What is Klein-gordon: Definition and 90 Discussions

The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction (with unknown interaction term in the Hamiltonian,) the practical utility is limited.
The equation can be put into the form of a Schrödinger equation. In this form it is expressed as two coupled differential equations, each of first order in time. The solutions have two components, reflecting the charge degree of freedom in relativity. It admits a conserved quantity, but this is not positive definite. The wave function cannot therefore be interpreted as a probability amplitude. The conserved quantity is instead interpreted as electric charge, and the norm squared of the wave function is interpreted as a charge density. The equation describes all spinless particles with positive, negative, and zero charge.
Any solution of the free Dirac equation is, for each of its four components, a solution of the free Klein–Gordon equation. The Klein–Gordon equation does not form the basis of a consistent quantum relativistic one-particle theory. There is no known such theory for particles of any spin. For full reconciliation of quantum mechanics with special relativity, quantum field theory is needed, in which the Klein–Gordon equation reemerges as the equation obeyed by the components of all free quantum fields. In quantum field theory, the solutions of the free (noninteracting) versions of the original equations still play a role. They are needed to build the Hilbert space (Fock space) and to express quantum fields by using complete sets (spanning sets of Hilbert space) of wave functions.

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  1. G

    The Divergence of the Klein-Gordon Energy-Momentum Tensor

    I've tried this problem so, so, so so so many times. Given the equations above, the proof starts easily enough: $$\partial_\mu T^{\mu\nu}=\partial_\mu (∂^μ ϕ∂^ν ϕ)-\eta^{\mu\nu}\partial_\mu[\frac{1}{2}∂^2ϕ−\frac{1}{2}m^2ϕ^2]$$ apply product rule to all terms $$=\partial^\nu \phi \cdot...
  2. BiGyElLoWhAt

    A Question about factoring the Klein-Gordon equation

    Take the Klein-Gordon equation: ##\Box^2 = m^2## Say we want to linearize this equation, we try to come up with a new operator that squares into ##\Box^2##. ##(A\partial_t - B\partial_x - C\partial_y - D\partial_z)^2 = \Box^2## So we need ##-A^2=B^2=C^2=D^2=I## as this gives back the 2nd...
  3. V

    I Klein-Gordon: Schwarzschild Metric, Physically Acceptable?

    Does this give solutions which are physically acceptable?
  4. Haorong Wu

    A Heisenberg equations of Klein-Gordon Field in Space-Time

    Hi, there. I am reading An Introduction to Quantum Field Theory by Peskin and Schroeder. I am confused about some equations in section 2.4 The Klein-Gordon Field in Space-Time. It computes the Heisenberg equations of ##\phi \left ( x \right )## and ##\pi \left ( x \right)## as (in page 25) ##...
  5. G

    A Correlation function of a Klein-Gordon field

    First, let me introduce the notation; given a Hamiltonian ##H## and a momentum operator ##\vec{P}##, and writing ##P=(H,\vec{P})##. Let ##|\Omega\rangle## be the ground state of ##H##, ##|\lambda_\vec{0}\rangle## an eigenstate of ##H## with momentum 0, i.e. ##\vec{P}|\lambda_\vec{0}\rangle=0##...
  6. J

    A Is the Klein-Gordon equation a quantization of classical particles?

    The Schrödinger equation can be derived from the path integral quantization of the Lagrangian of classical, non-relativistic particles. Can the Klein-Gordon (and maybe the Dirac) equation be derived from the path integral quantization of a given classical (supposedly relativistic) Lagrangian of...
  7. P

    A Dirac's solution to the Klein-Gordon equation

    Dirac wanted to fix the problems with the Klein-Gordon equation by seeking a new solution to it. He wanted a relativistic solution so it makes sense that the solution needed to satisfy Einstein's energy-momentum relation. But why did it need to be of first order in time- and...
  8. CharlieCW

