Supplementary Comments

Physics together with metaphysics developed in an attempt to understand the “workings of nature.” An overview of the way such an understanding evolved from ancient times to the present was provided in the first article of the series, “Perceptions of physical reality,” available here in the Education section. Elemental issues: Nature of interaction between “Nature” and “Observer;” Meaning of understanding; and the Structure of space-time complex underlying the “Workings of nature,” which are intertwined with the workings, were also discussed in this article. Modern classical theories were discussed in the articles #2 to 5, and the quantum theories, in articles #6 to 9.

Regarding the nature of interaction between the nature and observer, two basic approaches developed from the early times: Positivism and Realism. Positivism places observer in a determining position through its mind implying that reality is subjective. The view that involvement of the observer in defining the perceived reality can be even deeper is also included in positivism. Idealism, which considers reality only the perceptions, stretches positivism even further. Realism considers reality to be objective that exists independent of the observer. An understanding of the physical nature acquires meaning as its image in terms of the concepts of a suitable mathematical structure, essentially a construct of mind, but developed in coordination with the observations. A structure of the space-time develops as an understanding of the workings of nature develops. These concepts develop interactively together defining the perceived reality.

Modern theories in physics include two distinct eras: The classical, from the time of Galileo to the end of the nineteenth century, and the quantum, from 1900 to date. The present article outlines the physical reality as perceived in these theories. Some supplementary material is also discussed.

Classical Theories

In the pre-relativistic classical, Galileo-Newtonian, physics, motion of a particle is described by its location in space with respect to time, the space being defined by the relative positions of material bodies and time, by the order of events. Space was assumed to be equipped with the Euclidean geometry and time served as an independent parameter. Newton’s laws stated in terms of “force” naturally suggested a simple mathematical framework, which describes the particle motion along a definite trajectory. Existence of “force” is assumed, postulated, entered by the mind to accommodate the non-rectilinear motion. Force could be exerted by a body on the other separated by a distance, i.e., action at distance, which is a nonlocal effect.

Newton’s laws are supposed to hold in the inertial frames defined to be the frames moving with constant velocities with respect to each other. Newton’s definition of an inertial frame was the one that moves with a constant velocity with respect to a fixed, absolute frame. However, existence of an absolute frame of reference is excluded by Galileo’s principle of relativity, i.e., the laws of physics are the same in all inertial frames and thus no experiment can distinguish one inertial frame from the other. No supporting evidence has ever been found in favor of an absolute frame. Space and time in the Galileo-Newton formulation are considered independent entities and the rigid intervals in space and time remain the same in all inertial frames. The relativity principle supplemented by the Galilean transformations relates the observations in different inertial frames, which can differ, even though the laws remain the same. According to the transformation rules, time in all frames remains the same. Further, the velocity of a particle in two frames differs by the relative velocity of the frames, e.g., a train moving at 100 km/hr with respect to the ground, moves at 70 km/hr with respect to a car chasing it at 30 km/hr. The concept of force and the relativity principle had a profound impact on our perceptions of reality altering the earlier ones drastically.

The action principle: A particle takes the path such that the action along other paths arbitrarily close to this path is almost equal to the action along the path taken, that replaces Newton’s laws, introduces a metaphysical concept in the formulation of motion, and the force is deducible from a potential. Often this assigns the minimum action trajectory to the particle. In view of this, nature emerged as an economical and stable organization.

Nature and magnitude of the forces were determined by the response of particles acted upon by them, which were gravitational, electric and magnetic. Later, the concept of force was replaced by that of the field. A body exerting a force was assumed to create a field, and the field in the immediate vicinity of the affected body impacts upon it. This eliminated the concept of “action at a distance,” making the formulation into a local theory. Electric and magnetic fields were unified in one, electromagnetic, following Ampere’s and Faraday’s observations culminating in the corresponding laws, which developed into Maxwell’s equations with consequent electromagnetic waves, and a startling revelation that light is a form of the electromagnetic waves, altering the perception of light, which was considered to be made of tiny corpuscles.

