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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Dirac vs KG propagation amplitude

Can someone explain to me the physical meaning of $\bar{\psi}=\psi^\dagger\gamma^0$ in the Dirac equation? I understand it is obtained as one of the solutions of Dirac equation and it is used to build ...
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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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Are the quasinormal modes of a black hole actually Fourier modes?

The quasisnormal modes of charged black holes coupled to a charged massive scalar field is such that the equation of motion for the scalar field is the Klein-Gordon equation $$[\nabla^{\nu}-iqA^{\nu}]...
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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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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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Why does the annihilation operator acting on the ground state in Quantum Field Theory gives a zero?

One of the main motivations for Quantum Field Theory after Dirac Equation is that the Dirac equation predicts negative energy states which leads to the ground state being unbounded which ultimately ...
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Mass dependence of the Euclidean Klein-Gordon propagator

Consider the Euclidean propagator \begin{equation} \Delta_E (x_1-x_2)\equiv \int \frac{d^4p}{(2\pi)^4} \frac{e^{ip\cdot(x_1-x_2)}}{p^2+m^2}. \end{equation} I am a bit confused as to whether the 4-...
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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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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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Making sense of QFT

I don't get what is it we're trying to do in QFT. I'm currently in the beginning of the course and a clear picture of what we're trying to achiever hasn't been painted yet to me. From what I've been ...
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Eigenvalues of the Klein-Gordon operator

If I've understood what I've read correctly, the eigenvalues of the Klein-Gordon (KG) operator $\Box+m^{2}$ are $-p^{2}+m^{2}$, but how does one show this? Naively I assumed that the eigenfunctions ...
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Can the necessity of using anti-commutators for Dirac fields and commutator for Klein-Gorden be deduced from the field equations?

We all learned to use the commutator for quantizing the KG field and the anti-commutator for the Dirac field. We are told (which is correct) so that KG-excitations are bosons and Dirac-excitations ...
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$\phi^{4}$ theory

Consider a scalar field theory with a $\phi^{4}$ interaction term $$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4},$$ where $\lambda\ll 1$. I am ...
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How to find Bilinears of a theory?

I'm trying to understand how one finds the bilinears of a given theory. In most litterature the bilinears are not really derived but rather taken as fundamental. The dirac bilinears are of course: $$\...
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QFT: expanding the propagator in terms of Minkowski modes

I'm trying to understand the usual "Fourier transform" of the free scalar propagator $ G(x,y) = \int \frac{d^{4}k}{(2\pi)^{4}} \frac{1}{\omega_{\mathbf{k}}^{2} + k^{2}} e^{i k \cdot (x - y)}$. I'd ...
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How to get to negative-momentum solutions from negative-energy solutions of the Klein-Gordon equation?

I'm trying to derive the general solution to the (complex, classical) Klein-Gordon equation: $$ \phi = \int \frac{d^3k}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{k}}} \left(B(k)e^{ikx} + C(k)e^{-ikx}\right). ...
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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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Are all quantum scalar fields related to the Klein-Gordon field?

In the book "Quantum Field theory and the Standard Model" by Matthew Schwartz, the author states: In quantum field theory, we generally work in the Heisenberg picture, where all time dependence is ...
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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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Second quantization and Klein Gordon equation

This is what I understood from Klein Gordon equation : We start from $$E^2=p^2+m^2.$$ We quantize it replacing $E \rightarrow \partial_t$, $p \rightarrow -ih\nabla$, $m \rightarrow m$ Thus, we get ...
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How to fill the gaps in this QFT construction?

I've seem in several books and lecture notes the quantization of the free KG field, and perhaps because I'm a kind of person that feels umconfortable with "hand waving" constructions, I still feel the ...
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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}-\...
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Klein Gordon inner product

Define the Klein Gordon inner product as $$(\psi_1,\psi_2)_{KG} = i\int d^3x \, \psi_1^*\,\partial{t}\,(\psi_2) - \partial{t}\,(\psi_1^*)\,\psi_2 \, .$$ It can be shown that for the one particle ...
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Interpretation of the conserved current in classic Klein-Gordon and Dirac equations

The conserved current in KG is $$j^{\mu}=i(\phi^*\partial^{\mu}\phi-\phi\partial^{\mu}\phi^*) =2p^{\mu}|N|^2$$ where N is a normalization factor. This current can't be understood as a probability ...
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Klein gordon field and positive/negative energy solutions

I my course we calculated the Klein-Gordon field: $$ \phi(x)= \int \frac{d^3k}{(2 \pi)^3}\frac{1}{2k_0} ~ \left[a(\vec{k})e^{-ik.x}+b^*(\vec{k})e^{i kx}\right]$$ We said that the part $ a(\vec{k})e^{...
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477 views

Klein Gordon Field Quantization: why this is the correct way to express the field?

