# Osterwalder-Schrader reconstruction theorem Many properties of QFTs, such as causality and positivity of scattering cross section, only make sense in Lorentzian signature, but in practice we perform computations and study the theories in Euclidean signature. For this to make sense, one must establish a relation between these theories in Lorentzian and Euclidean signatures. It is known that we can analytically continue any unitary, Lorentz invariant and causal theory in Minkowski spacetime to Euclidean signature in which the new theory satisfies reflection positivity, crossing symmetry, and Euclidean invariance. But for the converse to be true, we need the Osterwalder-Schrader theorem. The **Osterwalder-Schrader reconstruction theorem** states and proves conditions that assure that the Wick rotation is a well-defined isomorphism of quantum field theories on Minkowski and on Euclidean spacetimes. Take Schwinger functions $\mathcal{S}$ at *non-coincidental points* satisfying conditions: - (E1) rotation invariance, translation invariance - (E3) permutation symmetry - (E2) reflection positivity - (E$0^\prime$) growth condition (subtle and introduced only in the second paper) - $|\int dx_1...dx_n\mathcal{S}_n(x_1,...,x_n)f_n(x_1,...,x_n)|\le\sigma_n||f||_{const\cdot n}$ where $\|f\|_{N}=\operatorname{max}_{x}\left(1+|x|^{2}\right)^{\frac{N}{2}} \max _{|\alpha| \leqslant N}\left|\partial^{\alpha} f(x)\right|$, $\sigma_n\le\alpha\cdot (n!)^\beta$ Then, one recovers Wightman functions $\mathcal{W_n}$ satisfying all [[0165 Wightman axioms|Wightman axioms]]. ## Refs - original - [[1973#Osterwalder, Schrader]] small error - [[1975#Osterwalder, Schrader]] fixed the error and remains state of the art - [[book_Haag]] - [nLab](https://ncatlab.org/nlab/show/Osterwalder-Schrader+theorem) - [[Rsc0014 Rychkov TASI2019 Lorentzian CFT]] - gives an overview - also explains why growth conditions is needed there ## Limitations - this works in general QFT, but it is very difficult to check in practice. it requires knowledge of *all* $n$-point function (even if you just want to analytically continue e.g. the 4-pt function) to check the growth condition - even for some interesting CFTs you cannot check this - can circumvent this in CFT by assuming a different set of axioms: bootstrap axioms