Concentration of Measure for Quantum States with a Fixed Expectation Value

Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors \({|\psi\rangle}\) that have a fixed expectation value \({\langle\psi|H|\psi\rangle=E}\) with respect to H. Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere.     

Stochastic differential equations anda posteriori states in quantum mechanics

In recent years a consistent theory describing measurements continuous in time in quantum mechanics has been developed. The result of such a measurement is atrajectoryfor one or more quantities observed with continuity in time. Applications are connected especially with detection theory in quantum optics. In such a theory of continuous measurements one can ask what is the state of the system given that a certain trajectory up to timet has been observed. The response to this question is the notion ofa posteriori states and afilteringequation governing the evolution of such states: this turns out to be a nonlinear stochastic differential equation for density matrices or for pure vectors. The driving noise appearing in such an equation is not an external one, but its probability law is determined by the system itself (it is the probability measure on the trajectory space given by the theory of continuous measurements).     

Classical statistical mechanics of identical particles and quantum effects

The identification of the phase space ofN classical identical particles with the equivalence class of points is of crucial importance for statistical mechanics. We show that the refined phase space leads to the correct statistical mechanics for an ideal gas; moreover, Gibbs's paradox is resolved and the Third Law of Thermodynamics is recovered. The presence of both induced and stimulated transitions is shown as a consequence of the identity of the particles. Other results are the quantum contribution to the second virial coefficient and the Bose-Einstein condensation. Photon bunching and Hanbury Brown-Twiss effect are also seen to follow from the classical model. The only element of quantum theory involved is the notion of phase cells necessary to make the entropy dimensionless. Assuming the existence of the light quantum or the phonon hypothesis we could derive the Planck distribution law for blackbody radiation or the Debye formula for specific heats respectively.     

Classical probabilities for Majorana and Weyl spinors

We construct a map between the quantum field theory of free Weyl or Majorana fermions and the probability distribution of a classical statistical ensemble for Ising spins or discrete bits. More precisely, a Grassmann functional integral based on a real Grassmann algebra specifies the time evolution of the real wave function q_τ(t) for the Ising states τ. The time dependent probability distribution of a generalized Ising model obtains as . The functional integral employs a lattice regularization for single Weyl or Majorana spinors. We further introduce the complex structure characteristic for quantum mechanics. Probability distributions of the Ising model which correspond to one or many propagating fermions are discussed explicitly. Expectation values of observables can be computed equivalently in the classical statistical Ising model or in the quantum field theory for fermions.     

Quantum mechanics and the direction of time

In recent papers the authors have discussed the dynamical properties of large Poincaré systems (LPS), that is, nonintegrable systems with a continuous spectrum (both classical and quantum). An interesting example of LPS is given by the Friedrichs model of field theory. As is well known, perturbation methods analytic in the coupling constant diverge because of resonant denominators. We show that this Poincaré catastrophe can be eliminated by a natural time ordering of the dynamical states. We obtain then a dynamical theory which incorporates a privileged direction of time (and therefore the second law of thermodynamics). However, it is only in very simple situations that this time ordering can be performed in an extended Hilbert space. In general, we need to go to the Liouville space (superspace) and introduce a time ordering of dynamical states according to the number of particles involved in correlations. This leads then to a generalization of quantum mechanics in which the usual Heisenberg's eigenvalue problem is replaced by a complex eigenvalue problem in the Liouville space.     

Connections Between the Thermodynamics of Classical Electrodynamic Systems and Quantum Mechanical Systems for Quasielectrostatic Operations

The thermodynamic behavior is analyzed of a single classical charged particle in thermal equilibrium with classical electromagnetic thermal radiation, while electrostatically bound by a fixed charge distribution of opposite sign. A quasistatic displacement of this system in an applied electrostatic potential is investigated. Treating the system nonrelativistically, the change in internal energy, the work done, and the change in caloric entropy are all shown to be expressible in terms of averages involving the distribution of the position coordinates alone. A convenient representation for the probability distribution is shown to be the ensemble average of the absolute square value of an expansion over the eigenstates of a Schrödinger-like equation, since the heat flow is shown to vanish for each hypothetical state. Subject to key assumptions highlighted here, the demand that the entropy be a function of state results in statistical averages in agreement with the form in quantum statistical mechanics. Examining the very low and very high temperature situations yields Planck's and Boltzmann's constants. The blackbody radiation spectrum is then deduced. From the viewpoint of the theory explored here, the method in quantum statistical mechanics of statistically counting the states at thermal equilibrium by using the energy eigenvalue structure, is simply a convenient counting scheme, rather than actually representing averages involving physically discrete energy states.     

