Questions and Answers of Modern Physics

The electromagnetic field tensor \(F^{\mu v}\) given in Eq. (14.14) is Lorentz covariant since it is a rank-2 Lorentz tensor. Show that it is also invariant under the local gauge transformation given
Consider spacetime paths between fixed endpoints \(A\) and \(B\). For a Lagrangian function \(L\left(x^{\mu}(\sigma), \dot{x}^{\mu}(\sigma)\right)\), where \(\sigma\) parameterizes the position on
Construct the matrix generator \(A_{\mu}(x)=\tau_{i} A_{\mu}^{i}(x)\) of Eq. (16.49) for an SU(3) YangMills field assuming the representation (8.2).Data from Eq. 16.49 AH (o, A) (A,A) = = j" = (p, j)
Evaluate the commutator \(\left[D_{\mu}, D_{v}\right]\), where \(D_{\mu}\) is the covariant derivative defined in Eq. (16.36). Show that for the U(1) electromagnetic field\[(i q)^{-1}\left[D_{\mu},
Prove that Eq. (16.52) follows from Eq. (16.51). Take \(A_{\mu}(x)=\tau_{j} A_{\mu}^{j}(x)\) and\[U(\boldsymbol{\theta}) \simeq 1-i \tau_{k} \theta^{k} \quad\left[\tau_{i}, \tau_{j}\right]=i f_{i j
Show that if Eq. (16.24) is true, the charge \(Q\) of Eq. (16.21) is conserved.where \(\boldsymbol{n}\) is an outward normal to \(S\) and \(d s\) is a surface differential element.Data from Eq.
Prove that if a symmetry is broken spontaneously, at least one generator of the symmetry group gives a non-zero value when applied to the vacuum state.
Use Eq. (17.1) to find the minima of Fig. 17.2 . Confirm Eq. (17.5) for \(\mu^{2})^{1 / 2}=\) \(\left(-2 \mu^{2}\right)^{1 / 2}\) after the symmetry is broken spontaneously.Data from Eq. 17.1Data
Assume the Lagrangian density (17.14) with \(i=1,2,3\) for an isovector Lorentz scalar field to be invariant under a global internal \(\mathrm{SO}(3)\) symmetry. Show that for \(\mu^{2}Data from Eq.
Show that the expansion (18.7) substituted into Eq. (18.1) gives a Lagrangian density for which there appear to be five degrees of freedom: one from a massive scalar \(\eta\), one from a massive
Prove that taking the 4-divergence of both sides of Eq. (18.11) leads to the constraint \(\partial_{\mu} A^{\mu}=0\), so that Eq. (18.11) reduces to Eq. (18.12).Data from Eq. 18.11Data from Eq. 18.12
Show that the current \(\boldsymbol{j}\) in Eq. (18.17) is invariant under the gauge transformation \(\boldsymbol{A} \rightarrow \boldsymbol{A}+\boldsymbol{abla} \chi \equiv \boldsymbol{A}^{\prime}\)
Verify the weak isospin and weak hypercharge quantum number assignments in Table 19.1 , given the charges \(Q\) in the last column. Verify the weak isospin and weak hypercharge assignments for the
Demonstrate that an explicit Dirac mass term in the electroweak Lagrangian of the form \(m \bar{\psi} \psi\) as in Eq. (16.10) would violate gauge symmetry. Show that a mass introduced by breaking
Show that the Weinberg angle \(\theta_{\mathrm{W}}\) is related to the coupling strengths \(g\) and \(g^{\prime}\) by\[\sqrt{g^{2}+g^{\prime 2}}=\frac{g}{\cos
Prove that the generators \(\left(\tau_{1}, \tau_{2}, K\right)\) of Eq. (19.17) for the local \(\mathrm{SU}(2)_{\mathrm{w}} \times \mathrm{U}(1)_{y}\) standard electroweak symmetry annihilate the
One possible exotic QCD hadronic structure is \(q q \bar{q} \bar{q}\). Assuming the quarks and antiquarks to transform according to the fundamental and conjugate representations of an
Assume the basis vectors of the fundamental representation for color \(\mathrm{SU}(3)\) to correspond to the "colors" \(r, g\), and \(b\). Write the properly symmetrized wavefunction corresponding to
Demonstrate that for colors \(r, g\), and \(b\), color \(\mathrm{SU}(3)\) two-quark states are of the form \(q q=\mathbf{3} \otimes \mathbf{3}=\mathbf{6} \oplus \overline{\mathbf{3}}\), withShow that
Verify the color SU(3) representations for combinations of three or fewer quarks and antiquarks given in Eq. (19.28).Data from Eq. 19.28 qq=303=108 qq 3 3 603, 999 3 3 3 = 36315, qqq 3 3 3 1088 10,
Prove that Eq. (19.34) gives the simplest multi-gluon and gluon-quark states that contain an \(\mathrm{SU}(3)\) color singlet in the decomposition.Data from Eq. 19.34 (GG)1: (88)1 (Gqq) : [8 (383)8]
Prove that the boosted right-handed spinor \(\psi_{\mathrm{R}}(\boldsymbol{p})\) is related to the corresponding rest spinor by Eq. (14.21).
