In particle physics, dimensional transmutation is a physical mechanism providing a linkage between a dimensionless parameter and a dimensionful parameter. In classical field theory, such as gauge theory in four-dimensional spacetime, the coupling constant is a dimensionless constant. However, upon quantization, logarithmic divergences in one-loop diagrams of perturbation theory imply that this “constant” actually depends on the typical energy scale of the processes under considerations, called the renormalization group (RG) scale. This “running” of the coupling is specified by the beta-function of the renormalization group.
Consequently, the interaction may be characterised by a dimensionful parameter Λ, namely the value of the RG scale at which the coupling constant diverges. In the case of quantum chromodynamics, this energy scale Λ is called the QCD scale, and its value 220 MeV supplants the role of the original dimensionless coupling constant in the form of the logarithm (at one-loop) of the ratio μ and Λ. Perturbation theory, which produced this type of running formula, is only valid for a (dimensionless) coupling g ≪ 1. In the case of QCD, the energy scale Λ is an infrared cutoff, such that μ ≫ Λ implies g ≪ 1, with μ the RG scale.
On the other hand, in the case of theories such as QED, Λ is an ultraviolet cutoff, such that μ ≪ Λ implies g ≪ 1. This is also a way of saying that the conformal symmetry of the classical theory is anomalously broken upon quantization, thereby setting up a mass scale.
Transmutation of species
Transmutation of species and transformism are terms often used to describe 19th-century evolutionary ideas for the altering of one species into another that preceded Charles Darwin’s theory of natural selection. The French Transformisme was a term used by Jean Baptiste Lamarck in 1809 for his theory, and other 19th century proponents of pre-Darwinian evolutionary ideas included Étienne Geoffroy Saint-Hilaire, Robert Grant, and Robert Chambers who anonymously published the book Vestiges of the Natural History of Creation. Opposition in the scientific community to these early theories of evolution, led by influential scientists like the anatomists Georges Cuvier and Richard Owen and the geologist Charles Lyell, was intense. The debate over them was an important stage in the history of evolutionary thought and would influence the subsequent reaction to Darwin’s theory. Transformation is another word used quite as often as transmutation in this context. These early 19th century evolutionary ideas played an important role in the history of evolutionary thought.
The proto-evolutionary thinkers of the 18th and early 19th century had to invent terms to label their ideas, but it was first Joseph Gottlieb Kölreuter who used the term “transmutation” to refer to species who have had biological changes through hybridization. The terminology did not settle down until some time after the publication of the Origin of Species. The word evolution was quite a late-comer: it can be seen in Herbert Spencer’s Social Statics of 1851, and there is at least one earlier example, but it was not in general use until about 1865-70.
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