The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.
We review the notion of complementarity of observables in quantum mechanics, as formulated and studied by Paul Busch and his colleagues over the years. In addition, we provide further clarification on the operational meaning of the concept, and present several characterisations of complementarity - some of which new - in a unified manner, as a consequence of a basic factorisation lemma for quantum effects. We work out several applications, including the canonical cases of position-momentum, position-energy, number-phase, as well as periodic observables relevant to spatial interferometry. We cl…
Niels Bohr introduced the concept of complementarity in order to give a general account of quantum mechanics, however he stressed that the idea of complementarity is related to the general difficulty in the formation of human ideas, inherent in the distinction between subject and object. The complementary descriptions approach is a framework for the interpretation of quantum mechanics, more specifically, it focuses in the development of the idea of complementarity and the concept of potentiality in the orthodox quantum formulation. In PART I of this article, we analyze the ideas of Bohr and pr…
Niels Bohr introduced the concept of complementarity in order to give a general account of quantum mechanics, however he stressed that the idea of complementarity is related to the general difficulty in the formation of human ideas, inherent in the distinction between subject and object. The complementary descriptions approach is a framework for the interpretation of quantum mechanics, more specifically, it focuses in the development of the idea of complementarity and the concept of potentiality in the orthodox quantum formulation. In PART I of this article, we analyze the ideas of Bohr and pr…
Classes of the nonlinear Schrodinger-type equations compatible with the Galilei relativity principle are described. Solutions of these equations satisfy the continuity equation.
A review of fundamentals and physical applications of fractional quantum mechanics has been presented. Fundamentals cover fractional Schrödinger equation, quantum Riesz fractional derivative, path integral approach to fractional quantum mechanics, hermiticity of the Hamilton operator, parity conservation law and the current density. Applications of fractional quantum mechanics cover dynamics of a free particle, new representation for a free particle quantum mechanical kernel, infinite potential well, bound state in δ-potential well, linear potential, fractional Bohr atom and fractional oscilla…
Niels Bohr introduced the concept of complementarity in order to give a general account of quantum mechanics, however he stressed that the idea of complementarity is related to the general dificulty in the formation of human ideas, inherent in the distinction between subject and object. Earlier, we have introduced a development of the concept of complementarity which constitutes a new approach to the interpretation of quantum mechanics. We argue that this development allows a better understanding of some of the paradigmatic interpretational problems of quantum theory. Within the scheme propose…
In previous work we have developed a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group that includes transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as is the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. A special feature of these previously constructed representations is that they al…
A recent concept in theoretical physics, motivated in string duality and M-theory, is the notion that not all quantum theories arise from quantising a classical system. Also, a given quantum model may possess more than just one classical limit. In view of these developments, we analyse some general properties that quantum mechanics must satisfy, if it is not to be formulated as a quantisation of a given classical mechanics. Instead, our approach to quantum mechanics is modelled on a statement that is close in spirit to the equivalence principle of general relativity, thus bearing a strong rese…
The Landauer principle asserts that the energy cost of erasure of one bit of information by the action of a thermal reservoir in equilibrium at temperature T is never less than $kTlog 2$. We discuss Landauer's principle for quantum statistical models describing a finite level quantum system S coupled to an infinitely extended thermal reservoir R. Using Araki's perturbation theory of KMS states and the Avron-Elgart adiabatic theorem we prove, under a natural ergodicity assumption on the joint system S+R, that Landauer's bound saturates for adiabatically switched interactions. The recent work of…
We present a formalism of Galilean quantum mechanics in non-inertial reference frames and discuss its implications for the equivalence principle. This extension of quantum mechanics rests on the Galilean line group, the semidirect product of the real line and the group of analytic functions from the real line to the Euclidean group in three dimensions. This group provides transformations between all inertial and non-inertial reference frames and contains the Galilei group as a subgroup. We construct a certain class of unitary representations of the Galilean line group and show that these repre…
Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulati…
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…
The aim of this paper is to review the classical limit of Quantum Mechanics and to precise the well known threat of chaos (and fundamental graininess)to the correspondence principle. We will introduce a formalism for this classical limit that allows us to find the surfaces defined by the constants of the motion in phase space. Then in the integrable case we will find the classical trajectories, and in the non-integrable one the fact that regular initial cells become "amoeboid-like". This deformations and their consequences can be considered as a threat to the correspondence principle unless we…
Working within the framework of parity-time-symmetric quantum mechanics we look into the possibility of entanglement generation and demonstrate that the feature of non-violation of no-signaling principle may hold for the simplest non-trivial case of bipartite systems. Basically our arguments are based on the computation of the reduced density matrix of one party to justify that the entropy of the other does not change.