Use the \(\gamma\)-matrices in the Weyl representation to show that the Dirac equation (14.31) is equivalent to Eq. (14.25).Data from Eq. 14.31Data from Eq. 14.25 (y"Pu-m)(p) = (iy" - m)(p) = 0
Prove the identity \((\sigma \cdot \boldsymbol{p})^{2}=\mathrm{I}^{(2)} p^{2}\), where \(\sigma=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)\) are the Pauli matrices, \(\boldsymbol{p}\) is the
Show that the relations (14.23) follow from Eqs. (14.21) and (14.22).Data from Eq. 14.21Data from Eq. 14.22Data from Eq. 14.23 VR (P) = E+m+o.p VR (0) 2m(E+m)
Verify that the Pauli-Dirac representation (14.43) satisfies Eq. (14.27).Data from Eq. 14.27Data from Eq. 14.43 {,} = + = 2
Verify the projection characteristics implied by equations (14.51).Data from Eq.14.51 2 In = (s + I) Da 1- 24(1-y's) = UR
Verify the results of Eq. (14.48) for the properties of the chiral projection operators.Data from Eq. 14.48 P = P+ P+ + P = 1 P_P+ P+P = 0 Py" = y P
Starting from Eq. (14.17), show that the equation for the 4-vector potential \(A^{\mu}\) takes the covariant form \(\square A^{\mu}=j^{\mu}\) of Eq. (14.13) in Lorenz gauge.Data from Eq. 14.13Data
Beginning with Eq. (14.50), prove Eq. (14.52), and derive the chiral decompositions given for \(\bar{\psi} \psi, \bar{\psi} \gamma^{\mu} \psi\), and \(\bar{\psi} \gamma^{\mu} \gamma^{5} \psi\)
Using the Dirac Hamiltonian \(H=-i \gamma_{0} \gamma^{i} p_{i}+\gamma_{0} m\) given in (14.45), show that the chirality operator \(\gamma_{5}\) commutes with this Hamiltonian only if the mass \(m
Show that in Fig. 14.2 , normal double \(\beta\)-decay conserves lepton number but neutrinoless double \(\beta\)-decay does not. Define a lepton number by \(L \equiv n_{\ell}-n_{\bar{\ell}}\), where
Show that a gauge transformation of the 4-vector potential \(A^{\mu}\) leaves the Maxwell field tensor \(F^{\mu v}\) invariant.
Show that the Maxwell equations written in the manifestly covariant form (14.17) and (14.18) are equivalent to the standard form given by Eqs. (14.1a)-(14.1d).Data from 14.17Data from 14.18Data from
Prove that Eq. (14.3) implies the Maxwell field tensor of Eqs. (14.14) and (14.15).Data from Eq. (14.3)Data from Eq. (14.14)Data from Eq. (14.15) B = V XA E = -VO- at
It is common to use the normalization \(u^{(\alpha) \dagger}(\boldsymbol{p}) u^{(\beta)}(\boldsymbol{p})=(E / m) \delta_{\alpha \beta}\) for a massive free Dirac spinor \(u^{(\alpha)}\), where \(E\)
In the Pauli-Dirac representation of Table 14.1, a suitable charge conjugation operator is \(C=i \gamma^{2} \gamma^{0}\). Show that \(C\) is given explicitly by the matrixin this representation.
