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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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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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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$D$-dimensional Coulomb problem, is the generalization from the 3D case supposed to be simple? (Gilmore's Lie algebra, chapter 14, problem 20)

I am solving problem 20 of chapter 14 of Robert Gilmore's Lie groups, physics and geometry: An Introduction for Physicists, Engineers and Chemists, which focuses on the $D$-dimensional Coulomb problem,...
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Significance of momentum density *also* satisfying Klein-Gordon equation for free scalar field?

In our QFT course we show that for a free scalar field: $$(\square + m^2) \phi(t,x) = 0\tag{1}$$ i.e. that the field operator satisfies the Klein-Gordon equation (as we expect). But also that: $$(\...
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Was Dirac's discovery of the Dirac equation by requiring a positive-definite probability density $j^0$ an unjustified coincidence?

Ever since I learned about the Dirac equation as an undergraduate student, it bothered me that it was introduced - both in my course and many others - on the back of concluding that the time component ...
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What is the significance of the Schrodinger Equation requiring complex solutions but the Klein-Gordon permitting purely real and imaginary solutions?

My intuition for the significance of complex solutions in QM initially was that the phase of wavefunction solutions was more 'rotating' than 'oscillating' (e.g. more like a rotating string (spiral) ...
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Non-relativistic limit to Klein-Gordon equation

I am having trouble understanding this topic fro the lecture notes I have been provided. Please suggest a book in which it is given clearly.
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Momentum Operator for a free Scalar Field

As $$\hat{P_i} = \int d^3x T^0_i,$$ and $$T_i^0=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi-\delta_i^0\mathcal{L}=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\...
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Solution to Klein-Gordon equation in terms of $\vec{p}$ and $\vec{k}$

A general solution to the Klein-Gordon equation can be written as: $$\phi = \int d^3k \frac{1}{(2 \pi)^3 \sqrt{2\omega_k}} \left(a(\vec{k})e^{-i(\omega_kx_o-\vec{k}\cdot \vec{x})}+a^{\dagger}(\vec{k})...
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What does it mean that the vacuum is unstable?

This book I'm reading states that the Klein-Gordon equation, for $m^2<0$ has an "unstable" vacuum $\phi=0$. It's talking about tachyons and how their presence is a sign of instability. ...
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Relationship between KG equation and Yukawa potential

If we start from Klein Gordon Lagrangian density and work through canonical quantization, we could arrive at field operators for scalar fields. Now, if we solve for the free propagator we arrive at (...
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Euclidean-signature Klein-Gordon propagator in curved space don't match with the one in flat space

The Klein-Gordon propagator in euclidean signature and flat space can be written as: \begin{equation} G(x,y)\propto m \frac{K_1(m\sqrt{s})}{\sqrt{s}} \tag{1} \end{equation} With $s=(x-y)^2$ and $m$ a ...
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The significance of the potential term in the Klein Gordon equation in the presence of an Electromagnetic Field

After replacing the derivative in the Klein-Gordon equation with the minimal coupling prescription i.e. $\partial^\mu \to D^\mu \equiv \partial^\mu+ieA^\mu$. The equation becomes: \begin{equation*} ((\...
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Why does the classic wave equation for a non-relativistic string look like the Klein-Gordon equation?

There is a very old equation known as the "wave equation". It's an ordinary classical non-relativistic differential equation which applies to just about every kind of ordinary wave you can ...
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Lagrangians for Non-Interacting Scalar Fields in QFT

I am currently taking a QFT class and we are using both canonical and path integral quantization to solve non-interacting scalar fields. We have seen the real scalar field with Lagrangian $$\mathcal{L}...
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Normalization of One-Particle States for Klein-Gordon Field Quantization

Peskin & Schroeder in their QFT textbook discusses how we may normalize one-particle states $|\textbf{p}\rangle$ for Klein-Gordon field quantization in pages 22-23. The excerpts are given below. ...
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Fall-off of Klein-Gordon massless field in flat spacetime (proof from Wald)

In Wald's General Relativity (1984) he devotes one of the last chapters to asymptotic flatness. He starts by showing how the conformal compactification of Minkowski spacetime can be used to determine ...
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Klein-Gordon Field Quantization and Bose-Einstein Statistics in Peskin & Schroeder

I am trying to understand how Klein-Gordon particles obey Bose-Einstein statistics from Peskin & Schroeder's QFT textbook (page no. 22). The excerpt is given below: From this passage it is clear ...
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Solution to two-dimensional PDE (wave/Klein-Gordon type equation)

I'm cross-posting from the Math SE as more people might have relevant knowledge here. I was playing with an optimization problem and ended up reducing it to solving the following PDE: $$ a^2 xy \frac{\...
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Time dependence of operators; Complex Klein Gordon Field

In Peskin and Schroeder chapter 2 they discuss the quantization of the real Klein-Gordon field. First, they do this in the Schrodinger picture and they assert: $\phi(x)$ and $\pi(y)$ obey the ...
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Physical interpretation of Klein-Gordon Equation conserved charge

In the Klein-Gordon Equation the conserved charge is: $$\rho = \frac{i \hbar}{2m} (\psi^* \frac{\partial \psi}{\partial t} - \frac{\partial \psi^*}{\partial t} \psi) $$ rather than the conserved (...
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Decay of the time derivative of solutions of the Klein-Gordon equation in decelerating expanding space-times

Suppose that we have a model of a universe* given by a flat FLRW metric.* In short, the model universe has $n\in\mathbb N$ dimensions, is homogeneous, isotropic and its expansion is governed solely by ...
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Hamiltonian Field Theory in Peskin & Schroeder

In Section 2.2 of their QFT textbook, Peskin & Schroeder introduce the Lagrangian and Hamiltonian field theories of a classical scalar field. While defining the action $S[\phi]$ and deriving the ...
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Derivation of Equation 2.27 from Peskin & Schroeder

In Section 2.3, Peskin & Schroeder discusses the quantization of real scalar field in Schrodinger picture. He writes Eq. (2.25) as follows. $$\phi(\textbf{x}) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{...
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