CONSERVATION LAWS

A conservation law is a statement of constancy in nature. One quantity (the conserved quantity) remains constant while other quantities may change. For example, in a collision of elementary particles, the total momentum is conserved (it is the same after the collision as before) while other quantities, such as the speeds and directions of the particles, and even the number of particles and their masses, may change.

The world of particles, like the larger-scale world around us, is a world of incessant change. Probing the small-scale world, one might not be surprised to find that everything changes, that nothing is constant. Yet scientists have, in fact, identified a limited number of conserved quantities. These quantities have special significance because constancy is an idea of such power and simplicity and because conserved quantities are related to symmetry principles in nature.

Conservation laws are not just keystones of theory, they are practical tools of analysis. They can be applied to processes whose complex details are beyond any capability of measurement or calculation. In a particle collision, conserved quantities such as energy, momentum, angular momentum, and electric charge are the same after the collision as before, even though incredible complexity, with countless interactions, may attend the process.

Conservation laws are tested not so much by measuring some quantity before and after an interaction to see if it is the same as by looking for, and not finding, evidence of a process, which, if it occurred, would violate the conservation law. For example, the decay of an electron into lighter neutral particles has never been observed. A process consistent with all known conservation laws except charge conservation is the decay of an electron into a neutrino of the electron type (called an electron neutrino) and a photon (or gamma ray), indicated by Energy, momentum, and angular momentum could all be conserved in this process, but not charge. The slash through the arrow means that the process does not occur—or, more accurately stated, has never been seen. We can say that the electron is stabilized by charge conservation. Since the electron is the lightest charged particle, its decay would necessarily violate the law of charge conservation. Experiment puts the lifetime of the electron at more than 4 × 10²⁴ years. This means, roughly, that in the lifetime of the universe, no more than one out of a million billion electrons could have decayed. It is on this basis (together with a theoretical underpinning) that we call charge conservation an absolute conservation law.

Conservation laws apply to isolated systems, those for which external influences are absent or too small to be significant. Particle processes are almost always isolated in this sense. Gravity, electric fields, magnetic fields, and neighboring atoms have no appreciable effect during the brief moment of a particle collision. (There are certain examples of nuclear gamma decay within an atom in a crystal where neighboring atoms do have an effect.)

Absolute and Partial Conservation Laws

Conservation laws may be absolute or partial. An absolute conservation law is one for which no confirmed violation has ever been seen and which is believed to be valid under all circumstances. By this definition, momentum, energy, angular momentum, and charge are absolutely conserved, as is the color charge of the strong interaction. (Although color charge is an attribute of strongly interacting particles only, its conservation can be considered universal because leptons and their associated bosons have zero color—they are "colorless.") Another absolutely conserved quantity is the combined symmetry called TCP, standing for time reversal, charge conjugation (or particle-antiparticle inversion), and parity (or left-right inversion). (The individual symmetries T, C, and P are only partially conserved.)

Baryon conservation occupies a special and ambiguous place. Baryons are a class of heavy particles (those made up of three quarks) that include the proton and the neutron. The law of baryon conservation, which states that the number of baryons minus the number of antibaryons never changes, is valid experimentally. Its most stringent test is the absence of proton decay. Since the proton is the lightest baryon, its decay would imply a change of baryon number, a violation of the law of baryon conservation. Experiment puts the lifetime of the proton at greater than 10³³ years. This leaves room for only an incredibly tiny probability of decay (much less even than the limit on electron decay probability), yet theorists are reluctant to call baryon conservation an absolute law. The law has no known theoretical basis, and indeed there is theoretical reason to expect a tiny but nonzero probability of proton decay. Searches for it continue.

Like baryon conservation, lepton conservation is absolute so far as experiment is concerned. The law of lepton conservation states that the total number of leptons of all types (electron type, muon type, and tau type) minus the number of antileptons never changes. No violation of this law has ever been reported. Yet there is no known theoretical reason for it, and physicists expect that in the end it will prove to be partial, not absolute. For many years, the numbers of leptons of the three individual types appeared to be separately conserved (from which it followed, of course, that the total of all lepton types is conserved). Because of recent evidence for the "oscillation" of one type of neutrino into another type, the conservation laws of individual lepton types are now recognized to be partial, not absolute (although the transformation of one type of charged lepton into another has yet to be seen).