By applying two successive Poincaré transformations (15.1), show that the Poincaré multiplication rule is given by Eq. (15.2). Data from 15.1Data from 15.2 "q+^x "V= "H =
Prove that the infinitesimal rank-2 tensor \(\epsilon_{\mu u}\) introduced in Eq. (15.7) is antisymmetric in its indices, \(\epsilon_{\mu v}=-\epsilon_{v \mu}\), by requiring that Eq. (15.7) be
Prove the result in Eq. (15.9) that the \(L_{\mu v}\) defined in Eq. (15.8) are generators of proper Lorentz transformations. Utilize \(\partial^{\alpha}
Prove the result in Eq. (15.13) that the \(P_{\mu}\) defined in Eq. (15.12) are generators of translations. Remember that \(\partial^{\alpha} x_{ho}=\eta_{ho}^{\alpha}=\delta_{ho}^{\alpha}\).Data
Verify \(\left[P^{2}, P_{\mu}\right]=0\) [Eq. (15.16)] and \(\left[P_{\mu}, L_{ho \sigma}\right]=i\left(\eta_{\mu ho} P_{\sigma}-\eta_{\mu \sigma} P_{ho}\right)\) [Eq. (15.14)].Data from Eq 15.14Data
Use Eq. (15.10) to prove that \(J_{1}=L_{23}, J_{2}=L_{31}\), and \(J_{3}=L_{12}\).Data from Eq. 15.10 1 Ji J = - Eijk Ljk K = Loi,
Prove Eq. (15.28) for lightlike particles. Use Eq. (15.26) for the standard vector and take note of the Minkowski metric, so \(p_{\mu}=\eta_{\mu u} p^{v}\) and \(L^{\mu u}=\eta^{\mu \lambda}
Prove Eq. (15.30) for massless Poincaré particles.Data from Eq. 15.30 W2 == W = WW=-w (P + P).
\(\mathrm{SO}(2,1)\) is the analog in two spatial dimensions of the Lorentz group \(\mathrm{SO}(3,1)\). Its generators \(\left(X_{1}, X_{2}, X_{3}\right)\) obey the Lie algebra \(\left[X_{i},
Show that the Lorentz group commutation relations (13.20) are satisfied by the choices \(K_{i}= \pm \frac{i}{2} \sigma_{i}\) and \(J_{i}=\frac{1}{2} \sigma_{i}\), where the \(\sigma_{i}\) are Pauli
Verify that the Lorentz transformation (13.16) leaves invariant the squared Minkowski line element (13.5).Data from 13.5Data from 13.16 T = F iF U = F6 iF7 V = F4 iF5.
Find the relationship of the eight SU(3) operators \(T_{ \pm}, V_{ \pm}, U_{ \pm}, T_{3}\), and \(Y\) defined in Eqs. (8.2) and (8.7)-(8.8), and the nine oscillator operators
Verify that the set of matrices (5.14) is closed under ordinary matrix multiplication.Data from Eq. 5.14 T(oc)= = 629 > - (+19) TOO) = (721) TO) = ( ). T(oa)= T(b) TO) -(11) T(4-(11) TO=(9) T(C3)= =
(a) Show that the most general \(2 \times 2\) unitary matrix with unit determinant can be parameterized as in Eqs. (6.76) and (6.77). (b) Take the group identity element \(U(1,0,0,0)\) to correspond
Use Eqs. (7.27)-(7.31) to verify the entries in Table 7.1.Data from Eq. 7.27Data from Eq. 7.28Data from Eq. 7.29Data from Eq. 7.30Data from Eq. 7.31Data from Table 7.1 n = 2 a.m = q- p.
Use the definition of the adjoint representation matrices (3.8), to compute the action of a generator \(X_{a}\) on a state \(\left|X_{b}\rightangle\) given in Eq. (7.3).Data from Eq. 3.8Data from Eq.
Show that the operators given in Eq. (7.20) have the SU(2) commutators (7.21).Data from Eq. (7.20)Data from Eq. (7.21) Eta a. H E = E3 E = |a| a
The group \(\mathrm{D}_{3}\) in Schoenflies notation (32 in international notation, which is read "three-two"; see Table 5.1 ) consists of the proper (those not reflections or inversions) covering
Derive the two-dimensional matrix representation Tic)=(2) Tin)=(3) Tex)=(37) (69) T(c2b)= 1 TO)-(71) 10-(11) TO=(9) = for the group D3, using the basis (e1, e2) defined in the following figure.
