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WannabeNewton

Science Advisor

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## Main Question or Discussion Point

Hi guys! So this question has been bugging me a bit and I can't seem to find any textbook (at least, restricted to physics textbooks) that talks about it.

In QM, the overarching formalism is clear. We have ##L^2(\mathbb{R})## and states are given by ##|\psi\rangle \in L^2(\mathbb{R})## such that ##\langle \psi |\psi\rangle = 1## (so really we consider the projective space of unit norm kets in ##L^2(\mathbb{R})## since ## |\psi\rangle \sim |\varphi \rangle## if they differ by a phase factor). Furthermore we have unbounded operators ##A: \text{dom}A\rightarrow L^2(\mathbb{R})## where ##\text{dom}A \subseteq L^2(\mathbb{R})## (if ##A## is bounded then of course ##\text{dom}A = L^2(\mathbb{R})## as usual; the continuity of ##A## in the operator norm topology will guarantee that we can always extend it as such). If ##\text{dom}A = \text{dom}A^{\dagger}## and ##A^{\dagger}|\psi \rangle = A |\psi \rangle## for all ##|\psi \rangle \in \text{dom} A## then as usual ##A## is self-adjoint (if ##A## is bounded then this just reduces to ##A = A^{\dagger}## of course) and these are the operators we care about in QM because they correspond to dynamical variables etc.

But the coherence of this formalism isn't apparent to me in QFT. For example for scalar quantum fields we start with a classical scalar field ##\varphi## and its conjugate momentum ##\pi = \frac{\partial \mathcal{L}}{\partial \dot{\varphi}}##, along with the Poisson bracket relation ##\{\varphi(\mathbf{x},t), \pi(\mathbf{x'},t)\} = \delta(\mathbf{x}-\mathbf{x'})## and promote the classical fields to operator fields satisfying ##[\varphi(\mathbf{x},t), \pi(\mathbf{x'},t)] = i\hbar\delta(\mathbf{x}-\mathbf{x'})##. But as mathematical objects, how do the operator fields ##\varphi## and ##\pi## actually work i.e. what domain and codomain do they map between? Is the Fock space formalism even useful in this context (I mention Fock space because we are interested in multiparticle states for creation and destruction operators to act on)?

Thanks in advance!

In QM, the overarching formalism is clear. We have ##L^2(\mathbb{R})## and states are given by ##|\psi\rangle \in L^2(\mathbb{R})## such that ##\langle \psi |\psi\rangle = 1## (so really we consider the projective space of unit norm kets in ##L^2(\mathbb{R})## since ## |\psi\rangle \sim |\varphi \rangle## if they differ by a phase factor). Furthermore we have unbounded operators ##A: \text{dom}A\rightarrow L^2(\mathbb{R})## where ##\text{dom}A \subseteq L^2(\mathbb{R})## (if ##A## is bounded then of course ##\text{dom}A = L^2(\mathbb{R})## as usual; the continuity of ##A## in the operator norm topology will guarantee that we can always extend it as such). If ##\text{dom}A = \text{dom}A^{\dagger}## and ##A^{\dagger}|\psi \rangle = A |\psi \rangle## for all ##|\psi \rangle \in \text{dom} A## then as usual ##A## is self-adjoint (if ##A## is bounded then this just reduces to ##A = A^{\dagger}## of course) and these are the operators we care about in QM because they correspond to dynamical variables etc.

But the coherence of this formalism isn't apparent to me in QFT. For example for scalar quantum fields we start with a classical scalar field ##\varphi## and its conjugate momentum ##\pi = \frac{\partial \mathcal{L}}{\partial \dot{\varphi}}##, along with the Poisson bracket relation ##\{\varphi(\mathbf{x},t), \pi(\mathbf{x'},t)\} = \delta(\mathbf{x}-\mathbf{x'})## and promote the classical fields to operator fields satisfying ##[\varphi(\mathbf{x},t), \pi(\mathbf{x'},t)] = i\hbar\delta(\mathbf{x}-\mathbf{x'})##. But as mathematical objects, how do the operator fields ##\varphi## and ##\pi## actually work i.e. what domain and codomain do they map between? Is the Fock space formalism even useful in this context (I mention Fock space because we are interested in multiparticle states for creation and destruction operators to act on)?

Thanks in advance!

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