Beyond Peaceful Coexistence

The following sections are included:

• Contents
• Foreword by Gerard ’t Hooft

A physical theory consists of a formal structure and one or more interpretations. The latter can come out from cultural and cognitive tension going far beyond any sound operational pact between theoretical constructs and empirical data. We have no reason to doubt about the consistency and efficacy of syntaxes if properly used in the right range. The formal side of Physics has grown in a strongly connected and stratified way through an almost autopoietic, self-dual procedure (let's think of the extraordinary success of the gauge theories), whereas the foundational debate is still blustering about the two pillars of such monumental construction. The general relativity (GR) and the quantum mechanics (QM), which still appear to be greatly incompatible and stopped in a limited peaceful coexistence between local causality in space–time and quantum non-locality [1]. The formidable challenges waiting for us beyond the Standard Model seem to require a new semantic consistency [2] within the two theories, so as to build a new way to look at them, to work and to relate them…

In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl–von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called ‘shadow phase spaces’. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton–Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.

We have explained in previous work the correspondence between the standard squeezed coherent states of quantum mechanics, and quantum blobs, which are the smallest phase space units compatible with the uncertainty principle of quantum mechanics and having the symplectic group as a group of symmetries. In this work, we discuss the relation between quantum blobs and a certain level set (which we call “Fermi blob”) introduced by Enrico Fermi in 1930. Fermi blobs allows us to extend our previous results not only to the excited states of the generalized harmonic oscillator in n dimensions, but also to arbitrary quadratic Hamiltonians. As is the case for quantum blobs, we can evaluate Fermi blobs using a topological notion, related to the uncertainty principle, the symplectic capacity of a phase space set. The definition of this notion is made possible by Gromov's symplectic non-squeezing theorem, nicknamed the “principle of the symplectic camel”.

The conceptual incompatibility between General Relativity and Quantum Mechanics is generally seen as sufficient motivation for the development of a theory of Quantum Gravity. If—so a typical argument goes — Quantum Mechanics gives a universally valid basis for the description of the dynamical behavior of all natural systems, then the gravitational field should have quantum properties, like all other fundamental interaction fields. And if General Relativity can be seen as an adequate description of the classical aspects of gravity and space–time — and their mutual relation — this leads, together with the rather convincing arguments against semi-classical theories of gravity, to a strategy which takes a quantization of General Relativity as the natural avenue to a theory of Quantum Gravity. And because in General Relativity, the gravitational field is represented by the space–time metric, a quantization of the gravitational field would in some sense correspond to a quantization of geometry. Space–time would have quantum properties…

Elementary Cycles Theory (ECT) is a novel exact formulation of quantum-relativistic mechanics. Here, we present an introduction to its basic quantum aspects. On the one hand, Newton's law of inertia states that every isolated particle has persistent motion, i.e. constant energy and momentum. On the other hand, undulatory mechanics associates, by means of the Planck constant, a recurrence in time and space to the energy and the momentum of an elementary particle, respectively. Paraphrasing these two fundamental principles of modern physics, ECT postulates that every elementary constituent of nature (every elementary particle) is characterized by persistent intrinsic periodicity (as long it does not interact) whose space–time duration determines its kinematical state (energy and momentum). In other words, undulatory mechanics is imposed as constraint “overdetermining” relativistic mechanics, with fundamental motivations on Einstein's proposal of unification of quantum and relativistic theories. Every free particle is a (de Broglie) “periodic phenomenon” which can also be regarded as a reference clock and every system is decomposable in modulated elementary cycles. Indeed, ECT introduces a cyclic nature to the ordinary relativistic space–time coordinates. The resulting classical-relativistic mechanics turns out to be fully consistent with relativity and, at the same time, reproduces exactly all the fundamental aspects of ordinary quantum-relativistic mechanics (without any explicit quantisation). Relativity only fixes the differential structure of space–time without giving any prescription about the boundary of space–time, and the constraint of covariant periodicity (or similar relativistic boundary conditions) is allowed by the variational principle for relativistic theories. The constraint of intrinsic periodicity enforces the local nature of relativistic space–time and the wave-particle duality. Besides such unified description of relativistic and quantum dynamics, ECT allows for a geometrodynamical formulation of gauge interaction which, similarly to gravity, is represented by modulations of space–time periodicities and brings novel elements to address fundamental open problems of modern physics.