    Coherent states for Klein-Gordon field

    Homework Statement Show that the coherent state ##|c\rangle=exp(\int \frac{d^3p}{(2\pi)^3}c(\vec{p})a^{\dagger}_{\vec{p}})|0\rangle## is an eigenstate of the anhiquilation operator ##a_{\vec{p}}##. Express it in terms of the states of type ##|\vec{p}_1...\vec{p}_N\rangle## Homework Equations...
  9. S

    I Klein-Gordon in QFT: Understanding the KG Equation

    Hello! I am a bit confused about the KG equation in the context of QFT. In QM, the KG equations describes the evolution of a wavefunction, ##\phi(x,t)##, in space and time (I will assume we have no potential). This function gives the probability of finding a particle described by this...
  10. N

    Quantizing the complex Klein-Gordon field

    I'm self-studying QFT and attempting exercise 2.2 on Peskin & Schroeder. First off, I'm a bit confused on the logic the authors use in the quantization process. They first expand the fields in terms of these ##a_{\vec{p}},a_{\vec{p}}^\dagger## operators which, if I understand correctly, is...
  11. N

    On deriving the standard form of the Klein-Gordon propagator

    I'm trying to make sense of the derivation of the Klein-Gordon propagator in Peskin and Schroeder using contour integration. It seems the main step in the argument is that ## e^{-i p^0(x^0-y^0)} ## tends to zero (in the ##r\rightarrow\infty## limit) along a semicircular contour below (resp...
  12. bleist88

    A The Lagrangian Density and Equations of Motion

    Can Lagrangian densities be constructed from the physics and then derive equations of motion from them? As it seems now, from my reading and a course I took, that the equations of motion are known (i.e. the Klein-Gordon and Dirac Equation) and then from them the Lagrangian density can be...
  13. C

    I Klein-Gordon propagator derivation

    I was reading about the classical Klein-Gordon propagator here: https://en.wikipedia.org/wiki/Propagator#Relativistic_propagators Basically they are looking for ##G##, that solves the equation $$(\square _{x}+m^{2})G(x,y)=-\delta (x-y).$$ So they take the Fourier transform to get...
  14. hilbert2

    A Constant Solutions of Real Scalar Field

    Suppose I have a self interacting real scalar field ##\phi## with equation of motion ##\partial^i \partial_i \phi + m^2 \phi = -A \phi^2 - B\phi^3##, and I attempt to find constant solutions ##\phi (x,t) = C## for it. The trivial solution is the zero solution ##\phi (x,t) = 0##, but there can...
  15. S

    I Does the Contour Integral for the Klein-Gordon Propagator Matter?

    Hello! I am reading Peskin's book on QFT and in the first chapter (pg. 30) he introduces this: ##<0|[\phi(x),\phi(y)]|0> = \int\frac{d^3p}{(2\pi)^3}\int\frac{dp^0}{2\pi i}\frac{-1}{p^2-m^2}e^{-ip(x-y)}## and then he spends 2 pages explaining the importance of choosing the right contour integral...
  16. S

    I Klein-Gordon in QFT: Wave Functions & Spins

    Hello! So in the Klein-Gordon equation you have a field ##\phi## which becomes an operator in QFT and when you apply it on the vacuum state ##|0>## you get a particle at position x: ##\hat{\phi}(x)|0>=|x>##. So if you look at this particle (in a non interaction theory) the wave function of this...
  17. F

    I Equivalent Klein-Gordon Lagrangians and equations of motion

    Suppose one starts with the standard Klein-Gordon (KG) Lagrangian for a free scalar field: $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}$$ Integrating by parts one can obtain an equivalent (i.e. gives the same equations of motion) Lagrangian...
  18. S

    I Causality preserved in Klein-Gordon equation

    Hello! I am reading Peskin's book on QFT and in chapter one he shows that ##[\phi(x), \phi(y)] = D(x-y) - D(y-x)##, with ##D(x-y)## being the propagator from ##x## to ##y##. He says that if ##(x-y)^2<0## we can do a Lorentz transformation such that ##(x-y) \to -(x-y)## and hence the commutator...
  19. LarryS

    I When can the Klein-Gordon Equation be used for a photon?

