Osterwalder-Schrader theorem



algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



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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 𝒟( d)\mathcal{D}'(\mathbb{R}^d), the space of distributions that is dual to the space of all smooth functions with compact support, 𝒟( d)\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 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 𝒟( d)\mathcal{D}(\mathbb{R}^d)

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

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

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:

ϕ E(x 1,t 1)ϕ E(x n,t n)=Ω,ϕ M(x 1,it 1)ϕ M(x n,it n)Ω \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

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


The original article is

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

Exposition is in

A textbook account is in