Prove that the matrix representation of \(\mathrm{D}_{3}\) worked out in Problem 5.6 is irreducible.Data from Problem 5.6Derive the two-dimensional matrix representation Tic)=(2) Tin)=(3) Tex)=(37)
Show that the group \(D_{3}\) has two 1D irreps in addition to the 2D irrep found in Problem 5.6 , and construct the character table.Derive the two-dimensional matrix representationData from
The matrix representation of \(\mathrm{C}_{3 \mathrm{v}}\) given in Eq. (5.14) was constructed with respect to the particular coordinate system defined by the unit vectors \(\boldsymbol{e}_{1}\) and
Show that the groups \(C_{3 v}\) and \(D_{3}\) have equivalent characters, but the basis functions corresponding to their irreps are different.
Prove that the action of the symmetry operations \(\sigma_{b}\) and \(\sigma_{c}\) on the basis vectors \(\boldsymbol{e}_{1}\) and \(\boldsymbol{e}_{2}\) in Fig. 5.10 are given by the matrix
If cyclic boundary conditions are imposed on a periodic 1D lattice by identifying the two ends with \(N\) cycles between the boundaries, the translation group becomes a cyclic group of order \(N\).
As shown in Problem 5.14 , the symmetry group for a finite 1D periodic lattice having cyclic boundary conditions with \(N\) periods between boundaries is the cyclic group of order \(N\), with irreps
Show that an arbitrary \(2 \times 2\) matrix with real entries that is orthogonal and has unit determinant can always be parameterized as in Eq. (6.3). Thus any \(\mathrm{SO}(2)\) matrix can be
Use Eq. (6.20) to show that Eq. (6.21) defines the SO(2) integration measure.
Show that the \(\mathrm{SO}(3)\) Clebsch-Gordan coefficients \(\left\langle j_{1} m_{1} j_{2} m_{2} \mid J M\right\rangle\) evaluate to (-1)j-m jmj'm'| 00): = 2j+1 djj' dm,-m'> for the special case J
Use the method of Section 6.3 .9 with stepping operators to find the Clebsch-Gordan coefficients for coupling \(j=1\) and \(j^{\prime}=\frac{1}{2}\) to good total angular momentum \(J\).
Prove Eqs. (6.42) and (6.43), thus showing that the dimension of the \(\mathrm{SO}(3)\) irrep labeled by \(j\) is the character \(2 j+1\) of the identity. Rearrange the sum and useThen take the limit
Derive Eqs. (6.55) and (6.57) for the commutators of tensor operators, beginning from the tensor transformation law given in Eq. (6.52).
(a) Prove that the position coordinate \(r\) transforms as a vector under 3D rotations; that is, show that it is an \(\mathrm{SO}(3)\) tensor of rank one. Hint: Begin by noting that the orbital
Derive the forms for the spherical tensor components given in Eq. (6.68).
Prove the spherical harmonic addition theorem(where \(\theta_{12} \equiv \theta_{1}-\theta_{2}\) ), by coupling two spherical harmonics to an \(\operatorname{SO}(3)\) scalar and invoking the
Use tensor methods to evaluate the reduced matrix element of the spherical harmonic \(Y_{L M}(\theta, \phi)\) between states of good angular momentum \(|J Mangle .
Define a quadrupole operator \(Q_{20}=r^{2} Y_{20}(\theta, \phi)\). The quadrupole moment \(Q\) for a state of good angular momentum \(|j mangle\) is conventionally defined as the expectation value
Prove that the operator \(a_{j m}^{\dagger}\) that acts on the vacuum as \(a_{j m}^{\dagger}|0angle=|j mangle\) to create a fermion with angular momentum \(j\) and magnetic quantum number \(m\)
Demonstrate the relationship between the groups \(\mathrm{SO}(3)\) and \(\mathrm{SU}(2)\) as follows.(a) Associate each 3D euclidean coordinate \(\boldsymbol{x}=\left(x_{1}, x_{2}, x_{3}\right)\)
Consider collisions of pions with nucleons. View the pions as a \(T=1\) isospin triplet \(\pi=\left(\pi^{+}, \pi^{0}, \pi^{-}\right)\), and the nucleon as a \(T=\frac{1}{2}\) isospin doublet, \(N=(p,
Derive the Clebsch-Gordan coefficients for the \(\mathrm{SU}(2)\) direct product \(\mathbf{2} \otimes \mathbf{2}\).
