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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Klein paradox for bosons and fermions

I am reading this paper about the Klein paradox, i.e. transmission of relativistic particles incident on a potential step of height $V_0 > E + mc^2 > 2mc^2$ with $E$ the energy of the incident ...
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447 views

Interpreting the Klein-Gordon Annihilation Operator Expression

I can derive $$a(k) = \int d^3 x e^{ik_{\mu} x^{\mu}} (\omega_{\vec{k}} \psi + i \pi)$$ for a free real scalar Klein-Gordon field in three ways mathematically: the usual Fourier transform way in ...
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349 views

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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418 views

Is there supersymmetry between Dirac and Klein Gordon solutions?

Usually supersymmetry is explained at the level of the action of a quantum field theory, and there are two ways to go down from QFT to relativistic quantum mechanics: either a non-covariant way where ...
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Probability density of Klein-Gordon equation

This may, perhaps, stir some healthy debate; at least I am having some "fun" thinking about it, hopefully I can solicit some outside views too. It is often regarded that the Klein-Gordon equation ...
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132 views

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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How to show that higher derivative theories (mostly) breaks unitarity

How to show that higher derivative theories (mostly) breaks unitarity? Spinor field $\psi_{a_{1}...a_{n}\dot {b}_{1}..\dot {b}_{m}} $, which refer to the $\left( \frac{n}{2}, \frac{m}{2} \right)$ ...
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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}...
3
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2answers
165 views

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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94 views

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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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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118 views

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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342 views

What is the real space causal Green's function for the Klein-Gordon equation in 5 dimensions?

I want to solve the Klein-Gordon Green's function equation $$\left[\partial_\mu\partial^\mu + m^2\right]G(x, x') = \delta(x - x') $$ in 5 space-time dimensions where the boundary conditions on $G(x,x')...
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1answer
152 views

Why does the Klein-Gordon propagator $D(x)$ depend on the sign of $x^0$?

In A. Zee's Quantum filed theory in a nutshell, it says the Klein-Gordon propagator depend on the sign of $x^0$. Here $x=(x^0,x^1,x^2,x^3)$. $$D(x)=-i\int \frac{d^3k}{(2\pi)^32\omega_k}[e^{-i(\...
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545 views

What exactly goes wrong when using the Klein-Gordon equation to calculate the spectrum of hydrogen?

In many textbooks and lecture notes, it says that the Klein-Gordon equation was discarded first because when interpreting it as an equation for a single-particle wave function and trying to calculate ...
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173 views

Is a solution to the Klein-Gordon equation homeomorphic (or even diffeomorphic) to a solution of an equation with a different covariance group?

Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
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Is the phase velocity of plane wave solutions of the Klein-Gordon equation larger than $c$?

The phase velocity is given by $$ v= \frac{\omega}{k} \, .$$ Using the usual dispersion relation $$ E^2 = p^2c^2+ m^2c^4 \leftrightarrow \omega^2 \hbar^2= k^2\hbar^2 c^2 + m^2c^4$$ yields $$ v= \frac{\...
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WKB solution in QFT: classical action and particle vs antiparticle case

Consider the theory of a complex scalar field $$S[\psi, \psi^\dagger] = -\int d^4x \left(\hbar \partial_\mu \psi^\dagger \partial^\mu\psi + \hbar^{-1} m^2 |\psi|^2\right)$$ giving the Klein-Gordon ...
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86 views

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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124 views

Number operator and, annihilation and creation operators

While reading Ryder chapter on quantization of Klein-Gordon field I got stuck at the following: It can be shown that, $$[a(k),a^{\dagger}(k')]=(2\pi)^32\omega_k \delta^3(\mathbf{k}-\mathbf{k'})$$ and ...
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174 views

Derivation of Klein Gordon propagator by Generating Functional

I am studying Generating functionals in the path integral formulation of QFT. The formalism is done via Gaussian integrals which require the inverse of a function defined as: $\int A(x,z) A^{-1}(z,y)...
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63 views

Comparison of vacua and annihilation operators of Klein-Gordon theory and phi-fourth theory

The ground state or vacuum of an interacting theory is, in general, different from the ground state or vacuum of a free theory. In what cases are the two vacuums the same as each other? Can an ...
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75 views

Klein Gordon equation of fields via the definition of the time ordered product

My question is as follows: Consider that, $$ (-\partial_1^2+m^2)\langle 0|T(\phi(x_1)\phi(x_2))|0\rangle $$ Due to the definition of the time ordered product one can get: $$ (-\partial_1^2+m^2)\bigg\...
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286 views

Is it possible extend Schrodinger theory in relativistic contexts with naive consideration?

