Sidney Coleman — Quantum Field Theory Lectures · Problem 4.1

October 2 - Constructing Fock space from the quantum field axiomatically; The Method of the Missing Box, Classical Particle Mechanics, Quantum Particle Mechanics, Classical Field Theory, Quantum Field Theory; Quantum field from free scalar theory — Sidney Coleman

Physics — QFT · creation-annihilation-operators-from-field

Problem

Given the expression ak⃗=12[2ωk⃗∫d3x(2π)3/2(ϕ(x⃗,0)+iωk⃗ϕ˙(x⃗,0))e−ik⃗⋅x⃗]a_{\vec{k}} = \frac{1}{2}\left[\sqrt{2\omega_{\vec{k}}}\int \frac{d^3x}{(2\pi)^{3/2}}\left(\phi(\vec{x},0) + \frac{i}{\omega_{\vec{k}}}\dot{\phi}(\vec{x},0)\right)e^{-i\vec{k}\cdot\vec{x}}\right] and the analogous expression for ak⃗†a^\dagger_{\vec{k}}: (a) Write out the double integral for [ak⃗,ak⃗′†][a_{\vec{k}}, a^\dagger_{\vec{k}'}] and use the equal-time commutation relation (property (3′')): [ϕ(x⃗,t),ϕ(y⃗,t)]=0,[∂0ϕ(x⃗,t),ϕ(y⃗,t)]=−iδ(3)(x⃗−y⃗)[\phi(\vec{x},t),\phi(\vec{y},t)] = 0, \quad [\partial_0\phi(\vec{x},t),\phi(\vec{y},t)] = -i\delta^{(3)}(\vec{x}-\vec{y}) to show that [ak⃗,ak⃗′†]=δ(3)(k⃗−k⃗′)[a_{\vec{k}}, a^\dagger_{\vec{k}'}] = \delta^{(3)}(\vec{k}-\vec{k}'). (b) From property (4), U(a)†ϕ(x)U(a)=ϕ(x−a),U(Λ)†ϕ(x)U(Λ)=ϕ(Λ−1x),U(a)^\dagger \phi(x) U(a) = \phi(x-a), \quad U(\Lambda)^\dagger \phi(x) U(\Lambda) = \phi(\Lambda^{-1}x), derive what U(a)†ak⃗U(a)U(a)^\dagger a_{\vec{k}} U(a) and U(Λ)†ak⃗U(Λ)U(\Lambda)^\dagger a_{\vec{k}} U(\Lambda) are. (Hint: it is easier to find the action of Lorentz transformations on a(k)≡(2π)3/22ωk⃗ ak⃗a(k) \equiv (2\pi)^{3/2}\sqrt{2\omega_{\vec{k}}}\, a_{\vec{k}}.) (c) Derive and state the analog of property (5) (completeness) for the operators ak⃗,ak⃗†a_{\vec{k}}, a^\dagger_{\vec{k}}.

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