RELATIVISTIC QUANTUM FIELD THEORY
Early quantum mechanics was not consistent with special relativity theory. Pioneering physicists such as Paul Dirac, Werner Heisenberg, Wolfgang Pauli, Pascual Jordan, and Victor Weisskopf during the 1930s developed a relativistic version of quantum mechanics that was eventually called relativistic quantum field theory. Their initial theory quantized Maxwell’s classical field equations and was called quantum electrodynamics. That is, it made Maxwell’s classical electromagnetic field equations consistent with the concept that energy comes in quantum packages the size of Planck’s constant multiplied by the frequency of the electromagnetic waves.
QED soon ran into difficulties because the calculation of the energy of the electron interacting with itself, called the self-energy, was infinite, which made calculations in QED meaningless. It was not until the advent of renormalization theory during the late 1940s and early 1950s—first proposed by Hendrik Kramers and later developed by Hans Bethe, Richard Feynman, Julian Schwinger, Sin-Itiro Tomonaga, and Freeman Dyson—that a method for removing the infinities in quantum field theory calculations was found.
Back in 1934, Enrico Fermi had developed a theory of the weak force, or beta decay. His theory explained the radioactive decays of nuclei by a simple formula involving effectively just the product of the fields of the leptons (e.g., the electron), including the one neutrino that was expected to exist at the time, and the proton and neutron fields. The constant determining the strength of the weak force was eventually named the Fermi coupling constant (GF), and it described the strength of the interactions of the particles and fields. This theory explained accurately at low energies such transmutations of matter as a neutron decaying into a proton plus an electron and an antineutrino (Figure 1.7). (The lifetime of the free neutron is about 11 minutes.)
As studies of Fermi’s beta decay theory continued, physicists recognized that it could not be fitted into the paradigm of a renormalizable quantum field theory, one with calculations of decay reactions of particles that do not produce meaningless infinities. A technical reason for this is that Fermi’s coupling constant GF is not a dimensionless number, but in certain units has dimensions of inverse mass squared. What this means in practice is that calculations of radioactive decay become infinite, and indeed these infinities cannot be removed by the technique of renormalization, as was done successfully for QED, in which the coupling constant, called the fine-structure constant, is a dimensionless number. QED’s fine-structure constant, alpha (α), which is equal to one divided by 137 and is dimensionless, does lead to a renormalizable quantum field theory, which allows for the calculation of finite quantities such as a scattering cross-section.
Figure 1.7 Feynman diagram for four-fermion interaction beta decay (a neutron decaying into a proton, electron, and antineutrino)
SOURCE: quantumdiaries.org
Theorists also discovered that if you quantize the gravitational field—interpreting the gravitational force between particles as being carried by the massless graviton—then the theory of quantum gravity is not renormalizable, just as Fermi’s theory of weak interactions is not renormalizable. Newton’s coupling constant GN, which measures the strength of the gravitational force, is not a dimensionless number either, but, as for Fermi’s constant GF, it also has dimensions of inverse mass squared, thus rendering any quantum gravity calculations as meaningless infinite quantities.
One of the founders of quantum field theory, Paul Dirac, was never happy with renormalization theory. He did not feel that the cancelation of infinities or ignoring infinite quantities that are supposed to be mathematically small in perturbation theory calculations could be the final answer for quantum field theory. In theoretical physics, we are faced primarily with trying to solve equations that do not have exact solutions. Therefore, we must solve problems by expanding the equations in a mathematical series with terms that are multiplied by successive powers of a small, dimensionless constant. The idea is that when you add up all the infinite terms in the series, you will obtain the exact solution of the equations. The perturbation series only works for dimensionless constants that are less than unity (one). Otherwise, the series will not converge to the exact solution. In particular, if we truncate the series at a certain term, we anticipate that the sum of the terms up to this point gives a good solution to the equations, which can then be compared with experiments.
Dirac expressed his disapproval of renormalization theory with its reliance on perturbation theory:
Most physicists are very satisfied with the situation. They say: “Quantum electrodynamics is a good theory and we do not have to worry about it anymore.” I must say that I am very dissatisfied with the situation, because this so-called “good theory” does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small—not neglecting it just because it is infinitely great and you do not want it!11
Richard Feynman, famous for his invention of Feynman diagrams for particle physics interactions and for his contributions to the development of QED, said in 1985:
The shell game that we play… is technically called “renormalization.” But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It’s surprising that the theory still hasn’t been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.12
Attempts have been made to put renormalization theory on a more rigorous mathematical basis. The published work by Nobel laureate Kenneth Wilson made a significant contribution to our understanding of renormalization theory.13 His work in condensed matter physics and relativistic quantum field theory further developed what was called renormalization group theory, which had been published originally in a seminal paper by Murray Gell-Mann and Francis Low in Physical Review in 1954.14 The work by Wilson made renormalization group theory more attractive to mathematicians and physicists, although the basic problem of infinite constants such as charge and mass was still present.
