Basic Interactions and Fundamental Forces
BASIC INTERACTIONS AND FUNDAMENTAL FORCES
One of the remarkable features of the fundamental forces governing the interactions among the constituents of matter is that they appear to act at a distance. For example, the force of gravity and the electrostatic force between two charged objects are both inversely proportional to the square of the distance between the bodies in question. Even in the case of the short-range forces between nucleons, the nuclear force appears to act in a nonlocal fashion. That is, a nucleon located in a given position exerts a force on another nucleon a distance away. In the case of the nuclear force, the strength of this interaction decreases exponentially with distance, whereas for gravity and electromagnetism, as well as for the effective van der Waal forces in molecules, the strength of the force only decreases as a power of the distance.
It is possible to understand the origin of these apparently nonlocal forces on a deeper level and view them as arising from local interactions between the constituents of matter-and force-carrying fields. Thus, for example, the electrostatic force between two charges arises as a result of the local interaction of each charge and the electrostatic potential established by the other charge at the location of the charge in question. Similarly, the gravitational forces of attraction between two massive bodies is the result of the local interaction between the energy density of each body and the gravitational potential set up by the other body.
These concepts are embedded in James Maxwell's theory for electromagnetic interactions and in Albert Einstein's theory of gravitation. According to these theories, electric charges and masses of particles are the sources of electromagnetic and gravitational fields, respectively. These fields permeate all space, and it is their local interactions with other charges or masses that result in the apparent action at-a-distance behavior of the electromagnetic and gravitational forces observed in nature. Similar but slightly more complex arguments can be made for other, less fundamental action at-a-distance forces. For instance, the van der Waal forces among the atoms in a molecule are the result of partially screening some of the electrostatic forces between the positively charged nuclei and the negatively charged electrons in the molecule.
These classical concepts may be naturally extended to the quantum domain. Indeed, in this context, they lead to an appealing physical interpretation, first advanced by Richard Feynman within the specific context of quantum electrodynamics (QED). All known matter appears to be made up of a few fundamental constituents, or elementary particles that are characterized by a set of intrinsic properties, such as their mass, electric charge, and possibly other (quantized) charges. These fundamental constituents have local interactions with force-carrying fields, each of which couples to one of these intrinsic attributes. It is also possible to associate particlelike excitations with each of these force-carrying fields, so that the local interactions between the fundamental constituents and the force-carrying fields can be viewed simply as an interaction between the fundamental constituents and the particles associated with the force-carrying fields. In this picture then, the forces between the fundamental constituents of matter result from the interchange of the force-carrying particles.
This concept is best illustrated by several examples. The electromagnetic force between two electrons can be viewed as resulting from the exchange of photons (the force-carrying particles associated with the electromagnetic field). The source of the electro-magnetic force is the local interaction of photons with the electric charge of the electrons. One can show that to reproduce the inverse square dependence of the electrostatic force with the distance between the two charges, it is necessary that the mass of the exchanged photon be precisely zero. Any finite mass would lead to a force with an exponential dependence on the distance, with the range being inversely proportional to the force-carrying particle's mass.
Hideki Yukawa's inference of the existence of π mesons as mediators of the nuclear force, with masses of the order of one-tenth the proton mass, originated precisely with this kind of argument, which was based on the known range of the nuclear force. Although this particle-exchange example for the nuclear force is essentially correct, it is nonetheless incomplete. The exchange of π mesons explains only a part of the nuclear force, since this force, like the van der Waal forces, results from a more complex residual interaction—in this case, between the quarks in each nucleon. Quarks have local interactions with gluons, the carriers of the strong force. These interactions are responsible for producing neutrons and protons, as well as π mesons, as quark bound states. They are also directly responsible for the nuclear force, with π meson exchange representing the simplest form of quark-antiquark exchange, leading to the formation of nuclei as nucleon bound states.
At the fundamental level, there appear to be only four forces: gravity, electromagnetism, the strong force, and the weak force. In fact, strong evidence exists that electromagnetism and the weak force responsible for weak decays—such as that of the neutron when it decays into a proton, an electron, and an electron-antineutrino—are part of a unified electroweak force. Furthermore, hints exist that at very high energies the strong and electroweak forces themselves may unify into a single grand unified force. Finally, through string theories, where point particles are replaced by vibrational modes of strings, it may be possible to put gravity on the same footing as the other forces and thereby achieve a unification of all forces.
