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[Azurite]

This isn't what "choosing a spacetime metric" means. A spacetime metric has units of length/time. It doesn't have units of 1/length or 1/time. If you have a thingy with units of 1/length and 1/time, it isn't a spacetime metric.
There is no such thing. If you have 1/length and 1/time, then you have length and time; you just take the reciprocals of your units. There's no way to not allow that if you're using math.

you said spacetime metric can't be 1/length or 1/time.. so if particle physicists use graph of them, then what they are called and how do you differentiate from spacetime diagram?

 

[PeterDonis]

if particle physicists use graph of them

Where did I say anything about a "graph"? I said particle physicists routinely switch between space/time units and momentum/energy units, which in quantum field theory with $\hbar = 1$ are reciprocals of each other.

Particle physicists use Feynman diagrams, which are most often drawn using momentum/energy, but I don't know if I would call those a "graph" in the sense you mean, since they're highly schematic and don't represent particles having any particular momentum or energy, but represent terms in integrals over a whole range of momenta and energies.

 

[Azurite]

Where did I say anything about a "graph"? I said particle physicists routinely switch between space/time units and momentum/energy units, which in quantum field theory with $\hbar = 1$ are reciprocals of each other.

Particle physicists use Feynman diagrams, which are most often drawn using momentum/energy, but I don't know if I would call those a "graph" in the sense you mean, since they're highly schematic and don't represent particles having any particular momentum or energy, but represent terms in integrals over a whole range of momenta and energies.

Ok thanks.
From a historical perspective. Einstein created spacetime with spatial metric concept even without quantum mechanics... this means even if quantum mechanics didn't exist.. spacetime is still valid.. so at that time Einstein never try to think about momentum space with momentum metric and this is just latter conjecture after qm and Planck h.. right?

So if momentum space metric is related to quantum mechanics.. this means spacetime with spatial metric is still more fundamental since one can propose this without quantum mechanics?

 

[PeterDonis]

even if quantum mechanics didn't exist.. spacetime is still valid

Yes.

at that time Einstein never try to think about momentum space with momentum metric

The concept of "momentum space", in terms of using momentum-energy units instead of space-time units, does not require quantum mechanics. Einstein did not look at things this way as far as I know, but then again he didn't initially pick up the spacetime concept either when Minkowski published it in 1907. It wasn't until Einstein realized that spacetime geometry was the concept he needed for a relativistic theory of gravity that he started thinking in terms of spacetime.

if momentum space metric is related to quantum mechanics

It isn't. See above. QM uses it, but that doesn't mean it's only valid in QM.

 

[Azurite]

Yes.
The concept of "momentum space", in terms of using momentum-energy units instead of space-time units, does not require quantum mechanics. Einstein did not look at things this way as far as I know, but then again he didn't initially pick up the spacetime concept either when Minkowski published it in 1907. It wasn't until Einstein realized that spacetime geometry was the concept he needed for a relativistic theory of gravity that he started thinking in terms of spacetime.
It isn't. See above. QM uses it, but that doesn't mean it's only valid in QM.

Also note spacetime metric doesn't hold in non-relativistic quantum mechanics in the Schrödinger equation. Therefore using 1/length and 1/space doesn't mean it's using spacetime metric... so for purposes of illustration is it not incorrect to use 1/length and 1/space graphs for depicting momentum and energy in non-relativistic QM? Does anyone make such graph? I'm not talking about relativistic qft Feynman diagram but non-relativistic QM.

 

[PeterDonis]

spacetime metric doesn't hold in non-relativistic quantum mechanics in the Schrödinger equation

Well, of course, since there is no "spacetime metric" in non-relativistic physics.

Therefore using 1/length and 1/space doesn't mean it's using spacetime metric

In non-relativistic physics, that's correct, you can adopt "natural" quantum units without using a spacetime metric.

so for purposes of illustration is it not incorrect to use 1/length and 1/space graphs for depicting momentum and energy in non-relativistic QM?

What would "1/length and 1/space graphs" look like?

Does anyone make such graph?

I don't know, since I don't know what you mean by those terms.