After some historical remarks concerning Schroedinger's discovery of wave mechanics, we present a unified formalism for the mathematical description of classical and quantum-mechanical systems, utilizing elements of the theory of operator algebras. We then review some basic aspects of quantum mechanics and, in particular, of its interpretation. We attempt to clarify what Quantum Mechanics tells us about Nature when appropriate experiments are made. We discuss the importance of the mechanisms of "dephasing" and "decoherence" in associating "facts" with possible events and rendering complementar…
We introduce an action principle for a class of integer valued cellular automata and obtain Hamiltonian equations of motion. Employing sampling theory, these discrete deterministic equations are invertibly mapped on continuum equations for a set of bandwidth limited harmonic oscillators, which encode the Schrödinger equation. Thus, the linearity of quantum mechanics is related to the action principle of such cellular automata and its conservation laws to discrete ones.
Quantum mechanics and classical mechanics are two very different theories, but the correspondence principle states that quantum particles behave classically in the limit of high quantum number. In recent years much research has been done on extending both quantum mechanics and classical mechanics into the complex domain. This letter shows that these complex extensions continue to exhibit a correspondence, and that this correspondence becomes more pronounced in the complex domain. The association between complex quantum mechanics and complex classical mechanics is subtle and demonstrating this …
For any ideal two-path interferometer it is shown that the wave-particle duality of quantum mechanics implies Heisenberg's uncertainty relation and vice versa. It is conjectured that complementarity and uncertainty are two aspects of the same general principle.
We study Sorkin's proposal of a generalization of quantum mechanics and find that the theories proposed derive their probabilities from $k$-th order polynomials in additive measures, in the same way that quantum mechanics uses a probability bilinear in the quantum amplitude and its complex conjugate. Two complementary approaches are presented, a $C^*$ and a Hopf-algebraic one, illuminating both algebraic and geometric aspects of the problem.
We argue that in a framework for emergent quantum mechanics, the weak equivalence principle is a consequence of concentration of measure in large dimensional spaces of $1$-Lipshitz functions. Furthermore, as a consequence of the emergent framework and the properties that we assume for the fundamental dynamics, it is argued that gravity must be a classical, emergent interaction.
Quantum mechanics is an extremely successful theory of nature and yet it lacks an intuitive axiomatization. In contrast, the special theory of relativity is well understood and is rooted into natural or experimentally justified postulates. Here we introduce an axiomatization approach to quantum mechanics which is very similar to special theory of relativity derivation. The core idea is that a composed system obeys the same laws of nature as its components. This leads to a Jordan-Lie algebraic formulation of quantum mechanics. The starting assumptions are minimal: the laws of nature are invaria…
We generalize the derivation of the Wallis formula for $π$ from a variational computation of the spectrum of the Hydrogen atom. We obtain infinite product formulas for certain combinations of gamma functions, which include irrational numbers such as $\sqrt 2$ as well as some nested radicals. We also derive Euler's reflection formula for reciprocals of positive even integers. We show that Bohr's correspondence principle allows us to derive our product formulas and the reflection formula without the need for the limit definition of the gamma function.
A quantum version of the action principle is considered in the case of a free relativistic particle. The classical limit of the quantum action is obtained.
We show that the exclusivity (E) principle singles out the set of quantum correlations associated to any exclusivity graph assuming the set of quantum correlations for the complementary graph. Moreover, we prove that, for self-complementary graphs, the E principle, by itself (i.e., without further assumptions), excludes any set of correlations strictly larger than the quantum set. Finally, we prove that, for vertex-transitive graphs, the E principle singles out the maximum value for the quantum correlations assuming only the quantum maximum for the complementary graph. This opens the door for …
In this paper we generalize the concept of Wigner function in the case of quantum mechanics with a minimum length scale arising due to the application of a generalized uncertainty principle (GUP). We present the phase space formulation of such theories following GUP and show that the Weyl transform and the Wigner function does satisfy some of their known properties in standard quantum mechanics. We utilise the generalized Wigner function to calculate the phase space average of the Hamiltonian of a quantum harmonic oscillator satisfying deformed Heisenberg algebra. It is also shown that average…
We present a concise derivation of Landauer's erasure principle from the postulates of statistical mechanics, along with a small number of additional but uncontroversial axioms.
