# Questions tagged [klein-gordon-equation]

The Klein-Gordon Equation or the Klein-Fock-Gordon Equation is an equation in quantum field theory which initially was discovered by Schrodinger but discarded by him soon after.

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### How to interpret the concept that a (scalar) field is a linear superposition of harmonic oscillators?

I've reading an introduction lecture on QFT, but I didn't started very well. I have troubles to understand some concepts of the very first example (But I think that they apply with the other fields): ...
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### Where do the particle and antiparticle wavefunctions originate from in the Klein Gordon equation?

In my textbook (Sakurai) it is given that $$\left(D_\mu D^\mu+m^2\right)\Psi(\mathbf{x},t)=0$$ where $D_\mu=\partial_\mu+ieA_\mu$ is the covariant derivative. It states that since it is a second ...
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### Quantization of Klein-Gordon field between two boundaries

Consider a real scalar $\phi(x,t)$ with mass $m$ in $1+1$ dimensional spacetime, described by the 2d free Klein-Gordon action. $\phi(x,t)$ lives on an interval $0 \leq x \leq L$, and is subject to ...
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### Obtaining mode decomposition of quantum field in a box with Dirichlet condition

I am considering the Klein Gordon Equation in a box with Dirichlet conditions (i.e., $\hat{\phi}(x,t)|_{boundary} = 0$). 1-D functions that obey the Dirichlet condition on interval $[0,L]$ are of the ...
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### Commutator of canonical fields in Quantum Field Theory

Let $\phi(\vec{x},t)$ denote the canonical fields and $\pi(\vec{x},t)$ denote the canonical impulses where they're given by: \begin{equation} \phi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2\omega_{\...
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### Lagragian density of a massless scalar field

I have seen in some books that the simplest Lagrangian density of a massless scalar field is $$\mathscr{L}=\dfrac{1}{2}\partial^\mu\phi\partial_\mu\phi=\dfrac{1}{2}\left(\partial_\mu\phi\right)^2.$$ ...
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### Klein Gordon equation in tensorial form

The problem is to show that $$\psi=\psi_0e^{iP_\mu x^\mu}$$ is a solution to Klein Gordon equation $$(\partial _\mu \partial ^ \mu +m^2)\psi=0$$ if and only if the 4-momentum $p _\mu$ satisfies the ...
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### Klein-Gordon and localizability

If we consider the theory of a single relativistic point particle, quantized using whatever appropriate method, the wavefunction simply obeys the Klein-Gordon equation, which allows for a fairly wide ...
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### Contradictory solutions to the Klein-Gordon equation as an initial value problem (canonical formulation)

I'm going to begin with a prelude about constructing the solution using the action/Lagrangian formalism in order to provide context and a point of comparison for the canonical one. The main question ...
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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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### Explicitly proving that the Hamiltonian is Lorentz covariant

I want to show explicitly that the Hamiltonian $$H = -\Omega V + \int d\tilde{\textbf{k}} \omega (a^\dagger(\textbf{k}) a(\textbf{k}) + b(\textbf{k}) b^\dagger(\textbf{k}) )$$ is Lorentz covariant....
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### Solution of Klein-Gordon field in Schwarzschild metric

I was trying to understand the behavior of scalar field in the vicinity of black hole and I set up the Klein-Gordon equation in the background of Schwarzschild metric. But I am having difficulty in ...
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### Vacuum energy of a real Klein-Gordon field

Hamiltonian for a Klein-Gordon field can be written as - $$H= \int \frac{d^3p}{(2\pi)^3 \omega_{\vec{p}}}[a^{\dagger}_\vec{p}a_\vec{p}+\frac{1}{2}(2\pi)^3\delta^{(3)}(0).\tag{1}]$$ In one of my ...
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### Time evolution operator of Klein-Gordon field

If $U(t)=e^{itH}$ is the time evolution operator. And $|\phi \rangle$ is a state of a field at particular time $t_1$ and $|\phi' \rangle$ is the state of a (free) Klein Gordon field at a time $t_2$. ...
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