nLab Osterwalder-Schrader theorem

Idea

The Osterwalder-Schrader theorem (Osterwalder-Schrader 73) states precise conditions under which Wick rotation between relativistic field theory and Euclidean field theory works.

Rough idea: The Wightman axioms describe how the algebra of observables of a quantum field theory on Minkowski spacetime is generated by quantum fields. The Wightman reconstruction theorem asserts that knowing all correlation functions of all fields in the vacuum state is equivalent to knowing the quantum fields. The Osterwalder–Schrader theorem states conditions that correlation functions on Euclidean spacetime have to satisfy to be equivalent to the correlation functions of a Wightman QFT on Minkowski spacetime.

In this sense the Osterwalder–Schrader 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 spacetime.

Axioms of euclidean field theory

The axioms of euclidean field theory are the euclidean analogue of the Wightman axioms on Minkowski spacetime. The axioms may be formulated for tempered distributions, but we follow the lines of Glimm and Jaffe and define them for $\mathcal{D}'(\mathbb{R}^d)$, the space of distributions that is dual to the space of all smooth functions with compact support, $\mathcal{D}(\mathbb{R}^d)$. In the original paper of Osterwalder and Schrader the axioms are given in terms of the Schwinger functions. Here the axioms are given in a form more directly related to the measure on field space and its characteristic function, rather than the Schwinger functions themselves. This form was first presented by Fröhlich. We define the generating functional on $\mathcal{D}(\mathbb{R}^d)$

$$S(f) := \integral e^{i \phi(f)} d\mu$$

as the inverse Fourier transform of a Borel probability measure $d\mu$ on $\mathcal{D}'(\mathbb{R}^d)$.

• OS0 (analyticity): For every finite set of test functions $f_1, f_2,...f_n$ and complex numbers $z:= (z_1, z_2, ...z_n)$ the function

  $$z \mapsto S(\sum_{k=1}^n z_k f_k)$$

  is entire analytic on $\mathbb{C}^n$.

• OS1 (regularity): For some p with $1 \le p \le 2$ and some constant c the following inequality holds for all test functions f:

  $$| S(f) | \le exp(c \| f \|_{L_1} + \| f\|^p_{L_p})$$

• OS2 (invariance): S is invariant under euclidean symmetries E of $\mathcal{R}^d$ (translations, rotations, reflections), that is S(f) = S(Ef) for all symmetries E and test functions f.

• OS3 (reflection positivity) We define exponential functionals on $\mathcal{D}'(\mathbb{R}^d)$ via

  $$A(\phi) := \sum_{k=1}^n c_k exp(\phi(f_k))$$

  Let $\mathcal{A}$ be the set of all these functionals, by axiom OS0 this is a subset of $L_2(d\mu)$. Euclidean symmetries act on $\mathcal{D}'(\mathbb{R}^d)$ via duality, that is $E\phi(f) = \phi(Ef)$, and thus define an unitary continuous action on $L_2(d\mu)$. Let $\mathcal{A}^+ \subset \mathcal{A}$ be the set of functionals with $supp(f_i) \subset \mathbb{R}^d_+$ where $\mathbb{R}^d_+ := \{(x, t): t \gt 0 \}$. Let $\theta: (x, t) \mapsto (x, -t)$ be the time reflection. Then the content of the axiom is:

  $$0 \le \langle \theta A, A\rangle_{L_2}$$

• OS4 (ergodicity): the time translation subgroup acts ergodically on the measure space $(\mathcal{D}'(\mathbb{R}^d), d\mu)$.

• theorem (Schwinger functions): A measure that satisfies OS0 has moments of all order, the nth moment has a density $S \in \mathcal{D}'(\mathbb{R}^{nd})$. These distributions are called Schwinger functions.

The theorem

One possible formulation: To every measure satisfying the axioms stated above there is a Wightman field such that the Schwinger and Wightman functions are related by:

$$\integral \phi_E(x_1, t_1) \cdots \phi_E(x_n, t_n) = \langle \Omega, \phi_M(x_1, i t_1) \cdots \phi_M(x_n, i t_n) \Omega \rangle$$

$\phi_E$ is a Schwinger function, $\phi_M$ is a Wightman field and $\Omega$ is the vacuum vector of the Wightman fields. See theorem 6.15 in the book by Glimm and Jaffe (see references).

Related theorems

• Reeh-Schlieder theorem

• spin-statistics theorem

References

The original article is

• Konrad Osterwalder, Robert Schrader, Axioms for Euclidean Green’s functions, Comm. Math. Phys. Volume 31, Number 2 (1973), 83-112 (Euclid:1103858969)

Discussion for compact/periodic Euclidean time, as needed for thermal quantum field theory is in

• Abel Klein, Lawrence Landau, Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle, Pacific Journal of Mathematics, Vol. 94, No. 2, 1981 (DOI: 10.2140/pjm.1981.94.341, pdf)

Exposition is in

• Dirk Schlingemann, From euclidean field theory to quantum field theory (arXiv:hep-th/9802035)

A textbook account is in

• James Glimm, Arthur Jaffe, Quantum physics: a functional integral point of view