• Home
  • Textbooks
  • Quarks And Leptons. An Introductory Course In Modern Particle Physics
  • Gauge Symmetries

Quarks And Leptons. An Introductory Course In Modern Particle Physics

Francis Halzen, Alan D. Martin

Chapter 14

Gauge Symmetries - all with Video Answers

Educators

Chapter Questions

03:37

Problem 1

If you are unfamiliar with this formalism, consult, for example, Goldstein (1977) or work through the example given in Sakurai (1967), page $3 .$

Carolina Acevedo
Carolina Acevedo
Numerade Educator
00:58

Problem 2

Verify that the Dirac equation follows from
$$
\mathcal{L}=i \bar{\psi} \gamma_{\bar{\mu}} \partial^{\mu} \psi-m \bar{\psi} \psi,
$$
where each of the four components of $\psi$ and $\bar{\psi}$ is regarded as an independent field variable.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
03:29

Problem 3

Show that the substitution of the Lagrangian
$$
\mathcal{L}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu}-j^{\mu} A_{\mu}
$$
into the Euler-Lagrange equation for $A_{\mu}$ gives the Maxwell equations, $(6.57)$
$$
\partial_{\mu} F^{\mu \nu}=j^{\nu}
$$
where $F^{\mu \nu} \equiv \partial^{\mu} A^{\nu}-\partial^{\nu} A^{\mu}$. Hence, show that the current is conserved, that is, $\partial_{\nu} j^{p}=0$.

Ameer Said
Ameer Said
Numerade Educator
01:53

Problem 4

With the addition of a term $\frac{1}{2} m^{2} A_{\mu} A^{\mu}$, show that the Lagrangian of (14.9) leads to an equation of motion
$$
\left(\square^{2}+m^{2}\right) A^{\mu}=j^{\mu}
$$
The new term is therefore a photon mass contribution. We shall see in Section $14.3$ that it is forbidden by gauge invariance. The photon is massless.

Manik Pulyani
Manik Pulyani
Numerade Educator
04:47

Problem 5

Show that } d Q / d t=0

Scott Stetson
Scott Stetson
Numerade Educator
03:13

Problem 6

{ Show that } U(1) \text { phase invariance of the Lagrangian }

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:21

Problem 7

Read about the Bohm-Aharonov effect. Suggested references are the Feynman Lectures on Physics, Volume 2, or Wu, T. T. and Yang, C. N. (1975) Phys. Rev. D12, $3845 .$

Suzanne W.
Suzanne W.
Numerade Educator
05:26

Problem 8

Show that det $U=e^{i \phi}$, where $\phi$ is real. We separate off such an overall phase by restricting the group transformations to those with $\operatorname{det} U=+1$, see Section $2.3$. We called this the group of special unitary $3 \times 3$ matrices $S U(3)$. Show that the requirement $\operatorname{det} U=+1$ implies $\operatorname{Tr}\left(T_{0}\right)=0$. Verify that $U^{\dagger}=U^{-1}$ requires

Wasim Sher
Wasim Sher
Numerade Educator
01:10

Problem 9

Show that the structure constants $f_{a b c}$ are antisymmetric under interchange of any pair of indices,

Manik Pulyani
Manik Pulyani
Numerade Educator
02:57

Problem 10

Due to the additional term in (14.38), $G_{\mu \nu}^{a}$ has a more complicated form than its counterpart in QED, (14.27). In order that the kinetic energy be invariant under (14.38), show that

Urvashi Arora
Urvashi Arora
Numerade Educator
17:54

Problem 11

Using the prescription for obtaining the Feynman rules from the Lagrangian that we mentioned in Section 14.1, show that the vertex factors for the quark-gluon and triple gluon vertices of Fig. 14.1 are, respectively,

Mahnoor Amin
Mahnoor Amin
Numerade Educator
02:23

Problem 12

The Lagrangian for three interacting real fields $\phi_{1}, \phi_{2}, \phi_{3}$ is

Alvar Garcia-Fernandez
Alvar Garcia-Fernandez
Numerade Educator
04:35

Problem 12

Rather than $(14.60)$, take instead $\phi$ to be an $S U(2)$ triplet of real scalar fields. For $\mu^{2}<0$ and $\lambda>0$, show that in this case two gauge bosons acquire mass but that the third remains massless.

Hint Verify, and use, $\left(T_{k}\right)_{i j}=-i \varepsilon_{i j k}$ for the triplet representation of $S U(2)$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator