GitHub

Lorentz transformations

Changing reference frame means a Lorentz transformation on $p^\mu$. This package implements active transformations: the frame stays fixed and the four-momentum is rotated or boosted. That matches the usual HEP habit of “rotating the vector into the $z$ axis” before a longitudinal boost.

All transforms preserve $m^2 = E^2 - |\vec{p}|^2$ (numerically, up to round-off).

using FourVectors
using Test

p = FourVector(0.5, 0.5, 0.5; M = 1.0)
4-element FourVector{Float64} with indices SOneTo(4):
 0.5
 0.5
 0.5
 1.3228756555322954

Spatial rotations (Rx, Ry, Rz)

Rotations act on $(p_x, p_y, p_z)$ only; $E$ is unchanged (orthogonal transformation of the spatial part). Angles are in radians, right-handed, about the lab axes:

  • Rx(p, α) — rotation about $x$
  • Ry(p, θ) — rotation about $y$ (often used to bring $\vec{p}$ into the $x$–$z$ plane)
  • Rz(p, φ) — rotation about $z$ (azimuthal alignment)

Composing with the opposite angle returns the original momentum:

α = 0.4
@test Rx(Rx(p, α), -α) ≈ p
@test Ry(Ry(p, α), -α) ≈ p
@test Rz(Rz(p, α), -α) ≈ p

p_ry = Ry(p, π / 4)
4-element FourVector{Float64} with indices SOneTo(4):
 0.7071067811865475
 0.5
 5.551115123125783e-17
 1.3228756555322954

Longitudinal boost (Bz)

Bz(p, γ) applies a boost along the lab $z$ axis with Lorentz factor $\gamma$. Use boost_gamma(p) (from LorentzVectorBase) when you need the $\gamma$ of an existing four-momentum.

Sign convention: negative $\gamma$ reverses the boost direction ($\beta \to -\beta$ at fixed $|\gamma|$). This is handy when chaining “boost to lab” and “boost back to rest” without recomputing $\beta$.

γ = 2.5
@test Bz(Bz(p, γ), -γ) ≈ p

p_boosted = Bz(p, γ)
4-element FourVector{Float64} with indices SOneTo(4):
 0.5
 0.5
 4.281088913245535
 4.452833062569698

Chaining transforms (|>)

Rx(α), Ry(θ), Rz(φ), and Bz(γ) can be partially applied to obtain unary maps, which compose cleanly in pipelines — the same pattern as in the π⁰ decay tutorial.

p_pipeline = p |> Ry(π / 6) |> Rz(0.2) |> Bz(1.5)
@test p_pipeline isa FourVector
Test Passed

rotate_to_plane — align a reference direction

rotate_to_plane(p, which_z, which_xplus) builds the rotation that:

  1. Rotates the spatial direction of which_z onto lab $+\hat{z}$ (Ry, then Rz), and
  2. Rotates about $z$ so which_xplus lies in the $x$–$z$ plane with $p_x \ge 0$.

Only the directions of the reference four-vectors matter (masses are ignored for the rotation axes). Use this when you want a standard frame where a beam or parent defines $z$ and a second vector fixes the $x$–$z$ plane.

which_z = FourVector(0.0, 0.0, 1.0; M = 2.0)
which_xplus = FourVector(1.0, 1.0, 0.0; M = 0.0)

p_plane = rotate_to_plane(p, which_z, which_xplus)
@test p_plane.py ≈ 0.0 atol = 1e-10
@test p_plane.px ≥ 0.0
Test Passed

transform_to_cmf — go to another particle’s rest frame

transform_to_cmf(p, which_cmf) transforms p into the rest frame of which_cmf: rotate which_cmf onto $+\hat{z}$, then boost along $z$ with $\gamma = \gamma(\text{which\_cmf})$ so that which_cmf has zero spatial momentum.

This is the frame where the total momentum of a two-body system is zero when which_cmf is chosen as one of the participants or as the total four-momentum.

beam = FourVector(1.0, 0.0, 0.0; E = 2.0)
p_cmf = transform_to_cmf(p, beam)
@test p_cmf isa FourVector

(collect(p), collect(p_ry), collect(p_boosted), collect(p_plane), collect(p_cmf))
([0.5, 0.5, 0.5, 1.3228756555322954], [0.7071067811865475, 0.5, 5.551115123125783e-17, 1.3228756555322954], [0.5, 0.5, 4.281088913245535, 4.452833062569698], [0.7071067811865475, 5.551115123125783e-17, 0.5, 1.3228756555322954], [-0.49999999999999994, 0.5, -0.18641234663634787, 1.238850097057134])