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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How does spin influence the dynamics of quantum mechanical systems?

I have just been introduced to the Klein-Gordon Equation and the Dirac Equation for the first time. The way they were explained to me, these equations govern the (relativistic) evolution of spin-0 and ...
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Derivation of the Klein-Gordon solution via Fourier Transforms

I recently graduate with a bachelor's in physics, and I've been trying to take the next steps toward learning QFT. To this end, I have been working through Peskin and Schroeder's textbook step-by-step....
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Quantization of field with other complete orthogonal system

I've learned the quantization of Klein-Gordon field using Fourier expansion. I understand that this process is kind of exchanging complex fourier coefficients to operator and makes it satisfying the ...
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Klein-Gordon equation with position-dependent mass [closed]

Does there exist a general solution for a differential equation like: $$\ddot{\phi}(x,t) - \partial^2_x\phi(x,t) + \phi(x,t)m^2(x) = 0,$$ where $m(x)$ is a known function.
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Is the phase velocity of plane wave solutions of the Klein-Gordon equation larger than $c$?

The phase velocity is given by $$ v= \frac{\omega}{k} \, .$$ Using the usual dispersion relation $$ E^2 = p^2c^2+ m^2c^4 \leftrightarrow \omega^2 \hbar^2= k^2\hbar^2 c^2 + m^2c^4$$ yields $$ v= \frac{\...
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Expanding about background field

I refer to this set of lecture notes by Hugh Osborn, equation 4.184 on p.70. We expand an action $S[\phi]$ around a background field $\varphi(x) = \phi(x) -f(x)$ If we expand the action $S[\phi]$ ...
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Is the scalar propagator an even function?

The scalar propagator for the Klein-Gordon Lagrangian is given by: $$D(x-y)=\int \frac{d^{4} k}{(2 \pi)^{4}} \frac{e^{i k(x-y)}}{k^{2}-m^{2}+i \varepsilon}$$ I need to know if it is an even ...
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WKB solution in QFT: classical action and particle vs antiparticle case

Consider the theory of a complex scalar field $$S[\psi, \psi^\dagger] = -\int d^4x \left(\hbar \partial_\mu \psi^\dagger \partial^\mu\psi + \hbar^{-1} m^2 |\psi|^2\right)$$ giving the Klein-Gordon ...
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Classical action is zero in Klein-Gordon theory for a particle wavepacket

I'm interested in rewriting actions in the form $$ S = -\int H dt + \int p_i dx^i, $$ (where $H$ is the Hamiltonian and the $p_i$ are conjugate momenta) and then evaluating them along a classical ...
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Equation in Current vector in a Klein Gordon Equation

I'm trying to get the current vector $J^\mu$ of a Klein-Gordon equation: $$\Psi^* \Box \Psi =\Psi^* \partial^{\mu} \partial_\mu \Psi= \partial^{\mu}(\Psi^*\partial_\mu \Psi)-\partial^\mu \Psi^*\...
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Momentum in complex scalar field

Consider a complex scalar field $\psi(x)$ with Lagrangian density $$ \mathcal{L} = \partial_\mu\psi^* \partial^\mu\psi - M^2\psi^*\psi. $$ Expand the complex field operator as a sum $$ \psi = \int \...
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How creation operator pops out while expanding field operator?

While doing QFT when we try to canonically quantize the Klein Gordon equation $\Box \phi =0$ we promote the $\phi $ to an operator field and impose the commutation rule $[\phi(x,t),\pi (y,t)]=i\hbar\...
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How can velocity and momentum be in opposite direction for antiparticles as given in the solutions of Klein Gordon Equation?

This is given in Greiner, Relativistic Quantum Mechanics For a free particle solution and antiparticle solution with momentum $\vec{p}$ the current is given by $e\frac{c^2\vec{p}}{E_p}$. The current ...
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How can energy be negative for antiparticles in the solutions of Klein Gordon equation?

Although similar questions have been asked before I'm still confused. This is from Greiner, Relativistic Quantum Mechanics $E^2=c^2\sqrt{\vec{p}^2+m_0^2c^2}$ Consequently, there exist two possible ...
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Mass of the fields in quantum field theory

I understand that if I have an action $$S=\int \phi(\Box + m^2 )\phi$$ Then the field $\phi$ has mass $m$ since this is the pole of the propagator of $\phi$. Now If I have an action $$S=\int \phi_1 \...
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Proving an equivalent [duplicate]

Can anyone help me prove the following equivalent found in Peskin & Schroeder on page 27. $$ \frac{1}{4\pi^2} \int_{m}^{+\infty} dE \sqrt{E^2-m^2} e^{-iEt} \sim_{t \to \infty} e^{-imt}$$ The ...
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Limit on speed of expansion of the bounded support interval of a position wave function in relativistic quantum mechanics

If the support of a quantum mechanical position wave function is a bounded interval, and that interval is expanding or contracting, then I think it cannot change in any direction faster than $c$. To ...
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Green's function for 1D Klein Gordon equation in position space

I want to derive the Green's function for the 1D Klein Gordon equation in position space. The Klein-Gordon equation in 1D: \begin{equation} (\partial_t^2-\partial_z^2+m^2)\phi=f(z,t) \end{equation} ...
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Question about the “wave function” on Relativistic Quantum Mechanics (RQM) and Quantum Field Theory (QFT)

I'm enrolled on a short and conceptual couse on RQM and QFT and the professor made a distinction about the Klein-Gordon (K-G) equation on RQM and the K-G equation on QFT. Roughly speaking, he said ...
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Commutation relations in QFT [duplicate]

So I have just started learning QFT. So you take a classical field and turn the degrees of freedom into operators. All fine, just like normal quantum. However I am confused about the commutation ...
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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 ...
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Did Dirac derive the correct equation for the wrong reasons? [closed]

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 ...
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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}...
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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 ...
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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 ...
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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)\...
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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}=\...
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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 ...
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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{...
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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 ...
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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 ...
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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?
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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 ...
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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{\...
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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 ...
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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 $\...
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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})^*...
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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^...
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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 ...
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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 $$ \Psi (x) = <x| \Psi>,\tag{2.83}$$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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$m$ 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 ...
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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(\...
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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 ...
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$\partial^{\nu} \partial_{\nu}$ 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 ...
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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 ...