What is a partial conservation law? At first, it sounds like a contradiction in terms, like partial pregnancy. A conservation law is called partial if the quantity it governs is conserved when certain interactions are at work but not conserved for all interactions. Stated differently, a partial conservation law is one obeyed by one or more kinds of interaction and violated by at least one kind of interaction. For example, conservation of the number of quarks of a particular "flavor" (up, down, strange, charm, top, or bottom—in each case counting particles minus anti-particles) is obeyed by the strong and electromagnetic interactions but not by the weak interaction. The weak interaction can cause one type of quark to turn into another type.

The weak interaction is recognized as a "symmetry-breaker" that prevents several conservation laws from being absolute. Another symmetry-breaker is the "Higgs interaction," an interaction between every particle and the Higgs boson. The Higgs boson (named after one of its inventors, Scotland's Peter Higgs) is the quantum of a still-hypothetical field believed to permeate all space and to account, through its interactions, for particle masses. The Higgs interaction, along with the strong and electroweak interactions, is incorporated into the so-called Standard Model of particles. Whether gravity, which lies outside the Standard Model, is an additional symmetry-breaker is unknown. If it is, its effects in the particle world will surely be tiny and hard to detect. The present status of conservation laws is shown in Table 1.

TABLE 1

Conservation Laws and Feynman Diagrams

Quantum theory replaces smooth change with explosive change. Every interaction is believed to be driven ultimately by the creation and annihilation of particles. Simple space-time diagrams, or Feynman diagrams (Figure 1), illustrate this idea and show how conservation laws work at the most fundamental level.

For all interaction types, Feynman diagrams have the same "three-prong" structure. At a space-time "vertex," one fermion world line ends, another fermion world line begins, and a boson world line begins or ends. No particle survives an interaction. (If an electron enters an interaction event and an electron leaves it, theory treats them as different particles, one being annihilated and one created.) What do survive an interaction are conserved quantities— whichever ones are preserved by the interaction in question (with a subtlety involving energy and momentum, to be discussed below). So conservation

FIGURE 1

laws apply not just to the before and after of a process in the laboratory. They are believed to apply to the before and after of every separate interaction. From that base, they reach out to all that happens, at whatever scale of size.

It turns out that each of the simple Feynman diagrams shown in Figure 1 represents only a "virtual," not a real, process because energy and momentum cannot be simultaneously conserved across the vertex. So real processes that conserve all relevant quantities involve a "daisy chain" of at least two such interaction vertices (and perhaps millions of them). Illustrating this point is a diagram for a real process, the decay of a negative muon, shown in Figure 2. The time sequence is from the bottom to the top of the diagram. In this process, muon family number, electron family number, and electric charge are conserved.

Examples of Absolute Conservation Laws

Momentum is a vector (directed) quantity. Following a particle collision, the vector sum of the momenta has the same direction and the same magnitude as before the collision. Nonrelativistically, the momentum of an object is its mass times its velocity (p = mv ). Relativistically, the definition is different but the conservation law remains valid. Momentum conservation is related to the homogeneity of space (that the laws of physics are the same at every point).

FIGURE 2

Energy is a scalar (numerical) quantity that takes many forms. Its relevant forms in the particle world are kinetic energy (energy of motion) and mass. The energy locked up in mass is E = mc², where c is the speed of light. Initial kinetic energy is required if new mass is to be created. If mass decreases, as in a decay process, kinetic energy increases. A simple implication of energy conservation is that all spontaneous decay processes must be "downhill" in mass—that is, the total mass of the products must be less than the mass of the initial particle. Energy and momentum conservation taken together prevent one particle from decaying into another single particle. If the initial particle is at rest, the final particle would have to be at rest, too, to preserve zero momentum, but then energy would not be conserved. And if the final particle moves in such a way as to conserve energy, momentum is not conserved. So every decay process results in at least two particles. Energy conservation is related to the homogeneity of time (nature's laws are the same at one time as another).

Angular momentum, like momentum, is a vector quantity. It measures the strength of rotational motion and is directed along the axis of rotation. Particles may have spin angular momentum and orbital angular momentum, roughly analogous to the spin of the Earth about its axis and the orbiting of the Earth around the Sun. Remarkably, a particle may possess spin even if it has no spatial extent. In units of Planck's constant h divided by 2?, written ?, fermions have half-odd-integer spin (1/2, 3/2, etc.) and bosons have integer spin (0, 1, 2, etc.). The spins of the fundamental particles are as follows: quarks and leptons, ½ gluons, photon, and the W and Z bosons, 1; the Higgs boson, 0. Orbital angular momentum is always integral (0, 1, 2, etc.). One consequence of angular momentum conservation is that if the number of fermions before a reaction is even, the number afterward must also be even, and if the number before is odd, the number after must be odd (Figure 2, for example, shows a process where one fermion decays into three fermions.) Angular momentum is related to the isotropy of space (nature's laws don't depend on direction).