That gravitation can be understood as purely metric phenomenon depends crucially on the validity of a number of hypotheses which are summarised by the Einstein Equivalence Principle, the least well tested part of which being the Universality of Gravitational Redshift. A recent and currently widely debated proposal (Nature 463 (2010) 926-929) to re-interpret some 10-year old experiments in atom interferometry would imply, if tenable, substantial reductions on upper bounds for possible violations of the Universality of Gravitational Redshift by four orders of magnitude. This interpretation, howe…
We propose six principles as the fundamental principles of quantum mechanics: principle of space and time, Galilean principle of relativity, Hamilton's principle, wave principle, probability principle, and principle of indestructibility and increatiblity of particles. We deductively develop the formalism of quantum mechanics on the basis of them: we determine the form of the Lagrangian that satisfies requirements of these principles, and obtain the Schroedinger equation from the Lagrangian. We also derive the canonical commutation relations. Then we adopt the following four guide lines. First,…
Quantum coherence and quantum correlations lie in the center of quantum information science, since they both are considered as fundamental reasons for significant features of quantum mechanics different from classical mechanics. We present a group of complementary relations for quantum coherence and quantum correlations; specifically, we focus on thermal discord and conditional information in scenarios of multiple measurements. We show that the summation of quantum coherence quantified in different bases has a lower bound, resulting from entropic uncertainty relations with multiple measurement…
In a classical world, simultaneous measurements of complementary properties (e.g. position and momentum) give a system's state. In quantum mechanics, measurement-induced disturbance is largest for complementary properties and, hence, limits the precision with which such properties can be determined simultaneously. It is tempting to try to sidestep this disturbance by copying the system and measuring each complementary property on a separate copy. However, perfect copying is physically impossible in quantum mechanics. Here, we investigate using the closest quantum analog to this copying strateg…
The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric Kaehler manifold. The classical principles of Quantum Mechanics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are discussed in this non linear geometrical context.
Many quantization schemes rely on analogs of classical mechanics where the connections with classical mechanics are indirect. In this work I propose a new and direct connection between classical mechanics and quantum mechanics where the quantum mechanical propagator is derived from a variational principle. This principle allows a physical system to have imperfect information, i.e., there is incomplete knowledge of the physical state, and many paths are allowed.
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…
The non-relativistic quantum mechanics with a generalized uncertainty principle (GUP) is examined in $D$-dimensional free particle and harmonic oscillator systems. The Feynman propagators for these systems are exactly derived within the first order of the GUP parameter.
On the basis of a manifestly covariant formalism of non-relativistic quantum mechanics in general coordinate systems, proposed by us recently, we derive general expressions for inertial forces. The results enable us further to discuss, and to explain the validity of, the equivalence principle in non-relativistic quantum mechanics.
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, the general…
We examine the visualization of quantum mechanics in phase space by means of the Wigner function and the Wigner function flow as a complementary approach to illustrating quantum mechanics in configuration space by wave functions. The Wigner function formalism resembles the mathematical language of classical mechanics of non-interacting particles. Thus, it allows a more direct comparison between classical and quantum dynamical features.
I show how quantum mechanics, like the theory of relativity, can be understood as a 'principle theory' in Einstein's sense, and I use this notion to explore the approach to the problem of interpretation developed in my book Interpreting the Quantum World (Cambridge: Cambridge University Press, 1999).
A qualitative but formalized representation of microstates is first established quite independently of the quantum mechanical mathematical formalism, exclusively under epistemological-operational-methodological constraints. Then, using this representation as a reference-and-imbedding-structure, the foundations of an intelligible reconstruction of the Hilbert-Dirac formulation of Quantum Mechanics is developed. Inside this reconstruction the measurement problem as well as the other major problems raised by the quantum mechanical formalism, dissolve.
The classical limit of the Schrodinger equation implies the orthodox statistical interpretation for degrees of freedom in finite-dimensional subspaces of the full Hilbert space, but the argument presented does not imply the Born interpretation for degrees of freedom that have a classical counterpart.
Quantum mechanics is based on a series of postulates which lead to a very good description of the microphysical realm but which have, up to now, not been derived from first principles. In the present work, we suggest such a derivation in the framework of the theory of scale relativity. After having analyzed the actual status of the various postulates, rules and principles that underlie the present axiomatic foundation of quantum mechanics (in terms of main postulates, secondary rules and derived `principles'), we attempt to provide the reader with an exhaustive view of the matter, by both gath…
Two central concepts of quantum mechanics are Heisenberg's uncertainty principle, and a subtle form of non-locality that Einstein famously called ``spooky action at a distance''. These two fundamental features have thus far been distinct concepts. Here we show that they are inextricably and quantitatively linked. Quantum mechanics cannot be more non-local with measurements that respect the uncertainty principle. In fact, the link between uncertainty and non-locality holds for all physical theories.More specifically, the degree of non-locality of any theory is determined by two factors -- the s…
The superposition principle forms the very backbone of quantum theory. The resulting linear structure of quantum theory is structurally so rigid that tampering with it may have serious, seemingly unphysical, consequences. This principle has been succesful at even the highest available accelerator energies. Is this aspect of quantum theory forever then? The present work is an attempt to understand the attitude of the founding fathers, particularly of Bohr and Dirac, towards this principle. The Heisenberg matrix mechanics on the one hand, and the Schrodinger wave mechanics on the other, are crit…
Quantum Action Principle formulated earlier is used as a ground for a probabilistic interpretation of one-particle relativistic quantum mechanics. In this new approach the probability "flows" in the Minkowsky space being dependent on an inner time parameter which we interpret as a particle life time. The life time is determined as a function of observable parameters of the real experiment by means of an additional condition of stationarity for the quantum action.
