Rotations in \(\mathbb{R}^3\). The symmetries of 3d Euclidean space \(\mathbb{R}^3\) are those that preserve the distance between any two points. This includes translations \[\vec{x}' = \vec{x} - \vec{a} ~,\] for some constant vector \(\vec{a}\). We can also consider linear transformations \[\vec{x}' = Q \vec{x} ~,\] where \(Q\) is a \(3\times3\) matrix. This preserves the distance between two points if it preserves the dot product \[\vec{x}'\cdot\vec{y}' = \vec{x}\cdot\vec{y} ~.\]

1. Show that this implies \[\begin{equation} \label{eq:so3constraint} Q^T I Q = I ~, \end{equation}\] where \(I\) is the \(3\times3\) identity matrix.

2. Show that the constraint \(\eqref{eq:so3constraint}\) implies \[\det Q = \pm 1 ~.\]

3. For those matrices with \(\det Q = 1\) we can parametrise \[Q = e^{\theta \mathcal{J}} ~,\] where \(\theta \in \mathbb{R}\) is a continuous parameter and \(\mathcal{J}\) is a \(3\times3\) matrix known as the generator of the transformation. Considering infinitesimal \(\theta\) show that the constraint \(\eqref{eq:so3constraint}\) implies \[\mathcal{J} + \mathcal{J}^T = 0 ~,\] i.e., that \(\mathcal{J}\) must be an antisymmetric \(3\times3\) matrix.

4. There are three independent antisymmetric \(3\times3\) matrices and a common basis is given by \[\begin{equation} \label{eq:so3basis} J_1 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix} ~, \qquad J_2 = \begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} ~, \qquad J_3 = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} ~. \end{equation}\] Show that these generate rotations about the \(x^1\), \(x^2\) and \(x^3\) axes respectively.

Lorentz transformations. We denote Minkowski space-time as \(\mathbb{R}^{1,3}\), which we can understand as \(\mathbb{R}^4\) endowed with the Minkowski metric \[\eta = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} ~.\] The symmetries of Minkowski space-time include space-time translations \[x' = x - a ~,\] for a constant vector \(a\). Recall that we have \(x = (x^0,x^1,x^2,x^3)\) and \(a = (a^0,a^1,a^2,a^3)\). The other symmetries are linear transformations known as Lorentz transformations \[x' = \Lambda x ~,\] which are required to preserve the scalar product of vectors \[\begin{equation} \label{eq:scalarproductminkowski} x'{}^T \eta y' = x^T \eta y ~. \end{equation}\]

1. Show that the condition \(\eqref{eq:scalarproductminkowski}\) implies \[\begin{equation} \label{eq:so13constraint} \Lambda^T \eta \Lambda = \eta ~. \end{equation}\]

2. Show also that \(\det \Lambda = \pm 1\).

3. Lorentz transformations with \(\det \Lambda = 1\) and \(\Lambda^0{}_0 \geq 1\) can be expressed in the form \[\Lambda = e^{\omega \mathcal{M}} ~,\] where \(\omega \in \mathbb{R}\) is a continuous real parameter. The \(4\times4\) matrix \(\mathcal{M}\) is the generator of the transformation. By considering infinitesimal transformations and expanding in small \(\omega\), show that the constraint \(\eqref{eq:so13constraint}\) implies \[\begin{equation} \label{eq:so13algebraconstraint} \mathcal{M}^T = -\eta \mathcal{M}\eta ~. \end{equation}\]

4. Show that there are six independent solutions to the constraint \(\eqref{eq:so13algebraconstraint}\) and explain why each independent solution can be labelled with a pair of antisymmetric indices \(M^{\rho\sigma} = - M^{\sigma \rho}\) with \(\rho,\sigma=0,1,2,3\) \[\begin{align}\label{eq:so13basis} & M^{01} = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} ~, \qquad && M^{02} = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} ~, \qquad && M^{03} = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{pmatrix} ~, \\ & M^{12} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} ~, \qquad && M^{13} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix} ~, \qquad && M^{23} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{pmatrix} ~. \end{align}\]

5. Writing the most general solution to the constraint \(\eqref{eq:so13algebraconstraint}\) as \[\begin{equation} \label{eq:so13general} \mathcal{M} = \omega_{\rho\sigma} M^{\rho\sigma} ~, \end{equation}\] for six arbitrary constants \(\omega_{\rho\sigma} = -\omega_{\sigma\rho}\), show that the solutions parametrised by \(\omega_{12}\), \(\omega_{13}\) and \(\omega_{23}\) generate rotations around the \(x^3\), \(x^2\) and \(x^1\) axes.

