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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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Decoupling of degrees of freedom in Klein-Gordon equation

In David Tong's notes in QFT he states that the degrees of freedom decouple in momentum space for the Klein-Gordon eq. He writes that this can be seen by using the Fourier transform (see picture below)...
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$m$ in Klein-Gordon Equation

The Klein-Gordon equation is given by $$ (\square + m^2) \phi(x) = 0 $$ where $\square$ is the d'Alembertian operator, $m \in \mathbb{R}$ and $\phi$ is a scalar field. Question: What is $m$ in the ...
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Question about Mode expansion of free compact boson

$(1+1)$-Dim free compact boson, Lagrangian is $$\mathcal{L}= \frac{1}{2}(\partial_\mu\phi(\sigma,t))^2$$ with $\phi(x,t)\sim\phi(x,t)+2\pi r$ and periodic boundary condition along $x$, i.e. $\phi(\...
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Time-independent Klein-Gordon PDE

Given the KG PDE: $$\psi_{tt} - \psi_{xx} + m^2 \psi = 0.$$ Wikipedia describes the time-independent variant of this as just setting $\psi_{tt}=0$. My question is this: For the Schrödinger ...
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$\partial^{\nu} \partial_{\nu}$ vs. $\partial_{\nu} \partial^{\nu}$

I was doing a problem regarding field theory. I am given the following lagrangian density: $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi_i\partial^\mu\phi_i-\frac{m^2}{2}\phi_i\phi_i$$ for three scalar ...
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Scalar particles are described by a real scalar field or by a complex one?

Well, in the title is already stated my main question. I know you can use a complex scalar field to describe two real scalar fields, by using just one that involves both of them. But, in the modern ...
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Sign mistake in the energy momentum tensor of the Klein-Gordon Equation

Recently I understood that the energy momentum tensor can be calculated by: \begin{equation} T_{\mu \nu}=\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{\mu \nu}}.\tag{1} \end{equation} So consider ...
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Hamilton equations of motion for matter fields coupled to general relativity in ADM formalism

Do you know what are the Hamiltonian formalism analogs of the Klein-Gordon equation and/or the Maxwell equations in general relativity? Showing how these equations of motion for matter in the ...
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Non-Relativistic Limit of Klein-Gordon Probability Density

In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as: $$ \begin{eqnarray} P & = & \dfrac{...
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156 views

Intuitive explanation for the free field Lagrangian?

The free field Lagrangian is $$\mathcal{L}=\frac 1 2 \partial^\mu\phi\partial_\mu\phi-\frac 1 2m^2\phi^2$$ with sign convention $(+,-,-,-)$. Plugging this into the Euler-Lagrange equations gives the ...
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Corresponding particle-antiparticle solutions for Klein-Gordon equation

For free particle solutions in a box, the following 4 solutions are possible(Not all 4 are independent though) as $$\psi_+=A_+ \exp{\frac{i}{\hbar}(px-Et)}\\\psi_+^*=A_+^* \exp{\frac{-i}{\hbar}(px-Et)}...
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Sine Gordon model in 3+1 Dimensions

I'm have read the publication of Neuenhahn, C. and Marquardt, F. (2015) ‘Quantum simulation of expanding space–time with tunnel-coupled condensates’, New Journal of Physics. IOP Publishing, 17(12), ...
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Can someone Tong got this equation in his QFT notes

Can someone explain how D.Tong got equation 2.18 in his QFT notes in chapter 2? I am lost from equation 2.5, can someone explain? Link to notes: http://www.damtp.cam.ac.uk/user/tong/qft.html Can ...
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E.L. Equations in QFT

In QFT, we use the Lagrangian to construct the Hamiltonian, and in the Interaction Picture (with regards to the Free Field Hamiltonian) use the full Hamiltonian to calculate the changes in the field (...
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Units of Klein-Gordon equation

I'm looking at the units of the Klein-Gordon equation $$u_{tt} - c^2\Delta u = -\frac{m^2c^2}{\hbar^2}u. $$ Disregarding the units of $u$, which are the same everywhere and so cancel, I get $seconds^{-...
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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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Solution to Klein-Gordon equation: real field condition and other questions

