I) Let us here prove the invariance of the two $3$-forms

$$\tag{1} p^0 dq^1 \wedge dq^2 \wedge dq^3\qquad \text{and}\qquad
\frac{dp_1 \wedge dp_2 \wedge dp_3}{p^0} $$

under (restricted) Poincare transformations. As a consequence, the volume-form $dq^1 \wedge dq^2 \wedge dq^3 \wedge dp_1 \wedge dp_2 \wedge dp_3$ is also an invariant.

Let $c=1$. Here we will assume:

The metric is $\eta_{\mu\nu}={\rm diag}(-1,+1,+1,+1)$.

The $q^{\mu}=(t, {\bf q})$ and $p^{\mu}=(p^0, {\bf p})$ transform under Poincare transformations as an affine and a linear $4$-vector, respectively.

All particles have the same rest-mass $m_0\geq 0$. In particular,
$$\tag{2} p^0 ~=~ \sqrt{{\bf p}^2+m_0^2 }.$$

The momentum is kinetic ${\bf p}~=~p^0{\bf v}$.

Since the two $3$-forms (1) are clearly invariant under translation and rotations, it is enough to consider a Lorentz-boost along the $q^1$-axis. This follows because

$$ \tag{3} p^{0}dq^1, \quad dq^2, \quad dq^3, \quad \frac{dp_1\wedge dp_2 \wedge dp_3}{p^0}, $$

are all invariant under boost along the $q^1$-axis. Only the invariance of the first item $p^{0}dq^1$ 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 $({\bf q}_{(0)},{\bf p}_{(0)})$ in phase space at $t=t_{(0)}=\overline{t}_{(0)}$. Because of translation symmetry, we may assume that the point ${\bf q}_{(0)}=\overline{\bf q}_{(0)}$ is a common origin for the two coordinate systems (one barred and one un-barred) of the Lorentz transformation at $t=t_{(0)}=\overline{t}_{(0)}$. Let us define

$$\tag{4} {\bf x}(\Delta t)~:=~{\bf q}(t)-{\bf q}_{(0)}, \qquad
\Delta t~:= t-t_{(0)}, $$

and
$$\tag{4'} \overline{\bf x}(\overline{\Delta t})~:=~\overline{\bf q}(t)-\overline{\bf q}_{(0)}, \qquad
\overline{\Delta t}~:= \overline{t}-\overline{t}_{(0)}. $$

We imagine that we observe an infinitesimally small space-time (and energy-momentum) region around the fixed point $(q^{\mu}_{(0)},p^{\nu}_{(0)})$. 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 $p^{\mu}=p^{\mu}_{(0)}$ (and velocity ${\bf v}={\bf v}_{(0)}$). Then

$$\tag{5} {\bf x}(\Delta t)~=~{\bf v}\Delta t+ {\bf x}_0 ,\qquad
{\bf v}~=~\frac{{\bf p}}{p^0}, \qquad
{\bf x}_0~=~d{\bf q}, $$

and
$$\tag{5'} \overline{\bf x}(\overline{\Delta t})~=~\overline{\bf v}\overline{\Delta t}+ \overline{\bf x}_0,\qquad
\overline{\bf v}~=~\frac{\overline{\bf p}}{\overline{p}^0},
\qquad \overline{\bf x}_0~=~d\overline{\bf q}. $$

The Lorentz transformation reads
$$ \overline{\Delta t}~=~\gamma(\Delta t-\beta x^1(\Delta t)), \qquad
\overline{x}^1(\overline{\Delta t})~=~\gamma(x^1(\Delta t)-\beta \Delta t), $$ $$\tag{6} \qquad \overline{x}^2(\overline{\Delta t})~=~x^2(\Delta t),
\qquad \overline{x}^3(\overline{\Delta t})~=~x^3(\Delta t), $$

and

$$\tag{7}p^0~=~\gamma(\overline{p}^0+\beta \overline{p}^1) , \qquad p^1~=~\gamma(\overline{p}^1+\beta \overline{p}^0), \qquad
p^2~=~\overline{p}^2, \qquad p^3~=~\overline{p}^3 .$$

Eqs. (5) and (5') can only both hold if the following well-known relativistic formulas hold

$$\tag{8} v^1~=~\frac{\beta+\overline{v}^1}{1+ \beta\overline{v}^1},
\quad v^2~=~\overline{v}^2,\quad v^3~=~\overline{v}^3,
\quad\text{(rel. velocity addition)} $$

and

$$\tag{9} x^1_0~=~\frac{\overline{x}^1_0}{\gamma(1+ \beta\overline{v}^1)},
\quad x^2_0~=~\overline{x}^2_0, \quad x^2_0~=~\overline{x}^2_0,
\quad \text{(length contraction)}. $$

On the other hand the first eq. in (7) yields

$$\tag{10} \frac{p^0}{\overline{p}^0}~=~\gamma(1+ \beta\overline{v}^1). $$

Combining the above equations yields the invariance of the first item $p^{0}dq^1=\overline{p}^{0}d\overline{q}^1$ 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
$$\tag{11} \omega ~=~\sum_{i=1}^3 dp_i \wedge dq^i $$

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