Waves, e.g., on the surface of water, arose out of the oscillations of particles of the medium supporting a wave. Light and thus, electromagnetic waves can travel in vacuum upsetting this view. Ether, the “substance that the vacuum was made of,” was postulated to maintain the prevailing concept of the waves. However, the efforts at detecting something like ether only revealed its nonexistence. Experiments also revealed that the velocity of light in vacuum is the same in all inertial frames. This observation invalidated the rules of Galilean transformations; they were replaced by the Lorentz transformation rules, which differ from the Galilean by a multiplicative factor for space, and a parallel rule for time transformation was developed. These developments presented an altered picture of reality: Space-time together form a 4-dimensional spacetime continuum, the Minkowski manifold. Lorentz contractions also imply that the moving rods shrink and moving clocks slow down as well as the result that mass and energy are equivalent. Non-relativistic formulation of Galileo and Newton was extended to the relativistic in a straightforward manner with time assuming the role of fourth dimension and the proper time replacing time as parameter.

In all of the above, physical reality was perceived in mechanical terms. A significant adjustment to this view resulted from Einstein’s formulation of gravitation in terms of the geometrical concepts spurred by Galileo’s observation of equivalence of the inertial and gravitational masses. The concept of force was replaced by the geometrical structure of spacetime, which was viewed as Riemannian, in general. A particle travels along natural pathways in this Riemannian spacetime. Earlier view that the space and time intervals are rigid under parallel transport, with an interval defined with respect to the structure of space, was maintained in this formulation.

In spite of periodic major adjustments to the descriptions, reality was considered objective and observer’s role was just to determine what it was, i.e., the approach that emerged out of these formulations was realist-ic. A particle was assigned a unique trajectory and a wave was described in terms of the oscillations of particles. Even though this description did not apply to the electromagnetic waves, the shortcoming was glossed over and they were studied in terms of the attributes pertaining to the waves in general. Physical observables, e.g., the position and momentum of a particle; and the intensity of a wave; are exactly determined; i,e, the formulation is deterministic. It is also local in space and time, spacetime, i.e., a physical system is affected only by the happenings in its immediate vicinity and it does not remember its history.

Although it was not fully appreciated at the time, realism, determinism and locality suffered a severe setback with Weyl’s suggestion that the rigid intervals in spacetime change as they are transported along trajectories. This change was quantified in terms of the gauge transformations. A component of this formulation is that a gauge, measure of the length can be assigned at points in spacetime essentially at will, introducing subjectivity. Weyl’s views were objected to mainly for their positivist-ic underpinnings. The theory also did not succeed in unifying the gravitational and electromagnetic fields, which was the initial intent of the formulation. However, Weyl’s these concepts went on to have a profound impact later. As an alternative to Weyl’s geometry, Kaluza-Klein developed a 5-dimensional Riemannian geometry. In view of these developments, physical reality came to be defined, perceived, in terms of the geometrical concepts.

Quantum Theories

Quantum mechanics is magic.
Magic is to enjoy; not to understand.….
(So)
Shut up and Calculate …. —  David Mermin

Quantization of the electromagnetic waves was introduced in Planck’s description of the black body radiation, but was not considered genuine. However, the concept took hold as it successfully described specific heat of solids and the photoelectric effect also; and coupled with Rutherford’s model of atom, spurred the formulation of Bohr`s quantized orbits. Dual nature of light, particle and wave, was already well established, although not understood. De Broglie proposed the dual nature for material bodies also. Arguments used were scientifically tenuous inspiring its dubbing as the French Comedy. However, observations in the double slit experiment confirmed this dual nature of all such entities. Each individual entity registers as a particle but a large number of them collected together, exhibit wave characteristics of each entity; if “which path” information is available, an entity shows its particle nature but if it is not, then its wave character is revealed.

In an empirical formulation of quantum mechanics that followed, the state of a system is represented by a wavefunction, which is a superposition of many states; however, a measurement finds it in one pure state, with an associated probability. Also, certain pairs of physical observables, e.g., the position and momentum of a particle, cannot be described exactly simultaneously; such measurements abide by the Heisenberg’s uncertainty principle. Structure of spacetime is Galilean in the non-relativistic quantum mechanics and Minkowskian, in the relativistic.

