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.

Filter by
Sorted by
Tagged with
5
votes
2answers
230 views

Is the ground state energy of a quantum field actually zero?

I start by outlining the little I know about the basics of quantum field theory. The simplest relativistic field theory is described by the Klein-Gordon equation of motion for a scalar field $\phi(\...
0
votes
0answers
355 views

Propagator solution to Klein-Gordon equation

We know that the Klein-Gordon operator is given by $(\partial^2+m^2)=(\partial_\mu\partial^{\mu}+m^2)$, which is used to describe the evolution of relativistic free particles. How can we show that ...
0
votes
1answer
60 views

can somebody explain how you get the second line from the first line in the picture?

I'm trying to understand the transition from the 1st line of the Lagrangian to the second. we substitute for $\eta$ but how is the multiplication happening here? if I multiply the terms into the ...
0
votes
1answer
83 views

Why is it necessary to introduce different sets of creation and annihilation operators to quantize the complex K-G field?

I am reading Peskin & Schroeder and in chapter 2 (p.21) he quantizes the real K-G field such as: $$\phi=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_p}}\left(a_pe^{ip·x}+ a^{\dagger}_pe^{-ip·x}\...
1
vote
1answer
107 views

Getting the relativistic inner product of Siegel's book

Last time I was discussing with a physicist about quantum field theory and how in the firsts chapters of textbooks it is often regarded that the Klein-Gordon equation does not have a positive definite ...
3
votes
0answers
44 views

Is there a mathematical singularity in the relativistic contraction of valence orbitals as a function of nuclear charge?

I'm looking at Figure 1 on p. 2 of Jansen, "Effects of relativistic motion of electrons on the chemistry of gold and platinum" (2005) (should be a free download). From the bottom curve, there is such ...
3
votes
0answers
39 views

Do relativistically-contracted electron states have the same energy and angular momentum values?

I've been reading that electron bound states are defined by four quantum numbers, $n$, $l$, $m_l$, and $m_s$, respectively the principal quantum number, the azimuthal quantum number, the magnetic ...
2
votes
3answers
936 views

Derivation of Klein-Gordon equation in General Relativity

I am trying to derive the Klein-Gordon equation for the case of GR using the action: $$S\left[ {\varphi ,{g_{\mu \nu }}} \right] = \int {\sqrt g {d^4}x\left( { - {1 \over 2}{g^{\mu \nu }}{\nabla _\mu ...
1
vote
0answers
279 views

How one can use the Klein-Gordon equation to deal with the hydrogen atom? [closed]

I'm not familiar with physics jargons compare to those who study physics professionally, but to remark my knowledge regarding this question, the Klein-Gordon equation is a relativisitc equation for ...
4
votes
1answer
378 views

Klein-Gordon Inner product being independent of the choice of spacelike hypersurface $\Sigma$ used to integrate it

Here I'll work in flat 4-dimensional Minkwoski space, but using arbitrary coordinates (described by some metric $g_{\mu\nu}$). Suppose we've got two complex-valued scalar functions $f$ and $g$ which ...
2
votes
2answers
154 views

Real Scalar Field: in what sense are the Minkowski modes complete?

When considering a real scalar field with Lagrangian $$\mathcal{L} = - \frac{1}{2}(\partial_{\mu} \phi)(\partial^{\mu} \phi) - \frac{1}{2}m^2\phi^2$$ the equation of motion is the Klein-Gordon ...
4
votes
0answers
165 views

What motivates the definition of the Klein-Gordon Inner Product? [duplicate]

I am following along Marc Casal's lecture slides "Quantum Field Theory in Curved Spacetime". For scalar functions $f$ and $g$ we define the Klein-Gordon inner product as follows: $$ <f,g> \ = \ ...
0
votes
0answers
37 views

Models of Inflation

I'm trying to get an overview of cosmological models which include inflation. My aim is to study the numerics of nonlinear Klein- Gordon equations which in fact are the Friedmann equations in the FRLW ...
3
votes
1answer
668 views