The following sections are included:

• Introduction
• Basic Concepts and Terms
• Particles
• Questions, Answers, and Laboratories
• Observers and Experiments
• The Stern–Gerlach Experiment
• The DS Experiment
• The Monitored DS Experiment
• The Contextuality of Space and Time
• Acknowledgment
• References

The following sections are included:

• Introduction: Physical Models and Structural Stability
• From Galilean to Relativistic Dynamics and From Classical to Quantum Mechanics
• Stabilizing the Heisenberg–Poincaré algebra
• Stability, Universality and Critical Phenomena
• Appendix A: Structural Stability in Dynamical Systems Theory
• Appendix B: Algebraic Deformation Theory: Basic Notions
• References

The ubiquitous nature of modern Information Retrieval (IR) and Virtual World give rise to new realities. To what extent are these ‘realities’ real? Which ‘physics’ should be applied to quantitatively describe them? In this chapter, I dwell on few examples. The first is adaptive neural networks, which are not networks and not neural, but still provide service similar to classical artificial neural networks (ANNs) in extended fashion. The second is the emergence of objects looking like Einsteinian space–time, which describe the behavior of an Internet surfer like geodesic motion. The third is the demonstration of nonclassical and even stronger-than-quantum probabilities in IR, their use…

One of the main challenges in theoretical physics in the past 50 years has been to define a theory of quantum gravity, i.e. a theory which consistently combines general relativity and quantum theory in order to define a theory of space–time itself seen as a fluctuating field. As such, a definition of space–time is of paramount importance, but it is precisely the attainment of such a definition which is one of the main stumbling blocks in quantum gravity. One of the striking features of quantum gravity is that although both general relativity and quantum theory treat space–time as a four-dimensional (4D) manifold equipped with a metric, quantum gravity would suggest that, at the microscopic scale, space–time is somewhat discrete. Therefore the continuum structure of space–time suggested by the two main ingredients of quantum gravity seems to be thrown into discussion by quantum gravity itself. This seems quite an odd predicament, but it might suggest that perhaps a different mathematical structure other than a smooth manifold should model space–time. These considerations seem to shed doubts on the use of the continuum in general in a possible theory of quantum gravity. An alternative would be to develop a mathematical formalism for quantum gravity in which no fundamental role is played by the continuum and where a new concept of space–time, not modeled on a differentiable manifold, will emerge. This is precisely one of the aims of the topos theory approach to quantum theory and quantum gravity put forward by Isham, Butterfield, and Doering and subsequently developed by other authors. The aim of this article is to precisely elucidate how such an approach gives rise to a new definition of space–time which might be more appropriate for quantum gravity.

Born's principle of reciprocity — the exchangeability of relativistic energy–momentum and time-position — can be seen as a discrete element of a continuous group of symmetry transformations which transcend relativity. Invariance under the semi-direct product of the Weyl–Heisenberg group H(4) of canonical commutation relations with the non-compact unitary group U(3, 1) — the so-called quaplectic group U(3, 1) ⋉ H(4) — has been considered by Low as an extension of Born reciprocity to a fundamental symmetry principle of ‘reciprocal relativity’ for the physics of non-inertial frames and high energy processes…

Based on recent results from general relativistic statistical mechanics and black hole information transfer limits, a space–time entropy-action equivalence is proposed as a generalization of the holographic principle. With this conjecture, the action principle can be replaced by the second law of thermodynamics, and for the Einstein–Hilbert action the Einstein field equations are conceptually the result of thermodynamic equilibrium. For non-equilibrium situations, Jaynes' information-theoretic approach to maximum entropy production is adopted instead of the second law of thermodynamics. As it turns out for appropriate choices of constants, quantum gravity is obtained. For the special case of a free particle the Bekenstein–Verlinde entropy-to-displacement relation of holographic gravity and thus the traditional holographic principle emerges. Although Jacobson's original thermodynamic equilibrium approach proposed that gravity might not necessarily be quantized, this particular non-equilibrium treatment might require it.

We show that a crystalline model with defects can reproduce Einstein's theory of gravity down to the Planck scale.

