By Moshe Carmeli; Shimon Malin

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Throughout the text the term representation stands for continuous representation (unless otherwise stated)]. It then follows that if x (t) and A (t) are vector and operator functions, respectively, which are continuous in a closed bounded set D, then the numerical functions I x (t) I and I A (t) I are bounded in that set. 3 can be generalized to infinite dimensions as follows. Let U be an isometric operator mapping R onto itself, then U is called a CHAPTER 2. REPRESENTATION THEORY 32 unitary operator in R.

This gives xi0 = x° cosh x'1 = x° sinh /i. 8) 40 CHAPTER 3. 10) c. Accordingly we obtain from Eq. 9) cos h 1 ,= 1 sink V' = (3 . 11b) Using these results in Eq. 7) then yields ( A= 1 -Q 1 - Qz -Q 1 _'32 - #2 l - Q2 1 0 1 0 0 0 oof 0 0 1 0 0 1 ( 3 . 12 ) for the matrix of the Lorentz transformation. 13) for the inverse matrix describing the inverse Lorentz transformation. 1. ELEMENTS OF SPECIAL RELATIVITY y =y, Z=z. We also obtain ct = x ct' +,3x' 1-,Q2 , x'+/3ct' y=y, z=z. 15a) ( 3 . 15c) for the inverse transformation from the coordinates x'µ back to xµ.

1 CHAPTER 3. THE LORENTZ AND SL(2,C) GROUPS Orthochronous Lorentz Transformation From Eq. 43) and taking a =,8 = 0, one obtains (A°°)2 - (A 0)2 - (A20)2 - (A'0)2 = 1. 45) in which case the transformation is called orthochronous, or A°0 < -1. 46) The aggregate of all orthochronous Lorentz transformations provides a sub- group of the Lorentz group. The aggregate of all proper, orthochronous, Lorentz transformations also provides a group which is a subgroup of the Lorentz group. In the following we will be concerned with the group of all proper, orthochronous, Lorentz transformations.