Conceptual Barriers to a Unified Theory of Physics 1 Introduction

Conceptual barriers to a uniﬁed theory of physics

Dennis Crossley∗ Dept. of Physics, University of Wisconsin-Sheboygan, Sheboygan, WI 53081 August 31, 2012

Abstract The twin pillars of twentieth-century physics, quantum theory and general relativity, have conceptual errors in their foundations, which are at the heart of the repeated failures to combine these into a single unified theory of physics. The problem with quantum theory is related to the use of the point-particle model, and the problem with general relativity follows from a misinterpretation of the significance of the equivalence principle. Correcting these conceptual errors leads to a new model of matter called the space wave model which is outlined here. The new perspective gained by space wave theory also makes it clear that there are conceptual errors in the two main thrusts of twenty-first- century theoretical physics, string theory and loop quantum gravity. The string model is no more satisfactory than the point-particle model and the notion that space must be quantized is, frankly, nonsensical. In this paper I examine all of these conceptual errors and suggest how to correct them so that we can once again make progress toward a unified theory of physics.

1 Introduction – the challenge of the uniﬁcation program

The goal of theoretical physics is to construct a single unified description of fundamental particles and their interactions, but flaws in the foundations of physics have prevented physi- cists from achieving this goal. In this essay I examine conceptual errors that have led to this impasse and propose alternatives which break this impasse and let us once again move forward toward the goal of a unified theory of physics. The history of theoretical physics in the twentieth century is characterized by two successes and one frustrating failure. The successes are, of course, relativity theory[1] and quantum theory[2]. The failure is the inability to combine these two theories into a single unified theory of physics. Built upon the foundations of quantum theory, the standard model of particle physics gives us a model of the elementary particles and their interactions (ex- cluding gravity)[3]. Built upon the foundations of special relativity, Einstein’s general theory of relativity gives us a model of the gravitational interaction but says nothing about elementary particles[4]. These two theories use profoundly different concepts and methods and all attempts to construct a unified theory by forming a direct link between them have been unsuccessful. As we will see below, the difficulty lies deeper than merely an incompatibility of the methods of these two theories; the fundamental concepts on which these theories are constructed contain inconsistencies which preclude their successful unification.

∗[email protected]

1 Early efforts, by Einstein and others, to construct a unified theory of physics involved attempts to incorporate the electromagnetic field into the formalism of general relativity[5]. The goal of this early work was a unified field theory, built on the foundation of general relativity, describing the gravitational and electromagnetic interactions of matter in a unified mathematical formalism. With the discovery of the nuclear interactions, it became acknowl- edged that the goal of the unified field theory program was too limited. With no apparent way of incorporating the nuclear interactions into the geometric approach of general relativity, this approach was abandoned. Subsequent efforts have, instead, taken quantum field theory as the foundation. With the successes of gauge theories to describe both the electromagnetic and the nuclear interactions[6], some efforts have been made to incorporate gravity into this formalism by expanding the gauge group to include the gauge group of general relativity[7]. As we enter the twenty-first century, these unification efforts have morphed into the programs of string theory[8] and loop quantum gravity[9]. String theory attempts to adapt quantum field theory using one-dimensional strings instead of point particles, while loop quantum gravity attempts to incorporate quantum concepts directly into general relativity. But neither of these programs addresses the conceptual problems underlying quantum theory and relativity theory upon which they are built, and therefore have little more chance of success than their twentieth-century predecessors. Thus, all efforts to date to construct a unified theory of physics have taken general relativity and quantum field theory, and the conceptual foundations upon which they are built, and have attempted to combine them in some way, using one or the other as a foundation. But a critical examination of the conceptual foundations of these theories exposes a fundamental error in the conceptual foundations of each of them, and it becomes clear that neither general relativity nor quantum field theory nor their extensions in the form of string theory and loop quantum gravity are adequate foundations upon which to build a unified theory of all interactions. Both general relativity and quantum field theory appear to be “correct” theories in the sense that they both give correct predictions for the behavior of matter, but correct predictions, though necessary, is not a sufficient criterion for choosing a final theory. In the context of the unification program, we must also consider the generalizability of a theory, and on this criterion both general relativity and quantum field theory fail. We begin our search for a new foundation for fundamental physics with a critical analysis of the fundamental concepts upon which our current theories are built.

