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Field Theories

Fields And Subatomic Particles

In 1905, Albert Einstein (1879–1955) disposed of the concept of the mechanical ether. Electromagnetic fields propagated in vacuo with the speed of light in all inertial frames. His special theory of relativity showed that the electric and magnetic fields could be represented by one (tensor) field, such that the effects that appear in a reference system as arising from a magnetic field appear in another system moving relative to the first as a combined electric field and magnetic field, and vice versa. The theory also engendered the conception of space and time as a four-dimensional continuum. To account for gravity as a field effect, Einstein formulated the general theory of relativity in 1915. Using tensor calculus and the non-Euclidean geometry of Bernhard Riemann (1826–1866), Einstein described gravitational fields as distortions of the space-time continuum. Meanwhile, following Maxwell, some physicists attempted to construe material particles as structures of fields, places where a field is concentrated. Einstein was among them, yet in 1905 he had proposed that light is composed of particles, "photons."

In the 1920s, Werner Heisenberg (1901–1976), Erwin Schrödinger (1887–1961), Max Born (1882–1970), and others formulated quantum mechanics. Its instrumental successes suggested the possibility of describing all phenomena in terms of "elementary particles," namely electrons, protons, and photons. The components of atoms were treated as objects with constant characteristics and whose lifetimes could be considered infinite. Protons and electrons were specified by their mass, spin, and by their electromagnetic properties such as charge and magnetic moment. Particles of any one kind were assumed to be indistinguishable, obeying characteristic statistics.

Quantum mechanics originally described nonrelativistic systems with a finite number of degrees of freedom. Attempts to extend the formalism to include interactions of charged particles with the electromagnetic field brought difficulties connected with the quantum representation of fields—that is, systems with an infinite number of degrees of freedom. In 1927, Paul Adrien Maurice Dirac (1902–1984) gave an account of the interaction, describing the electromagnetic field as an assembly of photons. For Dirac, particles were the fundamental substance. In contradistinction, Pascual Jordan (1902–1980) argued that fields were fundamental. Jordan described the electromagnetic field by operators that obeyed Maxwell's equations and satisfied certain commutation relations. Equivalently, he could exhibit the free electromagnetic field as a superposition of harmonic oscillators, whose dynamical variables satisfied quantum commutation rules. These commutation rules implied that in any small volume of space there would be fluctuations of the electric and magnetic fields even for the vacuum state, that is even for the state in which there were no photons present, and that the root mean square value of such fluctuations diverged as the volume element probed became infinitesimally small. Jordan advocated a unitary view of nature in which both matter and radiation were described by wave fields, with particles appearing as excitations of the fields.

The creation and annihilation of particles—first encountered in the description of the emission and absorption of photons by charged particles—was a novel feature of quantum field theory (QFT). Dirac's "hole theory," the relativistic quantum theory of electrons and positrons, allowed the possibility of the creation and annihilation of matter. Dirac had recognized that the (one-particle) equation he had devised in 1928 to describe relativistic spin 1/2 particles, besides possessing solutions of positive energy, also admitted negative energy solutions. Unable to avoid transitions to negative energy states, Dirac eventually postulated in 1931 that the vacuum be the state in which all the negative energy states were filled. The vacuum state corresponded to the lowest energy state of the theory, and the theory now dealt with an infinite number of particles. Dirac noted that a "hole," an unoccupied negative energy state in the filled sea, would correspond to "a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron" (p. 62). Physicists then found evidence that positrons exist.

Beta-decay was important in the field theoretic developments of the 1930s. The process wherein a radioactive nucleus emits an electron (-ray) had been studied extensively. In 1933, Enrico Fermi (1901–1954) indicated that the simplest model of a theory of -decay assumes that electrons do not exist in nuclei before -emission occurs, but acquire existence when emitted; in like manner as photons emitted from an atom during an electronic transition.

The discovery by James Chadwick (1891–1974) in 1932 of the neutron, a neutral particle of roughly the same mass as the proton, suggested that atomic nuclei are composed of protons and neutrons. The neutron facilitated the application of quantum mechanics to elucidate the structure of the nucleus. Heisenberg was the first to formulate a model of nuclear structure based on the interactions between the nucleons composing the nucleus. Nucleon was the generic name for the proton and the neutron, which aside from their differing electric charges were assumed to be identical in their nuclear interactions. Nuclear forces had to be of very short range, but strong. In 1935, Hideki Yukawa (1907–1981) proposed a field theoretic model of nuclear forces. The exchange of a meson mediated the force between neutrons and protons. In quantum electrodynamics (QED), the electromagnetic force between charged particles was conceptualized as the exchange of "virtual" photons. The massless photons implied that the range of electromagnetic forces is infinite. In Yukawa's theory, the exchanged quanta are massive. The association of interactions with exchanges of quanta is a feature of all quantum field theories.

