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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Why is it okay to replace the momentum operator with $p_x + im \omega x$?

In this paper the momentum operator was replaced with something that looked very similar to the ladder operator for the non relativistic harmonic oscillator. They started with the Klein Gordon ...
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Massless scalar propagator in Euclidean space and Green's equation

In this paper (Erickson et al, 2000), the authors claim in eq. (46) that the Green's equation corresponding to a bosonic propagator $\Delta(x)$ in $2\omega$ dimensions is: $$ - \partial^2 \Delta(x) = \...
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Why is it necessary to wrap our contour around the branch cut at $+ im$ in the spacelike Klein-Gordon propagator? (P&S)

This question is in reference to eq. (2.52) on the bottom of page 27 in Peskin and Schroeder. To evaluate the Klein-Gordon field propagator along a spacelike interval we wrap the contour around the ...
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Real scalar field units

I need to get the Lagrangian of a real scalar field in SI units. In all books and websites I have been consulting they do the typical $\hbar=c=1$ which is useless for me right now. I am trying to find ...
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How to show the (Klein-Gordon) retarded propagator satisfies its equation of motion?

The retarded propagator for a massless scalar field is $$ G_R(t,\mathbf{x} ;t',\mathbf{x}' ) = \frac{ \Theta(t-t') \delta\big( - (t-t')^2 + |\mathbf{x} - \mathbf{x}'|^2 \big)}{2\pi} \tag{1} $$ which ...
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The quantisation of the harmonic oscillator applied to the free Klein-Gordon field

In David Tong's lecture notes on quantum field theory, at the bottom of page 23, we are applying the quantisation of the harmonic oscillator to the field to obtain expressions for the field operators ...
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42 views

Hamiltonian density from Klein-Gordon field

In the solution for Peskin & Shroeder 2.2 where the Hamiltonian density obtained from the Klein-Gordon Lagrangian is given by: $$ H = \pi^* \pi + \nabla \phi \cdot \nabla \phi^* + m^2 \phi^* \phi ...
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Retarded vs Feynman Klein-Gordon Propagators

Although I follow all the manipulations -- Green's functions, choice of contour/i$\epsilon$ prescription, etc -- I seem to be struggling with too many trees. The forest remains blurry. In ...
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Fourier expansions of Klein Gordon field not Lorentz invariant?

I’m working in Peskin and Schroeders book on QFT and noticed that they expanded a solution to the Klein Gordon equation in a manner that seems to me not to be be Lorentz invariant even though the ...
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Fourier transform of field variables rearrangement

I’m working in Peskin and Schroeders book on QFT These are the Fourier transforms of the field solutions to the Klein-Gordon equation: I don’t understand how to get from (2.25) to (2.27) The logical ...
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Tensor Question (Klein–Gordon equation) [closed]

I have a question following the derivation of the Klein-Gordon equation from a lagrangian. From Eq. (13d), where does $\delta^\mu_\nu$ come from? I guess it's a conversion factor of some sort.
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Transparent Boundary Conditions 1D Klein Gordon

The wave equation is an example of an equation for which there are simple transparent boundary conditions. We can factorize the wave operator $$\partial_t^2 -\partial_x^2 = (\partial_t - \partial_x)(\...
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Can I write a complex field, in some cases, as a real field?

I am learning quantum field theory. Now I am considering this case: Suppose a spin-0 particle which obeys the Klein-Gordon field equation and its anti-particle obeying the same equation do not have ...
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Scalar fields and spinors in 0+1D

As part of learning about SUSY quantum mechanics, I am trying to get a grasp on the following Lagrnagians in 1 (temporal dimension): But since these early times the treatment and methods of field ...
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What's wrong with this “proof” that QFT violates causality?

In An Introduction to Quantum Field Theory, by Peskin and Schroeder, when discussing the quantized real Klein-Gordon field ($\phi=\phi^\dagger$), they show the commutator $[\phi(x),\phi(y)]$ vanishes ...
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LSZ derivation and contact terms [closed]

In the derivation of the LSZ reduction formula we write asymptotic states using ladder operators at $\pm \infty$. For example if we consider for a real scalar field $\phi$ then we have the formula $\...
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Phase factor in the equal time commutation relation in Klein-Gordon field

Two related questions regarding the equal-time commutation relation in the Klein-Gordon field (I suspect they have the same/very related answers): In the following notes http://www-thphys.physics.ox....
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Some basic concepts in quantum field theory part 2

This question is a sequel of Some basic concepts in quantum field theory part 1 In Peskin's book, the part How Not to Quantize the Dirac Field, the writer says,"...in analogy with Klein-Gordon ...
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Some basic concepts in quantum field theory part 1

I have some problems about basic concepts of quantum field theory. First let's look at Klein-Gordon field. Klein-Gordon equation has two branches of solutions, one of which is positive frequency and ...
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Is there a non-degenerate propagator for scalar fields?

