Observables of quantum or classical mechanics form algebras called quantum or classical Hamilton algebras respectively (Grgin E and Petersen A (1974) {\it J Math Phys} {\bf 15} 764\cite{grginpetersen}, Sahoo D (1977) {\it Pramana} {\bf 8} 545\cite{sahoo}). We show that the tensor-product of two quantum Hamilton algebras, each characterized by a different Planck's constant is an algebra of the same type characterized by yet another Planck's constant. The algebraic structure of mixed quantum and classical systems is then analyzed by taking the limit of vanishing Planck's constant in one of the component algebras. This approach provides new insight into failures of various formalisms dealing with mixed quantum-classical systems. It shows that in the interacting mixed quantum-classical description, there can be no back-reaction of the quantum system on the classical. A natural algebraic requirement involving restriction of the tensor product of two quantum Hamilton algebras to their components proves that Planck's constant is unique.
Existence and uniqueness results for the solution of the Gibbs-type formula from non-extensive mechanics are derived rigorously. A new conditional extremal problem is proposed to get in a more simple way the Gibbs-type formula itself.
In Bohmian mechanics the distribution $|ψ|^2$ is regarded as the equilibrium distribution. We consider its uniqueness, finding that it is the unique equivariant distribution that is also a local functional of the wave function $ψ$.
We show that the Bub-Clifton uniqueness theorem for 'no collapse' interpretations of quantum mechanics (Studies in the History and Philosophy of Modern Physics 27, 181-219 (1996)) can be proved without the 'weak separability' assumption.
I argue that the linearity of quantum mechanics is an emergent feature at the Planck scale, along with the manifold structure of space-time. In this regime the usual causality violation objections to nonlinearity do not apply, and nonlinear effects can be of comparable magnitude to the linear ones and still be highly suppressed at low energies. This can offer alternative approaches to quantum gravity and to the evolution of the early universe.
We discuss the systematic expansion of the solution of the Fokker-Planck equation with the help of the eigenfunctions of the time-dependent Fokker-Planck operator. The expansion parameter is the time derivative of the external parameter which controls the form of an external potential. Our expansion corresponds to the perturbative calculation of the adiabatic motion in quantum mechanics. With this method, we derive a new formula to calculate the irreversible work order by order, which is expressed as the expectation value with a pseudo density matrix. Applying this method to the case of the ha…
It is proved by a functional method that the conventional expression for the Dirac current is the only conserved 4-vector implied by the Dirac equation that is a function of just the quantum state. The demonstration is extended to derive the unique conserved currents implied by the coupled Maxwell-Dirac equations and the Klein-Gordon equation. The uniqueness of the usual Pauli and Schrodinger currents follows by regarding these as the non-relativistic limits of the Dirac and Klein-Gordon currents, respectively. The existence and properties of further conserved vectors that are not functions of…
Usual quantum mechanics requires a fixed, background, spacetime geometry and its associated causal structure. A generalization of the usual theory may therefore be needed at the Planck scale for quantum theories of gravity in which spacetime geometry is a quantum variable. The elements of generalized quantum theory are briefly reviewed and illustrated by generalizations of usual quantum theory that incorporate spacetime alternatives, gauge degrees of freedom, and histories that move forward and backward in time. A generalized quantum framework for cosmological spacetime geometry is sketched. T…
With the simplest proof ever, we justify the significance of quantum-gravity in non-relativistic quantum mechanics together with the related theories and experiments. Since the de Broglie wave length is inverse proportional to the mass, it would descend towards and below the Planck scale 10^{-33} cm for large masses even at slow non-relativistic motion. The tricky relationship between gravity and quantum mechanics ---well-known in the relativistic case--- shows up in non-relativistic motion of massive objects. Hence the gravity-related modification of their Schrödinger equation is m a n d a t …
In a recent work we have found a contraction semigroup able to correctly approximate a projected and perturbed one-parameter group of isometries in a generic Banach space, in the limit of weak-coupling. Here we study its generator by specializing to $W^*$-algebras: after defining a Physical Subsystem in terms of a completely positive projecting conditional expectation, we find that it generates a Quantum Dynamical Semigroup. As a consequence of uniqueness and strong generality (well defined dynamics, irrespective of the Physical Subsystem spectral properties or dimensions), its generator deser…
According to the widely accepted notion, the Schr{ö}dinger equation (SE) is not derivable in principle. Contrary to this belief, we present here a straightforward derivation of SE. It is based on only two fundamentals of mechanics: the classical Hamilton-Jacobi equation(HJE) and the Planck postulate about the discrete transfer of energy at micro-scales. Our approach is drastically different from the other published derivations of SE which either employ an ad hoc underlying assumption about the probabilistic or the statistical nature of the micro-scale phenomena, or rely on the prior knowledge …
We present arguments suggesting that deviations from the Born probability rule could be generated for trans-Planckian field modes during inflation. Such deviations are theoretically possible in the de Broglie-Bohm pilot-wave formulation of quantum mechanics, according to which the Born rule describes a state of statistical equilibrium. We suggest that a stable equilibrium state can exist only in restricted conditions: on a classical background spacetime that is globally hyperbolic or in a mild quantum-gravity regime in which there is an effective Schrödinger equation with a well-defined time p…
