Linked Questions

14
votes
1answer
2k views

Schrödinger wavefunctional quantum-field eigenstates

The reason that I have this problem is that I'm trying to solve problem 14.4 of Schwartz's QFT book, which've confused me for a long time. The problem is to construct the eigenstates of a quantum ...
4
votes
2answers
609 views

What is the analogy of $|x\rangle$ in quantum field theory?

Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ $|...
7
votes
2answers
328 views

Can quantum fields be viewed as superpositions of classical fields?

At the end of my undergraduate quantum mechanics class, we looked at phonons. You can let $x_i$ be the position operator of an nth quantum harmonic oscillator, and couple the harmonic oscillators with ...
7
votes
1answer
657 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 ...
4
votes
2answers
567 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 ...
4
votes
0answers
1k 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
2answers
197 views

$SU(2)$ symmetry of $\mathcal{L}=\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi - \Phi^{\dagger}M\Phi$

I'm considering a Lagrangian of two complex scalar field: $$\mathcal{L}=\partial_{\mu}\phi_1^{*}\partial^{\mu}\phi_1-m_1^2\phi_1^{*}\phi_1+\partial_{\mu}\phi_2^{*}\partial^{\mu}\phi_2-m_2^2\phi_2^{*}\...
3
votes
0answers
572 views

Continuous and discrete basis of Hilbert Space [duplicate]

Any state in Hilbert space $|\phi\rangle$ can be expressed in terms of a complete basis $\{| v_i\rangle, i=1,2,...\}$ as $$|\phi\rangle=\sum|v_i\rangle \langle v_i |\phi\rangle . $$ Now, if I ...
4
votes
1answer
268 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 ...
6
votes
0answers
266 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 ...
0
votes
1answer
243 views

How to show the field operator creates a particle at known position in Fock space

I'm very confused by something I saw in Susskind's Advanced Quantum Mechanics Lecture 6. He introduces Fock space $F$, defines the creation/annihilation operators $a^+_n,a^-_n$ on it (in terms of ...
0
votes
1answer
64 views

Eigenvalues of the momentum operator in position basis

We know that the definition of the momentum operator $\hat{P_x}$ in an state space $\mathcal{E}$ is: $$\hat{P_x}|\psi\rangle=P_x|\psi\rangle$$ where $P_x \in \mathbb{R}$. However we also know that ...
0
votes
0answers
79 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 ...