I am trying to work out a functional propagator $K[\phi_{in},\phi_{out};t]$ for a scalar field for a free Klein Gordon field. It must satisfy the Hamiltonian equation: $$\left(i\frac{\partial}{\...
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What are the symmetries of the Schrödinger functional?

Given the definition here. What are the symmetries of: $$\mathcal{S}[\phi_2,t_2;\phi_1,t_1]=\langle\,\phi_2\,|e^{-iH(t_2-t_1)/\hbar}|\,\phi_1\,\rangle.$$ which is the amplitude to go from the field ...
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Time dependence of ladder operators in QFT

I'm currently going through Matthew D. Schwartz book Quantum Field Theory and the Standard Model, p. 23. For free (non interacting) field theories we are able to quantise the field by expanding our ...
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$SU(2)$ doublets from the transformation law of a matrix of scalar fields

If we have a $2 \times 2$ $SU(2)_L$ and $SU(2)_R$ matrix $\Phi=\begin{bmatrix} a & c \\ b & d \end{bmatrix}$, where a, b, c and d are four complex Klein-Gordon fields, that under a gauge ...
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Is causality violated in QFT?

I'm studying QFT, and Peskin is his book takes a couple of paragraphs to talk about causality in QFT, using the Klein-Gordon field as an example. The book says on p. 28: To really discuss causality, ...
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Gauge invariance of Klein-Gordon equation in electromagnetic field

I would like to show, that the following two equations: are invariant under a local $U(1)$ transformation: Before coming to my question, I will show you what I did: I defined the local U(1) ...
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Generating Functional for Complex Scalar Theory

The generating functional for a free complex scalar field theory is given by: $$W[J,J^*]=\int D\phi D\phi^* \exp (i \int_{}^{} d⁴ x [(\partial_{\mu}\phi)^*(\partial^{\mu}\phi) -m^2\phi^*\phi + J^*\...
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Changing variables between two different metric ansatzes in the calculation of the Klein-Gordon equation

My question concerns changing variables in the calculation of the Klein-Gordon equation for a scalar field given two different "guesses" for the metric. I consider the following Einstein ...
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Form of the MTZ black hole hair from Klein-Gordon equation

In the MTZ paper "Exact black hole solution with a minimally coupled scalar field" by Martinez, Troncoso, & Zanelli, the scalar hair field is given by $$\phi(r)=\sqrt{\frac{3}{4\pi G_N}}\...
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Time evolution operator and Klein-Gordon-Equation

The basis of classical QM is the postulate of a time evolution operator $$|\alpha,t_0;t\rangle=U(t,t')|\alpha,t_0;t'\rangle$$ Is it correct to interpret this postulate as All future states are ...
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Why is the annihilation operator associated with the positive frequency solutions?

I am looking for a qualitative explanation, if possible, of the following passage from Peskin & Schroeder, bottom of page 26, concerning the quantised real Klein-Gordon field: A positive ...
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How to verify the Klein-Gordon field commutation relations, Peskin and Schroeder Equation (2.30)

I am trying to verify the commutation relation given in Peskin and Schroeder. In particular, I don't know how to go between these two lines: $$[\phi(\textbf{x}), \pi(\textbf{x}')] = \int \frac{d^3p ...
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Klein-gordon, Dirac equation and the spin of the particles [closed]

This question might seem basic, but how does one conclude that the Klein-Gordon equation describes spin zero particles but Dirac equation describes spin half particles. Thanks. EDIT: Adding more ...
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Quantising the energy-momentum relation

The energy-momentum relation in special relativity states $m^2 = E^2 - ||p||^2$ (in natural units). So $$ E = \pm\sqrt{\| p \|^2 + m^2}. $$ If we want to find a theory for a relativistic free ...
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KG on AdS space-time

In his TASI notes Oliver DeWolfe starts with the KG equation on the Poincaré patch metric $$ ds^2=\frac{r^2}{L^2}(-dt^2+dx^2)+\frac{L^2}{r^2}dr^2. $$ When we use the ansatz$$ \phi(r\rightarrow\infty,x,...
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Inflationary Klein Gordon Field, derivation of a result [closed]