2 Barrier #1: the particle model

The particle concept[10] has proven to be extremely useful in virtually all areas of classical physics, regardless of the intrinsic size of the object being modeled. For example, objects as large as the Sun and the Earth are adequately modeled as point particles in Newtonian

2 gravity theory, and elementary particles can be modeled as point particles in Maxwell’s electromagnetic theory (as long as the interaction distances are not too small). The point particle concept reaches beyond its range of applicability, however, when it is applied to the microscopic domain to describe fundamental particles. It is the adherence to the particle model which has necessitated the introduction of probability into quantum mechanics. Likewise, the Heisenberg uncertainty principle, which makes sense in the wave context in terms of Fourier transforms, becomes probabilistic nonsense in the particle context. The point-particle concept cannot account for the existence of intrinsic properties like spin and gauge degrees of freedom, both of which require internal structure, nor can it account for the existence of particle interactions (forces)[11]. The wave nature of fundamental particles becomes apparent when interaction distances become small. Quantum theory accounts for the wave nature of fundamental particles by simply tacking it onto the particle concept, resulting in a rather schizophrenic juxtaposition known as wave/particle duality. While wave/particle duality has brought us closer to a complete understanding of fundamental particles, it has brought us only half-way. Quantum theorists, building upon the legacy of classical physics, kept the point particle concept central and attempted to graft the newly discovered wave properties of matter onto it. They were naturally led to ask the wrong question: How can a particle have wave-like properties? The answer is, of course, that it can’t, so elementary particles must be strange objects, they concluded, which sometimes behave like particles and sometimes like waves. This dominance of the particle concept on our thinking has been a subtle but insurmountable barrier to the unification of theoretical physics. To get beyond barrier #1, we need to abandon the notion that the particle concept is fundamental and instead embrace the wave nature. There are strong reasons for believing that the wave properties of matter are more fundamental than the particle properties. This leads to a reversal of our logic. If we say that waves exist and that somehow (not so mysteriously) they have particle properties associated with them, the whole situation becomes much clearer. In the language of wave-particle duality we are then led to ask the right question: How can a wave have particle-like properties? This rephrasing may seem simple, but the consequences of it are profound since the question now has an answer: Waves can have particle-like properties by being localized in space. Such localized waves are commonly referred to as solitons and are familiar in a variety of contexts. This soliton model contains the key to a new approach to understanding fundamental particles and their interactions, but by itself is not the complete answer. Many questions present themselves, perhaps the most important of which is: What is the “medium” in which these localized waves propagate? I offer a surprisingly simple answer to this question in section 4, but first we must adjust our approach to gravitation theory so that it can be made compatible with the wave model of fundamental particles.

3 3 Barrier #2: the equivalence principle

What’s wrong with curved space? The weak equivalence principle of general relativity states that gravity can be described either by a nonzero gravitational interaction in Euclidean space or by curved (Riemannian) space with the gravitational interaction identically zero. Gen- eral relativity is based on the second formulation. The first formulation, which I will refer to as “flat gravity” is, unfortunately, a theory that does not exist. I am not referring to Newtonian gravity, which is a quasi-static theory comparable to Coulomb’s law in electromagnetic theory, but rather to a theory which includes the finite propagation velocity of the gravitational interaction, comparable to Maxwell’s theory. Some work in this direction was being done around the same time that Einstein developed general relativity but was put aside with Einstein’s “successful” formulation of gravity in terms of geometry. Both formulations would be mathematically equivalent in the sense that they would give identical experimental predictions. A criterion other than experimental prediction is necessary to choose between these two formulations. At present, the choice is based on aesthetic arguments, but I propose that the choice should instead be be based on which formulation leads more directly to the unification of gravity with the other fundamental interactions. Why is there a problem with describing gravity in terms of curved space? We ask the simple question, “Is general relativity a correct theory?” and the best answer is “Yes, but...”. General relativity does indeed give correct experimental predictions on the macroscopic scale of planets and stars, but it is universally recognized that it does not give us a microscopic model of gravity at the atomic level. So at the very least, general relativity is not a complete theory of gravity. But an even stronger claim can be made, that it is the formulation in terms of curved space which has imposed an insurmountable barrier to the unification of gravity with the other forces. We need to reexamine this formulation and ask ourselves if it really is the best one. It is important to recognize that general relativity, although a “correct” theory of gravity, is not the unique theory of gravity. In fact, this is implicit in Einstein’s discovery of the equivalence principle which led to his development of general relativity. His recollection of this event is interesting enough to justify a quote: “I was sitting in a chair in the patent office at Bern when all of a sudden a thought occurred to me: ‘If a person falls freely he will not feel his own weight.’ I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.”[4] This led Einstein to the notion of equivalence between the gravitational force and curved space. But “equivalence” really means equivalence. In other words, we can conceive of two different theories of gravity which are equivalent and related by a coordinate transformation. Hans Reichenbach has explained this best[12]. Mathematics proves that every geometry of the Riemannian kind can be mapped upon another one of the same kind. We obtain a statement about physical reality only if in addition to the geometry G of the space its universal force field F is specified. Only the combination of G + F is a testable statement. It seems that the coordinate definition F = 0 is simpler because then the expression G + F