6. Show that the solutions parametrised by \(\omega_{01}\), \(\omega_{02}\) and \(\omega_{03}\) correspond to Lorentz boosts in the \(x^1\), \(x^2\) and \(x^3\) directions.

7. Starting from the constraint \(\eqref{eq:so13algebraconstraint}\) show that the matrix \(\mathcal{M} \eta\) is antisymmetric.

8. Confirm that the general solution \(\eqref{eq:so13general}\) is an antisymmetric matrix when multiplied by \(\eta\).

Lorentz transformations of fields.

1. Consider an infinitesimal Lorentz transformation \[\begin{equation} \label{eq:lorentz} \Lambda = I + \omega_{\rho\sigma}M^{\rho\sigma} + \mathcal{O}(\omega^2) ~. \end{equation}\] Under this transformation the coordinate \(x\) transforms by an infinitesimal amount \(\delta x\) \[x \to x' = \Lambda x = x + \delta x + \mathcal{O}(\omega^2) ~.\] Show that \[\delta x = \omega_{\rho \sigma} M^{\rho\sigma} x ~,\] and write this equation in index notation.

2. Consider a scalar field. The field transforms under the Lorentz transformation \(\eqref{eq:lorentz}\) as \[\phi(x) \to \phi'(x) = \phi(x) + \delta \phi(x) + \mathcal{O}(\omega^2) ~.\] Show that \[\delta\phi(x) = \omega_{\rho\sigma} L^{\rho\sigma} \phi(x) ~,\] where \[L^{\rho\sigma} = - (M^{\rho\sigma} x) \cdot \frac{\partial}{\partial x} ~,\] and write this equation in index notation.

3. Using \[(M^{\rho\sigma})^\mu{}_\nu = \eta^{\sigma\mu}\delta^\rho_\nu - \eta^{\rho\mu} \delta^\sigma_\nu ~,\] show that \[L^{\rho \sigma} = x^\sigma \frac{\partial}{\partial x_\rho} - x^\rho \frac{\partial}{\partial x_\sigma} ~.\]

4. Show that \(L^{\rho\sigma}\) satisfies the Lorentz algebra \[\phantom{}[L^{\mu\nu}, L^{\rho\sigma}] = -\eta^{\nu\rho}L^{\mu\sigma} + \eta^{\mu\rho} L^{\nu\sigma} + \eta^{\nu\sigma} L^{\mu\rho} - \eta^{\mu\sigma}L^{\nu\rho} ~.\]

5. Consider an infinitesimal translation \[x\to x' = x-a ~.\] This induces an infinitesimal transformation on the field \(\phi\) \[\phi(x) \to \phi'(x) = \phi(x) + \delta\phi(x) + \mathcal{O}(a^2) ~.\] Show that \[\delta \phi(x) = a^\mu \frac{\partial}{\partial x^\mu} \phi(x) ~.\]

6. Identifying \[P_\mu = \frac{\partial}{\partial x^\mu} ~,\] show that \[\phantom{}[P^\mu,L^{\nu\rho}] = -\eta^{\mu\nu}P^\rho + \eta^{\mu\rho}P^\nu ~, \qquad [P^\mu,P^\nu] = 0 ~.\]

Energy-momentum tensor. Consider the action of a real scalar field \(\phi\) \[S[\phi] = \int d^4 x \, \Big(-\frac12 \partial_\mu\phi(x)\partial^\mu\phi(x) - \frac12 m^2 \phi(x)^2\Big) ~.\]

1. Show that this action is invariant under space-time translations \[x^\mu \to x'{}^\mu = x^\mu - a^\mu ~,\] where \(a^\mu\) is a constant vector.

2. What is the number of independent conserved currents associated with this symmetry?

3. Using Noether’s theorem derive the conserved currents and the associated conserved charges.

4. Explicitly show that all the currents are conserved on-shell and hence that the charges are independent of time.

Hamiltonians in gauge theory. Consider the action associated to the electromagnetic field \[S[A_\mu] = \int d^4x \, \Big(-\frac14 F_{\mu\nu}F^{\mu\nu} \Big) ~,\] where the field strength \(F_{\mu\nu}\) is defined in terms of the electromagnetic gauge potential \(A_\mu(x)\) as \[F_\mu\nu = \partial_\mu A_\nu - \partial_\nu A_\mu ~.\]

1. Starting from this action determine the Euler-Lagrange equations for the gauge potential \(A_\mu\).

2. Determine the expression for the Hamiltonian.

Noether’s theorem. Consider the following action of two real scalar fields \(\phi_1\) and \(\phi_2\) \[\begin{equation} \label{eq:acttwoscalar} S[\phi_1,\phi_2] = \int d^4x \, \Big(-\frac12 \partial_\mu \phi_1 \partial^\mu \phi_1 -\frac12 \partial_\mu \phi_2 \partial^\mu \phi_2 -\frac12 m^2 (\phi_1^2 + \phi_2^2) - \lambda(\phi_1^2+\phi_2^2)^2 \Big) ~. \end{equation}\]