Lorentz transformation of the Klein-Gordon equation

In the Lorentz transformation of the field $\partial_\mu\phi(x)$ (Peskin, p.36) \begin{eqnarray} \partial_\mu\phi(x)\to\partial_\mu(\phi(\Lambda^{-1}x))=(\Lambda^{-1})^\nu_{\phantom{\nu}\mu}(\...
0
votes
1answer
181 views

Quantization of Klein-Gordon Field

I have a question about the quantization procedure of the Klein-Gordon field as presented in Peskin&Schroeder. The field is expressed as a Fourier decomposition $$ \phi(x,t) = \int \frac{d^3p}{(2\...
0
votes
0answers
37 views

Eigen-Energy of Vibration on a Loop

I recently started this recreational project to find the energy of the modes on a loop. To obtain the eigen-energies $E_n$, I decided to solve the 1D wave equation on a loop of circumference $2\pi l$ ...
5
votes
1answer
617 views

Creation and Annihilation Operators in QFT

I have a general question about heuristic way of QFT to introduce creation and annihilation operators: The Klein-Gordon field is introduced as continuous interference of plane waves $\mathrm{e}^{i(\...
1
vote
0answers
59 views

How to interpret the concept that a (scalar) field is a linear superposition of harmonic oscillators?

I've reading an introduction lecture on QFT, but I didn't started very well. I have troubles to understand some concepts of the very first example (But I think that they apply with the other fields): ...
2
votes
1answer
64 views

Where do the particle and antiparticle wavefunctions originate from in the Klein Gordon equation?

In my textbook (Sakurai) it is given that $$\left(D_\mu D^\mu+m^2\right)\Psi(\mathbf{x},t)=0$$ where $D_\mu=\partial_\mu+ieA_\mu$ is the covariant derivative. It states that since it is a second ...
3
votes
1answer
141 views

Quantization of Klein-Gordon field between two boundaries

Consider a real scalar $\phi(x,t)$ with mass $m$ in $1+1$ dimensional spacetime, described by the 2d free Klein-Gordon action. $\phi(x,t)$ lives on an interval $0 \leq x \leq L$, and is subject to ...
1
vote
1answer
118 views

Obtaining mode decomposition of quantum field in a box with Dirichlet condition

I am considering the Klein Gordon Equation in a box with Dirichlet conditions (i.e., $\hat{\phi}(x,t)|_{boundary} = 0 $). 1-D functions that obey the Dirichlet condition on interval $[0,L]$ are of the ...
1
vote
1answer
412 views

Commutator of canonical fields in Quantum Field Theory

Let $\phi(\vec{x},t)$ denote the canonical fields and $\pi(\vec{x},t)$ denote the canonical impulses where they're given by: \begin{equation} \phi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2\omega_{\...
2
votes
1answer
206 views

Lagragian density of a massless scalar field

I have seen in some books that the simplest Lagrangian density of a massless scalar field is $$\mathscr{L}=\dfrac{1}{2}\partial^\mu\phi\partial_\mu\phi=\dfrac{1}{2}\left(\partial_\mu\phi\right)^2.$$ ...
0
votes
1answer
984 views

Klein-Gordon equation in curved space time

The Klein-Gordon equation in curved spacetime has the following form: $$\left (\square+m^2 \right)\Phi=\left[\frac{1}{\sqrt{-g}} \partial_{\mu}\left(\sqrt{-g}g^{\mu\nu} \partial_{\nu} \right)+m^2\...
2
votes
2answers
110 views

What wavefunctions do the creation operator of a massive real scalar free field create?

A real scalar free field of mass $m$ can be represented as: $$\hat\phi(\mathbf{x}) = \int \frac{d^3\mathbf{k}}{(2\pi)^3\sqrt{2\omega_{\mathbf{k}}}}\hat a_{\mathbf{k}}e^{i(\omega_{\mathbf{k}}t - \...
-1
votes
1answer
124 views

Klein Gordon equation in tensorial form

The problem is to show that $$\psi=\psi_0e^{iP_\mu x^\mu}$$ is a solution to Klein Gordon equation $$(\partial _\mu \partial ^ \mu +m^2)\psi=0$$ if and only if the 4-momentum $p _\mu$ satisfies the ...
2
votes
0answers
86 views

Klein-Gordon and localizability

If we consider the theory of a single relativistic point particle, quantized using whatever appropriate method, the wavefunction simply obeys the Klein-Gordon equation, which allows for a fairly wide ...
1
vote
1answer
156 views

Contradictory solutions to the Klein-Gordon equation as an initial value problem (canonical formulation)

I'm going to begin with a prelude about constructing the solution using the action/Lagrangian formalism in order to provide context and a point of comparison for the canonical one. The main question ...
0
votes
0answers
94 views

A step in derivation of Schrodinger eq. from scalar QFT

I've read the related questions and understand the following derivation except for one key step that involves a minus sign. (From Schwartz ch.2): $$ i \langle 0| \partial_t \phi(\vec x,t) | \psi\...
3
votes
1answer
799 views

Solving the Klein-Gordon Equation with a Fourier Transform

So I am trying to solve the Klein-Gordon equation using a Fourier transform of the spatial components only. The Klein-Gordon equation reads: $$ (\partial ^2 + m^2)\phi(x) = 0. $$ If I let $$ \phi(x)...
1
vote
1answer
335 views

Positive and negative energy modes of scalar field

If we have the normal KG scalar field expansion: $$ \hat{\phi}(x^{\mu}) = \int \frac{d^{3}p}{(2\pi)^{3}\omega(\mathbf{p})} \big( \hat{a}(p)e^{-ip_{\mu}x^{\mu}}+\hat{a}^{\dagger}(p)e^{ip_{\mu}x^{\mu}} ...
12
votes
5answers
1k views

Why can the Klein-Gordon field be Fourier expanded in terms of ladder operators?

Using the plane wave ansatz $$\phi(x) = e^{ik_\mu x^\mu}$$ the solution to the Klein-Gordon equation $(\Box + m^2) \phi(x) =0$ can be written as a sum of solutions, since the equation is linear and ...
0
votes
1answer
161 views

Regarding the commutator of ladder operator in QFT

I am trying to verify the computation of the commutator of the ladder operator for Klein-Gordon solutions, but it seems like I am unable to do it properly. Here is what I do: For, $$ \varphi(x^\mu)=\...
0
votes
1answer
367 views

Fourier transforming the Klein-Gordon equation

I'm aware of the fact that there are similar questions on this forum but I could not find an answer that fits my problem. Many textbooks state that a general solution to the Klein-Gordon equation \...
1
vote
0answers
236 views

Explicitly proving that the Hamiltonian is Lorentz covariant

I want to show explicitly that the Hamiltonian $$ H = -\Omega V + \int d\tilde{\textbf{k}} \omega (a^\dagger(\textbf{k}) a(\textbf{k}) + b(\textbf{k}) b^\dagger(\textbf{k}) ) $$ is Lorentz covariant....
1
vote
1answer
448 views

Solution of Klein-Gordon field in Schwarzschild metric

I was trying to understand the behavior of scalar field in the vicinity of black hole and I set up the Klein-Gordon equation in the background of Schwarzschild metric. But I am having difficulty in ...
0
votes
0answers
172 views

Transformation rule under translation of the Fourier transform of a scalar field

Let $\phi(x)$ be a real classical scalar field which satisfies the Klein-Gordon equation; without loss of generality it can be written as: $$ \phi(x) = \int \frac{d\vec k}{(2\pi)^{3/2} 2 \omega_k} \...
4
votes
1answer
161 views

Hermiticity property of “position” operators with Klein-Gordon inner product

Given an appropriate function space $H$, suppose $H_0$ to be the linear subspace spanned by the solutions of the Klein-Gordon equation and to equip that linear subspace with the inner product $$\...
3
votes
2answers
182 views

