From Classical Mechanics to Quantum Field Theory

222 From Classical Mechanics to Quantum Field Theory. A Tutorial

Letθbe the Euclidean-time reflection symmetry defined byθ(x,τ)=(x,−τ).
The action ofθonR^4 induces a transformation on the classical fields test functions
given byθf ̃(x)=f ̃(θx) and in the multivariable test functions ̃fn∈Sin a similar
wayθ ̃fn(x 1 ,x 2 ,...,xn)= ̃fn(θx 1 ,θx 2 ,...,θxn).

• E1(Regularity):The Schwinger functionsSnare tempered distributions
  inS, satisfying the reflection reality condition
  Sn( ̃fn)∗=Sn(θ ̃fn∗).


• E2(Permutation symmetry):The Schwinger functions are symmetric un-
  der permutations, i.e.
  Sn(f ̃σ(1),f ̃σ(2),...,f ̃σ(n))=Sn(f ̃ 1 ,f ̃ 2 ,...,f ̃n)

for any permutationσ∈Sn.

• E3(Euclidean invariance):The Schwinger functions are covariant under
  Euclidean transformations, i.e.
  Sn( ̃fn(Λ,a))=Sn( ̃fn)
  for any Euclidean transformation (Λ,a)∈E 4 =ISO(4).


• E4(Reflection positivity):For any family of multivariable functions, test
  functionsf ̃ni∈S(R^4 +ni),i=0, 1 , 2 ,...,nthe following inequality^12
  ∑n

i,j=0

Sni+nj(θ ̃fn∗i. ̃fnj)≥ 0 (3.51)

holds.

• E5(Cluster property).For any pair of multivariable functions test func-
  tions ̃fn∈S(R^4 n), ̃fm∈S(R^4 m), we have that

σlim→∞Sn+m( ̃fn. ̃fm(I,σ))=Sn( ̃fn)Sm( ̃fm),

where (I,σ) is the Euclidean time translation (I,σ)(x,τ)=(x,τ+σ).

These Euclidean principles follow from the field theory principles introduced
in section 3. The Euclidean principles E1-E3 are a straightforward consequence
of the Minkowskian principles. The permutation symmetry of Euclidean fields
(principle E2) follows from the fact that

[φE(f 1 ),φE(f 2 )] = 0,

(^12) The spaceR (^4) +={(x, τ)∈R (^4) ,τ≥ 0 }isthehalfofthen-dimensional Euclidean spaceR (^4) with
positive Euclidean time.