All Questions
18
questions
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
2
votes
0
answers
86
views
Overlap of Vacuum States (Problem 2.4 of Modern Quantum Field Theory by Thomas Banks) [closed]
Problem
Compute the overlap of the ground-state wave functions of a harmonic oscillator with two different frequencies. A free-bosonic field theory is just a collection of oscillators. Use your ...
1
vote
2
answers
157
views
Argument about identical particles
I am reading Schwartz, Quantum field theory and the standard model, p.207, 12.1 Identical Particles and some question arises (I think that I am beginner for quantum field theory and please understand):...
- 323
3
votes
2
answers
647
views
Lorentz transformation of annihilation operator
In Srednicki's Quantum Field Theory, chapter 4, the author claims that the Lorentz transformation for given a scalar field $\varphi(x)$,
\begin{align}
U(\Lambda)^{-1} \varphi(x) U(\Lambda) = \varphi(\...
- 1,540
0
votes
1
answer
77
views
How to calculate the kinetic energy in field theory when the state is not vacuum state?
I have puzzles in calculating kinetic energy in the language of second quantization
As we know, the operator of kinetic energy is
$$\hat{H}=-\hat{\phi}^{\dagger}(x)\nabla^{2}\hat{\phi}(x)$$
where $\...
- 776
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:
$$
\langle p_2|P^\mu|p_1\...
- 2,035
2
votes
1
answer
252
views
How can $⟨0|ϕ(x)|p⟩=e^{ip⋅x}$ be mathematically shown?
I was reading Peskin and Schroeder's quantum field theory and going through the book mathematically. Then I got stuck at one equation.
Consider a single, non-interacting real scalar field. The book ...
- 35
2
votes
1
answer
150
views
How is the state $|a_0 a_i\rangle$ physical?
For a state $|\psi\rangle$ to be physical we require that:
$$\langle\psi|a_0^\dagger a_0|\psi\rangle = \langle\psi|a_i^\dagger a_i|\psi\rangle$$
It is always said that physical state must contain ...
- 1,323
8
votes
1
answer
2k
views
How to derive completeness relation in quantum field theory with a Lorentz invariant measure?
$\bullet$ 1. For the one-particle states, the completeness relation is given in Peskin and Schroeder, $$(\mathbb{1})_{1-particle}=\int\frac{d^3\textbf{p}}{(2\pi)^{3}}|\textbf{p}\rangle\frac{1}{2E_\...
- 26.8k
0
votes
1
answer
106
views
Creation and Anihilation Operators [closed]
Consider the following expressions:
$$ \langle \Psi | a_{i_1} a_{i_2} a_{i_3}^* a_{i_4}^* | \Psi\rangle$$
$$ \langle \Psi | a_{i_1}^* a_{i_2} a_{i_3}^* a_{i_4} | \Psi\rangle$$
where in the first ...
- 11
3
votes
1
answer
2k
views
Time-ordered product of two normal-ordered products of fields
Suppose you have a scalar field theory with field operators $\phi(x)=\phi(x)_+ + \phi(x)_- $ that can be decomposed into terms of annihilation and destruction operators.
Let
$$
D(x-y) = <0|T(\phi(...
- 31
1
vote
0
answers
194
views
QFT: Ground State Momentum - Normalisation of States
In my notes I have,
$$
\left\langle \mathbf{p} \left| \mathbf{q} \right.\right\rangle
=
\left\langle 0 \left|
{a(\mathbf{p})}\
{a(\mathbf{q})}^{\dagger}
\right| 0 \right\rangle
$$
I am not sure how ...
5
votes
1
answer
2k
views
Wick's Theorem: Why is the vacuum expectation value of uncontracted operators zero?
I'm am right now reading Chapter 4.3 (Wick's Theorem) in Peskin & Schroeder. It is said that
In the vacuum expectation value, any term in which there remain uncontracted operators gives zero (...
- 379
3
votes
1
answer
143
views
How many particles in $\phi_0(x)^2|0\rangle$?
In Schwartz's "QFT and the standard model" on pg 22 he writes:
A two or zero particle state as in $\phi_0(x)^2\left|0\right>$.
I was wondering how this can be proved? I tried checking if $\...
- 2,094
0
votes
2
answers
193
views
Explanation on anticommutation relations
Setup
Given two states: $|K\rangle=a_i^+a_j^+|\rangle$ and $|L\rangle=a_k^+a_l^+|\rangle$.
Evaluating the overlap: $\langle K|L\rangle=\langle|a_ja_ia_k^+a_l^+|\rangle$
Introducing: $a_ia_k^+=\delta_{...
- 139