- (a) Show that Noether's theorem for global \(S U(3)_{\text {color }}\) symmetry in QCD leads to the conserved color current \(j_{\mu}^{a}=\bar{\psi}_{i} \gamma_{\mu} T_{i j}^{a} \psi_{j}-f^{a b c}
- Calculate the \(\overline{\mathrm{MS}}\) counterterm for the quark-gluon vertex in an arbitrary covariant gauge and obtain the result for \(\delta_{1}\) in Eq. (9.2.28).
- Draw the diagrams contributing to the \(\mathcal{O}\left(g^{2}\right)\) (one-loop) corrections to the tree-level ghostgluon, three-gluon and four-gluon vertices. Write down the loop integrals for the
- Define the BRST transformation of some field \(\phi\) as \(Q \phi\), where under the BRST transformation \(\phi \rightarrow \phi+\delta \phi\) with \(\delta \phi \equiv \theta Q \phi\). For example,
- Starting from Eq. (9.4.2), complete the derivation of the Lagrangian densities \(\mathcal{L}_{\text {gauge }}\) and \(\mathcal{L}_{\text {Higgs }}\) in terms of the physical fields ( \(W_{\mu}^{
- Show that the fermion kinetic contribution \(\mathcal{L}_{\text {kin-ferm }}\) in Eq. (9.4.20) can be rewritten in terms of the physical vector boson fields as Eq. (9.4.22) with the electroweak
- (a) Use the definition of the Fermi coupling constant in Eq. (5.2.7) and the low-momentum limit of the tree-level contribution to the scattering \(e^{-} \bar{u}_{e} \rightarrow \mu^{-}
- (a) Calculate to lowest order the decay rate \(h \rightarrow b \bar{b}\).(b) Calculate the decay rate for \(h \rightarrow g g\). (Hint: This problem requires a one-loop calculation and some care and
- Consider an \(S U(3)\) nonabelian gauge theory with coupling to eight real scalar fields in the adjoint representation, \(\phi^{a}\) for \(a=1, \ldots, 8\). The covariant derivative is then
- A cylinder of mass \(M\), moment of inertia \(I\) about the cylindrical axis and radius \(R\) rolls on a horizontal surface without slipping.(a) Express the no-slip constraint in differential form.
- A block of mass \(m\) slides without friction on a larger block of mass \(M\) and is attached to a pin in this block by a massless spring with spring constant \(k\). All motion is in one dimension.
- A uniform circular wire of radius \(R\) is forced to rotate about a fixed vertical diameter at constant angular velocity \(\omega\). A bead of mass \(m\) experiences gravity, is smoothly threaded on
- A bead of mass \(m\) is threaded without friction on a massless wire hoop of radius \(R\) that is forced to oscillate vertically in a fixed vertical plane at angular frequency \(\omega\) and with
- Two point masses \(m\) and \(M\) are connected by a massless rod of length \(\ell\) and placed on a horizontal table. The mass \(m\) is also connected to a fixed point \(P\) on the table by a spring
- A mass- \(m\) particle moves along the \(x\)-axis subject to a potential energy \(V(x)=\lambda\left[\left(x^{2}-a^{2}\right)^{2}+\right.\) \(\left.2 a x^{3}\right]\), where \(\lambda\) and \(a\) are
- A uniform thin rod of length \(\ell\) and mass \(m\) is suspended from fixed points \(A\) and \(B\) by two identical massless springs with zero natural length and spring constant \(k\). At
- Show that the Lagrangian \(L=\left(q_{1} \dot{q}_{2}-q_{2} \dot{q}_{1}\right)^{2}-a\left(q_{1}-q_{2}\right)^{4}\) is invariant under the transformationsand construct the corresponding conserved
- A system with a single degree of freedom is governed by the Lagrangian \(L=e^{\beta t}\left(\frac{1}{2} m \dot{q}^{2}-\right.\) \(\frac{1}{2} k q^{2}\).(a) Write down the equation of motion. What
- A particle of mass \(m\) moves in one dimension in a potential \(V(q)=-C q e^{-\lambda q}\) where \(C\) and \(\lambda\) are positive constants. Write down the Lagrangian and find the point of stable
- The Lagrangian of a particle of mass \(m\) and charge \(e\) in a uniform, static magnetic field \(\mathbf{B}\) is given by \(L(\mathbf{x}, \dot{\mathbf{x}})=\frac{1}{2} m \dot{\mathbf{x}}^{2}+(e / 2
- (a) Show that the relativistic Lagrangian \(L=-(1 / \gamma) m c^{2} e^{U / c^{2}}\) reduces to the Newtonian Lagrangian for a particle of mass \(m\) in a gravitational potential \(U\) in the slow
- Consider a differentiable function \(V(q)\) of a generalized coordinate \(q\). Consider the Lagrangian \(L=(1 / 12) m^{2} \dot{q}^{4}+m \dot{q}^{2} V(q)-V^{2}(q)\). Show that this system is
- (a) A particle with rest mass \(M\) has kinetic energy \(T\). Show that the magnitude of its momentum is (b) A particle \(A\) decays at rest into a lighter particle \(B\) and a photon \(\gamma\).