ENTER THE W PARTICLE
The electromagnetic force between two electrons or between a positron and an electron is produced by the charged particles exchanging massless photons. In 1938, Swedish physicist Oskar Klein proposed the equivalent of the photon for the weak force. This boson carrier particle came to be known as the intermediate vector boson. It is a spin-1 particle that was eventually called the W boson. In weak interactions, the W interacts with quarks and leptons, and the strength of this interaction is measured by a dimensionless coupling constant, g. Fermi’s coupling constant GF is proportional to the square of g divided by the square of the W mass. Making the W mass heavy enough makes the weak-interaction theory with the intermediate vector boson W reduce at low energies to Fermi’s original weak-interaction theory with Fermi’s coupling constant GF (Figure 1.8).
Because the weak interaction is much weaker than the electromagnetic interaction, the W particle must be heavy, physicists reasoned—between 50 GeV and 130 GeV. Because of the heavy mass of the W, in theory it would decay into other particles such as leptons and have a short lifetime. This means that the range of interaction of the W particle carrying the weak force is very short compared with the infinite range of interaction of the electromagnetic force carried by the massless photon.
Figure 1.8 Feynman diagram for neutron decay into proton, electron, and antineutrino, mediated by the W boson.
SOURCE: pfnicholls.com
However, it was soon found that the difficulty of obtaining finite calculations in weak-interaction theory was not much improved. The W boson turned out to have similar problems with mathematical infinities as those in Fermi’s theory. Quantum field theory calculations for the intermediate vector boson model also produced infinities. In addition, a s
erious failure of probabilities to add up to 100 percent, in perturbation theory calculations for the scattering of particles involving the W, occurred at an energy of about 1.7 TeV. This meant that the calculated probability of scattering of particles would exceed 100 percent, which of course was impossible.
Such was the situation with the weak-interaction force during the late 1950s and early 1960s. How could one make sense of calculations with weak interactions in quantum field theory? Many attempts were made to resolve this conundrum; but for one reason or another, all failed.
GAUGE THEORY AND THE W PARTICLE
Back in 1918, Hermann Weyl attempted to unify gravity and electromagnetism by generalizing the geometry of Einstein’s gravity theory. This geometry is Riemannian, invented by Bernhard Riemann during the 19th century, and describes a non-Euclidean curved space, such as the surface of a balloon or Einstein’s spacetime. In electromagnetism, the electromagnetic fields are described mathematically as vector fields. These vector fields can be compared in their direction and magnitude at different points arbitrarily distant from each other in spacetime.
Weyl introduced the idea of the “gauge” of a vector field in the geometry of spacetime, similar to the different “gauges” of railroad tracks. In Einstein’s spacetime, a locomotive can travel on any railroad track, whereas in Weyl’s more generalized spacetime this is not true. In his theory, Weyl was able to unify gravity with electromagnetism by generalizing the Riemannian geometry on which Einstein’s theory of gravity was based. In addition to having a metric tensor field, which in Riemannian geometry is used to determine the infinitesimal distance between two points in spacetime, Weyl included a vector field into the geometry. Together, these fields gave rise to both gravity and electromagnetism.
When a vector, a field with a direction in space, is transferred from one point to another distant point, Weyl claimed that it no longer maintained its integrity or its gauge, unlike the case in Einstein’s gravity. However, Einstein criticized Weyl’s ideas, claiming that nature did not work this way, and eventually Weyl abandoned his theory. The theory simply was not in accord with what you get when you measure the distance between two points in spacetime.
But Weyl did not give up on his idea of gauge in spacetime. In 1929, he attacked the problem again from a different angle. Maxwell’s equations of the electromagnetic field remained the same—that is, they are invariant under a mathematical transformation of what is known as the vector potential field. This invariance was called gauge invariance and had been discovered during the development of QED during the 1930s, soon after Dirac’s discovery of his wave equation for the electron. What Weyl discovered is that there is a fundamental connection between the electromagnetic field and the phase of Dirac’s wave function for the electron.15 More is said about gauge theory in Chapter 3.
As we recall, Oskar Klein predicted the existence of the W bosons. He published a paper16 in which he extended the higher dimensional spacetime unified theory of gravity and electromagnetism of Theodor Kaluza. Later, in 1938, Klein introduced into this theory a massive intermediate vector boson. This was the first appearance in particle physics of what we now call a nonabelian gauge field, representing a vector particle. The properties of this nonabelian gauge theory are also discussed in Chapter 3. It must be appreciated that, at that time, during the 1930s, experimentalists had never observed a massive boson with spin 1, which the theorists were predicting.
GENERALIZING MAXWELL’S EQUATIONS
In 1954, two physicists, Chen-Ning Yang and Robert Mills, published their important generalization of Maxwell’s equations of electromagnetism.17 They introduced the idea of isotopic spin space.18 Instead of the electric charge on particles, they concentrated on the isotopic spin charge of protons and neutrons. When this isotopic spin charge was conserved, then a proton and a neutron would appear to be the same spin-½ particle. When the proton and neutron are subject to strong interactions, they interact as if they are the same particle even though the proton is positively charged and the neutron has zero charge. The small difference in their masses resulting from the interaction of the proton with electromagnetism is not important when they interact through strong interactions. This leads to the isospin symmetry of strong interactions.
