# Origin of Symmetries

This is exactly the book the a physics student need.

More specifically, the introductory presentation of symmetry and Lie Groups makes this ordinarily complex topic clear and understandable! I also appreciate the level of presentation — no boring topics such as ropes and pulleys, inclined planes with sliding boxes, etc. Plus an Appendix reviewing Calculus and Linear Algebra.

The book contains much more than these topics, but this gives an indication of the level and scope involved. Overall, an excellent, almost magical,mtext covering all the major areas of physics in an accessible manner. Highly recommended! Most books, as well as lectures, introduce Quantum Physics from Classical Physics, deriving microscopical from macroscopical laws. Didactically this may be correct although sometimes certain rules and formulae pop up out from nowhere , but in fact the macroscopic exists through certain limits of the microscopic.

This is a book in which this way was chosen, deriving physics from only a few assumptions. For the interested reader with prior knowledge about modern physics I can strongly recommend this book to further deepen your understanding of the matter. This book gives very easy and intuitive access to the main ideas of group theory with very little assumptions made and foreknowledge required. If you have an exam exclusively about Quantum Physics next week and only time to read one book, you probably should better read a standard textbook.

But if you are looking for a deeper understanding and the link between things, or if you just started your studies and want an intuitive introduction into the world of physics using symmetry — this is the perfect book for you. I have to do two of these for my rather strange M. The research project is at the Lorentz Instute in Leiden, and contrary to delft they do deal with theoretical physics, specifically quantum field theory, group theory and general relativity.

Delft mostly focuses on nano-devices in the master program, so sadly these topics are missing. I decided to hit the library for an e-book that was better fit for me. Looking into it, I found a very modern website, an interesting preface and decided to download the book. I particularly like going from simple definition to abstract mathematical definition and back. Today Springer released the digital version of Physics from Symmetry. The print version should be available soon. The book can be bought and downloaded as a PDF document at the […].

## Symmetry About a Point

All corrections and change request have been incorporated. Now the production can begin and the digital and print versions should be available soon. The book will be part of the Undergraduate Lecture Notes in Physics series and be published next year. Skip to content Physics from Symmetry. Why This Book?

Jakob Schwichtenberg, January Contents Special Relativity The book starts with a short chapter about special relativity, which is the foundation for everything that follows. We derive the most important concepts like the Minkowski metric and the famous Lorentz transformations. In addition, we discuss why there is an upper speed limit for everything in physics.

Group Theory The second part of the books develops the mathematics required to utilize symmetry ideas in a physical context. Most of these mathematical tools come from a branch of mathematics called group theory. A completely self-contained introduction is included and all concepts that are needed to understand the modern theories of physics are explained.

The Lagrangian Formalism The introduction starts with the basic idea that we get the correct equations of nature by minimizing something, which is commonly called Lagrangian. Demanding that physics is the same in all frames of reference enables us then to derive the fundamental equations of Quantum Field Theory, Quantum Mechanics, Special Relativity, Electrodynamics and Classical Mechanics. Gauge Theory In addition to spacetime symmetries, we can have internal symmetries.

Spacetime symmetries enable us to derive equations that describe free particles. Internal symmetries enable us to derive equations that describe interacting particles. The framework that we use to derive these equations from internal symmetries is called Gauge Theory. Quantum Mechanics The quantum formalism and the probabilistic interpretation of Quantum Mechanics are explained and motivated. Quantum Field Theory The fundamental equations that are derived in the second part of the book are used most naturally in Quantum Field Theory. The basic idea how quantum fields can describe particles and particle interactions is described in great detail.

Afterwards an outline is presented how this idea can be used to calculate numbers that can be compared with experiments. Electrodynamics We derive the inhomogeneous and homogeneous Maxwell equations and discuss how the notions electric and magnetic field come about. In addition, the Lorentz force law and Coulomb's potential are derived. Appendices The appendices contain many explanations of important mathematical concepts that may be unfamiliar to beginner students.

For example, the Taylor series, the Kronecker Symbol and the matrix exponential function are explained. Leave a review on Amazon and receive a free bonus chapter! What Readers Say Congratulations on an inspiring book! Thank you so much for writing this book! GREAT book! Thank you so much! I must say that you have done the current and future generations of undergraduates, as well as interested self-studying amateurs such as myself, a tremendous service by applying the logical structure and clarity of your writing to this often difficult topic.

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.

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## Unit Circle Symmetries

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.

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.

### Please note:

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?

## Symmetry and Graphs | Purplemath

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.

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. Physicists once again look to symmetries for the answer. Consider an equilateral triangle drawn on an elastic sheet. This is an analogy for a symmetric world in which there are three identical charged leptons, with one side of the triangle representing each lepton.

The symmetry of the triangle under a rotation of degrees shows that the three leptons are identical. If the triangle is stretched, the three sides are no longer equal, analogous to our world where the electron, muon, and tau have very different masses. As discussed earlier, physicists believe that a symmetrical. The symmetry is not manifest directly, but its presence is inferred from the stretched forms that have broken the symmetry. To understand why nature looks the way we find it, we have to understand broken symmetries. Even if we are able to uncover these broken symmetries of nature, physicists ask why the symmetry was there in the first place: Why start with an equilateral triangle?