Level Dynamics and Universality of Spectral Fluctuations

The spectral fluctuations of quantum (or wave) systems with a chaotic classical (or ray) limit are mostly universal and faithful to random-matrix theory. Taking up ideas of Pechukas and Yukawa we show that equilibrium statistical mechanics for the fictitious gas of particles associated with the parametric motion of levels yields spectral fluctuations of the random-matrix type. Previously known clues to that goal are an appropriate equilibrium ensemble and a certain ergodicity of level dynamics. We here complete the reasoning by establishing a power law for the dependence of the mean parametric separation of avoided level crossings. Due to that law universal spectral fluctuations emerge as average behavior of a family of quantum dynamics drawn from a control parameter interval which becomes vanishingly small in the classical limit; the family thus corresponds to a single classical system. We also argue that classically integrable dynamics cannot produce universal spectral fluctuations since their level dynamics resembles a nearly ideal Pechukas–Yukawa gas.     

Resonances in the Rigged Hilbert Space and Lax-Phillips Scattering Theory

The rigged Hilbert space formalism of quantum mechanics provides a framework in which one can identify resonance states and obtain the typical exponential decay law. However, there remain questions of the interpretation and extraction of physical information through the calculation of expectation values of observables. The Lax-Phillips scattering theory provides a mathematical construction in which resonances are assigned with states in a Hilbert space, thus no such difficulties arise. The original Lax-Phillips structure is inapplicable within standard nonrelativistic quantum theory. Through the powerful theory of H ^p spaces certain relations between the two theories are uncovered, which suggest that a search for a unifying framework might prove useful.     

Instead of particles and fields: A micro realistic quantum “smearon” theory

A fully micro realistic, propensity version of quantum theory is proposed, according to which fundamental physical entities—neither particles nor fields—have physical characteristics which determine probabilistically how they interact with one another (rather than with measuring instruments). The version of quantum smearon theory proposed here does not modify the equations of orthodox quantum theory: rather it gives a radically new interpretation to these equations. It is argued that (i) there are strong general reasons for preferrring quantum smearon theory to orthodox quantum theory; (ii) the proposed change in physical interpretation leads quantum smearon theory to make experimental predictions subtly different from those of orthodox quantum theory. Some possible crucial experiments are considered.     

On the Role of Density Matrices in Bohmian Mechanics

It is well known that density matrices can be used in quantum mechanics to represent the information available to an observer about either a system with a random wave function (statistical mixture) or a system that is entangled with another system (reduced density matrix). We point out another role, previously unnoticed in the literature, that a density matrix can play: it can be the conditional density matrix, conditional on the configuration of the environment. A precise definition can be given in the context of Bohmian mechanics, whereas orthodox quantum mechanics is too vague to allow a sharp definition, except perhaps in special cases. In contrast to statistical and reduced density matrices, forming the conditional density matrix involves no averaging. In Bohmian mechanics with spin, the conditional density matrix replaces the notion of conditional wave function, as the object with the same dynamical significance as the wave function of a Bohmian system.PACS number:03.65.Ta (foundations of quantum mechanics)     

Quantum probabilities,operators of state preparation,and the principle of superposition

An integrated view concerning the probabilistic organization of quantum mechanics is first obtained by systematic confrontation of the Kolmogorov formulation of the abstract theory of probabilities with the quantum mechanical representationand its factual counterparts. Because these factual counterparts possess a peculiar space-time structure stemming from the operations by which the observer produces the studied states (operations of state preparation) and the qualifications of these (operations of measurement), the approach brings forth probability-trees, complex constructs with treelike space-time support. Though it is strictly entailed by confrontation with the abstract theory of probabilities as it now stands, the construct of a quantum mechanical probability treetransgresses this theory. It indicates the possibility of an extended abstract theory of probabilities: Quantum mechanics appears to be neither a normal probabilistic theory nor an abnormal one, but a pioneering particular realization of afuture extended abstract theory of probabilities. The integrated perception of the probabilistic organization of quantum mechanics removes the current identifications of spectral decompositions of one state vector, with superpositions of several state vectors. This leads to the definition of operators of state preparation and of the calculus with these and to a clear understanding of the physical significance of the principle of superposition. Furthermore, a complement to the quantum theory of measurements is obtained.     