Multiple path formulations of quantum mechanics were developed in time. The first one among them, Feynman’s, assumes that a particle observed in a spacetime region reached there from everywhere along all paths and the paths possess wave-like coherence, rendering the wavefunction a superposition of infinitely many wave amplitudes. Wheeler developed a parallel formulation in terms of Weyl’s gauge transformations. Present author carried this concept forward, which resulted in elimination of wave-particle duality; instead the entities were taken to be purely particles, with their paths determined by the action-gauge principle, which is an extension of the classical action principle. Assigned gauge appears explicitly in the formulation. According to this principle, a particle follows a path such that the length of a vector transported along it has the same length at its terminal point as at the original point, with the classical action being the generator of Weyl’s essential gauge and thus, of the change. These paths, termed the physical paths, are constituted of the elementals defined to be the smallest monotonic physical trajectories; i.e., a periodic structure with period equal to one elemental is defined on the physical paths, which imparts wave-like coherence to the physical paths. Each physical trajectory is a union of randomly selected elementals. Consequent to this formulation, macroscopic path of a particle was shown to be almost classical and at a microscopic scale, a particle shows wave-like behavior as the allowed set of elementals coupled with randomness conforms to the Huygens-like structure for waves.

Although a significant improvement over the original empirical formulation of quantum mechanics, the essential content of Feynman’s path integral formulation is the same. The multiple path formulation in its final form, which is formally quite similar to Feynman’s, radically altered the content of quantum mechanics. Firstly; this formulation is based on purely particle picture eliminating the concept of wave-particle duality. The wavefunction appears as an aggregate in the framework of the Weyl geometry. Then, the wavefunction appears as a representative of a class of systems, being the same for all assigned gauges, which is the same as the one with assigned gauge being equal to one everywhere, not the representative of a specific physical system. To determine the wavefunction for a specific state, it is required that the assigned gauge be introduced by additional considerations. In this formulation, the assigned gauge is chosen by the observer and thus physical reality is determined by “nature” and the observer interactively. An experimental arrangement determines the gauge values in the underlying manifold impacting upon its geometrical structure. Perceived physical reality is defined by the totality of geometry of the underlying manifold. Further, in Feynman’s formulation, wave-like coherence on the trajectories and Born’s probability rule are externally assumed, which are deduced in the action-gauge formulation within the confines of the action-gauge principle.

Although described correctly by quantum mechanics, some of the observations in experiments involving microscopic systems are counterintuitive and anomalous in terms of the classical thinking. The first one, behavior of the entities in the double slit experiment, discussed in article #7, which Feynman considered the only anomaly, spurred the development of quantum mechanics including its multiple path formulations. Some of the others are discussed in the article #8, namely: Passage around a hole in space; Passage through a barrier; Entanglement; and Schrödinger’s cat. In case of the passage around a hole, two beams of electrons from a common source, neither passing through a magnetic field but enclosing it, are affected by the field when observed on the other side. In case of a barrier, a particle can penetrate a classically forbidden barrier and reflected from a penetrable one. Behavior of the entangled particles is peculiar in that a measurement on one particle affects the other separated from it. In case of Schrödinger’s cat, it is both dead and alive until observed.

As mentioned above, in the quantum mechanical formulation, the state function is expressed as a superposition of many “pure” states but a measurement reveals it in one of the states with a probability attached to it. Earliest “explanation” of this occurrence was the Copenhagen interpretation: The wavefunction collapses into one of the states by an act of observation. However, this explanation reveals its weakness in various observations, and mechanism of collapse is not clear. Also, the collapse can be erased. This interpretation is most widely believed most of the time although it has been called meaningless. Pilot wave interpretation views an entity as a particle following a definite trajectory but it is determined by a wave and the trajectories are hidden. This interpretation has not taken a strong hold, although it has its fan following. According to the relative state, equivalently the many worlds, many histories, interpretation, with each act of observation, the state of a system is understood to branch off in two non-communicating “worlds,” each containing one of the possible observable realities. In case of Schrödinger’s cat, as the cat is observed, it is dead in one of the worlds and alive in the other. The observation depends on which world the observer enters. This interpretation has been the most favorite one at times. Many worlds and the Copenhagen interpretations have been the major competitors most of the times. However, “scientific truth” is not to be determined by the popularity contests but by logical deductions.

