For which values of lambda the euclidean two-point function $(p^2 +m^2)^{-\lambda}$ is reflection positive

Question

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..

Answer 1

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.