Quantum Mechanics for Mathematicians

wave equation will in general be reducible. Irreducible such representations will
be the objects corresponding to elementary particles. This chapter will deal
with the Lorentz group itself, chapter 41 with its representations, and chapter
42 will move on to the Poincar ́e group and its representations.

40.1 Minkowski space

Special relativity is based on the principle that one should consider space and
time together, and take them to be a four dimensional spaceR^4 with an indef-
inite inner product:

Definition(Minkowski space). Minkowski spaceM^4 is the vector spaceR^4
with an indefinite inner product given by

(x,y)≡x·y=−x 0 y 0 +x 1 y 1 +x 2 y 2 +x 3 y 3

where(x 0 ,x 1 ,x 2 ,x 3 )are the coordinates ofx∈R^4 ,(y 0 ,y 1 ,y 2 ,y 3 )the coordi-
nates ofy∈R^4.

Digression.We have chosen to use the−+ ++instead of the+−−−sign
convention for the following reasons:

• Analytically continuing the time variablex 0 toix 0 gives a positive definite
  inner product.


• Restricting to spatial components, there is no change from our previous
  formulas for the symmetries of Euclidean spaceE(3).


• Only for this choice will we have a real (as opposed to complex) spinor
  representation (sinceCliff(3,1) =M(4,R) 6 = Cliff(1,3)).


• Weinberg’s quantum field theory textbook [99] uses this convention (al-
  though, unlike him, we’ll put the 0 component first).
  This inner product will also sometimes be written using the matrix

ημν=









−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1









as

x·y=

∑^3

μ,ν=0

ημνxμyν

Digression(Upper and lower indices).In many physics texts it is conventional
in discussions of special relativity to write formulas using both upper and lower
indices, related by

xμ=

∑^3

ν=0

ημνxν=ημνxν