Preface The preparation of these notes began in 2008 when I taught the first offering of a newly designed class, Particles and Symmetries. This class was created to give undergraduate physics students, early in their studies, an introduction to the fundamental constituents of matter and the symmetries which characterize their interactions. The presentation begins with an overview of special relativity, and then moves into an examination of the building blocks of the current Standard Model of particle physics. The material, by design, takes advantage of the fact that a remarkable amount of particle physics may be understood quantitatively using relatively few basic concepts. Students are assumed to have had introductory physics and at least one quarter of quantum mechanics introducing state vectors (bras and kets), quantum time evolution, observables and expectation values, spin-1/2 and related two-state systems, and quantized angular momentum. Facility with calculus, linear alge- bra, and basic mathematical methods is also assumed. Brief appendices on basic mathematics and quantum mechanics summarize some of this needed background material. Prior exposure to special relativity, or particle physics, is not required. This version of these notes incorporates or adapts a number of suggestions due to my colleague, Stephen D. Ellis, who has taught Particles and Symmetries multiple times starting in 2011. His contributions are gratefully acknowledged. Some words regarding conventions: Arrows are used to indicate three-dimensional spatial vectors, such as ~x. Components of spatial vectors are written as xi, with a Latin index (such as i) which runs from 1 to 3. Four-dimensional spacetime vectors, which are introduced in chapter 2, are not marked with a vector sign, but their meaning should be clear from context. Components of a spacetime vector are written as xµ, with a Greek index (such as µ) running from 0 to 3. Sadly, there are two different conventions in common use in the physics community for defining the dot product of spacetime vectors, differing by an overall minus sign. These notes use the only sensible choice (in the opinion of this author), which makes the dot product of spacetime vectors having vanishing time component the same as the usual three-dimensional dot product, and allows plane waves in space and spacetime to have the same eik·x form. Pay no attention to anyone urging use of the other convention! Laurence G. Yaffe March 2015 1 Introduction As we start this study of Particles and Symmetries it is appropriate to begin with a description of the overall goal of the course, which is to provide an introduction to an area of physics that has seen dramatic progress in the last 50 years | elementary particle physics. A central tool underlying this progress has been the exploitation of symmetries, reflected in the interactions of particles, hence the title of this course. The understanding which has emerged is encoded in the so-called Standard Model of particle physics, which identifies the fundamental particles and interactions among these particles relevant for describing nearly all of the physical universe. When one includes collective behavior (quarks bound in nuclei, electrons bound in atoms, atoms bound in solid matter) plus classical gravity, the result is a nearly complete explanation for the physics observed from the largest distance scales, e.g., the evolution of the universe from very early times, down to the shortest distances probed at particle accelerators. Full command of this fundamental understanding requires some tools not at our disposal, namely quantum field theory. However, using only special relativity, basic quantum mechanics, and symmetries, one can understand a surprisingly large portion of particle physics in an accessible and relatively quantitative fashion. From a pedagogical perspective, this endeavor provides an opportunity to discuss special relativity in detail, and practice using it to describe the kinematics of particle collisions at high energy. Key concepts will include 4-dimensional momentum conservation (which is itself associated with the in- variance of physics under translations in space and time), and the universal speed limit set by the speed of light. Developing practical facility with 4-vector notation and the transformations (boosts) that relate quantities in different inertial reference frames will be emphasized. From quantum me- chanics, heavy use will be made of the uncertainty principle, and the key role played by simultaneous eigenstates of mutually commuting operators. You should have seen some of this structure in the context of quantized angular momentum. We will use symmetries, and a little bit of the underlying mathematics of group theory, to make many