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 Klein-Gordon Equation violate postulate of quantum mechanics?

In QFT for gifted amateur, authors stated the probability current may be negative, a negative probability violates the Copenhagen interpretation. However, in the book QFT by Srednicki mentioned the ...
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Waves in quantum field theory

There are two sources of 'waviness" in quantum field theory: waves in the underlying classical field, and the Schrodinger equation. I'm learning QFT using the notes by David Tong (I believe they ...
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Why does the Klein-Gordon harmonic oscillator have non-normalizable states?

The equation given here is the vector Klein-Gordon equation for the Harmonic oscillator potential. I have read about it in some papers. As far as I know, it gives the correct energy eigenvalues but ...
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Validity of Euler-Lagrange Equation in Quantum Theory [duplicate]

Lagrangian density for a single-spin 0-real-bosonic field ($\phi$) is given by, $$\mathcal{L}=-\frac{1}{2}\partial_\mu \phi \partial^\mu \phi-\frac{m^2}{2}\phi^2$$ Now if we formulate the Euler ...
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AQFT: Microscopic causality (local commutativity) of the free Klein-Gordon field

Let us consider the positive and negative frequency part of the free Klein-Gordon field operator $$ \hat\phi(x) = \hat\phi^-(x) + \hat\phi^+(x) \tag{1} $$ where $$ \begin{aligned} \hat\phi^-(x) = \hat\...
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Solution of Klein-Gordon equation admit no probability interpretation

Let's consider the solutions $\psi$ of the Klein-Gordon equation: $$\bigg{(}\frac{\partial^2}{\partial t^2}-\Delta + m^{2}\bigg{)}\psi(x) = 0$$ and define: $$\rho = \frac{i}{2m}\bigg{(}\psi^{*}\frac{\...
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Fourier mode expansion of relativistic scalar field: Negative frequency modes and positive energy requirement

Let $\phi(x)$ be a Klein-Gordon field, i.e., a field satisfying the Klein-Gordon equation $$ \left(\partial^2+m^2\right)\phi(x)=0\tag{1}, $$ then the Fourier expansion of the Klein-Gordon field is $$ \...
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Derivative of d'Alembertian (KG Equation proof)

I'm trying to obtain the Klein-Gordon equation using a specific lagrangian: $$\mathcal{L} = -\frac{1}{2} \phi(\Box+\mu^2)\phi$$ and the generalized Euler-Lagrange equation: $$\frac{\partial L}{\...
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Evaluating the operator of free photon lagragian

I want to evaluate the operator from a free photon Lagrangian as a exercise and arrive into something with the form: $$L = A_{\mu}O^{\mu \nu}A_{\nu};$$ parting from the free photon lagrangian: $$L = -\...
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Non-locality of pre-Klein-Gordon equation

In Relativistic Quantum Mechanics, Bjorken and Drell state that expanding the square root in the equation $$-\hbar^2\frac{\partial^2\psi}{\partial t^2}=\sqrt{-\hbar^2c^2\boldsymbol{\nabla}^2+m^2c^4}\...
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Quantum field creates particle interpretation

In Peskin & Schroeder's book, the authors say that if $\phi(x)$ is the (quantum) Klein-Gordon field operator in Schrödinger's picture and $|0\rangle$ is the vacuum state, the application $$\phi(x)|...
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Simplest possible solution to the Klein-Gordon field equation has a (KG) norm which is not constant in time

It is a fact that the Klein-Gordon inner product must be constant for all $t>0$, where the Klein-Gordon product is defined by $$ \langle f, g \rangle \ := \ i \int d^3x \; \left[ f^{\ast}(t,\mathbf{...
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Issue with a sign in the commutator calculation of field operators for a real scalar field

In the derivation for the commutator (real scalar field, Klein-Gordon equation) $$[\phi(x),\phi(y)]=0$$ I have solved up to $$[\phi(x),\phi(y)]=\frac{1}{2(2\pi)^3}\int \frac{d^3 p}{\omega_p}[e^{ip(x-y)...
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Problem with derivation of Dirac equation

In my book on QFT (Lancaster & Blundell) while deriving the Dirac equation, they arrive at the following result: $$(\partial ^2 +m^{2})=(\not\!\partial -im)(\not\!\partial+im)$$ They then state: &...
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The gradient of the d'Alembertian Green's function

So I have to prove that the d'alembertian of the associated green's function $G(t,t',\vec{r},\vec{r}')$ is equal to zero when given that $\vec{r}\neq\vec{r}'$ $$\left(\frac{1}{c^2}\partial^2 t-\Delta\...
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Is there a “square root” of Klein–Gordon equation which is different from Dirac equation?

