Bz in fact it makes problem completely intractable

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Unformatted text preview: So how does it stick them together? It s like electromagnetism, but with matrices for fields. the Ez Bx ￿ B = By Right?! Going from numbers to matrices shouldn t make too much difference. Bz In fact it makes problem completely intractable!! the Bx ￿ It s because the world is quantum, not classical. Recall the path integral from ￿ B = By the first lecture. You should integrate over allProb ∼ paths that a particle￿) possible exp (iS/ Bz this translates to the fact that you should integrate takes. In particle physics, all fields over all possible configurations of the electric and magnetic fields. Prob ∼   ￿ exp (iS/￿) all fields S= ￿ ￿ ￿ ￿ d4 x Tr E 2 − B 2 References vectors...but not when the We can do this sum when the fields are normal vectors... ￿ elements of the vectors are matrices! ￿ ￿ ￿ ￿ S = [1] d4 x E 2 andB 2 Gibbons − Rychenkova, 9608085 ￿ QCD is Hard!       Clay Mathematics problem: $1 million Serious Supercomputers Even the simplest question is immensely difficult: what does empty space look like? Prob ∼ Confinement     ￿ exp (iS/￿) all fields 4 ￿ ￿2 ￿2 ￿ S = d x Tr E − B The quarks can never escape the proton or neutron. If you try to pull a quark away, a long string forms pulling it back in. The force between two quarks is linear: F ∼ r . This is called confinement References [1] Gibbons and Rychenkova, 9608085 [2] Kapustin and Strassler, 9902033     This is why quarks stick together in so many ways. To prove confinement from the equations of QCD is one of the most important open problems in theoretical physics. ￿ ￿ ￿ ￿ ￿ An Aside:4 xStrings B 2 QCD S = d Tr E 2 − in     Homework exercise: consider two relativistic particles with separation r, and an attractive force F=r. Show that whenF ∼ r the angular momentum, J, is large, the energy of a the spinning particles, attached by a string, scales as E∼ √ J But since E=mc2, this gives a relationship between mass and angular momentum of particles.   This the beginning of string c...
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