Why not a square or something else? The next section describes work of the past decade that has grappled with such questions. The energy scale at which the strengths of the three forces are predicted by assuming supersymmetry to become equal, as shown in Figure 3. Above this energy scale, the symmetry becomes so encompassing that quarks and leptons become unified; this is called grand unification. This energy scale is intriguingly close to the famous Planck energy scale, which is about a hundred times larger.

Unfortunately, this energy is so high that it can be reached only by theoretical speculation. The Planck scale of energy is where the gravitational force becomes strong. At low energies, gravity is an incredibly weak force, noticeable only in the influence of large objects such as planets and stars. The gravitational force depends on a particle's mass when at rest, but on its energy when in motion, so that it increases with increasing energy and at the Planck scale is competitive with the other forces of nature.

The Planck scale is determined by the three fundamental units of nature: the maximum speed that of light, c , the quantum of action Planck's constant, h , and the gravitational coupling constant Newton's gravitational constant, G. A mathematical combination of these three constants to yield a term whose units are those of energy results in the Planck energy scale. Given the fundamental nature of these constants it is reasonable to suppose that the Planck scale is the fundamental scale of physics. Indeed there is a disparity of 17 orders of magnitude between these scales.

Not only is a new symmetry needed to govern the weak scale, but the question arises as to why the energy associated with this new symmetry is so low. If the purported unification scale is so close to the Planck length, then gravity must be treated on an equal footing with the other forces of nature. This is a long-outstanding problem of theoretical physics. Einstein's theory of gravity is remarkably successful at low energies, yet it gives rise to deep problems and inconsistencies at high-energy.

These problems suggest that it must be replaced by a more fundamental theory, reinforcing the view that new physics will appear close to the Planck length that might unify all the forces of nature, including gravity. Fortunately, there exists a theory that appears to have the potential of achieving these goals—string theory. What is string theory? String theory is a natural generalization of previous theories of particles but represents a radical departure from the tradition initiated by Thales of Miletus. In uncovering string theory about 25 years ago, physicists set out on a path whose end we can still barely conceive, one that has led to a trail of theoretical surprises—including supersymmetry—without obvious historical parallel.

String theory not only eliminates the contradiction between gravity and quantum mechanics but in a sense explains why just this combination exists in nature. String theory also automatically generates all of the ingredients that seem to be needed as building blocks of the Standard Model. In these and other ways, string theory provides potential answers to many of the puzzles posed by the Standard Model. Two major revolutions in physics have already occurred in this century: relativity and quantum mechanics.

These were associated with two of the three really basic parameters of physics: the velocity of light and Planck's quantum of action. Both revolutions involved major conceptual changes in the framework of physical thought. In each case, the new theory was totally different from the old in its basic tools and concepts, but it reduced approximately to the old one when the appropriate parameter could be considered small. The last parameter of this sort is Newton's gravitational constant. A third revolution appears to be likely, and string theory—which reduces to more familiar theories at large distances—may be the key.

Perhaps this third revolution will lead to a final theory or perhaps only to a next theory that will lead to new questions. The present state of theoretical physics is reminiscent of the days of confusion that preceded the birth of quantum theory in the mids, when it was clear that a new theory was coming but not at all clear what this theory was. In the present case, a whole host of theoretical insights clearly point toward a basic. One should not underestimate the likely scope of this change.

String theory is now in the midst of intense theoretical development. Although it appears to have the potential of reproducing the Standard Model and explaining its structure and parameters, the understanding is too primitive to be able to make complete predictions about details of the Standard Model; however, the main qualitative properties of the Standard Model have been derived from string theory in a strikingly elegant way. Moreover, string theory requires the existence of both quantum mechanics and gravity, whereas previous theories in physics make it impossible to have both together; other general predictions of string theory are gauge invariance, which has been seen to be the bread and butter of the Standard Model, and supersymmetry, which is one of the main targets in the worldwide enterprise of particle physics.

Many deep problems remain to be solved before the theory can be compared directly with experiment. Nonetheless string theory is testable by experiment. It would be easy for new experimental discoveries that did not fit into a straightforward extrapolation of the Standard Model to provide evidence that string theory is the wrong theory to follow. Conversely, the discovery of supersymmetry would be an important validation for string theory. In addition, this discovery would provide invaluable clues as to the mechanism of supersymmetry breaking that could help in unraveling the predictions of string theory.

Thus, we have the beginnings of a new theory of fundamental physics—string theory—whose full elucidation could be as revolutionary as the discovery of quantum mechanics or relativity. Part of the Physics in a New Era series of assessments of the various branches of the field, Elementary-Particle Physics reviews progress in the field over the past 10 years and recommends actions needed to address the key questions that remain unanswered.

It explains in simple terms the present picture of how matter is constructed. As physicists have probed ever deeper into the structure of matter, they have begun to explore one of the most fundamental questions that one can ask about the universe: What gives matter its mass? A new international accelerator to be built at the European laboratory CERN will begin to explore some of the mechanisms proposed to give matter its heft.

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