First in 1992 and then later in 1995, in collaboration with my student Darius Tatarski, I proposed what is now called the void cosmology.4,5 I pictured that we on earth, and in our galaxy, sit inside a large cosmic void. To describe this—which has also been called the Hubble bubble—I used an exact solution of Einstein’s inhomogeneous field equations of cosmology to describe a spherically symmetric bubble. Light coming in from galaxies and passing through this void would be influenced by a weaker gravitational field inside the bubble because of the relative lack of matter compared with the outside, which abounded in galaxies and clusters of galaxies. The spacetime inside the void would therefore be expanding faster than the spacetime outside, so that distant galaxies and supernovae would appear to be dimmer than would be expected in a standard homogeneous isotropic universe, such as in the Friedmann–Robertson–Walker standard model. The observer would interpret this dimming of light as evidence for the acceleration of the expansion of the universe. In the void cosmology, the void bubble is embedded in a much larger universe which is not globally accelerating. In contrast, the standard LambdaCDM model says that the expansion of the universe is accelerating and that this is the result of the repulsive nature of “dark energy.” In the void cosmology, there is no dark energy, Einstein’s cosmological constant is zero or negligible in size, and the expansion of the universe is not accelerating. Numerous papers6 have been published showing that the void cosmology can fit cosmological data as well as the standard LambdaCDM model, although this still remains a controversial issue today. In a recent paper, astronomers Ryan Keenan, Amy Barger, and Lennox Cowie claim they have observational evidence, from a determination of the average mass density of the local universe, for the existence of a very large void, which contains our Milky Way galaxy.7
The third alternative area on which I was working during the early 1990s was modified gravity. One of the biggest mysteries of modern physics and cosmology is dark matter—the need to strengthen the gravitational force with invisible matter in galaxies, clusters of galaxies, and large-scale cosmology to fit the observational data showing much stronger gravity than is expected in Newton’s and Einstein’s gravity theories. Dark-matter particles have not been detected so far, yet the standard model claims that about 25 percent of matter in the universe is in the form of invisible dark matter. (Almost 70 percent is dark energy; visible matter constitutes only about 5 percent of the total matter–energy budget of the universe in the standard model.) An alternative to dark matter is to modify Newton and Einstein’s gravity theories, to strengthen gravity in the presence of ordinary observed matter, which makes dark matter unnecessary. From 1995 onward, I published versions of a modified theory of gravity, which I came to call MOG (modified gravity). These were fully relativistic theories of gravity. My collaborators Joel Brownstein, Viktor Toth, and Sohrab Rahvar and I have succeeded in explaining most of the astrophysical and cosmological data currently available in the literature without any dark matter.
A NEW NONLOCAL QUANTUM FIELD THEORY
The problem with mathematical infinities in quantum field theory—our tool for understanding the basic relativistic interactions of matter—has never really gone away during the past seven decades of research in particle physics. Modern refinements of renormalization theory made quantum field theory a more acceptable tool for particle physicists who, unlike Dirac, were prepared to ignore the infinities still lurking in the undergrowth. A nonlocal quantum field theory can raise the issue of whether causality is maintained at the microscopic level of particle physics. Strict causality in quantum field theory demands that, within the tiny distances of particle physics, you will never have an effect before a cause. This requirement is an axiom in quantum field theory called microcausality. The other basic axioms of quantum field theory are unitarity and invariance of the theory under Lorentz transformations (special relativity).
Isaac Newton was not happy with his need to postulate an absolute space and time. Nor was he pleased that the gravity acting between bodies in his theory was instantaneous—the so-called “action at a distance.” As Newton himself complained:
That Gravity should be innate, inherent and essential to Matter, so that one Body may act upon another at a Distance through a Vacuum, without the mediation of anything else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity, that I believe no Man who has in philosophical Matters a competent Faculty of thinking, can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain Laws, but whether this Agent be material or immaterial, I have left to the Consideration of my Readers.8
Experiments have shown that quantum mechanics is a nonlocal theory. This means that in the case, for example, of quantum entanglement between particles such as photons, there has to be an instantaneous communication between the two quantum entangled particles, quite like Newton’s gravitational “action at a distance.” This violates our understanding of “local” classical physics, which means that distant entities in the universe do not influence us locally. However, in standard quantum mechanics, it is understood that special relativity is not violated, because no actual “information” is communicated between the entangled particles with a speed greater than light. Obviously, there is a contradiction. How can entangled particles communicate, but not communicate information? This is a conundrum that still faces quantum mechanics today.
In Einstein’s gravity theory, gravity propagates at a finite speed, the speed of light, and there is no instantaneous action at a distance between gravitating bodies. In a famous paper by Einstein, Boris Podolsky, and Nathan Rosen,9 they concluded that what Einstein called the “spooky” action at a distance in quantum mechanics meant that it was not a complete theory. In summary, nonlocality means that two objects, such as elementary particles separated in space, can interact without an intermediate agency, and they interact instantaneously. This is Newton’s action at a distance. Locality, on the other hand, means that two objects interact through a mediated influence such as a force with a finite speed. In special relativity, this speed of influence cannot exceed the speed of light. Classical physics is strictly local whereas the microscopic objects of quantum mechanics influence one another in a nonlocal manner.
