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Homework Statement
Recall that ##T(a)^{-1}\varphi(x)T(a) = \varphi(x - a)## where ##T(a) = e^{-iP^{\mu}a_{\mu}}## is the space-time translation operator and ##P^{\mu}## is the 4-momentum operator.
(a) Let ##a^{\mu}## be infinitesimal and derive an expression for ##[P^{\mu},\varphi]##.
(b) Show that the time component of your result is equivalent to the Heisenberg equation of motion ##\dot{\varphi} = i[H,\varphi]##.
(c) For a free field, use the Heisenberg equation to derive the Klein-Gordon equation.
(d) Define a spatial momentum operator ##\mathbf{P} = -\int d^{3}x (\pi \nabla\varphi)##. Use the canonical commutation relations to show that ##\mathbf{P}## obeys the relation you derived in part (a).
(e) Express ##\mathbf{P}## in terms of ##a(\mathbf{k})## and ##a^{\dagger}(\mathbf{k})##.
Homework Equations
The Hamiltonian density is given by ##\mathcal{H} = \frac{1}{2}\pi^2 + \frac{1}{2}(\nabla \varphi)^2 + \frac{1}{2}m^2 \varphi^2## where ##\pi## is the field conjugate momentum as usual.
The equal time commutation relations for the field and the field conjugate momentum are ##[\varphi(x,t),\varphi(x',t)] = 0##, ##[\pi(x,t),\pi(x',t)] = 0##, and ##[\varphi(x,t),\pi(x',t)] = i\delta^{3}(\mathbf{x} - \mathbf{x'})##.
You probably already noticed but ##\varphi## is a real field. Also I'll use the notation ##\varphi' \equiv \varphi(x',t)## with ##\pi'## and ##\nabla'## defined similarly just to make things cleaner.
The Attempt at a Solution
(a) ##\varphi(x^{\mu} - a^{\mu}) = \varphi(x^{\mu}) - a_{\mu}\partial^{\mu}\varphi +O(a^2)## and ##T(a^{\mu}) = I - ia_{\mu}P^{\mu}+O(a^2)## so ##T^{-1}\varphi T = (I + ia_{\mu}P^{\mu})(\varphi - ia_{\mu}\varphi P^{\mu}) = \varphi(x^{\mu}) + ia_{\mu}[P^{\mu},\varphi]+O(a^2)## hence ##i[P^{\mu},\varphi] = -\partial^{\mu}\varphi## since ##a_{\mu}## was arbitrary.
(b) ##\mu = 0## gives ##i[H,\varphi] = -\partial^{0}\varphi = \dot{\varphi}##
(c) We have ##2[\varphi,H] = [\varphi,\int d^3{x'}(\pi'^2 + (\nabla'\varphi')^2 + m^2 \varphi'^2)] = \int d^{3}x'([\varphi,\pi'^2] + [\varphi,(\nabla'\varphi')^2] + m^2[\varphi,\varphi'^2])##; the equal time commutator can be pulled into the integral since the commutator is evaluated at the field point ##x## whereas the integral is over ##x'##.
Now ##[\varphi,\nabla'\varphi'] = \nabla'[\varphi,\varphi'] = 0## hence ##[\varphi,(\nabla'\varphi')^2] = 0##. Similarly ##[\varphi,\varphi'^2] = 0##. This leaves us with ##2[\varphi,H] = \int d^{3}x'[\varphi,\pi'^2] ##.
We have ##[\varphi,\pi'^2]= i\delta^{3}(\mathbf{x} - \mathbf{x}')\pi' + (\pi' \varphi)\pi' - (\pi' \varphi) \pi' + i\delta^{3}(\mathbf{x} - \mathbf{x}')\pi' = 2i\delta^{3}(\mathbf{x} - \mathbf{x}')\pi'## hence ##[\varphi,H] = i\int d^{3}x' \delta^{3}(\mathbf{x} - \mathbf{x}')\pi' = i\pi##. The Heisenberg equation of motion thus gives ##\dot{\varphi} = \pi##.
Similarly, ##2[\pi,H] = \int d^{3}x'([\pi,\pi'^2] + [\pi,(\nabla'\varphi')^2] + m^2[\pi,\varphi'^2])##; ##[\pi,\pi'] = 0## so ##[\pi,\pi'^2] = 0## and ##[\pi,\varphi'^2] = - 2i\delta^{3}(\mathbf{x} - \mathbf{x'})\varphi' ##.
Furthermore ## [\pi,(\nabla'\varphi')^2] = (\pi \nabla'\varphi')\cdot\nabla'\varphi' - \nabla'\varphi'\cdot((\nabla'\varphi')\pi) \\= (\nabla'\varphi')\cdot\pi (\nabla'\varphi') - i(\nabla'\delta^{3}(\mathbf{x} - \mathbf{x'}))\cdot\nabla'\varphi' - (\nabla'\varphi')\cdot\pi(\nabla'\varphi') - i\nabla'\varphi'\cdot(\nabla'\delta^{3}(\mathbf{x} - \mathbf{x'})) \\= -2i(\nabla'\delta^{3}(\mathbf{x} - \mathbf{x'}))\cdot\nabla'\varphi'##
This leaves us with ##[\pi,H] = -i\int d^{3}x'\delta^{3}(\mathbf{x} - \mathbf{x'})\varphi' - i\int d^{3}x'(\nabla'\delta^{3}(\mathbf{x} - \mathbf{x'}))\cdot\nabla'\varphi' = i(\nabla^2\varphi - m^2\varphi)##.
So now here's the part that I'm not sure about. The Heisenberg equation of motion derived above is valid for any scalar quantum field ##\varphi## right? In the derivation itself I think all I used was the fact that ##\varphi## is an arbitrary scalar field, making no reference to a specific scalar field. So it works for the field conjugate momentum ##\pi## as well? If so then we get ##i[\pi,H] = -\dot{\pi} = -\ddot{\varphi}## and at the same time ##i[\pi,H] = -(\nabla^2\varphi - m^2\varphi)## so that ##\ddot{\varphi} - \nabla^2\varphi + m^2\varphi = -\partial^{\mu}\partial_{\mu}\varphi+m^2\varphi = 0## as desired.
Does this all check out i.e. are all my calculations sound? I just want to make sure the calculations are sound so that I can move on to parts (d) and (e). Also if you know of a faster calculation to get the KG equation from the Heisenberg equation of motion then I'd appreciate it if you could show it. Thanks in advance!