Linked Questions

2
votes
2answers
2k views

Definition of conjugate momentum in QFT

My lecture notes define the conjugate momentum of a scalar field via: $$\pi = \dot{\psi}$$ Where $$\psi = \int \frac{d^3p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_p}}\left(a_p e^{i\vec{p}\cdot \vec x} + a_p^\...
4
votes
1answer
296 views

Is Lorentz invariant differential measure arbitrary?

In Srednicki, we chose a function $f(\mathbf k)$ to make $d^3\mathbf k/f(\mathbf k)$ Lorentz invariant. The way to do this is to first start from a 4 dimensional measure and multiply it by a Dirac ...
1
vote
2answers
209 views

What is the precise formal correspondance between an oscillator and a quantum field?

A common route of introduction to quantum field theory is to note a similarity between the mathematical structure of a quantum harmonic oscillator and of a quantum field "at a point". The quantised ...
0
votes
1answer
506 views

Lorentz invariant integration measure and Heaviside step function

I'm currently studying Klein-Gordon fields and I ran onto the concept of the Lorentz invariant integration measure, namely: \begin{equation} \frac{d^3k}{(2\pi)^32E_k} \end{equation} where $E_k=\sqrt{\...
1
vote
1answer
408 views

Lorentz invariance of volume element from the four-volume element: why on-shell?

In Srednicki's QFT book, in chapter 3 (eqn. 3.16 onwards) he talks about the lorentz invariance of the volume element. For this he writes $d^3k/f(k)$ should be invariant under lorentz transformations. ...
2
votes
2answers
300 views

Solution to the Klein-Gordon Equation

Most textbooks solve the Klein-Gordon equation with the ansatz $$\varphi(\mathbf{x},t) = \int \frac{\mathrm{d}^3\mathbf{k}}{f(k)}\left(a(\mathbf{k})\, \mathrm{e}^{i k x} + b(\mathbf{k})\,\mathrm{e}^{-...
0
votes
1answer
333 views

Fourier transforming the Klein-Gordon equation

I'm aware of the fact that there are similar questions on this forum but I could not find an answer that fits my problem. Many textbooks state that a general solution to the Klein-Gordon equation \...
2
votes
1answer
145 views

Fourier transforming the wave equation twice

The wave equation $$\nabla^2 u(r,t)-\frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}(r,t)=0$$ can be Fourier transformed with respect to time, using $\frac{\partial}{\partial t}=i\omega$, to obtain the ...
1
vote
3answers
287 views

Mode Expansion in Klein-Gordon QFT

I have a confusion regarding the mode expansion of the Klein-Gordon field theory. I am following Peskin and Schroeder. My questions are about how we formally get to the expansion of the KG QFT in ...
3
votes
1answer
285 views

Lorentz Covariant formula for Noether Charges in QFT

I'm looking for a Lorentz covariant expression of Noether charges and I found this article: https://arxiv.org/abs/hep-th/0701268, section II-A in particular. Consider specifically eq. (20-21), they ...
3
votes
2answers
210 views

Quantizing Klein Gordon Field: Sign Problem

I'm trying to re-derive the Quantization of the Klein Gordon Field but I'm running into sign problems. My starting point is: $$ \phi(x,t) = \frac{1}{(\sqrt{2 \pi})^3} \int \tilde{\phi}(k,t) e^{i kx}...
2
votes
1answer
238 views

How to derive this expression for the free scalar field in QFT? (Peskin & Schroeder)

In the introductory text to quantum field theory by Peskin & Schroeder, they state that in analogy to the simple harmonic oscillator in quantum mechanics, the free scalar field can be expressed as:...
1
vote
2answers
108 views

Can we derive free field expansion formula for the spin-1/2 Dirac field?

The Dirac field has the expansion $$\Psi(x)=\int\frac{d^3p}{\sqrt{(2\pi)^32E_p}}\sum\limits_{s=1,2}\Big(b_s(p)u^s(p)e^{-ip\cdot x}+d^\dagger_s(p)v^s(p)e^{+ip\cdot x}\Big)$$ where $b_s$ and $d_s$ are ...
0
votes
0answers
133 views

Free Field Klein-Gordon Equation

The free field Klein-Gordon equation $$(\Box+m^{2})\phi(t,\mathbf{x})=0$$ may be solved to give $$\phi(t,\mathbf{x})=\int d\omega d\mathbf{k}\widetilde{\phi}(\omega,\mathbf{k})\delta(\omega^{2}-\...
0
votes
1answer
77 views

Why is it necessary to introduce different sets of creation and annihilation operators to quantize the complex K-G field?

I am reading Peskin & Schroeder and in chapter 2 (p.21) he quantizes the real K-G field such as: $$\phi=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_p}}\left(a_pe^{ip·x}+ a^{\dagger}_pe^{-ip·x}\...

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