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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Schrodinger equation from Klein-Gordon?

One can view QM as a 1+0 dimensional QFT, fields are only depending on time and so are only called operators, and I know a way to derive Schrodinger's equation from Klein-Gordon's one. Assuming a ...
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A question on using Fourier decomposition to solve the Klein Gordon equation

Given the Klein Gordon equation $$\left(\Box +m^{2}\right)\phi(t,\mathbf{x})=0$$ it is possible to find a solution $\phi(t,\mathbf{x})$ by carrying out a Fourier decomposition of the scalar field $\...
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Localized field quanta?

After the canonical quantization of the Klein-Gordon field (for example), we interpret the quantum of excitations of the fields with definite energy and momentum as particles. But our mental image of ...
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What is the relationship between vibration of the field and quantum fluctuation?

Consider a free field like the KG equation. I see that why $$\tilde \phi(\mathbf{p},t)$$ a momentum-dependent quantity, is an oscillator, vibrating at a frequency because when we apply the Fourier ...
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single-particle wavepackets in QFT and position measurement

Consider a scalar field $\phi$ described by the Klein-Gordon Lagrangian density $L = \frac{1}{2}\partial_\mu \phi^\ast\partial^\mu \phi - \frac{1}{2} m^2 \phi^\ast\phi$. As written in every graduate ...
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Why is there a minus sign in this wave equation derivation?

My book on quantum mechanics suggests a derivation of the wave equation $$\left(\Delta - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right) \psi(\bar{r},t) = 0$$ from the photon energy-impulse ...
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Negative probabilities in quantum physics

Negative probabilities are naturally found in the Wigner function (both the original and its discrete variants), the Klein paradox (where it is an artifact of using a one-particle theory) and the ...
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How to derive the theory of quantum mechanics from quantum field theory?

I have read the book on quantum field theory for some time, but I still do not get the physics underline those tedious calculations. The thing confused me most is how quantum mechanics relates to ...
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Why would Klein-Gordon describe spin-0 scalar field while Dirac describe spin-1/2?

The derivation of both Klein-Gordon equation and Dirac equation is due the need of quantum mechanics (or to say more correctly, quantum field theory) to adhere to special relativity. However, excpet ...
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Lorentz transformation of classical Klein–Gordon field

I'm trying to see that the invariance of the Klein–Gordon field implies that the Fourier coefficients $a(\mathbf{k})$ transform like scalars: $a'(\Lambda\mathbf{k})=a(\mathbf{k}).$ Starting from the ...
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Bessel function representation of spacelike KG propagator

Preliminaries: In their QFT text, Peskin and Schroeder give the KG propagator (eq. 2.50) $$ D(x-y)\equiv<0|\phi(x)\phi(y)|0> = \int\frac{d^3p}{(2\pi)^3}\frac{1}{2\omega_\vec{p}}e^{-ip\cdot(x-y)...
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Noether's Theorem and scale invariance

Noether's theorem usually considers coordinate/field transformations which leave the Lagrangian invariant up to a divergence term, i.e. $$\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$$ ...
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Does relativistic quantum mechanics really violate causality?

The Hamiltonian $H=\sqrt{p^2+m^2}$ defines a one-particle quantum mechanics in the usual way. Let us call this theory RQM for short. Peskin and Schroeder claim that RQM violates causality because ...
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Delocalization in the square root version of Klein-Gordon equation

In this Wikipedia article a relativistic wave equation is derived using the Hamiltonian $$H=\sqrt{\textbf{p}^2 c^2 + m^2 c^4}$$ Substituting this into the Schrödinger equation gives the square root ...
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Where do the quantum fields encode the spin information?

I know basically the difference between Klein-Gordon and Dirac field is spin. But I am not sure where we need to implement this info. The solutions of both equations are the wave packets which ...
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Does the Lorentz invariance of equation of motion guarantee the Lorentz invariance of the solutions?

If I have a Lorentz invariant equation of motion, like Klein-Gordon equation, is the solution automatically guaranteed to be Lorentz invariant? I ask this question because of the discussion from Mark ...
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How does QFT interpret the Negative probability problem of the real scalar fields' Klein-Gordon equation?

