Why time ordering (and not space ordering)?

Question

I am trying to self-learn QFT and thermal field theory, starting from Srednicki's book. Eqns-8.14-8.15 of Srednicki's book on QFT shows the following.

$$\langle 0 | T \phi(\mathbf{x}_1) \phi(\mathbf{x}_2) | 0 \rangle = \frac{1}{\imath} \frac{\delta }{\delta J(\mathbf{x}_1)} \frac{1}{\imath} \frac{\delta }{\delta J(\mathbf{x}_2)} Z(J) |_{J=0} .\tag{8.15}$$

It seems to me that there is some implicit assumption that is not mentioned in the discussions preceding this result. The result is fundamental, as it is linked to the ubiquity of time ordering in the rest of the book. Therefore I want to ensure that I understand the arguments correctly. More importantly, I want to understand the origin of $\tau$ ordering (imaginary time) in thermal field theory using the discussion here.

The right hand side does not appear to treat time distinctly from position. So why does the $time$ ordering appear on the left hand side ? The explanation given by the author is that time ordering appears in analogy with the Harmonic oscillator, but this does not help me to understand the expression.

I have a few guesses as to why it could occur. Although I am not able to clearly identify the correct arguments.

1) We are treating time specially on the left hand side, we work with the ground state of the Hamiltonian (time like component of four momentum). This would tell me that the left hand side could also have been $\langle 0_{p_x} | T_x \phi(\mathbf{x}_1) \phi(\mathbf{x}_2) | 0_{p_x} \rangle $, where $p_x$ is the x component of the momentum operator and $|0_{o_x}\rangle$ is the 'ground state' of $p_x$, $T_x$ is the x-ordering. Is this correct ?

2) Other possibility is that it arises from causality which is imposed through multiplying $\mathcal{H}_0$ by $(1+\imath 0^+)$, as discussed in the beginning of the chapter. This would tell me that time ordering appears on the left hand side because, the Lagrangian is defined on the right hand side in a biased manner so as to enforce causality.

I understand that there has been multiple different questions on time ordering in this forum, especially in the context of the LSZ formula.

Time-ordered operator in Srednicki

I believe I am asking a question in the complementary context, as to why time ordering appears in the path integral formalism.

Answer 1

The role of the time-order $T$ is correlated with the standard choice of Cauchy surface to be an equal-time surface.

Other choices of world-line parameter (let's call it $\lambda$) are possible, cf. e.g. this Phys.SE post. Then the ordering is wrt. $\lambda$.

E.g. in light-cone quantization, the world-line parameter $\lambda=x^+$ is light-cone time.

The spatial coordinate $x$ is usually not used as a world-line parameter, because the $x$-coordinate is a bad world-line parameter to describe particles/fields at rest or moving backwards along the $x$-axis.

Answer 2

It could be that I am missing some deep point, but I was under the impression that when we calculate two point correlation functions we are choosing two points in space and ask how the quantum correlations vary over that distance. Now we could have chosen any other two points but that is why we integrate over all space on the right hand side. But anyways, once we choose two points i.e there is no ambiguity the field operators have to act at those two points. On the other hand there is an ambiguity as to which one is applied first and second. Path ordering does come up in physics so for example non-abelian geometric phases. If there are parameters in a Hamiltonian that I can change I am essentially picking a path in parameter space, but I have to be careful because the Hamiltonian may not commute with itself for different parameter values so in parameter space there is path ordering.