In QFT, The two-point correlator (Green function) for operators $O_A, O_B$ is given by $\langle O_AO_B \rangle $. Why is not defined by $ \langle O_A O_B\rangle -\langle O_A\rangle \langle O_B\rangle $, which is also the covariance in statistics? Isn't this definition captures the notion of correlation better than just the $\langle O_A O_B\rangle $? Is it because one point function like $\langle O_A\rangle$ is trivial?

# [Physics] Definition of two point correlators in QFT

correlation-functionsgreens-functionsquantum-field-theory

#### Related Solutions

The quantum fields in the interaction picture evolve according to the free field equations – the evolution is given just by the quadratic part of the Hamiltonian, without the interactions – so Green's functions constructed from these interaction-picture field operators would be those of the free field theory, too. It wouldn't be terribly interesting. We would only "learn" the Wick's theorem and things about the free field theory.

The normal Green's ($n$-point) functions are supposed to include all the interactions given by Feynman's vertices etc., so they need to be evaluated from the operators in the normal picture, i.e. the Heisenberg picture. The interaction picture is just a "fudged" compromise between the Heisenberg picture and the Schrödinger picture – a compromise that is useful and convenient but in no way fundamental. It's the Heisenberg picture and correlators in it that reduce to classical physics in the $\hbar\to 0$ classical limit.

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.

## Best Answer

$\langle O_1O_2\rangle$ is called the

fullcorrelation function. $\langle O_1O_2\rangle-\langle O_1\rangle\langle O_2\rangle$ is called theconnectedcorrelation function. They are both important concepts, and they are both used in practice. It is true that the second one is, in general, more useful, because (as is noted in the OP), it satisfies the clustering property, which reflects the locality of the corresponding object.It should be noted, though, that it is typically the case that $\langle O_i\rangle=0$, in which case these functions are identical, and there is no need to specify which one we are using.