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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Dimensional analysis of quantized Klein-Gordon Field

For the free Klein-Gordon Lagrangian density: $$\mathcal{L}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu} \phi-m^2\phi^2 .$$ Since we need the dimension of Lagrangian density equal to 4 (in this case ...
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Klein-Gordon equation coupled to scalar curvature

Consider the Klein-Gordon equation of the form $$\square_g \psi - m^2 \psi - \xi R \psi \enspace = \enspace 0 \quad .$$ This equation describes the relativistic propagation of a scalar field with mass ...
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Relation between low-energy modes of phonon and infinite oscillator limit

The Klein-Gordan equation is often described as coupled oscillators, taking the limit $l\rightarrow \infty $ to reduce to wave equation with linear dispersion relation. I'm considering here the ...
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Peculiar calculation of the Klein-Gordon Propagator

I am reading Peskin & Schroeder's QFT textbook (page 29~30). Here, to calculate Klein-Gordon Propagator, author computes following integral. $$\left< 0 | [\phi(x), \phi(y)]|0\right> = \int \...
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Construction of the Klein-Gordon field theory - what is missing?

Many references I know on QFT start the discussion of the Klein-Gordon field theory with some discussion about harmonic oscillators. One such reference is Folland's Quantum Field Theory book. The idea ...
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What's the rationale of replacing the Fourier coefficients in a field expansion by operators?

Let's take a look on the particular case of the Fourier expansion of the Klein-Gordon field: $$\psi (x,t) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_0(p)}[a(p)e^{i(E_0(p)t-px)}+a^\star (p)e^ {-i(E_0(p)t-...
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How can I find an operator originally expressed in terms of raising and lowering operators in terms of the field operators?

I'm following this book on QFT called "Quantum Field Theory of Point Particles and Strings" by Brian Hatfield. After the end of the scalar field theory section on Exercise 3.6, it asks us to ...
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Noether charge on complex scalar field

For complex scalar field, we write the Lagrangian as: $$ \mathcal{L}=\partial_{\mu}\phi^{*}\partial^{\mu}\phi-m^2 \phi^{*}\phi $$ with the $U(1)$ symmetry, and under infinitesimal transformation: $$ \...
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Field shift in free Klein-Gordon theory

I am reading Peskin & Schroeder Ch9 and am stuck on a calculation going from equation 9.36. The problem is essentially a change of variable of a Klein-Gordon field. Beginning, we have an integral ...
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Massive and massless modes in the Klein-Gordon equation

I am studying the massive Klein-Gordon equation in 3+1 dimensions with a given scalar source. The equation I am looking at is $$ (\Box +m^2)\phi(x)=J(x) \ , $$ and am interested in the following ...
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On the Integral Representation of Greens Functions

In Birrell & Davies book "QFT in Curved Spacetime" the authors discuss in chapter 2.7 that all Greens Functions (which are the vacuum expectation values of products of fields) can be ...
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Expression of Klein-Gordon field in Heisenberg picture

In Schrodinger picture, the scalar field is $$ \phi(\vec{x}) = \int \frac{d^3 p}{2E(\vec{p})} \left( a(\vec{p}) e^{i\vec{p}\cdot\vec{x}} + a(\vec{p})^{\dagger} e^{-i\vec{p}\cdot\vec{x}} \right). \...
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How to understand the term $\frac{1}{-2E(\vec{p})} e^{-ip(x-y)}$ of Klein-Gordon propagator in Peskin & Schroeder's book?

I am reading Peskin & Schroeder's book on Chapter 2. I have a question about how to get the term $\frac{1}{-2E(\vec{p})} e^{-ip(x-y)}$. The original equation for propagator is $$ \langle 0 | [\phi(...
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Klein-Gordon equation from general relativity?

I am trying to derive the Klein-Gordon equation from Einstein's field equation, since the energy momentum tensor for the Klein-Gordon equation is defined as: $$T^{\mu\nu} =\partial^{\mu}\phi \ \...
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Peskin and Schroeder confusion on promoting Classical Klein-Gordon equation to quantum field equation

I am reading "An introduction to quantum field theory" by Peskin and Schroeder and I am confused. I appreciate your help. Here's the context to my question: In chapter 2, the book introduces ...
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Path integral in a boundary QFT

I'm trying to compute the following path integral \begin{equation} Z = \int\mathcal{D}\phi\exp\left(-\int_{\mathbb{R}^d_+}\frac{d^dx}{2}\phi(-\partial_\mu^2 + m^2)\phi \right) \propto \frac{1}{\sqrt{\...
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Klein-Gordon solution's Fourier image

