# 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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### Space and Time on equal footings for QFT "Equations of Motion"? [closed]

How can we write the equations of motion for the free field (Spin 0) in QFT, which put space and time on an equal footing? (Canonical Quantization) In this setting, is said that space and time are on ...
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### Spectrum of Klein-Gordon operator in AdS Black Hole

I'm working on obtaining the spectrum of the Klein-Gordon operator in $AdS_2$ for black hole coordinates. To accomplish that, I first consider the problem in hyperbolic space $H_2$ and then Wick ...
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### Dirac Equation and the Klein-Gordon Equation

I am trying to solve an exercise in Halzen and Martin's Quarks and Leptons book and got stuck on doing some math. The Dirac equation reads $$i \gamma^{\mu} \partial_{\mu} \psi - m\psi = 0.$$ Now, I ...
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In quantum field theory most observables $A$ do not have a definite value in the ground state (vacuum). For an observable $A$, a reasonable measure of the spread in the ground state is its variance $\... 0 votes 0 answers 52 views ### Ricci Scalar Curvature under conformal transformation Consider the Klein-Gordon equation in curved spacetime with metric$g$$$\square_g \phi - \xi R \phi = 0$$ and consider a conformal transformation $$g \longmapsto \tilde{g} = \Omega^{2} g \quad , \... -2 votes 1 answer 137 views ### Does the QFT Klein-Gordon equation describe the state of the field or the field operator? In the canonical quantization of QFT we talk about: states representing a field. field operators. The quantum Klein-Gordon equation is expressed in terms of the field φ. Is φ (in the equation) the ... 1 vote 0 answers 65 views ### Why is the Klein-Gordon equation a 2nd derivative equation? Note: I am not asking about why time is in 2nd derivative: that makes perfect intuitive sense given the relativistic need to treat space and time in equal footing. We often hear how the Klein-Gordon ... 2 votes 1 answer 151 views ### Transformation law of interacting potential in non-inertial frame? I haven't seen this addressed in most lecture notes but how does the interacting potential of (say) the Klein-Gordon equation transform in a non-inertial frame? (Feel free to answer more generally as ... 0 votes 0 answers 39 views ### Solution to the Klein-Gordon equation in a generic metric Is there a way to solve the Klein-Gordon equation in a generic spacetime for a massless scalar field? \begin{equation} \frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi)=0. \end{... 0 votes 0 answers 45 views ### Klein-Gordon and Green's Function I want to prove the following relation:$$(\partial_x^2 + m^2)\langle0|T\phi(x)\phi(x_1)|0\rangle = -i\delta^{(4)}(x - x_1).$$My Approach: Consider LHS$$(\partial_x^2 + m^2)\langle0|T\phi(x)\phi(x_1)... 0 votes 0 answers 16 views ### Scalar Axion Field Amplitude Calculation in localised laboratory / Earth This question concerns the paper "Axion Dark Matter: What is it and Why Now?", in the Appendix A.3 regarding equations related to the Axion Field. It states that by the Friedmann Equation, $$... 0 votes 0 answers 61 views ### Confusion related to creation and annihilation operators I'm studying QFT from Peskin and from the book by Ashok Das, and there seems to be a disagreement between the creation and annihilation operators, for the scalar Klein Gordon field theory. In Das, we ... 0 votes 0 answers 66 views ### Total momentum operator for the KG field This question pertains to Equation (2.33) in Peskin and Schroeder:$$ \hat{\vec P}=-\int d^3\!x\,\hat\pi(\vec x)\vec\nabla\hat\phi(\vec x)=\int d^3\!p\,\vec p\,\hat a_{\vec p}^\dagger\,\hat a_{\vec p} ... 