- A relativistic positron \(e^{+}\)with kinetic energy \(T\) annihilates a stationary electron \(e^{-}\)and produces two photons \(e^{+}+e^{-} \rightarrow \gamma+\gamma\).(a) At what angles are the
- We define a free complex scalar field as \(\phi(x)=(1 / \sqrt{2})\left[\phi_{1}(x)+i \phi_{2}(x)\right] \in \mathbb{C}\) in terms of two real free scalar fields \(\phi_{1}(x), \phi_{2}(x) \in
- Let \(\phi\) be a Lorentz-scalar field.(a) Show that the quantities \(T_{\mu u}=\partial_{\mu} \phi \partial_{u} \phi-\frac{1}{2} g_{\mu u}\left\{(\partial \phi)^{2}-\kappa^{2} \phi^{2}\right\}\)
- Consider \(n\) real equal-mass scalar fields \(\phi_{1}, \ldots, \phi_{n}\) with a quartic interaction of the form \(\mathcal{L}_{\text {int }}=-(\lambda / 4
- A Lagrangian density with two complex scalar fields, \(\phi\) and \(\chi\), is given bywhere \(m_{\phi}, m_{\chi}, \lambda_{\phi}, \lambda_{\chi}, g\) are real constants.(a) Using the transformations
- Let \(F^{\mu u}\) be the electromagnetic tensor and consider the symmetric stress-energy tensor defined in Eq. (3.3.9), \(\bar{T}^{\mu}{ }_{u}=-F^{\mu \tau} F_{u \tau}+\frac{1}{4} \delta^{\mu}{ }_{u}
- Let \(A^{\mu}\) be the electromagnetic four-vector potential and define the four-vector \(K_{\mu}=\epsilon_{\mu u ho \sigma} A^{u} \partial^{ho} A^{\sigma}\). Show that T = 0 becomes T = (1/c)jF.
- A general real scalar Lagrangian density at most quadratic in the four-vector potential is \(\mathcal{L}=-\frac{1}{2}\left(\partial_{\mu} A_{u} \partial^{\mu} A^{u}+ho \partial_{\mu} A_{u}
- Scalar electrodynamics is the gauge invariant minimal coupling of a charged scalar field and the electromagnetic field. It has the Lagrangian density \(\mathcal{L}=-\frac{1}{4} F_{\mu u} F^{\mu
- Show that the symmetric \(n\)-boson basis states in Eq. (4.1.149) are normalized; i.e., show \(\left\langle b_{i_{1}} \cdots b_{i_{n}} ; S \mid b_{i_{1}} \cdots b_{i_{n}} ; S\rightangle=1\).
- For any vector operator \(\hat{\mathbf{V}}\) in quantum mechanics we have \(\left[\hat{V}^{i}, \hat{J}^{j}\right]=i \hbar \epsilon^{i j k} \hat{V}^{k}\) from Eq. (4.1.114). It will be helpful to also
- A scalar particle with mass \(m\) and charge \(q\) interacts with a time-independent electromagnetic potential given by \(q A^{0}=0\) for \(z0\) for \(z \geq 0\). A particle plane wave is moving in
- Using the properties of the Dirac \(\gamma\)-matrices verify that the matrices \(\left(\frac{1}{2} \sigma^{\mu u}\right)\) satisfy the Lie algebra of the Lorentz transformations in Eq. (1.2.179).
- Explain why the hermiticity of \(\gamma^{0}\) and \(\gamma^{5}\) and the antihermiticity of \(\gamma\) are preserved in any representation unitarily equivalent to the Dirac representation.
