U101 Lorentz Transformation

This routine transforms momentum and energy of a particle from one Lorentz-frame to another.

Seen from the reference system $\Sigma$ , the other system $\Sigma \text{'}$ has the velocity $\beta$ , with $\eta =\; \gamma \beta$ .

If a rest mass M is tied to system $\Sigma \text{'}$ , with energy E and momentum $P$ , we have:

$\beta =P/E,\eta =P/M,\; \gamma =\; E/M.$

The momentum and energy of a particle with mass m is

in system $\Sigma$	$p$	and	$e\; =p2+m2$ ,
in system $\Sigma \text{'}$	$p\text{'}$	and	$e\text{'}\; =p\text{'}2+m2$ .

Structure:

SUBROUTINE subprogram
User Entry Names: LOREN4

Usage:

    CALL LOREN4(S,A,X)

calculates the transformed 4--vector $X=\; (p\text{'},e\text{'})$ .
LOREN4 contains one square-root to derive M from P and E.

Method:

If we split $p=p$_L+ p_T

into components parallel and normal to $\beta$ , where

$p$_L= {p ηη²} η, p_T= p-p_L,

we can write the transformations as

$p\text{'}$_L= γ p_L-η e, p'_T= p_T, e' = γ e - η p

and get

$p\text{'}$	$=$	$p+\; (\gamma -1)p$L- e η
	$=$	$p+\eta \hspace{0.17em}((\gamma -1)p\hspace{0.17em}\eta /\eta 2-\; e)$
	$=$
	$=$	$p+P\hspace{0.17em}(p\hspace{0.17em}P/(E+M)\; -e)/M,$
$e\text{'}$	$=$	$\gamma e\; -\eta \hspace{0.17em}p$
	$=$	$(\; eE-p\hspace{0.17em}P)/M.$

$•$

U102