Each had to be located in internal space. Apparent asymmetries in the distribution of known particles within internal space would appear if one had only part of the picture. In this way the existence of particles necessary to complete the symmetric pattern, and some of their properties, could be predicted. Particles in the same row have similar masses: particles with the same electric charge shown by superscripts also lie on straight lines.

The most profound and powerful symmetries of physical law established at present are so-called local or gauge symmetries. Such symmetries underlie both Einstein's theory of gravity also called general relativity and the Standard Model of strong, weak, and electromagnetic interactions. Most of the matter on Earth is made up of just two quarks, the so-called up and down quarks, denoted by u and d. A glossary to describe many of these terms is included in the Appendix at the end of this report. Because of internal symmetries, each of these quarks comes in three varieties, which are labeled by colors: u r , u g , and u b , represent the red, green, and blue up quarks.

Physical laws are invariant if quark colors are interchanged, for example, if u r , and u g are switched. In fact, a local symmetry means that physical laws are unchanged even when different interchanges are made at different locations in space. For example, one might switch u r with u g in one part of the laboratory and u r with u b in another part. There are an infinite number of such local operations, and requiring the laws to be unchanged under any of them is extremely constraining.

Local internal symmetries actually require the existence of particles called force carriers whose interactions are the origin of the forces. The local symmetry that acts on the three colors leads to the strong force that binds quarks into nuclei. Insight into this basis for the understanding of forces can be gained by returning to the example of the collision of two electrons in Figure 3.

The circular blob represents the actual interaction between particles and is highly constrained if it arises in a theory with a local internal symmetry. If one could look inside this blob at high magnification, such a theory would dictate that the interaction results from exchange of a force particle, which for an electromagnetic interaction is called the photon, as illustrated in Figure 3. Furthermore, the interaction of a photon with two electrons is itself greatly constrained by local symmetry.

No matter what the speeds, directions, and spins of the par-. This is the same single parameter that enters all electromagnetic interactions of the electron, for example, the bending of the path of an electron in a magnetic field and the electron's binding to an atomic nucleus. The electromagnetic interaction is represented diagramatically in Figure 3. In these figures, straight lines represent matter particles and wavy lines represent force particles.

Local symmetries are also called gauge symmetries, and the resulting force particles, such as the photon, are known as gauge bosons. It is startling to realize that the apparent infinite variety of chemical properties and reactions of atoms and molecules all result fundamentally from this single electromagnetic vertex. The very existence of the photon, as well as the. It is clear that symmetries are the most powerful tool physicists have for understanding the properties and interactions of particles, yet only by careful experimentation can we learn which symmetries nature possesses. Many symmetries have been proposed, but measurements provide the only sure guide.

Future experiments will continue the quest to uncover more of nature's symmetries, and theoretical physics will struggle further to understand why nature has chosen these symmetries. A major development in theoretical physics this century was the construction of what are called quantum field theories—theories of particles and their interactions that incorporate the probabilistic laws of quantum mechanics, special relativity, and the symmetries discussed above. This enterprise began soon after the discovery of quantum mechanics in the late s. The quantum field theory of electromagnetism, describing the electron and the photon, reached its final form in the late s, but theories involving larger local internal symmetries were not fully understood until the early s.

Quantum field theories are the basic tool for theoretical particle physicists. There are many such theories, and the great variety of phenomena they can describe is the subject of continuing research. The Standard Model is a quantum field theory that provides a concise and accurate description of all known particle phenomena. This discussion relies on the ideas of symmetry and interaction vertices introduced in the previous section. Three local internal symmetries have been discovered in nature: They are called strong, weak, and electromagnetic, after the three forces to which they give rise.

Strong symmetry leads to force particles of the strong interactions—the gluons, g. The matter particles that feel this force are called up and down quarks, u and d, and come in red, green and blue varieties. The gluon vertex for the up quark is illustrated in Figure 3.

A quark of one color goes into the interaction and comes out as a quark of a different color, but its other properties are not changed.

The mathematical theory of quarks and gluons that underlies this vertex is called ''quantum chromodynamics," or QCD for short. The strength of the gluon interaction is called g 3. It is large, making this QCD interaction strong. The matter particles that do not feel this strong force are leptons: the electron, e, and its neutrino, v e , as well as their second- and third-generation counterparts, the muon and tau, and their respective neutrinos.

