From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 199

space rotations, e.g.

x′μ=Λμνxν, Λμν=

⎛

⎜⎜

⎝

10 00

0cosθ sinθ 0

0 −sinθcosθ 0

00 01

⎞

⎟⎟

⎠, with 0≤θ≤^2 π (3.10)

and Lorentz transformations, e.g.

x′μ=Λμνxν, Λμν=

⎛

⎜⎜

⎝

γ −γv 00

−γv γ 00

0010

0001

⎞

⎟⎟

⎠, withv∈Rand γ=

√

1 −v^2 (3.11)

which are linear transformations that leave the Minkowski metric invariant

ημν=ΛσμηστΛτν.

There are some extra discrete symmetries which play an important role in field
theory. They are generated by time reversal,

T:(x,t)→(x,−t)

and parity

P:(x,t)→(−x,t).

The whole group of space-time symmetries is the Poincar ́egroup

P=ISO(3,1) ={(Λ,a); Λ∈O(3,1),a∈R^4 },

which contains all these continuous and discrete symmetries and has four connected
components.
The first attempts to make the quantum theory compatible with the theory of
relativity where based on covariant equations of motion like the Maxwell equations
of classical electrodynamics. This leadsto the discovery of Klein-Gordon equation
for scalar fields and the Dirac equation for spinorial fields. Wigner introduced a
completely different approach. If the quantum theories are defined in a Hilbert
space and the relativity is based on Poincar ́e invariance, he conjectured that ele-
mentary quantum particles as the most elementary quantum systems must arise
from irreducible projective representations of the Poincar ́e group. This program
was achieved by Wigner and Mackey and their results show that the simplest ir-
reducible representations are characterized by two numbers which represent the
massm∈R+and the spins∈N/2 of the particle. The case of spin zero corre-
spond to a scalar field satisfying the Klein-Gordon equation. The case of spin^12
corresponds to spinor fields satisfying the Dirac equation, and the case of massless