20201021 Quantum Mechanics II Special Relativity Preparatory Course Teaching Assistant: Oz Davidi October 27-28, 2019 Disclaimer: These notes should not replace a course in special relativity, but should serve as a reminder. If some of the topics here are unfamiliar, it is recommended to read one of the references below or any other relevant literature. Notations and Conventions 1. We use τ as a short for 2π.1 References There exist lots of references about the subject. Many books about general relativity include good explanations in their first chapters. Other sources are advanced books on mechanics and electromagnetism. Here is a list of some examples which covers the subject from those different points of view. In each of them, look for the relevant chapters. 1. Classical Mechanics, H. Goldstein. 2. Classical Electrodynamics, J. D. Jackson. 3. A First Course in General Relativity, B. F. Schutz. 4. Gravitation and Cosmology, S. Weinberg. 1See https://tauday.com/tau-manifesto for further reading. 1 2 INDEX NOTATION 1 Motivation One of the main topics of the Quantum Mechanics II course is to develop a (special) relativistic treatment of quantum mechanics, which is done in the framework of quantum field theory. We will learn how to quantize (relativistic) scalar and fermionic fields, and about their interactions. For this end, a basic knowledge in special relativity is needed. 2 Index Notation We will find that index notation is the most convenient way to deal with vectors, matrices, and tensors in general. Let us focus on tensors of rank 2 and below. T • For a vector ~v = v1 v2 ··· vn , we denote the i's component by vi. 0 1 m11 m12 ··· B C • For a matrix M =Bm21 m22 ···C, we denote the [ij]'s entry by Mij. @ . A . .. T Notice: In general, Mij 6= Mji, but Mji = M ij. • When we multiply a vector by a matrix from the left, we get a new vector ~u = M~v. The P i's component of the new vector is given by ui = [M~v]i = j Mijvj. From now on, we will use Einstein's Summation Convention: 1. If an index appears twice, we sum over it X Mijvj ≡ Mijvj : (2.1) j 2. An index will NEVER appear more then twice! • What about multiplying by a matrix from the right, ~vT M? Again, we get a new vector T T T T T ~w = ~v M. In index notation ~w i = [~w]i = wi = ~v M i = ~v j Mji = vjMji. Here is an example why this is so useful: Example 2.1. Prove that ~u ×(~v × ~w) = ~v (~u · ~w) − (~u · ~v) ~w. 2 3 FAST INTRODUCTION TO SPECIAL RELATIVITY Proof. By using index notation [~u ×(~v × ~w)]i = ijkuj [~v × ~w]k = ijkklmujvlwm = ijklmkujvlwm = (δilδjm − δimδjl) ujvlwm = viujwj − ujvjwi = [~v (~u · ~w) − (~u · ~v) ~w]i : 3 Fast Introduction to Special Relativity 3.1 Defining Special Relativity (B. F. Schutz: 1.1, 1.2) At first, Einstein's theory of special relativity was understood algebraically, as a set of (Lorentz) transformations that move us from one inertial observer's system to another. Special relativity can be deduced from two fundamental postulates: 1. Principle of Relativity (Galileo): No experiment can measure the absolute velocity of an observer; the results of any experiment performed by an observer do not depend on his speed relative to other observers who are not involved in the experiment. 2. Universality of the Speed of Light (Einstein): The speed of light relative to any unacceler- 8 ated (inertial) observer is 3×10 m=s, regardless of the motion of the light's source relative to the observer. Let us be quite clear about this postulate's meaning: two different iner- tial observers measuring the speed of the same photon will each find it to be moving at 8 c = 3 × 10 m=s relative to themselves, regardless of their state of motion relative to each other. But what is an \inertial observer"? An inertial observer is simply a coordinate system for spacetime, which makes an observation by recording the location x y z and time t of any event. This coordinate system must satisfy the following three properties to be called inertial: 1. The distance between point P1 = x1 y1 z1 and point P2 = x2 y2 z2 is indepen- dent of time. 3 3.2 Transformation Rules 3 FAST INTRODUCTION TO SPECIAL RELATIVITY 2. The clocks that sit at every point, ticking off the time coordinate t, are synchronized and all run at the same rate. 3. The geometry of space at any constant time t is Euclidean. 