Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
2answers
51 views

Invariance of Klein-Gordon equation under gauge transformations

I'm sure this is really simple, and I might be right; it's just that I'm not sure. I'm asked to prove that the Klein-Gordon equation it's invariant under global gauge transformations. In Greiner's ...
0
votes
2answers
76 views

Dirac equation implies Klein Gordon equation?

In Dirac field (Peskin and Schroeder), there is one equation in which it multiples the Dirac operator $(-i\gamma^{\mu}\partial_{\mu}-m )$ with $(i\gamma^{\nu}\partial_{\nu}-m )$ to get $\partial^2+...
2
votes
1answer
110 views

Problems of Klein Gordon equation

Consider the Klein-Gordon equation $$(\square+m^2)\varphi=0.$$ People usually claim that $\varphi^* \varphi$ cannot be interpreted as a probability density because $\int d^3\vec{x}\varphi(t,\vec{x})^*...
0
votes
1answer
46 views

Divergent integral problem

When expanding the scalar field vacuum energy $$\sum_k \frac{1}{2} \omega_k = \frac{1}{2} (L/2\pi)^{n-1} \int \omega(k) d^{n-1}k = \frac{(L^2/4\pi)^{(n-1)/2}}{\Gamma(\frac{n-1}{2})} \int_0^\infty (k^...
2
votes
1answer
112 views

Lagrangian of Klein Gordon equation

Consider the following Lagrangian density $$ \mathcal{L}(\Phi,\partial_\mu\Phi)=-\frac{1}{2}\partial_\mu\Phi\partial^\mu\Phi-\frac{m\Phi^2}{2}. $$ I want to calculate the equation of motion using the ...
0
votes
0answers
37 views

QM limit of QFT in Schwartz [duplicate]

In Matthew Schwartz's QFT text, he derives the Schrodinger Equation in the low-energy limit. I got lost on one of the steps. First he mentions that $$ \Psi (x) = <x| \Psi>,\tag{2.83}$$ ...
0
votes
2answers
117 views

Finding the expression for probability density (the Klein Gordon equation)

Source: Quantum Field Theory for the Gifted Amateur by Tom Lancaster, Stephen J. Blundell. I am struggling to understand the logical step from the outline of the 'proof' in the footnote, to the fact ...
1
vote
0answers
85 views

Where this interpretation for the field modes comes from?

I'm reading the book "Modeling Black Hole Evaporation" by Alessandro Fabbri and Jose Navarro-Salas, and in section 3.3.2 they talk about wavepackets at $\mathscr{I}^+$. It all starts like this: one ...
1
vote
1answer
57 views

Event horizon and the existence of point particles

In this paper by David Kuap that first introduced the concept of Boson stars, he states that when the Einstein-Klein-Gordon system of equations is solved, the solutions obtained do not account for an ...
1
vote
1answer
32 views

Is it possible for me to use the Pauli matrices to show that they can give you the Klein-Gordon equation even though the KG equation isn't a matrix?

Start with general wave equation $${\partial \over \partial t}\Psi=\pm \vec \alpha\cdot \vec \nabla\Psi$$ Show that the choice of $\alpha_i=\sigma_i$ ($\sigma_i$ are the Pauli matrices) and squaring ...
1
vote
0answers
48 views

Decoupling of degrees of freedom in Klein-Gordon equation

In David Tong's notes in QFT he states that the degrees of freedom decouple in momentum space for the Klein-Gordon eq. He writes that this can be seen by using the Fourier transform (see picture below)...
5
votes
5answers
200 views

$m$ in Klein-Gordon Equation

The Klein-Gordon equation is given by $$ (\square + m^2) \phi(x) = 0 $$ where $\square$ is the d'Alembertian operator, $m \in \mathbb{R}$ and $\phi$ is a scalar field. Question: What is $m$ in the ...
1
vote
1answer
52 views

Question about Mode expansion of free compact boson

$(1+1)$-Dim free compact boson, Lagrangian is $$\mathcal{L}= \frac{1}{2}(\partial_\mu\phi(\sigma,t))^2$$ with $\phi(x,t)\sim\phi(x,t)+2\pi r$ and periodic boundary condition along $x$, i.e. $\phi(\...
1
vote
0answers
42 views

Time-independent Klein-Gordon PDE

Given the KG PDE: $$\psi_{tt} - \psi_{xx} + m^2 \psi = 0.$$ Wikipedia describes the time-independent variant of this as just setting $\psi_{tt}=0$. My question is this: For the Schrödinger ...
2
votes
2answers
131 views

$\partial^{\nu} \partial_{\nu}$ vs. $\partial_{\nu} \partial^{\nu}$

I was doing a problem regarding field theory. I am given the following lagrangian density: $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi_i\partial^\mu\phi_i-\frac{m^2}{2}\phi_i\phi_i$$ for three scalar ...
0
votes
1answer
66 views

Scalar particles are described by a real scalar field or by a complex one?

