Mind the gap

Monday, 18 January 2016

An international team of researchers have demonstrated that the spectral gap problem is algorithmically undecidable. The spectral gap, or the difference between the energy of the ground state and the first excited state, is fundamental to understanding the properties of a quantum many-body system. The question posed in their paper is: given a quantum many-body system, is it gapped or gapless? And they find out that that question is undecidable in the same form encountered in Gödel’s incompleteness theorem.

An important consequence of this study is that it has been demonstrated that there exists a physical problem where the reductionist approach is not possible, that is, the macroscopic properties of a system cannot be derived from its detailed microscopic description.  Even if we can perfectly describe all the parts, the properties of the whole cannot be predicted. Nature seems to dodge our immature science.

Emergent properties belong to the realm of complex systems. As Anderson puts it, “More is different”. There seems to exist a hierarchy in the description of nature, and that there is not straightforward way of going from one to the other.

This hierarchic structure has been recently found in our brains when we process language. Apparently, we keep track of different abstract linguistic structures (words, phrases and sentences) at different timescales. The authors of the research found thus a basis for Chomsky’s ideas about how we have a grammar in our head, which underlies our processing of language.

A similar idea permeates the Schenkerian analysis of a passage of music. This analysis shows the hierarchical relationships among its pitches, drawing conclusions about the structure of the analysed passage. Music is another kind of language, with different compositional time scales. I wouldn’t be surprised if a similar kind of analysis for music would led to finding a musical grammar in our heads.

Music, the ultimate abstract language, the one that Kandinsky tried to translate into visual art. To completely disembody lines and colours from their figurative meaning. Just a feast to the eyes.

A complex work of art, created by a complex system, the painter, apprehended by a complex system, us humans. How could ever the beauty of a piece of art be deduced from something so simple as an equation?

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