Contents 1 Natural Units

PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 02 Wednesday, August 26, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Contents 1 Natural Units 1 2 Spacetime 3 2.1 4-Vectors . .3 2.2 Lorentz Transforms . .4 2.3 The Minkowski Metric . .5 3 Relativistic Dynamics 6 3.1 The 4-Velocity . .6 3.2 The 4-Momentum . .7 4 Special Relativity and Electromagnetism 8 1 Natural Units Let us use a convention known as natural units. We know that the speed of light is approximately c ≈ 3 × 108 m=s: (1) The (reduced) Planck’s constant is −34 ~ ≈ 1:05 × 10 J · s: (2) In the convention of natural units we set c = ~ = 1: (3) PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 Using natural units, we can express all quantities as energy to some power. For example, mass is [m] = [E]1: (4) This is obvious, since E = mc2. In energy units the mass of a proton is approximately 2 −27 8 2 mpc = 1:67 × 10 kg × 3 × 10 m=s = 15:03 × 10−11 kg · m2=s2 = 15:03 × 10−11 J = 15:03 × 10−11 × 6:24 × 1018 eV = 93:78 × 107 eV = 938 MeV: (5) Similarly, we can show that the mass of an electron is 0.511 MeV. In natural units we have [L] = [E]−1: (6) We can see this from the Compton wavelength of a particle λ = ~ : (7) C mc The physical interpretation of the Compton wavelength is that it is the smallest scale on which a single particle can be identified. On smaller scales, the energy goes up, and particles can be created out of the vacuum. Large scales are low energies and vice versa. Since distance and time have the same units, we have [T ] = [E]−1: (8) One particularly interesting case involves Newton’s constant G. In natural units, G has a value of 1 G = 2 ; (9) EP l where EP l is known as the Planck energy. It represents the energy scale on which quantum mechanics and gravity are both important. Planck energy has value r c E = ~ ≈ 1:22 × 1019 GeV: (10) P l G In Table 1 we write out a few more useful conversions to natural units. We plot a range of energy scales in Fig. 1. 2 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 mu mp mn mW EPl Electronic energies md mμ mτ mH GUT m me ν mt Electroweak 10−10 10−5 100 105 1010 1015 1020 10−6 10−11 10−16 10−21 10−26 10−31 10−36 Visible Hydrogen Atomic Planck light atom nuclei length Length (m) Figure 1: Characteristic energy/length ranges in nuclear and particle physics. Unit Natural Units 1 kg 5:63 × 1026 GeV 1 m (1:97 × 10−16 GeV)−1 1 s (6:58 × 10−25 GeV)−1 19 EP l 1:22 × 10 GeV Table 1: Conversion of MKS Units to Natural Units. 2 Spacetime 2.1 4-Vectors In 1908, Herman Minkowski made the following statement. “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” Address to the 80th Assembly of German Natural Scientists and Physicians, (Sep 21, 1908) The union of the two is what we now call spacetime. In relativity, we define positions and other vectorial quantities in terms of a 4-vector 0 1 t B C µ BxC x = B C : (11) ByC @ A z In the above, µ (and other Greek-letter indices) may take values 0; 1; 2; 3. We use the convention in which Roman indices take the values i = 1; 2; 3. All vectors in Euclidean space have three components: labelled as ~x. 3 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 All well-defined vectors in Minkowski spacetime have four components and labelled as x. 2.2 Lorentz Transforms Immediately after the result of the Michelson-Morley experiment, George FitzGerald (1889) and Hendrik Lorentz (1892) attempted to explain the constancy of the speed of light. Their radical proposal was that measurement equipment was deformed in a particular way when traveling through a hypothetical entity called aether. This aether was to be the medium through which electromagnetic radiation propagated. As the history progressed, the concept of aether was abandoned. However, the mathematical grounding developed by FitzGerald, Lorentz and others gives the relationship between the 4-vector coordinates of two frames in relative motion. These relationships are known as Lorentz transformations. For the simplest case of a relative speed of v in the x-direction we have 0 1 γ vγ 0 0 B C µ¯ Bvγ γ 0 0C Λ (v) = B C : (12) µ B 0 0 1 0C @ A 0 0 0 1 For the inverse, the sign of the velocity is simply switched. Boosts