1. y x z S y' x' z' S'S' v x  = (x 0, x 1, x 2, x 3 ) (ct, x, y, z) Relativistic (4-vector) Notation compare: x i = (x 1, x 2, x 3 ) = (x, y, z)  =  = v/c 1  1  2 t' =  (t-vx/c 2 ) x' =  (x-vt) y' = y z' = z which can be recast as: ct' =  ct  x x' =  x  v/c)ct x o =  x o  x 1 x 1 =  x 1  x o which should be compared to the coordinate-mixing of rotations: x' = xcos   ysin  y' = xcos   ysin 

2. Noticing: we write: x0'x1'x2'x3'x0'x1'x2'x3' compared to rotations about the z-axis: x0'x1'x2'x3'x0'x1'x2'x3' x0 x1 x2 x3x0 x1 x2 x3     0 0     0 0 0 0 1 0 0 0 0 1 = x0 x1 x2 x3x0 x1 x2 x3   0 0 0 cos  sin  0 0  sin  cos  0 0 0 0 1 = exactly same form!

3. World line of particle moving in straight line along the x-direction event ct x vt 1 ct 1 ct ´ x´x´  The Lorentz transformation is not exactly a ROTATION, but mechanically like one. We will consider it a generalized rotation. But now, the old “dot product” will no longer do. It can’t guarantee invariance for many of the quantities we know should be invariant under such transformations. The fix is simple and obvious…

4. So that under a Lorentz transformation

6. To help keep track of the sign conventions, we introduce the metric tensor:   0 0   0 0 0 0  1 0 0 0 0  = g  Then our “dot product” becomes and a lowered index means the metric tensor has been applied Notice we argue: or contra-variant 4-vector co-variant 4-vector xxxx

7. summed over   The lowered index just means that  is in the appropriate form to “dot” into a vector Since the  is raised, the above multiplication gives (x´ )  Notice: means  has been multiplied by the metric tensor!

8. And just what does g  look like?

9. Now notice that  ( g  ) = = g

10. Also notice:  T =  which means g T = g or but (g  ) T = g  This is because: (g  ) T =  T g T =  g = g 

11. So as an exercise in using this notation let’s look at The indices indicate very specific matrix or vector components/elements. These are not matrices themselves, but just numbers, which we can reorder as we wish. We still have to respect the summations over repeated indices!  ( g  ) = g And remember we just showed  i.e. All dot products are INVARIANT under Lorentz transformations.

12. even for ROTATIONS as an example, consider rotations about the z-axis

13. The relativistic transformations: suggest a 4-vector that also transforms by so should be an invariant!

14. EcEc In the particle’s rest frame: p x = ? E = ? p  p  = ? 0 mc 2 m2c2m2c2 In the “lab” frame: =   mv =  =  mc so

15. Limitations of Schrödinger’s Equation 1-particle equation 2-particle equation: mutual interaction But in many high energy reactions the number of particles is not conserved! n  p+e + + e n+p  n+p+3  e  + p  e  + p + 6  + 3 