4 reduces to G. But even this result is not essential, since in this case simplicity is not a criterion for truth. So what are appropriate criteria for choosing the “best” formulation of gravity? We consider four criteria:

1. Experimental prediction - of no use here, equivalent theories give the same predictions.

2. Aesthetics - useful for making tentative choices, but not a test of truth.

3. Physical insight - neither passes this test: “Flat gravity” does not tell us how matter produces the gravitational interaction; general relativity does not tell us how matter curves space.

4. Generalizability/uniﬁability - general relativity has repeatedly failed this test; “ﬂat gravity”...?

We are at a stalemate. Criteria 1 and 2 do not apply and we are left with the puzzle of how to apply criteria 3 and/or 4. How can we make progress toward a microscopic model of gravity and/or uniﬁcation of gravity with the other interactions? Why has progress been so diﬃcult? The problem lies not only with the choice of Riemannian geometry to describe gravity, but also with the point-particle model of elementary particles.

4 Beyond the barriers – three-dimensional space as a dynamic continuum

To understand what an elementary particle is, we first need to understand what light is. In spite of all that we have learned about light, it remains one of nature’s mysteries. Its dual nature as both wave and particle is well known, but even before addressing this mystery, consider the better-known wave aspect of light. It is often said that light does not need a material medium, that it is merely a wave in the electromagnetic field, but this is utter nonsense. If we suspend for the moment our prejudices about what we think we know about light, we can follow a simple chain of logic which leads to a new way to view light. Light is a physical wave (we measure its wave properties in the laboratory). A physical wave requires a physical medium (you can’t have a wave without something waving). Light propagates through empty space (nothing there but space itself). The simple conclusion is that space is the medium through which light propagates. In other words, light is a wave in the geometry of space. In this context it is natural to identify the speed of light c as the intrinsic wave speed of space (as a medium). Furthermore, the significance of electromagnetic and gravitational interactions both propagating at the speed of light becomes apparent. The common speed of propagation implies a common mode of wave motion, which cries out for space wave formulations of the electromagnetic and gravitational interactions as well.

5 Even more subtle, however, is the connection to matter. The cosmic speed limit of c for the motion of matter is widely recognized, but no one has ever given a physical reason for why this speed limit exists. Special relativity justifies this claim only in terms of mathematical consistency, showing that certain physical quantities approach infinity as the speed of an object approaches the speed of light, but does not explain why this happens. This common speed for both matter and light suggests that they are two aspects of the same phenomenon, both modes of wave motion in the geometry of space. Indeed, if this simple conclusion is true, then we should expect to see transformations from one mode to the other, from light to matter and vice versa; and of course we do in the phenomena of particle creation (such as γ → e+e−) and particle-antiparticle annihilation. The wave-like behavior of matter as described by quantum mechanics is a further indication that we are on the right path. Although the relationship between the quantum- mechanical wave function and the space waves being proposed here is not simple and needs to be worked out, the simple fact that electrons and other particles have wave-like properties is a strong indication that these “particles” are waves. The space-wave model of elementary particles offers a natural explanation of the mystery of wave-particle duality[13]. Summarizing our simple chain of logic, starting with the hypothesis that light is a wave in the geometry of space, and observing the clues that (i) light and matter both demonstrate wave properties, (ii) light and matter share a common speed c, and (iii) light can be trans- formed into localized lumps of matter, leads us to the conclusion that light and matter are two forms of the “same stuff.” Our hypothesis on light then leads to the important hypothesis on the nature of matter, that elementary particles are localized waves in the geometry of space. The fundamental units of matter have a purely wave nature. They acquire their particle properties by being localized in space. In other words, the fundamental units of matter are solitons. This simple concept is the foundation for a new unified theory of fundamental particles and their interactions which I call the space wave theory of matter. The fundamental hypotheses of this theory are:

1. Three-dimensional space is a dynamic continuum on the microscopic scale.

2. The allowed motions of three-dimensional space are those of propagating displacement waves.

3. These displacement waves can, as a result of nonlinear self-interactions, propagate around small closed paths. We identify these localized soliton-like waves as elementary particles.

