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. J

    Poincare Recurrence and the Klein-Gordon Equation

    There exists Green's Functions such that the solutions appear to be retro-causal. The Klein-Gordon equation allows for antiparticles to propagate backwards in time. Does this mean the future can influence the past and present? Then again The Poincare Recurrence Theorem states that over a...
  2. I

    Dimensions of klein-gordon field

    Does anyone know how to write the classical solution to the Klein-Gordon equation in NON natural units? Where do all the c's and h's go?
  3. N

    Renormalizing solutions of the Klein-Gordon equation

    It is said that the solutions of the Klein-Gordon equation cannot be interpreted as probability densities since the norm isn't conserved in the time evolution. Now a pretty evident idea seems to be to renormalize the solution at each moment so that it is renormalized (and hence interpretable...
  4. J

    Klein-Gordon for a massless particle

    So I'm trying to find a solution of the Klein-Gordon equation for a massless particle. I reached the Klein-Gordon from the total energy-momentum equation. Then for a massless particle i get to this equation: $${ \partial^2 \psi \over \partial t^2 } = c^2 \nabla^2 \psi$$How do I solve for psi? I...
  5. P

    Klein-Gordon equation and factorization

    Hi! I read a text were some kind of "Schroedinger-equation" for a neutrino field is being derived. But there is a particular step I do not understand. Consider a Dirac field \psi(t, \vec{x}) of a neutrino, satisfying the Klein-Gordon equation: \left( \partial_{t}^{2} + \vec{k}^{2} +...
  6. Matterwave

    Klein-Gordon Equation: Derivation of Density Relativistic Transform

    Hello, I have a question. In Ryder chapter 2, he develops the KG equation and says something along the lines of "the density, in order to be relativistic, must transform like the time component of a 4 vector" and he immediately gives: \rho=\frac{i\hbar}{2m}\left(\phi^*\frac{\partial...
  7. J

    Klein-Gordon field is spin-0: really?

    Usually in the first sentence of the definition of the Klein Gordon equation is the statement that it describes spin-0 particles. Similarly, in the first sentence of the definition of the Dirac equation is the statement that it describes spin-1/2 particles. But then comes the bit that got...
  8. N

    Do higher spin particles obey Klein-Gordon or Dirac equations?

    Please teach me this: We know that 0-spin particles obey Klein-Gordon equation and 1/2spin particles obey Dirac equation.But I do not know whether higher integer spin particles obey Klein-Gordon equation or not.Similarly,do higher half integer spin particles obey Dirac equation?Because if we...
  9. Y

    The Klein-Gordon field as harmonic operators

    I am reading through 'An Introduction to QFT' by Peskin & Schroeder and I am struggling to follow one of the computations. I follow writing the field \phi in Fourier space ϕ(x,t)=∫(d^3 p)/(2π)^3 e^(ip∙x)ϕ(p,t) And the writing the operators \phi(x) and pi(x) as ϕ(x)=∫(d^3 p)/(2π)^3...
  10. G

    Quantization of Klein-Gordon Field

    I was reading the book written by Peskin about QFT when I found that the following equation: (\frac{\partial}{\partial t^2}}+p^2+m^2)\phi(\vector{p},t)=0 has as solutions the solutions of an Harmonic Oscillator. From what I know about harmonic oscillators, the equation describing them should...
  11. O

    Derive expression for Klein-Gordon annihilation operator

    [/tex]1. Homework Statement [/b] Given: \phi(x)=\phi^+(x)+\phi^-(x) Where: \phi^+(x)=\sum_{\mathbf{k}} \sqrt{\frac{\hbar c^2}{2 V \omega_{\mathbf{k}}}} a(\mathbf{k}) \mathrm{e}^{-\mathrm{i}kx} and \phi^-(x)=\sum_{\mathbf{k}} \sqrt{\frac{\hbar c^2}{2 V \omega_{\mathbf{k}}}} a^\dagger...
  12. Pengwuino

    Klein-Gordon Finite Well

    Homework Statement The problem is basically solving the Klein-Gordon equation for a finite well for a constant potential under the condition V > E + mc^2 Homework Equations V = 0 -a<x<a V = V_o elsewhere KG Equation: [\nabla^2 + \left({{V-E} \over {\hbar c}}\right)^2 - k_c^2]\phi(x) = 0...
  13. C

