Physics Notes: The Standard Model

Lecture 3: Jan 25, 2010 Back to PHY30

Responses to questions

Question about changes in values of coupling constants since big bang -

e is within one part in of the big bang values, so it doesn’t appear to be changing.

(I don’t remember the referenced measurement, but probably spectral data from early galaxies)

Reminders about particle/anti-particle fields

For electrons (and positrons)

Or mixing differently

The point is that the field for the anti-particle is the Hermitian conjugate of the field for the particle.

Charge conservation -> all terms in Lagrangian have an equal # of and so that each individual process preserves charge.

Symmetry operation – multiply by and , then in Lagranian terms we will always have , leaving the field unchanged.

Introducing group theory:

Let be such a phase.

We have an identify element g(0) = I, the identity element in the group

Every element has an inverse .

In this case .

This simple group has the name U(1). U stands for Unitary.

A Two State System

This one will be known as Z(2)

Take the system where

P|u> -> |d>, and P|u> -> |d>

In field terms

Our group has two members (I, P)

Groups have encodings as matrices.

Trivially, U(1) is encoded as the 1x1 matrix set

For Z(2) we can encode the group as a set of 2x2 matrix values which have the desired properties under multiplication.

For example:

another option is:

Both work for representing the group. The second representation has an advantage in that it couples well with the column vector state encoding. We would like

If we take a system of 2 quarks, then we have 4 states: dd, ud, du, uu

Suppose we want a transformation that exchanges d’s with u’s, so

We still have only two members of our group, the identity element and an element that swaps d for u . We have to change the encoding of the group to be compatible with the 4 element column vectors.

Rotations

Side Note: Comment about continuous symmetries corresponding to Lie groups.

Each rotation can be characterized by 3 parameters: longitude and latitude of the vector to rotate about plus the angle of rotation.

Rotations can be combined in an associative fashion.

There is an identity and rotations have inverses.

Color of quarks is a rotation group

This rotation group can be encoded in a matrix

Using Einstein summation notation

What do we know about our rotations R?

(Lapsing to reals here)

so: because these are the only terms allowed in the product

Another way to say this is that

, This is the “Unitary” condition

How many parameters are there in a rotation?

, when applied to spinor

The determinant has to be 1 so that the magnitude of the vector is unchanged

, which removes one parameter, taking us back to the original 3 parameters needed to encode a rotation.

I struggled a moment with the idea of applying a rotation to a spinor. It clicked when I went back to the notion of an electron in a prepared state. The spinor represents the spin state along a particular axis. If the electron (or the coordinate system) is rotated, then the state along that axis is transformed as well.

The group of NxN unitary matrices is U(n).

With determinate=1 we have the Special Unitary Group or SU(n)

The group we apply to spinors is SU(2).