We review the properties of discrete and integer-valued, hence “natural”, cellular automata (CA), a particular class of which comprises “Hamiltonian CA” with equations of motion that bear strong similarities to Hamilton's equations, despite presenting discrete updating rules. The resulting dynamics is linear in the same sense as unitary evolution described by the Schrödinger equation. Employing Shannon's Sampling Theorem, we construct an invertible map between such CA and continuous quantum mechanical models which incorporate a fundamental discreteness scale. This leads to one-to-one correspondence of quantum mechanical and CA conservation laws. In order to illuminate the all-important issue of linearity, we presently introduce an extension of the class of CA incorporating nonlinearities. We argue that these imply non-local effects in the continuous quantum mechanical description of intrinsically local discrete CA, enforcing locality entails linearity. We recall the construction of admissible CA observables and the existence of solutions of the modified dispersion relation for stationary states, besides discussing next steps of the deconstruction of quantum mechanical models in terms of deterministic CA.

Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck scale, one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of our ordinary continuum physics and mathematics. We regard these continuum concepts and continuum space–time (S-T) in particular as being emergent, coarse-grained and derived relative to an underlying erratic and disordered microscopic substratum which is expected to play by quite different rules. A central role in our analysis is played by a geometric renormalization group which creates (among other things) a kind of sparse translocal network of correlations in classical continuous space–time and underlies in our view such mysterious phenomena as holography and the black hole entropy-area law. The same point of view holds for quantum theory which we also regard as a low-energy, coarse-grained continuum theory, being emergent from something more fundamental.

The aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint operators for suitable Observables in non-relativistic and relativistic quantum mechanics (QM), and in quantum electrodynamics. More specifically, this work deals with: (i) the Hermitian (but not self-adjoint) Time operator in non-relativistic QM and in quantum electrodynamics; (ii) idem, the introduction of Time and Space operators; and (iii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physical applications of non-self-adjoint (and even non-Hermitian) operators are briefly discussed. We mention how non-Hermitian operators can indeed be used in physics [as it was done, elsewhere, for describing Unstable States]; and some considerations are added on the cases of the nuclear optical potential, of quantum dissipation, and in particular of an approach to the measurement problem in QM in terms of a chronon. This paper is largely based on work developed, along the years, in collaboration with V.S. Olkhovsky, and, in smaller parts, with P. Smrz, with R.H.A. Farias, and with S.P. Maydanyuk.

It has been shown that a nonlinear Schrödinger equation in 2+1 dimensions equipped with an SU(N) Chern–Simons gauge field can provide an exact description of certain self-dual Einstein spaces in the limit N−=∞. Ricci flat Einstein spaces can then be viewed as arising from a quantum pairing of the classical self-dual and anti-self-dual solutions. In this chapter, we will outline how this theory of empty space–time might be generalized to include matter and vacuum energy by transplanting the nonlinear Schrödinger equation used to construct Einstein spaces to the 25+1-dimensional Lorentzian Leech lattice. If the distinguished 2 spatial dimensions underlying the construction of Einstein spaces are identified with a hexagonal lattice section of the Leech lattice, the wave-function becomes an 11 × 11 matrix that can represent fermion and boson degrees of freedom (DOF) associated with 2-form and Yang–Mills gauge symmetries. The resulting theory of gravity and matter in 3+1 dimensions is not supersymmetric, which provides an entry for a vacuum energy. Indeed, in the case of a Lemaitre cosmological model, the emergent space–time will naturally have a vacuum energy on the order of the observed cosmological constant.

Basic foundational aspects of both quantum physics and relativity suggest that space–time may have the fundamental stochastic nature of a turbulent fluid. After reviewing the basic theoretical motivations, we have compared this picture with the phenomenological pattern observed in the ether-drift experiments. To this end, we have performed numerical simulations in which the parameters of the macroscopic Earth's cosmic motion are only used to fix the limiting boundaries for a microscopic velocity field which has instead an intrinsic stochastic nature. In this framework, both classical and modern experiments become consistent with the type of cosmic Earth's motion which today is used to describe the CMB anisotropy. The need for confirmations with a new generation of dedicated experiments is finally emphasized.

In closed systems, energy is conserved. The origin of the time axis is completely arbitrary due to the invariance under continuous time-translations. The flowing of time swallows those fictitious origins one might assign on its axis, as Kronos ate his sons. Dissipation breaks such a scenario. It implies a non-forgettable origin of time. Open systems need their complement (their “double”) in order to become, together, a closed system. Time emerges as an observable measured by the evolution of the open system complement, which acts as a clock. The conservation of the energy–momentum tensor in electrodynamics is considered and its relation with dissipative systems and self-similar fractal structures is discussed. The isomorphism with coherent states in quantum field theory (QFT) is established and the generator of transitions among unitarily inequivalent representations of the canonical commutation relations (CCR) is shown to provide sequences in time of phases, which defines the arrow of time. Merging properties of electrodynamics, fractal self-similarity, dissipation and coherent states point to an integrated vision of Nature.

