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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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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Quantization of complex scalar field

I'm learning Peskin's qft now and I'm a little confused about problem 2.2 . Suppose I write the field $\phi(x)$ as: $\phi(x) =\int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{p}}} (a_{p}e^{-ipx}+b_{p}e^...
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Commutator of canonical fields in Quantum Field Theory

Let $\phi(\vec{x},t)$ denote the canonical fields and $\pi(\vec{x},t)$ denote the canonical impulses where they're given by: \begin{equation} \phi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2\omega_{\...
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Quantization of the Klein-Gordon equation, sign problem

In Peskin and Schroeder, they quantize the Klein-Gordon field in the following way. They write the Fourier transform of $\phi(x,t)$ $$ \phi(x,t)=\int \frac{d^3 p}{(2\pi)^3}e^{ipx}\phi(p,t) $$ after ...
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Problem with Klein-Gordon equation derivation

In Notes for a course on Classical Fields by R. ALdrovandi, one the the exercises in page 94 is to derive the klein Gordon equation $(\Box + m²)\phi = 0$ from the following lagrangian density \begin{...
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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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Why does the Klein-Gordon propagator $D(x)$ depend on the sign of $x^0$?

In A. Zee's Quantum field theory in a nutshell p. 24, it says the Klein-Gordon propagator depends on the sign of $x^0$. Here $x=(x^0,x^1,x^2,x^3)$ with Minkowski sign convention $(+,-,-,-)$. $$D(x)=-...
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Separation of Klein-Gordon-/Dirac-equation (Bohmian-mechanics)

With the function $R{ e }^{ \frac { i }{ \hbar } S }$ one can separate the Schrödinger equation $$i \hbar \frac{\partial \psi}{\partial t}=\left(-\frac{\hbar^{2}}{2 m} \nabla^{2}+V\right) \psi$$ into ...
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Quick question on Deriving Klein–Gordon equation from Dirac equation

On page 172 of Schwatz’s QFT book, he derives the Klein–Gordon equation from Dirac equation as following: $$(i \not\partial +m) (i \not\partial -m)\psi=(-\frac{1}{2} \partial_\mu \partial_\nu {\gamma^...
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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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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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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}...
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Ladder operators in terms of Klein Gordon real field

an exercise asks me to explicit the functions ${a}(k)$ and $a^{*}(k)$ in term of the real field $\phi$ and its temporal derivate $\partial_0 \phi$ if the general solution of the Klein-Gordon equation ...
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What motivates the definition of the Klein-Gordon Inner Product? [duplicate]

I am following along Marc Casal's lecture slides "Quantum Field Theory in Curved Spacetime". For scalar functions $f$ and $g$ we define the Klein-Gordon inner product as follows: $$ <f,g> \ = \ ...
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Continuous plane wave solutions to Klein Gordon Field Equation

The continuous plane wave solution to the Klein Gordon Field Equation can be written as $\phi(x) = \int\frac{d^3\vec{k}}{\sqrt{2(2\pi)^3w_\vec{k}}} a(\vec k) e^{-ikx} + \int\frac{d^3\vec{k}}{\sqrt{2(...
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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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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 | \...
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How can a Dirac field $\psi$ satisfy KG equation? What's wrong in my derivation?

The covariant form Dirac equation $(i\gamma^\mu\partial_\mu-m)\psi(x)=0$ can be multiplied from the left with the operator $(i\gamma^\nu\partial_\nu+m)$ and 4xpanding it out, to get $$(i\gamma^\nu\...
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Mass dimension of Klein Jordan field [closed]

I like to know about dimension of KG fields, Wikipedia searches don't give me a satisfied answer Can any one please help me?
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How can I find out the 'Klein-Gordon' equation by using Path Integral techniq? [closed]

I get the following relation from eqn. (6.52) in Sakurai(2nd edition) $$\Psi +\Delta t \frac{\partial \Psi}{\partial t}+\Delta t^2\frac{1}{2!}\frac{\partial^2\Psi}{\partial t^2}+...=\lim_{\Delta t\to ...
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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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Klein-Gordon Hamiltonian commutator with annihilation (creation) operator

Probably I'm missing something trivial here. When calculating a commutator of Klein Gordon Hamiltonian with annihilation/creation operator it seems that the operators are inserted under the integral, ...
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Interpretation of the quantum field in light of the interpretation of propagators

In Page 38 of David Tong's QFT notes and Page 27 (Chapter 2.4 The Klein-Gordon Field in Space-Time under the heading Causality) of Peskin and Schroeder's Introduction to Quantum Field Theory, the ...
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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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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 ...
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When can we quantise with ladder operators?

So I am now nearing the end of my first QFT course and in it we quantised the KleinGordon and Dirac fields using ladder operators, however this method seems very specific to these fields. We ...
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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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Is $\gamma_\mu \gamma^\mu$ a unit operator?