0 votes 0 answers 58 views ### Derive the KG field operator in terms of ladder operators In Peskin and Schroeder, they skip a few steps to arrive at the KG field operator in Equation (2.25): $$\hat\phi(\vec x)=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2\omega_{\vec p}}}\left(\hat a_{\vec ... -1 votes 1 answer 90 views ### Making background curvature variable in QFT on curved spacetime In QFT on curved spacetime one may start with a Klein-Gordon like equation$$(\square_g - m^2 j) \phi = j,$$where \square_g := g^{\mu \nu} \nabla_\mu \nabla_\mu is the D'Alembertian operator, and ... -1 votes 2 answers 95 views ### Square root of the wave operator How are the Dirac matrices the square root of the wave operator? I keep seeing it mentioned as such but never explained. 0 votes 1 answer 168 views ### What am I solving when I use Klein-Gordon equation? [closed] \newcommand{\Ket}{\left|#1\right>} I began to study QFT using David Tong's notes and I have a doubt. This is what the notes says (http://www.damtp.cam.ac.uk/user/tong/qft.html page 21/22). I ... 2 votes 0 answers 85 views ### Why Klein-Gordon equation before QFT? [closed] Note : This is an edit of the original question. Previously (with some more elaboration of what I had in mind): Peskin's QFT book starts with Klein Gordon equation, which is little taught for most ... 0 votes 0 answers 49 views ### How do we interpret the second-order differential operator in the QFT path integral? For the free scalar field theory, the path integral has a differential operator term in the exponent,$$ Z[J] = \int \mathcal{D}\phi \, \exp\left( i \left[ -\frac{1}{2} \int d^d x \, \phi(x) A \phi(x) ... 1 vote 0 answers 76 views ### Problematic Factor of 2 in Klein-Gordon Propagator Derivation I want to derive the Klein-Gordon Green function equation $$(\Box_b + m^2) D_F(x_b - x_a) = - i \delta^4(x_b - x_a)$$ by using the same steps taken when fixing the 'exact' Green function of the non-... 5 votes 1 answer 154 views ### Fourier transformation of the inverse Klein-Gordon propagator On Peskin & Schroeder's QFT, page 30, the scalar field propagator as the retarded Green function is defined as $$(\partial^2+m^2)D_R(x-y)=-i\delta^4(x-y) \tag{2.56}$$ The Fourier transformation is ... 3 votes 1 answer 203 views ### Problem obtaining Klein-Gordon equation solutions I am having some problems and also some questions regarding how can one get the general solution to the Klein-Gordon equation, which usually appears in the literature as $$\phi(t,\mathbf{x})=\int\... 1 vote 1 answer 65 views ### Why only \phi=\pm1 are considered "vacuum states" in the Klein-Gordon model with \phi^4 potential, and not \phi=0? I am reading "Kink Moduli Spaces — Collective Coordinates Reconsidered," by Manton, Oleś, Romańczukiewicz, and Wereszczyński (arXiv version), where they consider the Klein-Gordon equation,$$... 1 vote 1 answer 57 views ### Can a black hole in a quasi-de Sitter universe have scalar hair? Any static and axially symmetric, asymptotically flat black hole spacetime cannot support scalar hair. The universe is not asymptotically flat since it is quasi-de Sitter. Could we find a black hole ... 1 vote 2 answers 154 views ### Negative energy solutions not a problem for Klein-Gordon equation? I already posed this question Negative energy solutions in Klein-Gordon and Dirac equations but I am not satisfied with the answers. Trying to be very sharp: does Klein-Gordon equation have negative ... -2 votes 1 answer 67 views ### Klein-Gordon in Schwarzschild Curved Spacetime [closed] Given such a Schwarzschild metric, the covariant Klein-Gordon equation for a mass$m$takes the form $$\left[\frac{1}{g_{00}} \frac{\partial^2}{\partial t^2 }-\frac{1}{r^2} \frac{\partial}{\partial r}... 0 votes 0 answers 20 views ### Field equations of parametrized field theory I have problems calculating the field equations of parametrized field theory. Parametrized field theory is essentially the Klein-Gordon field \phi but, considering the "inertial coordinates&... 