Algebra of four-vectors

Four-momentum is conserved as a four-vector, not as separate sums of energy and three-momentum in an arbitrary frame. Because FourVector subtypes FieldVector{4}, Julia’s +, -, and scalar * act component-wise — exactly what you need for building total momenta, decay kinematics, and linear combinations in matrix elements.

The Minkowski inner product (invariant scalar product) is dot, with the same $(+,-,-,-)$ signature as the mass formula.

using FourVectors
using LinearAlgebra
using Test

p = FourVector(10.0, 0.0, 0.0; E = 11.0)   # nearly collinear with +x
q = FourVector(-3.0, 1.0, 2.0; M = 2.0)    # on-shell particle of mass 2

4-element FourVector{Float64} with indices SOneTo(4):
 -3.0
  1.0
  2.0
  4.242640687119285

Addition and subtraction

p + q adds each component: $(p+q)^\mu = p^\mu + q^\mu$. In a decay or splitting $P \to p_1 + p_2 + \cdots$, momentum conservation reads $\sum_i p_i^\mu = P^\mu$.

Important: $m(P) \neq m(p_1) + m(p_2)$ in general. Only the invariant $m^2 = p\cdot p$ is computed from the full four-vector; adding masses is not meaningful physics.

sum_pq = p + q
diff_pq = p - q

@test sum_pq.px ≈ p.px + q.px
@test sum_pq.E ≈ p.E + q.E
@test diff_pq ≈ p - q
@test (p - q) + q ≈ p

Test Passed

Typical pattern: one daughter is reconstructed, the other is inferred by subtraction in the same frame:

parent = FourVector(1.0, 1.0, 30.0; M = 0.135)   # e.g. π⁰ in GeV
daughter = FourVector(0.2, 0.1, 5.0; E = 5.01)   # photon or track candidate
other = parent - daughter

@test parent ≈ daughter + other

Test Passed

Here other is the four-momentum of the complementary system; it is on-shell only if daughter was exact.

Scalar multiplication

Multiplying by a real number scales all components. A common trick for massless particles is to start from a light-like direction $(\hat{n}, 1)$ in arbitrary units and multiply by an energy scale so that $m^2 \approx 0$ after boosts (see the π⁰ decay tutorial).

scaled = 0.5 * parent
@test scaled.px ≈ 0.5 * parent.px
@test scaled.E ≈ 0.5 * parent.E
@test parent * 2 ≈ 2 * parent

Test Passed

Minkowski inner product

dot(p, q) = E_1 E_2 - \vec{p}_1\cdot\vec{p}_2 (ordinary Euclidean dot for the spatial parts). For one four-vector, dot(p, p) = m^2(p) — this is the on-shell condition checker.

The product is Lorentz invariant: the same number in every inertial frame, unlike $E$ or $p_z$ alone.

@test dot(p, p) ≈ mass2(p)
@test dot(p, q) ≈ p.E * q.E - (p.px * q.px + p.py * q.py + p.pz * q.pz)

Test Passed

Illustration: reconstructing the parent mass from a sum of daughters only works when the daughters sum to the parent four-vector.

(mass(parent), mass(daughter + other), dot(parent, parent), scaled.E)

(0.13499999999968806, 0.13499999999968806, 0.01822499999991578, 15.016809123445633)