Good afternoon, I have a problem whilst studying an Inflationary Klein Gordon field theory. The main problem arises when I try to derive the following result: what I have is: $$\ddot\phi + 3H\dot\...
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Schrodinger equation using relativistic kinetic energy [duplicate]

I have seen work with introducing relativistic corrections to the momentum term of the Hamiltonian by taking higher order terms in the Taylor series, but do people work with $H = \sqrt{P^2 c^2 - m^2 c^...
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Writing the Klein-Gordon in terms of the gamma matrices

Hello I have a quick question regarding the Dirac gamma matrices or Dirac equation and the Klein-Gordon equation. Recall that the Klein-Gordon is given by the following \begin{equation} \left(\Box + ...
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How can I prove $\hat{a}^\dagger(\vec{k})$ is a complex scalar field?

I want to prove $\hat{a}^\dagger(\vec{k})$, the creation operator for real Klein-Gordon bosons transforms like a complex scalar field under Lorentz transformations, so $$\exp\left\{-\frac{\mathrm{i}}{...
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Relationship between the Klein-Gordon equation and Poincaré invariance

Derivations of the Klein-Gordon equation such as the one given by Phoenix in here, are based on studying the wave equation of the wave function of a relativistic particle. In this case, the Klein-...
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Quantization of complex scalar field

I'm learning Peskin's qft now and I'm a little confused about problem 2.2 . Suppose I write the field $\phi(x)$ as: $\phi(x) =\int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{p}}} (a_{p}e^{-ipx}+b_{p}e^...
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Quantization of the Klein-Gordon equation, sign problem

In Peskin and Schroeder, they quantize the Klein-Gordon field in the following way. They write the Fourier transform of $\phi(x,t)$ $$ \phi(x,t)=\int \frac{d^3 p}{(2\pi)^3}e^{ipx}\phi(p,t) $$ after ...
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Problem with Klein-Gordon equation derivation

In Notes for a course on Classical Fields by R. ALdrovandi, one the the exercises in page 94 is to derive the klein Gordon equation $(\Box + m²)\phi = 0$ from the following lagrangian density \begin{...
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Separation of Klein-Gordon-/Dirac-equation (Bohmian-mechanics)

With the function $R{ e }^{ \frac { i }{ \hbar } S }$ one can separate the Schrödinger equation $$i \hbar \frac{\partial \psi}{\partial t}=\left(-\frac{\hbar^{2}}{2 m} \nabla^{2}+V\right) \psi$$ into ...
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Quick question on Deriving Klein–Gordon equation from Dirac equation

On page 172 of Schwatz’s QFT book, he derives the Klein–Gordon equation from Dirac equation as following: $$(i \not\partial +m) (i \not\partial -m)\psi=(-\frac{1}{2} \partial_\mu \partial_\nu {\gamma^...
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Ladder operators in terms of Klein Gordon real field

an exercise asks me to explicit the functions ${a}(k)$ and $a^{*}(k)$ in term of the real field $\phi$ and its temporal derivate $\partial_0 \phi$ if the general solution of the Klein-Gordon equation ...
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Continuous plane wave solutions to Klein Gordon Field Equation

The continuous plane wave solution to the Klein Gordon Field Equation can be written as $\phi(x) = \int\frac{d^3\vec{k}}{\sqrt{2(2\pi)^3w_\vec{k}}} a(\vec k) e^{-ikx} + \int\frac{d^3\vec{k}}{\sqrt{2(...
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How can a Dirac field $\psi$ satisfy KG equation? What's wrong in my derivation?

The covariant form Dirac equation $(i\gamma^\mu\partial_\mu-m)\psi(x)=0$ can be multiplied from the left with the operator $(i\gamma^\nu\partial_\nu+m)$ and 4xpanding it out, to get $$(i\gamma^\nu\...
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Mass dimension of Klein Jordan field [closed]

I like to know about dimension of KG fields, Wikipedia searches don't give me a satisfied answer Can any one please help me?
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How can I find out the 'Klein-Gordon' equation by using Path Integral techniq? [closed]

I get the following relation from eqn. (6.52) in Sakurai(2nd edition) $$\Psi +\Delta t \frac{\partial \Psi}{\partial t}+\Delta t^2\frac{1}{2!}\frac{\partial^2\Psi}{\partial t^2}+...=\lim_{\Delta t\to ...
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Interpretation of the quantum field in light of the interpretation of propagators

In Page 38 of David Tong's QFT notes and Page 27 (Chapter 2.4 The Klein-Gordon Field in Space-Time under the heading Causality) of Peskin and Schroeder's Introduction to Quantum Field Theory, the ...

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