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### Energy eigeinstates written in the field operator eigenstates basis

For an harmonic oscillator we can write the Hamiltonian eigenstates in the basis of the amplitude eigenstates : for example the ground state is a gaussian : $⟨x|0⟩=a.e^{-b.x^{2}}$. I was wondering ...
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### Exercise on bosonic vacuum

Consider bosonic canonical transformation, generated by operator $S = e^{\lambda (a^{\dagger})^2}$. Show, that $$b \equiv SaS^{-1} = a - 2\lambda a^{\dagger}.$$ ...
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### Fock space vs. wavefunctionals

There are at least two representations of the Hilbert spaces of quantum field theory. For a scalar field, we have The Fock space representation, such that every state is represented as the Fock ...
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### What is the overlap $\langle \phi | 0 \rangle$ for a scalar field?

Consider a massive free real scalar field $\hat{\Phi}$ (with $\mathcal{L}[\Phi] = \partial_{\mu}\Phi\partial^\mu \Phi - \tfrac{1}{2} m^2 \Phi^2$). I was wondering what is the overlap for the ...
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### What is the wave field functional?

I was reading on some QFT and I came across the following paragraph: In the same way that a generic state $|\psi\rangle$ of a particle can be described by giving its overlap with all the possible ...
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### How does one write eigenstates of field operators in terms of particle states in scalar field theory?

I am reading the first paper in Schwinger's QED anthology, where he discusses his action principle. In this, he writes down states that are simultaneous eigenkets of the field operators at all points ...
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### Intuitive understanding in QFT

I recently read a bit about the Schrodinger picture in QFT and wavefunctionals, see e.g. Polchinski's String Theory lectures, and I wanted to ask if the intuitive understanding of QFT I got is "right"...
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### Position and momentum eigenstates in terms of creation and annihilation operator? [closed]

Consider a simple harmonic oscillator; the position operator is $\hat{x}=(a^\dagger+a)/\sqrt{2}$ and the momentum operator is $\hat{p}=-i(a-a^\dagger)/\sqrt{2}$. One may verify that the eigenstates ...
We can construct the unitary transformation for change of basis from $x$ to number operator $n$ in harmonic oscillator by using $a|0\rangle=0$ and then multiply $\langle x|$ to the both side and ...