What does mathematical equivalence means here? - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-09-15T09:50:30Z https://physics.stackexchange.com/feeds/question/183617 https://creativecommons.org/licenses/by-sa/4.0/rdf https://physics.stackexchange.com/q/183617 1 What does mathematical equivalence means here? Beyond-formulas https://physics.stackexchange.com/users/79444 2015-05-12T18:29:45Z 2015-05-12T18:51:00Z <p>On Motls blog, <a href="http://motls.blogspot.com/2012/06/on-importance-of-conformal-field.html" rel="nofollow">http://motls.blogspot.com/2012/06/on-importance-of-conformal-field.html</a>, while I was trying to understand what dimensional transmutation means, he said:</p> <blockquote> <p><em>I said that by omitting the mass terms, we only get "classically" scale-invariant theories. What does "classically" mean here? Well, such theories aren't necessarily scale-invariant at the quantum level. The mechanism that breaks the scale invariance of classically invariant theories is known as the dimensional transmutation. It has a similar origin as the anomalous dimensions mentioned above. Roughly speaking, the Lagrangian density of QCD, \$−Tr(F_{μν}F^{μν})/2g^2\$, no longer has the units of \$mass^4\$ but slightly different units, so a dimensionful parameter \$M^{δΔ}\$ balancing the anomalous dimension has to be added in front of it. In this way, the previously dimensionless coefficient \$1/2g^2\$ that defined mathematically inequivalent theories is transformed into a dimensionful parameter M – which is the QCD scale in the QCD case – and the rescaling of the coefficient may be emulated by a change of the energy scale. So different values of the coefficient are <strong>mathematically equivalent</strong> after the dimensional transmutation because the modified physics may be "undone" by a change of the energy scale of the processes.</em></p> </blockquote> <p>I did not understand how are the coefficients mathematically equivalent by this analysis?</p>