of quantum mechanics is that the Hamiltonian is Hermitian, as this guarantees that the eigenvalues are real. Recently there has been an interest in asking if $H^\dagger = H$ is a necessary condition, and has lead to the development of PT-symmetric quantum mechanics. This note shows that any finite physically acceptable non-Hermitian Hamiltonian is equivalent to doing ordinary quantum mechanics in a non-orthogonal basis. In particular, this means that there is no experimental distinction between PT-symmetric quantum mechanics and ordinary quantum mechanics for finite syste…
Recently, it has been argued that quantum mechanics is complete, and that quantum states vectors are necessarily in one-to-one correspondence with the elements of reality, under the assumptions that quantum theory is correct and that measurement settings can be freely chosen. In this work, we argue that the adopted form of the free choice assumption is stronger than needed. We unveil hidden assumptions underlying these results, which limit their range of validity. We support our argument by a model for the bipartite two-level system, reproducing quantum mechanics, in which the free will assump…
In this paper, we show that a qutrit version of ZX-calculus, with rules significantly different from that of the qubit version, is complete for pure qutrit stabilizer quantum mechanics, where state preparations and measurements are based on the three dimensional computational basis, and unitary operations are required to be in the generalized Clifford group. This means that any equation of diagrams that holds true under the standard interpretation in Hilbert spaces can be derived diagrammatically. In contrast to the qubit case, the situation here is more complicated due to the richer structure…
Besides their use for efficient computation, quantum computers are a base for studying quantum systems that create valid physical theories using mathematics and physics. An essential part of the validation process for quantum mechanics is the development of a coherent theory of mathematics and quantum mechanics together. Such a theory should combine mathematical logical concepts with quantum mechanics. That this might be possible is shown here by defining truth, validity, consistency, and completeness for a quantum mechanical version of a simple classical expression enumeration machine describ…
It is shown that probabilistic treatment of quantum mechanics can be coordinated with causality of all physical processes. The physical interpretation of quantum-mechanical phenomena such as process of measurement and collapse of quantum state is given.
There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an interpretation functor to the dagger-SMC of (finite dimension) Hilbert spaces. In the recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of the extens…
The ZX-Calculus is a graphical language for quantum mechanics. An axiomatisation has recently been proven to be complete for an approximatively universal fragment of quantum mechanics, the so-called Clifford+T fragment. We focus here on the expressive power of this axiomatisation beyond Clifford+T Quantum mechanics. We consider the full pure qubit quantum mechanics, and mainly prove two results: (i) First, the axiomatisation for Clifford+T quantum mechanics is also complete for all equations involving some kind of linear diagrams. The linearity of the diagrams reflects the phase group structur…
This paper analyses quantum mechanics in multiply connected spaces. It is shown that the multiple connectedness of the configuration space of a physical system can determine the quantum nature of physical observables, such as the angular momentum. In particular, quantum mechanics in compactified Kaluza-Klein spaces is examined. These compactified spaces give rise to an additional angular momentum which can adopt half-integer values and, therefore, may be identified with the intrinsic spin of a quantum particle.
