Most problems in physics cannot be solved exactly.
The few that that can be are solved by exploiting symmetries. The original example is the Kepler problem of a planet moving around the Sun under the inuence
of gravity. The symmetry und
1. C OMPACT L IE A LGEBRAS
1.1. An inner product on a real vector space is a positive bilinear function ., . : V V R. That is
< u, v >=< v, u >
< u, u > 0,
< u, u >= 0 = u = 0.
< 1 u1 + 2 u2 , v >= 1 < u1 , v > +2 < u2 , v >
1.2. A Lie algebra i
1. R ELATIVITY
1.1. It is an astonishing physical fact that the speed of light is the same
for all observers. This is not true of other waves. For example, the speed
of sound measured by someone on standing on Earth is different from that
1. YANG -M ILLS T HEORY
1.1. Yang-Mills Theory is the foundation of the theory of elementary
particles. It describes the self-interaction of spin 1 particles: the photon,
Z,W and the gluons. The principle of gauge invariance also determines
S ITTER G EOMETRY
1.1. Gravity modies the geometry of space-time. Particles move along
geodesics (lines of least length) instead of straight lines. The departure of
the geomery from Minkowski space manifests itself as the bending of the
1. M AXWELL S E QUATIONS
The book by Jackson on Classical Electrodynamics has become a standard reference. The second volume of the series by Landau and Lifshitz
Classical Theory of Fields shows greater physical insight.
1.1. All magnetic elds
,; t"r 4- T'l
jgltgy SE:!- z
= T"t .Tt'+ z T.t'
= Tot 5' t 2 tod* t e 5n rr' + zl.J/
r-lt> = J )
r.,tt)= L l+>
Jzlt>= I It)
xt! f4 si M A.A 3 r. = 6 "k"*'e" un LV.
Jo W W,* ,-r,-btr*. .,4cfw_v.+a-c'^h"bl- u<' &o ^? .WX^br".-L
h"A,& *"d, .lL h. lr-eltU n*b'* .*"L x,(/"6r,* ",k'A *otl(
f rtr^- ,f
Lecture 8 1. ROTATIONS T HROUGH 2 Recall that the the rotation group is SO 3 , the set of orthogonal matrices of positive determinant. Closely related is the group SU 2 of unitary matrices of determinant one. We saw that innitesimally they are the same: t
Lecture 7 1. A DDITION OF A NGULAR M OMENTUM 1.1. If we add two vectors of lengths r and r the sum can have any length between r r and r r . In particular if we combine two classical systems with angular momenta j and j the combined system can have any an
1. I DEAL I NCOMPRESSIBLE F LUIDS
Euler not only gured out the equations of a rigid body but also those of
a uid. An ideal uid is a model of uid motion in which we ignore the
effects of friction: in the real world uids lose energy when different
The Rigid Body
If the distance between any two points of a body is xed as
it moves, it is a rigid body.
Thus a rigid body can move by a translation of its center of mass; and a rotation
around its center of mass. Imagine throwing a book in
1. T HE R EPRESENTATIONS OF o(3)
The book Principles of Quantum Mechanics by R. Shankar has a more
detailed discussion of this topic.
Symmetry transformations form a group. Innitesimal transformations
form a Lie algebra. In quantum mechanics, sy
1. L IE A LGEBRAS
1.1. A Lie algebra is a vector space along with a map
a b c a c b c bi linear
bc a ca b
We will only think of real vector spaces. Even when we talk of matrices
1. G ROUPS
Once you understand the basic structure of a physical or mathematical
theory, it is useful to summarize the basic laws as axioms: independent
facts from which all others can be derived. This was rst achieved for plane
geometry by Eucl
1. R ANDOM M ATRICES
The main reference is the book Random Matrices by M. L. Mehta.
More modern developents are in the book by D. Gioev and P. Deift.
1.1. A matrix whose entries are random variables is a random matrix.
Example 1. This idea was