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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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Retarded Green's function in Peskin & Schroeder

In an Introduction to Quantum Field Theory by M. E. Peskin & D. V. Schroeder (eq. 2.56 on page 30) the following relation for the retarded Green's function was established: $$(\partial^2 + m^2) ...
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Relativistic Schrödinger Equation: How is it relativistic and can it be useful? [duplicate]

As is well known, the usual Schrödinger equation, $$\mathrm{i}\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^2}{2m}\Delta\psi+V\psi,$$ is not relativistic. It can be derived formally by applying ...
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Equation for real/complex $\phi^4$ theory

On wikipedia (see this link), the Lagrangians of the $\phi^4$ equation for real AND complex scalar fields are given. One may derive the Klein-Gordon equation by inserting into the Euler-Lagrange-...
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Why does the mass term not violate particle number conservation in a free theory?

The Lagrangian of a free real scalar field theory is $$ \mathcal{L} = \frac{1}{2} \partial_{\mu} \phi\; \partial^{\mu} \phi \; - \frac{1}{2} m^2 \phi^2. $$ If we decompose $\phi$ in terms of the ...
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What does it mean for particles to be created?

I am right now studying scalar (semi) quantum electrodynamics, i.e. a charged Klein Gordon field coupled to a background classical electric field in 1+1 spacetime dimensions. The setup is as follows: ...
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In the path integral formulation of QFT, how do we get quantized particles out of a field?

Every QFT textbook starts by basically postulating that we have discrete states connected by creation and annihilation operators. In Quantum Mechanics, we started from a differential equation and ...
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Quantum fields can leak out of the light cone? [duplicate]

So the transition amplitude for a free Klein-Gordon field for a space-like interval is finite and non-vanishing (decays exponentially). What does one make of this physically, i.e. what is the meaning ...
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The Klein-Gordon Propagator According to Peskin and Schroeder (Derivation of *Retarded* Green's Function)

On page 29 of Peskin and Schroeder's An Introduction to Quantum Field Theory, the authors write that the propagator is given by: $$\begin{align} \langle 0|[\phi(x),\phi(y)]|0\rangle&=\int{d^3p\...
Albertus Magnus's user avatar
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Restrict Fourier Transform to a Hypersurface

Assume a field evolves according to the Klein-Gordon equation $$(\Box+m^2)\phi=0$$ The general solution to this is given by Fourier mode decomposition $$\phi(t,\vec{x})=\int\frac{\text{d}^3p}{(2\pi)^3}...
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Ehrenfest's theorem in QFT

In quantum mechanics, for a free particle, we know that the expectation value of its position travels in a straight line in the direction of the expectation value of the momentum (we get this from ...
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Deriving Klein-Gordon equation from Euler-Lagrangian equation: Taking partial derivative inside [duplicate]

Lagrangian for Klein-Gordon equation is given by $$L=\frac{1}{2}\partial_\mu \phi \partial^\mu\phi - m^2\phi^2/2.$$ To derive Klein-Gordon Equation I have to Compute derivative in Euler-Lagrange ...
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Studying behaviour of a Klein-Gordon field inmersed in a classical electric field

I want to study the behavior of a 1+1 dimensional Klein-Gordon field immersed in a classical constant electric field, in the context of backreaction, i.e. I calculate the solution to the Klein-Gordon ...
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Dervation of the first-order Klein-Gordon equation

How to derive the first-order perturbed Klein-Gordon equation: $$ \square \phi=\left[\frac{1}{\sqrt{-g}} \partial_{\mu}\left(\sqrt{-g}g^{\mu\nu} \partial_{\nu} \right) \right]\phi=0$$ For a first-...
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Total momentum operator of the Klein-Gordon field (before limit to the continuum)

I'm following K. Huang's QFT: From Operators to Path Integrals book. In the second chapter, he introduces the Klein-Gordon equation (KGE), and its scalar field $\phi(x)$, which satisfies this equation....
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Fourier transform of $\phi$ [closed]

I was reading through David Tong's Lectures Notes on Quantum Field Theory and I was wondering how, on page 22, he derives that the Fourier transform of $\phi(\vec{x}, t), \tilde{\phi}(\vec{p}, t)$, ...
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Ground state of an harmonic chain and Klein-Gordon vacuum

Consider the lagrangian of a system of classical coupled harmonic oscillators of mass $M$, connected with springs with elastic constant $\chi$ and connected to the background with springs of elastic ...
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Yukawa potential as the time integral of 4D retarded Green's function

I am attending an advanced QFT course, and trying to verify the instructor's claim that the retarded Green's function $$ G_{\text{ret}}^{(4D)}(t,\mathbf{x}) = \theta(t) \left[ \frac{1}{2\pi}\delta(\...
Hyeongmuk LIM's user avatar
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Time ordered correlator from path integral: equation of motion?

Consider a Lagrangian $L(\phi)$ for a field $\phi$ (assume it is a free real scalar for simplicity). Then the time ordered propagator can be expressed as a path integral $$ \langle\Omega|T\{ \phi(x) \...
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Understanding derivation of Klein-Gordon equation from Dirac equation

Above is Tong's notes which shows how the Klein-Gordon equation is derived from Dirac equation. But I don't get why: $\gamma^{\mu}\gamma^{\nu}\partial_{\mu}\partial_{\nu} = \frac{1}{2} \{\gamma^{\mu},\...
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How to interpret QFT fields (in relation with QM)? [duplicate]

In QM we deal with the Schrödinger equation:1 $$i\frac{\partial}{\partial t}\psi = H \psi$$ the wave function $\psi(x)$ is the main object of interest: it can be interpreted as a scalar field, in the ...
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What particles are described by the Klein-Gordon Equation?

The Klein-Gordon equation $$\left(\frac{\partial ^2}{\partial t^2} - |\nabla|^2 + m^2\right)\phi = 0\tag{1}$$ should describe non interacting particles without spin. So what particles in the standard ...
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The Klein-Gordon equation and the sign of the mass term

A derivation of the Klein-Gordon equation starts with the following lagrangian for a scalar field ϕ: $$ L=\frac{1}{2}g^{ab}(∇_a\phi)(∇_b\phi)-V(\phi) $$ If we plug this lagrangian in the Euler-...
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How can the Klein-Gordon equation have negative-energy solution if its Hamiltonian is positive-definite?

In a lesson about the introduction of classical field theory it was mentioned the Klein-Gordon equation $$(\Box + m^2) \phi(x) = 0. \tag{1}$$ But before we got this equation, we studied the ...
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Does a normal ordered Hamiltonian admit the same solutions as the non-normal ordered Hamiltonian

I'm reading "Lectures of Quantum Field Theory" by Ashok Das, where I encountered for the first time the normal ordering of an Hamiltonian (Chapter 5.5). In the book, the Hamiltonian for the ...
Yotam Ohad's user avatar
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What does a quantized field in QFT do? [duplicate]

I'm studying for an exam called Introduction to QFT. One of the main topics in this class is the quantized free fields. I can now find the fields that solve the Klein-Gordon equation and the Dirac ...
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Why do different contours give different answers in the limit $\epsilon \rightarrow 0$ when calculating propagators?

Let $\phi$ denote the Klein-Gordon field. Then its propagator $\langle 0 \mid [\phi(x), \phi(y)] \mid 0 \rangle$ can be calculated as $$\int \frac{d^4}{(2\pi)^3} \frac{-e^{-ip(x-y)}}{p^2 -m ^2}. \tag{...
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Second quantization of hamiltonian of the Klein-Gordon field [closed]

Good day everyone. When I try to do a second quantization on the hamiltonian, I end up with the following equation, $$ H = \int \frac{d^3p}{(2\pi)^3} \omega_{\vec{p}} {a_{\vec{p}}}^{\dagger} {a_{\vec{...
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Is there an equivalent to the Klein-Gordon and Dirac Equations for Vector and other fields?

The Klein-Gordon equation describes a scalar field, and the Dirac Equation describes a spinor field. Is there an equivalent equation for a vector field? As well as spin 3/2 and spin 2 tensor fields? ...
zion does math weird's user avatar
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Why covariant quantization is not enough for a relativistic quantum mechanics theory?

It is often said in quantum field theory books that a quantum theory of fields is needed because every other attempt to develop a quantum-mechanical theory compatible with the principles of relativity ...
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Multi-particle Hamiltonian for the free Klein-Gordon field

The text I am reading (Peskin and Schroeder) gives the Hamiltonian for the free Klein-Gordon field as: $$H=\int {d^3 p\over (2\pi)^3}\; E_p\; a^{\dagger}_{\vec p}a_{\vec p}$$ This does not seem to be ...
Albertus Magnus's user avatar
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On creation annihilation operators of the free Klein-Gordon field [closed]

I want to calculate multiparticle states like $|\vec p,\vec p\rangle$ from $|0\rangle$. It seems that I would need to compute from things like: $a^{\dagger}_{\vec p}a^{\dagger}_{\vec p}|0\rangle$? It ...
Albertus Magnus's user avatar
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How does the non-relativistic limit of the Klein Gordon equation simplify to become the Schrodinger equation? [duplicate]

For the complex Klein-Gordon Lagrangian density in the non-relativistic limit, we can decompose the complex scalar field into the form $$\phi=\frac{1}{\sqrt{2m}}e^{-imt}\psi.$$ When substituting the ...
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Question about Witten 0+1 quantum gravity

I am trying to follow Witten’s article as much as I can. What every physicist should know about string theory. Physics Today 68 (11), 38–43 (2015); my knowledge is very basic GR and QFT. I am really ...
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Klein-Gordon mode functions in curved spacetime

