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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Hamiltonian density of classical Klein-Gordon field

I am working my way through Peskin and Schroeder section 2.2 and trying to show that $T^{00}$ is equivalent to the expression $\frac{1}{2}\pi^2-\frac{1}{2}(\nabla \phi)^2-\frac{1}{2}m^2\phi^2$ in ...
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Is there a scalar field that is not a lorentz scalar if we begin with Lorentz invariant Lagrangian?

In Quantum Field Theory by Mark Srednicki chapter 3 and 4, he constructs Lorentz invariant theory for scalar field by assuming that the scalar field transforms by ...
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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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Every Relativistic Field Satifies the Klein-Gordon Equation?

I've read that every relativistic scalar field (and in some sense, any field) satisfies the Klein-Gordon equation. Is the reasoning for this just based on the quantum mechanical substitution of $E\to ...
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Why is the Klein Gordon equation of second order in time?

I was wondering if there is any way to interpret the fact that the Klein Gordon equation is a 2nd order PDE in time. I mean, normally you would expect that as soon as you fix the initial wavefunction, ...
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Transformation of the scalar field under Lorentz Boost [closed]

Assume a Lorentz transformation $\Lambda$ is to be implemented as the unitary operator $U(\Lambda)$ in the Hilbert space of quantum states of the Fock representation upon which the scalar Klein-Gordon ...
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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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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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Klein Gordon for spin-1 particle photon

If Klein Gordon equation is for spin-0 particles, I write massless fields as $\square A=0$, how can I say $A_\mu=\epsilon^\mu e^{-ikx}$ as a wave function of polarized photon (spin-1) ?
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How to derive the conserved current of the Klein Gordon equation?

Similarly to the probability current in non-relativistic quantum mechanics, there is a conserved current for the Klein Gordon equation, however a different one. I'm trying to calculate that. The KG ...
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47 views

Relation between $a_{-p}$ and $a^{\dagger}$? for real scalar field

Fast question. Consider the real Klein-Gordon field. Is there a way to relate $a_{-p}$ with $a_p^{\dagger}$?
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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 Lagrangian of electromagnetism with Lorenz Gauge evolve Klein Gordon equation?

Simply Lagrangian without a source for Maxwell equation is $$ L = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} $$ Also Lorenz Gauge condition is $$ \partial_{\mu}A^{\mu}=0 $$ and if so I can briefly add this ...
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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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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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Lagrangians with higher derivatives than Klein-Gordon [duplicate]

Has anyone ever tried to work with Lagrangians involving higher derivatives? The Klein-Gordon Lagrangian only involves $(\frac{\partial}{\partial t})^2$ and $\nabla^2$ terms, what about third and ...
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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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Interpretation of the four-vector $k$ in scalar QFT

I'm studying the canonical quantization of the Klein-Gordon real scalar quantum field theory, given by the classical Lagrangian density $$\mathscr L = {1\over ...
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breitenlohner freedman stability condition

I am looking for a simple way to derive the breitenlohner-freedman bound. Actually I can't understand why we have stability above the BF bound and instability below the BF bound,while both have ...
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Question about the foundation of part I in A. Zee's book

Zee says in Section I.3 of QFT in a nutshell: The functional integral $$Z = \int D \varphi e^{i \int d^4 x [\frac{1}{2} (\partial \varphi)^2 - V(\varphi) + J(x) \varphi (x)]} \tag{11} $$ is ...
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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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Quantum field theory: field operators in terms of creation/annihilation operators

I am learning Quantum Field Theory and there is a step in my notes that I do not really understand. It starts with the classical definitions of position $q$ and momentum $p$: $$ q = ...
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How is the integrand concluded to be identically zero?

In expanding the classical Klein-Gordon field in Fourier space to write it in terms of $\phi(\mathbf{p})$ instead of $\phi(\mathbf{x})$, I reached the following result. $$\int ...
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Can a relativistic quantum particle be completely confined into a finite hole?

If we write the Klein-Gordon equation in this form \begin{equation*} c^2 \hbar^2 \nabla^2 \Psi = \hbar^2 \ddot{\Psi} + 2i\hbar (U - mc^2) \dot{\Psi} + U (2mc^2 - U) \Psi \end{equation*} we have a ...
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Is it true that the Schrödinger equation only applies to spin-1/2 particles?

