# Lorentz transformation of delta function

• Chenkb
In summary, the conversation discusses the use of a delta function to describe the momentum in a two body decay in the CM frame. The question is then posed about the Lorentz transformation of the delta function when transitioning to the lab frame. The conversation also touches on the properties of the delta function and its application in representing distributions in momentum space. The distinction between 3-momentum and 4-momentum is also mentioned.
Chenkb
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?

## 1. What is the Lorentz transformation of a delta function?

The Lorentz transformation of a delta function is a mathematical expression that describes how the delta function (also known as the Dirac delta function) changes when viewed from different reference frames in special relativity. It allows us to understand how measurements of space and time are affected by the relative motion between two observers.

## 2. Why is the Lorentz transformation important in physics?

The Lorentz transformation is important in physics because it is a fundamental tool in understanding the effects of special relativity on the laws of physics. It allows us to reconcile the different observations of space and time between different reference frames, and has been used to make accurate predictions in fields such as particle physics and electromagnetism.

## 3. How is the Lorentz transformation of a delta function derived?

The Lorentz transformation of a delta function is derived using the Lorentz transformation equations, which describe how the coordinates (time and position) of an event change between two reference frames in relative motion. By applying these equations to the delta function, we can derive its transformation between frames.

## 4. Can the Lorentz transformation of a delta function be visualized?

Yes, the Lorentz transformation of a delta function can be visualized using mathematical graphs and diagrams. One common visualization is the Minkowski diagram, which shows how space and time are affected by the transformation. It is a useful tool in understanding the concepts of length contraction and time dilation in special relativity.

## 5. Are there any applications of the Lorentz transformation of a delta function?

Yes, there are many applications of the Lorentz transformation of a delta function in physics and engineering. It is used in particle accelerators to calculate the trajectories of particles moving at near-light speeds, and in the design of GPS systems to account for the effects of relativity on time measurements. It is also used in cosmology to understand the expansion of the universe and in the study of black holes.

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