Linked Questions

4
votes
2answers
600 views

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 ...
1
vote
2answers
97 views

Exercise on bosonic vacuum

Consider bosonic canonical transformation, generated by operator $S = e^{\lambda (a^{\dagger})^2}$. Show, that \begin{equation} b \equiv SaS^{-1} = a - 2\lambda a^{\dagger}. \end{equation} ...
7
votes
1answer
731 views

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 ...
6
votes
1answer
1k views

Creating a QM state of definite position in Fock space

I'm wondering if somebody could help me to finish a simple calculation. Let me first provide motivation for the question below: I would like to write a QM amplitude in the 'QFT-style', as $$\langle \...
4
votes
1answer
327 views

Eigenstates in QFT and amplitude of a field operator

I've seen in different posts (such as here) that given a field $\hat{\phi}(x)$, its eigenstates $|\phi\rangle$ are of the form: $$|\phi\rangle\ = e^{\int dx\phi(x)\hat{\phi}(x)}|0\rangle\tag1$$ I ...
3
votes
1answer
150 views

Coherent state of second quantized hamiltonian

In my preparation for the exam I tried to solve the exercise 2.4 in Coleman's Introduction to Many-Body Physics. I like diagonalizing Hamiltonian, so I picked this problem. Also to learn more about ...
3
votes
1answer
166 views

Eigenvalues of a quantum field

In the book 'Quantum field theory for the Gifted Amateur", the following is stated, cf. 9.3: "A quantum field $\hat{\phi}(x)$ takes a position in spacetime and returns an operator whose eigenvalues ...
2
votes
1answer
204 views

What are the orthogonal eigenstates of the field operator?

In Peskin & Schroeder section 9.2, they derive the two-point function in the path integral formalism: $$\langle \Omega | \mathcal{T} \left\{ \hat{\phi}(x_1)\hat{\phi}(x_2)\right\} | \Omega \...
2
votes
1answer
107 views

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 ...
1
vote
1answer
122 views

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 ...
6
votes
0answers
278 views

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 ...
4
votes
0answers
358 views

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"...
4
votes
0answers
2k views

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 ...
2
votes
0answers
284 views

Field operator eigenstate vs. single-particle state

I would like to make sure I understand some basic QFT. My understanding so far is that field operators measure field intensity and their Fourier transform measure intensity of field oscillation. In ...
0
votes
0answers
91 views

Transition from phi basis to occupation number in quantum field theory

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 ...

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