# Heisenberg UP and space homogeneity - an intuitive reasoning

• lightarrow
In summary, the conversation discusses the relationship between the position and momentum of a free particle and the implications of the Heisenberg uncertainty principle. It is noted that if the uncertainty in position is small, the uncertainty in momentum is large and this may violate the conservation of momentum. However, it is argued that the homogeneity of space implies the conservation of momentum and that the uncertainty in initial conditions allows for the uncertainty in final states. The concept of virtual particles is also mentioned as a means of explaining the conservation of momentum during interactions. The conversation concludes with a discussion on how Poincare symmetry is a fundamental aspect of quantum field theory and the role of virtual particles in understanding Feynman diagrams and path integrals.
lightarrow
Let's consider the position x and momentum p of a free particle.

$$\Delta\ x\Delta\ p\geq \hbar/2$$ so, if $$\Delta \ x$$ is little enough, $$\Delta \ p$$ is big enough.

The fact $$\Delta \ p$$ is big implies that we cannot say the momentum conservation law is valid anylonger.

But:

space is homogeneus --> momentum conservation,
non conservation of momentum --> space is not homogeneus

So, for $$\Delta \ x$$ very little, that is, considering space at very small distances, space itself must be non-homogeneus.

Consider that this is a very rough reasoning with no pretence at all.

Can it have any meaning?

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It will be valid, it's just you weren't sort of momentum before or after an interaction (within certain limits obviously).

Giving wildly unrealistic numbers here consider particles A and B interacting. A has momentum somewhere in the range [5,6] and B is somewhere in [6,7]. If after interactioning A has momentum somewhere in [7,8] then you'd expect B to be in [4,5].

This is totally unrigorous and I half expect someone much more versed in QM to come along and correct me, but that's how I would reconcile the HUP and conservation of momentum.

Yes, that's right. There is in principle exact momentum conservation, but it are the uncertainties on the initial conditions which allow for the uncertainties on the final states. This can in fact be illustrated by using Bohmian mechanics where this is explicit.

In QFT this issue deals with real/virtual particles versus conservation/nonconservation of 4-momentum.

Daniel.

P.S. Nowadays we prefer the interpretation: there are virtual particles because at any point in spacetime the 4 momentum is conserved.

dextercioby said:
In QFT this issue deals with real/virtual particles versus conservation/nonconservation of 4-momentum.

Daniel.

P.S. Nowadays we prefer the interpretation: there are virtual particles because at any point in spacetime the 4 momentum is conserved.
You mean this is assumed as a postulate? And, of course, we are talking about SR only (not GR).

Does that mean that during an interaction (a) "virtual particle(s)" is created which carries off the missing momentum to maintain the conservation law?

I was under the impression that a virtual particle was just a tool to make it easier to understand Feynman diagrams and path integrals.

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lightarrow said:
You mean this is assumed as a postulate? And, of course, we are talking about SR only (not GR).

In QFT on Minkowski background, Poincare' symmetry is a must, so you can take it an an axiom.

Jheriko said:
Does that mean that during an interaction (a) "virtual particle(s)" is created which carries off the missing momentum to maintain the conservation law?)

That's right.

Daniel.

## 1. What is the Heisenberg Uncertainty Principle (UP)?

The Heisenberg Uncertainty Principle is a fundamental principle of quantum mechanics that states it is impossible to simultaneously know the precise position and momentum of a particle. This means that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa.

## 2. How does the Heisenberg UP relate to space homogeneity?

The Heisenberg UP and space homogeneity are related in the sense that space homogeneity ensures that the laws of physics, including the Heisenberg UP, are the same at all points in space. This means that the uncertainty in position and momentum of a particle is constant throughout space.

## 3. Can you explain the intuitive reasoning behind the Heisenberg UP?

The intuitive reasoning behind the Heisenberg UP is that in order to measure the position of a particle, we need to interact with it, and this interaction will inevitably change its momentum. Therefore, it is impossible to know both the position and momentum of a particle with absolute certainty.

## 4. How does the Heisenberg UP affect our understanding of the physical world?

The Heisenberg UP has significant implications for our understanding of the physical world. It shows that at the subatomic level, the behavior of particles is inherently unpredictable and uncertain. This challenges our traditional understanding of causality and determinism in the physical world.

## 5. Is there any way to get around the Heisenberg UP and make precise measurements of position and momentum?

No, the Heisenberg UP is a fundamental principle of quantum mechanics and cannot be circumvented. However, through the use of advanced technologies and techniques, scientists have been able to make increasingly precise measurements while still adhering to the principles of the Heisenberg UP.

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