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

updated
Thursday 25
th
March, 2010
09:54
1

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

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

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