I am totally a beginner in QFT, here's the problem that I got: for the real scalar fields, are there any elementary particles descriped by them. If so, how to understand the negative probability ...
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Why don't the De Broglie dispersion relation contain a constant term?

Wikipedia says that the dispersion relation for a non-relativistic particle is: $$ \omega = \frac{\hbar k^2}{2m}. $$ But when I tried to calculate it myself, I seem to get a constant term in that ...
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Why must the Dirac equation multiplied by its complex conjugate give the KG equation?

This may be a simple question. I can show this is the case mathematically but cannot explain why it happens. It was only when asked why this happens when I realised I couldn't explain it intuitively/...
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Difference between field and wavefunction

Can someone give me a clear explanation of what is the difference between a classical field, a wave function of a particle and a quantum field? I haven't find a clear explanation. For example for ...
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Wavefunction in quantum mechanics and locality

Every wavefunction of a form $\Psi(x)$ can be described as a superposition of multiple free particle solutions. We can see the following Fourier transform: $$ \psi(x) = \int e^{ik\cdot x} \psi(k) dk $...
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Solving the Klein-Gordon equation via Fourier transform

I have been writing a personal set of notes on QFT and I'm currently writing up a section on solving the Klein-Gordon (K-G) equation. I many texts that I've read, the author starts by expressing the ...
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Regarding a small step in the derivation of the LSZ formula

I'd like to prove the LSZ formula, but there is a specific step that is bugging me a lot. I know there are many subtleties in its derivation, but I'm not worrying about this right now: I'm trying to ...
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Why can't the Klein-Gordon equation explain the hydrogen atom but the Dirac equation does?

Why can't the Klein-Gordon equation with a Couloumb potential describe the hydrogen atom? Why can the first order Dirac equation explain it? What are the failures?
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Quantization of a free field: Klein-Gordon case

I am a beginner and reading this course text on QFT. The author first introduces the KG equation: $$\partial_\mu\partial^{\mu}\phi+m^2\phi=0$$ [with Minkowski signature $(+,-,-,-)$]. Then the ...
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What would change if we add a linear term in the Klein-Gordon Lagrangian?

The usual Klein-Gordon Lagrangian reads \begin{equation} \mathscr{L}= \frac{1}{2}( \partial _{\mu} \Phi \partial ^{\mu} \Phi -m^2 \Phi^2) \, . \tag1\end{equation} Without additional symmetry ...
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Why use Fourier expansion in Quantum Field Theory?

I have just begun studying quantum field theory and am following the book by Peskin and Schroeder for that. So while quantising the Klein Gordon field, we Fourier expand the field and then work only ...
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What's wrong with the square root version of the Klein-Gordon equation?

The Wikipedia article has a derivation of the Klein-Gordon equation. It gets to this step: $$\sqrt{\textbf{p}^2 c^2 + m^2 c^4} = E$$ and inserts the QM operators to get $$\left( \sqrt{ (-i \hbar \...
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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(\...
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Deriving Schrodinger equation from Klein-Gordon QFT with the definition $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$

In the book "Quantum Field theory and the Standard Model" by Matthew Schwartz, page 23-24, the position space wavefunction is defined as $$\psi(x)=\langle 0|\phi(x)|\psi\rangle, \tag{2.82+2.83}$$ ...
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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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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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Equation for relativistic electron and two-component spinor

Recently I heard that there is some "alternate" equation for the Dirac one. It can be introduced if we refuse some properties of the theory describes the electron, which Dirac used in his original ...
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Is Maxwell's field the wave function of the photon?

In his ArXiv paper What is Quantum Field Theory, and What Did We Think It Is? Weinberg states on page 2: In fact, it was quite soon after the Born–Heisenberg–Jordan paper of 1926 that the idea ...
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Contradictory result for scalar-field propagator from Feynman rules and LSZ formula

I am trying to learn how to calculate scattering amplitudes in a Klein-Gordon theory. I am getting stuck with the simplest of the examples: $\phi\to\phi$ in a free scalar-field theory. If I calculate ...
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Evaluating propagator without the epsilon trick

Consider the Klein–Gordon equation and its propagator: $$G(x,y) = \frac{1}{(2\pi)^4}\int d^4 p \frac{e^{-i p.(x-y)}}{p^2 - m^2} \; .$$ I'd like to see a method of evaluating explicit form of $G$ ...
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Is the Dirac equation equivalent to the Klein-Gordon equation for its left handed component?

