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### 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 \...
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### 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 ...
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### 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 ...
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### 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 ...
I'm trying to re-derive the Quantization of the Klein Gordon Field but I'm running into sign problems. My starting point is: $$\phi(x,t) = \frac{1}{(\sqrt{2 \pi})^3} \int \tilde{\phi}(k,t) e^{i kx}... 1answer 238 views ### How to derive this expression for the free scalar field in QFT? (Peskin & Schroeder) In the introductory text to quantum field theory by Peskin & Schroeder, they state that in analogy to the simple harmonic oscillator in quantum mechanics, the free scalar field can be expressed as:... 2answers 108 views ### Can we derive free field expansion formula for the spin-1/2 Dirac field? The Dirac field has the expansion$$\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)$$where b_s and d_s are ... 0answers 133 views ### Free Field Klein-Gordon Equation The free field Klein-Gordon equation$$(\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}-\...
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}\...