- Provide detailed proofs of the trace identities:(a) traces of products of \(\gamma\) matrices (with no \(\gamma^{\mu \dagger}\) ) are representation independent (same for \(\gamma^{\mu}\) and
- Prove the following identities:(a) \(\gamma_{\mu} \gamma_{u} \gamma_{ho}=g_{\mu u} \gamma_{ho}-g_{\mu ho} \gamma_{u}+g_{u ho} \gamma_{\mu}+i \epsilon_{\mu u ho \sigma} \gamma^{\sigma}
- Prove each of the identities: \(\gamma^{\mu} \sigma^{ho \sigma} \gamma_{\mu}=0, \sigma^{\mu u} \sigma_{\mu u}=12, \gamma^{\mu} \sigma^{ho \sigma} \gamma^{\lambda} \gamma_{\mu}=2 \gamma^{\lambda}
- Consider the sixteen matrices \(\left\{\Gamma_{1}, \Gamma_{2}, \ldots, \Gamma_{16}\right\} \equiv\left\{I, \gamma^{\mu},\left.\sigma^{\mu u}\right|_{\mu
- (a) Prove that if \(\psi(x)\) is a solution of the Dirac equation, Eq. (4.4.59), then it is also a solution of the Klein-Gordon equation, Eq. (4.3.12).(b) Prove that \(\partial_{\mu}\) and
- Using the standard representation of the Pauli spin matrices, show that complex conjugation with a rotation of \(\pi\) about the \(y\)-axis reverses the orientation of the spin; i.e., show that
- Prove directly from the definition of \(M(\Lambda)\) and \(\bar{M}(\Lambda)\) in Eq. (4.6.14) that \(i \sigma^{2} \bar{\chi}^{*}\) transforms with \(M(\Lambda)\) and \(-i \sigma^{2} \psi^{*}\)
- Show that the area of two other of the six nontrivial unitarity triangles of the CKM matrix are equal to half of the Jarlskog invariant, \(J\).
- In some circumstances the contribution of one neutrino flavor can be approximately neglected; e.g., in \(u_{\mu} \rightarrow u_{\tau}\) atmospheric oscillations the role played by the \(u_{e}\) is
- Show that the set of permutations of \(n\) symbols \(S_{n}\) satisfies the properties of a group. Recall that the four properties of a group are closure, associativity, existence of the identity and
- Draw the weight diagram for the \((6,3)_{D}\) multiplet of \(S U(3)\) and indicate the multiplicities of each layer. Use this to calculate the number of states in the irrep/multiplet. Draw the
- Draw the Young diagram for the \((1,0,1)_{D}\) irrep/multiplet of \(S U(4)\). Draw all standard Young tableaux for this irrep/multiplet and construct the corresponding Young symmetrizers.
- Prove the \(S U(2)\) results in Eq. (5.3.28): \(\mathbf{2} \otimes \mathbf{2}=\mathbf{3} \oplus \mathbf{1} ; \mathbf{3} \otimes \mathbf{3}=\mathbf{5} \oplus \mathbf{3} \oplus \mathbf{1}\); and
- Prove the \(S U(3)\) results in Eq. (5.3.29): \(\mathbf{3} \otimes \overline{\mathbf{3}}=\mathbf{8} \oplus \mathbf{1} ; \mathbf{3} \otimes \mathbf{3}=\mathbf{6} \oplus \overline{\mathbf{3}} ;
- Prove Eq. (5.3.38) for \(S U(4): \mathbf{4} \otimes \overline{\mathbf{4}}=\mathbf{1 5} \oplus \mathbf{1}\).
- Prove Eq. (5.3.41) for \(S U(6): \mathbf{6} \otimes \mathbf{6} \otimes \mathbf{6}=\mathbf{5 6} \oplus \mathbf{7 0} \oplus \mathbf{7 0} \oplus \mathbf{2 0}\). Explain using Young diagrams why the
- For \(S U(3)_{\text {flavor }}\) recall that we defined \(\left|i_{1}\rightangle=|uangle,\left|i_{2}\rightangle=|dangle,\left|i_{3}\rightangle=|sangle\) and
- A beam of \(\pi^{-}\)mesons is directed at a proton target and produces a variety of final states. For each of the particle states listed below propose a reaction that leads to a final state
- Consider the following reactions. Which reactions are allowed? If allowed, explain what the main reaction mechanism is using a labeled diagram showing quark, lepton and boson lines. Explain whether
- Substitute Eq. (6.2.14) for \(\hat{\phi}_{s}(\mathbf{x})\) and \(\hat{\pi}_{s}(\mathbf{x})\) into Eq. (6.2.16) and hence show that the expressions given for the annihilation and creation operators,
- Show that the commutation relations of \(\hat{a}_{\mathbf{p}}\) and \(\hat{a}_{\mathbf{p}}^{\dagger}\) in Eq. (6.2.17) reproduce each of the Schrödinger picture canonical commutation relations in
- Obtain the result for the total three-momentum operator \(\mathbf{P}\) for a free scalar field in terms of annihilation and creation operators as given in Eq. (6.2.34).
