Lecture 6

Reading:

Martin & Shaw, chapter 1

Recall from last lecture:

Summation Notation

An alternative notation is often used for the transformation. This notation makes the equations a bit more compact and easier to manipulate. It comes about by realizing that we can organize the 4-vector as a 1-column by 4-row matrix. Number the rows of the matrix from 0 to 3, with (x⁰, x¹, x², x³) = (ct, x, y, z), then the Lorentz transformation is:

x^m = L^m_n xⁿ

where the indices run from 0 to 3 and summation over the repeated index n is implied.

To be precise, the quantities in the above expression are tensors, not column vectors and matrices. However, we can interpret the expression as matrix multiplication with L given by the 4×4 matrix:

L =	g	-gb	0	0
	-gb	g	0	0
	0	0	1	0
	0	0	0	1

Written this way, the Lorentz transformation resembles the rotation matrix about the z-axis in 3-dimensions:

R =	cos q	-sin q	0
	sin q	cos q	0
	0	0	1

We consider the Lorentz transformation to be an extension of rotations into 4-dimensional space-time. Also, by rotating about the y and z axes, we can produce a Lorentz transformation that boosts in an arbitrary direction from the one given above that boosts along the x axis.

Finally, the dot product of two 4-vectors is written in this notation as:

x·y = g_μν x^m yⁿ

where g is the metric of the space. We will always work in flat space where the metric is:

g =	1	0	0	0
	0	-1	0	0
	0	0	-1	0
	0	0	0	-1

The Standard Model of Particle Physics

The Standard Model (SM) of particle physics emerged in the 1970's when quantum chromodynamics (QCD) and the electroweak theory (the two major components of the standard model Lagrangian) emerged from a throng of alternatives as best able to explain experimental observations. It has passed rigorous experimental confirmation since. We will discuss many of the important results in the coming weeks.

The Fundamental Fermions

In the SM, all matter is constructed from a small number of fermions (spin ½ particles). These are the six quarks and six leptons, as shown in the table below.

particle	charge	flavor
quarks	+2/3	u	c	t
quarks	-1/3	d	s	b
leptons	-1	e	μ	τ
leptons	0	ν_e	ν_μ	ν_τ

The existence of all of the fundamental particles has now been verified. The last two to be verified were the top quark, t (1995), and the tau neutrino, ν_τ (2001). The four rows of the table group particles of the same electric charge with similar properties, while the three columns divide the particles into generations. While the matter around us is built using just the particles of the first generation (column), the existence of three generations has important consequences in the Standard Model. But why the number is three is not understood.

The leptons carry integral electric charge. The electron, e, is the familiar component of atoms. The muon, μ, and tau, τ are heavier versions of the electron. The neutral leptons are called neutrinos, the ethereal and nearly massless particles of the SM. Paired with each charged lepton is a different neutrino.

The quarks carry fractional charge +2/3 and -1/3 times the proton charge, denoted by |e|.