1. Derive the Euler-Lagrange equations for the fields \(\phi_1\) and \(\phi_2\).

2. Show that the action \(\eqref{eq:acttwoscalar}\) is invariant under the continuous transformation \[\begin{equation} \begin{split} \phi_1 & \to \phi_1' = \phi_1 \cos\alpha - \phi_2 \sin\alpha ~, \\ \phi_2 & \to \phi_2' = \phi_1 \sin\alpha + \phi_2 \cos\alpha ~, \end{split}\label{eq:so2symmetry} \end{equation}\] where \(\alpha\) is a constant parameter.

3. Determine the conserved current and charge associated to the symmetry \(\eqref{eq:so2symmetry}\).

4. Show that the current is conserved on-shell, i.e., if the Euler-Lagrange equations are satisfied.

We can assemble two real scalar fields \(\phi_1\) and \(\phi_2\) with the same mass \(m\) into a single complex scalar field \(\Phi = \frac{1}{\sqrt{2}}(\phi_1 + i \phi_2)\).

5. Show that the action \(\eqref{eq:acttwoscalar}\) written in terms of the complex scalar field \(\Phi\) reads \[\begin{equation} \label{eq:actcomplexscalar} S[\Phi,\Phi^*] = \int d^4x \, \big(-\partial_\mu\Phi^* \partial^\mu\Phi - m^2 \Phi^* \Phi - 4\lambda(\Phi^*\Phi)^2 \big) ~. \end{equation}\]

6. Derive the Euler-Lagrange equations for the field \(\Phi\) and its conjugate \(\Phi^*\).

7. Show that the action \(\eqref{eq:actcomplexscalar}\) is invariant under the continuous transformation \[\begin{equation} \label{eq:u1symmetry} \Phi \to \Phi' = e^{i\alpha} \Phi ~, \qquad \Phi^* \to \Phi'{}^* = e^{-i\alpha} \Phi^* ~, \end{equation}\] where \(\alpha\) is a constant parameter, and that this action is equivalent to the transformation \(\eqref{eq:so2symmetry}\) of \(\phi_1\) and \(\phi_2\).

8. Determine the conserved current and charge associated to the symmetry \(\eqref{eq:u1symmetry}\).

9. Show that the current is equivalent to that associated to the symmetry \(\eqref{eq:so2symmetry}\).

Many harmonic oscillators. A system of \(N\) decoupled complex simple harmonic oscillators with frequencies \(\omega_c\), \(c=1,\dots,N\), has the action \[S[q_c] = \int dt \, \sum_{c=1}^N \big(\dot q_c^*(t) \dot q_c(t) - \omega_c^2 q_c^*(t) q_c(t) \big) ~,\] and Euler-Lagrange equations \[\ddot{q}_c(t) + \omega_c^2 q_c(t) = 0 ~, \qquad \ddot{q}_c(t) + \omega_c^2 q_c(t) = 0 ~, \qquad c=1,\dots,N ~.\] The general solution to these equations can be written as \[\begin{split} q_c(t) & = \frac{1}{\sqrt{2\omega_c}} \big(a_c e^{-i\omega_c t} + b_c^* e^{i\omega_c t} \big) ~, \\ q_c^*(t) & = \frac{1}{\sqrt{2\omega_c}} \big(b_c e^{-i\omega_c t} + a_c^* e^{i\omega_c t} \big) ~. \end{split}\]

1. Starting from the Lagrangian formulation of the complex simple harmonic oscillators, show that the corresponding Hamiltonian takes the form \[\begin{equation} \label{eq:shohamiltonianclassical} H = \sum_{c=1}^N \big(p_c(t) p_c^*(t) + \omega_c^2 q_c^*(t) q_c(t) \big) ~. \end{equation}\]

2. Choosing the order of the operators as they appear in the Hamiltonian \(\eqref{eq:shohamiltonianclassical}\), show that upon canonical quantization the Hamiltonian takes the following form in terms of \(\hat a_c\), \(\hat a_c^\dagger\), \(\hat b_c\) and \(\hat b_c^\dagger\) \[\hat H = \sum_{c=1}^N \omega_c\big(\hat a_c^\dagger \hat a_c + \hat b_c \hat b_c^\dagger \big) = \sum_{c=1}^N \omega_c\big(\hat a_c^\dagger \hat a_c + \hat b_c^\dagger \hat b_c + 1 \big) ~.\]