4. These localized displacement waves are, in turn, the sources of ripples which propagate outward in all directions to inﬁnity. These ripples we identify as the force ﬁelds generated by the elementary particles.

6 The fourth hypothesis takes us beyond the description of elementary particles to a simple explanation for their interactions. In the space-wave theory of matter the fundamental interactions are reduced to interference phenomena involving the (nonlinear) superposition of localized displacement waves (particles) and their outward propagating ripples (ﬁelds). The localized displacement waves are intrinsically nonlinear while their outward propagating ripples quickly reduce to linear waves as they leave their sources. The fundamental interactions can thus be categorized as follows:

1. The superposition of two sets of ripples involves the familiar phenomenon of linear superposition in which two linear waves pass through each other unimpeded. In the context of force ﬁelds, the electromagnetic and gravitational interactions pass through each other unchanged, and the interaction produced by multiple sources is the linear sum of the interactions of the individual sources.

2. The superposition of a (nonlinear) localized displacement wave with the (linear) ripples from another source produces an interference phenomenon which shifts the path of the localized displacement wave causing it to drift from its original position. This is the essence of the electromagnetic and gravitational interactions. These interactions are inﬁnite in range because the ripples propagate out to inﬁnity.

3. The overlapping of two localized displacement waves causes a much stronger interference phenomenon in the region where both waves are nonlinear. This nonlinear-nonlinear interference is identiﬁed as the nuclear interactions. (The weak nuclear interaction also involves a topology change in the path of one or more of the localized displacement waves which manifests itself as parity non-conservation.) The nuclear interactions are short-range because the regions of nonlinear displacement are conﬁned to the immediate vicinities of the localized displacement waves.

The argument I find most convincing in support of this ripple model of particle interactions goes like this: In quantum field theory, particle interactions are described in terms of exchange particles. The concept of exchange particles may work for understanding the results of particle physics experiments in which a single exchange particle is transferred between two other particles, but extending this to a description of a long-range interaction like the electromagnetic force stretches the imagination beyond any reasonable limit. If two particles interact via the electric force by exchanging virtual photons, then each individual charged particle must be constantly emitting an infinite number of virtual photons to account for the infinite range of the electric force. So much simpler is the idea of a rotating displacement wave in three-dimensional space sending out ripples in all directions, propagating outward to infinity, just as surely as twirling a twig in a pond sends out ripples that cover the pond as they travel to distant shores. So the choice is between an infinite number of virtual particles or a single ripple field.

7 5 Barrier #3: the string model

The space wave model outlined in the previous section sheds light on the current efforts to develop string theory and helps us identify flaws in the conceptual foundations of this theory. String theorists like to point out that instead of the large number of “fundamental” particles of the standard model, string theory posits that all particles are just different modes of a single fundamental string. But what is this string? What is it made of? It is clearly not a material substance like a “strand of spaghetti,” for this would have to be a skinny three- dimensional object, and we could ask about the pieces of matter that make up this strand. A somewhat better answer that is often given is that a string is a “vibrating strand of energy.” But this replaces one unknown with another, because we really don’t know what energy is, as Feynman has so clearly explained[14]. So we are left with this ill-defined notion of a vibrating one-dimensional object. But the space wave model doesn’t require strings any more than it requires point particles. The vibrations that string theorists talk about are, unbeknownst to them, vibrations in three-dimensional space, and the one-dimensionality of strings is identified with the one- dimensional path through space that the circulating space waves follow. In other words, the one-dimensional string is really a mathematical object representing the path of a physical wave in three-dimensional space. What’s more, the hidden spatial dimensions that are essential to string theory[15] are exposed by space wave theory to not be spatial dimensions at all. They are dimensions in a mathematical solution space containing the allowed solitonic motions of space waves. In terms of spatial dimensions, there are only three, and as localized space waves follow their one-dimensional paths they form various loops with various topologies. The simplest of these, representing the electron, is easiest to visualize. It is simply a circular path in space, exhibiting the familiar U(1) symmetry of electromagnetism. The paths followed by waves representing mesons and baryons are more complicated and not as easy to visualize, but we know the symmetries that these paths must have, SU(2) and SU(3), to match the standard model. The important point is that the hidden dimensions of string theory (and the internal degrees of freedom of the standard model) are not curled up physical dimensions of space, but are rather dimensions in the mathematical solution space of the motions of three-dimensional space waves.

