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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Space and Time on equal footings for QFT "Equations of Motion"? [closed]
How can we write the equations of motion for the free field (Spin 0) in QFT, which put space and time on an equal footing? (Canonical Quantization)
In this setting, is said that space and time are on ...
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Proving that Retarded K-G Propagator is Green function (Peskin & Schroeder 2.56) [closed]
I am trying to derive Peskin & Schroeders expression $2.56$:
$$(\partial^2 +m^2)D_R(x-y)=-i\delta^{(4)}(x-y)\tag{2.56}$$
with $$D_R(x-y)=\theta(x^0-y^0)\langle 0|[\phi(x),\phi(y)]|0\rangle.\tag{2....
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How to choose contour of integration prescription Klein- Gordon Propagator? [duplicate]
I am going through the complex integral in peskin & Schroeder's intro to QFT (equation 2.54, deriving the Free Klein-Gordon Propagator):
$$\langle0|[\phi(x),\phi(y)]|0\rangle=\int \frac{d^3p}{(2\...
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Spectrum of Klein-Gordon operator in AdS Black Hole
I'm working on obtaining the spectrum of the Klein-Gordon operator in $AdS_2$ for black hole coordinates. To accomplish that, I first consider the problem in hyperbolic space $H_2$ and then Wick ...
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Non-vanishing amplitude outside light cone doesn't violate causality? [duplicate]
I am following Peskin & Schroeder's QFT book. And on equation 2.51, we get an expression for the free Klein-Gordon propagator for timelike intervals $x^0-y^0=t$, $x-y=0$:
$$D(x-y) \sim e^{-imt}\...
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Understanding derivation Klein-Gordon equation
The Klein-Gordon Equation is given as follows, using natural units ($\hbar \to1, c \to 1$):
$-\frac{\partial^2 \Psi}{{\partial t}^2} = -\nabla^2 \Psi + m^2 \Psi $
The way I tried to derive this is as ...
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On the continuity condition of the Klein-Gordon equation
I have to show that $\partial_\mu j^\mu = 0$ for the four-current $j^\mu = \frac{i}{2m}\left(\phi^*\partial^\mu\phi - \phi\:\partial^\mu\phi^*\right)$.
Using the Leibniz rule, one gets to
$$\partial_\...
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Dirac Equation and the Klein-Gordon Equation
I am trying to solve an exercise in Halzen and Martin's Quarks and Leptons book and got stuck on doing some math. The Dirac equation reads $$i \gamma^{\mu} \partial_{\mu} \psi - m\psi = 0.$$ Now, I ...
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Variance of Observable in Klein-Gordon field theory
In quantum field theory most observables $A$ do not have a definite value in the ground state (vacuum). For an observable $A$, a reasonable measure of the spread in the ground state is its variance $\...
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Ricci Scalar Curvature under conformal transformation
Consider the Klein-Gordon equation in curved spacetime with metric $g$
$$\square_g \phi - \xi R \phi = 0$$
and consider a conformal transformation
$$g \longmapsto \tilde{g} = \Omega^{2} g \quad , \...
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Does the QFT Klein-Gordon equation describe the state of the field or the field operator?
In the canonical quantization of QFT we talk about:
states representing a field.
field operators.
The quantum Klein-Gordon equation is expressed in terms of the field φ. Is φ (in the equation) the ...
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Why is the Klein-Gordon equation a 2nd derivative equation?
Note: I am not asking about why time is in 2nd derivative: that makes perfect intuitive sense given the relativistic need to treat space and time in equal footing.
We often hear how the Klein-Gordon ...
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Transformation law of interacting potential in non-inertial frame?
I haven't seen this addressed in most lecture notes but how does the interacting potential of (say) the Klein-Gordon equation transform in a non-inertial frame? (Feel free to answer more generally as ...
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Solution to the Klein-Gordon equation in a generic metric
Is there a way to solve the Klein-Gordon equation in a generic spacetime for a massless scalar field?
\begin{equation}
\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi)=0.
\end{...
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Klein-Gordon and Green's Function
I want to prove the following relation:
$$(\partial_x^2 + m^2)\langle0|T\phi(x)\phi(x_1)|0\rangle = -i\delta^{(4)}(x - x_1).$$
My Approach: Consider LHS
$$(\partial_x^2 + m^2)\langle0|T\phi(x)\phi(x_1)...
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Scalar Axion Field Amplitude Calculation in localised laboratory / Earth
This question concerns the paper "Axion Dark Matter: What is it and Why Now?", in the Appendix A.3 regarding equations related to the Axion Field.
It states that by the Friedmann Equation,
$$...
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Confusion related to creation and annihilation operators
I'm studying QFT from Peskin and from the book by Ashok Das, and there seems to be a disagreement between the creation and annihilation operators, for the scalar Klein Gordon field theory.