I'm reading a book in QFT and the first thing tackled is the quantization of the Klein Gordon Field. The classical Klein Gordon field satisfies the partial differential equation $$(\partial^\mu\...
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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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Klein Gordon equation in Schwarzschild spacetime (spherical harmonic mode expansion)

My Question: In his GR text, Robert Wald claims that solutions, $\phi$, to the Klein-Gordon equation, $\nabla_a\nabla^a \phi = m^2 \phi$, in Schwarzschild spacetime can be expanded in modes of the ...
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Fourier transform in Klein-Gordon equation [duplicate]

I'm Solving (or at least trying to solve) 4 Klein Gordon equation with sources (Maxwell equations) with the Green function method, therefore I have $$ \Box G(x,x')= \delta^4(x-x') \tag{1}$$ Where $G$...
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159 views

Do the particle and anti-particle solutions of the Klein-Gordon equation live in different Hilbert spaces?

Do the particle and anti-particle solutions of the Klein-Gordon equation live in different Hilbert spaces? Our professor said that this is true because the integral of their respective probability ...
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Is the conserved current probability or charge in Klein-Gordon and Dirac equations?

The conserved currents in KG and Dirac are: K-G : $j^{\mu}=i(\phi^*\partial^{\mu}\phi-\phi\partial^{\mu}\phi^*)$ Dirac: $j^{\mu}=\bar{\psi}\gamma^{\mu}\psi$ $j^0$ is positive definite in Dirac's ...
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Misconceptions about the Klein-Gordon and Dirac equations and their problems [duplicate]

In my course on particle physics the KG and Dirac equations were explained in an historical way. However, it was more confusing than clarifying. The problems with the KG equation were The probability ...
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Interacting Fields in QFT

I am trying to work through Peskin and Schröder and am a little stuck in Chapter 4 [section 4.2 p. 83 below eq. (4.13)], when he first treats interacting fields. The subject is the quartic interaction ...
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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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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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The Klein-Gordon field

The Klein-Gordon field is said to be "scalar". But when we write an expansion of it in terms of creation and annihilation operators or when it and its conjugate field satisfy a commutation relation, ...
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What is the significance of $\hbar$ appearing in classical equation of motion?

Books on QFT treats, any quantum field as quantized classical fields. For example, the Klein-Gordon field is first treated as a classical field $\phi(x)$ obeying classical Euler-Lagrange equation $$\...
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On Schrödinger equation and $U= 2mV$ potential

If we consider $\hbar=c=1$ we can write stationnary Schrödinger equation for a free particle as $$ \left(\frac{k^2}{2m} + V - \frac{k_0^2}{2m}\right)\Psi = 0$$ I see often the use of a reduced ...
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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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Lorentz invariant integration measure and Heaviside step function

I'm currently studying Klein-Gordon fields and I ran onto the concept of the Lorentz invariant integration measure, namely: \begin{equation} \frac{d^3k}{(2\pi)^32E_k} \end{equation} where $E_k=\sqrt{\...
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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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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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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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Show that this propagator is NOT Lorentz Invariant

For the free Klein-Gordon theory: If I change the definition of time-ordering such that: $\mathcal{T} A(x)B(y) = A(x) B(y) \Theta( x^{0} - y^{0} ) - B(y) A(x) \Theta(y^{0}-x^{0})$ In the above $\...
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Creation and Annihilation operators of negative momenta in QFT

I'm studying the free Klein-Gordon field, $\phi$. You can write this field as: $\phi(\mathbf{x}) = \int \frac{ d^{3} \mathbf{p} }{(2\pi)^{3}} \frac{1}{\sqrt{ 2 E_{\mathbf{p}} }} \left[ a_{\mathbf{p}} ...
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Discrepancy in the form of the free Klein-Gordon field

When solving for the Klein-Gordon field $\phi$, most texts and online resources that I look at say that: $$\phi(x) = \int \frac{ d^{3} p }{ ( 2 \pi )^{3} } \frac{1}{\sqrt{ 2 E_{\mathbf{p}} }} \left[ ...
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409 views

Klein-Gordon field $\phi$ and a Lorentz transformation

I'm supposed to consider a Lorentz transformation of the form $$\Lambda^{\mu}_{\ \nu} = \delta^{\mu}_{\ \nu} + \omega^{\mu}_{\ \nu},$$ where $\omega$ is some tensor. Since Lorentz tranformations ...
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How precisely the Klein-Gordon equation is derived?

In various articles (and books) such as the wiki article of the Klein-Gordon equation wrote: "The Klein-Gordon equation is a "quantized" version of the relativistic energy-momentum relation"; In ...
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753 views

How to explain the field result of Klein Gordon field in QFT?

I derived the equations of Klein Gordon field, and I find a statement like this: In quantum field theory, the wave functions that could have had both positive and negative probabilities are used as ...
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333 views

Notation with covariant/contravariant derivative with product rule

I have a question that should be rather simple, but i simply can not find enough information about it. I have searched and found a lot of related material, but not exactly like my problem. I am ...

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