From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 223

for any pair of functionsf 1 ,f 2 which follows from the commutation property of
Minkowskian fields for test functions with space-like separated supports (local
causality property P7) and Euclidean rotation invariance. The third Euclidean
principle E3 follows from the positivity of the norm of the state

∑n

i=0

φ( ̃fni)| 0 〉.

Finally the cluster property is a consequence of the uniqueness of the vacuum
assumed in the fourth Minkowskian principle P4.
What is more interesting is that the relativistic QFT can be fully recon-
structed from the Euclidean principles. The proof was achieved by Osterwalder
and Schrader in the early seventies. We will not elaborate in the proof that can
be found in the Simon[ 23 ]and Glimm-Jaffe[ 13 ]books.

3.8 ConformalInvariantTheories

There is a subfamily of field theories which besides the above principles are invari-
ant under a larger symmetry group: the conformal group. Conformal transforma-
tions are space-time transformations which leave the Minkowski metric invariant
up to a scale factor

x′=c(x); ημ,ν′ (x′)=Ω^2 c(x)ημ,ν(x). (3.52)

The group of conformal transformations is an extension of the Poincar ́e symme-
try. Besides the translations (3.9), rotations (3.10) and Lorentz transformations
(3.11) two new type of transformations preserve the Minkowski metric up to a
constant scale factor (3.52):

(i)dilations

x′μ=eσxμ,

defined by any real numberσ;and

(ii)special conformal transformations

x′μ= x

μ−aμx 2

1 − 2 a·x+a^2 x^2

,

defined by any vectoraof Minkowski space-time.

In conformal invariant theories, the 2-point Schwinger function is of the form

S 2 (f 1 ,f 2 )=(f 1 ,Δ−kf 2 ),