Each of these four forces couples to certain attributes of matter, and each force has an associated force-carrying particle. For example, the carrier of the electromagnetic force, the photon, couples to electromagnetic current. Remarkably, the specific form of this coupling is dictated by a symmetry principle: the freedom of being able to make local phase transformations of the fields associated with charged particles, in conjunction with an associated gauge transformation of the electromagnetic field. Electromagnetic interactions arise precisely by requiring that nature be invariant under these local U(1)em transformations, which requires only one parameter.
The carriers of the strong force, the gluons, and their interactions can be understood in the same manner as the electromagnetic interactions. The details are a bit more complicated for the carriers of the strong force, but the principles involved are quite similar. There are, in fact, eight gluons because the strong interactions, instead of being invariant under U(1)em transformations, are invariant under an SU(3) symmetry group of transformations, which requires more than one parameter. To be invariant under SU(3) symmetry transformations, eight separate force carriers, one for each independent parameter of the SU(3) group of transformations, are needed. Similar considerations are involved for the weak force that has three carriers: the W+, W-, and Z . These three particles, along with the photon, are needed to be able to also associate the electroweak force with an underlying symmetry group of transformations—in this case, the four-parameter group SU(2) × U(1). Finally, the gravitational force is more akin to electromagnetism with only one carrier of the gravitational force, called the graviton.
At its most basic level, matter appears to be composed of two kinds of excitations: quarks and leptons. Quarks feel all four forces, whereas leptons (of which the electron and the electron neutrino are examples) are not subject to the strong force. The specific interactions of quarks and leptons with the strong and electroweak force carriers are dictated by the way these states transform under the full symmetry group connected with these interactions: SU(3) × SU(2) × U(1). For example, quarks are triplets  under SU(3), whereas antiquarks are antitriplets [3̄]. The interactions giving rise to the strong force must respect the SU(3) group of transformations, since this is the symmetry of the theory. This is achieved by coupling the octet current of quarks and antiquarks [3 × 3 = 8 + 1] in an invariant manner to the eight gluons. The resulting theory, known as quantum chromodynamics (QCD), is a generalization of quantum electrodynamics in which the electron-positron electromagnetic current couples to the photon.
The weak force is more complex. First, the weak bosons couple asymmetrically to matter states, depending on their polarization. Thus, for example, only the left-handed component of electrons and electron neutrinos couples to the SU(2) force-carrying particles. Second, only the electromagnetic piece of the electroweak force is associated with a massless excitation (the photon) and gives rise to long-range forces. The remaining excitations of the electroweak interactions (the W+, W-, and Z weak bosons) actually acquire heavy masses—of order 100 times the proton mass. This arises as the result of the spontaneous breakdown of the SU(2)× U(1) symmetry to the phase symmetry connected to electromagnetism, which is associated with another Abelian symmetry group U(1)em. Hence, weak interactions are very short-range. This is the reason why, at first, there did not seem to be any connection between the weak forces and electromagnetism. However, at energies much above the masses of the W and Z bosons, the distinction between electromagnetism and the weak interactions ceases to be important and the under-lying SU(2) × U(1) symmetry is clear. If in nature, indeed, a grand unified theory does exist, it may well be that the distinction between the strong and electroweak forces, so apparent now, may also, in fact, disappear at superhigh energies.
The particles associated with the strong and electroweak forces all carry spin 1, whereas the graviton has spin 2. In the static limit, it is easy to show that the exchange of a spin-2 particle gives rise to an attractive force, while spin-1 exchange leads to a repulsive force. Since gravitons couple to mass, and this is always positive, one understands why gravity is attractive. In contrast, photon exchange is only attractive among charges of opposite sign but is repulsive for charges of the same sign. Because the gluons and the weak bosons carry a non-Abelian charge, the channels that are attractive or repulsive will depend in detail on the group structure of the matter states. Furthermore, because these force-carrying particles have non-trivial quantum numbers under the SU(3) × SU(2) × U(1) group, they themselves will feel these forces. So, in effect, for gluons and weak bosons the distinction between matter states and force-carrying particles blurs, since these states both carry and feel the forces.
Perhaps the most far-reaching and important feature of the basic interactions is that their form is fixed purely by symmetry considerations. As discussed, the theories that describe these interactions are invariant under a group of symmetry transformations. In fact, the invariance exists under local transformations of the symmetry group, in which each point in space-time can be subject to an independent transformation of the symmetry group. This is a much stronger requirement, and to achieve this invariance, which is known as gauge invariance, it is necessary to introduce compensating fields at each space-time point. These compensating fields are precisely those associated with the fundamental forces, and their interaction with the matter constituents is entirely fixed by the requirement of gauge invariance. Thus, remarkably, symmetry principles fully determine the forces of nature.
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