The main raison d’être of nonlocal quantum field theory was to create a more natural theory that removed the troublesome infinities that occurred in quantum field theory calculations. The infinities could be traced to the assumption that the particles and fields must satisfy the axiom of locality and strict causality at the microscopic level of particle physics. This requirement of locality in quantum field theory would seem to contradict the well-known need for non-relativistic quantum mechanics to be a nonlocal theory. This situation brings into relief the tension between special relativity, quantum mechanics, and relativistic quantum field theory. In the standard interpretation of the phenomenon of quantum entanglement, spacetime has no influence whatsoever. The whole program of renormalization theory was tied to this idea that particles interact locally at a point in spacetime, which means they cannot affect one another over a region of space.
In Richard Feynman’s celebrated paper of 1949 in Physical Review,10 in which he first developed his approach to QED, he traced the origin of the troublesome infinities in the QED calculations to the axiom of microcausality. In technical terms, when he was confronted with the divergent (infinite) quantum radiative corrections in the form of Feynman loop integrals, he introduced into the calculations a “fudge factor” when integrating over the momentum of the virtual particles to infinity. This fudge factor effectively removed microcausality from the theory and made his quantum field theory calculations nonlocal. However, his choice of the “smearing function,” which removed the pointlike interactions between the photons and electrons, violated unitarity, gauge invariance, and the Lorentz invariance of special relativity!
In many attempts to develop a nonlocal quantum field theory over several decades
, including unpublished attempts by Feynman, the serious obstacles of destroying unitarity, gauge invariance, and special relativity prevented any significant progress. It began to seem that making nonlocal quantum field theory a convincing alternative to the standard local and microcausal quantum field theory was not going to work. The violation of gauge invariance, in particular, generated unphysical negative energy modes in the interaction of particles.
In 1989/1990, I began attempting to develop a consistent, nonlocal quantum field theory. Like many before me, including Dirac and Feynman, I was not happy with the infinities in quantum field theory calculations. At the time, I realized that the increasingly popular string theory was, in fact, a nonlocal theory. The string is pictured as a one-dimensional object, whereas a point is a zero-dimensional object, so that when the strings interact, they do not interact at a point. This gave me a clue about how to proceed with quantum field theory in four-dimensional spacetime. Somehow, if string theory was self-consistent, as was claimed, it could not violate gauge invariance, unitarity, or special relativity through the nonlocal interactions of particles. However, the violation of special relativity could only be avoided by situating string theory in more than three spatial dimensions.
In a paper published in 198911 based on research done while on sabbatical leave in 1987 in Paris, I hypothesized that there was an infinite “tower” of particles or fields in nature with increasing spin values, from zero all the way to infinity. I demonstrated that when these infinite towers, which I called superspin fields or superspin particles, interacted with one another, that interaction would be nonlocal. However, this formulation of quantum field theory in four-dimensional spacetime also ran into problems with potential unphysical negative energy modes, resulting from the violation of gauge invariance and unitarity. However, I believed that future work on this theory could possibly produce a physical, self-consistent model of particle physics and unification with gravity by introducing new symmetries for each higher particle spin, which would remove unphysical negative energy modes.
At the time, I chose not to extend the gauge symmetries of this model, and I set aside the idea of infinite spin towers of particles and fields to concentrate on a more standard approach to field theory. I made use of the seminal research on nonlocal field theory published by Russian physicist Gary Efimov during the 1960s.12 In 1990, I published a paper13 in which I proposed a method of solving the gauge invariance and unitarity problems in nonlocal quantum field theory by adding compensating field contributions produced by photons and electrons at each term in the perturbation expansion of the fields. These canceled out the problematic gauge-breaking field contributions that had impeded many earlier attempts. I applied this technique to QED and nonabelian gauge theory.
In this paper, I proposed a way to maintain gauge invariance in a nonlocal electroweak model. I included a scalar Higgs boson with the Higgs mechanism to break the electroweak symmetry. Promoting a nonlocal energy scale (or length scale) as a basic constant of nature, I showed that it was possible to solve the Higgs mass hierarchy problem and make the standard model including a Higgs boson a “natural” theory of particle physics. The paper concluded with a section devoted to a nonlocal quantum gravity that was finite and unitary to all orders of perturbation theory. This paper would prove to be of possible fundamental significance almost a quarter of a century later, when the possible existence of a Higgs-like boson was discovered at the LHC.
In 1990, I invited Richard Woodard from the University of Florida at Gainesville to come to Toronto to give lectures on his work on string theory. Richard was, at the time, an active string theorist, and because there was widespread interest in string theory in the physics community, I wanted to learn more about the subject from him. (Later, Richard abandoned string theory and claimed that the years he spent working on it wasted a significant part of his physics career.) During lunch at the University of Toronto graduate school cafeteria, Richard asked me what research I was currently pursuing. I told him about my superspin field theory ideas and the need for a nonlocal quantum field theory to avoid meaningless infinities in the calculations of particle scatterings. He became upset and told me these ideas were nonsensical and would lead nowhere. Nonetheless, despite Richard’s negative reaction, and my worries about it, I published my 1990 paper on nonlocal quantum field theory.
