advanced_tools:expectation_values
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advanced_tools:expectation_values [2017/12/04 08:01] |
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| + | ====== Expectation Values ====== | ||
| + | |||
| + | <tabbox Why is it interesting?> | ||
| + | |||
| + | <tabbox Layman> | ||
| + | |||
| + | <note tip> | ||
| + | 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. | ||
| + | </note> | ||
| + | | ||
| + | <tabbox Student> | ||
| + | |||
| + | <blockquote> | ||
| + | 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(r), or spin | ||
| + | vector, S(r), 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(r) | ||
| + | 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 [[advanced_notions:critical_exponent]] (or critical index) $d - 2 + \eta$. | ||
| + | |||
| + | <cite>Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao</cite> | ||
| + | </blockquote> | ||
| + | |||
| + | |||
| + | <note tip> | ||
| + | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | ||
| + | </note> | ||
| + | |||
| + | <tabbox Researcher> | ||
| + | |||
| + | <note tip> | ||
| + | The motto in this section is: //the higher the level of abstraction, the better//. | ||
| + | </note> | ||
| + | |||
| + | --> Common Question 1# | ||
| + | |||
| + | |||
| + | <-- | ||
| + | |||
| + | --> Common Question 2# | ||
| + | |||
| + | |||
| + | <-- | ||
| + | | ||
| + | <tabbox Examples> | ||
| + | |||
| + | --> Example1# | ||
| + | |||
| + | |||
| + | <-- | ||
| + | |||
| + | --> Example2:# | ||
| + | |||
| + | |||
| + | <-- | ||
| + | | ||
| + | <tabbox History> | ||
| + | |||
| + | </tabbox> | ||
| + | |||
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