Relativistic kinematics

Question 1

Lorentz transformations

What’s a proper orthocronous Lorentz transformation?

Answer 1

A Lorentz transformation that doesn’t change either the direction of time nor the orientation of space and that consists of 3D rotations and boosts.

• orthocronous: Λ^0(0) ≥ 1, doesn’t change the sign of the time coordinate
• proper: detΛ = 1, leaves the orientation of space unchanged

Question 2

Mandelstam variables

What are Mandelstam variables?

Answer 2

A convenient set of variables to describe the kinematics of 2 → 2 scattering processes:

s ≡ (p(a) + p(b))^2 = (p(c) + p(d))^2 ,
t ≡ (p(a) − p(c))^2 = (p(b) − p(d))^2 ,
u ≡ (p(a) − p(d))^2 = (p(b) − p(c))^2 .

• Lorentz invariant by construction
• only two of the three variables can be independent at once
• s + t + u = m(a)^2 + m(b)^2 + m(c)^2 + m(d)^2
• they satisfy various bounds that determine the physical region in which s, t, u can take values for a physical proces (graphical rep of the allowed range of values: Mandelstam plane (triangle))

Question 3

Mandelstam variables

What is crossing symmetry?

Answer 3

The scattering amplitude for the processes

• a + b → c + d: s-channel,
• a + ¯c → ¯b + d: t-channel,
• a + ¯d → c + ¯b: u-channel,

are part of a single analytic function extending beyond the physical domain of the Mandelstam variables. These are the three wedges outside the Mandelstam triangle. The crossing-symmetry relations: A(s)(s, t, u) = A(t)(t, s, u) = A(u)(u, t, s), that are the scattering amplitudes of the corresponding processes.

Question 4

Invariant phase space

What is the invariant phase space of particles?

Answer 4

The possible states of a spinless particle of mass m are characterised by the four-momenta p^μ that satisfy p^2 = m^2 with positive energy, p0 ≥ m > 0. The phase space is the corresponding domain in R^4: Φ = {p ∈ R^4|p^2 − m^2 = 0 , p0 > 0}.

• infinitesimal phase space: dΩp = d^3p/[(2π)^3 2ε(p)], where ε(p) ≡ √(p^2 + m^2), invariant measure under orthocronous Lorentz trafos
• for n particles: dΦ(n) = ∏(j = 1,n) dΩ(p(j)) (2π)^4 δ^(4)[p(tot) − ∑(j = 1, n) p(j)]
• the two-body phase space: dΦ(2) = dΩ(CM)/(32π^2) √λ(s, m1, m2)/s, where λ(s, m1, m2) = (s − (m1 + m2)^2)(s − (m1 − m2)^2)