Several weeks after Richard’s visit, I received an e-mail message from him claiming that he had had an “epiphany” about my ideas on nonlocal quantum field theory. He understood that the nonlocal field theory offered a way to avoid infinities in QED, and suggested that we pursue these ideas together, in collaboration with my post doc Dan Evens and his graduate student Gary Kleppe. I was certainly willing to go along with this because Richard is one of the world’s experts in performing calculations in quantum field theory. Indeed, during the next few weeks, between Toronto and Gainesville, we wrote a long and detailed paper together in which we solved the problems of gauge invariance and unitarity in the new nonlocal quantum field theory while maintaining the Lorentz invariance of special relativity. The paper established that, by introducing the nonlocal behavior of the fields into QED in a special way, we could obtain a theory that was finite to all orders of perturbation theory. That is, the theory did not suffer from the usual infinities in calculations in quantum field theories. In the paper, we extended the theory to nonabelian gauge theories and quantum gravity. It was possible to view the nonlocal extension of quantum field theory as either a regularization scheme simply to make the calculations finite, or as a fundamental theory, depending on whether one kept finite a fundamental energy constant in the nonlocal field theory or let it become infinite at the end of the calculations. We published the paper in 1991.14
A special kind of mathematical function called an entire function enters into the nonlocal quantum field theory calculations. This function plays a special role in what is called complex variable theory, in that it has no singular points anywhere in the finite complex plane. There are only singularities occurring at infinity in the complex plane.15 The use of entire functions was a key element in my nonlocal quantum field theory, which I had recognized in my 1990 paper and in the paper published in 1991 with Woodard, Evens, and Kleppe.
APPLYING THE NONLOCAL QUANTUM FIELD THEORY TO PARTICLE PHYSICS
After our collaborative effort, I speculated that the nonlocal field theory could be applied to the electroweak unification of weak and electromagnetic interactions, which is normally a local field theory. This formulation would be without a Higgs particle and would generate both a finite electroweak theory and the masses of the elementary particles.
One of the compelling reasons for believing in the standard model of particle physics with its Higgs particle/field was spontaneous symmetry breaking—explaining how the universe went from a supposed state of massless particles to one in which all the elementary particles have distinctive masses, including the zero masses of the photon and gluon. As we recall, according to the widely accepted standard electroweak theory, the Higgs mechanism was responsible for generating the masses of the W and Z bosons and the quarks and leptons. I came up with the idea that the masses of the elementary particles could be generated by natural quantum field theory processes rather than by a Higgs boson. In technical terms, the particle masses could be generated by quantum field theory self-energy calculations without any additional scalar degree of freedom corresponding to the spontaneous symmetry breaking of the vacuum or the resulting prediction of a Higgs particle.
You should understand that, at this time, during the 1990s, it was far from clear that a Higgs boson would ever be detected at the high-energy accelerators, so there was a significant body of literature investigating how a theory could be developed without a Higgs boson. Moreover, one of the chief motivations for introducing a Higgs boson into the electroweak theory is to guarantee the theory is renormalizable and finite as a local quantum field theory. On the other hand, having a nonlocal quantum field theory th
at is intrinsically finite removes the need for a local, renormalizable quantum field theory.
In the standard electroweak theory, the mass of the Higgs boson is generated by its so-called self-interaction. In turn, the Higgs boson is responsible for generating the masses of the other standard-model elementary particles—namely, the W and Z bosons and the leptons and quarks. My proposal was that the masses of the elementary particles were all generated by their self-interactions, which meant there was no need for a Higgs boson. In quantum field theory, an elementary particle can interact with itself through fields, which produces a contribution to the mass of the particle. In a nonlocal quantum field theory, this self-energy is finite and it can be made to constitute the total mass of the particle. In a local quantum field theory, you can do this as well, but the self-energy calculation is divergent, or infinite, and has to be made finite by regularization and renormalization methods.16 Without a Higgs boson in my theory I, of course, had no Higgs mass hierarchy problem, which had plagued the standard model for decades. Another problem that my theory removed by not having a Higgs boson was the extremely large vacuum energy density produced by the Higgs field in the standard model, which seriously disagreed with observation when it was linked to Einstein’s cosmological constant.
It was clear to me that because I possessed a finite quantum field theory based on nonlocal interactions of particles in the standard model, I could accommodate a finite renormalization of the masses and charges of the particles. To establish a fundamental nonlocal field theory, it is necessary to introduce a fundamental length or energy scale. Such a constant would take on as fundamental a role as Planck’s constant in quantum mechanics. But, removing the Higgs boson from the theory left the unitarity problem of scattering amplitudes still to be resolved.
Cracking the Particle Code of the Universe Page 18