4 votes 2 answers 849 views ### Derivation of the Klein-Gordon equation [closed] My question arises from studying section 8.1.3 of Sakurai. I am confused on the way Klein–Gordon equation is created. In Sakurai, the book said that it was derived by taking another time derivative to ... 0 votes 1 answer 43 views ### Dependence of Klein-Gordon solution only on spatial coordinates I am studying QFT with the Peskin and Schroeder textbook, and I am new to this area of physics. I'm struggling with the solution of the Klein-Gordon equation using Fourier integral as a continuum set ... 0 votes 1 answer 150 views ### Particle-Antiparticle of Klein-Gordon equation When we solve Klein-Gordon equation, it gives both positive energy solution and negative energy solution. These two solutions are responsible for positive and negative value of \rho which was ... 0 votes 1 answer 113 views ### How to transition from a Green function to vacuum expectation value in quantum field theory for a scalar field? When considering the scalar field that solves the Klein-Gordon equation, one can use Green's functions to identify a propagator. This can be constructed from first principles, and can be left as an ... 0 votes 0 answers 98 views ### Basic concept clarification about the expectation value formula in quantum field theory In introducing quantum field theory, the field solution to the Klein-Gordon equation is$$\phi(x^{\mu}) = \int \frac{a_{\bf{k}} e^{-i(k^{\mu}x_{\mu})} + a^{\dagger}_{\bf{k}} e^{i(k^{\mu}x_{\mu})}}{\... 7 votes 1 answer 459 views ### What is the missing part of the argument needed to justify the claim of (2.52) in Peskin and Schroeder's QFT textbook? [This paragraph has been added to make clear that this is not a homework question having been branded as such by a mod of some kind. The question is attempting to the core of a very important question ... 1 vote 0 answers 78 views ### Is this theory renormalizable? If so, is there any proof? [closed] Consider a kinetic lagrangian of 2 Klein-Gordon-Foch fields$\varphi$and$\chi$with interaction term $$\mathcal{L}_I=g^2\bar{\chi}\chi\bar{\varphi}\varphi.$$ Is theory with such interaction ... 4 votes 0 answers 61 views ### Derivation of the Bessel function representation of the Green function of the inhomogeneous Klein-Gordon equation I will link the following question, as it is partly related to the problem I am trying to deal with. Green's function for the inhomogenous Klein-Gordon equation As you can read from this User´s ... 0 votes 0 answers 46 views ### Description of a Classical Klein-Gordon Field with Momentum Distribution So if I want to solve the free KG equation, the solution is of the form $$\phi(x) = \int_{p\in \mathbb{R}^4}\frac{1}{2\omega_\vec{p}} \left( a(\vec{p})e^{-ipx} +b(\vec{p})e^{ipx} \right),$$ where the ... 0 votes 1 answer 97 views ### Energy spectrum in Klein-Gordon equation in general relativity I know that the Klein-Gordon equation in general relativity takes the form (a massless field)$\nabla_\mu \nabla^\mu \phi=\sum_{a,b} \frac{1}{\sqrt{-g}}\partial_a(\sqrt{-g}g^{ab}\partial_b\phi) =0$... 0 votes 0 answers 53 views ### Can we interpret the number of particles in a state as an amplitude of oscillation TL;DR If we have a classical field expressed in a Fourier expansion, is the amplitude of each mode related to the particle number in that specific mode? Attempt to formalize We know particles should ... 2 votes 2 answers 333 views ### Does there exist a square root of Euler-Lagrange equations of a field? (Factorization) Does there exist a square root of Euler-Lagrange equations$\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0$in the sense that$(x+... 151 views

### Show that Majorana equation implies Klein-Gordon equation

Show that the Majorana equation $i \bar{\sigma}\cdot\partial\chi -im\sigma^2\chi^* = 0$ for 2-component spinors $\chi$ implies the Klein-Gordon equation $(\partial^2+m^2)\chi$. This is part of an ...
I have seen in many texts that the form of the solution to the Klein-Gordon equation for Schwarzschild metric is: $\phi(r,t,θ,φ)=r^{−1}f(r,t)Y_{lm}(θ,φ)$ where does the term $1/r$ comes from? I can ...