    Consider the double-slit experiment done with photons from a laser. If one was interested only in computing position (vertical) probability amplitudes and did not care about spin/helicity, could the Klein-Gordon Equation (with mass set to zero) be used? Thanks in advance.
  20. joebentley10

    I Applying Euler-Lagrange to (real) Klein-Gordon Lagrangian

    I'm currently studying Quantum Field Theory and I have a confusion about some mathematics in page 30 of Mandl's Quantum Field Theory (Wiley 2010). Here is a screenshot of the relevant part: https://www.dropbox.com/s/fsjnb3kmvmgc9p2/Screenshot%202017-01-24%2018.10.10.png?dl=0 My issue is in...
  21. P

    Gauge symmetry for massless Klein-Gordon field

    Homework Statement I need to gauge the symmetry: \phi \rightarrow \phi + a(x) for the Lagrangian: L=\partial_\mu\phi\partial^\mu\phi Homework EquationsThe Attempt at a Solution We did this in class for the Dirac equation with a phase transformation and I understood the method, but...
  22. T

    I Klein-Gordon Equation: Forming a Conservation of Charge

    Suppose φ is solution to Klein-Gordon equation, Multiplying it by -iφ* we get iφ^*\frac{\partial^2φ}{\partial t^2}-iφ^*∇^2φ+iφ^*m^2=0 .....(5) Taking the complex conjugate of the Klein-Gordon equation and multiplying by -iφ we get iφ\frac{\partial^2φ^*}{\partial t^2}-iφ∇^2φ^*+iφm^2=0].....(6)...
  23. hilbert2

    A Approximating a QF with finite-dimensional Hilbert space

    Is it possible to approximately calculate the dynamics of a "phi-fourth" interacting Klein-Gordon field by using a finite dimensional Hilbert state space where the possible values of momentum are limited to a discrete set ##-p_{max},-\frac{N-1}{N}p_{max},-\frac{N-2}{N}p_{max}...
  24. carllacan

    I Why do Dirac spinors obey the Klein-Gordon equation?

    The solutions to the Dirac equation are also solutions of the Klein-Gordon equation, which is the equation of motion for the real scalar field. I can see that the converse is not true, but why do spinors follow the equation for real-field particles? Is there any physical meaning to it?
  25. S

    A Hamiltonian of the quantised Klein-Gordon theory

    The Klein-Gordon field ##\phi(\vec{x})## and its conjugate momentum ##\pi(\vec{x})## is given, in the Schrodinger picture, by ##\phi(\vec{x})=\int \frac{d^{3}p}{(2\pi)^{3}}...
  26. S

    Energy-Momentum Tensor for the Klein-Gordon Lagrangian

    Homework Statement The energy-momentum tensor ##T^{\mu\nu}## of the Klein-Gordon Lagrangian ##\mathcal{L}_{KG} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}## is given by $$T^{\mu\nu}~=~\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}.$$ Show...
  27. S

    A Proof of Lorentz invariance of Klein-Gordon equation

    I would like to prove the Lorentz invariance of the Klein-Gordon equation by proving the invariance of the action ##\mathcal{S} = \int d^{4}x\ \mathcal{L}_{KG}## under a Lorentz tranformation. I would like to do this by first proving the Lorentz invariance of the ##\mathcal{L}_{KG}## and then...
  28. S

    Lorentz invariance of Klein-Gordon eqn & Maxwell Lagrangian

    Homework Statement 1. Show directly that if ##\varphi(x)## satisfies the Klein-Gordon equation, then ##\varphi(\Lambda^{-1}x)## also satisfies this equation for any Lorentz transformation ##\Lambda##. 2. Show that ##\mathcal{L}_{Maxwell}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}## is invariant under...
  29. D

    A How to derive general solution to the Klein-Gordon equation

    I understand that the ansatz to $$(\Box +m^{2})\phi(\mathbf{x},t)=0$$ (where ##\Box\equiv\partial^{\mu}\partial_{\mu}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}##) is of the form ##\phi(\mathbf{x},t)=e^{(iE_{\mathbf{k}}t-\mathbf{k}\cdot\mathbf{x})}##, where...
  30. S