    Quantizing the Klein-Gordon equation

    Hi all. I'm taking my first foray into QFT, and have a question which is hopefully pretty basic. I think I understand the concept itself, I just don't quite get how the math works out. I'm right at the beginning, in the discussion of how to set up the creation/annihilation operators for a...
  14. bcrowell

    Massless Klein-Gordon equation not conformally invariant?

    massless Klein-Gordon equation not conformally invariant?? Wald discusses conformal transformations in appendix D. He shows that the source-free Maxwell's equations in four dimensions are conformally invariant, and this makes sense to me, since with photons all you can do is measure the...
  15. W

    What is the hamiltonian in Klein-Gordon equation?

    since the time derivative is second order, the KG equation can not be put in the form i \dot{\psi}= H \psi so there is no H in KG equation? and no Heisenberg picture for KG equation?
  16. C

    A question about Lorentz invariance for Klein-Gordon field

    Homework Statement Hi everyone, in Peskin & Schroeder, P36, the derivative part of KG field is transformed as eqn (3.3). But why does the partial derivative itself not transform? Homework Equations \partial_{\mu} \phi (x) \rightarrow \partial_{\mu} ( \phi ( \Lambda^{-1} x) ) = (...
  17. Š

    Klein-Gordon propagator ill-defined?

    Hi, I've had the following problem in elementary quantum field theory. The propagator for the Klein-Gordon scalar field takes the form D(x-y)=\int\frac{\textrm{d}^3\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}e^{-ip\cdot(x-y)} I was interested what the propagator looks like for...
  18. Y

    Solving Klein-Gordon PDE w/ Change of Variables

    Hi. I'm following the solution of a Klein-Gordon PDE in a textbook. The equation is \begin{align} k_{xx}(x,y) - k_{yy}(x,y) &= \lambda k(x,y) \\ k(x,0) &= 0 \\ k(x,x) &= - \frac{\lambda}{2} x \end{align} The book uses a change of variables $\xi = x+y$, $\eta = x-y$ to write \begin{align}...
  19. F

    Newtonian limit of Gravitational Klein-Gordon Equation

    Hey guys, Was wondering if anyone has seen this done? Essentially I've tried plugging in the Schwarzschild exterior metric and getting a radial wave equation then taking a series expansion for small M (gravitating mass) and comparing that to the KG radial wave equation in a radial potential...
  20. A

    The gravitational Klein-Gordon Equation

    Hi Following a request by FunkyDwarf (I don't know if dwarfs have odor!) in https://www.physicsforums.com/showpost.php?p=2556673&postcount=11" thread, regarding how one can get the Klein-Gordon equation (KGE) for free particles in a gravitational field, i.e. the equation -g^{\mu \nu}...
  21. W

    Global symmetry of an N-component Klein-Gordon theory?

    The Lagrangian is given by, \sum_{a=1}^N \left[(\partial^{\mu}\phi_{a}^{\ast})(\partial_{\mu}\phi_{a})-m^{2}\phi_{a}^{\ast}\phi_{a}\right]. Is the symmetry SO(2N), SU(N) or U(N)? It seemed quite obvious to me and some of my friends that such theory has an SO(2N) symmetry. If we view...
  22. pellman

    Dirac conserved current vs Klein-Gordon conserved current

    The conserved current for a field \phi obeying the Klein-Gordon equation is (neglecting operator ordering) proportional to i\phi^{\dag}\partial_\mu \phi-i\phi\partial_\mu \phi^{\dag}. The conserved current for a four component field \psi obeying the Dirac equation is...
  23. L

    Conserved current; Klein-Gordon

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  24. W

    Deriving Klein-Gordon from Heisenberg

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  25. W

    How to Derive the Klein-Gordon Equation from its Lagrangian Density?

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  26. pellman

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  27. J

    Does the Klein-Gordon Lagrange Density Determine the Solution of the Equation?