The force particles of the weak interaction, the W and Z bosons. The electromagnetic interaction, on the other hand, has a massless force particle, the photon, with a corresponding range of interaction that is infinite, allowing us to see to the edge of the universe. A single parameter g 2 describes the strength of the weak interactions see Figure 3.

The weak symmetry has a very peculiar property. Only counterclockwise spinning left-handed quarks and leptons feel the weak force. The reason nature treats left-handed and right-handed objects differently is one of the many questions about the nature of forces for which we have as yet no adequate answers. As indicated earlier, symmetries dictate both the forces and the so-called multiplet structure of particles that feel these forces.

Table 3. For strong and weak forces, the entries represent the size of the multiplet of particles that interacts with the corresponding force particles. The strong force acts among triplets of quarks three colors , changing one into the other; the weak force acts between quark and lepton doublets, again changing one into the other. An entry "I" implies that there is no interaction, since there is nothing to change into.

Electromagnetic force acts on all particles except neutrinos not changing their nature , and the entry in Table 3.

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There are several questions that this knowledge raises, questions that are not answered by the Standard Model: Why is it that some particles feel the strong force and some do not? Why are the weak interactions left-handed? Why are there not multiplets having more than three components? In short, why are the matter particles what they are, and why do they interact with force particles in the way shown in Table 3. This table offers a mystifying array of numbers—how can it be understood? The four matter particles discussed so far u,d, e, v e are the members of the first family, or generation, of particles.

Three such generations of particles have been found, as shown in Table 2. The only known difference between the three generations is their mass-in particular, the force particle vertices of the heavier generations are identical to those of Figures 3. This replication of particles suggests to some that there is a new internal symmetry to be discovered that is responsible for the different generations. Physicists believe that some deeper understanding of the three. In contrast, the particles of a single generation cannot be grouped into subgroups or periods of particles with similar properties- Table 3.

Whereas interactions of the force particles are restricted by the three local symmetries, the observed masses of the quarks are restricted by the strong symmetry. For example, although u r , u g , and u b have the same mass, members of weak doublets, such as v e and e, do not. The nonzero masses of elementary particles are said to break electroweak symmetries i. This seems unsatisfactory—surely all aspects of a theory should have the same symmetry.

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In fact, physicists believe that the equations of the theory do initially possess electroweak symmetries but that something within the theory causes the solutions to the equations to break the symmetry. This important phenomenon of spontaneous symmetry breaking can be illustrated by the examples of the square and circle mentioned earlier. Recall that these shapes are symmetrical when rotated about their centers, by 90 degrees for the square and any angle for the circle.

If a square and a circle are drawn on an elastic sheet and the sheet is stretched in one direction so that these shapes are elongated into a rectangle and an oval, we have broken the symmetry. Now, if the rectangle is rotated by 90 degrees, it does not match the shape corresponding to its original position. This stretching is a simple analogy for the spontaneous symmetry breaking that occurs in theories of particle physics.

The first step is to infer from data what stretching is occurring which is understood quite well—and the next step is to understand what is causing the stretching. Here there are ideas, but the correct answer is not yet known. When the sheet is stretched, the symmetries of the square and circle are not completely broken: The resulting rectangle and oval are both symmetric with respect to rotations about their centers by an angle of degrees.

Similarly, not all of the electroweak symmetries are broken—the local electromagnetic symmetry discussed in the last section is unbroken. An important consequence of an exact local symmetry is that it requires the mass of the corresponding force particle to vanish. This explains why gluons and photons are massless. On the other hand, W and Z particles, which correspond to the broken parts of the electroweak symmetry, are not constrained to be massless.

In fact, they are so heavy that only in the s did accelerators attain sufficient energies to produce them. The origin of electroweak symmetry breaking, which leads to masses for W and Z particles as well as for quarks and leptons, is a crucial problem of particle physics. What is doing the stretching? The stretching must be generated by some new interactions of the theory-the known interactions illustrated in Figures 3. In the Standard Model, a hypothetical particle, called the Higgs boson, is introduced and given interactions, which allow the elementary particles to become heavy.

The Higgs boson is quite unlike either a matter or a force particle. When physicists say that the Standard Model has been verified in thousands of experiments, they are referring to all the processes that result from the force particle vertices of Figures 3. The Higgs boson is still a matter of speculation, lacking solid experimental support. Nevertheless, something must generate particle masses, and physicists know that this physics is inextricably linked to the mass scale of the W and Z particles.

