# What is the proper definition of Schwinger functions?

In the book Mathematical Aspects of Quantum Field Theory (2010) on page 159-160 in chapter 6.6 Wick rotations and axioms for Euclidean QFT the following is stated:

We have seen earlier in this chapter that the Wightman functions $$W(x_1,...,x_n) = ⟨Ω,φ(x_1)···φ(x_n)Ω⟩$$ can be used to completely reconstruct a scalar bosonic field theory. These functions, as it turns out, can be analytically continued to the so-called Schwinger points $z_j = (ix^0_j , x^1_j , x^2_j , x^3_j )$, provided the points $x_j ∈ \mathscr{R}^4$ are such that $x_j\ne x_j$ if $j\ne k$. The map $(\mathbb{R}^4)^n → (i\mathbb{R}×\mathbb{R}^3)^n$ given by $(x_1,x_2 ...,x_n) → (z_1,z_2,...,z_n)$ is called a Wick rotation. The functions $$S_n(x_1,x_2,...,x_n) = W(z_1,z_2,...,z_n)$$ are called Schwinger functions.

(I did not make the typos it is as it is in the book)

But this makes it seems like the Wightman functions are from $(\mathbb{R}^4)^n$ to $\mathbb{C}$ whereas previously in the book they were introduced as the following from page 138

Definition 6.2 The $n$-th Wightman correlation function of a quantum field theory satisfying the Wightman axioms is a function $W_n : \mathscr{S} (\mathbb{R}^4)×\mathscr{S} (\mathbb{R}^4)× ···\mathscr{S}(\mathbb{R}^4) → \mathbb{C}$ given by $$W_n(f_1,f_2,...,f_n) = ⟨ψ_0,φ(f_1)φ(f_2)···φ(f_n)ψ_0⟩,$$ for all $f_j ∈ \mathscr{S} (\mathbb{R}^4)$, where $ψ_0$ is the vacuum vector of the theory and $φ$ is its field operator.

## 1 Answer

The Wightman functions are distributions. They are similar to the Dirac delta, in that they only make sense as functions "inside an integral". However, one can show that these distributions are the boundary values of genuine, complex analytic functions defined on some open subset of $$(\mathbb{C}^4)^n$$. Restricting these analytic functions to $$(i\mathbb{R} \times \mathbb{R}^3)^n$$ yields the Schwinger functions.

See e.g. https://journals.aps.org/pr/pdf/10.1103/PhysRev.101.860 for an explanation of how the Wightman functions can by analytically continued using a Fourier-Laplace transform.