Canonical quantization of a free massive complex scalar field. The Hamiltonian of a free massive complex scalar field takes the form \[\begin{equation} \label{eq:complexscalarhamiltonian} H = \int d^3x \, \big( \Pi \Pi^* + \vec{\nabla}\Phi^* \cdot \vec{\nabla}\Phi + m^2 \Phi^* \Phi \big) ~, \end{equation}\] where \(\Pi(t,\vec{x}) = \partial_t \Phi^*(t,\vec{x})\). \(\Phi\) admits the Fourier mode expansion \[\Phi(t,\vec{x}) = \int \frac{d^3k}{(2\pi)^3} \, \frac{1}{\sqrt{2\omega_{\vec{k}}}}\big(a_{\vec{k}} e^{- i\omega_{\vec{k}}t + i\vec{k}\cdot\vec{x} } + b_{\vec{k}}^* e^{+ i\omega_{\vec{k}}t -i\vec{k}\cdot\vec{x} } \big) ~,\] where \(\omega_{\vec{k}}^2 = \vec{k}^2 + m^2\).

1. Show that \[\begin{split} \Phi^*(t,\vec{x}) & = \int \frac{d^3k}{(2\pi)^3} \, \frac{1}{\sqrt{2\omega_{\vec{k}}}}\big(b_{\vec{k}} e^{- i\omega_{\vec{k}}t + i\vec{k}\cdot\vec{x}} + a^*_{\vec{k}} e^{+ i\omega_{\vec{k}}t-i\vec{k}\cdot\vec{x}} \big) ~, \\ \vec{\nabla}\Phi(t,\vec{x}) & = \int \frac{d^3k}{(2\pi)^3} \, \frac{i\vec{k}}{\sqrt{2\omega_{\vec{k}}}}\big(a_{\vec{k}} e^{- i\omega_{\vec{k}}t + i\vec{k}\cdot\vec{x} } - b_{\vec{k}}^* e^{+ i\omega_{\vec{k}}t -i\vec{k}\cdot\vec{x} } \big) ~, \\ \vec{\nabla}\Phi^*(t,\vec{x}) & = \int \frac{d^3k}{(2\pi)^3} \, \frac{i\vec{k}}{\sqrt{2\omega_{\vec{k}}}}\big(b_{\vec{k}} e^{- i\omega_{\vec{k}}t + i\vec{k}\cdot\vec{x}} - a^*_{\vec{k}} e^{+ i\omega_{\vec{k}}t-i\vec{k}\cdot\vec{x}} \big) ~, \\ \Pi^*(t,\vec{x}) & = \int \frac{d^3k}{(2\pi)^3} \, (-i)\sqrt{\frac{\omega_{\vec{k}}}{2}}\big(a_{\vec{k}} e^{- i\omega_{\vec{k}}t + i\vec{k}\cdot\vec{x} } - b_{\vec{k}}^* e^{+ i\omega_{\vec{k}}t -i\vec{k}\cdot\vec{x} } \big) ~, \\ \Pi(t,\vec{x}) & = \int \frac{d^3k}{(2\pi)^3} \, (-i)\sqrt{\frac{\omega_{\vec{k}}}{2}} \big(b_{\vec{k}} e^{- i\omega_{\vec{k}}t + i\vec{k}\cdot\vec{x}} - a^*_{\vec{k}} e^{+ i\omega_{\vec{k}}t-i\vec{k}\cdot\vec{x}} \big) ~. \end{split}\]

2. Promote the field \(\Phi\) and its conjugate \(\Phi^*\) to operators using the procedure of canonical quantization.

3. Choosing the order of the operators as they appear in the Hamiltonian \(\eqref{eq:complexscalarhamiltonian}\), show that \[\hat H = \int \frac{d^3k}{(2\pi)^3} \, \omega_{\vec{k}} \big(\hat a_{\vec{k}}^\dagger \hat a_{\vec{k}} + \hat b_{\vec{k}} \hat b_{\vec{k}}^\dagger \big) ~.\]

4. Using \([\hat b_{\vec k}, \hat b^\dagger_{\vec k'}] = (2\pi)^3 \delta^{(3)}(\vec{k} - \vec{k}')\), show that \[\hat H = \int \frac{d^3k}{(2\pi)^3} \, \omega_{\vec{k}} \big(\hat a_{\vec{k}}^\dagger \hat a_{\vec{k}} + \hat b_{\vec{k}}^\dagger \hat b_{\vec{k}} + (2\pi)^3 \delta^{(3)}(0) \big) ~.\]

The number operator. Consider the operator \[\hat{\mathcal{N}}_A = \int \frac{d^3k}{(2\pi)^3} \, \hat a_{\vec{k}}^\dagger \hat a_{\vec{k}} ~,\] in the quantum field theory for a free massive complex scalar.