In the action gauge formulation, an experimental arrangement determines the assigned gauge distribution in the underlying manifold. Various phenomena are described in terms of the physical trajectories, using their properties, which depend on the assigned gauge. Consequently, the observer participates in defining the perceived physical reality. In the framework of this formulation, the above anomalies are explained more clearly from the properties of the physical trajectories that are part of the formulation in contradistinction with the other interpretations, which use some concepts external to the empirical and Feynman’s path-integral formulations of quantum mechanics. As mentioned above, this formulation expands the original content of quantum mechanics as a consequence of an inclusion of the assigned gauge into the formulation, which shows also in the explanations of the anomalies,

Space-Time and Fields

Understanding of the structure of space-time progressed from the absolute independent rigid entities in the non-relativistic classical thinking to a composite 4-dimensional Minkowskian spacetime in the relativistic formulation. Quantum concept did not alter these pictures. Weyl’s radical departure from these concepts by abandoning the rigidity of intervals under parallel transport with length conforming to the underlying geometry of spacetime, has been incorporated in the path-integral formulation of quantum mechanics with a significant impact. Some recent investigations in the framework of multiple path formulations have affected the earlier understanding of space-time, together with a novel formulation of quantum fields. These concepts are discussed in brief below referring specifically to the electromagnetic fields for simplicity, which has all the essential ingredients for the treatment of the other gauge fields.

As discussed before, classical electromagnetic field is a collection of infinitely many oscillators. The field is quantized by replacing the classical oscillators by the quantum oscillators. This formulation has been very successful in describing the quantum electromagnetic phenomena, except for one problem as follows. Minimum energy of a classical oscillator is zero but minimum energy of a quantum oscillator is strictly positive implying that vacuum has infinite energy. Experiments in laboratories do not determine the absolute value of energy; they determine the energy differences, which remain the same as if the vacuum energy was zero or anything else. However, the absolute value of energy cannot be ignored as energy is equivalent to mass and mass produces proportionate gravitational field implying that vacuum produces infinite gravitational field making the existence of universe impossible as there is some vacuum in the universe; in fact a great deal. Non-zero vacuum energy is also conceptually unsettling. Various phenomena have been successfully described by assuming infinite vacuum energy but one contradiction is sufficient to invalidate a theory. Therefore, quantum electromagnetism is a flawed theory.

Consider a trajectory in a free 3-Dimensional Euclidean manifold, which can be parameterized by its arclength, instead of an external parameter; i.e., coordinates of a moving body with respect to the distance travelled specifies the path completely. This enables one to determine the “velocities” and “accelerations” with respect to this parameter, arclength. Now, one can use the action principle to determine the classically prescribed path for a particle; the resulting trajectory is a straight line, regardless of its mass. Clearly, the distance travelled, i.e., the shortest distance between its end points, along a straight line is equal to its arclength, i.e., the particle travels with unit velocity. The magnitude of the velocity of light is equal to one in natural units. Thus, a particle travelling in 3-dimensional Euclidean space with arclength of its path identified with time is identified with a photon. Energy of the photon is arbitrary, i.e, the photons of all frequencies travel classically along straight lines with unit speed.

In the multiple path formulations, all physical trajectories passing through a point can facilitate the passage of a particle from the corresponding initial points to that point. Arclength of each trajectory at the point under consideration is different from the other, in general. Thus, arclegth acquires significance as a parameter independent of the trajectories. In article #4, just as the musings of a mathematician, this parameter was attached to the 3-dimensional Euclidean space as an added dimension and all 3-dimensional trajectories were described in this 4-dimensional manifold. This naturally yielded the light cone together with the 4-dimensional Minkowski manifold with the parameter arclength identified with time. The straight lines in 3-dimensional space are mapped on the surface of the cone and the curved ones in its interior. Exterior of the cone accommodates the trajectories with arclength less than the shortest distance between the points, i.e., the tachyons in the relativistic formulation. Coupled with the concept of a physical particle and the action principle, a mathematical musing becomes physically real, a photon as a classical particle, existing on the surface of the light cone. Quantum particle description of photon is now obtained from the multiple path formulation.