I'm solving Klein-Gordon equation in order to get scalar field expression. $$(\partial^2 + m^2)\phi=0$$ I expand solution $\phi$ into Fourier integral in momentum space: $$\phi=\int\frac{d^4p}{(2\pi)^...
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Solving the Klein-Gordon Equation and Enforcing Causality

Perhaps this is too basic a question, but I am running into trouble attempting to solve the Klein-Gordon equation with a simple spherically symmetric source. Consider, for instance $$ \square \phi = A(...
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Far field approximation for massive Klein-Gordon equation in 3+1D

For a massless scalar, one has the familiar Green's function $$ G(t,r) = \frac{\delta(t - r)}{4\pi r}\,, $$ and one may take the far-field approximation in a rather straight-forward way: $$ \int d t d^...
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How does the lagrangian for a Klein-Gordon field changes?

For K-G field, the lagrangian density $$L=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2$$ In ryder quantum field theory book i saw that in next step he writes $$L=-\frac{1}{2}\phi(\...
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Klein-Gordon equation with imaginary mass

I was wondering if someone could provide me the name of the following equation $$\square \varphi - \tilde{m}^2 \varphi = 0,$$ where $\square := \partial_t^2 - \nabla^2$. I am specifically seeking ...
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Could the Dirac Eq./Klein-Gordon Eq. handle solutions with finite extinction times?

I was reading in Wikipedia about how the Dirac Equation and the Klein-Gordon Equation where built to introduce in the Schrödinger equation the relativistic description of the Energy–momentum relation: ...
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Obtaining the KG equation from Action

After solving the field equation for $$S = \int \sqrt{-g}dx^4[f(\phi)R + h(\phi)g^{\mu \nu}\nabla_{\mu}\phi\nabla_{\nu}\phi - V(\phi)]$$ I have obtained $$2h\square \phi + \frac{\partial h}{\partial \...
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Does this relativistic generalization of the Schrodinger equation make sense? [duplicate]

So I'm aware that the correct relativistic approach to quantum mechanics is through quantum fields, but I'm still interested in the question that follows. We know the Schrodinger equation in free ...
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Derive interaction lagrangian for KG equation in QED

The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$ By ...
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Energy-Momentum tensor in the non-relativistic limit of Klein-Gordon Field

Assume we have a real Klein Gordon field $\phi(x,y,z,t)$, and we do the non-relativistic expansion of it in terms of a complex field $\psi(x,y,z,t)$ $$\phi=\frac{1}{\sqrt{2m}}(\psi e^{-imt}+\psi^* e^{...
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Why do scalars and fermions have a different result in a Lagrangian?

Consider the Lagrangian for Yukawa theory: $$ \mathcal{L} =i\bar{\psi}\not{\partial}\psi- \bar{\psi}m_F \psi +\frac{1}{2} \partial_\mu \phi \partial^{\mu} \phi - \frac{1}{2}m_s^2 \phi^2 + \mathcal{L}_{...
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Why don't we just say that the Klein Gordon equation describes a two component complex function?

These vectors form the basis vectors of the field that the KG equation describes: (for each $\vec{p}$ in $R^3$): $$|e^{i\vec{p} \cdot \vec{x}} , E=+\sqrt {p^2+m^2}\rangle$$ $$|e^{i\vec{p} \cdot \vec{x}...
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Why didn't the Klein-Gordon equation suggest antimatter like the Dirac equation did?

I have heard the story that the Dirac equation suggested the existence of antimatter due to the existence of negative energy solutions. The Klein-Gordon equation also has negative energy solutions. ...
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What is the interpretation of the quantum field operator solving the Klein-Gordon equation?

Does the quantum field operator $\hat \psi^\dagger(x)$ solving the KGE mean that we should think that every quantum field configuration evolves under a KGE field equation. Or do we just understand it ...
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Can we solve the Klein-Gordon equation in the Schrodinger picture?

In QFT, the Klein-Gordon equation is solved with the field operator $\hat \psi(x)$/$\hat \psi^\dagger(x)$ in the Heisenberg picture, and (as I understand it) gives the evolution of a single on-mass-...
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The two ways to get Schrodinger equation from Klein-Gordon equation

We can take the Klein-Gordon equation describing the evolution of a complex scalar field. Taking the non-relativistic limit yields a classical wave equation that is identical in form to the ...
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Does Heisenberg picture only work for time-dependent Schrödinger equation not Klein-Gordon equation?