Sorry for the lengthy question, pretty much the whole text is the standard derivation of the solution of the KG equation which I included to illustrate my doubts, and some questions are at the end. ...
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Klein Gordon equation in the nonrelativistic and semiclassical limit in a Wigner approach

I would like to analyse the semiclassical and nonrelativistic limit of the Klein-Gordon equation, \begin{equation} \frac{1}{c^2} \partial_t^2 \phi - \Delta \phi + \frac{M^2 c^2}{\hbar^2} \phi =0. ...
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Using fourier analysis of the Klein Gordon equation

This question is more about a mathematical detail, and I am undoubtedly missing something very obvious. And note, I have sifted through the numerous questions on Fourier transform (FT) and the Klein-...
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283 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}^{-...
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Tensors and the Klein-Gordon Equation

Consider the Klein-Gordon equation: \begin{equation} \frac{\partial^2 \psi}{\partial t^2} = c^2 \Delta \psi - \frac{m^2 c^4}{\hbar^2} \psi, \end{equation} and define for each one of its solutions $\...
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Sign confusions in solution to Klein Gordon's equation

I have two basic questions on the solution of the Klein Gordon equation. The Lagrangian of the Klein Gordon field is $$\mathcal{L}=\frac12\partial_\mu\phi\partial^{\mu}\phi-\frac12m^2\phi^2 $$ ...
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Using a time-like boundary as a computer?

Question and Summary Using classical calculations and the Robin boundary condition I show that one calculate the anti-derivative of a function within time $2X$ (I can compute an integral below) $$\...
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284 views

Nonexistence of a Probability for the Klein-Gordon Equation

David Bohm in his wonderful monograph Quantum Theory, in Section 4.6 discusses the difficulties one encounters in trying to develop a relativistic quantum mechanics. He starts from the relation \...
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Klein-Gordon quantization and SHO analogy

I understand that the procedure to quantize Klein-Gordon's field is to manipulate in a such a way to bring up the simple harmonic oscillator behavior of the field. This is done by Fourier transforming ...
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Unexpected symmetry of wave equations in momentum representation

In the $x$-representation, the translational invariance implies that $$ \mathcal{D}[\psi(\vec{x},t)]=0\quad \Longrightarrow\quad \mathcal{D}[\operatorname{e}^{i\vec{a}\hat{\vec{P}}}\psi(\vec{x},t)]=0 $...
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Can a second-order Schrödinger equation preserve the norm?

Suppose we lived in a universe in which the Schrödinger equation contains second order time derivatives, $$i\hbar \partial_t^2|\varphi(t)\rangle = \mathbb{H} | \varphi(t)\rangle.$$ Would it be true ...
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Klein-Gordon-Equation contains no Spin

I have a question about an argument used in Schwabl's "Advanced Quantum Mechanics" concerning the properties of the Klein-Gordan-Equation (see page 120): Since the eigenenergies of free solutions are ...
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193 views

Interaction-Picture Field as Solution of Klein-Gordon Equation

I am following a problem in a QFT textbook (Srednicki) which asks us to show that the interaction-picture field $$\phi_I(\textbf{x},t)=e^{iH_0 t}\phi(\textbf{x},0)e^{-iH_0 t}$$ obeys the Klein-...
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Klein-Gordon-Equation provides a Scalar Theory that doesn't contain Spin [duplicate]

I'm reading actually Schwabl's "Advanced Quantum Mechanics" and encounter an understanding problem while considering the Klein-Gordan-equation. Here the excerpt: Obviously, since the eigen energies ...
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Is the inverse of the Klein-Gordon equation ever used in physics?

The Klein-Gordon equation (scaling constants) is $$\square u = -m^2 u.$$ I am wondering if the equation $$\square u = m^2 u.$$ for real $m$ ever shows up in the physical literature?
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Replacing a squared potential by a position-dependent mass

I'm studying the solutions of the Klein-Gordon and Dirac equations for a relativistic particle in a potential of the form $$V(x)=\left\lbrace\begin{array}{ll} 0, & x\in[0,L]\\V_0, & x\not\in[0,...
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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:...
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Is the ground state energy of a quantum field actually zero?