The 2D rotation matrix \(R(\phi)\) defined in Eq. (6.3) is a reducible representation of \(\mathrm{SO}(2)\). Diagonalize \(R(\phi)\) to give the eigenvalues \(\lambda_{ \pm}=e^{ \pm i \phi}\) and Eq.
Construct the rotation matrix \(d_{m m^{\prime}}^{l}\) for \(l=1\). Check your results against the entries in Table C. 1 of Appendix C. Hint: You can save time by considering the product \(d^{1 / 2}
Show that if \(x\) and \(z\) are positive real numbers and \(y\) is an arbitrary real number, the matricesform a group under matrix multiplication but the naive group integration measure \(d g=d x d
Starting from Eq. (6.27) and similar expressions for rotations around the \(x\) and \(y\) axes, show that the generators of 3D rotations are given by Eq. (6.28). Hint: Assume group elements to be
Use \(\mathrm{SO}(3)\) group characters to show that in Eq. (6.44) the coefficients \(c_{J}\) are all zero or one [so \(\mathrm{SO}(3)\) is simply reducible], which leads to the \(\mathrm{SO}(3)\)
Find the simple roots for the rank-2 Lie algebras \(\mathrm{G}_{2}\) and \(\mathrm{SO}(5)\).
Show that the angular momentum Lie algebra \(\left[J_{i}, J_{j}\right]=i \epsilon_{i j k} J_{k}\) can be put in the form\[\left[X_{1}, X_{2}\right]=X_{3} \quad\left[X_{2}, X_{3}\right]=X_{1}
Beginning from the Dynkin diagram for the \(\mathrm{SU}(3)\) algebra, construct the complete root diagram.
For the group \(\mathrm{SO}(3)\), find the metric tensor (7.19) and show that \(\mathrm{SO}(3)\) is compact and semisimple. Use the metric tensor to construct the Casimir operator. Hint: The SO(3)
Prove the result of Eq. (7.13) that the \(E_{ \pm \alpha}\) act as raising and lowering operators within the weight space.
Use the Dynkin diagramto construct the simple roots, and from those all roots, for the algebra \(\mathrm{G}_{2} .
From Problem 7.11 , suitably normalized simple roots for the algebra \(\mathrm{G}_{2}\) are \(\alpha_{1}=\) \((0, \sqrt{3})\) and \(\alpha_{2}=\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\). What is
Use the \(\mathrm{SU}(3)\) algebra to prove that \(T_{ \pm}, V_{ \pm}\), and \(U_{ \pm}\)have the raising and lowering properties in the \(\left(T_{3}, Y\right)\) plane that we have ascribed to them.
Prove that for \(\mathrm{SU}(2)\) symmetry \(\mathbf{2} \otimes \mathbf{2} \otimes \mathbf{2}=\mathbf{4} \oplus \mathbf{2} \oplus \mathbf{2}\), while for \(\mathrm{SU}(3)\) symmetryWhat is the irrep
Determine the \(\mathrm{SU}(3)\) irreps appearing in the direct product \((2,1) \otimes(3,0)\). Label them by their \((\lambda, \mu)\) quantum numbers and dimensionality.
Use the method of Young diagrams to find the irrep content of \(\mathbf{6} \otimes \mathbf{1 0}\) for \(\mathrm{SU}(3)\).
Show that the \(\mathrm{SU}(3)\) quadratic Casimir operator (8.9) can be written aswhere the integers \(p\) and \(q\) labeling representations are defined in Section 8.3 .2. (F) = (p + pq + q ) + p
A representation is said to be complex if it is equivalent to its complex conjugate representation. Show that the \(\mathbf{2}\) and \(\overline{\mathbf{2}}\) representations are equivalent for
Use the graphical method described in Section 8.11 to find the direct product \(\mathbf{3} \otimes \overline{\mathbf{3}}\) for \(\mathrm{SU}(3)\).
Use the fundamental quark triplet and Young diagrams to construct the quark content of the \(\Delta\) decuplet illustrated in Fig. 9.4 .
Use the Young diagram method to deduce the SU(2) isospin content of the SU(3) flavor representations \(\mathbf{6}\) and \(\mathbf{2 7}\).
Use Young diagrams to deduce the \(\mathrm{SU}(2)\) isospin irrep content of the flavor \(\mathrm{SU}(3)\) irrep . (A, ) = (2,1)

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