I'm fairly new to the subject of quantum field theory (QFT), and I'm having trouble intuitively grasping what a n-point correlation function physically describes. For example, consider the 2-point correlation function between a (real) scalar field $\hat{\phi}(x)$ and itself at two different space-time points $x$ and $y$, i.e. $$\langle\hat{\phi}(x)\hat{\phi}(y)\rangle :=\langle 0\rvert T\lbrace\hat{\phi}(x)\hat{\phi}(y)\rbrace\lvert 0\rangle\tag{1}$$ where $T$ time-orders the fields.

Does this quantify the correlation between the values of the field at $x=(t,\mathbf{x})$ and $y=(t',\mathbf{y})$ (i.e. how much the values of the field at different space-time points covary, in the sense that, if the field $\hat{\phi}$ is excited at time $t$ at some spatial point $\mathbf{x}$, then this will influence the "behaviour" of the field at later time $t'$ at some spatial point $\mathbf{y}$)? Is this why it is referred to as a correlation function?

Furthermore, does one interpret $(1)$ as physically describing the amplitude of propagation of a $\phi$-particle from $x$ to $y$ (in the sense that a correlation of excitations of the field at two points $x$ and $y$ can be interpreted as a "ripple" in the field propagating from $x$ to $y$)?


Yes, in scalar field theory, $\langle 0 | T\{\phi(y) \phi(x)\} | 0 \rangle$ is the amplitude for a particle to propagate from $x$ to $y$. There are caveats to this, because not all QFTs admit particle interpretations, but for massive scalar fields with at most moderately strong interactions, it's correct. Applying the operator $\phi({\bf x},t)$ to the vacuum $|0\rangle$ puts the QFT into the state $|\delta_{\bf x},t \rangle$, where there's a single particle whose wave function at time $t$ is the delta-function supported at ${\bf x}$. If $x$ comes later than $y$, the number $\langle 0 | \phi({\bf x},t)\phi({\bf y},t') | 0 \rangle$ is just the inner product of $| \delta_{\bf x},t \rangle$ with $| \delta_{\bf y},t' \rangle$.

However, the function $f(x,y) = \langle 0 | T\{\phi(y) \phi(x)\} | 0 \rangle$ is not actually a correlation function in the standard statistical sense. It can't be; it's not even real-valued. However, it is a close cousin of an honest-to-goodness correlation function.

If make the substitution $t=-i\tau$, you'll turn the action $$iS = i\int dtd{\bf x} \{\phi(x)\Box\phi(x) - V(\phi(x))\}$$ of scalar field theory on $\mathbb{R}^{d,1}$ into an energy function $$-E(\phi) = -\int d\tau d{\bf x} \{\phi(x)\Delta\phi(x) + V(\phi(x))\}$$ which is defined on scalar fields living on $\mathbb{R}^{d+1}$. Likewise, the oscillating Feynman integral $\int \mathcal{D}\phi e^{iS(\phi)}$ becomes a Gibbs measure $\int \mathcal{D}\phi e^{-E(\phi)}$.

The Gibbs measure is a probability measure on the set of classical scalar fields on $\mathbb{R}^{d+1}$. It has correlation functions $g(({\bf x}, \tau),({\bf y},\tau')) = E[\phi({\bf x}, \tau)\phi({\bf y},\tau')]$. These correlation functions have the property that they may be analytically continued to complex values of $\tau$ having the form $\tau = e^{i\theta}t$ with $\theta \in [0,\pi/2]$. If we take $\tau$ as far as we can, setting it equal to $i t$, we obtain the Minkowski-signature "correlation functions" $f(x,y) = g(({\bf x},it),({\bf y},it'))$.

So $f$ isn't really a correlation function, but it's the boundary value of the analytic continuation of a correlation function. But that takes a long time to say, so the terminology gets abused.

  • $\begingroup$ By "amplitude", is it meant to be a transition amplitude? Also, what is the associated correlation function describing? Is it simply a measure of how the field values at separate points covary? $\endgroup$ – user35305 Aug 11 '16 at 19:39
  • $\begingroup$ 1) Yes. It's the inner product of two states, so an amplitude or a transition amplitude, or whatever you wish to call it. 2) Yes. The associated correlation function is doing exactly what you say: it's measuring how the values of the stochastic fields in the associated statistical field theory covary as we move around the points where we sample the field values. $\endgroup$ – user1504 Aug 11 '16 at 19:58
  • $\begingroup$ So the use of the phrase "correlation function" in QFT is a slight abuse of terminology, but is consistent in the sense that you describe in your answer?! Does this analogy hold for n-point correlation functions? $\endgroup$ – user35305 Aug 11 '16 at 20:12
  • $\begingroup$ Yes, it's also true for n-point functions (of composite operators, even). It's not guaranteed that a Euclidean statistical theory exists for every theory, though. The existence of spinors depends on spacetime signature, for example, as does the gauge invariance of the Chern-Simons action. $\endgroup$ – user1504 Aug 11 '16 at 20:59
  • $\begingroup$ Can you also say something about the relation of the retarded and advanced correlator? Does the fact that the QFT correlator is a Greens function also somehow carry over to statistics? $\endgroup$ – tonydo Oct 28 '16 at 16:41

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