The Dirac equation $$(i\gamma^a\partial_a - m)\psi=0\tag{0}$$ is given by a first order operator acting on a Dirac spinor, which is the direct sum of a left handed spinor and a right handed spinor. ...
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Fourier Transforming the Klein Gordon Equation

Starting with the Klein Gordon in position space, \begin{align*} \left(\frac{\partial^2}{\partial t^2} - \nabla^2+m^2\right)\phi(\mathbf{x},t) = 0 \end{align*} And using the Fourier Transform: $\...
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Quantizing a complex Klein-Gordon Field: Why are there two types of excitations?

In most references I've seen (see, for example, Peskin and Schroeder problem 2.2, or section 2.5 here), one constructs the field operator $\hat{\phi}$ for the complex Klein-Gordon field as follows: ...
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Why can the Klein-Gordon field be Fourier expanded in terms of ladder operators?

Using the plane wave ansatz $$\phi(x) = e^{ik_\mu x^\mu}$$ the solution to the Klein-Gordon equation $(\Box + m^2) \phi(x) =0$ can be written as a sum of solutions, since the equation is linear and ...
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How to obtain the explicit form of Green's function of the Klein-Gordon equation?

The definition of the green's function for the Klein-Gordon equation reads: $$ (\partial_t^2-\nabla^2+m^2)G(\vec{x},t)=-\delta(t)\delta(\vec{x}) $$ According to these resources: Green's function ...
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What is the Interpretation of This Field Operator: $\vec{X} = \int \pi \vec{x} \phi \operatorname{d}^3 x$?

In scalar quantum field theory the field momentum operator is constructed from the canonical field operators, $\phi$ and $\pi$, in the equation: $$P_j = -\int \pi \partial_j \phi \operatorname{d}^3x.$$...
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Can't the Negative Probabilities of Klein-Gordon Equation be Avoided?

I came across these notes of Dyson on Relativistic Quantum Mechanics. There on p. 3, he mentions that the issue with the Klein-Gordon equation is that the only way to relate $\psi$ with a probability ...
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Green's function for the inhomogenous Klein-Gordon equation

I'm trying to solve the massive Klein-Gordon equation in good old Minkowski space-time: $$(\square + m^2) \phi = \rho(t,\mathbf{x})$$ where $\square = \partial_{\mu} \partial^{\mu} = \partial_{t}^2 - \...
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What is $\phi(x)|0\rangle$?

Suppose for instance that $\phi$ is the real Klein-Gordon field. As I understand it, $a^\dagger(k)|0\rangle=|k\rangle$ represents the state of a particle with momentum $k\,.$ I also learned that $\phi^...
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Can the Klein-Gordon Equation represent Particles with non-zero spin?

Every Solution of the Dirac Equation is also a solution of the Klein-Gordon equation. So the K-G equation does not necessarily represent particles with non-zero spin. Would it be incorrect to ...
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Propagator and probability amplitude that a particle propagates

My QFT knowledge has very much rusted and i got confused by these few lines from Peskin and Schroeder: p.27: " [..] the amplitude for a particle to propagate from $y$ to $x$ is $\langle 0| \phi(x) \...
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fitting free QFTs into the Haag-Kastler algebraic formulation

Has the free Klein-Gordon quantum field theory been fitted into the Haag-Kastler algebraic framework? (Actually, John Baez told me "yes", and he should know.) If so, can you describe the basic ...
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Massless limit of the Klein-Gordon propagator

I am working with the propagator associated to the Klein-Gordon equation, as derived in "Quantum Physics a functional integral point of view", James Glimm, Arthur Jaffe or as derived here: http://www....
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Klein-Gordon inner product

Studying the scalar field and Klein-Gordon equation in quantum field theory I came across this definition for the inner product in the space of the solutions of the K.G. equation: $$\langle \Phi_1 | \...