All Questions
30
questions
2
votes
1
answer
88
views
Why does the mass term not violate particle number conservation in a free theory?
The Lagrangian of a free real scalar field theory is
$$ \mathcal{L} = \frac{1}{2} \partial_{\mu} \phi\; \partial^{\mu} \phi \; - \frac{1}{2} m^2 \phi^2. $$
If we decompose $\phi$ in terms of the ...
- 61
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
75
views
What does a quantized field in QFT do? [duplicate]
I'm studying for an exam called Introduction to QFT. One of the main topics in this class is the quantized free fields.
I can now find the fields that solve the Klein-Gordon equation and the Dirac ...
- 19
-3
votes
2
answers
107
views
Multi-particle Hamiltonian for the free Klein-Gordon field
The text I am reading (Peskin and Schroeder) gives the Hamiltonian for the free Klein-Gordon field as:
$$H=\int {d^3 p\over (2\pi)^3}\; E_p\; a^{\dagger}_{\vec p}a_{\vec p}$$
This does not seem to be ...
- 4,399
-2
votes
1
answer
74
views
On creation annihilation operators of the free Klein-Gordon field [closed]
I want to calculate multiparticle states like $|\vec p,\vec p\rangle$ from $|0\rangle$. It seems that I would need to compute from things like: $a^{\dagger}_{\vec p}a^{\dagger}_{\vec p}|0\rangle$?
It ...
- 4,399
2
votes
2
answers
132
views
Commutator of conjugate momentum and field for complex field QFT
In Peskin & Schroeder's Introduction to QFT problem 2.2a), we are asked to find the equations of motion of the complex scalar field starting from the Lagrangian density. I want to show that:
$$i\...
- 538
0
votes
1
answer
66
views
Clarification Needed for The Klein-Gordon Field Acting on the Vacuum State (Peskin and Schroeder)
In Peskin and Schroesder's Introduction to Quantum Field Theory, section 2.3, the Klein Gordon Field has the expression
$$
\phi(x,t) := \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2\omega_{p}}} [a_{...
- 23
-2
votes
1
answer
144
views
Does the QFT Klein-Gordon equation describe the state of the field or the field operator?
In the canonical quantization of QFT we talk about:
states representing a field.
field operators.
The quantum Klein-Gordon equation is expressed in terms of the field φ. Is φ (in the equation) the ...
- 454
0
votes
0
answers
64
views
Confusion related to creation and annihilation operators
I'm studying QFT from Peskin and from the book by Ashok Das, and there seems to be a disagreement between the creation and annihilation operators, for the scalar Klein Gordon field theory.
In Das, we ...
- 1,791
0
votes
0
answers
111
views
Total momentum operator for the KG field
This question pertains to Equation (2.33) in Peskin and Schroeder:
$$
\hat{\vec P}=-\int d^3\!x\,\hat\pi(\vec x)\vec\nabla\hat\phi(\vec x)=\int d^3\!p\,\vec p\,\hat a_{\vec p}^\dagger\,\hat a_{\vec p}
...
- 713
0
votes
0
answers
59
views
Derive the KG field operator in terms of ladder operators
In Peskin and Schroeder, they skip a few steps to arrive at the KG field operator in Equation (2.25):
$$
\hat\phi(\vec x)=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2\omega_{\vec p}}}\left(\hat a_{\vec ...
- 713
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:
$$\mathbf{P}=\int \frac{d^3p}{(2\pi)^3}\...
- 538
2
votes
0
answers
78
views
QFT Formalism, Relation between different POVs
A Klein-Gordon field on a Minkowski background can be written in the following expansion
$$ \hat{\phi}(x) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}} (\hat{a}_p e^{-ip x} + \hat{a}^\dagger_p e^{...
- 136
0
votes
1
answer
472
views
How does the Klein-Gordon equation represent a classical field? [duplicate]
This is what I know about the Klein-Gordon equation so far. Suppose we are working with natural units such that $c = 1$. Then we may obtain the Klein-Gordon equation by considering any 4-vector $p^\mu ...
- 3,350
1
vote
1
answer
91
views
What's the rationale of replacing the Fourier coefficients in a field expansion by operators?
Let's take a look on the particular case of the Fourier expansion of the Klein-Gordon field:
$$\psi (x,t) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_0(p)}[a(p)e^{i(E_0(p)t-px)}+a^\star (p)e^ {-i(E_0(p)t-...
- 500