in other directions can be developed by inspection. The “gamma factor” is defined as 1 γ ≡ p : (13) 1 − v2 The Lorentz transforms are at the center of relativistic physics. Equations are said to be Lorentz invariant if they are identical in any rotated or boosted frame. In physics, we are supposed to derive only quantities and expressions that are Lorentz invariant. In the non-relativistic limit, γ ' 1, the Lorentz transforms approach the Galilean transforms. Lorentz transforms reveal to us an important manifestation of reality: clocks run slow by a factor of γ compared with their stationary counterparts. This is not simply an optical illusion. Any measurement of time will display the same time dilation: the ticking of a clock, the beating of a heart, or even the decay of particles. Example: Muons (a type of charged elementary particles) created in the upper atmosphere survive to the surface of the earth instead of being destroyed, despite their short mean lifetime of 1.4 µs. The explanation was that muons travel at relativistic speeds, their internal “clocks” are slowed relative to the earth. Thus their effective decay time is effectively increased. The Lorentz transforms guarantee that massive particles move at sublight speeds in all frames. 4 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 If a photon travels at the speed of light in one frame, then it quickly follows that it moves at the speed of light in all boosted frames. 2.3 The Minkowski Metric The Minkowski metric helps us find distances in spacetime. We have 0 1 1 0 0 0 B C B0 −1 0 0 C gµν = B C : (14) B0 0 −1 0 C @ A 0 0 0 −1 It allows us to tell the distance between two nearby event separated by a 4-vector dxµ 2 µ ν dτ = gµνdx dx : (15) This distance is known as the interval. Supposing the interval is positive, τ is known as the proper time between two events. The proper time is the flow of time measured by an observer in their own frame. The interval is a Lorentz scalar. It is a Lorentz invariant quantity dt2 − dl2 = dt¯2 − d¯l2: (16) We can also write the interval as 2 µ dτ = dx dxµ = dx · dx: (17) Since the indices are contracted, if does not matter which is upstairs and which is downstairs. The sign of the interval gives us some useful information 1. dτ 2 = 0: Lightlike or null separation. Particles traveling at the speed of light will have a lightlike separation. The emission and observation of a photon will always be separated by an interval zero. That is, time does not pass for a photon or any other massless particle. 2. dτ 2 < 0: Spacelike separation. Two events that do not have a causal connection are spacelike separated. 3. dτ 2 > 0: Timelike separation. A positive interval means that two events are causally connected. 5 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 The Minkowski metric itself does not change upon a Lorentz boost, just as the Euclidean metric did not change upon coordinate rotation. That is µ ν gµ¯ν¯ = Λ µ¯(v)Λ ν¯(v)gµν (18) will again produce an identical copy of the Minkowski metric. This result is a consequence of Einstein’s first postulate of special relativity. 3 Relativistic Dynamics 3.1 The 4-Velocity Upon using our new language, let us construct dynamical quantities that will be useful in under- standing particles and their interactions. In Newtonian mechanics we have the 3-velocity dxi vi = : (19) dt This is clearly not Lorentz invariant since time and space are not being treated on the same footing. The t coordinate is no longer fixed in special relativity, so derivatives with respect to time are no longer well determined. In special relativity, we can define a velocity. The definition of 4-velocity is dxµ uµ ≡ ; (20) dτ where τ is the proper time. The 4-velocity allows us to take derivatives of an arbitrary function f with respect to the proper time in a convenient way df @f dxµ = = (@ f)uµ: (21) dτ @xµ dτ µ Note that derivatives generate a downstairs index. For a particle at rest, the velocity is 0 1 1 B C µ B0C u = B C : (22) (rest) B0C @ A 0 We have µ ν u · u = gµνu u = 1: (23) 6 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 The dot products of vectors are Lorentz invariant.

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