The generalized Weyl transform of index $α$ is used to implement the time-slice definition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter $α$. We succeed in proving that the $α$-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian involves products of noncommuting operators originating from the non-commutativity. The antisymme…
We show that one-dimensional Bohmian mechanics is unique, in that, the Bohm trajectories are the only solutions that conserve total left (or right) probability. In Brandt et al., Phys. Lett. A, 249 (1998) 265--270, they define quantile motion--unique trajectories are solved by assuming that the total probability on each side of the particle is conserved. They argue that the quantile trajectories are identical to the Bohm trajectories. Their argument, however, fails to notice the gauge freedom in the definition of the quantum probability current. Our paper sidesteps this under-determinedness of…
The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing $h_0$. For the second of these two extreme values, introduced operatorial algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to t…
The uniqueness of the Bohmian particle interpretation of the Kemmer equation, which describes massive spin-0 and spin-1 particles, is discussed. Recently the same problem for spin-1/2 was dealt with by Holland. It appears that the uniqueness of boson paths can be enforced under well determined conditions. This in turn fixes the nonrelativistic particle equations of the nonrelativistic Schrodinger equation, which appear to correspond with the original definitions given by de Broglie and Bohm only in the spin-0 case. Similar to the spin-1/2 case, there appears an additional spin-dependent term i…
The discrete phase space continuous time representation of relativistic quantum mechanics involving a characteristic length $l$ is investigated. Fundamental physical constants such as $\hbar$, $c$, and $l$ are retained for most sections of the paper. The energy eigenvalue problem for the Planck oscillator is solved exactly in this framework. Discrete concircular orbits of constant energy are shown to be circles $S^{1}_{n}$ of radii $2E_n =\sqrt{2n+1}$ within the discrete (1 + 1)-dimensional phase plane. Moreover, the time evolution of these orbits sweep out world-sheet like geometrical entitie…
Due to its connection to the diffeomorphism group, nonlinear quantum mechanics may play an important role in quantum geometry. The Doebner-Goldin nonlinearity (arising from representations of the diffeomorphism group) amplifies nonlocal signaling effects under extreme localization, suggesting that even if greatly suppressed at low energies, such effects may be significant at the Planck scale. This offers new perspectives on Planck-scale physics.
On 14 December 1900 Max Planck first formulated the idea of energy quanta related to a new universal constant now known as Planck's constant. Despite the following progress of thus initiated "quantum mechanics", the physical origin of both quantization and universality of Planck's constant remains mysterious, as well as other "peculiar" properties of quantum dynamics. In this paper we review a recent causal extension of quantum mechanics consistently explaining all its "mysteries" by irreducibly complex, "dynamically multivalued" behaviour of the underlying system of two interacting protofield…
The possibility is discussed of inferring or simulating some aspects of quantum dynamics by adding classical statistical fluctuations to classical mechanics. We introduce a general principle of mechanical stability and derive a necessary condition for classical chaotic fluctuations to affect confined dynamical systems, on any scale, ranging from microscopic to macroscopic domains. As a consequence we obtain, both for microscopic and macroscopic aggregates, dimensional relations defining the minimum unit of action of individual constituents, yielding in all cases Planck action constant.
In the paper the Schrodinger equation (SE) with gravity term is developed and discussed. It is shown that the modified SE is valid for particles with mass m<M_P, M_P is the Planck mass, and contains the part which, we argue describes the pilot wave. For m \to M_P the modified SE has the solution with oscillatory term, i.e. strings. Key words: Schrodinger-Newton equation; Planck time; pilot wave.
The effect of Kolmogorov-Arnold-Moser (KAM) theorem in quantum systems is manifested in dividing eigenstates into regular and irregular states. We propose an effective method based on Wannier basis in phase space to illustrate this division of eigenstates. The quantum kicked-rotor model is used to illustrate this method, which allows us to define the area and effective dimension of each eigenstate to distinguish quantitatively regular and irregular eigenstates. This Wannier basis method also allows us to define the length of a Planck cell in the spectrum that measures how many Planck cells the…
Recently we showed that the postulated diffeomorphic equivalence of states implies quantum mechanics. This approach takes the canonical variables to be dependent by the relation p=\partial_q S_0 and exploits a basic GL(2,C)-symmetry which underlies the canonical formalism. In particular, we looked for the special transformations leading to the free system with vanishing energy. Furthermore, we saw that while on the one hand the equivalence principle cannot be consistently implemented in classical mechanics, on the other it naturally led to the quantum analogue of the Hamilton-Jacobi equation, …
The paper discuss the structure of quantum mechanics and uniqueness of its postulates. The Born rule for quantum probabilities is fixed by requirement of nonexistence of quantum telepathy. Von Neumann projection postulate describes the transformation of quantum state under the condition of no-interaction measurement. Projection postulate could be considered as transition to conditional probability under the condition of a certain result of quantum measurement.
