For which values of lambda the euclidean two-point function $(p^2 +m^2)^{-\lambda}$ is reflection positive The case $\lambda=1$  is  well  known  free field kernel. What  about  $\lambda$ in  between  0  and  1 ??
... for  $\lambda>1$  I have  a proof  that  the  kernel is  not  reflection  positive ,
but  for  $\lambda<1$  there  is the  so called  fractional Brownian  motion  representation  of  the  corresponding  semigroup describing the fractional  diffusion  generalized ( if  space  dimension  is  greater  zero )  processes..
 A: Good question, about something which is not very well known.
For $\lambda=0$, the two-point function is a delta function and the random field corresponding to the Euclidean functional integral is called white noise. The model is trivially reflection positive, i.e., this is a free (Gaussian) unitary QFT, albeit an ultralocal one which is not very interesting.
For $0<\lambda<1$, the model is also reflection positive, i.e., this is a unitary Euclidean QFT, often called a generalized free field.
The reason for this unitarity property is that one has a convergent integral representation
$$
\frac{1}{(p^2+m^2)^{\lambda}}=
\frac{1}{C_{\lambda}}\times
\int\limits_{0}^{\infty} \frac{1}{p^2+m^2+u}\times \frac{du}{u^{\lambda}}
$$
where
$$
C_{\lambda}=\int\limits_{0}^{\infty}\frac{du}{u^{\lambda}(u+1)}\ .
$$
The above formula is an explicit Källén-Lehmann representation.
So, reflection positivity for the fractional case $0<\lambda<1$ follows, by continuous superposition, from the reflection positivity for the usual case $\lambda=1$.
For more details, see Theorem 7.1 of the article "A PDE construction of the Euclidean $\Phi_{3}^{4}$ quantum field theory" by Gubinelli and Hofmanová.
