Lorentz transformation of delta function

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Discussion Overview

The discussion revolves around the Lorentz transformation of a delta function in the context of particle decay, specifically focusing on how this transformation applies when transitioning from the center of mass (CM) frame to the lab frame. The conversation touches on mathematical properties and implications of the delta function in momentum space.

Discussion Character

  • Technical explanation, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant introduces the concept of using a delta function to describe the momentum in a two-body decay scenario, suggesting that it fixes the momentum to a constant value.
  • Another participant questions the initial claim about the delta function, seeking clarification on its mathematical application in this context.
  • A later reply reiterates the need for clarity, emphasizing the mathematical aspect of the Lorentz transformation of the delta function.
  • One participant points out a fundamental property of the delta function related to volume elements and suggests considering how these elements transform under Lorentz transformations.
  • Another participant raises a question about whether the discussion pertains to a distribution function in momentum space and whether it involves 3-momentum or 4-momentum considerations.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of the delta function in this context, and there is no consensus on the specifics of the Lorentz transformation or the nature of the momentum being discussed.

Contextual Notes

There are unresolved assumptions regarding the definitions of momentum space and the specific conditions under which the delta function is applied. The discussion also lacks clarity on whether the focus is on 3-momentum or 4-momentum.

Chenkb
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For two body decay, in CM frame, we know that the magnitude of the final particle momentum is a constant, which can be described by a delta function, ##\delta(|\vec{p^*}|-|\vec{p_0^*}|)##, ##|\vec{p_0^*}|## is a constant.
When we go to lab frame (boost in z direction), what's the Lorentz transformation of the delta function?
regards!
 
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What do you mean by "which can be described by a delta function" ?
 
maajdl said:
What do you mean by "which can be described by a delta function" ?

I mean that we can use a delta function to fix the momentum i.e. p=p0*.
Maybe my example of two body decay is not so suitable, but my question is just for mathematics, that is the Lorentz transformation of ##\delta(|\vec{p}|-|\vec{p_0^*}|)##
 
Chenkb said:
When we go to lab frame (boost in z direction), what's the Lorentz transformation of the delta function?
One of the basic properties of the delta function is that ∫δ3(x) d3x = 1. So write down how the volume element transforms under a Lorentz transformation (hint: x is Lorentz contracted) and you will have it.
 
Chenkb,

Are talking about a distribution function in the momentum space,
and about how this function might evolve with an interaction?
Are you considering 3-momentum or 4-momentum?
 

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