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# Weird new maths does not commute

It's a strange world, in which the laws of classical physics break down. A hundred years after quantum theory was formulated, scientists are still grappling with mathematical systems to describe the sub-atomic realm.

Australian National University mathematicians, Alan Carey and Adam Rennie, are in the forefront of the race to develop a powerful new system of geometry to apply to quantum mechanics. Already, the system, non-commutative geometry, is showing promise in handling the theoretical framework of quantum physics. It can also describe the weird behaviour of "quantum states of matter", such as low temperature conductors, at the centre of a worldwide research effort.

The seventeenth century scientific revolution was marked by the use of mathematics to represent the physical world. However, the intuitive, Euclidean geometry and classical mechanics used by Galileo and Newton to describe the macro-world in terms of length, mass and time could not cope with modern physics.

"From the early twentieth century, physics faced the inadequacy of classical mechanics to describe matter and forces," says Rennie, of the Mathematical Sciences Institute. "The introduction of quantum mechanics severed the close link between geometry and physics. New branches of mathematics sprang up, some inspired by the new problems in quantum mechanics. Geometry continued to grow, but was no longer the language of physics."

In recent years, non-commutative geometry, pioneered by the mathematician Alain Connes, and based on the ideas of Sir Michael Atiyah and Isodore Singer, has been used to interpret the counter-intuitive quantum world, in which deterministic laws evaporate and the act of observing changes what is being observed. In this space the "commutative laws" of mathematics do not hold, so A times B does not equal B times A.

"We see the world on a very large scale," says Carey, director of the MSI. "We don't perceive what happens at a very small scale. It's for this reason that quantum phenomena appear very strange. That's why we need new mathematical tools."

The new geometry is being used to refine quantum field theory, which extends quantum mechanics. It also squares with observed quantum phenomena, including the "mysterious" behaviour of electrical conductors at low temperatures, says Carey.

"This is the new frontier of physics - to understand these new states of matter discovered in the last 20 years and to use them in some way."

A big strength of the new system is that it is so general it can be reconciled with classical geometry, which can be recovered under certain conditions as a special case. "Classical geometry can be written down in this new language," says Rennie.

Carey, Rennie and colleagues recently caused a stir in the international mathematical community when they came up with a vast generalisation of the "local index formula", central to non-commutative geometry. Their approach unified and extended many of the discoveries made in the field since the 1960s.

So how do the mathematicians cross between classical and non-commutative spaces? "You get used to thinking quantum mechanically," says Rennie. "If you're used to that sort of formalism, it's not so difficult. Somehow you're working between the two most of the time."

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