    A Solution of the classical Klein-Gordon equation

    The classical Klein-Gordon equation is ##(\partial^{2}+m^{2})\varphi(t,\vec{x})=0##. To solve this equation, we need to Fourier transform ##\varphi(t,\vec{x})## with respect to its space coordinates to obtain ##\varphi(t,\vec{x}) = \int...
  31. loops496

    Klein-Gordon Hamiltonian commutator

    Homework Statement Consider the quantum mechanical Hamiltonian ##H##. Using the commutation relations of the fields and conjugate momenta , show that if ##F## is a polynomial of the fields##\Phi## and ##\Pi## then ##[H,F]-i \partial_0 F## Homework Equations For KG we have: ##H=\frac{1}{2} \int...
  32. A

    Klein-Gordon eqn: why dismiss messages at phase velocity

    Hi All, I've heard it said that the superluminal phase velocity of the KG eqn is not a problem for relativistic causality because signals travel at the packet/group velocity, which is the inverse of the phase velocity (c being 1). I'm a bit skeptical of this. We can strip away all the quantum...
  33. Telemachus

    Klein-Gordon equation with fields

    Homework Statement I was just studying the Klein Gordon equation with fields. In particular I was reviewing the continuity equation. In the derivation for it, the usual approach is to take the klein-gordon equation (I'm using 4-vector covariant notation), multuply by the complex conjugate of...
  34. D

    Showing that the real Klein-gordon lagrangian is Lorentz invariant

    Homework Statement Hey guys! So this question should be simple apparently but I got no idea how to do it. Basically I have the following Lagrangian density \mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)-\frac{m}{2}\phi^{2} which should be invariant under Lorentz...
  35. D

    Klein-Gordon operator on a time-ordered product

    Homework Statement Hey guys, So here's the problem I'm faced with. I have to show that (\Box + m^{2})<|T(\phi(x)\phi^{\dagger}(y))|>=-i\delta^{(4)}(x-y) , by acting with the quabla (\Box) operator on the following...
  36. D

    Commutator of complex Klein-Gordon solution with total momentum

    Homework Statement Hey guys, So I have to show the following: [P^{\mu} , \phi(x)]=-i\partial^{\mu}\phi(x), where \phi(x)=\int \frac{d^{3}k}{(2\pi)^{3}2\omega(\vec{k})} \left[ a(\vec{k})e^{-ik\cdot x}+b^{\dagger}(\vec{k})e^{ik\cdot x} \right] and P^{\mu}=\int...
  37. ShayanJ

    Lorentz invariance of Klein-Gordon Lagrangian

    I want to prove the invariance of the Klein-Gordon Lagrangian \mathcal{L}=\frac 1 2 \partial^\mu \phi \partial_\mu \phi-\frac 1 2 m^2 \phi^2 under a general Lorentz transformation \Lambda^\alpha_\beta but I don't know what should I do. I don't know how to handle it. How should I do it? Thanks
  38. B

    Momentum operator of the quantized real Klein-Gordon field

    Homework Statement a+(k) creates particle with wave number vector k, a(k) annihilates the same; then the Klein-Gordon field operators are defined as ψ+(x) = ∑_k f(k) a(k) e^-ikx and ψ-(x) = ∑_k f(k) a+(k) e^ikx; the factor f contains constants and the ω(k). x is a Lorentz four vector, k is a...
  39. Greg Bernhardt

    What is the Klein-Gordon equation

    Definition/Summary In this library item, some properties and interpretations of the Klein-Gordon equation (KG) will be covered. We will first focus on its usage in Relativistic Quantum Mechanics (RQM) and then examine it in Quantum Field Theory (QFT). The Klein-Gordon equation is...
  40. C