Legend has it that Dirac arrives at the Dirac equation (all equations are in Planck units $c= \hbar = 1$ in this post): $$ i\gamma^\mu\partial_\mu\psi - m\psi =0 $$ by taking the square root of the ...
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Why does the 3-point function of a real scalar field vanish?

$$\langle0\lvert T\hat\phi(x_1)\hat\phi(x_2)\hat\phi(x_3)\rvert0\rangle$$ I'm looking for an intuition to it, if not an actual interpretation. Otherwise, I know how to get the result 0 using Wick's ...
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Lorentz-invariant measure in Klein-Gordon field

What is the exact reason why the solution of the classical Klein-Gordon equation is written as a mode expansion with a Lorentz invariant measure and after that the coefficients are promoted to ...
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Multiplication of symmetric and antisymmetric tensors times a vector, a question about indices

I am studying the Klein-Gordon equation that is invariant under Lorentz transformation, the infinitesimal transformation gives us: $$ \delta x^\mu = \epsilon^\mu_\nu x^\nu $$ $$ \delta \phi = \frac{1}...
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Dirac equation with minimal coupling derivation from Klein-Gordon equation

I am wondering if the form of the Dirac equation given in the case of minimal coupling can be "squared" to give back the corresponding Klein-Gordon equation as in normally done in the field-...
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Derivation of Klein-Gordon inner product

I've been studying the Klein-Gordon equation but I can't seem to wrap my head around this problem. The KG Equation is a second order differential equation that has solutions in the form $$ \psi= Ae^{\...
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Derivation of operator version of the classical wave equation

I have the following summarised derivation for the operator version of the classical wave equation for massless and material particle. My question is about the statement: However, a problem is that ...
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Peskin and Schroeder, where is the mass in the denominator of the simple harmonic oscillator Hamiltonian?

This relates to page 20 of Peskin and Schroeder. They state that the Fourier transform of the Klein-Gordon field satisfies the following: $$\left[\frac{\partial^2}{\partial t^2}+(|\vec p|^2+m^2)\right]...
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Derivation of the Dirac spinor expressed as a Fourier transform

Introduction For the Klein Gordon Field, the equations of motion are described by the equation $$(\partial_{\mu}\partial^{\mu} + m^2)\phi(\vec{x},t)=0$$ Which when the field is expressed as a Fourier ...
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What is the non-relativistic limit of the quantised electromagnetic field?

I’m not a physicist so this question may be naive ... For a real scalar field, quantisation yields the Klein-Gordon equation and the non-relativistic limit of this gives the Schrödinger field. What is ...
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Why is the $i\epsilon$-prescription necessary in the Klein-Gordon propagator?

When evaluating the Klein-Gordon propagator, in the book by P&S, p. 31, I see that, it is customary to shift the poles and add $i\epsilon$ in the denominator. I don't understand, why this is ...
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Why can you deform the contour in the integral expression for the Klein-Gordon propagator to get the Euclidean propagator?

I'm trying to understand the use of the Euclidean correlation functions in QFT. I chased down the problems I was having to how they manifest in the simplest example I could think of: the two-point ...
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What is the unit of Klein Gordon field?

Normally I don't care about units in the derivations on relativity or QM. Just set $\hbar = c = 1$. But learning about the energy momentum tensor for the Klein Gordon equation, I couldn't make $T^{00}$...
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Physical Interpretation of Field Operator in Quantum Field Theory and Mode Expansion

I'm struggling to understand the physical interpretation behind the field operators $ \phi(\mathbf x)$ and $\phi ^\dagger (\mathbf x)$ in quantum field theory. My understanding is $ \phi ^\dagger (\...
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Why in 3+1D QFT a scalar field has (mass) dimension $1$ but in 3+1D QM the wavefunction has dimension $3/2$?

In this question we assume as usual that $\hbar = c =1$, so the word "dimension" means the dimension in mass. From the fact that an action has the same unit as $\hbar$, or dimensionless ...
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Does special relativity limit quantum field propagation?

For the classical Klein-Gordon field, the motion of a wavepacket is constrained slower than the speed of light for $m^2 >0$ and constrained to exactly the speed of light for $m = 0$ (the equation ...
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2nd Quantization of KG Field

The space of solutions of Klein Gordon-equation Klein Gordon-equation is spanned by eigenbasis consisting of plane wave solutions $A\mathrm{e}^{-i(\vec{p}\cdot\vec{x} \pm E_pt)}$ which correspond to ...
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One dimensional Klein-Gordon harmonic oscillator

Some context: Let us consider a spinless charged massive particle of mass $m$, charge $q$ in an electrostatic potential $V(x) = \frac{m}{2q}\omega^2x^2$. The corresponding stationary Klein-Gordon ...
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Deriving Klein-Gordon from Hamilton's equations for fields using functional derivatives

I have found two different ways of doing this and I am seeking commentary on the fine nuance. Suppose there is a Hamiltonian $$ H=\frac{1}{2}\int\!d^3x \left[ \pi^2+(\nabla\varphi)^{\!2}+m^2\varphi^...
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Is the Klein-Gordon Equation an equation for a Classical field or a Quantum Field?

When Canonically Quantising the Klein-Gordon Field you usually start with the Klein-Gordon Equation, from which you can guess a corresponding Lagrangian Density. Then utilising this information along ...
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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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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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