Preamble Let's consider a generic sinusoidal wave $\Psi (\mathbf{r},t) = A e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t + \phi)}$ and let's insert it into Schroedinger equation (please note that $ \...
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322 views

From Dirac to Klein-Gordon in curved spacetime

Is there an easy/elegant way of showing that "squaring" the Dirac equation in curved spacetime yields the Klein-Gordon equation, just like it happens in Minkowski space? A brute force approach would ...
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50 views

Conserved charge commutation relation under $SU(2)$ symmetry in two complex Klein-Gordon fields

I'm trying to show that conserved charges of two complex equal-mass Klein-Gordon fields under $SU(2)$ transformation fulfill the following commutation relation: $$ [Q^j, Q^k]=i\epsilon^{jkl}Q^l .$$ I ...
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48 views

Quantization of field with other complete orthogonal system

I've learned the quantization of Klein-Gordon field using Fourier expansion. I understand that this process is kind of exchanging complex fourier coefficients to operator and makes it satisfying the ...
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44 views

How can energy be negative for antiparticles in the solutions of Klein Gordon equation?

Although similar questions have been asked before I'm still confused. This is from Greiner, Relativistic Quantum Mechanics $E^2=c^2\sqrt{\vec{p}^2+m_0^2c^2}$ Consequently, there exist two possible ...
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1answer
37 views

Klein-Gordon equation propagators: intersection with the support of the source

Let $(M,g)$ be a globally hyperbolic. Let $P = \Box - m^2$ be the Klein-Gordon differential operator. Following Fewster's notes, we may define the retarded/advanced propagators $$E^\pm : C^\infty_0(M)\...
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87 views

Where this interpretation for the field modes comes from?

I'm reading the book "Modeling Black Hole Evaporation" by Alessandro Fabbri and Jose Navarro-Salas, and in section 3.3.2 they talk about wavepackets at $\mathscr{I}^+$. It all starts like this: one ...
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71 views

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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1answer
68 views

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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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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43 views

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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54 views

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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1answer
125 views

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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227 views

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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1answer
388 views

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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83 views

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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158 views

Free complex scalar field - showing operators are creation and annihlation?

Using the method of canonical quantization we can show that for a free scalar field we have: $$\phi(x)=\int d\tilde p (a(\vec p) e^{-ipx}+b^\dagger(\vec p) e^{ipx})$$ where $a(\vec p)$ and $b^\dagger(\...
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504 views

Gauge Invariance for the Klein Gordon Equation

Maybe it's the four vector notation that is throwing me off, but can someone explain why after substituting the primed quantities in and applying the four-momentum to the phase term the four momentum ...
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142 views

Hadamard expansion of interacting Klein Gordon 2-point function

Context: There is an algorithm due to Hadamard, I believe, for constructing local bi-distributional solutions to elliptic and hyperbolic equations for the purpose of proving existence and uniqueness ...
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2answers
616 views

Hamiltonian of Klein-Gordon Field

The Hamiltonian of the Klein-Gordon Field may be written $$H=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{2\omega_{\mathbf{p}}}\omega_{\mathbf{p}}\left(a^{\dagger}(p)a(p)+\frac{1}{2}(2\pi)^{3}2\omega_{\...
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Intuition about this derivation on QFT

I've found on nLab this post on Wightman axioms which in particular contains a nice example about the quantization of the Klein-Gordon Field. This is a remarkably clean approach from the point of view ...
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Divergences in the Klein Gordon equation?

Question Where are the UV and Infrared divergences present in the Klein Gordon equation: $$ \hat E^2 |\psi> = (\hat p^2 + m^2)|\psi> $$ Before diagonalizing the operators (substituting with ...
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476 views

Green's function for the Klein-Gordon operator

I'm not very sure I get the right way of verifying Feynman's propagator is indeed the Green's function for KG operator. The propagator is $$\Delta_F(x-y)=\int \frac{d^4p}{(2\pi)^4}\frac{ie^{ip\cdot(x-...
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382 views

Simplest Hamiltonian for 2d wave equation with periodic boundary conditions

I want to use the 2d wave equation ($\frac{\partial^2u}{\partial t^2}=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}$) with periodic boundary conditions as a simple toy model of a ...
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186 views

QFT basics for Klein-Gordon fields

I am teaching myself QFT from Peskin for next years maths course and I have two questions: What is a c-number? Is it a complex number, and if so why does it mean, $[\hat{\phi}(x),\hat{\phi}(y)]~=~<...
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Can local rotations lead to a gauge theory?

I was reading about the relation between electromagnetism and the complex Klein-Gordon field. The KG field had a global $U(1)$ symmetry and upon demanding that even a local phase transformation must ...