Quantization of Klein-Gordon Field

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Discussion Overview

The discussion revolves around the quantization of the Klein-Gordon field, specifically examining the relationship between its equation of motion and the harmonic oscillator. Participants explore the implications of this relationship in both classical and quantum contexts.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that the equation from Peskin's book resembles the equation of a harmonic oscillator, suggesting a connection between the two.
  • Another participant clarifies that the variable in question is \(\phi\), leading to a formulation that aligns with the classical harmonic oscillator equation.
  • A third participant questions the reasoning behind expressing \(\phi\) in terms of ladder operators, implying a need for further exploration of this representation.
  • It is mentioned that the equation also corresponds to the quantum harmonic oscillator in the Heisenberg picture, indicating a duality in interpretation.

Areas of Agreement / Disagreement

Participants express varying perspectives on the implications of the harmonic oscillator analogy, with some agreeing on the mathematical form while others question the application of ladder operators. The discussion remains unresolved regarding the appropriateness of these representations.

Contextual Notes

Participants do not fully explore the assumptions underlying the use of ladder operators or the implications of the quantum mechanical perspective, leaving these aspects open for further discussion.

go quantum!
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I was reading the book written by Peskin about QFT when I found that the following equation:
[tex] (\frac{\partial}{\partial t^2}}+p^2+m^2)\phi(\vector{p},t)=0[/tex]

has as solutions the solutions of an Harmonic Oscillator.

From what I know about harmonic oscillators, the equation describing them should have, for instance in 1-d, a second derivative and squared multiplication term with respect to the same variable, let's say x.

Thanks for your kind help-
 
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In this case the variable is [itex]\phi[/itex]:
[tex] \ddot \phi = -\left(p^2 + m^2 \right) \phi[/tex]
 
So it's the classical HO's equation. Why then write [tex]\phi[/tex] in terms of ladder operators (the famous a's)?
 
It is also the quantum HO equation in the Heisenberg picture.
 

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