# nLab Euclidean field theory

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

### General

In field theory one speaks of Euclidean field theory if the underlying spaces on which the fields are defined are Riemannian manifolds, as opposed to Lorentzian spacetimes used in relativistic field theory, hence locally are Euclidean spaces instead of Minkowski spacetimes, whence the name “Euclidean field theory.”

Concretely this means that in Euclidean field theory the locality condition on the net of quantum observables requires observables $A, B$ to commute as soon as their spacetime supports are disjoint at all

$supp(A) \cap supp(B) = \emptyset \;\;\;\;\Rightarrow\;\;\;\; [A,B] = 0 \,.$

This is in contrast to the analogous condition in relativistic field theory whose causal locality requires this implication only if the two supports are in addition spacelike-separated.

Equivalently this means that the n-point functions of Euclidean field theories are distributions of several variables with singularities on the fat diagonal (instead of on all of the relative light cones, as for relativistic field theory). This means that Euclidean $n$-point functions restrict to non-singular distributions on the configuration space of n points, allowing to express correlators as differential forms on configuration spaces of points. Systematic discussion of perturbative renormalized Euclidean field theory from this perspective is due to Bergbauer-Brunetti-Kreimer 09, Berghoff 14a, Berghoff 14b.

This Euclidean locality property applies in particular in statistical mechanics, where the “fields” of the field theory are not thought of as encoding the spacetime-behaviour of fundamental particles as governed by quantum physics, but instead the spatial expectation values (at any given time) of equilibrium thermodynamic-processes governed by classical physics.

An archetypical example of a Euclidean field theory in this thermodynamic sense is the Ising model. Generally, most 2d conformal field theories considered are Euclidean field theories and (should) have the interpretation of describing thermodynamical systems “at criticality”.

However, other Euclidean 2d CFTs are not necessarily regarded as thermodynamical systems, notably the worldsheet-field theories defining a string perturbation series in perturbative string theory.

Another broad class of examples of Euclidean field theories are topological quantum field theories after their diffeomorphism-symmetry is partially gauge fixed by a choice of Riemannian metric. The archetypical example here is perturbative Chern-Simons theory, see there for more.

### Wick rotation to Relativistic Field theory

Despite this superficially stark contrast between Euclidean and relativistic field theory, the two turn out to be tightly related to each other, at least under some conditions, in a subtle way that involves and generalizes the concept of analytic continuation from complex analysis, here this is called Wick rotation.

Roughly this says that propagators and hence n-point functions of relativistic field theory on Minkowski spacetime $\mathbb{R}^{d,1}$ (may) have analytic continuation to complex values of the time-coordinates, such that replacing real time with imaginary time turns these n-point functions into those of a Euclidean field theory on $\mathbb{R}^{d+1}$, and vice versa.

Precise formulation of the conditions that go into this Wick rotation between relativistic field theory and Euclidean field theory is the content of the Osterwalder-Schrader theorem.

### Temporal compactification to Thermal relativistic field theory

In fact this relation goes deeper still: Under suitable conditions the Euclidean field theory not on $\mathbb{R}^{d+1}$ but on $\mathbb{R}^d \times S^1_\beta$, with the circle-factor $S^1_{\beta}$ of length $\beta$, corresponds to relativistic field theory on Minkowski spacetime $\mathbb{R}^{d,1}$ in a vacuum state that represents thermal equilibrium at temperature $T \coloneqq 1/\beta$. (The previous case of Euclidean field theory on $\mathbb{R}^{d+1}$ may be thought of as the special case $\beta \to \infty$, hence $T \to 0$.)

This curious relation of Wick rotation with “compact peridodic Euclidean time” makes, when it applies, Euclidean field theory be a unification of relativistic field theory with statistical mechanics/thermodynamics, then called thermal quantum field theory or quantum statistical field theory or similar.

$\array{ \left. \array{ \text{relativistic field theory} \\ \text{on Minkowski spacetime} \\ \mathbb{R}^{d,1} \\ \text{in a thermal equilibrium state} \\ \text{at temperature}\; T } \right\} & \;\;\;\; \overset{ \text{Wick rotation} }{\leftrightarrow} \;\;\;\; & \left\{ \array{ \text{Euclidean field theory} \\ \text{on Euclidean space} \\ \mathbb{R}^d \times S^1_{\beta} \\ \text{with compact/periodic Euclidean time} \\ \text{of length} \; \beta = 1/T } \right. \\ \phantom{A} \\ \underset{ { \text{equal-time n-point function} \atop \text{of relativistic fields} } \atop \text{ in thermal equilibrium state } \; \vert T\rangle }{ \underbrace{ \left\langle T\vert :\mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t) : \vert T \right\rangle_{\mathbb{R}^{d,1}} }} &\;=\;& \underset{ \text{correlator of Euclidean fields} \atop \text{ for "Euclidean time" of periodicity}\; \beta = 1/T }{ \underbrace{ \left\langle 0 \vert \mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t) \vert 0 \right\rangle_{\mathbb{R}^{d} \times S^1_{\beta}} }} }$

graphics grabbed form Frolov-Zelnikov 11

Notice that the evident breaking of Lorentz symmetry on the right side of this correspondence is perfectly consistent with what happens on the left hand: A thermal vacuum state in Minkowski spacetime also singles out a preferred Lorentz frame.

The basic idea of this relation seems to go back to Bloch 58. The physics literature often states this suggestively but informally in terms of path integral-imagery, see e.g. Moore 03, section 1.1.

The first precise formulation seems to be due to Høegh-Krohn 74 (in 1+1 dimensions) and a more comprehensive discussion in view of the Osterwalder-Schrader theorem for compact Euclidean time is due to Klein-Landau 81.