In Das, we ...
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Total momentum operator for the KG field
This question pertains to Equation (2.33) in Peskin and Schroeder:
$$
\hat{\vec P}=-\int d^3\!x\,\hat\pi(\vec x)\vec\nabla\hat\phi(\vec x)=\int d^3\!p\,\vec p\,\hat a_{\vec p}^\dagger\,\hat a_{\vec p}
...
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Derive the KG field operator in terms of ladder operators
In Peskin and Schroeder, they skip a few steps to arrive at the KG field operator in Equation (2.25):
$$
\hat\phi(\vec x)=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2\omega_{\vec p}}}\left(\hat a_{\vec ...
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Making background curvature variable in QFT on curved spacetime
In QFT on curved spacetime one may start with a Klein-Gordon like equation
$$(\square_g - m^2 j) \phi = j,$$
where $\square_g := g^{\mu \nu} \nabla_\mu \nabla_\mu$ is the D'Alembertian operator, and $...
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Square root of the wave operator
How are the Dirac matrices the square root of the wave operator? I keep seeing it mentioned as such but never explained.
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What am I solving when I use Klein-Gordon equation? [closed]
$\newcommand{\Ket}[1]{\left|#1\right>}$
I began to study QFT using David Tong's notes and I have a doubt. This is what the notes says (http://www.damtp.cam.ac.uk/user/tong/qft.html page 21/22).
I ...
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Why Klein-Gordon equation before QFT? [closed]
Note : This is an edit of the original question.
Previously (with some more elaboration of what I had in mind): Peskin's QFT book starts with Klein Gordon equation, which is little taught for most ...
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How do we interpret the second-order differential operator in the QFT path integral?
For the free scalar field theory, the path integral has a differential operator term in the exponent,
$$
Z[J] = \int \mathcal{D}\phi \, \exp\left( i \left[ -\frac{1}{2} \int d^d x \, \phi(x) A \phi(x) ...
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Problematic Factor of 2 in Klein-Gordon Propagator Derivation
I want to derive the Klein-Gordon Green function equation
$$(\Box_b + m^2) D_F(x_b - x_a) = - i \delta^4(x_b - x_a)$$
by using the same steps taken when fixing the 'exact' Green function of the non-...
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Fourier transformation of the inverse Klein-Gordon propagator
On Peskin & Schroeder's QFT, page 30, the scalar field propagator as the retarded Green function is defined as
$$(\partial^2+m^2)D_R(x-y)=-i\delta^4(x-y) \tag{2.56}$$
The Fourier transformation is ...
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Problem obtaining Klein-Gordon equation solutions
I am having some problems and also some questions regarding how can one get the general solution to the Klein-Gordon equation, which usually appears in the literature as
$$
\phi(t,\mathbf{x})=\int\...
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Why only $\phi=\pm1$ are considered "vacuum states" in the Klein-Gordon model with $\phi^4$ potential, and not $\phi=0$?
I am reading "Kink Moduli Spaces — Collective Coordinates Reconsidered," by Manton, Oleś, Romańczukiewicz, and Wereszczyński (arXiv version), where they consider the Klein-Gordon equation,
$$...
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Can a black hole in a quasi-de Sitter universe have scalar hair?
Any static and axially symmetric, asymptotically flat black hole spacetime cannot support scalar hair.
The universe is not asymptotically flat since it is quasi-de Sitter. Could we find a black hole ...
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Negative energy solutions not a problem for Klein-Gordon equation?
I already posed this question Negative energy solutions in Klein-Gordon and Dirac equations but I am not satisfied with the answers.
Trying to be very sharp:
does Klein-Gordon equation have negative ...
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Klein-Gordon in Schwarzschild Curved Spacetime [closed]
Given such a Schwarzschild metric, the covariant Klein-Gordon equation for a mass $m$ takes the form
$$\left[\frac{1}{g_{00}} \frac{\partial^2}{\partial t^2 }-\frac{1}{r^2} \frac{\partial}{\partial r}...
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Field equations of parametrized field theory
I have problems calculating the field equations of parametrized field theory.
Parametrized field theory is essentially the Klein-Gordon field $\phi$ but, considering the "inertial coordinates&...
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Derivation of the Klein-Gordon equation [closed]
My question arises from studying section 8.1.3 of Sakurai. I am confused on the way Klein–Gordon equation is created. In Sakurai, the book said that it was derived by taking another time derivative to ...
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Dependence of Klein-Gordon solution only on spatial coordinates
I am studying QFT with the Peskin and Schroeder textbook, and I am new to this area of physics. I'm struggling with the solution of the Klein-Gordon equation using Fourier integral as a continuum set ...