The idea of isotopic spin space was originally introduced by Werner Heisenberg in 1934. The fundamental paper that Yang and Mills published explored the possibility of the gauge invariance of the interactions of protons and neutrons under isotopic spin rotations in the isotopic spin space. Using the analogy of the gauge invariance of QED and Maxwell’s equations, they stated that all physical processes are invariant under local, spacetime-dependent, isotopic spin gauge transformations. Their paper, underappreciated at the time, would become a significant mathematical building block for the future of particle physics.
Ronald Shaw, a research student at Trinity College Cambridge, who was supervised by Abdus Salam, a fellow at St. John’s College Cambridge, discovered this mathematical theory of isotopic spin interacting with nucleons (protons and neutrons) independently; it was part of his PhD thesis. Shaw was two years ahead of me in his PhD research when I arrived at Trinity College as a student in 1954. During discussions with Shaw, I learned Salam’s opinion about his PhD project. Because, at the time, the physical significance of the isospin generalization of Maxwell’s equations was not appreciated, Salam told Shaw that he could not see any future significance for physics in this mathematical development. Unfortunately, Shaw took Salam’s advice and did not attempt to publish this idea from his thesis as a paper. If he had published a paper, he would have been as well-known for this significant contribution to physics as Yang and Mills.
Around 1955, Yang gave a talk at the Institute for Advanced Study in Princeton about the theory he had developed with Mills. Wolfgang Pauli was in the audience, and with his usual perspicacity, he asked a question about the presumed mass of the force-carrying W particle because, already at this time, issues had begun to arise about the nature of the hypothesized intermediate vector particle responsible for nuclear radioactive decay. Moreover, Pauli had also speculated on a generalization of Maxwell’s equations similar to the Yang–Mills generalization. However, he had not published his results because of his concern about the issue of the mass of the gauge boson that carries the force between isotopic spin charges. Gauge boson is a term for a boson associated with a field that satisfies a gauge symmetry (see Chapter 3). This isotopic spin gauge boson eventually became identified with the charged W intermediate vector boson.
Pauli asked Yang early during his presentation, “What about the mass of your gauge boson?” Yang was unable to provide a satisfactory answer. Later during the talk, Pauli again asked the question, “What about the mass of the gauge boson responsible for carrying the force between the nucleons?” Again, Yang was not able to answer the question, but he became so disturbed by this line of questioning that he refused to continue with his talk, and sat down. However, Robert Oppenheimer, who was chairing the seminar, urged Yang to continue his talk, which he did.19
As it turned out, Pauli foresaw all the problems that were going to arise in the theory of weak interactions associated with the charged intermediate vector boson because, unlike the photon carrying the electromagnetic force, this boson had a mass. The mass of the charged intermediate vector boson W caused serious difficulties for particle physicists trying to understand the nature of the weak interaction. Almost 60 years after Yang’s talk at the institute, how to fit the intermediate vector boson into the standard model of particle physics remains a mystery, particularly if the large hadron collider (LHC) does not confirm the existence of the Higgs particle. The introduction of the Higgs boson and field into weak-interaction theory during the mid 1960s—and later into the unified theory of electromagnetism and weak interactions, electroweak theory—provided a way to resolve the problem of the W boson mass. The so-called Higgs mechanism gave the W boson a mass and, in so doing, led to a finite, renormalizable, and se
lf-consistent theory of electroweak interactions.
Despite the experimental successes of QED, some theorists were still not happy with the theoretical foundations of quantum field theory and QED. Such great physicists as Lev Landau, Paul Dirac, and Gunnar Källen did not like having to use renormalization theory to save the calculations of QED from meaningless infinities. Källen and Landau published papers during the 1950s claiming that QED was fundamentally incorrect, because the “renormalization constants” that were used to make the mass and charge in the theory finite were intrinsically infinite, and therefore QED was fundamentally inconsistent as a physical theory.20
In their theory, Yang and Mills replaced the photon of electromagnetism with a triplet of bosons that carried the isotopic spin force, one electrically neutral and two oppositely charged. In contrast to Maxwell’s electromagnetic theory, in which photons do not interact, or couple, with themselves, the Yang–Mills gauge bosons do interact with each other. If these particles were made massless, the Yang–Mills theory was fully gauge invariant, which means that it could be renormalizable, avoiding the pesky infinities. In their paper, they were concerned that these intermediate vector particles would probably have to have a mass, although they did not explicitly include mass contributions from these bosons in their theory. They recognized that, when you included the protons and neutrons interacting with the triplet of intermediate vector bosons, then this would introduce a mass dimension into the theory, and the triplet of vector bosons would have to be massive. However Yang and Mills already recognized that putting in the gauge boson masses by hand would break gauge invariance in the theory. This would cause the theory to be beset with infinities, and, unlike QED, it would not be renormalizable.
Cracking the Particle Code of the Universe Page 4