The law of charge conservation goes back to Benjamin Franklin in the eighteenth century. It has withstood the test of time. It is related now to the masslessness of the photon through a principle called gauge invariance. In units of the proton charge e, leptons have charges 0 and -1 (antileptons 0 and +1); quarks have charges +2/3;; and -1/3 (anti quarks -2/3 and +1/3); gluons, the photon, the Z boson, and the Higgs boson have charge 0; and the W boson has charge -1 (its antiparticle +1). As noted above, the most salutary effect of charge conservation is to prevent the decay of the electron.

The conservation of "color" (or "color charge") is much like the conservation of electric charge. Both color charge and electric charge are quantized properties of particles that are preserved in every interaction, even when no particle survives the interaction. (Think of runners in a relay race as particles and the baton they carry as color charge or electric charge. When one runner stops and another starts, the baton continues on.) Quarks may have any one of three "colors," arbitrarily called red, green, and blue, so there are really eighteen different quarks, not six. There are no "colorless" quarks. Anti-quarks are said to have anticolor. Gluons carry a color-anticolor mixture such as red-antiblue, of which there are eight independent combinations. Figure 1a shows an example of a color-conserving quark-gluon interaction.

Lepton conservation means that the number of leptons minus the number of antileptons is preserved in every interaction. Figure 2 illustrates the principle with a decay process in which one lepton turns into two leptons and an antilepton. Another example is the decay of the neutron into a proton, an electron, and an antineutrino, The lepton number is zero before and after. This process illustrates several other conservation laws as well: charge conservation (0 to 0), baryon conservation (1 to 1), energy conservation ("downhill" in mass), and angular momentum conservation (odd number of fermions to an odd number of fermions). This process and the one in Figure 2 also show conservation of the individual lepton families (electron family and muon family), for which there are no known exceptions involving charged leptons.

Examples of Partial Conservation Laws

The concept of isospin was introduced in the 1930s to describe the similarities of the proton and neutron—they have nearly the same mass and appeared to have the same strong interaction. The proton and neutron came to be regarded as two states of a single underlying particle, the nucleon, to which the mathematics of a spin-½ particle with its two orientations of spin could be applied (thus the name isospin, which otherwise has nothing to do with spin). Later other particle "multiplets" were found, such as the pion triplet and the xi-particle doublet (and some singlets such as the lambda particle). The law of isospin conservation states that the strong interaction is identically the same for the members of each multiplet. Isospin conservation is clearly violated by the electromagnetic interaction because particles within a given multiplet don't have the same charge (and also differ slightly in mass).

Isospin conservation is now recognized to be a consequence of "flavor invariance." The six quark flavors are called up, down, charm, strange, top, and bottom; flavor invariance means that the strong interactions do not distinguish among flavors. Thus replacing an up quark by a down quark (which could mean changing a proton to a neutron), or even replacing a strange quark by a top quark, does not change any property of the strong interaction. The electroweak interaction does, however, depend on flavor, or quark type—that is, it violates the law of flavor invariance. This is attributed, ultimately, to effects of the Higgs boson.

The strong interaction goes further and conserves each quark flavor separately. If the conservation of quark flavor were an absolute law, there would be no transformations among different quark types. An observed decay process that violates the law of flavor conservation is the decay of a lambda particle into a proton and a negative pion, Using u for up, d for down, and s for strange, this process is represented in terms of its quark constituents by Balancing the books here requires that a strange quark be transformed into a down quark. This is a process forbidden by the strong interaction but allowed by the weak interaction. The mean life of the lambda is many orders of magnitude greater than if the process occurred through the strong interaction, confirming that the flavor violation is provided by the weak interaction.

According to the reductionist view that dominates modern science, conservation laws in the large-scale world result from the action of such laws in the particle world. There is extensive evidence that this is the case. But how fascinating it will be if future discovery reveals large-scale regularity not attributable to small-scale laws.

See also:Energy; Momentum; Noether, Emmy; Symmetry Principles

Bibliography

Bernstein, J. The Tenth Dimension (McGraw-Hill, New York, 1989).

Bernstein, J.; Fishbane, P.; and Gasiorowicz, S. Modern Physics (Prentice Hall, Upper Saddle River, NJ, 2000).

Feynman, R. The Character of Physical Law (MIT Press, Cambridge, MA, 1967).

Pagels, H. The Cosmic Code: Quantum Physics as the Language of Nature (Simon & Schuster, New York, 1982).

Kenneth W. Ford