For a Klein-Gordon field, our QFT lecture notes say we use the following relationship to define the Heisenberg picture. $$i \frac{dQ}{dt} = [Q,H]$$ which leads to $$Q(t) = e^{iHT}Q(0)e^{-iHt}$$ ...
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Klein-Gordon Hamiltonian in terms of Fourier transformed variables

The Klein-Gordon Hamiltonian density is a function of four complex variables $\psi , \psi ^* , \pi , \pi ^*$. Suppose we make the change to Fourier transformed variables. Then the Fourier expansions ...
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Symmetry arguments and derivation for product of gamma matrices and derivatives

I am trying to work with the Dirac equation and the solution for the Klein-Gordon equation for some derivation and I stomped on the following problem in my derivation. $\gamma^{\mu} \gamma^{\nu} \...
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Ingoing wave boundary condition and outgoing wave boundary condition

In solving wave propagation equation, for example, solving Klein-Gordon equation in some complicated spacetime Geometry, usually equipped with a horizon. I usually encountered with jargon like "...
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Why is the Propagator given by the Green's Function for a General Field in Canonical Quantisation?

In canonical quantisation, it is taught that the propagator for the Klein-Gordon field is defined as $$\Delta_F(\vec x - \vec y) \equiv \left < 0 \right | \overleftarrow{\mathcal T} \phi(\vec x) \...
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Commutation relations interacting fields

I am reading Schwartz's "Quantum field theory and the standard model". I have a question on how he derives the Feynman rules for an interacting scalar field from a Lagrangian formalism. In ...
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In calculation of Hamiltonian of real scalar field (Quantum field theory Srednicki)

I'm now reading the Mark Srednicki, Quantum field theory, p.27 I'm now trying to understand the Third step in the calculation of $H$. Through the integration over $k'$ involving the delta functions $...
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Euler-Lagrangian equation of motion of quantum fields in QFT

A canonical way of doing quantum field theory is by starting with some Lagrangian, for example, that of free scalar field $$L=\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi-\frac{1}{2}m\phi^2$$ Then ...
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How is the expectation value defined in relativistic quantum mechanics?

Since the norm of a wavefunction in relativistic quantum mechanics is defined as: $$|\psi|^2=i\int\left(\psi^*\frac{\partial \psi}{\partial t}-\frac{\partial \psi^*}{\partial t}\psi\right)dx$$ How is ...
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Feynman propagator for spacelike points

When I calculate the feynman propagator for spacelike points for free scalar quantum field it is not zero. How do I interpret this result. Since it seems to me that it violates causality.
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2 votes
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Is the retarded propagator exactly the Green's function?

I am trying to prove that, for the real scalar field $\phi(x)$, the retarded propagator, which is defined as $$ D_{R}(x-y)=\theta(x^0-y^0)\langle 0 |[\phi(x),\phi(y)]|0\rangle $$ is the Green's ...
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In what sense is $\sqrt{ {\bf p}^2c^2 +m^2 c^4}$ the Hamiltonian of special relativity?

This Hamiltonian is used in the derivation of the Klein Gordon equation. How is this a Hamiltonian when it doesn't even have a position-dependent potential term? Is this the free-particle Hamiltonian? ...
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Klein-Gordon Inner Product for Real Wavefunctions

Inner product in Klein-Gordon equation in one dimension is written this way : $$(\psi_1,\psi_2) = i\int dx \, \psi_1^*\,\partial_t\,(\psi_2) - \partial_t\,(\psi_1^*)\,\psi_2 \,$$ Suppose $\psi_i$ are ...
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Linearization of the Klein-Gordon equation and decoupling of ''spinors''! [closed]

We know that the K-G equation is deduced from the Einstein relation: $E^{2}=m^{2} +\vec{p}^{2} \;\;\;\;$ (with $c=1$) It is known that :$E^{2}=\frac{m^{2}}{1-\beta^{2}}=\left(\frac{m}{1-\beta}\...
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Is there a derivation of the classical free scalar lagrangian?

In my particle physics course notes, I see that the Lagrangian (density) for free scalars is given by $$ \mathcal{L} = \frac{1}{2} g^{\mu\nu} \partial_\mu\phi \partial_\nu \phi - \frac{1}{2}m^2\phi^2 $...
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Why is the scalar field Lorentz invariant?

I have the following solution for the KG equation (real scalar field): $$\phi (x) = \int \frac{d^3p}{(2\pi)^3\sqrt{2E_p}} [a_p e^{-ipx}+ a_p^\dagger e^{(ipx)}]$$ In my course we have rewritten it as $$...
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Klein-Gordon equation and Dirac equation

I am facing hardships understanding these equations mainly due to the confusing terminologies used in books. Can anyone suggest an easy to read explanation and then one which has mathematically ...
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Is this a manifestation of some infinite-dimensional Cayley-Hamilton theorem?

In classical field theory, when you have a free real scalar field $\phi$ with Lagrangian (density): $$ L = \frac{1}{2} \, \eta^{\mu \nu} \, \partial_{\mu} \phi \,\partial_{\nu} \phi - \frac{1}{2} m^2 \...
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