Motivated by the question what it is that makes quantum mechanics a holistic theory (if so), I try to define for general physical theories what we mean by `holism'. For this purpose I propose an epistemological criterion to decide whether or not a physical theory is holistic, namely: a physical theory is holistic if and only if it is impossible in principle to infer the global properties, as assigned in the theory, by local resources available to an agent. I propose that these resources include at least all local operations and classical communication. This approach is contrasted with the well…
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,…
Within the traditional Hilbert space formalism of quantum mechanics, it is not possible to describe a particle as possessing, simultaneously, a sharp position value and a sharp momentum value. Is it possible, though, to describe a particle as possessing just a sharp position value (or just a sharp momentum value)? Some, such as Teller (Journal of Philosophy, 1979), have thought that the answer to this question is No -- that the status of individual continuous quantities is very different in quantum mechanics than in classical mechanics. On the contrary, I shall show that the same subtle issues…
The paper reviews and discusses four ideas scattered in previous papers of the author. First, objective properties of quantum systems are not associated with observables but are defined by preparations. Second, measurable results of classical theories are not sharp. All such results can be obtained as high-entropy limits of quantum mechanics. Third, a careful study of exchange symmetry requires a modification of the theory of measurement. Fourth, screens and detectors are closely associated with state reduction. A new, rather improved understanding of quantum mechanics is achieved without any …
It is argued that although quantum theory isn't an absolutely deterministic theory, it is partially deterministic. The approach followed here is in the framework of the standard (Copenhagen interpretation of) quantum mechanics without any additional assumption or alternative interpretation. The argument is based on the conceptual meaning of determinism and by means of some well-known phenomena in the quantum world (measurement of a quantum particle spin, energy values and spatial states of an electron in atoms, the so-called wavefunction collapse problem) which are usually considered in reject…
The standard axiomatization of quantum mechanics (QM) is not fully explicit about the role of the time-parameter. Especially, the time reference within the probability algorithm (the Born Rule, BR) is unclear. Using a plausible principle P1, about the role of probability in a physical theory, and a second principle P2 affording a most natural way to make BR precise, a logical conflict with the standard expression for the completeness of QM can be derived. Rejecting P1 is implausible. Rejecting P2 leads to unphysical results and to a conflict with a generalization of P2, a principle P3. It is t…
I propose a version of quantum mechanics featuring a discrete and finite number of states that is plausibly a model of the real world. The model is based on standard unitary quantum theory of a closed system with a finite-dimensional Hilbert space. Given certain simple conditions on the spectrum of the Hamiltonian, Schrödinger evolution is periodic, and it is straightforward to replace continuous time with a discrete version, with the result that the system only visits a discrete and finite set of state vectors. The biggest challenges to the viability of such a model come from cosmological con…
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.
I argue that quantum mechanics is fundamentally a theory about the representation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles. The notion of quantum information is to be understood as a new physical primitive -- just as, following Einstein's special theory of relativity, a field is no longer regarded as the physical manifestation of vibrations in a mechanical medium, but recognized as a new physical primitive in its own right.
The physical condition that the expectation values of physical observables are real quantities is used to give a precise formulation of PT-symmetric quantum mechanics. A mathematically rigorous proof is given to establish the physical equivalence of PT-symmetric and conventional quantum mechanics. The results reported in this paper apply to arbitrary PT-symmetric Hamiltonians with a real and discrete spectrum. They hold regardless of whether the boundary conditions defining the spectrum of the Hamiltonian are given on the real line or a complex contour.
We introduce a method to construct wave packets with complete classical and quantum correspondence in one-dimensional non-relativistic quantum mechanics. First, we consider two similar oscillators with equal total energy. In classical domain, we can easily solve this model and obtain the trajectories in the space of variables. This picture in the quantum level is equivalent with a hyperbolic partial differential equation which gives us a freedom for choosing the initial wave function and its initial slope. By taking advantage of this freedom, we propose a method to choose an appropriate initia…
Generally a central role has been assigned to an unavoidable physical interaction between the measuring instrument and the physical entity measured in the change in the wave function that often occurs in measurement in quantum mechanics. A survey of textbooks on quantum mechanics by authors such as Dicke and Witke (1960), Eisberg and Resnick (1985), Gasiorowicz (1974), Goswami (1992), and Liboff (1993) supports this point. Furthermore, in line with the view of Bohr and Feynman, generally the unavoidable interaction between a measuring instrument and the physical entity measured is considered r…
Quantum mechanics, one of the most successful theories in the history of science, was created to account for physical systems not describable by classical physics. Though it is consistent with all experiments conducted thus far, many of its core concepts (amplitudes, global phases, etc.) can not be directly accessed and its interpretation is still the subject of intense debate, more than 100 years since it was introduced. So, a fundamental question is why this particular mathematical model is the one that nature chooses, if indeed it is the correct model. In the past two decades there has been…
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…
