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21
questions
0
votes
1
answer
165
views
Completeness relation for Hilbert space in quantum field theory
I'm studying chapter 7 section 1 of Peskin and Schroeder. On page 212, we have the one particle Hilbert space $$\tag{7.1} (1)_{\text{1-particle}}=\int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}|p\rangle\...
- 789
1
vote
0
answers
79
views
Rigorous definition of the energy-momentum operator in QFT
Given a Hilbert space $H$, let $\Gamma(H)$ denote the associated Fock space. Let $a^*$ be the standard creation operator-valued distribution on the Fock space $\Gamma(L^2(\mathbb{R}^3))$, i.e.
$$
a^*(\...
- 567
6
votes
1
answer
172
views
Does the Fock space for a free QFT decompose into a tensor product of Fock spaces for each momentum?
Take a free relativistic QFT, say for a real scalar field $\phi$ with the Lagrangian density
$$
\mathscr{L} = \frac{ \partial_{\mu} \phi \partial^\mu \phi - m^2 \phi^2}{2} \ .
$$
After quantization we ...
- 3,719
2
votes
1
answer
214
views
What is the proper translation of a field operator?
I am trying to write the correct expression for a translated quantum field operator. There appear to be conflicting expressions given in this PSE post, and this one. In the former linked PSE post, the ...
- 5,371
1
vote
0
answers
112
views
Weinberg's normalization convention for momentum eigenstates
In this answer https://physics.stackexchange.com/a/376193/274751 two different conventions for the normalization of momentum eigenstates are mentioned.
This convention amounts to the choice of $N(p)$ ...
2
votes
1
answer
189
views
Is $|i\rangle=\sqrt{2\omega_1}\sqrt{2\omega_2}a^\dagger_{p_1}(-\infty)a^\dagger_{p_2}(-\infty)|\Omega\rangle$ a momentum eigenstate?
Define an asymptotic state in the far past as $$|i\rangle=\sqrt{2\omega_1}\sqrt{2\omega_2}a^\dagger_{{\vec p}_1}(-\infty)a^\dagger_{{\vec p}_2}(-\infty)|\Omega\rangle$$ where $|\Omega\rangle$ is the ...
- 11.8k
5
votes
1
answer
971
views
Weinberg QFT 1 Normalization one 1 particle states p. 66
I encounter a question regarding the derivation of the normalization of 1 Particle states in Weinbergs book (Formula 2.5.14).
Similar questions were asked in A question on page 65 of Weinberg's ...
- 420
2
votes
2
answers
97
views
Quantum operator calculations [closed]
We define the quantum operator
$$
P^\mu=\int{\frac{d^3p}{(2\pi)^3}}p^\mu a_p^\dagger a_p
$$
Now how can I calculate
$$
\langle p_2|P^\mu|p_1\rangle~?
$$
My attempt:
$$
\langle p_2|P^\mu|p_1\...
- 2,035
2
votes
1
answer
198
views
Is the field-momentum operator ambiguous?
In this question I asked wether the definition of the momentum operator as an operator that has to generate translations by satisfying the canonical commutation relations was ambiguous. The answer to ...
- 6,763
6
votes
0
answers
189
views
What's the momentum-space vacuum wave-functional of a fermion?
In the Schrödinger picture, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\phi\rangle=\phi(\mathbf x)|\phi\...
- 970
1
vote
2
answers
607
views
How does the momentum operator act on a multi-particle state?
My issue is about the proper development of the action of the momentum operator $P^{\mu}$ - the generator of spacetime translations - on multi-particle states. I'm a little clueless on this, so I'm ...
- 5,371
2
votes
1
answer
368
views
Contraction between scalar field and momentum eigenstate
In solving the $\phi^4$ theory in Peskin, we define the contraction (on pg. 110 eq. 4.94)
$$C\left(\phi(x)|p \rangle \right)=\phi(x)|p \rangle . $$
That is, the contraction is just the action of $\...
- 4,780
2
votes
1
answer
141
views
What justifies the use of asymptotic momentum state?
The LSZ scattering approach starts with initial and final asymptotic momentum states. But we know that $\langle k' | k\rangle = \delta^3(k'-k)$, which means that it is not a properly normalizable ...
- 327
0
votes
1
answer
166
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Confusion in normalization of position space and momentum space
It seems that in LSZ formalism approach, or just Feynman diagram approach, we can compute scattering amplitude of $\langle x_{out} | y_{in}\rangle$ (position space) and $\langle p_{out} | p_{in}\...
- 327
1
vote
1
answer
101
views
Doubt regarding field expansion in Schwartz
In section 2.3.1 he says that annihilation and creation operator satisfy $$[a,a^{\dagger}]=1$$ fine no problem this is the basic definition after all. He then says on generalizing it we get $$[a_k,a_p^...
- 3,043