3.2 Transformation Rules Let us derive the transformation rules of special relativity in 1 + 1 dimensions (1 space and 1 time dimensions). • Imagine two systems, O and O0, with respective velocity v between them. • We choose x = x0 = 0 at t = t0 = 0. • The position of a wave-front in system O is measured to be x = ct : (3.1) • We would like to see how this wave form is seen (parametrized) in system O0. We take the transformation to be linear x0 = ax + bt, where a (which is dimensionless) and b (which has dimensions of velocity) will be found below. The physical reason is that we want O −! O0 −! O00 to be identical to O −−−−−! O00 (you can compare it to rotations).2 T1 T2 T1\+"T2 • The origin of O0 in the O system is given by x = vt, hence 0 = (av + b) t =) b = −av =) x0 = a(x − vt) : (3.2) • The inverse transformation is given by changing the sign of the velocity, namely x = a(x0 + vt0) : (3.3) • Plugging x0 into x gives (1 − a2) x t0 = at + : (3.4) av • Now, we demand that a wave-front in O, i.e. x = ct, is also a wave-front in O0, i.e. x0 = ct0 (here, we demand that the speed of light is the same for all observers). By using 2We will make this statement more precise once we study group theory. 4 3.2 Transformation Rules 3 FAST INTRODUCTION TO SPECIAL RELATIVITY the expression for x0, Eq. (3.2), and the expression for t0, Eq. (3.4), and substituting x = ct, we get (1 − a2) c2 a(c − v) = ca + : (3.5) av Solving for a, one gets 1 a ≡ γ = : (3.6) q v2 1 − c2 To summarize, the transformation rules are x0 = γ(x − vt) ; (3.7) v t0 = γ t − x : (3.8) c2 As an important side note: Always check the dimensions of the quantities you look for. Indeed, a turned out to be dimensionless, and b has the dimensions of velocity. An immediate result is that the time coordinate is not universal! This is depicted in Fig.1. In classical mechanics, an event A = tA xA yA zA shares the same time with an infinite number of events B = tA xB yB zB . They all have the same time, meaning that events that happen simultaneously at one inertial system, also happen at the same time in another. On the other hand, the same event A, under special relativity, has a unique \now". Other events have their own \now", hence different observers may not agree on the relative time between events. A trajectory x(t) of a particle for example, is called a world line. A world line must cross a constant time slice once (and only once), but the crossing point can be at any point, depending on the observer. The slope of a world line is the velocity reciprocal, v−1 =x _ −1. Because the velocity is bounded from above by the same constant value c in all reference frames, at each point of the trajectory x(t), one can draw a light-cone, and all inertial observers will agree that the trajectory is within this light-cone. We will sometime use β = v=c, and from now on, we set the speed of light to 1 c = 1 : (3.9) Using the general 3 + 1 dimensional transformation rules, one can show that while different 5 4 THE METRIC Figure 1: Spacetime structure in classical mechanics (top) and special relativity (bottom). In classical me- chanics, a universal time slice exists, while in special relativity, each event defines a light-cone. (B. F. Schutz: 1.6) inertial observers determine different world-lines for the same particle, they agree that (∆t)2 − (∆x)2 = (∆t0)2 − (∆x0)2 : (3.10) We take the infinitesimal limit, and define the interval ds2 ds2 ≡ dt2 − dx2 = dt02 − dx02 : (3.11) The interval is a Lorentz scalar - it is invariant under Lorentz transformations (to be discussed in Sec.5). 4 The Metric Minkowski pointed out that space (~x) and time (t) should be treated all as coordinates of a four- dimensional space, which we now call spacetime. We thus define the spacetime four-vector xµ = t ~x . The index µ 2 f 0; 1; 2; 3 g is called the Lorentz index. The Minkowski spacetime is not Euclidean. In order to measure distances, we define the 6 4 THE METRIC metric as a symmetric function which maps two four-vectors to R g(v1; v2) = g(v2; v1) 2 R : (4.1) Note that we did not define g to be positive definite. In special relativity, we parametrize 3 the metric by the rank-2 tensor ηµν = diag(1; −1; −1; −1). The metric with upper indices is identical ηµν = diag(1; −1; −1; −1). Einstein's Summation Convention (Revisited): 1. If an index appears twice, once as a lower and once as an upper index, we sum over it.