# Expectation Values

## Layman

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

## Student

To that end, consider a locally defined microscopic variable which I will denote $\psi(r)$. In a ferromagnet this might well be the local magnetization, M®, or spin vector, S®, at point r in ordinary d-dimensional (Euclidean) space; in a fluid it might be the deviation $\delta p(r)$, of the fluctuating density at r from the mean density. In QFT the local variables $\psi(r)$ are the basic quantum fields which are 'operator valued.' For a magnetic system in which quantum mechanics was important, M® and S(x) would, likewise, be operators. However, the distinction is of relatively minor importance so that we may, for ease, suppose $\psi(r)$ is a simple classical variable. It will be most interesting when $\psi$ is closely related to the order parameter for the phase transition and critical behavior of concern.

By means of a scattering experiment (using light, x rays, neutrons, electrons, etc.) one can often observe the corresponding pair correlation function (or basic 'two-point function') $$G(r)=\langle \psi(0) \psi(r)\rangle$$ where the angular brackets $\langle \cdot \rangle$ denote a statistical average over the thermal fluctuations that characterize all equilibrium systems at nonzero temperature. (Also understood, when $\psi(r)$ is an operator, are the corresponding quantum-mechanical expectation values.) Physically, $G(r)$ is important since it provides a direct measure of the influence of the leading microscopic fluctuations at the origin $0$ on the behavior at a point distance $r = |r|$ away. But, almost by definition, in the vicinity of an appropriate critical point - for example the Curie point of a ferromagnet when $\psi = \vec M$ or the gas-liquid critical point when $\psi = \delta p$ - a strong "ordering" influence or correlation spreads out over, essentially, macroscopic distances. As a consequence, precisely at criticality one rather generally finds a power-law decay, namely, $$G(r) \approx D/d^{d-2+\eta}$$ as $r \to \infty$ which is characterized by the Critical Exponents (or critical index) $d - 2 + \eta$.

Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao

In this section things should be explained by analogy and with pictures and, if necessary, some formulas.

## Researcher

The motto in this section is: the higher the level of abstraction, the better.
Common Question 1
Common Question 2

Example1
Example2:

## History

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