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Particle-Antiparticle of Klein-Gordon equation
When we solve Klein-Gordon equation, it gives both positive energy solution and negative energy solution. These two solutions are responsible for positive and negative value of $\rho$ which was ...
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How to transition from a Green function to vacuum expectation value in quantum field theory for a scalar field?
When considering the scalar field that solves the Klein-Gordon equation, one can use Green's functions to identify a propagator. This can be constructed from first principles, and can be left as an ...
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Basic concept clarification about the expectation value formula in quantum field theory
In introducing quantum field theory, the field solution to the Klein-Gordon equation is $$\phi(x^{\mu}) = \int \frac{a_{\bf{k}} e^{-i(k^{\mu}x_{\mu})} + a^{\dagger}_{\bf{k}} e^{i(k^{\mu}x_{\mu})}}{\...
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What is the missing part of the argument needed to justify the claim of (2.52) in Peskin and Schroeder's QFT textbook?
[This paragraph has been added to make clear that this is not a homework question having been branded as such by a mod of some kind. The question is attempting to the core of a very important question ...
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Is this theory renormalizable? If so, is there any proof? [closed]
Consider a kinetic lagrangian of 2 Klein-Gordon-Foch fields $\varphi$ and $\chi$ with interaction term
$$\mathcal{L}_I=g^2\bar{\chi}\chi\bar{\varphi}\varphi.$$
Is theory with such interaction ...
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Derivation of the Bessel function representation of the Green function of the inhomogeneous Klein-Gordon equation
I will link the following question, as it is partly related to the problem I am trying to deal with.
Green's function for the inhomogenous Klein-Gordon equation
As you can read from this User´s ...
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Description of a Classical Klein-Gordon Field with Momentum Distribution
So if I want to solve the free KG equation, the solution is of the form
$$\phi(x) = \int_{p\in \mathbb{R}^4}\frac{1}{2\omega_\vec{p}} \left( a(\vec{p})e^{-ipx} +b(\vec{p})e^{ipx} \right),$$
where the ...
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Energy spectrum in Klein-Gordon equation in general relativity
I know that the Klein-Gordon equation in general relativity takes the form (a massless field)
$\nabla_\mu \nabla^\mu \phi=\sum_{a,b} \frac{1}{\sqrt{-g}}\partial_a(\sqrt{-g}g^{ab}\partial_b\phi) =0$
...
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Can we interpret the number of particles in a state as an amplitude of oscillation
TL;DR
If we have a classical field expressed in a Fourier expansion, is the amplitude of each mode related to the particle number in that specific mode?
Attempt to formalize
We know particles should ...
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Does there exist a square root of Euler-Lagrange equations of a field? (Factorization)
Does there exist a square root of Euler-Lagrange equations $\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}-\frac{\partial \mathcal{L}}{\partial \phi} = 0$ in the sense that $(x+...
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Show that Majorana equation implies Klein-Gordon equation
Show that the Majorana equation
$i \bar{\sigma}\cdot\partial\chi -im\sigma^2\chi^* = 0$
for 2-component spinors $\chi$ implies the Klein-Gordon equation
$(\partial^2+m^2)\chi$.
This is part of an ...
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Klein-Gordon equation for Schwarzschild
I have seen in many texts that the form of the solution to the Klein-Gordon equation for Schwarzschild metric is:
$\phi(r,t,θ,φ)=r^{−1}f(r,t)Y_{lm}(θ,φ)$
where does the term $1/r$ comes from? I can ...
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Why do we drop the renormalization term in momentum Klein-Gordon Field Theory?
I'm following Peskin & Schroeder's book on QFT. I managed to prove expression (2.33) which gives us the 3-momentum operator for the Klein-Gordon Theory:
$$\mathbf{P}=\int \frac{d^3p}{(2\pi)^3}\...
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What is the right way to treat the vacuum energy?
I am reading the QFT book by Peskin and Schroeder. They compute the Hamiltonian for a free quantum scalar field and find
\begin{equation}
H = \int \frac{d^3p}{(2\pi)^3}\omega_p (a_p^\dagger a_p + \...
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Why Klein-Gordon and Dirac's waves have a phase velocity of $c$?
From De Broglie’s relations and the energy-momentum dispersion relation one finds:
$$v_p=\frac{w}{k}=\frac{E}{p}=\frac{\gamma mc^2}{\gamma mv}=\frac{c^2}{v} \tag{1}$$
Where, $v_p$ is the phase speed ...
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Solving Klein-Gordon-Foch equation with initial conditions
Suppose wave function $\psi=\psi(\vec{r},t)$ satisfies Klein-Gordon-Foch equation $$\square\psi=-m^2\psi.$$ Also suppose initial conditions $\psi_0=\psi(\vec{r},0)$ and $\psi_{t0}=(\partial_t\psi)(\...
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