    Peskin and Schroeder derivation of Klein-Gordon propagator

    In page 30 of book "An introduction to quantum field theory" by Peskin and Schroeder in the derivation of Klein-Gordon propagator, why p^0=-E_p in the second step in equation (2.54). and why change "ip(x-y)" to "-ip(x-y)"? I thought a lot time, but get no idea. Thank you for your giving me an...
  41. G

    Green's function of the Klein-Gordon operator

    Again, from the Peskin and Schroeder's book, I can't quite see how this computation goes: See file attached The thing I don't get is how the term with (\partial^{2}+m^{2})\langle 0| [\phi(x),\phi(y)] | 0 \rangle vanishes, and also why they only get a \langle 0 | [\pi(x),\phi(y)] | 0 \rangle...
  42. G

    Why does the Klein-Gordon propagator have a negative i in the exponential?

    Hello, I'm looking at the following computation from the Peskin and Schroeder's book: See file attached In the second page, the second term that's being integrated, I don't understand why it has a negative i in the exponential, that'll keep the energy term the same, but will swap the sign...
  43. N

    How does the Klein-Gordon Lagrangian relate to the equations of motion?

    Hi, I hope I put this in the right place! I'm having trouble with some of the calculus in moving from the Klein-Gordin Lagrangian density to the equations of motion. The density is: L = \frac{1}{2}\left[ (\partial_μ\phi)(\partial^\mu \phi) - m^2\phi ^2 \right] Now, to apply the...
  44. P

    Velocity of Klein-Gordon particles of mass m

    Homework Statement Show that ψ(x,t)=Ae^{i(kt-ωt)} is a solution to the Klein-Gordon equation \frac{∂ ^2ψ(x,t)}{∂x^2}-\frac{1}{c^2}\frac{∂^2ψ(x,t)}{∂t^2}-\frac{m^2c^2}{\hbar^2}ψ(x,t)=0 if ω=\sqrt{k^2c^2+(m^2c^4/\hbar^2)} Determine the group velocity of a wave packet made up of waves satisfying...
  45. S

    Imposing Klein-Gordon on Dirac Equation

    Hey, My question is on the Dirac equation, I am having a little confusion with the steps that have been taken to get from this form of the Dirac equation: i\frac{\partial \psi}{\partial t}=(-i\underline{\alpha}\cdot \underline{\nabla}+\beta m)\psi to -\frac{\partial^2 \psi}{\partial...
  46. snoopies622

    Interpretation of Klein-Gordon equation

    Comparing the Klein-Gordon equation to the equation of motion for a classical harmonic oscillator, I notice that for a particle of mass m, \frac {mc^2}{\hbar} is a frequency. Does this frequency have a physical meaning?
  47. S

    Klein-Gordon Equation & Continuity Equation

    Hello, My question is on the Klein-Gordon equation and it's relation to the continuity equation, so for a Klein-Gordon equation & continuity equation of the following form, I have attained the following probability density and probability current relations (although not normalised correctly...
  48. S

    The Klein-Gordon equation with a potential

    Hello, My question concerns the Klein-Gordon Equation under some potential of the form (and refers to a higgs-like interaction, i assume as that's what we're researching): \delta V= \lambda \Psi^{*}\Psi For substitution into the Klein-Gordon equation: (\frac{\partial^2 }{\partial...
  49. R

    Green's function for Klein-Gordon equation in x-space

    I'm trying to derive the x-space result for the Green's function for the Klein-Gordon equation, but my complex analysis skills seems to be insufficient. The result should be: \begin{eqnarray} G_F(x,x') = \lim_{\epsilon \rightarrow 0} \frac{1}{(2 \pi)^4} \int...
  50. I

    Particle number and resonance with Klein-Gordon waves.

    Hi all, I'm hoping this will be a quickly solved question. In Peskin and Schroeder (2.66), when dealing with source terms in the Klein-Gordon equation, ##(\partial^2+m^2)\phi(x) = j(x)##, they have $$\int d N =\int \frac{d^3 p}{(2\pi)^3}\frac{1}{2E_p}|\tilde{j}(p)|^2\quad...
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