Lecture 40.

Handouts.

SU(3) representations.

As an example of the entirely general power you now have to
understand symmetry generator eigenstates, we used the step operator
construction to build the representations of SU(3).

We quickly reviewed the strategy previewed last time, for the step
operator construction for a general Lie algebra: first we find the
Cartan generators H_i, then the step operators E_\alpha such that
[H_i, E_\alpha] = \alpha_i E_\alpha. There is some nomenclature:
E_\alpha acting on simultaneous eigenstates of H_i raises the
vector of H_i eigenvalues \mu (called a ``weight vector'') by the
vector amount \alpha (called a ``root'').

We mentioned many facts about roots, most of which had 1-line proofs.
These were: 1) \alpha must be real (since H_i are Hermitian); 2)
\alpha is a root implies -\alpha is a root (with associated step
operator E_\alpha^\adjoint; 3) if a) \alpha, \beta are roots,
b)\alpha+\beta doesn't equal 0 and c) E_\alpha and E_\beta don't
commute, then \alpha + \beta is a root with associated step operator
\pm \ [E_\alpha, E_\beta] (note \pm means plusorminus; \mp means
minusorplus); and 4) we can choose the normalization of E_\alpha such
that [E_\alpha, E_{-\alpha} ] = \alpha \dot H. This last fact was the
only one not fully proven to you; you can verify it for SU(3) below,
though.

The last fact, along with the definition of the step operators
E_{\pm\alpha}, means that the three objects E_{\pm\alpha} and
\alpha \dot H form a closed subalgebra among themselves: that is,
their commutators always give back multiples of one of the 3
complexified generators. We showed that proper rescaling of these 3
gave commutation relations between them that were exactly those of the
complexified su(2) subalgebra J_\pm, J_z. We were thus able to
lift results we derived for su(2): building a positive definite
operator analogous to J^2 that commutes with the full subalgebra and
whose eigenvalues are j(j+1) for half-integer j. As for su(2),
1) our highest weight state |\mu> is annihilated by the
raising operator (here proportional to E_\alpha); 2) this state
|\mu> has an eigenvalue (here of \alpha \dot H/ |\alpha|^2)
equal to j; 3) the highest weight state |\mu> can be
lowered 2j times, here by E_{-\alpha}; and 4) the state obtained by
lowering |\mu> p times has H_i eigenvalues \mu_i -
p\alpha_i. Thus, for every allowed highest weight \mu (which must
have half-integer \alpha \dot \mu/ |\alpha|^2), we can build an
``\alpha-ladder'' of H_i eigenstates.

Since we have many roots \alpha, we set about identifying some
minimal set (called the ``simple'' roots \alpha^i) whose combined
\alpha-ladders include the effects of lowering by all possible
roots. (We expect this set will be small because many of the roots are
linearly dependent.) To do this we defined the notion of a
``positive'' root (one whose leftmost nonzero component is positive)
and of a simple root (those positive roots which are not sums of other
positive roots). These definitions always give us rank G (that is,
the number of commuting H_i) linearly independent roots, which give
all other roots as integer linear combinations.

Since all highest weights must have integer 2\alpha^i \dot \mu/
|\alpha^i|^2 (indicating the number of times they can be lowered by
\alpha^i), we find it convenient to define a basis of allowed
``fundamental weights'' \mu^j. We defined \mu^j by 2\alpha^i
\dot \mu^j/ |\alpha^i|^2 = \delta^{ij}, which means that \mu^j can
be lowered exactly once by the simple root \alpha^j, and not by any
other simple roots. Any highest weight with integer 2\alpha^i \dot
\mu/ |\alpha^i|^2 for all i must then be a sum of fundamental
weights \mu^j with integer coefficients.

We now have everything in place to define our algorithm for
constructing nonabelian group representations: 1) choose a highest
weight \mu, which will be a nonnegative integer linear combination
of fundamental weights; 2) lower \mu by the simple roots \alpha^i
as many times as 2\alpha^i \dot \mu/ |\alpha^i|^2 indicates; 3)
check to see how many times the lowered weights can still be lowered,
and lower them; 4) when each weight has been lowered as many times as
allowed, you're done. You now know the allowed quantum numbers (the
weights) of the representation. The only ambiguity occurs when a given
weight can be reached by multiple paths; that is, by different
sequences of lowering operations. In that case, you have to determine
how many of those paths give linearly independent states: this will
give the number of distinct particles in the representation with the
given quantum numbers.

Finally, we applied this algorithm to SU(3), identifying the Cartan
subalgebra generators T_3 and T_8 and the 6 step operators
E_{\pm(1,0)} = \frac{1}{\sqrt{2}}\ (T_1 \pm i T_2),
E_{\pm(\frac{1}{2}, \frac{\sqrt{3}}{2}\ )} = \frac{1}{\sqrt{2}}\ (T_4
\pm i T_5), and E_{\mp(\frac{1}{2},-\ \frac{\sqrt{3}}{2}\ )} =
\frac{1}{\sqrt{2}}\ (T_6 \pm i T_7). The positive roots are (1,0),
(\frac{1}{2}, \frac{\sqrt{3}}{2}\ ), and (\frac{1}{2}, -\
\frac{\sqrt{3}}{2}\ ), with simple roots \alpha^1 = (\frac{1}{2},
\frac{\sqrt{3}}{2}\ ) and \alpha^2 = (\frac{1}{2}, -\
\frac{\sqrt{3}}{2}\ ). This determines the fundamental weights \mu^1
= (\frac{1}{2}, \frac{1}{2\sqrt{3}}\ ), and \mu^2 = (\frac{1}{2}, -\
\frac{1}{2\sqrt{3}}\ ). We then built representations with various
choices of highest weight, showing the quantum number plots predicted
by representation theory.

We noted that some of the predicted representations --- particularly
the 8 and the 10 --- predicted quantum number patterns that looked
quite familiar to particle physicists when Gell-Mann and Neeman
pointed them out. In fact, it is sensible to interpret T_3 as the
I_z of isospin symmetry --- noting that T_1, T_2, and T_3
define an SU(2) subgroup in the 12-plane that we may identify with
isospin. We can then identify \frac{2}{\sqrt{3}}\ T_8 with
hypercharge: as desired, it commutes with the entire SU(2) isospin
subgroup. With these identifications, the observed quantum numbers of
baryon octets and decuplets, and meson octets, agreed exactly with
those predicted for SU(3) representations 8 and 10. We will discuss
Gell-Mann and Zweig's explanation for why these representations arise
--- the quark model --- next time.


NOTE: The non-ascii version of this on the class web-page is much
easier reading.


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KB