All Questions
30
questions
1
vote
1
answer
55
views
Total momentum operator of the Klein-Gordon field (before limit to the continuum)
I'm following K. Huang's QFT: From Operators to Path Integrals book. In the second chapter, he introduces the Klein-Gordon equation (KGE), and its scalar field $\phi(x)$, which satisfies this equation....
- 11
1
vote
0
answers
43
views
Conjugate momenta in Radial Quantization
When we radially quantize a conformal field theory, is there at least formally a notion of a conjugate momentum $\Pi$ to the primary fields $O$ which would satisfy an equal radius commutation relation ...
- 482
2
votes
2
answers
314
views
CFT Radial Quantization Raising and Lowering Operators Sign Question [closed]
Following Slava Rychkov (Page 41, Or here on arxiv page 39), I am trying to show that the momentum operator raises the scaling dimension.
I've seen the other related questions on this matter by Y. ...
- 194
0
votes
1
answer
167
views
Are creation and annihilation operators functions of momentum?
In QFT, we usually write creation and annihilation operators in the following way: ${a^s_{\textbf{p}}}^\dagger$, $a^r_{\textbf{q}}$, where $r,s$ denote the spins and $p,q$ the three-momenta of ...
- 280
0
votes
0
answers
111
views
Total momentum operator for the KG field
This question pertains to Equation (2.33) in Peskin and Schroeder:
$$
\hat{\vec P}=-\int d^3\!x\,\hat\pi(\vec x)\vec\nabla\hat\phi(\vec x)=\int d^3\!p\,\vec p\,\hat a_{\vec p}^\dagger\,\hat a_{\vec p}
...
- 713
0
votes
0
answers
56
views
Adjoint of the Dirac equation, and hermiticity of the momentum operator
I'm trying to derive the adjoint of the Dirac equation in standard relativistic quantum mechanics. We have the Dirac equation as follows :
$$(i\gamma^{\mu}\partial_\mu -m)\psi=0$$
To find it's adjoint,...
- 1,791
2
votes
3
answers
170
views
Why do we drop the renormalization term in momentum Klein-Gordon Field Theory?
I'm following Peskin & Schroeder's book on QFT. I managed to prove expression (2.33) which gives us the 3-momentum operator for the Klein-Gordon Theory:
$$\mathbf{P}=\int \frac{d^3p}{(2\pi)^3}\...
- 538
0
votes
1
answer
103
views
Operators localized in space and $\hat{q}$ position operator
I am reading the "Quantum Field Theory lectures of Sidney Coleman". In the first chapter (subchapter 1.2), the author talks about translation invariance. In particular, he states that $U(a)=...
- 3,992
2
votes
0
answers
359
views
Momentum of the Dirac field in terms of creation/annhilation operators [closed]
In Peskin & Schroeder, the Momentum operator of the Dirac field is given as:
$$
{\bf P}=\int d^3x \psi^\dagger \left(-i\nabla\right) \psi=\int \frac{d^3p}{(2\pi)^3}\sum_s {\bf p}(a_p^{s\dagger}a_p^...
- 129
0
votes
1
answer
351
views
Expressing the four-momentum operator in terms of field operators
There are a series of problems in chapter 3 of the book Quantum Field Theory of Point Particles and Strings by Hatfield that lead to a proof of Lorentz invariance in the canonical formulation of ...
- 785
0
votes
1
answer
179
views
Weinberg Chapter 10: Sign convention for momentum operator
Weinberg says that translational invariance produces a conserved momentum, i.e., $P^\mu$, such that (Eq. 10.1.1):
\begin{align*}
[P_\mu, O(x)] = +i\hbar \frac{\partial}{\partial x^\mu} O(x). \tag{...
- 433
1
vote
0
answers
79
views
Rigorous definition of the energy-momentum operator in QFT
Given a Hilbert space $H$, let $\Gamma(H)$ denote the associated Fock space. Let $a^*$ be the standard creation operator-valued distribution on the Fock space $\Gamma(L^2(\mathbb{R}^3))$, i.e.
$$
a^*(\...
- 567
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
0
votes
2
answers
69
views
Doubt concerning the definition of $p$ and $-p$ in quantum field theory
We can define the field in term of the ladder operators as:
$$
\phi(\vec{x}) \propto \int d^3p \left( a_{\vec{p}}e^{i\vec{p}\cdot\vec{x}} + a^\dagger_{\vec{p}}e^{-i\vec{p}\cdot\vec{x}} \right)
$$
...
- 297
12
votes
3
answers
1k
views
Does the concept of a "momenton" make sense?
Looking at the QED-Lagrangian
$$\mathcal L = -\bar\psi(\not\!p + e\not\!\!A + m)\psi -\frac14 F_{\mu\nu} F^{\mu\nu} $$
I was wondering: While $\bar\psi\not\!\!A\psi$ describes the interaction between ...
- 6,820