Is the term: $$γ^μ γ_μ$$ An identity matrix? Since,if we start with both the Dirac equation, $$(iγ^μ ∂_μ-m)Ѱ=0$$ We find that, $$iγ^μ ∂_μ=m$$ If we square both sides, we get, $$-γ^μ γ_μ∂^μ ∂_μ=m^{2}$$...
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How to write Lagrangian of field with vector and scalar potential

I am beginner of quantum field theory. I have a basic question. Lagrangian of a electron in the potential has the form of $${1\over2}m{\bf v}^2-e\Phi+e \bf v \cdot A$$ Lagrangian in quantum field ...
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Commutation relations following from quantization of a complex scalar field

As someone who has recently started doing QFT I have some (algebraic) confusion about the following derivation. Starting with the Lagrangian of a complex scalar field $$\mathcal{L} =\partial_\mu \psi^...
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Coupled Oscillator's Stiffness and speed of light

In Schwabl book (Advanced Quantum Mechanics) page 258, in his triumph to show the relation between the coupled oscillators and Klein-Gordon equation he finds the following relation which is the ...
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What's a physical interpretation of the two terms that appear in the mode expansion of solutions of the Klein-Gordon equation?

For simplicity, let's say we consider a real scalar field in a purely classical context. A general solution of the Klein-Gordon equation can be written as: \begin{align} \phi(x)&= \int \frac{{\...
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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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Mode Expansion in Klein-Gordon QFT

I have a confusion regarding the mode expansion of the Klein-Gordon field theory. I am following Peskin and Schroeder. My questions are about how we formally get to the expansion of the KG QFT in ...
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How can $⟨0|ϕ(x)|p⟩=e^{ip⋅x}$ be mathematically shown?

I was reading Peskin and Schroeder's quantum field theory and going through the book mathematically. Then I got stuck at one equation. Consider a single, non-interacting real scalar field. The book ...
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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 does spin influence the dynamics of quantum mechanical systems?

I have just been introduced to the Klein-Gordon Equation and the Dirac Equation for the first time. The way they were explained to me, these equations govern the (relativistic) evolution of spin-0 and ...
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Derivation of the Klein-Gordon solution via Fourier Transforms

I recently graduate with a bachelor's in physics, and I've been trying to take the next steps toward learning QFT. To this end, I have been working through Peskin and Schroeder's textbook step-by-step....
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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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Klein-Gordon equation with position-dependent mass [closed]

Does there exist a general solution for a differential equation like: $$\ddot{\phi}(x,t) - \partial^2_x\phi(x,t) + \phi(x,t)m^2(x) = 0,$$ where $m(x)$ is a known function.
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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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Expanding about background field

I refer to this set of lecture notes by Hugh Osborn, equation 4.184 on p.70. We expand an action $S[\phi]$ around a background field $\varphi(x) = \phi(x) -f(x)$ If we expand the action $S[\phi]$ ...
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Is the scalar propagator an even function?

The scalar propagator for the Klein-Gordon Lagrangian is given by: $$D(x-y)=\int \frac{d^{4} k}{(2 \pi)^{4}} \frac{e^{i k(x-y)}}{k^{2}-m^{2}+i \varepsilon}$$ I need to know if it is an even ...
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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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Classical action is zero in Klein-Gordon theory for a particle wavepacket

I'm interested in rewriting actions in the form $$ S = -\int H dt + \int p_i dx^i, $$ (where $H$ is the Hamiltonian and the $p_i$ are conjugate momenta) and then evaluating them along a classical ...
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Derivation of Klein-Gordon equation in General Relativity

I am trying to derive the Klein-Gordon equation for the case of GR using the action: $$S\left[ {\varphi ,{g_{\mu \nu }}} \right] = \int {\sqrt g {d^4}x\left( { - {1 \over 2}{g^{\mu \nu }}{\nabla _\mu ...
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Equation in Current vector in a Klein Gordon Equation

I'm trying to get the current vector $J^\mu$ of a Klein-Gordon equation: $$\Psi^* \Box \Psi =\Psi^* \partial^{\mu} \partial_\mu \Psi= \partial^{\mu}(\Psi^*\partial_\mu \Psi)-\partial^\mu \Psi^*\...
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Momentum in complex scalar field

Consider a complex scalar field $\psi(x)$ with Lagrangian density $$ \mathcal{L} = \partial_\mu\psi^* \partial^\mu\psi - M^2\psi^*\psi. $$ Expand the complex field operator as a sum $$ \psi = \int \...
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How creation operator pops out while expanding field operator?

While doing QFT when we try to canonically quantize the Klein Gordon equation $\Box \phi =0$ we promote the $\phi $ to an operator field and impose the commutation rule $[\phi(x,t),\pi (y,t)]=i\hbar\...
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How can velocity and momentum be in opposite direction for antiparticles as given in the solutions of Klein Gordon Equation?

This is given in Greiner, Relativistic Quantum Mechanics For a free particle solution and antiparticle solution with momentum $\vec{p}$ the current is given by $e\frac{c^2\vec{p}}{E_p}$. The current ...

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