1. Show that it counts the number of particles in a multiparticle state.

2. Write down the operator that counts the number of antiparticles in a multiparticle state.

The charge operator and normal ordering. Consider the charge operator \[\hat Q = - i \int d^3x \, \big(\hat\Pi \hat\Phi - \hat\Phi^\dagger \hat\Pi^\dagger \big) ~,\] in the quantum field theory for a free massive complex scalar.

1. Show that the operator is hermitian, i.e., satisfies \(\hat Q^\dagger = \hat Q\).

2. Show that its normal ordered version is \[\,:\!\hat Q\!:\, = \int \frac{d^3k}{(2\pi)^3} \, \big( \hat a_{\vec{k}}^\dagger \hat a_{\vec{k}} - \hat b_{\vec{k}}^\dagger \hat b_{\vec{k}}\big) ~.\]

Canonical quantization of a free massive real scalar field. Canonically quantize a free massive real scalar field with the action \[S[\phi] = \int d^4 x \, \Big(-\frac12 \partial_\mu \phi \partial^\mu \phi - \frac12 m^2\phi^2 \Big) ~,\] and Hamiltonian \[H = \int d^3x \, \Big( \frac12\pi^2 + \frac12\vec{\nabla}\phi \cdot \vec{\nabla}\phi + \frac12 m^2 \phi^2 \Big) ~,\] following the steps below.

1. Verify that the Fourier expansion for the field operator takes the form \[\hat \phi(t,\vec{x}) = \int \frac{d^3k}{(2\pi)^3} \, \frac{1}{\sqrt{2\omega_{\vec{k}}}} \big( \hat a_{\vec{k}} e^{-i \omega_{\vec{k}} t + i \vec{k}\cdot \vec{x}} + \hat a_{\vec{k}}^\dagger e^{+i \omega_{\vec{k}} t - i \vec{k}\cdot \vec{x}} \big) ~.\]

2. Prove that the commutation relation of the creation and annihilation operators \(\hat a_{\vec{k}}\) and \(\hat a_{\vec{k}}^\dagger\) is \[\phantom{}[\hat a_{\vec{k}},\hat a_{\vec{k}'}^\dagger] = (2\pi)^3 \delta^{(3)}(\vec{k} - \vec{k}') ~.\]

3. Show that the normal ordered Hamiltonian is given by \[\,:\!\hat H\!:\, = \int \frac{d^3k}{(2\pi)^3} \, \omega_{\vec{k}} \hat a_{\vec{k}}^\dagger \hat a_{\vec{k}} ~.\]

Momentum of single particle states. For a free massive real scalar field the Noether charge associated to symmetry under space translations is given by \[\vec{P} = - \int d^3x \, \partial_t \phi \vec{\nabla}\phi ~.\]

1. Canonically quantizing, show that the normal ordered expression for the corresponding operator in terms of creation and annihilation operators is given by \[\,:\!\hat{\vec{P}}\!:\, = \int \frac{d^3k}{(2\pi)^3} \, \vec{k} \hat a_{\vec{k}}^\dagger \hat a_{\vec{k}} ~.\]

2. The operator \(\,:\!\hat{\vec{P}}\!:\,\) measures the momentum of a state. Show that a particle created by \(\hat a_{\vec{k}}^\dagger\) has momentum \(\vec{k}\).

Propagators in 1+1d. In this problem we work out the form of the Feynman propagator in 1+1d, i.e., one time and one space dimension. We start from the following expression for the Feynman propagator \(G(y-x)\) if \(y^0 > x^0\) \[G(y-x) = \int_{-\infty}^{+\infty} \frac{dk}{(2\pi)2\omega_k} \, e^{-i\omega_{k} (y^0-x^0) + i k (y^1-x^1)} ~, \qquad \omega_k = \sqrt{k^2+m^2} ~,\] where \(k\) is the single component of the spatial momentum.

1. Evaluate this integral for timelike and spacelike separations of \(x\) and \(y\).

   Hint: You can use the following integral identities \[\int_{-\infty}^{+\infty} d\eta \, e^{-i\beta\cosh\eta} = - i\pi H_0^{(2)}(\beta) ~, \qquad \int_{0}^{+\infty} d\eta \, \cos(\beta\sinh\eta) = K_0 (\beta) ~,\] where \(H_0^{(2)}\) and \(K_0\) are the Hankel function of the second kind and the modified Bessel function of the second kind respectively. It may also be helpful to use the substitution \(k = m \sinh\eta\) and use Lorentz covariance to simplify the form of the integrand.