Well, in the title is already stated my main question. I know you can use a complex scalar field to describe two real scalar fields, by using just one that involves both of them. But, in the modern ...
0
votes
1answer
118 views

Sign mistake in the energy momentum tensor of the Klein-Gordon Equation

Recently I understood that the energy momentum tensor can be calculated by: \begin{equation} T_{\mu \nu}=\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{\mu \nu}}.\tag{1} \end{equation} So consider ...
1
vote
0answers
51 views

Hamilton equations of motion for matter fields coupled to general relativity in ADM formalism

Do you know what are the Hamiltonian formalism analogs of the Klein-Gordon equation and/or the Maxwell equations in general relativity? Showing how these equations of motion for matter in the ...
4
votes
2answers
263 views

Non-Relativistic Limit of Klein-Gordon Probability Density

In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as: $$ \begin{eqnarray} P & = & \dfrac{...
1
vote
2answers
93 views

Intuitive explanation for the free field Lagrangian?

The free field Lagrangian is $$\mathcal{L}=\frac 1 2 \partial^\mu\phi\partial_\mu\phi-\frac 1 2m^2\phi^2$$ with sign convention $(+,-,-,-)$. Plugging this into the Euler-Lagrange equations gives the ...
1
vote
2answers
113 views

Corresponding particle-antiparticle solutions for Klein-Gordon equation

For free particle solutions in a box, the following 4 solutions are possible(Not all 4 are independent though) as $$\psi_+=A_+ \exp{\frac{i}{\hbar}(px-Et)}\\\psi_+^*=A_+^* \exp{\frac{-i}{\hbar}(px-Et)}...
0
votes
0answers
25 views

Sine Gordon model in 3+1 Dimensions

I'm have read the publication of Neuenhahn, C. and Marquardt, F. (2015) ‘Quantum simulation of expanding space–time with tunnel-coupled condensates’, New Journal of Physics. IOP Publishing, 17(12), ...
1
vote
1answer
94 views

Can someone Tong got this equation in his QFT notes

Can someone explain how D.Tong got equation 2.18 in his QFT notes in chapter 2? I am lost from equation 2.5, can someone explain? Link to notes: http://www.damtp.cam.ac.uk/user/tong/qft.html Can ...
1
vote
2answers
177 views

E.L. Equations in QFT

In QFT, we use the Lagrangian to construct the Hamiltonian, and in the Interaction Picture (with regards to the Free Field Hamiltonian) use the full Hamiltonian to calculate the changes in the field (...
0
votes
1answer
44 views

Units of Klein-Gordon equation

I'm looking at the units of the Klein-Gordon equation $$u_{tt} - c^2\Delta u = -\frac{m^2c^2}{\hbar^2}u. $$ Disregarding the units of $u$, which are the same everywhere and so cancel, I get $seconds^{-...
0
votes
0answers
134 views

Mode Expansion in Klein-Gordon QFT

I have a confusion regarding the mode expansion of the Klein-Gordon field theory. I am following Peskin and Schroeder. My questions are about how we formally get to the expansion of the KG QFT in ...
2
votes
1answer
67 views

Solution to Klein-Gordon equation: real field condition and other questions

Sorry for the lengthy question, pretty much the whole text is the standard derivation of the solution of the KG equation which I included to illustrate my doubts, and some questions are at the end. ...
3
votes
2answers
104 views

Klein Gordon equation in the nonrelativistic and semiclassical limit in a Wigner approach

I would like to analyse the semiclassical and nonrelativistic limit of the Klein-Gordon equation, \begin{equation} \frac{1}{c^2} \partial_t^2 \phi - \Delta \phi + \frac{M^2 c^2}{\hbar^2} \phi =0. ...
0
votes
0answers
72 views

Using fourier analysis of the Klein Gordon equation

This question is more about a mathematical detail, and I am undoubtedly missing something very obvious. And note, I have sifted through the numerous questions on Fourier transform (FT) and the Klein-...
2
votes
2answers
220 views

