From Classical Mechanics to Quantum Field Theory

226 From Classical Mechanics to Quantum Field Theory. A Tutorial

regularity, symmetry and Euclidean covariance principles. Concerning the reflec-
tion positivity property, it is not so evident. Let us check that this is the case to
illustrate the subtleties of the very special Osterwalder-Schrader’s property. Let
us consider the case of an one-particle state with a complex functionf ̃∈S(R^4 +,C)

S 2 (θf ̃∗,f ̃)=

∫

S′(R^4 )

δφe−SE(φ)θφ(f ̃)∗φ(f ̃)=

1

2

(θf, ̃(−∇^2 +m^2 )−^1 f ̃).

Let us defineφ=(−∇^2 +m^2 )−^1 f ̃.Sinceθcommutes with (−∇^2 +m^2 )wehave

(θf, ̃(−∇^2 +m^2 )−^1 f ̃)=(θf,φ ̃ )=((−∇^2 +m^2 )θφ,φ), (3.55)

and since the support off ̃is contained inS(R+^4 ), that ofθf ̃is inS(R^4 −)^13 ,thus,
we can restrict the integral in Eq. (3.55) toS(R^4 −),

(θf,φ ̃ )=(θf,φ ̃ )−=((−∇^2 +m^2 )θφ,φ)−.

By the same reason, (θφ,(−∇^2 +m^2 )φ)−=(θφ,f ̃)−=0and

(θf,φ ̃ )=((−∇^2 +m^2 )θφ,φ)−.

Integrating by parts and using Stokes theorem one gets[ 3 ]

(θf,φ ̃ )=(θφ,(−∇^2 +m^2 )φ)−−

∫

R^3

d^3 x(∂nθφ)∗φ+

∫

R^3

d^3 xθφ∗∂nφ,

where∂nφdenotes the normal derivative ofφat the boundary∂R^4 − =R^3 at
Euclidean timeτ=0ofR^4 −. Now, at the boundaryτ=0,θφ=φand∂nθφ=
−∂nφ,thus,

(θf,φ ̃ )=2Re

∫

R^3

d^3 xθφ∗∂nφ.

Finally, by integrating by parts back we get

Re

∫

R^3

d^3 xθφ∗∂nφ=(∇φ,∇φ)−+(φ,∇^2 φ)−,

and since∇^2 φ=∇^2 (−∇^2 +m^2 )−^1 f ̃=−f ̃+m^2 φ,

(θf,φ ̃ )=2‖∇φ‖^2 −+2m^2 ‖φ‖^2 −≥ 0.

(^13) The spaceR (^4) −={(x, τ)∈R (^4) ,τ≤ 0 }isthehalfofthen-dimensional Euclidean spaceR (^4) with
negative Euclidean time.