2. From the known asymptotic behaviour of the Hankel and Bessel functions, determine the asymptotic behaviour of \(G(y-x)\) at large timelike and spacelike separations.

Feynman, retarded and advanced propagators. The Green’s function for the Klein-Gordon operator can be written as \[-i \int \frac{d^4k}{(2\pi)^4} \, \frac{e^{+i k\cdot(y-x)}}{k^2 + m^2} = -i \int \frac{d^3k}{(2\pi)^3} \int_{-\infty}^\infty \frac{dk^0}{2\pi} \, \frac{e^{-ik^0 (y^0-x^0) + i \vec{k}\cdot(\vec{y}-\vec{x})}}{-(k^0)^2 + \omega_{\vec{k}}^2} ~.\]

1. Briefly explain why this integral is not well-defined as an ordinary real integral.

2. Letting \(k^0\) be a complex variable, taking \(y^0 > x^0\) and closing the contour in the lower half plane, compute the three integrals \[\begin{split} I(\vec{k}) & = -i \int_{-\infty}^\infty \frac{dk^0}{2\pi} \, \frac{e^{-ik^0 (y^0-x^0) + i \vec{k}\cdot(\vec{y}-\vec{x})}}{-(k^0)^2 + \omega_{\vec{k}}^2 - i\epsilon} ~, \\ I_R(\vec{k}) & = -i \int_{-\infty}^\infty \frac{dk^0}{2\pi} \, \frac{e^{-ik^0 (y^0-x^0) + i \vec{k}\cdot(\vec{y}-\vec{x})}}{-(k^0 + i\epsilon)^2 + \omega_{\vec{k}}^2} ~, \\ I_A(\vec{k}) & = -i \int_{-\infty}^\infty \frac{dk^0}{2\pi} \, \frac{e^{-ik^0 (y^0-x^0) + i \vec{k}\cdot(\vec{y}-\vec{x})}}{-(k^0 - i\epsilon)^2 + \omega_{\vec{k}}^2} ~, \end{split}\] to leading order in the positive infinitesimal quantity \(\epsilon\).

3. Taking \(x^0 > y^0\) and closing the contour in the upper half plane, compute the same three integrals.

Dyson’s formula. The unitary time evolution operator \(U(t,t_0)\) obeys the following differential equation \[\begin{equation} \label{eq:timeevolution} i\frac{d}{dt} U(t,t_0) = \epsilon H_I(t) U(t,t_0) ~, \end{equation}\] where we have rescaled \(H_I(t)\) by a parameter \(\epsilon\) to allow us to consider a series expansion. Taking \(t>t_0\), the solution to this equation with boundary condition \(U(t_0,t_0) = 1\) is \[U(t,t_0) = \mathrm{T}\hspace{-1pt}\overset{\longleftarrow}{\exp}\big(-i \epsilon \int_{t_0}^t dt' \, H_I (t')\big) ~, \qquad t > t_0 ~,\] where \(\mathrm{T}\hspace{-1pt}\overset{\longleftarrow}{\exp}\) denotes the time ordered exponential with operators evaluated at later times placed to the left and operators evaluated at earlier times placed to the right.

1. Consider the operator \[\exp\big(-i\epsilon\int_{t_0}^t dt' \, H_I (t')\big) ~.\] Show that at up to \(\mathcal{O}(\epsilon)\) this satisfies the differential equation \(\eqref{eq:timeevolution}\), but that this is no longer the case at the next order in \(\epsilon\).

2. Show that this is resolved for \(t>t_0\) if the exponential is replaced by a time ordered exponential.

3. Let \[U(t_2,t_1) = \mathrm{T}\hspace{-1pt}\overset{\longleftarrow}{\exp}\big(-i \epsilon \int_{t_1}^{t_2} dt' \, H_I (t')\big) ~, \qquad t_2 > t_1 ~,\] where both \(t_2\) and \(t_1\) can vary. Show that \(U(t_3,t_2)U(t_2,t_1) = U(t_3,t_1)\) for \(t_3 > t_2 > t_1\).

Real scalar fields.

1. Show that the time ordered product \(\overset{\leftarrow}{\mathrm{T}}\big(\phi(y)\phi(x)\big)\) and the normal ordered product \(\,:\!\phi(y)\phi(x)\!:\,\) are both symmetric under the interchange of \(x\) and \(y\).

2. Taking \(\phi\) to be a real scalar field, show that the Feynman propagator \(G(y-x)\) has the same symmetry property, i.e. \(G(y-x) = G(x-y)\).