The interactions that generate the quark and lepton masses play a role in a small but very significant property of the weak force. This is an interaction that causes the generation of a particle to change, as illustrated in Figure 3. The regular and faint lines represent smaller pieces of the weak force, which are called flavor-changing interactions and are described by three parameters. Experiments have not uncovered any flavor-changing interactions of leptons.

Are the laws of physics invariant under the interchange of particles and antiparticles? If so, there would be a new symmetry of nature, known as CP. The masses and interactions of the particles are nearly identical to those of the antiparticles, but there is a small difference-CP is not an exact symmetry of nature. Breaking of the CP symmetry has been observed as a very small difference in neutral K meson decay probabilities. Within the Standard Model, it is the interactions of the Higgs boson that break CP symmetry, an origin for CP breaking that must be viewed as speculative.

This breaking is described by a single extra parameter that enters the flavor-changing vertices of weak interactions. The parameter is capable of describing all of the CP violation observed to date. New experiments studying K and B mesons will soon test whether the generation-changing parts of the weak interaction, illustrated in Figure 3.

A good theory allows calculations that predict many phenomena in terms of just a few free parameters, which must be measured. The Standard Model has been used to calculate thousands of phenomena in terms of the 18 independent parameters listed in Table 3. These are the few quantities that cannot be calculated within the Standard Model. There is a limit to the accuracy of predictions resulting from calculations in the Standard Model. Frequently this is just because high-accuracy calculations are lengthy. In these cases, great effort can produce extraordinarily precise predictions.

For example, the motion of electrons in magnetic fields has been successfully predicted to one part in a trillion. The measurement is also a great effort!

Calculations of processes induced by the weak force have been com-. For the strong force, calculations are more difficult. As yet, these calculations are far from yielding a quantitative understanding of more complex phenomena such as the detailed structure of the proton, but many important calculations are under way. The Standard Model represents an astonishing synthesis of our understanding of the properties and interactions of elementary particles. The next two sections describe how physicists, inspired by its success, are attempting to understand fundamental laws at a deeper level, with greater conviction than ever before that new symmetries remain to be discovered.

Two questions are paramount in furthering an understanding of particles and their interactions, and both of these involve the masses of the particles. The first question involves the masses of force carriers. The massless photon can be understood in terms of the electromagnetic symmetry, and the mass of the proton follows from the dynamics generated by the strong symmetry.

Without such a symmetry, it is not just that the mass scale of weak interactions cannot be determined by theory; rather, the theory naturally makes the mass scale huge, many orders of magnitude larger than observed in nature. The theory can be made to agree with observation only if several large contributions to the weak mass scale are made to cancel, which is an unnatural fine-tuning. The second question involves the pattern of masses and interactions of the matter particles, shown in Tables 2. What determines this structure and the values of these parameters? Could a larger symmetry be responsible for grouping the particle in generations, and could such a symmetry provide an understanding of the pattern of interaction strengths and particle masses'?

The first question, which is about how symmetries break, is considered now. Physicists are sure that there are new forces responsible for symmetry breaking, and these new forces should themselves be governed by a new symmetry. One possibility is an extension of space-time symmetry, known as supersymmetry. A second is another local internal symmetry, which physicists call technicolor sym-. Rotation symmetry leads to electrons with both left- and right-handed spin.

Symmetry under velocity changes leads to a further doubling of the particles, with the electron partnered with its antiparticle. Supersymmetry would lead to still one more doubling: The electron would be partnered with its superpartner. Another possibility is that it could be some new scheme that has yet to be invented. Supersymmetry adds new dimensions to space-time with coordinates that are not ordinary numbers but have a quantum mechanical character.

The breaking of such a symmetry could provide an origin for the weak scale. As indicated, as space-time symmetries get larger, the number of states associated with a particle, such as an electron, also increases. It is therefore no surprise that the further extension of space-time symmetries to include supersymmetry leads to a further doubling of the kinds of particles, as illustrated in Figure 3. For example, the electron has a superpartner, called the selectron.

Moreover, Higgs particles are required. Technicolor, if it exists, would be a new strong force-similar in many ways to the known strong, or color, force. In the same way that the strong force is responsible for the masses of the proton and other hadrons, so the strong technicolor force could provide masses for the W and Z particles. As elementary-particle physicists look beyond the Standard Model, they expect to discover a new force. The symmetries for the new forces differ greatly in their predictions: For example, supersymmetry incorporates the Higgs particle of the Standard Model as the origin for quark and lepton masses, whereas in technicolor theories there is no Higgs particle.