This is a comment to the paper, Limitations on the superposition principle: superselection rules in non-relativistic quantum mechanics by C Cisneros et al 1998 Eur. J. Phys. 19 237. doi:10.1088/0143-0807/19/3/005. The proof that the authors construct for the limitation on the superposition of state vectors corresponding to different sectors of the Hilbert space, partitioned by a superoperator has a flaw as outlined below.
A quantum particle moving in a gravitational field may penetrate the classically forbidden region of the gravitational potential. This raises the question of whether the time of flight of a quantum particle in a gravitational field might deviate systematically from that of a classical particle due to tunnelling delay, representing a violation of the weak equivalence principle. I investigate this using a model quantum clock to measure the time of flight of a quantum particle in a uniform gravitational field, and show that a violation of the equivalence principle does not occur when the measurem…
I draw attention to the fact that three recently proposed physical principles, namely "local orthogonality", "global exclusive disjunction", and "compatible orthogonality" are not new principles, but different versions of a principle that Ernst Specker noticed long ago. I include a video of Specker stating this principle in 2009 in the following terms: "Do you know what, according to me, is the fundamental theorem of quantum mechanics? (...) That is, if you have several questions and you can answer any two of them, then you can also answer all of them". I overview some results that suggest tha…
Reduction is shown to be a possible consequence of the basic principles of quantum mechanics, involving no branching of the quantum state of the universe. The key feature of a measurement is attributed to the creation of macroscopic germs of ionized tracks, bubbles, sparks and so on, in a superposed quantum state, the main mechanism being a breaking and a restoration of ``classicality'' in nearby macroscopic objects and their small patterns. A new significance of quantum probabilities, involving no absolute randomness, is also proposed.
The fundamental principle of quantum mechanics is that the probabilities of physical outcomes are obtained from the intermediate states and processes of the interacting particles, considered as happening concurrently. When the interaction is described by a potential, the total energy of the particle is equal to its total kinetic plus potential energies. We derive the Schrodinger and Dirac equations as the conditions the wavefunction must satisfy at each point in order to fulfill the corresponding energy equation. In our approach quantum theory is tangible, experimentally justified and theoreti…
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…
The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schroedinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton's equations also give the Schroedinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order derivatives. For a time-independent system the Principle of Least Action reduces to the energy variational principle. For time-dependent systems the Principle of Least Action gives time-dependent approxi…
I propose a new and direct connection between classical mechanics and quantum mechanics where I derive the quantum mechanical propagator from a variational principle. This variational principle is Hamilton's modified principle generalized to allow many paths due to the non-local existence of particles in phase space. This principle allows a physical system to evolve non-locally in phase space while still allowing a representation that uses many classical paths. Whereas a point in phase space represents a classical system's state, I represent the state of a non-local system by a mixed trajector…
A role of the superposition principle is discussed for the quantum-mechanical Carnot engine introduced by Bender, Brody, and Meister [J. Phys. A 33, 4427 (2000)]. It is shown that the efficiency of the engine can be enhanced by superposition of quantum states. A finite-time process is also discussed, and the condition of the maximum power output is presented. Interestingly, the efficiency at the maximum power is lower than that without superposition.
The correspondence of a new form of quantum mechanics based on a quantum version of the action principle, which was proposed earlier [arXiv:0807.3508], with the ordinary quantum mechanics is established. New potentialities of the quantum action principle in the interpretation of quantum mechanics are considered.
We consider the Bohr correspondence limit of the Schrodinger wave function for an atomic elliptic state. We analyse this limit in the context of Nelson's stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler's laws of planetary motion in a quantum mechanical setting. In this quantum mechanical setting, local mild instabilities occur in the Kelperian orbit for eccentricities greater than 1/\sqrt{2} which do not occur classically.