3. Consider the following action for two real scalar fields \(\phi_1\) and \(\phi_2\) of mass \(m_1\) and \(m_2\) respectively \[S[\phi_1,\phi_2] = \int d^4x \, \Big(-\frac12 \partial_\mu\phi_1\partial^\mu\phi_1 - \frac12 m_1^2 \phi_1^2 -\frac12 \partial_\mu\phi_2\partial^\mu\phi_2 -\frac12 m_2^2 \phi_2^2 \Big) ~.\] The Feynman propagators for the fields \(\phi_1\) and \(\phi_2\) are given by \[\begin{split} G_1 (y-x) & = -i \int \frac{d^4k}{(2\pi)^4} \, \frac{e^{+i k\cdot(y-x)}}{k^2 + m_1^2 - i\epsilon} ~, \\ G_2 (y-x) & = -i \int \frac{d^4k}{(2\pi)^4} \, \frac{e^{+i k\cdot(y-x)}}{k^2 + m_2^2 - i\epsilon} ~. \end{split}\] Show that \[\int d^4z \, G_1 (y-z) G_1(z-x) = - \int \frac{d^4k}{(2\pi)^4} \, \frac{e^{+i k\cdot(y-x)}}{(k^2 + m_1^2 - i\epsilon)^2} ~.\]

4. Show that \[\begin{split} \int d^4z_1 \int d^4z_2 \hspace{1ex} \dots \int d^4z_{N-1} \int d^4z_N \, & G_1(y-z_1) G_1(z_1-z_2) \dots G_1(z_{N-1}-z_{N}) G_1(z_N - x) \\& \hspace{10ex} = (-i)^{N+1} \int \frac{d^4k}{(2\pi)^4} \, \frac{e^{+i k\cdot(y-x)}}{(k^2 + m_1^2 - i\epsilon)^{N+1}} ~. \end{split}\]

5. Using Wick’s theorem, evaluate the expression \[\langle 0|\overset{\leftarrow}{\mathrm{T}}\big(\phi_2(x_6)\phi_2(x_5)\phi_2(x_4)\phi_2(x_3)\phi_1(x_2)\phi_1(x_1)\big)|0\rangle ~,\] in terms of the Feynman propagators \(G_1(y-x)\) and \(G_2(y-x)\).

   Hint: There is no contraction between \(\phi_1\) and \(\phi_2\).

6. Evaluate the expression \[\langle 0|\overset{\leftarrow}{\mathrm{T}}\big(\phi_2(x_{103})\phi_2(x_{102})\phi_2(x_{101})\phi_2(x_{100})\phi_1(x_{99})\dots\phi_1(x_2)\phi_1(x_1\big))|0\rangle ~.\]

Feynman diagrams.

1. Determine the symmetry factors for the following Feynman diagrams that contribute to an interacting theory of a real scalar field with \(\phi^3\) and \(\phi^4\) interactions.

   

   Hint: The rules for computing the symmetry factor are the same as for \(\phi^4\) theory.

2. Consider the vacuum-to-vacuum amplitude in \(\phi^4\) theory \[\langle 0|\mathrm{T}\hspace{-1pt}\overset{\longleftarrow}{\exp}\big(-\frac{i\lambda}{4!} \int d^4z \, \phi_I^4(z) \big) |0\rangle ~.\] Using Wick’s theorem, identify the different contributions to the amplitude up to and including \(\mathcal{O}(\lambda^2)\).

More real scalar fields. Consider the following action for two real scalar fields \(\phi_1\) and \(\phi_2\) \[S[\phi_1,\phi_2] = \int d^4x \, \Big(-\frac12 \partial_\mu\phi_1\partial^\mu\phi_1 - \frac12 m_1^2 \phi_1^2 -\frac12 \partial_\mu\phi_2\partial^\mu\phi_2 -\frac12 m_2^2 \phi_2^2 - \lambda \phi_1 \phi_2 \Big) ~,\] where \(\lambda\) is a coupling constant. Treat the term proportional to \(\lambda\) as an interaction and consider perturbation theory in \(\lambda\). In this question we omit the subscripts \(I\) and \(H\) on interaction and Heisenberg picture fields.

1. Write down the Feynman rules for time ordered correlation functions in this theory.

   Hint: Note that there are two scalar fields each with their own Feynman propagator. You therefore need two types of lines, e.g., solid and dashed, in order to differentiate between the two propagators.

2. Consider the vacuum-to-vacuum amplitude \[\langle 0| \mathrm{T}\hspace{-1pt}\overset{\longleftarrow}{\exp}\big(-i\lambda\int d^4z \, \phi_1(z)\phi_2(z)\big)|0\rangle ~,\] and expand up to and including \(\mathcal{O}(\lambda^4)\). Apply Wick’s theorem at each order to find an expression in terms of Feynman propagators and draw the corresponding Feynman diagrams.

