198 From Classical Mechanics to Quantum Field Theory. A Tutorial
wherex 0 =ctdenotes the time-like coordinate and
ημν=⎛
⎜⎜
⎝
10 0 0
0 −10 0
00 − 10
00 0− 1
⎞
⎟⎟
⎠.
From now on we shall assume that the speed of lightcis normalized to unit.
The geodesic which connects two points x, yof Minkowski space-time is a
straight line inR^4 and the Minkowski distance betweenxandyis
d(x, y)=(y 0 −x 0 )^2 −(y 1 −x 1 )^2 −(y 2 −x 2 )^2 −(y 3 −x 3 )^2.When this distance vanishes, the geodesic line represents the trajectory of a light
ray connecting the two points.
The Minkowski metric introduces a causal structure in the space-time. The
line connecting two pointsx, yis time-like ifd(x, y)>0; space-like ifd(x, y)<0;
light-like ifd(x, y) = 0; Two pointsx, yare causally separated iffd(x, y)≥0; and
spatially separated iffd(x, y)<0; A causal line connectingx, yis future oriented
ify 0 −x 0 >0; past oriented ify 0 −x 0 <0 (See Figure 3.1).
The space-time symmetries of a relativistic theory are space-time translations,
x′=x+a (3.9)
Fig. 3.1 Light cone and causal structure of Minkowski space-time.