# Euclidean QFT commutator vanishes for all spacetime separations?

Yes, Euclidean commutators vanish.

This is pretty obvious in the path integral formalism: You can compute the matrix elements of the commutator from the correlation functions: $\langle \hat{\phi}(x)\hat{\phi}(y) ... \rangle = \int \phi(x)\phi(y) ... e^{-S(\phi)}d\phi$. The integral involves only commuting variables, so you can switch $x$ and $y$.

The connection between Euclidean correlation functions and Minkowski ones is slightly subtle. The Euclidean correlation functions are analytic functions of several variables $x$, $y$, and can be extended to the complexification of a Cartesian product of several copies of Euclidean space. The corresponding Cartesian products of Minkowski space also sit within this complexification. The Minkowski correlation functions are, in fact, boundary values of the analytic extension of the Euclidean correlation functions. But which time-ordering of Minkowski correlation functions you get depends on how you approach the boundary.

I doubt that your reasoning goes through without caveats. It's true that when you analytically continue the Minkowski spacetime *two-point function* to imaginary time, you get a function (the two-point Schwinger function $S(x,y)$) which is symmetric in the two Euclidean spacetime points and is covariant with respect to Euclidean isometries. So if you *define*
$$
[\phi (x),\phi(y)]:=S(x,y)-S(y,x)
$$
for $x$, $y$ being Euclidean spacetime points, then you have $[\phi (x),\phi(y)]=0$.

However, it is not clear to me that the left-hand side of the definition above can be interpreted as the commutator of two well-defined operators (or operator-valued distributions). For example, let $\phi$ be a free scalar field, defined initially in Minkowski spacetime by its mode expansion $$\phi(t,\mathbf{x})=\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{\sqrt{2\omega(\mathbf{k})}}\left(a(\mathbf{k})e^{-i\omega t+i\mathbf{k}\cdot\mathbf{x}}+a^{\dagger}(\mathbf{k})e^{i\omega t-i\mathbf{k}\cdot\mathbf{x}}\right).$$

If we formally continue this to $t=-i\tau$, we see that the modes of positive (negative) frequency diverge badly for $\tau<0$ ($\tau>0$). (Note: This divergent behavior of the modes for imaginary times also puts your manipulations of the integral representation of the commutator into question.)

One can still make sense of expressions like $\langle 0|\phi (-i\tau_2,\mathbf{x}_2)\phi(-i\tau_1,\mathbf{y}_1) |0\rangle$ as long as $\tau_2>\tau_1$ and the momentum integrals are understood to be taken only at the end. Using this formalism, one can equate this expression to a path integral in Euclidean spacetime with two "operator insertions": $$ \langle 0|\phi (-i\tau_2,\mathbf{x}_2)\phi(-i\tau_1,\mathbf{x}_1) |0\rangle = \frac{\int \mathcal D \phi \exp{\left(-S_E[\phi]\right)\phi(x_2)\phi(x_1)}}{\int \mathcal D \phi \exp{\left(-S_E[\phi]\right)}} $$

The path integral automatically takes care of the ordering in imaginary time and we can therefore use it to extend the meaning of the left-hand side to arbitrary $\tau$-ordering, which gives us the Schwinger two-point function for this field expressed as a path integral. Still, this does not mean that we have made sense of the field operator by itself for non-zero imaginary times.

The automatic Euclidean time-ordering done by the path integral prevents us from using it to calculate the general matrix elements of the field commutator for unequal Euclidean times. Maybe this is the reason why one usually does not find a calculation of this commutator for Euclidean spacetime in textbooks. (At least I found none in the texts I have available.)

For 2-dimensional conformal field theories one finds calculations of commutators $[T,\phi(y)]$ between the field $\phi$ and a conserved charge $T$ making use of the operator product expansion (OPE) in Euclidean spacetime ([1], [2]). Such commutators can be non-zero even if the operator $T$ is built from the field and its derivatives. Your conclusion that the commutator $[\phi(x), \phi(y)]$ is identically zero on Euclidean spacetime would also imply that all commutators of quantities built out of the fields and their derivatives vanish on Euclidean spacetime, wouldn't it?

Schwinger mentioned the *formal* possibility to construct fully (anti)commuting field operators for Euclidean spacetime at the end of [3]. I'm not sure what to make of that.

A related question about field operators and two-point functions in Euclidean spacetime has been discussed on mathoverflow: https://mathoverflow.net/q/237647

[1] K. Becker, M. Becker, J. H. Schwarz. String theory and M-theory: A modern introduction. Cambridge University Press, 2007. p. 64f.

[2] J. Polchinski. String Theory. Vol. 1 An Introduction to the Bosonic String. Cambridge University Press, 1998. p. 55.

[3] J. Schwinger, 1958. Four-dimensional euclidean formulation of quantum field theory. Proceedings, Conference: C58-06-30, p.134-140.