6 Barrier #4: quantized spacetime

The loop quantum gravity program is perhaps even more speculative than the string theory program, and here too the space wave model sheds light on the conceptual foundations and ﬁnds them wanting. First of all, loop quantum gravity is usually presented as an attempt to quantize general relativity, but as discussed in section 3, general relativity is not the correct foundation upon which to build a new theory. But even more fundamental is the notion at the heart of loop quantum gravity that

8 space at the microscopic level must be quantized, that there must be a smallest quantum of space. (A similar argument is also made for time, but I will focus on space here.) But this follows from a naive application of the concept of quantization – the idea that everything must be quantized. In fact, quantization is less of a mystery than is usually thought, and it always follows from the imposition of boundary conditions in periodic motion. This is why an electron bound in an atom can have only discrete energies but an electron traveling in free space can have a continuum of energies. The same is true of a photon emitted from an atom compared to a photon in free space. The quantization of fundamental properties such as mass and charge in elementary particles only seems more mysterious because of the use of the point-particle model as described in section 2 because the boundary conditions are hidden from view by the point particle model, but here too the space wave model easily accounts for the quantization of these particle properties in terms of periodic boundary conditions of the circulating waves which represent the fundamental particles. Coming back to the question of space, it is the circulating waves in space that are subject to boundary conditions and the quantization of particle properties is completely consistent with space being a true continuum. It is therefore meaningless to say that space must be quantized or to argue that there must be a minimum quantum of space. The entire loop quantum gravity program is thus built on a faulty foundation.

7 Conclusion – a glimpse of the future

Point particles and strings need to be replaced with waves in the three-dimensional continuum of flat Euclidean space. This is the basis for a new approach to fundamental physics which I call the space wave theory of matter and its interactions. The concepts and pictures of the space theory of matter are so rich that it is impossible to discuss all aspects of it in this short paper. There are many more properties of elementary particles to be accounted for by the space theory of matter such as mass, intrinsic spin, and parity, just to mention a few. It is my belief that these also have natural interpretations in terms of the displacement wave model of elementary particles. Indeed, a truly unified theory of physics must account for all aspects of physics, a rather daunting task. Here my goal has been more modest, to present enough of the fundamental concepts of the space theory of matter to persuade the reader that they offer better alternatives to the foundational concepts that have dominated physics through the twentieth century and into the twenty-first century. Space wave theory lays a new foundation for the unification of fundamental physics that is both simple and comprehensive. In the search for the smallest building blocks of nature, the space wave theory of matter is the ultimate reduction. There are no particles! The fundamental building blocks of the universe are (1) three-dimensional space and (2) its allowed motions (propagating displacement waves). Everything else, including the existence of elementary particles and their interactions, all of the properties associated with elementary particles, and even their unusual quantum and relativistic behaviors, follow from these two modest building blocks.

9 References

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[4] Pais, Abraham, ‘Subtle is the Lord...’: The Science and the Life of Albert Einstein, Oxford University Press, New York, 1982.

[5] Vizgin, Vladimir P., Unified Field Theories in the first third of the 20th century, BirkhäuserVerlag, Switzerland, 1994.

[6] O’Raifeartaigh, Lochlainn, The Dawning of Gauge Theory, Princeton University Press, 1997.

[7] Sardanashvili, G. A., Gauge Gravitation Theory, World Scientiﬁc, Singapore, 1992.

[8] Peat, F. David, Superstrings and the Search for the Theory of Everything, Contemporary Books, Chicago, 1988.

[9] Rovelli, Carlo, Quantum Gravity, Cambridge University Press, Cambridge, UK, 2004.

[10] Falkenburg, Brigitte, Particle Metaphysics: A Critical Account of Subatomic Reality, Springer, New York 2007.

[11] Crossley, Dennis, “On the dynamic nature of charge quantization,” arXiv:1201.0747v1, 2012.

[12] Reichenbach, Hans, The Philosophy of Space and Time, Dover, 1958. (Originally pub- lished in 1927 as Philosophie der Raumzeit-Lehre.)

[13] Crossley, Dennis, “Solving the mystery of wave/particle duality–the road to a uniﬁed theory of physics,” http://fqxi.org/data/essay-contest-ﬁles/Crossley Essay.pdf, 2009.

[14] Feynman, Richard P., Robert B. Leighton and Matthew Sands, Feynman Lectures on Physics, Addison-Wesley, Boston, 1977, chapter 4.

[15] Yau, Shing-Tung, and Steve Nadis, The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions, Basic Books, New York, 2010.