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The Millennium Prize Problems are seven mathematics problems laid out by the Clay Mathematics Institute in 2000. They’re not easy-_ a correct solution to any one results in a US$1,000,000 prize being awarded by the institute_.
Russian mathematician Grigori Perelman was awarded the Prize on March 18 last year for solving one of the problems, the Poincaré conjecture-as yet the only problem that’s been solved. Famously, he turned down the $1,000,000 Millennium Prize.
One of the outstanding discoveries made in the early part of the last century was that of the quantum behaviour of the physical world. At very short distances, such as the size of an atom and smaller, the world behaves very differently to the “classical” world we are used to.
Typical of the quantum world is so-called wave-particle duality: particles such as electrons behave sometimes as if they are point particles with a definite position, and sometimes as if they are spread out like waves.
This strange behaviour is not just of theoretical interest, since it is underpins much of our modern technology. It is fundamental to the behaviour of semiconductors in all our electronic devices, the behaviour of nano-materials, and the current rise of quantum computing.
Quantum theory is fundamental. It must govern not just the very small but also the classical realm. That means physicists and mathematicians have had to develop methods not just for understanding new quantum phenomena, but also for replacing classical theories by their quantum analogues.
When we have a finite number of degrees of freedom, such as for a finite collection of particles, although the quantum behaviour is often counter-intuitive, we have a well-developed mathematical machinery to handle this quantization called quantum mechanics.
This is well understood physically and mathematically. But when we move to study the electric and magnetic fields where we have an infinite number of degrees of freedom, the situation is much more complicated. With the development of so-called quantum field theory, a quantum theory for fields, physics has made progress that mathematically we do not completely understand.
What’s the problem?
Many field theories fall into a class called gauge field theories, where a particular collection of symmetries, called the gauge group, acts on the fields and particles. In the case that these symmetries all commute, so-called abelian gauge theories, we have a reasonable understanding of the quantization.
This includes the case of the electromagnetic field, quantum electrodynamics, for which the theory makes impressively accurate predictions.
The Yang-Mills theory describes elementary particles, which are particles with no substructure (quarks, leptons, Higgs boson), using algebraic objects. Specifically, non-abelian Lie groups are used to unify electromagnetic and weak forces. While the predictions of Quantum Yang-Mills theory have been verified by experiments, the mathematical foundation is still not well understood.
The discoverer would have to prove the following in order for their solution to be considered:
Mass of the Least Massive Particle is Strictly Positive
The problem asks to prove that the mass of the least massive particle is strictly positive. Specifically, there must be some positive constant such that every excitation of the vacuum has energy that is at least the magnitude of the constant.
Quark Confinement
The theory also must have what is called quark confinement, namely that transform under SU(3) are SU(3) invariant, where SU(3) is the 8-dimensional simple Lie group of 3x3 unitary matrices with determinant 1. This means quarks can not be isolated from their parent hadron without producing new hadrons.
Chiral Symmetry
Finally, it must have chiral symmetry breaking. This means that as the quark bare-mass vanishes the vacuum is potentially invariant under a specific subgroup of the full symmetry group that acts on the quark field.
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क्यों 1,000,000 $ के इस सवाल से डरते है वैज्ञानिक | yang mills and mass gap million dollar question
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