Solution to the Klein-Gordon Equation

Most textbooks solve the Klein-Gordon equation with the ansatz $$\varphi(\mathbf{x},t) = \int \frac{\mathrm{d}^3\mathbf{k}}{f(k)}\left(a(\mathbf{k})\, \mathrm{e}^{i k x} + b(\mathbf{k})\,\mathrm{e}^{-...
1
vote
1answer
128 views

Tensors and the Klein-Gordon Equation

Consider the Klein-Gordon equation: \begin{equation} \frac{\partial^2 \psi}{\partial t^2} = c^2 \Delta \psi - \frac{m^2 c^4}{\hbar^2} \psi, \end{equation} and define for each one of its solutions $\...
1
vote
1answer
92 views

Sign confusions in solution to Klein Gordon's equation

I have two basic questions on the solution of the Klein Gordon equation. The Lagrangian of the Klein Gordon field is $$\mathcal{L}=\frac12\partial_\mu\phi\partial^{\mu}\phi-\frac12m^2\phi^2 $$ ...
7
votes
0answers
297 views

Using a time-like boundary as a computer?

Question and Summary Using classical calculations and the Robin boundary condition I show that one calculate the anti-derivative of a function within time $2X$ (I can compute an integral below) $$\...
1
vote
1answer
276 views

Nonexistence of a Probability for the Klein-Gordon Equation

David Bohm in his wonderful monograph Quantum Theory, in Section 4.6 discusses the difficulties one encounters in trying to develop a relativistic quantum mechanics. He starts from the relation \...
3
votes
1answer
91 views

Klein-Gordon quantization and SHO analogy

I understand that the procedure to quantize Klein-Gordon's field is to manipulate in a such a way to bring up the simple harmonic oscillator behavior of the field. This is done by Fourier transforming ...
3
votes
0answers
93 views

Unexpected symmetry of wave equations in momentum representation

In the $x$-representation, the translational invariance implies that $$ \mathcal{D}[\psi(\vec{x},t)]=0\quad \Longrightarrow\quad \mathcal{D}[\operatorname{e}^{i\vec{a}\hat{\vec{P}}}\psi(\vec{x},t)]=0 $...
4
votes
3answers
249 views

Can a second-order Schrödinger equation preserve the norm?

Suppose we lived in a universe in which the Schrödinger equation contains second order time derivatives, $$i\hbar \partial_t^2|\varphi(t)\rangle = \mathbb{H} | \varphi(t)\rangle.$$ Would it be true ...
3
votes
1answer
173 views

Klein-Gordon-Equation contains no Spin

I have a question about an argument used in Schwabl's "Advanced Quantum Mechanics" concerning the properties of the Klein-Gordan-Equation (see page 120): Since the eigenenergies of free solutions are ...
1
vote
1answer
128 views

Interaction-Picture Field as Solution of Klein-Gordon Equation

I am following a problem in a QFT textbook (Srednicki) which asks us to show that the interaction-picture field $\phi_I(\textbf{x},t)=e^{iH_0 t}\phi(\textbf{x},0)e^{-iH_0 t}$ obeys the Klein-Gordon ...
0
votes
0answers
53 views

Klein-Gordon-Equation provides a Scalar Theory that doesn't contain Spin [duplicate]

I'm reading actually Schwabl's "Advanced Quantum Mechanics" and encounter an understanding problem while considering the Klein-Gordan-equation. Here the excerpt: Obviously, since the eigen energies ...
0
votes
1answer
66 views

Is the inverse of the Klein-Gordon equation ever used in physics?

The Klein-Gordon equation (scaling constants) is $$\square u = -m^2 u.$$ I am wondering if the equation $$\square u = m^2 u.$$ for real $m$ ever shows up in the physical literature?
1
vote
0answers
41 views

Replacing a squared potential by a position-dependent mass

I'm studying the solutions of the Klein-Gordon and Dirac equations for a relativistic particle in a potential of the form $$V(x)=\left\lbrace\begin{array}{ll} 0, & x\in[0,L]\\V_0, & x\not\in[0,...
2
votes
1answer
158 views

How to derive this expression for the free scalar field in QFT? (Peskin & Schroeder)

In the introductory text to quantum field theory by Peskin & Schroeder, they state that in analogy to the simple harmonic oscillator in quantum mechanics, the free scalar field can be expressed as:...
4
votes
2answers
179 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
246 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
65 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
61 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 ...
2
votes
0answers
38 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
37 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 ...