All Questions
Tagged with quantum-field-theory hilbert-space
681
questions
0
votes
2
answers
85
views
The vanishing of vacuum expectation value
I have some difficulty understanding why the vacuum expectation value vanishes. As illustrated in my notes, we can split the field into two parts:
$$
\phi(x) = \phi^+(x) + \phi^-(x),
$$
where $\phi^+(...
- 41
7
votes
2
answers
406
views
States created by local unitaries in QFT
In quantum field theory, consider acting on the vacuum with a local unitary operator that belongs to the local operator algebra associated with a region. In such a way, can we obtain a state that is ...
- 1,221
2
votes
0
answers
167
views
Unitary representations of a Lorentz transformation
In QFT we have an action of the restricted Lorentz group which is implemented via a unitary transformation. In other words, if $\Lambda\in SO(1,3)^\uparrow$, then the corresponding unitary operator is ...
- 145
0
votes
1
answer
90
views
Greiner´s Field Quantization question [closed]
I upload a screenshot of Greiner´s book on QFT. I don´t understand one step. I need help understanding equation (3), what are the mathematical steps in between?
Greiner, Field Quantization, page 245 (...
- 15
3
votes
0
answers
78
views
Why do the Canonical Commutation Relations hold in Interacting Theories? [duplicate]
The canonical commutation relations for a scalar field theory stating
$$
[\phi(\vec x, t), \partial_t\phi(\vec x', t)] = i \hbar \delta^3(\vec{x} -\vec{x}').\tag{7.4}
$$
Schwartz in Section 7.1 in his ...
- 173
2
votes
1
answer
119
views
Understanding mathematically the promotion of field/observable to operator in QFT
First, I know it "worked", in physics sense.
My question is what happened in the math sense.
When promoting something, such as a field, to an operator, am I essentially mapping the field to ...
- 63
5
votes
1
answer
227
views
Lorentz generators on Fock space
Consider a free massive relativistic scalar field in $d+1$ dimensions. Its Hilbert space can be taken to be the bosonic Fock space on $\mathfrak h = L^2(\mathbb R^d)$:
$$\mathcal F = \bigoplus_{n=0}^{+...
- 5,989
3
votes
2
answers
363
views
Proof that asymptotic particle states are free
In quantum field theory, It’s often said that the interacting annihilation operator (defined by the Klein Gordon inner product between the interacting field and a plane wave) behaves like the free ...
user310742
3
votes
1
answer
176
views
How to normalize the states in the continuous limit?
In quantum field theories we can perform the continuous limit, where we take the limit $V\rightarrow\infty$ for the volume system. In quantum optics, we can start by absorbing a factor $\left(\frac{L}{...
- 865
2
votes
1
answer
179
views
Existence and uniqueness of vacuum of fermion or boson operators
Suppose I have a set of boson (or fermion) annihilation operators $\{a_i\}$ defined on a Hilbert space. These operators satisfy the canonical (anti-)commutation rules
$$
\text{boson:} \quad [a_i, a^\...
- 660
2
votes
1
answer
224
views
Ground state of Bogoliubov quasi-particles (simpler version)
This is a simplified version of one of my previous questions. Let $b_1, b_2$ be two boson operators; their vacuum is denoted as $|0\rangle$, i.e. $b_i |0\rangle = 0$. We can make a canonical ...
- 660
1
vote
1
answer
332
views
How is the interacting vacuum defined in QFT?
I have seen this in a couple of textbooks (Schwartz and Zee), where the author would use the interacting vacuum $|\Omega \rangle$ in a calculation, but would never mention how the state is defined.
...
- 1,896
2
votes
0
answers
60
views
How to perform the limit of infinite time in the LSZ approach?
I am computing the scattering matrix using the LSZ reduction formula in a semiclassical limit. The result that I am getting has the following form:
$$
S = \lim_{t_i \to - \infty} \lim_{t_f \to \infty} ...
- 81
1
vote
0
answers
134
views
Clarification on interaction picture in QFT
Say we want to calculate $\langle f(t_2)|O|i(t_1)\rangle$. Where $O$ is an arbitrary operator. We can treat the states as stationary and then evolve the operator
$$\langle f(0)|O(t)|i(0)\rangle\\O(t) =...
user310742
2
votes
2
answers
867
views
What does sandwiching with an unitary operator and its inverse imply?
I am following the book "An introduction to quantum field theory" by Peskin and Schroeder. In the section 'Discrete symmetries of the Dirac theory', it is written,
$P a^s _p P^{-1} = \eta_a ...
- 399