The use of a “periodic Euclidean time coordinate” is also known as Matsubara formalism (e.g. Landsman-vanWert 87, section 2.3.1) and specifically the condition that the periodicity has to be $\beta \coloneqq 1/T$ is known as the KMS conditions for a KMS state (For Kubo-Martin-Schwinger, due to Kubo 57, Martin-Schwinger 59 with its final form due to Haag-Hugenholtz-Winnink 67, see Fulling-Ruijsenaars 87, section 3.1).

Beware that literature discussing the KMS-condition often does not make the periodicity of Euclidean time explicit, and vice versa. This is clarified in Fulling-Ruijsenaars 87, sections 2 and 3.

General introduction to Euclidean and thermal field theory includes Thoma 00, section 2.2, Peeters-Zamaklar 09, section 1.3.

## References

### General

General introduction to Euclidean field theory includes

A systematic development of Euclidean renormalized perturbative quantum field theory with correlators as differential forms on configuration spaces of points is described in

### Thermal quantum field theory

A good introduction is in

• S.A. Fulling, S.N.M. Ruijsenaars, Temperature, periodicity and horizons, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 (pdf, doi:10.1016/0370-1573(87)90136-0)

The idea of Wick rotating thermal relativistic field theory to compact periodic Euclidean time apparently goes back to

• Claude Bloch, Sur la détermination de l’état fondamental d’un système de particules, Nucl. Phys. 7 (1958) 451

This has maybe first been made precise, for the case of 1+1 dimensions, in

• Raphael Høegh-Krohn, Relativistic Quantum Statistical Mechanics in two-dimensional Space-Time, Communications in Mathematical Physics 38.3 (1974): 195-224 (pdf)

A systematic discussion of the Osterwalder-Schrader theorem on Wick rotation for the case of thermal field theory/periodic Euclidean time 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)

The formulation of the KMS condition is due to

• R. Kubo Statistical-Mechanical Theory of Irreversible Processes I. General Theory and Simple Applications to Magnetic and Conduction Problems, Journal of the Physical Society of Japan 12, 570-586 1957

• Paul C. Martin, Julian Schwinger, Theory of Many-Particle Systems. I, Physical Review 115, 1342-1373 (1959)

and found its final, now generally accepted, form in

• Rudolf Haag, N. M. Hugenholtz, M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. Volume 5, Number 3 (1967), 215-236 (euclid:1103840050)

Review of thermal field theory via Euclidean field theory includes

Further discussion:

• Christian D. Jäkel, The Reeh–Schlieder property for thermal field theories, Journal of Mathematical Physics 41, 1745 2000 (doi:10.1063/1.533208)

• Christian D. Jäkel, Florian Robl, The relativistic KMS condition for the thermal $n$-point functions of the $P(\phi)_2$ model (arXiv:1103.3609)

With an eye towards lattice gauge theory:

• Guy Moore, Informal lectures on lattice gauge theory, 2003 (pdf)

In application to black hole thermodynamics:

Discussion of thermal Wick rotation on global anti-de Sitter spacetime (which is already periodic in real time) is in

The expansion of thermal field theory around the infinite-temperature-limit (i.e. around $\beta = 1/T = 0$, i.e. KK-reduction in compact/periodic Euclidean time) is discussed in

• Paul Ginsparg, First and second order phase transitions in gauge theories at finite temperature, Nuclear Physics B Volume 170, Issue 3, 15 December 1980, Pages 388-408 (doi:10.1016/0550-3213(80)90418-6)

• Thomas Appelquist, Robert Pisarski, High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics, Phys. Rev. D 23, 2305 (1981) (doi:10.1103/PhysRevD.23.2305)

• Sudhir Nadkarni, Dimensional reduction in finite-temperature quantum chromodynamics, Phys. Rev. D 27, 917 (1983) (doi:10.1103/PhysRevD.27.917)

• Sudhir Nadkarni, Dimensional reduction in finite-temperature quantum chromodynamics. II, Phys. Rev. D 38, 3287 (1988) (doi:10.1103/PhysRevD.38.3287)

• Alexander N Jourjine, Quantum field theory in the infinite temperature limit, Annals of Physics Volume 155, Issue 2, July 1984, Pages 305-332 (doi:10.1016/0003-4916(84)90003-4)

• Klaas Landsman, Limitations to dimensional reduction at high temperature, Nuclear Physics B Volume 322, Issue 2, 14 August 1989, Pages 498-530 (doi:10.1016/0550-3213(89)90424-0)

• T. Reisz, Realization of dimensional reduction at high temperature, Z. Phys. C - Particles and Fields (1992) 53: 169 (doi:10.1007/BF01483886)

• Eric Braaten, Solution to the Perturbative Infrared Catastrophe of Hot Gauge Theories, Phys. Rev. Lett. 74, 2164 (1995) (doi:10.1103/PhysRevLett.74.2164)

• K. Kajantiea, M. Laine, K. Rummukainen, M. Shaposhnikov, Generic rules for high temperature dimensional reduction and their application to the standard model, Nuclear Physics B Volume 458, Issues 1–2, 1 January 1996, Pages 90-136 (doi:10.1016/0550-3213(95)00549-8)

and specifically with an eye to discussion of the quark-gluon plasma in

• Jean-Paul Blaizot, Edmond Iancu, Anton Rebhan, Thermodynamics of the high temperature quark gluon plasma, Quark–Gluon Plasma 3, pp. 60-122 (2004) (arXiv:hep-ph/0303185, spire:615570)

General discussion for QCD:

• Jacopo Ghiglieri, Aleksi Kurkela, Michael Strickland, Aleksi Vuorinen, Perturbative Thermal QCD: Formalism and Applications (arXiv:2002.10188)