# How are Schwinger and Wightman functions used in practice?

In Reed & Simon's Methods of Mathematical Physics Volume II, they define a (Hermitian scalar) quantum field theory to be the quadruple $$\langle \mathcal{H}, U, \varphi, D\rangle$$ that satisfies the Wightman axioms. Here $$\mathcal{H}$$ is a separable Hilbert space, $$U$$ is a strongly continuous unitary representation of $$\mathcal{H}$$ of the restricted Poincare group, $$D \subset \mathcal{H}$$ is dense, and $$\varphi: \mathscr{S}(\mathbb{R}^4) \rightarrow T(\mathcal{H})$$ where $$\mathscr{S}$$ is the Schwartz space and $$T(\mathcal{H})$$ is the set of all (unbounded) operators on $$\mathcal{H}$$.

They define the Wightman functions as the operator valued distributions $$\{\mathcal{W}_n\}$$ where $$\mathcal{W}_n (f_1, \ldots, f_n) = \big(\psi_0, \varphi(f_1), \cdots \varphi(f_n)\psi_0\big)$$ and $$\psi_0$$ is the vacuum state in $$\mathcal{H}$$.

From what I have read, Schwinger functions $$\{\mathcal{S}_n\}$$ are defined are just the analytic continuation of $$\{\mathcal{W}_n\}$$ to imaginary time. Most resources present both sets of distributions as functions on $$\mathbb{R}^4$$ so the Wick rotation is easy to visualize, but this is less obvious when the domain is a function space and not space-time.

Based on the definition given by Reed and Simon it seems we are often given $$\varphi$$ so $$\{\mathcal{W}_n\}$$ is easily defined. When I first started studying axiomatic QFT my impression was that the axioms are used to construct the quantum field $$\varphi$$ from scratch. However, looking at these axioms, my impression now is that one instead constructs $$\varphi$$ using traditional means (for example, from a Lagrangian), and then checks if all the axioms are satisfied.

My questions can be summarized as follows:

1. Given that both $$\mathcal{W}_n$$ and $$\mathcal{S}_n$$ are distributions, how does one Wick rotate between them?

2. In reference to the last paragraph above, how does this work in practice? Where does $$\varphi$$ come from?

3. Lastly, is there some sort of uniqueness result for the Wightman functions? That is, if I have two sets of Wightman functions $$\{\mathcal{W}_n\}$$ and $$\{\tilde{\mathcal{W}}_n\}$$, do they correspond to the same scalar field (up to unitary equivalence)? And thus does are Wightman functions only applicable to scalar fields?

• Feb 16 at 6:04

1. Let's write $$x=(\mathbf{x},t)$$ for spacetime points in $$\mathbb{R}^d$$ so the first $$d-1$$ coordinates $$(x_1,\ldots,x_{d-1})=\mathbf{x}$$ pertain to space and the last coordinate $$x_d=t$$ pertains to time. Then the $$n$$-point Wightman function is $$W_n(\mathbf{x}_1,t_1;\ldots;\mathbf{x}_n,t_n):= \langle\Omega|e^{it_1 H}\phi(\mathbf{x}_1,0) e^{-i(t_1-t_2)H}\phi(\mathbf{x}_2,0)e^{-i(t_2-t_3)H}\cdots e^{-i(t_{n-1}-t_n)H}\phi(\mathbf{x}_n,0)e^{-it_n H}|\Omega\rangle$$ $$=\langle\Omega|\phi(\mathbf{x}_1,0) e^{-i(t_1-t_2)H}\phi(\mathbf{x}_2,0)e^{-i(t_2-t_3)H}\cdots e^{-i(t_{n-1}-t_n)H}\phi(\mathbf{x}_n,0)|\Omega\rangle$$ where $$H$$ is the Hamiltonian and $$|\Omega\rangle$$ is the vacuum state, of the possibly interacting theory. The exponentials near the vacuum disappeared at both ends because $$\Omega$$ is an eigenvector of $$H$$ for the eigenvalue zero.
On the other hand, and assuming now that $$\tau_1<\cdots<\tau_n$$, the Schwinger functions are $$S_n(\mathbf{x}_1,\tau_1;\ldots;\mathbf{x}_n,\tau_n):=W_n(\mathbf{x}_1,i\tau_1;\ldots;\mathbf{x}_n,i\tau_n)$$ $$=\langle\Omega|\phi(\mathbf{x}_1,0) e^{-(\tau_2-\tau_1)H}\phi(\mathbf{x}_2,0)e^{-(\tau_3-\tau_2)H}\cdots e^{-(\tau_{n}-\tau_{n-1})H}\phi(\mathbf{x}_n,0)|\Omega\rangle$$ In general, analytic continuation of functions in the complex domain is one of those riddles wrapped in a mystery etc. See the Voronin Universality Theorem for a cautionary example. To do analytic continuation one needs some very strong structural property of the function at hand. More often than not, this is a positivity property. For example if $$X$$ is a random variable with moments growing as $$C^n n!$$ then the characteristic function $$\mathbb{E}(e^{izX})$$ is obviously analytic in a disc $$|z|<\varepsilon$$ but what is remarkable is that it is then automatically analytic in the strip $$|\Im z|<\varepsilon$$. The reason is the elementary inequality $$\left|\int f(\omega)\ {\rm d}\mu(\omega)\right|\le \int |f(\omega)|\ {\rm d}\mu(\omega)$$ which holds because the measure $$\mu$$ is positive. Here the needed positivity property is that of the self-adjoint operator $$H$$.
2. What you said in that last paragraph is correct. Axiomatic QFT gives you zero information as to how to produce/construct the interacting field. You need to do that with completely different methods. The area of mathematics with tackles this problem is Constructive QFT. A noteworthy example where one has a "construction of the field" $$\varphi$$ is the 2d Ising CFT. There the Schwinger functions are given explicitly by $$S_n(x_1,\ldots,x_n)=\sqrt{\sum_{q} \prod_{1\le i where $$q=(q_1,\ldots,q_n)$$ is summed over all "charge-neutral" elements of $$\{-1,1\}^{n}$$, i.e., those satisfying $$\sum_{i=1}^{n}q_i=0$$. As far as I know, nobody knows how to prove the Axioms, specifically Osterwalder-Schrader positivity, from the above formula.