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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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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Interpretation Klein-Gordon 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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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 ...
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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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Scalar QFT Fock Space

I want to demostrate the following relation of the normal ordered product: $\Omega\equiv:\exp{\left(-\int d^3k~a^{\dagger}(k)a(k)\right)}:=|0\rangle\langle0|.$ I proved the commutation relation ...
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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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Klein-Gordon propagator integral

DISCLAIMER: I edited the question to include the integrals and the picture I'm stuyding Klein-Gordon field in Peskin & Schroeder An Introduction to Quantum Field Theory There is an integral that ...
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Non-relativistic limit of complex scalar field

In page 42 of David Tong's lectures on Quantum Field Theory, he says that one can also derive the Schrödinger Lagrangian by taking the non-relativistic limit of the (complex?) scalar field Lagrangian. ...
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Deriving the Hamiltonian density for a free scalar field

I'm working through my old notes on QFT (cf. Ref 1) and I'm not quite sure how to approach the derivation of the Hamiltonian density for a free scalar field (question 2.3 on page 19) and the ...
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Problem of Klein-gordon Equation

Ryder in his QFT book writes in eqn (2.20): Probability density, $\rho = \frac{i\hbar}{2m}(\phi^*\frac{\partial \phi}{\partial t} - \phi \frac{\partial \phi^*}{\partial t})$ Then in the next ...
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What happens to the wave function after applying the D'Alembert operator?

Is the result of applying the D'Alembert operator on a wave function always zero? Or are there exceptions?
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Noether current for a local gauge transformation for the Klein-Gordon Lagrangian

The Noether current corresponding to the transformation $\phi \to e^{i\alpha} \phi$ for the Klein-Gordon Lagrangian density $$\mathcal{L}~=~|\partial_{\mu}\phi|^2 -m^2 |\phi|^2$$ by finding ...
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Solving Klein-Gordon equation in the Rindler coordinates - the Unruh effect

I am reading 't Hooft's notes on Black Holes. I want to find the solutions of the Klein-Gordon equation $(\tilde{x},\tilde{y}, \rho, \tau)$ in the Rindler coordinates which are $$x=\tilde{x}\,\,\,\,\ ...
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Diagonalizing/eigenvalues of the infinite dimensional matrix of N harmonic oscillators on a ring

I have trying to show that the continuum limit of N quantum harmonic oscillators gives rise the the klein-gordon field. However, instead of a usual finite string, I want to do it on a ring. Hence, my ...
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Difference between vector and pseudo-scalar

In physics, a pseudo-scalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as improper rotations while a true scalar does not. Can someone show me ...
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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, ...
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The Klein–Gordon equation

As we know that the Schrödinger equation presents basis of Quantum Mechanics and analogy with Newton second law in Classical Mechanics, I thought that relativistic interpretation of Schrödinger ...
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Lorentz invariance of positive energy solutions to the Klein-Gordon equation

I am reading Arthur Jaffe's Introduction to Quantum Field Theory. (You can find it here.) There is an interesting question posed in Exercise 2.5.1: Solutions to the Klein-Gordon equation propagate ...
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Lagrangian formulation for relativistic quantum fields

The Lagrangian for a real scalar field is: $$\mathcal{L}=\frac{1}{2}\eta^{\mu \nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}m^2\phi^2 $$ How can I derive the dynamics of this field from this ...
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Annihilation and creation operator - $\phi$ and $\pi$ for Klein-Gordon Field

Introduction and Notation Let $\phi(\vec{x})$ be the real Klein-Gordon (quantum) field, written as: ...
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Klein-Gordon Canonical Commutation Relation (CCR)

In the complex Klein-Gordon field we regard as dynamical variables the field $\phi$, the complex conjugate of the field $\phi^*$, and the momenta $\pi$, $\pi^*$. I can't see how should arise the ...
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Regarding real field Klein Gordon Equations

Here are 2 doubts: If we change the sign of the mass term in the free massive KG Lagrangian to get: $L = \frac{1}{2}\partial^\mu\phi\partial_\mu\phi + \frac{1}{2}m^2\phi^2$, What would be the ...