I) Let us here prove the invariance of the two 3-forms
p0dq1∧dq2∧dq3anddp1∧dp2∧dp3p0
under (restricted) Poincare transformations. As a consequence, the volume-form dq1∧dq2∧dq3∧dp1∧dp2∧dp3 is also an invariant.
Let c=1. Here we will assume:
The metric is ημν=diag(−1,+1,+1,+1).
The qμ=(t,q) and pμ=(p0,p) transform under Poincare transformations as an affine and a linear 4-vector, respectively.
All particles have the same rest-mass m0≥0. In particular,
p0 = √p2+m20.
The momentum is kinetic p = p0v.
Since the two 3-forms (1) are clearly invariant under translation and rotations, it is enough to consider a Lorentz-boost along the q1-axis. This follows because
p0dq1,dq2,dq3,dp1∧dp2∧dp3p0,
are all invariant under boost along the q1-axis. Only the invariance of the first item p0dq1 on the list (2) is not completely obvious or well-known, so let us concentrate on that one. The derivation essentially follows Ref. 1. Consider an arbitrary fixed point (q(0),p(0)) in phase space at t=t(0)=¯t(0). Because of translation symmetry, we may assume that the point q(0)=¯q(0) is a common origin for the two coordinate systems (one barred and one un-barred) of the Lorentz transformation at t=t(0)=¯t(0). Let us define
x(Δt) := q(t)−q(0),Δt :=t−t(0),
and
¯x(¯Δt) := ¯q(t)−¯q(0),¯Δt :=¯t−¯t(0).
We imagine that we observe an infinitesimally small space-time (and energy-momentum) region around the fixed point (qμ(0),pν(0)). Since we are only interested in first-order variations in positions, it is enough to work to zero-order variations in momentas. In other words, we can imagine all particles travel with the same constant energy-momentum pμ=pμ(0) (and velocity v=v(0)). Then
x(Δt) = vΔt+x0,v = pp0,x0 = dq,
and
¯x(¯Δt) = ¯v¯Δt+¯x0,¯v = ¯p¯p0,¯x0 = d¯q.
The Lorentz transformation reads
¯Δt = γ(Δt−βx1(Δt)),¯x1(¯Δt) = γ(x1(Δt)−βΔt), ¯x2(¯Δt) = x2(Δt),¯x3(¯Δt) = x3(Δt),
and
p0 = γ(¯p0+β¯p1),p1 = γ(¯p1+β¯p0),p2 = ¯p2,p3 = ¯p3.
Eqs. (5) and (5') can only both hold if the following well-known relativistic formulas hold
v1 = β+¯v11+β¯v1,v2 = ¯v2,v3 = ¯v3,(rel. velocity addition)
and
x10 = ¯x10γ(1+β¯v1),x20 = ¯x20,x20 = ¯x20,(length contraction).
On the other hand the first eq. in (7) yields
p0¯p0 = γ(1+β¯v1).
Combining the above equations yields the invariance of the first item p0dq1=¯p0d¯q1 on the list (2).
II) Comments:
Part I discusses the local Poincare invariance. An integrated version therefore also exists (with appropriate change of integration regions under Poincare transformations).
Part I concerns systems consisting of particles of a single kind only. The generalization to mixtures is e.g. partially discussed in Ref. 2.
Perhaps surprisingly, a similar proof as in part I shows that the symplectic 2-form
ω = 3∑i=1dpi∧dqi
is not invariant under restricted Poincare transformations.
References:
- J. Goodman, Topics in High-Energy Astrophysics, 2012, p.12-13.
- S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, Relativistic kinetic theory, 1980.
This post imported from StackExchange Physics at 2014-04-08 05:11 (UCT), posted by SE-user Qmechanic