Bits,time, carriers,and matter

The formal structure of quantum information theory is based on the well-founded concepts and postulates of quantum mechanics. In the present contribution, I am inverting the usual approach presented in textbooks by beginning with the use of bit states as basic and fundamental units of information and establish a dynamical map for them. The condition of reversibility, imposed on an ordered sequence of actions operating on a bit state, introduces, by necessity, the unitarity property of actions. I also verify that the uniformity of time, as a parameter for ordering events, is due to the admission of a composition law for the actions. In the limit of infinitesimal intervals between actions, a reversible and linear equation arises for the dynamical changes in time of a qubit (superposition of bit states). The admission that a bit of information is stored or carried by a massive particle necessarily leads to the Schrödinger–Pauli equation (SPE); the bit is associated to a spin 1/2. Within this approach, I verify that the particle dynamical equation becomes “enslaved” by the spin dynamics. In other words, the bit (or spin) precedes in status the particle dynamical evolution, being at the root of the quantum character of the standard Schr¨odinger equation, even when spin and spatial degrees of freedom are uncoupled.     

The Dynamics of Difference

A proposal is made for a fundamental theory, in which the history of the universe is constituted of diverse views of itself. Views are attributes of events, and the theory’s only be-ables; they comprise information about energy and momentum transferred to an event from its causal past. A dynamics is proposed for a universe constituted of views of events, which combines the energetic causal set dynamics with a potential energy based on a measure of the distinctiveness of the views, called the variety (Smolin in Found Phys 46(6):736–758, 2016). As in the real ensemble formulation of quantum mechanics (Barbour and Smolin in Variety, complexity and cosmology, arXiv: hep-th/9203041), quantum pure states are associated to ensembles of similar events; the quantum potential of Bohm then arises from the variety.     

Relativity of topology and dynamics

Recent developments in quantum set theory are used to formulate a program for quantum topological physics. The world is represented in a Hilbert space whose psi vectors represent abstract complexes generated from the null set by one bracket operator and the usual Grassmann (or Clifford) product. Such a theory may be more basic than field theory, in that it may generate its own natural topology, time, kinematics and dynamics, without benefit of an absolute timespace dimension, topology, or Hamiltonian. For example there is a natural expression for the quantum gravitational field in terms of quantum topological operators. In such a theory the usual spectrum of possible dimensions describes only one of an indefinite hierarchy of levels, each with a similar spectrum, describing nonspatial infrastructure. While c simplices have no continuous symmetry, the q simplex has an orthogonal group 0(m, n). Because quantum theory cannot take the universe as physical system, we propose a third relativity:The division between observer and observed is arbitrary. Then it is wrong to ask for the topology and dynamics of a system, in the same sense that it is wrong to ask for the the psi vectors of a system; topology and dynamics, like psi vectors, are not absolute but relative to the observer.     

Eddington and Uncertainty

Arthur Stanley Eddington (1882-1944) is acknowledged to be one of the greatest astrophysicists of the twentieth century, yet his reputation suffered in the 1930s when he embarked on a quest to develop a unified theory of gravity and quantum mechanics. His attempt ultimately proved to be fruitless and was regarded by many physicists as misguided. I will show, however, that Eddingtons work was not so outlandish. His theory applied quantum-mechanical uncertainty to the reference frames of relativity and actually foreshadowed several later results. His philosophy regarding determinism and uncertainty also was quite orthodox at the time. I first review Eddingtons life and philosophy and then discuss his work within the context of his search for a theory of quantum gravity.     

Generalization of quantum statistics in statistical mechanics

We propose a generalization of quantum statistics in the framework of statistical mechanics. We derive a general formula which involves a wide class of equilibrium quantum statistical distributions, including the Bose and Fermi distributions. We suggest a way of evaluating the statistical distributions with the help of many-particle partition functions and apply it to studying some interesting distributions. A question on the statistical distribution for anyons is discussed, and the term following the Boltzmann one in the expansion of this distribution in powers of the Boltzmann factor, exp[(-_i)], is estimated. An ansatz is proposed for evaluating the statistical distribution forquons (particles whose creation and annihilation operators satisfy theq-commutation relations). We also treat non-equilibrium statistical mechanics, obtaining unified expressions for the entropy of a nonequilibrium quantum gas and for a collision integral which are valid for a wide class of statistics.     

The reinterpretation of wave mechanics

The author begins by recalling how he was led in 1923–24 to the ideas of wave mechanics in generalizing the ideas of Einstein's theory of light quanta. He made himself at that time a concrete physical picture of the coexistence of waves and particles and, in 1927, attempted to give them precise form in his theory of the double solution. As other ideas prevailed at the time, he abandoned the development of his conception. But for the past twenty years, once again convinced, like Einstein, that present-day quantum mechanics is only a statistical theory and does not give a true picture of physical reality, he has again taken up his old ideas and developed them considerably. He has in particular introduced an element of randomness into the theory and has thus attained to a hidden thermodynamics of particles, the results of which appear to be very interesting.