Klein-gordon Definition and 87 Threads

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...

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...

Does this give solutions which are physically acceptable?

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) ##...

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##...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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)...

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}...

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?

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}}...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

[SIZE="4"]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...

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...

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...

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...

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...

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...

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...

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?

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...

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...

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...

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...