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
0
votes
1answer
33 views

Show that time derivative of creation-annihilation operators of Klein-Gordon field are zero

For example, for the annihilation operator \begin{equation} a(\vec{k}) = C \int d^3x e^{i k\cdot x}\partial^ \leftrightarrow _t\phi(x), \end{equation} where C is a constant that I will ignore, the ...
-4
votes
2answers
145 views

Did Dirac derive the correct equation for the wrong reasons?

Did Dirac derive the correct equation for the wrong reasons? This is a question about the historical discovery of the Dirac equation and how it was deduced. Looking back at that discovery with our ...
3
votes
2answers
131 views
+50

Quantizing Klein Gordon Field: Sign Problem

I'm trying to re-derive the Quantization of the Klein Gordon Field but I'm running into sign problems. My starting point is: $$ \phi(x,t) = \frac{1}{(\sqrt{2 \pi})^3} \int \tilde{\phi}(k,t) e^{i kx}...
0
votes
1answer
57 views

Question about a point in Srednicki's QFT book

On page 6, Sredniciki says (taking into account the erratum), that the "simplest possibility is for Alice and Bob to agree on the value of the wave function at a particular space-time point". This ...
0
votes
2answers
126 views

Does the Schrodinger Equation care about spin?

I have taken the non-relativistic limit of the Klein-Gordon and Dirac equation, and both have brought me to the Schrodinger equation. The Klein-Gordon equation describes spin 0 particles, and the ...
1
vote
1answer
34 views

Klein-Gordon equation propagators: intersection with the support of the source

Let $(M,g)$ be a globally hyperbolic. Let $P = \Box - m^2$ be the Klein-Gordon differential operator. Following Fewster's notes, we may define the retarded/advanced propagators $$E^\pm : C^\infty_0(M)\...
0
votes
1answer
46 views

Klein-Gordon Inner Product from Greiner's book doubt

I was working on free field theory from Greiner's book "Field Quantization" In chapter 4, he introduces these phase functions: $$ u_{p}(\boldsymbol{x}, t)=N_{p} \mathrm{e}^{-\mathrm{i} p \cdot x}=\...
0
votes
1answer
61 views

QFT Klein-Gordon Equation “trick”

Both in the Wald and Parker/Toms texts on QFT in curved space time, when introducing QFT in flat space time first, they solve the Klein Gordon equation over the whole real line by placing the “field ...
0
votes
0answers
40 views

Expection values of the hamiltonian of Klein-Gordon field

The hamiltonian of the quantized Klein-Gordon field $\phi(\textbf{x},t)$ can be writting using the creation and annihilation operators: $$\hat{H} = \frac{1}{2} \int d^{3}\textbf{p} \ \omega_{p} (\hat{...
1
vote
1answer
42 views

Solutions of relativistic wave equations compared to classical wave functions

In classical quantum mechanics, absolute square of the wave function (i.e. $|\psi|²$) means probability density of particle's location, so when we integrate this over certain volume we get the ...
0
votes
0answers
38 views

Klein-Gordon/Maxwell Equation: dissipative or dispersive?

In Aspects of Symmetry, Coleman says (p. 185) ''Most of the simple field theories with which we are familiar have the property that all of their non-singular solutions of finite total energy are ...
0
votes
0answers
14 views

Units of the Klein-Gordon Propagator in SI Units

What are the SI units of the momentum-space propagator of the Klein-Gordon equation for a free particle?
0
votes
0answers
66 views

Transition from phi basis to occupation number in quantum field theory

We can construct the unitary transformation for change of basis from $x$ to number operator $n$ in harmonic oscillator by using $a|0\rangle=0$ and then multiply $\langle x|$ to the both side and ...
0
votes
2answers
68 views

Rewrite an equation by deriving the Schrondinger eigenvalue equation (linear momentum)

I want to rewrite the equation as follows: $$\frac{\partial^2\psi(x,t)}{\partial x^2}=-\bigg(\frac{2\pi}{\lambda}\bigg)^2\psi(x,t)$$ The initial equation is as follows: $$-i\frac{h}{2\pi}\frac{\...
0
votes
2answers
58 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 ...
1
vote
2answers
113 views

Derivation of Klein Gordon equation from Dirac equation; what does it mean?

In Dirac field (Peskin and Schroeder), there is one equation in which it multiples the Dirac operator $$(-i\gamma^{\mu}\partial_{\mu}-m )$$ by $$(i\gamma^{\nu}\partial_{\nu}-m ),$$ obtaining $\...
2
votes
1answer
154 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
150 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
129 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
87 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
61 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
33 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
53 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
215 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
53 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(\...
4
votes
2answers
174 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
137 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
78 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
150 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
56 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
329 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
107 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
126 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
101 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
187 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
46 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
152 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
74 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. ...
4
votes
2answers
133 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
248 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
132 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
95 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
318 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
279 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
96 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
94 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 $...