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.
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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 t µ x x = . (11) y 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.
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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 γ vγ 0 0 µ¯ vγ γ 0 0 Λ (v) = . (12) µ 0 0 1 0 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 ≡ √ . (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.
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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 1 0 0 0 0 −1 0 0 gµν = . (14) 0 0 −1 0 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.
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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 1 µ 0 u = . (22) (rest) 0 0
We have µ ν u · u = gµνu u = 1. (23)
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The dot products of vectors are Lorentz invariant. If the above equation is true in one frame, it is true in all. The individual components of the 4-velocity will change, even if the magnitude of the 4-velocity does not. Boosted by an arbitrary velocity, the 4-velocity can be computed as γ 1 µ γv u = . (24) γv2 γv3
The component x0 t u0 = = (25) dτ dτ tells us how quickly the particle is moving through time. This is simply γ, the time dilation factor.
3.2 The 4-Momentum
A 4-momentum of a particle of mass m is defined as
pµ = muµ. (26)
We have p · p = m2. (27)
The timelike (zeroth) component of the 4-momentum is just the energy. We have E mγ 1 1 µ p mγv p = = . (28) p2 mγv2 p3 mγv3 We also have
p2 = p · p µ = gµνp pν = (p0)2 − |~p|2 = E2 − |~p|2 = m2. (29)
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If we insert c to get the correct dimensionality we have
E2 = (mc2)2 + (|~p|c)2. (30)
This equation immediately tells us that light has no mass but it still carries momentum. The idea that light carries momentum was suggested by Johannes Kepler (1619) who noted that the tail of a comet points away from the sun. For a massless particle the above equation tells us
E = ±|~p|. (31)
What is the meaning of particles with negative energy? Logic tells us that only the positive energy case is valid. Quantum field theory (QFT) provides the consistent explanation of negative energy states.
4 Special Relativity and Electromagnetism
In electromagnetism we have the scalar potential φ (or φ/c) and the vector potential A~ = (A1,A2,A3). Let us define a 4-vector ! φ Aµ ≡ , (32) A~ and also its lower-index cousin Aµ ≡ φ −A~ . (33)
We know that, in general, the electric field and magnetic field change with time. That is, E~ = E~ (~x,t) and B~ = B~ (~x,t). Let us look at the two source-free Maxwell’s equations
∂B~ ∇ × E~ + = 0, (34) ∂t ∇ · B~ = 0. (35)
We can use scalar and vector potentials to solve both of these equations
∂A~ E~ = −∇φ − , (36) ∂t B~ = ∇ × A.~ (37)
Upon using the 4-derivative ∂ ∂ = , ∇ (38) µ ∂(ct)
8 / 10 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 and the 4-vector φ A = , −A~ , (39) µ c we can construct an anti-symmetric tensor
Fµν = ∂µAν − ∂νAµ. (40)
This tensor is known as the electromagnetic tensor or Faraday tensor. We can compute the components of this tensor explicitly
1 ∂(−A ) ∂φ/c E F = x − = x , 01 c ∂t ∂x c ∂(−A ) ∂(−A ) F = y − x = −B . (41) 12 ∂x ∂y z . . .
Thus we have the electromagnetic tensor
Ex Ey Ez 0 c c c Ex − c 0 −Bx By Fµν = . (42) − Ey B 0 −B c z x Ez − c −By Bx 0
We also have Ex Ey Ez 0 − c − c − c Ex µν µα νβ c 0 −Bx By F = g g Fαβ = . (43) Ey B 0 −B c z x Ez c −By Bx 0 Under Lorentz transformations this tensor transforms as
0µν µ ν αβ F = Λ αΛ βF . (44)
Maxwell’s equations can be rewritten as two tensor equations. We have the Maxwell’s equations
ρ ∇ · E~ = (Gauss’ law), (45) 0 ! ∂E~ ∇ × B~ = µ J~ + (Ampere’s law), (46) 0 0 ∂t ∇ · B~ = 0, (Gauss’s law for magnetism), (47) ∂B~ ∇ × E~ = − (Faraday’s law). (48) ∂t
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Combining two inhomogeneous Maxwell’s equations (Gauss’ law and Ampere’s law) we get
αβ β ∂αF = µ0J (Gauss-Ampere law). (49)
Similarly, combining two homogeneous equations (Gauss’ law for magnetism and Faraday’s law of induction) we get 1 ∂ αβγδF = 0 (Gauss-Faraday law). (50) α 2 γδ
In the above J β is the 4-current, J β = (cρ, J~) and αβγδ is the Levi-Civita symbol. Each of the above tensor equations corresponds to four scalar equations, one for each value of β.
References
[1] Dave Goldberg, The Standard Model in a Nutshell, Princeton University Press (2017).
[2] David Griffiths, Introduction to Elementary Particles, 2nd edition, Wiley-VCH (2008).
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