Theoretical difficulties in constructing complete technicolor theories of nature have led many physicists to see supersymmetry as the most likely option. If supersymmetry does provide the key to the weak scale, then the early decades of the twenty-first century will be a time of great discoveries for particle physics: many new particles, the superpartners of particles, and observations of many new effects in rare processes.

The most exciting prospect is that measurements of the masses and interactions of the new superpartner particles will shed light on another great puzzle—the pattern of quark and lepton masses. If technicolor forces are discovered, the future will be even more interesting. As well as a whole new hadron spectroscopy, additional new forces of nature must be present to generate masses for. As experiments reach toward the answer to the great question of how the weak symmetry is broken, physicists anticipate the possibility of dramatic developments in the future direction of the field.

The second question introduced at the beginning of this section concerns the origin of the multiplicity of particles, forces, and masses. Progress can be made by a conceptually straightforward extension of the use of local internal symmetries. If a generation is considered in more detail, including the colors of the quarks, one finds that it has 15 particle components.

The three local symmetries of the Standard Model distinguish between these components: Some feel strong and weak forces, whereas others do not, so it is natural to arrange these components into five groups see Table 3.

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Is it possible that the local symmetries of the Standard Model are just fragments of a much larger, grand unified symmetry? Remarkably, there is such a symmetry that treats all 15 particles of a generation as components of a single fundamental object. The most remarkable aspect is that the properties of this symmetry lead precisely to each and every number in Table 3. Grand unified symmetries provide an understanding of the patterns of particles. If there is a single large local symmetry treating all members of a generation in an equivalent symmetrical way, why does one not observe a single force acting identically on equal-mass particles u, d, e, and v e?

The grand unified symmetry must break at an energy scale that is larger than has been probed by accelerators. In the same way that the electromagnetic force is the low-energy relic from the breaking of electroweak forces, so the three forces of the Standard Model could be the low-energy remnant of a force based on a larger broken symmetry at higher energies. However, as physicists try to understand nature by introducing larger symmetries, the issue of how these symmetries are broken becomes even more important.

In gauge theories, the force between two particles, governed by the interaction strength g, depends slightly on the energy at which the particles collide. However, at lower energies, where today's experiments are performed, the grand unified symmetry is broken and the three interaction strengths have different dependencies on particle energy, as shown in Figure 3. A combination of g 1 and g 2 measured at the energy scale of weak interactions is predicted to be in the range 0.

Thus, grand unified theories can precisely predict this quantity, which in the Standard Model could take any value in the range 0 to 1. The strengths of the three forces g 1 , g 2 , and g 3 depend on the energy at which measurements are made. This dependence has been observed experimentally and can be calculated theoretically. Values for g 1 , g 2 , and g 3 , measured at the energy scale of weak interactions, can be extrapolated theoretically to high energies where, if the theory is supersymmetric, they are found to meet, providing a visual picture of the unification of the three forces.

Perhaps the most dramatic prediction of grand unification is that protons—a fundamental building block of all matter—are not stable, but decay into lighter particles. The simplest nonsupersymmetric theories have been excluded by experiments that searched for, but did not find, proton decay. The supersymmetric theory predicts a longer life for the proton—only a few in a hundred thousand tons of matter equivalent to a large battleship will decay each year.

Other phenomena could also probe the structure of these supersymmetric grand unified theories: Neutrinos may have mass, and muons may be converted to electrons when they are close to an atomic nucleus. The muon and tau are identical to the electron, except that they are much heavier. Why should these heavy copies of the electron exist?

Why are there three generations of matter as shown in Table 2. We have a dedicated site for Germany.

## Quantum field theory - Wikipedia

Soon after the discovery of quantum mechanics, group theoretical methods were used extensively in order to exploit rotational symmetry and classify atomic spectra. And until recently it was thought that symmetries in quantum mechanics should be groups. But it is not so. There are more general algebras, equipped with suitable structure, which admit a perfectly conventional interpretation as a symmetry of a quantum mechanical system.

In any case, a "trivial representation" of the algebra is defined, and a tensor product of representations. But in contrast with groups, this tensor product needs to be neither commutative nor associative. Quantum groups are special cases, in which associativity is preserved. Lattice models provide many examples of quantum theories with quantum symmetries. They were also covered at the school. Some such models and their nonlocal conserved currents were discussed.