I start by outlining the little I know about the basics of quantum field theory. The simplest relativistic field theory is described by the Klein-Gordon equation of motion for a scalar field $\phi(\...
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Propagator solution to Klein-Gordon equation

We know that the Klein-Gordon operator is given by $(\partial^2+m^2)=(\partial_\mu\partial^{\mu}+m^2)$, which is used to describe the evolution of relativistic free particles. How can we show that ...
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can somebody explain how you get the second line from the first line in the picture?

I'm trying to understand the transition from the 1st line of the Lagrangian to the second. we substitute for $\eta$ but how is the multiplication happening here? if I multiply the terms into the ...
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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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Getting the relativistic inner product of Siegel's book

Last time I was discussing with a physicist about quantum field theory and how in the firsts chapters of textbooks it is often regarded that the Klein-Gordon equation does not have a positive definite ...
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Is there a mathematical singularity in the relativistic contraction of valence orbitals as a function of nuclear charge?

I'm looking at Figure 1 on p. 2 of Jansen, "Effects of relativistic motion of electrons on the chemistry of gold and platinum" (2005) (should be a free download). From the bottom curve, there is such ...
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Do relativistically-contracted electron states have the same energy and angular momentum values?

I've been reading that electron bound states are defined by four quantum numbers, $n$, $l$, $m_l$, and $m_s$, respectively the principal quantum number, the azimuthal quantum number, the magnetic ...
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Derivation of Klein-Gordon equation in General Relativity

I am trying to derive the Klein-Gordon equation for the case of GR using the action: $$S\left[ {\varphi ,{g_{\mu \nu }}} \right] = \int {\sqrt g {d^4}x\left( { - {1 \over 2}{g^{\mu \nu }}{\nabla _\mu ...
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How one can use the Klein-Gordon equation to deal with the hydrogen atom? [closed]

I'm not familiar with physics jargons compare to those who study physics professionally, but to remark my knowledge regarding this question, the Klein-Gordon equation is a relativisitc equation for ...
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Klein-Gordon Inner product being independent of the choice of spacelike hypersurface $\Sigma$ used to integrate it

Here I'll work in flat 4-dimensional Minkwoski space, but using arbitrary coordinates (described by some metric $g_{\mu\nu}$). Suppose we've got two complex-valued scalar functions $f$ and $g$ which ...
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Real Scalar Field: in what sense are the Minkowski modes complete?

When considering a real scalar field with Lagrangian $$\mathcal{L} = - \frac{1}{2}(\partial_{\mu} \phi)(\partial^{\mu} \phi) - \frac{1}{2}m^2\phi^2$$ the equation of motion is the Klein-Gordon ...
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What motivates the definition of the Klein-Gordon Inner Product?

I am following along these lecture slides. For scalar functions $f$ and $g$ we define the Klein-Gordon inner product as follows: $$ <f,g> \ = \ i \int_{\Sigma} d^{3}\mathbf{x}\ \left[ f^{\ast}(t,...
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Models of Inflation

I'm trying to get an overview of cosmological models which include inflation. My aim is to study the numerics of nonlinear Klein- Gordon equations which in fact are the Friedmann equations in the FRLW ...
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Lorentz transformation of the Klein-Gordon equation

In the Lorentz transformation of the field $\partial_\mu\phi(x)$ (Peskin, p.36) \begin{eqnarray} \partial_\mu\phi(x)\to\partial_\mu(\phi(\Lambda^{-1}x))=(\Lambda^{-1})^\nu_{\phantom{\nu}\mu}(\...
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Quantization of Klein-Gordon Field

I have a question about the quantization procedure of the Klein-Gordon field as presented in Peskin&Schroeder. The field is expressed as a Fourier decomposition $$ \phi(x,t) = \int \frac{d^3p}{(2\...
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Eigen-Energy of Vibration on a Loop

I recently started this recreational project to find the energy of the modes on a loop. To obtain the eigen-energies $E_n$, I decided to solve the 1D wave equation on a loop of circumference $2\pi l$ ...
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Creation and Annihilation Operators in QFT

I have a general question about heuristic way of QFT to introduce creation and annihilation operators: The Klein-Gordon field is introduced as continuous interference of plane waves $\mathrm{e}^{i(\...