- #1
CAF123
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The definition I have for a 2-point correlator is $$\langle \phi_1(x_1)\phi_2(x_2)\rangle = \frac{1}{Z} \int \mathcal D \phi \,\,\phi_1(x_1)\phi_2(x_2) \exp-S[\phi],$$ where ##Z = \int \mathcal D \phi \,\,\exp-S[\phi]##. I am trying to understand the overall physical meaning of such a quantity and the motivation for why we define an object like this.
The LHS looks to be the expectation value of the product of two fields at two different positions. So it is a measure of the average value the product of these two states would have if we sampled the product over all possible ##x_i## in space? I don't think this is quite correct because it does not really make the RHS sensible. There, we seem to be integrating over a configuration of fields. Perhaps ##Z## is the integral over all permissible classical field configurations satisfying some classical theory and each configuration brings with it its effect on the action that is manifest in the ##\exp-S[\phi]## term. (i.e there exists a configuration where the action is minimized).
The definition looks analogous to how the expectation of the product of two random or stochastic variables would be defined, with the ##\exp-S[\phi]## being the weighting factor.
Thanks for any clarity.
The LHS looks to be the expectation value of the product of two fields at two different positions. So it is a measure of the average value the product of these two states would have if we sampled the product over all possible ##x_i## in space? I don't think this is quite correct because it does not really make the RHS sensible. There, we seem to be integrating over a configuration of fields. Perhaps ##Z## is the integral over all permissible classical field configurations satisfying some classical theory and each configuration brings with it its effect on the action that is manifest in the ##\exp-S[\phi]## term. (i.e there exists a configuration where the action is minimized).
The definition looks analogous to how the expectation of the product of two random or stochastic variables would be defined, with the ##\exp-S[\phi]## being the weighting factor.
Thanks for any clarity.