It should be clarified that the trajectories are mapped, not copied in the 4-dimensional structure. However, if two sets can be mapped into each other in a one to one manner both ways, then mathematically they are essentially equivalent.

Now consider the other, curved, trajectories which are mapped in the interior by decomposing it in its components: A straight line and its arclength. The same particle as considered above moving along a curved path has the corresponding trajectory in the interior of cone, which have velocities less than one. Now, the action principle applied to the Minkowski space constructed above assigns a trajectory to this particle, which has its curved counterpart in the Euclidean base. Some simple manipulations show that the particle acquires a rest mass equal to the photon energy. With the underlying space being Minkowskian, the results of special relativity are the natural outcome, without its usual assumptions.

Multiple path formulation yields a wavefunction. By translating the initial points along the trajectories, one can generate a one parameter family of wavefunctions, translation parameter being the defining parameter. When the translation is equal to one elemental on each trajectory, the wavefunction is again the same as the original one. Thus, this family is periodic with respect to the translation parameter. Since the translation is along the arclengths, it is identified with the parameter arclength. This together with the equation it satisfies, the periodic family is identified with the electromagnetic potential with the arclength parameter identified with time. The quantized Maxwell’s field equations are easily deduced from the equation for the potential and the photon is understood as the quantum of this field. The resulting vacuum energy in this formulation is zero, as it should be.

Repeating the above program in the Minkowski manifold with the massive particle, both generated in the process, one obtains the classical particle description, which is relativistic; quantum particle description; and its quantum field description; generating in the process a 5-dimensional Minkowskian manifold and a massive-like particle in it. The process continues ad infinitum.

Concluding Remarks

This article presents an outline of the modern theories of physics, which were covered in detail in the earlier articles of this series. In addition, a recently developed understanding of time together with that of the quantized fields is briefly described. In this new understanding, time is identified with arclength of the totality of curves in the Euclidean space rendering it a construct of the three dimensional Euclidean space, instead of an independent fundamental entity. Thus, the spacetime structure results from a natural logical mental construct out of the three-dimensional Euclidean space, to formulate our experiences in.

Answer to the question: How the particle came where it is spotted; would be: We do not know. All theories are conjectures, i.e., essentially they are postulates, assumptions. Based on such postulates, we deduce some results. In mathematics, deduced results are just what the assumptions imply; they do not have any more essential content than the original assumptions; process of deductions is just to delineate the contents of the assumptions. The action principle and all the other elements of the theory are basically our assumptions. Thus, a straight line path for a particle in a free 3-dimensional space and other similar results are the implications of our assumptions. Same is the case with the other theories. Thus, the above manifestations of the behavior of particles are the outcomes of certain assumptions that we use to develop suitable mathematical frameworks to describe our observations.

With the above background: If a particle is assumed to have followed a definite straight line path, it manifests itself as a classical photon; if it is assumed to follow a curved trajectory with an appropriate image defined by the action principle in the Minkowski space, it manifests itself as a massive relativistic particle. As discussed in article #7, if the particle is assumed to have come from somewhere out of everywhere along a path out of all, determined by a series of random selections of elementals, it conforms to its quantum mechanical description. Properties of the physical paths naturally describe the particles as the quantum of the electromagnetic field. Also, the observations are affected by the observer assigned gauge, imparting the role of a participant in determining the perceived physical reality. Repeat of the same program, describes the generated massive particle quantum mechanically and as the quantum of the corresponding field, generating parallel entities in yet another higher dimension, making this a never ending construction.

Yet again, the thought springs to mind: Perhaps our perception of physical reality is really a subjective manifestation of our ignorance of it, defined mostly by the constructs of mind.

I leave it to the reader to decide what our perception of physical reality is.