testable predictions. Important applications will involve approximate symmetries, where there is not exact invariance under some transformation, but rather the transformation induces \small' perturbations. This will allow us to separate big effects from small effects, or more formally organize perturbative expansions in these small effects | another essential tool in physics. Throughout the course, one emphasis will be honing your skill for making order-of-magnitude estimates, i.e., quickly estimating a rough value for some physical quantity even when fine details are not known. Do not be overly concerned if some of these concepts are unclear on first reading | clarity should improve as the course progresses. c Laurence G. Yaffe, 2008, 2009, 2010, 2012, 2013, 2015 2 Chapter 1 Special relativity 1.1 Galilean relativity Newton's laws of motion, d~p d~x ~p = F;~ = ; (1.1.1) dt dt m retain the same form if one substitutes ~x ! ~x 0 + ~ut ; ~p ! ~p 0 + m ~u; (1.1.2) for any velocity ~u which is constant (independent of time). In other words, equations (1.1.1) and (1.1.2) imply that d~p 0 d~x 0 ~p 0 = F;~ = : (1.1.3) dt dt m This shows that changing coordinates to those of a moving (inertial) reference frame does not affect the form of Newton's equations. In other words, there is no preferred inertial frame in which Newton's equations are valid; if they hold in one frame, then they hold in all inertial frames. This is referred to as Galilean relativity. It is an example of an invariance, a change in the description of a system (in this case, a change in the coordinate system) which preserves the form of the equations of motion. An intrinsic aspect of Galilean relativity is the assumption that time has the same meaning in all inertial frames, so t represents time as measured by any good (and synchronized) clock, regardless of whether that clock is moving. Consider a particle, or wave, which moves with some velocity ~v when viewed in the unprimed frame, so that the position of the particle (or crest of the wave) is given by ~x(t) = ~x0 + ~vt. In the primed 0 frame, using (1.1.2), the location of the same particle or wave-crest is given by ~x (t) = ~x0 + (~v − ~u) t. Hence, when viewed in the primed frame, the velocity of the particle or wave is given by ~v 0 = ~v − ~u: (1.1.4) This shift in velocities upon transformation to a moving frame is completely in accord with everyday experience. For example, as illustrated in Figure 1.1 , if a person standing on the ground sees a c Laurence G. Yaffe, 2008, 2009, 2010, 2012, 2013, 2015 3 Particles and Symmetries CHAPTER 1. SPECIAL RELATIVITY −80 kph 100 kph 20 kph 80 kph Figure 1.1: A moving train and car, as seen from the ground (left), and from the train (right). car moving at 100 kph (kilometers per hour) parallel to a train moving at 80 kph, then a person sitting in the train will see that car moving with a relative velocity of 20 kph = (100 − 80) kph, while the person on the ground recedes from view at a velocity of −80 kph. Similarly, a sound wave propagating at the speed of sound vs (in a medium), as seen by an observer at rest with respect to 0 the medium, will be seen (or heard) as propagating with speed v = vs − u by an observer moving in the same direction as the sound wave with speed u (with respect to the medium). Consequently, the frequency f 0 = v0/λ heard by the moving observer (i.e., the number of wave fronts passing the observer per unit time) will differ from the frequency f = vs/λ heard by the stationary observer, v − u u f 0 = s = f 1 − : (1.1.5) λ vs This is the familiar Doppler shift for the case of a moving observer and stationary source with respect to the medium. Recall that the medium plays an important role here. If, with respect to the medium, it is the observer who is stationary while the source moves away from the observer at speed u, then the result for the Doppler shift becomes . u f 0 = f 1 + : (1.1.6) vs The two results coincide to first order in u=vs (i.e., (1.1.6) approaches (1.1.5) for ju=vsj 1), but as u approaches vs (so that u=vs ! 1) the two expressions are very different. This reflects the fact that for sound, there is a physically distinguished special reference frame, the rest frame of the medium through which the sound propagates. 1.2 Constancy of c When applied to light (electromagnetic radiation), the Galilean relativity velocity transformation (1.1.4) predicts that observers moving at different speeds will measure different propagation velocities for light coming from a given source (perhaps a distant star). This conclusion is wrong. Many experiments, including the famous Michelson-Morley experiment, have looked for, and failed to find, any variation in the speed of light as a function of the velocity of the observer (or source).