A transactional account of the emergence of space–time events from a quantum substratum is presented. In this account, space–time is not a substantive manifold that becomes occupied with events; rather, space–time itself exists only in virtue of specific actualized events. This implies that space–time is discrete rather than continuous, and that properties attributed to space–time based on the notion of a continuum are idealizations that do not apply to the real physical world. It is further noted that the transactional picture of the emergence of space–time can provide the quantum dynamics that underlie the causal set approach as proposed by Sorkin and others.

We use graphical field gradients in an adynamical, background independent fashion to propose a new approach to quantum gravity (QG) and unification. Our proposed reconciliation of general relativity (GR) and quantum field theory (QFT) is based on a modification of their graphical instantiations, i.e. Regge calculus and lattice gauge theory (LGT), respectively, which we assume are fundamental to their continuum counterparts. Accordingly, the fundamental structure is a graphical amalgam of space, time, and sources (in parlance of QFT) called a “space–time source element”. These are fundamental elements of space, time, and sources, not source elements in space and time. The transition amplitude for a space–time source element is computed using a path integral with discrete graphical action. The action for a space–time source element is constructed from a difference matrix K and source vector J on the graph, as in lattice gauge theory. K is constructed from graphical field gradients so that it contains a non-trivial null space and J is then restricted to the row space of K, so that it is divergence-free and represents a conserved exchange of energy–momentum. This construct of K and J represents an adynamical global constraint (AGC) between sources, the space–time metric, and the energy–momentum content of the element, rather than a dynamical law for time-evolved entities. In this view, one manifestation of quantum gravity becomes evident when, for example, a single space–time source element spans adjoining simplices of the Regge calculus graph. Thus, energy conservation for the space–time source element includes contributions to the deficit angles between simplices. This idea is used to correct proper distance in the Einstein–de Sitter (EdS) cosmology model yielding a fit of the Union2 Compilation supernova data that matches ΛCDM without having to invoke accelerating expansion or dark energy. A similar modification to LGT results in an adynamical account of quantum interference.

We argue that not all the theoretical content of the Bohr model has been captured by the “definitive” quantum formalism currently in use. In particular, the notion of “quantum leap” seems to refer to non-dynamic features, closely related to non-locality, which have not yet been formalized in a satisfactory way.

The Theory of Archaic Universe, based on the de Sitter Projective Relativity, and the transactional interpretation of Quantum Mechanics merge into a new Archaic Holographic Principle which provides a unitary description of physical processes whose central idea consists of the finite exchange of information in microinteractions.

A Schrödinger-evolving forward-lightcone-interior “Milne” universe (“SMU”) is governed by “centered-Lorentz” (CL) symmetry — that of a 9-parameter Lie group with a 6-parameter SL(2,c) “exterior” and a 3-parameter “quality-space” center. “Reality” resides in current densities of electric charge and energy–momentum — the Dalembertian of an SMU-ray-specified classical retarded Lorentz-tensor field with 2² electro-magnetic and 3² gravitational components…

It is shown that the generators of Clifford algebras behave as creation and annihilation operators for fermions and bosons. They can create extended objects, such as strings and branes, and can induce curved metric of our space–time. At a fixed point, we consider the Clifford algebra Cl(8) of the 8D phase space, and show that one quarter of the basis elements of Cl(8) can represent all known particles of the first generation of the Standard model, whereas the other three quarters are invisible to us and can thus correspond to dark matter.

We show how the geometric data for commutative and Non-commutative NC models for Ricci solitons and Einstein spaces can be encoded equivalently in terms of almost symplectic geometries and/or Lagrange–Finsler spaces uniquely determined by the metric and/or almost Kähler structures. Such generic off-diagonal configurations are quantized following methods of deformation quantization (DQ). Using almost Kähler variables, an associative star-product can be introduced for NC generalizations of Ricci solitonic equations. There are new classes of exact solutions with NC variables describing NC black ellipsoid and solitonic waves generated as Ricci solitons. We conclude that the methods of non-holonomic almost Kähler–Finsler geometry allows us to formulate a unified formalism for Ricci flows/solitons and DQ and NC generalizations of such theories.

The following sections is included:

• Index