- Explain why the expressions for the \(n\)-boson identity operator \(\hat{I}_{n}\) in Eqs. (6.2.63) and (6.2.64) are identical and verify that \(\hat{I}_{n}\left|\mathbf{p}_{1} \cdots
- Use Eq. (6.2.129) and symmetry to explain why \(\hat{\mathbf{P}}\) and \(\hat{M}^{i j}\) do not need normal-ordering. Show that for a scalar boson fieldwhere \(\hat{L}^{\mu u}\) is the coordinate
- In Eq. (6.2.168) we showed, using its vacuum expectation value form, that \(D_{F}(x-y)\) satisfied \(\left(\partial_{x}^{2}+m^{2}\right) D_{F}(x-y)=-i \delta(x-y)\). Using their vacuum expectation
- Prove that (a) the only nonzero commutation relations for the annihilation and creation operators of a charged scalar field \(\left(\hat{f}_{\mathbf{p}}, \hat{f}_{\mathbf{p}}^{\dagger},
- Prove Eq. (6.3.197) for writing the fermion creation and annihilation operators in terms of the field operators.
- Show that the fermion operators \(\hat{P}^{\mu}\) and \(\hat{M}^{\mu u}\) are the generators of the Poincaré group by proving Eqs. (6.3.214), (6.3.212) and (6.3.213). Show that \(\hat{P}^{\mu}\) and
- In the covariant quantization approach:(a) Prove Eq. (6.4.148); i.e., the addition of \(-\frac{1}{2 \xi}\left(\partial_{\mu} A^{\mu}\right)^{2}\) to the Maxwell langrangian gives the equations of
- Generalize the arguments for the neutral vector field and obtain some of the key results for the charged vector field case in analogy with the extension of the neutral scalar field to the charged
- For a massive vector boson field:(a) Prove Eq. (6.5.51), which is the result that \(\hat{A}^{0}(x)=-\boldsymbol{abla} \cdot \hat{\mathbf{E}}(x) / m^{2}\) for a free massive vector boson field.(b)
- (a) Define \(\left\{V_{1}, V_{2}, \ldots, V_{16}\right\} \equiv \frac{1}{2}\left\{I, \gamma^{0}, i \gamma^{j}, i \sigma^{0 j},\left.\sigma^{j k}\right|_{j
- Consider the proton-proton collisions in the Large Hadron Collider. Assume that: improved beam optics gave an improved luminosity reduction factor \(F_{\text {red }}=0.88\) and a transverse beam size
- Sketch three examples of connected four-point \((n=4)\) Feynman diagrams in \(\phi^{3}\) theory contributing to the left-hand side of Eq. (7.5.37). Show that each diagram can be uniquely identified
- Sketch three examples of connected two-point \((n=2)\) Feynman diagrams with 10 vertices in \(\phi^{4}\) theory that contribute to \(D_{F}(p)\). Show that these can be uniquely identified with
- Consider a Yukawa theory with the Lagrangian density \[\begin{equation*}\mathcal{L}=\frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi-\frac{1}{2} m^{2} \phi^{2}+\bar{\psi}(i ot \partial-M) \psi-g
- By drawing on the key steps in the derivation of the LSZ formalism for scalars and fermions, outline the key steps in the derivation of the LSZ formalism for charged scalar bosons. There is no need
- Write down the version of the photon LSZ formula of Eq. (7.5.125) that is appropriate for massive vector Proca bosons. Justify your answer by describing the key steps needed to arrive at this result.