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  28. P

    Field Operators in Klein-Gordon theory

    Currently I am working through a script concerning QFT. To introduce the concept of canonical filed quantisation one starts with the (complex valued) Klein-Gordon field. I think the conept of quantising fields is clear to me but I am not sure if one can claim that the equations of motion for the...
  29. K

    Vacuum state of the Klein-Gordon field

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  30. D

    Klein-Gordon Momentum Question

    Dear all, I'd be very grateful for some help on this question: "The momentum operator is defined by: \displaystyle P = - \int_{0}^{L} dz \left(\frac{\partial \phi}{\partial t}\right) \left( \frac{\partial \phi}{\partial z} \right) Show that P can be written in terms of the operators a_n...
  31. D

    Klein-Gordon Approximation Question

    I'd be greatful for a bit of help on this question, can't seem to get the answer to pop out: A particle moving in a potential V is described by the Klein-Gordon equation: \left[-(E-V)^2 -\nabla^2 + m^2 \right] \psi = 0 Consider the limit where the potential is weak and the energy is...
  32. A

    Evaluate the Klein-Gordon action

    [SOLVED] Evaluate the Klein-Gordon action I'm interested in evaluating the Klein-Gordon action in P&S, p. 287. It goes as follows S_0 = \frac{1}{2} \int d^4 x \! \phi \left( - \partial^2 -m^2 \right) \phi + \left(\text{surface term} \right) The surface terms drops out, that's fine. I...
  33. L

    Klein-Gordon equation for electro-magnetic field?

    Can we imagine electro-magnetic field in vakuum as a massless particle that respects Klein-Gordon equation (instead of Electromagnetic wave equation)? It seems to me that both equations are the same, except that the electro-magnetic field can have 2 possible polarizations (then we count them...
  34. W

    Understanding the Klein-Gordon Propagator and its Satisfying Equation

    Homework Statement Homework Equations Show that the KG propagator G_F (x) = \int \frac{d^4p}{(2\pi)^4} e^{-ip.x} \frac{1}{p^2-m^2+i\epsilon} satsify (\square + m^2) G_F (x) = -\delta(x) The Attempt at a Solution I get (\square + m^2) G_F (x) = - \int \frac{d^4p}{(2\pi)^4} (p^2-m^2)...
  35. F

    Klein-Gordon Causality calculation

    [SOLVED] Klein-Gordon Causality calculation Homework Statement In Peskin and Schroeder on page 27 it is stated that when we compute the Klien-Gordon propagator in terms of creation and annihilation operators the only term that survived the expansion is...
  36. A

    Klein-Gordon signal propagation

    Given Klein-Gordon equation for a particle of mass m in covariant notation \left[ \partial_{\mu} \partial^{\mu} + \frac{m_0^2 c^2}{\hbar^2} \right] \phi = 0 show that the solution preserves causality, i.e. signals have a velocity not higher than c. HINT: You can build up a quantity...
  37. M

    What is the Significance of Substituting p for -p in the Klein-Gordon Equation?

    I'm just reading the schroeder/peskin introduction to quantum field theory. On Page 21 there is the equation \phi(x)=\int\frac{d^3 p}{(2\pi)^3}\frac{1}{ \sqrt{2\omega_{\vec{p}}} } (a_{\vec{p}} e^{i \vec{p} \cdot \vec{x}} +a^{+}_{\vec{p}} e^{-i \vec{p} \cdot \vec{x}} ) and in the...
  38. J

    Klein-Gordon Field QM: Replacing Wave Function with Wave Functional?

    Is it correct to think, that with a scalar complex Klein-Gordon field the wave function \Psi:\mathbb{R}^3\to\mathbb{C} of one particle QM is replaced with an analogous wave functional \Psi:\mathbb{C}^{\mathbb{R}^3}\to\mathbb{C}? Most of the introduction to the QFT don't explain anything like...
  39. E

    Klein-Gordon Equation: Understanding Scalar & Vector

    Homework Statement I do not understand how the Klein-Gordin equation can hold when you have a del operator on one side and a partiall derivative on the other. Doesn't the del operator give a vector and the partial derivative operator yield a scalar? Homework Equations The...
  40. E

    Mathematica Schroedinger, Klein-Gordon & Dirac Propagators

    Let be the propagators for the Schroedinguer,Klein-Gordon and Dirac?...are they hermitian operators?..are their eigenfunctions ortogonal?...
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