I recently came across a claim that the Schrödinger equation only describes spin-1/2 particles. Is this true? I realize that the question may be ill-posed as some would consider the general ...
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First-order and second-order wave equations, versus the uncertainty principle

In classical physics, we have second-order equations like Newton's laws, so we need to specify both position (zeroth order) and velocity (first order) of a particle as initial conditions, in order to ...
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Solution to Klein-Gordon equation

I have a sound grounding on ODE's, not that much on PDE's, i've read many books on QFT and most if not all come to the conclusion that the solution to the Klein-Gordon equation ...
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Fourier expansion of the Klein-Gordon field

Is there a reason(both physical and mathematical) why the Klein-Gordon field is represented as a fourier expansion in the second quantization as opposed to other mathematical expansions? Be gentle ...
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What is a superfluid in field theoretic terms?

I'm wondering how one precisely defines a superfluid in terms of the effective field theory description. In Nicolis's paper http://arxiv.org/abs/1108.2513 there seems to be an extremely simple ...
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Another Power Counting/ mass dimension question

Are the mass dimension of the Dirac field different from those of the Klein-Gordon field, or is this just another issue of "cannonical normalization?" For instance if $\mathcal{L}_{KG}=\int ...
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Naive unification of scalar QFT and GR is possible?

I am thinking on the Klein-Gordon equation with curved (non-diagonal) metrics. Is it possible? Doesn't have it some inherent contradiction? If yes, what? If no, what is this combined formula?
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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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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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Fourier Coefficents in general solution to Klein-Gordon Dirac-equation?

The most general solution to the Klein-Gordon equation is written as \begin{equation} \Phi(x)= \int \mathrm{d }k^3 \frac{1}{(2\pi)^3 2\omega_k} \left( a(k){\mathrm{e }}^{ -i(k x)} + a^\dagger(k) ...
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Free Field theory to Interacting Field theory

Free field theory: Why is it said that different Fourier modes in case of a free field (say, real Klein-Gordon field) are independent of each other? Interacting field theory: How exactly does the ...
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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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wave functions of Klein Gordon particles

It is known that the Klein Gordon equation does not admit a positive definite conserved probability density. Nonetheless, in Wikipedia (for example), you can read that with the $\textit{appropriate}$ ...
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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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What do people mean by gauge invariance of the normalization of field?

Lets have the scalar Klein-Gordon field interacting with EM field: $$ L = \partial_{\mu}\varphi \partial^{\mu}\varphi - m^2\varphi \varphi^{*} - j_{\mu}A^{\mu} + q^{2} A_{\mu}A^{\mu}\varphi ...
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Nonlinear Klein Gordon equation

For the Klein Gordon nonlinear equation, $$ u_{tt}- \Delta u +f(u)=0,$$ how could I use Noether's theorem to prove that there is a conserved quantity? I.e., $$ (\Pi _{k} )_{t} - \rm div(j_{k})=0 $$ ...
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How to think about antiparticles in KG equation?

I am a beginner to study QFT and have a problem. I know, in Dirac equation, thanking to the Pauli exclusion principle and believing that the vacuume is the state that all the negative energy states ...
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Non-Locality of Space - QFT (Srednicki's book)

I was going through Mark Srednicki's book on QFT. It says in the relativistic limit the Schrodinger equation becomes something like : $$ i\hbar\frac{\partial}{\partial t} \psi(\vec x,t) = ...
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What are the problems in trying to interpret the Klein-Gordon equation as a single particle equation?

What is the problem if we try to interpret KG equation as a single particle equation? Also I wish to know whether the born interpretation of wavefunction is applicable in relativistic quantum ...
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363 views

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 ...
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Regarding state of Klein-Gordon field

In regular quantum mechanics of particles, I have the Schrodinger evolution picture for a general state $$ i\hbar \frac{d}{dt} \left|\psi(t)\right> = \hat H \left|\psi(t)\right> $$ then we ...
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Parity of annihilation and creation operator - Real Klein-Gordon field

In calculating the total-momentum operator of the real Klein-Gordon field, I end up with an equation like $$ \vec P = \frac{1}{(2\pi)^3}\int d^3p \mspace{9mu} \vec p\Big(a_pa_{-p} + ...
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Why aren't four-vectors used in the definition of a Klein-Gordon quantum field?

I am a beginner who is learning QFT. When I was going through the quantisation of Klein-Gordon real-field. I got confused about something: The solution to Klein-Gordon equations are of the form $ ...
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Second quantization of Klein Gordon field

Does the second quantization of the Klein Gordon field which involves using the harmonic oscillator paradigm ultimately lead to the conclusion that electromagnetic field is nothing but photons(bosons) ...
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Analog of Klein-Gordon equation from Proca action

What would be the general form of Lagrange Equation when instead of a scalar field we have a vector potential? has anyone derived the klein gordon equation for a corresponding vector potential ...
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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 ...