- Consider a theory with three interacting scalar fields \(\Phi, \phi_{a}\) and \(\phi_{b}\) with masses \(M, m\) and \(m\), respectively, with \(3 m>M>2 m\) and with the Lagrangian density
- Consider a physical system described by the Lagrangian density \[\begin{equation*}\mathcal{L}=\frac{1}{2}\left[\partial_{\mu} \phi \partial^{\mu}\phi+\partial_{\mu} \chi \partial^{\mu}
- The tree-level invariant amplitude for the equivalent of Compton scattering in scalar QED was calculated in Eq. (7.6.54). Use this to calculate the unpolarized differential cross-section \(d \sigma /
- Consider a gauge theory of charged scalars and charged fermions interacting with photons, which is a combined form of QED and scalar QED, \[\begin{equation*}\mathcal{L}=\mathcal{L}_{\text
- Use the equal-time canonical commutation and anticommutation relations to prove Eq. (8.3.22). Then for either a boson or fermion operator \(\hat{X}(y)\) generalize Eq. (6.2.166) and show that
- Using the Feynman rules for a \(\phi \bar{\psi} \psi\) Yukawa theory, write down the one-loop contribution to the \(\phi^{4}\) proper vertex. Show that this one-loop contribution is divergent and
- Prove Eq. (8.7.36).
- Prove the results in Eq. (8.7.53) in the chiral representation or any other representation that you care to use.
- Complete the detailed steps to arrive at the scattering of a charged fermion from a static electromagnetic field given in Eq. (8.7.48).
- Consider the pseudoscalar Yukawa theory in Eq. (7.6.11) with a \(\phi^{4}\) interaction, \[\begin{equation*}\mathcal{L}=\frac{1}{2}\left(\partial_{\mu} \phi \partial^{\mu} \phi-m_{\phi}^{2}
- Calculate a one-loop contribution to the \(\gamma \gamma \gamma 1\) PI vertex in QED. Show that it vanishes as expected from Furry's theorem and so needs no renormalization. (Hint: Eq. (A.3.29) may
- In the discussion of the Casimir effect we placed the parallel plates in the middle of the box. Show that the Casimir effect depends only on the plate separation \(a\) and not on plate location as
- A muon is a more massive version of an electron and has a mass of \(105.7 \mathrm{MeV} / \mathrm{c}^{2}\). The dominant decay mode of the muon is to an electron, an electron antineutrino and a muon
- The diameter of our Milky Way spiral galaxy is 100,000-180,000 light years and our solar system is approximately 25,000 light years from the galactic center. Recalling the effects of time dilation,
- Consider two events that occur at the same spatial point in the frame of some inertial observer \(\mathcal{O}\). Explain why the two events occur in the same temporal order in every inertial frame
- Consider any two events that occur at the same time in the frame of an inertial observer \(\mathcal{O}\). Show that by considering any Lorentz transformation there is no limit to the possible time
- Two narrow light beams intersect at angle \(\theta\), where \(\theta\) is the angle between the outgoing beams. The beams intersect head on when \(\theta=180^{\circ}\). Using the addition of
- A ruler of rest length \(\ell\) is at rest in the frame of inertial observer \(\mathcal{O}\) and is at angle \(\theta\) with respect to the \(+x\)-direction. Now consider an inertial observer
- If a spaceship approaches earth at \(1.5 \times 10^{8} \mathrm{~ms}^{-1}\) and emits a microwave frequency of 10 \(\mathrm{GHz}\), what frequency will an observer on Earth detect?
- An inertial observer observes two spaceships moving directly toward one another. She measures one to be traveling at \(0.7 c\) in her inertial frame and the other at \(0.9 c\) in the opposite
- A light source moves with constant velocity \(v\) in the frame of inertial observer \(\mathcal{O}\). The source radiates isotropically in its rest frame. Show that in the inertial frame of
- Write down the Lorentz transformation rule for an arbitrary \((3,2)\) tensor \(A^{\mu u ho}{ }_{\sigma \tau}\). Hence show that any double contraction of this tensor leads to a contravariant vector,
- A rocket of initial rest mass \(M_{0}\) has a propulsion system that accelerates it by converting matter into light with negligible heat loss and directs the light in a collimated beam behind it. The
- Two identical spaceships approach an inertial observer \(\mathcal{O}\) at equal speeds but from opposite directions. A second observer \(\mathcal{O}^{\prime}\) traveling on one of the spaceships
- An observer on Earth observes the hydrogen- \(\beta\) line of a distant galaxy shifted from its laboratory measured value of \(434 \mathrm{~nm}\) to a value of \(510 \mathrm{~nm}\). How fast was the

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