I'm currently tackling QFT in curved spacetimes for the first time, mainly using "Quantum fields in curved space" by Birrell and Preskill's notes on QFT in curved spaces, to get a general ...
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Cauchy problem for the Klein-Gordon equation

Let $(M, g_{\mu\nu})$ be a globally hyperbolic spacetime and let $\Sigma$ be a spacelike Cauchy surface. The covariant Klein-Gordon equation has a well-posed initial value formulation, in the ...
Ric's user avatar
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Commutator of conjugate momentum and field for complex field QFT

In Peskin & Schroeder's Introduction to QFT problem 2.2a), we are asked to find the equations of motion of the complex scalar field starting from the Lagrangian density. I want to show that: $$i\...
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Canonical quantization of relativistic particle using Fourier transform

Suppose I want to quantize the Hamiltonian of a relativistic particle on space-time $\mathbb{R}^{4}$. Setting $c=1$ for simplicity, the energy of the particle is given by $w(p) = \sqrt{|p|^{2}+m^{2}}$,...
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Choice of hypersurface in Klein-Gordon inner product

Let $M$ be a globally hyperbolic spacetime, with metric $g_{\mu\nu}$. Consider the covariant Klein-Gordon equation $$(g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}+m^{2})\phi=0$$ Define the following indefinite ...
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Deriving Klein-Gordon equation in curved spacetime [closed]

I try to drive The Klein-Gordon equation for a massless scalar field in case of FRW metric: $$ ds^2= a^2(t) [-dt^2 + dx^2] $$ So I start by: $$\left(\frac{1}{g^{1/2}}\partial_{\mu}(g^{1/2}g^{\mu\nu}\...
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Orthonormality of the mode functions of the Klein-Gordon field in a globally hyperbolic space

In chapter 3 of "Quantum fields in curved space" of Birrell and Davies, the authors make the following statements. Consider a real Klein-Gordon field $\phi$ in a globally hyperbolic ...
Ric's user avatar
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Symplectic basis for the real solution space of the covariant Klein-Gordon equation

In lecture 12 of his course on "Quantum field theory for cosmology", that can be found for free on the web, professor Kempf makes the following statements. Consider a real Klein-Gordon field ...
Ric's user avatar
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Symplectic form for real solutions to the Klein-Gordon equation in curved spacetime

Consider a real Klein-Gordon field $\phi$ in a globally hyperbolic spacetime, with metric $g_{\mu\nu}$. The covariant Klein-Gordon equation is $$(g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}+m^{2})\phi=0$$ Let $...
Ric's user avatar
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Feynman Propagator in presence of boundary terms

It is known that when a Klein-Gordon (KG) field is expressed in plane wave basis, then the Feynman propagator defined as $$\Delta_F(x-y)=\langle 0|T\{\phi(x)\phi(y)\}|0\rangle$$ equals the Green's ...
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Definition of the left-right derivative symbol in the Klein-Gordon scalar product [duplicate]

At the start of QFT, studying the Klein-Gordon scalar field, it is often mentioned that the following is the definition of the scalar product in the space of the solutions: $$\langle f _{\vec{k}}|f_{\...
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How can the linear combination of infinite normalized Klein-Gordon fields be a normalizable field?

In the context of a Klein-Gordon field with charge $e$, mass $m$, immersed in an external classical electric field $A_\mu = (A_0(z), 0)$, I am asked to calculate the charge density of the field ...
dolefeast's user avatar
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Check of the Green function for the Klein-Gordon equation [closed]

In order to derive the Green function for the Klein-Gordon equation, one considers $$ \Box G(x-x')+m^2G(x-x')=\delta^4(x-x') $$ where $\delta^4(x)=\delta(x_0)\delta(x_1)\delta(x_2)\delta(x_4)$. The ...
Jon's user avatar
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In QFT what is the frequency of oscillation in time of a scalar field mode? Does it depend on the number of particles in the mode?

In classical field theory, for a given real scalar field $\phi$, each mode $\vec{k}$ vibrates in time at a frequency $\omega = \sqrt{\vec{k}^2 + \mu^2}$ with $\mu$ being the mass. In quantum field ...
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Question on Peskin & Schroeder's QFT: Noether's Theorem; finding conserved currents

So, I am trying to learn how to find conserved currents using what I see in P&S's "Intro to QFT"; sec.2.2 pg.17-18. Particularly, I am working through the example in the text for the ...
Albertus Magnus's user avatar
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Integration step when deriving position-space expression of Feynman propagator

I am looking at the various ways to derive explicit position-space expressions of the Green's function with certain boundary conditions (Feynman propagator) of the Klein-Gordon equation. In this ...
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Eigenvalue in the Energy Derivation from Klein-Gordon Equation

I was watching this video tutorial for the derivation of Energy (chapter 3) from the Klein-Gordon equation where they had taken the two equations of $$(P^\mu P_\mu-m^2)\psi=0$$ and $$(-(K^0)^2+\...
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