t6 - updated Thursday 25th March 2010 09:54 1 L =(i D q q Mq q G Ga(v a 5 q q q 4 a quark field q vector D = igTa G igTa G =[D D a a in 3D color space

t6 - updated Thursday 25th March 2010 09:54 1 L =(i D q q...

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updated Thursday 25 th March, 2010 09:54 1
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L = ( i ¯ q 6 D q - ¯ q M q q ) - 1 4 G μν a G aμν - ¯ q γ μ ( v μ + a μ γ 5 ) q quark field q vector in 3D color space + 6D flavor space D μ = μ + igT a G μ a is “covariant derivative” igT a G μν a = [ D μ , D ν ] G μν a is “gluon field–strength” T a 3 × 3 traceless Hermitian matrices in color space Tr( T a T b ) = 1 2 δ ab M q diagonal in flavor space u, d, c, s, t, b color gauge symmetry T a G μ a UT a G μ a U - i g U∂ μ U q Uq G μν UG μν U T a ’s are “color” charges like EM charge in QED binds quarks and antiquarks into color–neutral combinations like photon exchange binds charged particles into electrically neutral atoms — color–neutral combinations are ¯ qq mesons and ² jkl q j q k q l baryons v μ = v μ α t α and a μ = a μ α t α sources for the “vector” and “axial vector” currents where t α are 6 × 6 hermitian flavor generators 2
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quantizing a non-abelian gauge theory like QCD — general issues in a gauge theory (like QED) propagator is not well defined in perturbation theory because there is a separate symmetry at each point in space and so there are components of the field that don’t show up in the Lagrangian and so the kinetic energy term cannot be inverted to get the propagator related to the physical fact that only the transverse components of the field correspond to propagating states in QED you fixed a gauge and showed that nothing depends on the unphysical components of the gauge field not trivial because coupling to charged fields depends on A μ — not just on F μν (Aharonov-Bohm effect) — but OK because of the gauge invariant form of coupling A μ to matter — Ward identities the gluons carry the charges to which they couple — so unlike the situation in EM, the unphysical components of the gauge field can be important to the dynamics if you are not very careful with gauge fixing — “ghosts” — non-covariant or “unitary” gauges 3
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quantizing a non-abelian gauge theory like QCD — general issues in a gauge theory (like QED) propagator is not well defined in perturbation theory because there is a separate symmetry at each point in space and so there are components of the field that don’t show up in the Lagrangian and so the kinetic energy term cannot be inverted to get the propagator related to the physical fact that only the transverse components of the field correspond to propagating states in QED you fixed a gauge and showed that nothing depends on the unphysical components of the gauge field not trivial because coupling to charged fields depends on A μ — not just on F μν (Aharonov-Bohm effect) — but OK because of the gauge invariant form of coupling A μ to matter — Ward identities the gluons carry the charges to which they couple — so unlike the situation in EM, the unphysical components of the gauge field can be important to the dynamics if you are not very careful with gauge fixing — “ghosts” — non-covariant or “unitary” gauges 4
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