QED, Fermi's theory of -decay, and Yukawa's theory of nuclear forces established the model upon which subsequent developments were based. It postulated impermanent particles to account for interactions, and assumed that relativistic QFT was the proper framework for representing processes at ever-smaller distances. Yet relativistic QFTs are beset by divergence difficulties manifested in perturbative calculations beyond the lowest order. Higher orders yield infinite results. These difficulties stemmed from a description in terms of local fields, a field defined at a sharp point in space-time, and the assumption that the interaction between fields is local (that is, occurs at localized points in space-time). Local interaction terms implied that photons will couple with (virtual) electron-positron pairs of arbitrarily high momenta, and electrons and positrons will couple with (virtual) photons of arbitrary high momenta, all giving rise to divergences. Proposals to overcome these problems failed. Heisenberg proposed a fundamental unit of length, to delineate the domain where the concept of fields and local interactions would apply. His S-matrix theory, developed in the early 1940s, viewed all experiments as scattering experiments. The system is prepared in a definite state, it evolves, and its final configuration is observed afterwards. The S-matrix is the operator that relates initial and final states. It facilitates computation of scattering cross-sections and other observable quantities. The success of nonrelativistic quantum mechanics in the 1920s had been predicated on the demand that only observable quantities enter in the formulation of the theory. Heisenberg reiterated that demand that only experimentally ascertainable quantities enter quantum field theoretical accounts. Since local field operators were not measurable, fundamental theories should find new modes of representation, such as the S-matrix.

During the 1930s, deviations from the predictions of the Dirac equation for the level structure of the hydrogen atom were observed experimentally. These deviations were measured accurately in molecular beam experiments by Willis Eugene Lamb, Jr. (b. 1913), Isidor Isaac Rabi (1898–1988), and their coworkers, and were reported in 1947. Hans Albrecht Bethe (b. 1906) thereafter showed that this deviation from the Dirac equation, the Lamb shift, was quantum electrodynamical in origin, and that it could be computed using an approach proposed by Hendrik Kramers (1894–1952) using the technique that subsequently was called "mass renormalization." Kramers's insight consisted in recognizing that the interaction between a charged particle and the electromagnetic field alters its inertial mass. The experimentally observed mass is to be identified with the sum of the charged particle's mechanical mass (the one that originally appears as a parameter in the Lagrangian or Hamiltonian formulation of the theory) and the inertial mass that arises from its interaction with the electromagnetic field.

Julian Schwinger (1918–1994) and Richard P. Feynman (1918–1988) showed that all the divergences in the low orders of perturbation theory could be eliminated by re-expressing the mass and charge parameters that appear in the original Lagrangian, or in equations of motions in which QED is formulated, in terms of the actually observed values of the mass and the charge of an electron—that is, by effecting "a mass and a charge renormalization." Feynman devised a technique for visualizing in diagrams the perturbative content of a QFT, such that for a given physical process the contribution of each diagram could be expressed readily. These diagrams furnished what Feynman called the "machinery" of the particular processes: the mechanism that explains why certain processes take place in particular systems, by the exchange of quanta. The renormalized QED accounted for the Lamb shift, the anomalous magnetic moment of the electron and the muon, the radiative corrections to the scattering of photons by electrons, pair production, and bremsstrahlung.

In 1948, Freeman Dyson (b. 1923) showed that such renormalizations sufficed to absorb all the divergences of the scattering matrix in QED to all orders of perturbation theory. Furthermore Dyson demonstrated that only for certain kinds of quantum field theories is it possible to absorb all the infinities by a redefinition of a finite number of parameters. He called such theories renormalizable. These results suggested that local QFT was the framework best suited for unifying quantum theory and special relativity. Yet experiments with cosmic rays during the 1940s and 1950s detected new "strange" particles. It became clear that meson theories were woefully inadequate to account for all properties of the new hadrons being discovered. The fast pace of new experimental findings in particle accelerators quelled hopes for a prompt and systematic transition from QED to formulating a dynamics for the strong interaction.

For some theorists, the failure of QFT and the super-abundance of experimental results seemed liberating. It led to generic explorations where only general principles such as causality, cluster decomposition (the requirement that widely separated experiments have independent results), conservation of probability (unitarity), and relativistic invariance were invoked without specific assumptions about interactions. The American physicist Geoffrey Chew rejected QFT and attempted to formulate a theory using only observables embodied in the S-matrix. Physical consequences were to be extracted without recourse to dynamical field equations, by making use of general properties of the S-matrix and its dependence on the initial and final energies and momenta involved.

By shunning dynamical assumptions and instead using symmetry principles (group theoretical methods) and kinematical principles, physicists were able to clarify the phenomenology of hadrons. Symmetry became a central concept of modern particle physics. A symmetry is realized in a "normal" way when the vacuum state of the theory is invariant under the symmetry that leaves the description of the dynamics invariant. In the early 1960s, it was noted that in systems with infinite degrees of freedom, symmetries could be realized differently. It was possible to have the Lagrangian invariant under some symmetry, yet not have this symmetry reflected in the vacuum. Such symmetries are known as spontaneously broken symmetries (SBS). If the SBS is global, there will be massless spin zero bosons in the theory. If the broken symmetry is local, such bosons disappear, but the bosons associated with broken symmetries acquire mass. This is the Higgs mechanism.

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