nLab Osterwalder-Schrader theorem

Idea

The Osterwalder-Schrader theorem (Osterwalder & Schrader 1975) 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 & Jaffe 1981 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 Jürg Fröhlich. We define the generating functional on $\mathcal{D}(\mathbb{R}^d)$

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

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 $z \coloneqq (z_1, z_2, ...z_n)$ 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 $S(f) = S(E f)$ for all symmetries $E$ and test functions $f$.

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

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

  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 6.1.5 (p. 98) in Glimm & Jaffe 1981.

Related theorems

• Reeh-Schlieder theorem

• spin-statistics theorem

Related concepts

• lattice quantum field theory

• constructive quantum field theory

References

The original article, whose proof turned out to be flawed:

• Konrad Osterwalder, Robert Schrader: Axioms for Euclidean Green’s functions, Comm. Math. Phys. 31 2 (1973) 83-112 [doi:10.1007/BF01645738, euclid:1103858969]

and the followup article fixing the proof at the cost of a further assumption (cf. Rychkov 2019):

• Konrad Osterwalder, Robert Schrader: Axioms for Euclidean Green’s functions II, Commun. Math. Phys. 42 (1975) 281–305 [doi:10.1007/BF01608978]

The version for fermion/spinor fields:

• Jürg Fröhlich, Konrad Osterwalder: Is There a Euclidean Field Theory for Fermions, Helv. Phys. Acta 47 (1975) 781 [doi:10.5169/seals-114595, spire:1330, pdf]

Further discussion:

• Yu. M. Zinoviev: Equivalence of the Euclidean and Wightman Field Theories, Commun. Math. Phys. 174 (1995) 1-28 [arXiv:hep-th/9408009, doi:10.1007/3-540-59190-7_25]

Exposition:

• Dirk Schlingemann: From euclidean field theory to quantum field theory, Rev. Math. Phys. 11 (1999) 1151-1178 [arXiv:hep-th/9802035, doi:10.1142/S0129055X99000362]

Monograph in the context of constructive field theory:

• James Glimm, Arthur Jaffe; Sections 6.1 and 19 in: Quantum physics: a functional integral point of view, Springer (1981, 1987) [doi:10.1007/978-1-4612-4728-9]

  (use stronger axioms, cf. Ftn. 81 on p. 80 in KQS21)

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

• Abel Klein, Lawrence Landau: Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle, Pacific Journal of Mathematics 94 2, (1981) [doi:10.2140/pjm.1981.94.341, pdf]

Highlighting that the OS theorem depends on a linear growth condition which may be hard to verify (such as in conformal bootstrap applications), and reproof for conformal field theories without this condition:

• Petr Kravchuk, Jiaxin Qiao, Slava Rychkov: Distributions in CFT II. Minkowski Space, J. High Energ. Phys. 2021 94 (2021) [doi:10.1007/JHEP08(2021)094, arXiv:2104.02090]

Review with further critical comments on the state of the literature, going back to the original OS ‘73:

• Slava Rychkov: CFT Osterwalder-Schrader theorem, talk at Simons Collaboration on the Nonperturbative Bootstrap Annual Meeting 2019 [video]