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Expectation Values

Why is it interesting?


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.


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.


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




Contributing authors:

Jakob Schwichtenberg
advanced_tools/expectation_values.txt · Last modified: 2017/12/04 07:01 (external edit)