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物理中的近代数学 Modern mathematics in physics

已有 13632 次阅读 2014-3-31 13:28 |个人分类:科普|系统分类:观点评述

It happens several times in the history of physics that a truly new phenomenon requires a new mathematics to describe it. In fact, the appearance of new mathematics is a sign of truly new discovery. For example, Newton's theory of mechanics requires calculus (which was invented for that purpose). Einstein's theory of general relativity requires Riemannian geometry. Quantum theory requires linear algebra. Newton's case is the only time when the physicists are ahead of mathematiciants. When Newton was developing his theory, the required mathematics has not been invented yet. Newton had to invent calculus to describe his theory.




Right now, we are facing a similar situation. It is an exciting time in physics.


For a long time, we thought that different phases of matter are described by symmetry breaking patterns, which are classified by group theory. Now we know that there are many new quantum phases of matter that are beyond symmetry breaking. Those new phases are described by the patterns of many-body quantum entanglement.


In condensed matter physics, many-body entanglement is the origin of many new states of quantum matter (known as topologically ordered states, such as spin liquids and quantum Hall states), which host emergent gauge fields, and emergent Fermi or fractional statistics. It might be possible that our vacuum is a particular (long-range) entangled system whose emergent gauge fields and fermions are the elementary particles in the standard model. Such an emergent picture represent an unification of gauge interaction and fermions (see the blog http://blog.sciencenet.cn/blog-1116346-736093.html  ).


Many-body entanglement is a truly new phenomenon. It requires a new mathematics to describe it. But what mathematics describes pattern of many-body entanglement (and classify topologically ordered phases)? Have mathematiciants already found the right mathematics for many-body entanglement, or we need to invent such mathematics by ourselves?


To answer such question, we like to point out that, recently, it was realized that quantum many-body states (or many-body entanglement) can be divided into short-range entangled states and long-range entangled states.


The quantum phases with long-range entanglement correspond to topologically ordered phases. In two spatial dimensions, we have a quite good understanding of such long-range entangled states, which can be described by  unitary modular tensor category theory. 2D topological orders with gappable boundary can be described by unitary fusion category theory (see cond-mat/0404617). Topological order in higher dimensions is much less understood, which may need higher category to describe them (see   arXiv:1405.5858).


Recently, we obtain a systematic understanding of the quantum phases with short-range entanglement and symmetry G in any dimensions (Those phases are called symmetry protected trivial (SPT) phases (see wiki http://en.wikipedia.org/wiki/Symmetry_protected_topological_order  ). A large subset of SPT states can be classified by Borel group cohomology theory of the symmetry group G, ie by H^d[GxSO,U(1)].


The   quantum phases with short-range entanglement that break the symmetry are the familar Landau symmetry breaking states, which can be described by group theory.


So, to understand the symmetry breaking states, physicists have been forced to learn group theory. It looks like to understand patterns of many-body entanglement that correspond to topological order and SPT order, physicists will be forced to learn tensor category theory and group cohomology theory. In modern quantum many-body physics and in modern condensed matter physics, tensor category theory and group cohomology theory will be as useful as group theory. The days when physics students need to learn tensor category theory and group cohomology theory are coming, may be soon.


Have mathematicians already found the right mathematics for many-body entanglement? Well, some related mathematics has been found, but we do not have a full story. To classify patterns of many-body entanglement in bosonic systems in any dimensions, we do need to develop new mathematics (and this is one of the frontiers in Mathematics). To classify patterns of many-body entanglement in  fermionic systems, we surely need to develop new mathematics (and this may become one of the frontiers in Mathematics). It is an exciting time in physics and mathematics.


I like to stress that the long-range entanglement and the new mathematics (such as higher category theory) is not just a game. We know new phases of matter can come from it. We also known that gauge interaction and fermions (such as electrons and quarks) can come from the long-range entangled qubits. Many people believe that many-body entanglement --> geometry which may lead to quantum gravity. Thus

long-range entangled qubits tell us: matter <--> Information--> geometry.

(Quantum theory tells us: energy <--> 1/time, momentum<--> 1/length;

Relativity tells us: time <-->length.

Yes, arrows connecting objects <--> a categorytheory.)

It is an exciting time in physics and mathematics.




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