Asymptotic freedom, dimensional transmutation, and an infrared conformal fixed point for the δ-function potential in one-dimensional relativistic quantum mechanics

We consider the Schrödinger equation for a relativistic point particle in an external one-dimensional δ-function potential. Using dimensional regularization, we investigate both bound and scattering states, and we obtain results that are consistent with the abstract mathematical theory of self-adjoint extensions of the pseudodifferential operator H=p2+m2−−−−−−−√. Interestingly, this relatively simple system is asymptotically free. In the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point. Thus it can be used to illustrate nontrivial concepts of quantum field theory in the simpler framework of relativistic quantum mechanics.

Impact of range expansions on current human genomic diversity

The patterns of population genetic diversity depend to a large extent on past demographic history. Most human populations are known to have gone recently through a series of range expansions within and out of Africa, but these spatial expansions are rarely taken into account when interpreting observed genomic diversity, possibly because they are difficult to model. Here we review available evidence in favour of range expansions out of Africa, and we discuss several of their consequences on neutral and selected diversity, including some recent observations on an excess of rare neutral and selected variants in large samples. We further show that in spatially subdivided populations, the sampling strategy can severely impact the resulting genetic diversity and be confounded by past demography. We conclude that ignoring the spatial structure of human population can lead to some misinterpretations of extant genetic diversity.

Models of hybridization during range expansions and their application to recent human evolution

Several lines of genetic, archeological and paleontological evidence suggest that anatomically modern humans (Homo sapiens) colonized the world in the last 60,000 years by a series of migrations originating from Africa (e.g. Liu et al., 2006; Handley et al., 2007; Prugnolle, Manica, and Balloux, 2005; Ramachandran et al. 2005; Li et al. 2008; Deshpande et al. 2009; Mellars, 2006a, b; Lahr and Foley, 1998; Gravel et al., 2011; Rasmussen et al., 2011). With the progress of ancient DNA analysis, it has been shown that archaic humans hybridized with modern humans outside Africa. Recent direct analyses of fossil nuclear DNA have revealed that 1–4 percent of the genome of Eurasian has been likely introgressed by Neanderthal genes (Green et al., 2010; Reich et al., 2010; Vernot and Akey, 2014; Sankararaman et al., 2014; Prufer et al., 2014; Wall et al., 2013), with Papua New Guineans and Australians showing even larger levels of admixture with Denisovans (Reich et al., 2010; Skoglund and Jakobsson, 2011; Reich et al., 2011; Rasmussen et al., 2011). It thus appears that the past history of our species has been more complex than previously anticipated (Alves et al., 2012), and that modern humans hybridized several times with local hominins during their expansion out of Africa, but the exact mode, time and location of these hybridizations remain to be clarifi ed (Ibid.; Wall et al., 2013). In this context, we review here a general model of admixture during range expansion, which lead to some predictions about expected patterns of introgression that are relevant to modern human evolution.