Argument for discrete space

[DannyTR]

Please note this is rather speculative…

I was wondering about information. Is it reasonable to regard the position of particles as information about the volume of space containing the particles? If yes, then there seems to be a simple argument for space being discrete:

 

- Assume space is continuous

- Implies particles have infinite positional precision. We might not be able to measure with infinite precision; but the underlying system is continuous so possesses infinite precision

- So there is an infinite amount of information in a spacial volume of 1 cubic unit

- And there is also an infinite amount of information in a spacial volume of 10000 cubic units

- Both infinities are the same kind and have the same cardinality

- But this is a logical contradiction, there must be more information in the larger volume.

- So there must be a false assumption in the argument; space must be discrete

- (or maths treatment of infinity is not rich enough)

 

I was reading about the Bekenstein bound, which also seems to suggest discreteness:

https://en.wikipedia.org/wiki/Bekenstein_bound

 

[mathematic]

When talking about small distances you cannot ignore quantum theory.

[DannyTR]

1 minute ago, mathematic said:

When talking about small distances you cannot ignore quantum theory.

I believe particles have a definite position when measured. When in wave form, the wave must have a centre (not sure thats the right term) at a specific position? Presumably where the probability of the particle being is highest?

QM does also suggest that matter is discrete... 

 

[studiot]

1 hour ago, DannyTR said:

I believe particles have a definite position when measured. When in wave form, the wave must have a centre (not sure thats the right term) at a specific position? Presumably where the probability of the particle being is highest?

QM does also suggest that matter is discrete... 

 

P orbitals have a minimum probability at the centre.

What do you mean when in wave form?

Physical phenomena exhibit some characteristics of pure particulate nature (but never all) and some characteristics of a wave nature (again but never all).
They do not shape shift from one to the other.

[swansont]

12 hours ago, DannyTR said:

Please note this is rather speculative…

I was wondering about information. Is it reasonable to regard the position of particles as information about the volume of space containing the particles? If yes, then there seems to be a simple argument for space being discrete:

A position in space is arbitrary, because our choice of coordinate system is arbitrary.

Quote

- Assume space is continuous

- Implies particles have infinite positional precision. We might not be able to measure with infinite precision; but the underlying system is continuous so possesses infinite precision

They don't. QM doesn't just say we can't know/measure the position to arbitrary accuracy (in conjunction with momentum), it says that a quantum particle doesn't have one until it's measured. An electron in an atom is everywhere when you aren't measuring it. If it had a location, there would be a dipole moment, but neutral atoms don't have one. There would also have to be a trajectory, and the dipole would oscillate, and the atom would have to radiate. But it doesn't.

 

[DannyTR]

31 minutes ago, swansont said:

A position in space is arbitrary, because our choice of coordinate system is arbitrary.

But would a specific region of space count as a 'system' in information terms?

IE A system with N states depending on all the possible positions of the particles (and other factors) and then the information content of the system is then log(N)?

The coordinates would be chosen separately for each specific region of space for which we are calculating information content.

31 minutes ago, swansont said:

They don't. QM doesn't just say we can't know/measure the position to arbitrary accuracy (in conjunction with momentum), it says that a quantum particle doesn't have one until it's measured. An electron in an atom is everywhere when you aren't measuring it. If it had a location, there would be a dipole moment, but neutral atoms don't have one. There would also have to be a trajectory, and the dipole would oscillate, and the atom would have to radiate. But it doesn't.

When not measured, the wave form must have a centre. Could this could be used in place of the particles position, when calculating information content?

 

 

[swansont]

59 minutes ago, DannyTR said:

But would a specific region of space count as a 'system' in information terms?

IE A system with N states depending on all the possible positions of the particles (and other factors) and then the information content of the system is then log(N)?

A region of space is not a particle, or the state of a particle.

59 minutes ago, DannyTR said:

The coordinates would be chosen separately for each specific region of space for which we are calculating information content.

When not measured, the wave form must have a centre. Could this could be used in place of the particles position, when calculating information content?

 You only know where it is if you measure it. I don't see how that's different from simply knowing the particle exists. How is the information different if you know that it is in one region vs being in some other region? 

[DannyTR]

3 minutes ago, swansont said:

A region of space is not a particle, or the state of a particle.

What I mean is the information in a region of space is the sum of the information of the particles within that region.

4 minutes ago, swansont said:

You only know where it is if you measure it. I don't see how that's different from simply knowing the particle exists. How is the information different if you know that it is in one region vs being in some other region? 

But the particle has a position independent of any measurement we do, IE the centre of its wave? So the positional information exists, its just not measurable by us. So a region of space has information in it whether we measure or not.

If space is truly continuous then a 1 meter cubed region of space is graduated as finely as a 1 light-year region (implicit from the definition of continuous). That seems contradictory by itself: suggests the small region contains as many distinct (infinite) states (therefore information) as the large region.

 

[swansont]

1 hour ago, DannyTR said:

What I mean is the information in a region of space is the sum of the information of the particles within that region.

And that says nothing about  "the volume of space" carrying any information

1 hour ago, DannyTR said:

But the particle has a position independent of any measurement we do, IE the centre of its wave? So the positional information exists, its just not measurable by us. So a region of space has information in it whether we measure or not.

No, it does not exist until measured by us.

1 hour ago, DannyTR said:

If space is truly continuous then a 1 meter cubed region of space is graduated as finely as a 1 light-year region (implicit from the definition of continuous). That seems contradictory by itself: suggests the small region contains as many distinct (infinite) states (therefore information) as the large region.

If it's zero, then the issue is moot. But even if not, it is not a contradiction.

Same concept can be applied to the real numbers between any two values, and any two larger values. Both are infinite, but one is not larger than the other.