What is Klein gordon field: Definition and 13 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. PhysicsRock

    Meaning of coefficients in polynomial potential for scalar field

    For the solution of the equation of motion, we take a plane wave ##\phi(x) = e^{ik_\mu x^\mu}##. Plugged in, we obtain $$ -(k_0)^2 + (\vec{k})^2 = 2c_2 \Rightarrow k_\mu k^\mu = 2c_2 $$ One can then find the group velocity (using ##(k_0)^2 = \omega^2##) to be $$ \vec{v}_g =...
  2. Samama Fahim

    Total Momentum Operator for Klein Gordon Field

    As $$\hat{P_i} = \int d^3x T^0_i,$$ and $$T_i^0=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi-\delta_i^0\mathcal{L}=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi=\pi\partial_i\phi.$$ Therefore, $$\hat{P_i} = \int d^3x \pi\partial_i\phi.$$ However...
  3. dRic2

    I Non-relativistic limit of Klein Gordon field

    From Wikipedia: Which should be conceptually similar of what happen in the non-relativistic limit of the Dirac equations when you see that the solutions decouple. Do you have any reference that I can look up where the derivation for the KG field is performed? Thanks in advance!
  4. J

    Linear momentum of the Klein Gordon field

    The correct answer is: #P = \int \frac{dp^3}{(2\pi)^3}\frac{1}{2E_{\vec{p}} \big(a a^{\dagger} + a^{\dagger}a\big)# But I get terms which are proportional to ##aa## and ##a^{\dagger}a^{\dagger}## I hereunder display the procedure I followed: First: ##\phi = \int...
  5. F

    I How to compute second-order variation of an action?

    Starting with the action for a free scalar field $$S[\phi]=\frac{1}{2}\int\;d^{4}x\left(\partial_{\mu}\phi(x)\partial^{\mu}\phi(x)-m^{2}\phi^{2}(x)\right)=\int\;d^{4}x\mathcal{L}$$ Naively, if I expand this to second-order, I get $$S[\phi+\delta\phi]=S[\phi]+\int\;d^{4}x\frac{\delta...
  6. L

    Hamiltonian in terms of creation/annihilation operators

    Homework Statement Consider the free real scalar field \phi(x) satisfying the Klein-Gordon equation, write the Hamiltonian in terms of the creation/annihilation operators. Homework Equations Possibly the definition of the free real scalar field in terms of creation/annihilation operators...
  7. L

    I Understanding the scalar field quantization

    I am getting started with QFT and I'm having a hard time to understand the quantization procedure for the simples field: the scalar, massless and real Klein-Gordon field. The approach I'm currently studying is that by Matthew Schwartz. In his QFT book he first solves the classical KG equation...
  8. 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...
  9. U

    Yukawa potential and the Klein Gordon

    Homework Statement a)Show that the yukawa potential is a valid static-field euation b)Show this solution also works Homework EquationsThe Attempt at a Solution Part (a) Using the relation given, I got LHS = \frac{e^{-\mu r}}{r} \left[ (m^2 - \mu^2) - \frac{2\mu}{r} - \frac{2}{r^2}...
  10. T

    Computation of propagation amplitudes for KG field

    Note: I'm posting this in the Quantum Physics forum since it doesn't really apply to HEP or particle physics (just scalar QFT). Hopefully this is the right forum. In Peskin and Schroeder, one reaches the following equation for the spacetime Klein-Gordon field: $$\phi(x,t)=\int...
  11. G

    Klein-Gordon Field: Understanding Eq. (1)

    Hi everyone! Im' a new member and I'm studying Quantum Field Theory. I read this: "The interpretation of the real scalar field is that it creates a particle (boson) with momentum p at the point x." and : \phi\left(x\right) \left|0\right\rangle = \int \frac{d^3p}{(2\pi)^3(2\varpi_p)}...
  12. maverick280857

    Exploring the Meaning of Positive and Negative Energy in the Klein Gordon Field

    Hi everyone The Hamiltonian of the Klein Gordon field can be written as H = \frac{1}{2}\int d^{3}E_p \left[a^{\dagger}(p)a(p) + a(p)a^{\dagger}(p)\right] and we have [H,a(p')] = -E_{p'}a(p') [H,a(p')] = +E_{p'}a^{\dagger}(p') The book I'm reading states that What does this mean? Thanks.
  13. maverick280857

    Convergence of vacuum state of Klein Gordon field in a box

    Hi I've been reading through the book "Quantum Field Theory: A Tourist Guide for Mathematicians" by George B. Folland. On page 101, he describes the construction of a scalar field "in a box" \mathbb{B}: \left[-\frac{1}{2}L,\frac{1}{2}L\right]^3 in \mathbb{R}^3. Here \bf{p} lies in the...