Vacuum energy of a real Klein-Gordon field

Hamiltonian for a Klein-Gordon field can be written as - $$H= \int \frac{d^3p}{(2\pi)^3 \omega_{\vec{p}}}[a^{\dagger}_\vec{p}a_\vec{p}+\frac{1}{2}(2\pi)^3\delta^{(3)}(0).\tag{1}]$$ In one of my ...
1
vote
0answers
87 views

Time evolution operator of Klein-Gordon field

If $U(t)=e^{itH}$ is the time evolution operator. And $|\phi \rangle$ is a state of a field at particular time $t_1$ and $|\phi' \rangle$ is the state of a (free) Klein Gordon field at a time $t_2$. ...
1
vote
1answer
225 views

Schwinger-Dyson eqation for a Klein-Gordon-Field

I have a question about the usage of the Schwinger-Dyson-Equation for the Klein-Gordon-Field. $$ i <0|T (\delta S / \delta \phi(x) ) \phi(x_1)\ldots |0>+<0|T\delta(x-x_1)\ldots|0>=0 .\tag{...
3
votes
1answer
979 views

Derivation of the Klein Gordon Propagator

I'm learning some quantum field theory. I'm currently using the book An Introduction to Quantum Field Theory by Peskin and Schroeder. I have a problem that I can't figure out at the moment. It is ...
0
votes
0answers
90 views

Do we need to know first time derivatives of Klein-Gordon fields to make future predictions?

In the quantisation of the Klein Gordon field, because it has a second derivative, we need to know the values of the field and also the first derivative in order to predict future values of the field. ...
2
votes
1answer
509 views

“Deriving” Lorenz Gauge Condition from the Lagrangian

I am studying QFT with Mandl & Shaw, Quantum Field Theory and ran into Problem 2.3 (page 37 in the Second Edition) that says: Problem 2.3: Show that the Lagrangian density $$\mathscr{L} = -...
2
votes
2answers
488 views

What would change if we add a linear term in the Klein-Gordon Lagrangian?

The usual Klein-Gordon Lagrangian reads \begin{equation} \mathscr{L}= \frac{1}{2}( \partial _{\mu} \Phi \partial ^{\mu} \Phi -m^2 \Phi^2) \, . \tag1\end{equation} Without additional symmetry ...
1
vote
0answers
161 views

Free complex scalar field - showing operators are creation and annihlation?

Using the method of canonical quantization we can show that for a free scalar field we have: $$\phi(x)=\int d\tilde p (a(\vec p) e^{-ipx}+b^\dagger(\vec p) e^{ipx})$$ where $a(\vec p)$ and $b^\dagger(\...
3
votes
1answer
298 views

Lorentz Covariant formula for Noether Charges in QFT

I'm looking for a Lorentz covariant expression of Noether charges and I found this article: https://arxiv.org/abs/hep-th/0701268, section II-A in particular. Consider specifically eq. (20-21), they ...
2
votes
1answer
317 views

Proof of a solution of Klein-Gordon equation with Killing vector

In this (very good) notes: http://people.physics.tamu.edu/pope/geomlec.pdf it is set as an exercise to proof that if $u_i$ solves the Klein-Gordon equation: $$(\Box -m^2 )u_i = 0$$ then you can ...
9
votes
2answers
560 views

Does relativistic quantum mechanics really violate causality?

The Hamiltonian $H=\sqrt{p^2+m^2}$ defines a one-particle quantum mechanics in the usual way. Let us call this theory RQM for short. Peskin and Schroeder claim that RQM violates causality because ...
3
votes
3answers
697 views

Two-point correlation function in Peskin's book [duplicate]

I am reading Peskin's book on QFT and I reached a part (in chapter 4) where he is analyzing the two-point correlation function for $\phi^4$ theory. At a point he wants to find the evolution in time of ...

1 2
3
4 5
8