3. Consider \[\begin{split} & \langle 0| \overset{\leftarrow}{\mathrm{T}}\Big(\phi_2(y)\phi_1(x)\exp\big(-i\lambda\int d^4z \, \phi_1(z)\phi_2(z)\big)\Big) |0\rangle ~, \\ & \langle 0| \overset{\leftarrow}{\mathrm{T}}\Big(\phi_1(y)\phi_1(x)\exp\big(-i\lambda\int d^4z \, \phi_1(z)\phi_2(z)\big)\Big)|0\rangle ~, \end{split}\] and expand up to and including \(\mathcal{O}(\lambda^4)\). Apply Wick’s theorem at each order to find an expression in terms of Feynman propagators and draw the corresponding Feynman diagrams.

4. Consider the time ordered two-point correlation functions \[\begin{split} \langle\Omega| \overset{\leftarrow}{\mathrm{T}}\big(\phi_2(y)\phi_1(x)\big) |\Omega\rangle & = \frac{\langle 0| \overset{\leftarrow}{\mathrm{T}}\Big(\phi_2(y)\phi_2(x)\exp\big(-i\lambda\int d^4z \, \phi_1(z)\phi_2(z)\big)\Big) |0\rangle}{\langle 0| \mathrm{T}\hspace{-1pt}\overset{\longleftarrow}{\exp}\big(-i\lambda\int d^4z \, \phi_1(z)\phi_2(z)\big) |0\rangle} ~, \\ \langle\Omega| \overset{\leftarrow}{\mathrm{T}}\big(\phi_1(y)\phi_1(x)\big) |\Omega\rangle & = \frac{\langle 0| \overset{\leftarrow}{\mathrm{T}}\Big(\phi_1(y)\phi_1(x)\exp\big(-i\lambda\int d^4z \, \phi_1(z)\phi_2(z)\big)\Big) |0\rangle}{\langle 0| \mathrm{T}\hspace{-1pt}\overset{\longleftarrow}{\exp}\big(-i\lambda\int d^4z \, \phi_1(z)\phi_2(z)\big) |0\rangle} ~. \end{split}\] Recall that the effect of the denominator is the cancel the vacuum bubble diagrams. Determine the contributions at \(\mathcal{O}(\lambda^{2k})\) and \(\mathcal{O}(\lambda^{2k+1})\) and draw the corresponding Feynman diagrams.

Interacting fields. Consider the following action for two real interacting scalar fields \(\phi_1\) and \(\phi_2\) \[S[\phi_1,\phi_2] = \int d^4x \, \Big(-\frac12\partial_\mu\phi_1\partial^\mu\phi_1 - \frac12 m_1^2\phi_1^2 -\frac12\partial_\mu\phi_2\partial^\mu\phi_2 - \frac12 m_2^2\phi_2^2 - \frac{\lambda}{4} \phi_1^2 \phi_2^2 \Big) ~.\]

1. Write down the Feynman rules for time ordered vacuum expectation values in this theory.

2. Evaluate the vacuum-to-vacuum amplitude \[\langle 0|\mathrm{T}\hspace{-1pt}\overset{\longleftarrow}{\exp}\big(-\frac{i\lambda}{4}\int d^4x' \, \phi_{1I}^2(x') \phi_{2I}^2(x') \big)|0\rangle ~,\] in position space up to and including \(\mathcal{O}(\lambda^2)\).

   Hint: You can express your answer in terms of integrals over the position space propagators for the fields \(\phi_1\) and \(\phi_2\).

3. Evaluate \[\langle\Omega|\overset{\leftarrow}{\mathrm{T}}\big(\phi_1(y)\phi_1(x)\big)|\Omega\rangle ~, \qquad \langle\Omega|\overset{\leftarrow}{\mathrm{T}}\big(\phi_2(y)\phi_2(x)\big)|\Omega\rangle ~,\] in position space up to and including \(\mathcal{O}(\lambda^2)\).

   Hint: You can express your answer in terms of integrals over the position space propagators for the fields \(\phi_1\) and \(\phi_2\).

\(n\)-point interactions. Consider the following action for a real scalar field \[S[\phi] = \int d^4x \, \Big(-\frac12\partial_\mu\phi_1\partial^\mu\phi_1 - \frac12 m_1^2\phi_1^2 - \frac{\lambda}{n!}\phi^n \Big) ~,\] where \(n\) is a positive integer.

1. Write down the Feynman rules for time-ordered vacuum expectation values in this theory.

2. For \(n=3\), draw all vacuum bubble diagrams up to and including \(\mathcal{O}(\lambda^2)\) and write down the corresponding expressions in terms of the position space propagator.