A view of emission and absorption of photons by atoms, which was proposed by Max Planck, is recalled. A direct experiment that could corroborate or refute that view is defined.
Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a complex structure on classical phase space. In this letter we analyse the role played by generalised complex geometry in the classical and quantum mechanics of a finite number of degrees of freedom. We identify generalised complex geometry as an appropriate geometrical setup for dual…
Quantum mechanics has so far not been tested for mechanical objects at the scale of the Planck mass $\sqrt{\hbar c/ G} \simeq 22\,μ\mathrm{g}$. We present an experiment where a 1 mm quartz micropillar resonating at 3.6 MHz with an effective mass of 30 $μ$g is cooled to 500 mK with a dilution refrigerator, and further optomechanically sideband-cooled to an effective temperature of 3 mK, corresponding to a mode thermal occupancy of 20 phonons. This nearly 1000-fold increase in the mass of an optomechanical system with respect to previous experiments near the quantum ground state opens new perspe…
Here we present some results that would possibly have attracted the attention of the physics community in the early days of quantum mechanics in such a way that its development could have been different from what we see today. We will first present a derivation of Planck's blackbody spectrum radiation without the hypothesis of quantized energy levels of the oscillator, the only additional hypothesis to the classical theory being the existence of zeropoint fluctuations in the electromagnetic field, and this would stand as a nontrivial vacuum which could be inferred by classical means. After doi…
Relaxation according to Fokker-Planck equations is a standard scenario in classical statistical mechanics. It is however not obvious how such an equilibration may emerge within a closed, finite quantum system. We present an analytical and numerical analysis of a system comprising sixteen spins in which spatial inhomogeneities of the magnetization relax approximately in accord with a standard Fokker-Planck equation for a Brownian particle in a parabolic potential.
It is shown that in the model [3,4] of quantum mechanics besides probability amplitudes, the Planck constant and the Fock space, the cosmological constant also appear in the natural way. The Poisson brackets are generalized for the case of kinetics.
Within the context of the usual semi classical investigation of Planck scale Schwarzchild Black Holes, as in Quantum Gravity, and later attempts at a full Quantum Mechanical description in terms of a Kerr-Newman metric including the spinorial behaviour, we attempt to present a formulation that extends from the Planck scale to the Hubble scale. In the process the so called large number coincidences as also the hitherto inexplicable relations between the pion mass and the Hubble Constant, pointed out by Weinberg, turn out to be natural consequences in a consistent description.
We derive fundamental limits on measurements of position, arising from quantum mechanics and classical general relativity. First, we show that any primitive probe or target used in an experiment must be larger than the Planck length, $l_P$. This suggests a Planck-size {\it minimum ball} of uncertainty in any measurement. Next, we study interferometers (such as LIGO) whose precision is much finer than the size of any individual components and hence are not obviously limited by the minimum ball. Nevertheless, we deduce a fundamental limit on their accuracy of order $l_P$. Our results imply a {\i…
We provide an answer to the long standing problem of mixing quantum and classical dynamics within a single formalism. The construction is based on p-mechanical derivation (quant-ph/0212101, quant-ph/0304023) of quantum and classical dynamics from the representation theory of the Heisenberg group. To achieve a quantum-classical mixing we take the product of two copies of the Heisenberg group which represent two different Planck's constants. In comparison with earlier guesses our answer contains an extra term of analytical nature, which was not obtained before in purely algebraic setup. Keywords…
One of the main challenges in physics today is to merge quantum theory and the theory of general relativity into a unified framework. Various approaches towards developing such a theory of quantum gravity are pursued, but the lack of experimental evidence of quantum gravitational effects thus far is a major hindrance. Yet, the quantization of space-time itself can have experimental implications: the existence of a minimal length scale is widely expected to result in a modification of the Heisenberg uncertainty relation. Here we introduce a scheme that allows an experimental test of this conjec…
The rules of quantum mechanics require a time coordinate for their formulation. However, a notion of time is in general possible only when a classical spacetime geometry exists. Such a geometry is itself produced by classical matter sources. Thus it could be said that the currently known formulation of quantum mechanics pre-assumes the presence of classical matter fields. A more fundamental formulation of quantum mechanics should exist, which avoids having to use a notion of time. In this paper we discuss as to how such a fundamental formulation could be constructed for single particle, non-re…