 

[Azurite]

Well, of course, since there is no "spacetime metric" in non-relativistic physics.
In non-relativistic physics, that's correct, you can adopt "natural" quantum units without using a spacetime metric.
What would "1/length and 1/space graphs" look like?
I don't know, since I don't know what you mean by those terms.

Reference frames can be anything from basic 4 dimensional Einstein spacetime to the 6 compactified dimensions in superstring theory, etc.. How about non-relativistic reference frames.. is Galilean space also a reference frame? What are other non-relativistic reference frames?

 

[PeterDonis]

Reference frames

The things you are describing are not reference frames. They are manifolds.

 

[Azurite]

The things you are describing are not reference frames. They are manifolds.

Space, reciprocal space are reference frames.
Spacetime, calabi-yau compactified dimensions are manifolds.

Right?

So "reference frames" are used for non-relativistic space? So Galilean space is a reference frame and not manifold. Right?

 

[Azurite]

Space, reciprocal space are reference frames.
Spacetime, calabi-yau compactified dimensions are manifolds.

Right?

So "reference frames" are used for non-relativistic space? So Galilean space is a reference frame and not manifold. Right?

I found out there are many meaning of reference frames:

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

"The need to distinguish between the various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in Cartesian frame of reference. Sometimes the state of motion is emphasized, as in rotating frame of reference. Sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference.[1]"

I want to ask about reference frames as used in Newtonian space or non-relativistic quantum mechanics..

Does it make sense or semantically correct to say the reference frame of particles are direct space (x,y,z,t).. while the reference frame of waves are reciprocal space (with axis kx, ky, kz, kt)? Any instructor teach in this manner (to undergraduate for instance) or it is ok for laymen?

 

[PeterDonis]

Space, reciprocal space are reference frames.
Spacetime, calabi-yau compactified dimensions are manifolds.

Right?

No. "Space" is a manifold. I don't know what you mean by "reciprocal space".

So "reference frames" are used for non-relativistic space? So Galilean space is a reference frame and not manifold. Right?

Wrong. "Galilean space" is a manifold.

I found out there are many meaning of reference frames

Yes, but none of them correspond to the way you are using the term.

I want to ask about reference frames as used in Newtonian space or non-relativistic quantum mechanics

Then you should just ask, instead of making claims that are wrong and then asking if they are right.

Does it make sense or semantically correct to say the reference frame of particles are direct space (x,y,z,t).. while the reference frame of waves are reciprocal space (with axis kx, ky, kz, kt)?

No.

 

[Azurite]

No. "Space" is a manifold. I don't know what you mean by "reciprocal space".
Wrong. "Galilean space" is a manifold.
Yes, but none of them correspond to the way you are using the term.
Then you should just ask, instead of making claims that are wrong and then asking if they are right.
No.

When one creates a universe with spacetime. It automatically has momentum because there is a position, right?

But should it automatically produce the quantum configuration space or is this an additional thing?

If it is an additional thing. What kind of manifold can house the quantum configuration space.. so maybe one adds 2 manifolds together.. the spacetime manifold and the extra quantum configuration space manifold?

 

[PeterDonis]

When one creates a universe with spacetime. It automatically has momentum because there is a position, right?

I don't know what you mean by this.

should it automatically produce the quantum configuration space or is this an additional thing?

I don't know what you mean by this either.

As I said before, you seem to me to be speculating without a good conceptual basis.

 

[Azurite]

I don't know what you mean by this.

I just meant whenever there was a universe with spacetime.. it should automatically have position and momentum. Correct?

But should it also somehow contain the quantum configuration space? Or is this extra?

I don't know what you mean by this either.

As I said before, you seem to me to be speculating without a good conceptual basis.

 

[PeterDonis]

I just meant when there was a universe with spacetime.. it should automatically have position and momentum. Correct?

Spacetime doesn't have position and momentum. Particular objects in spacetime have position and momentum.

should it also somehow contain the quantum configuration space? Or is this extra?

I still don't know what you mean. We don't create the universe, we just discover what it's like. We don't get to pick whether it has a quantum configuration space or not, or whether it's a spacetime, or whether objects have position and momentum. So your questions seem meaningless to me.

 

[Azurite]

Spacetime doesn't have position and momentum. Particular objects in spacetime have position and momentum.
I still don't know what you mean. We don't create the universe, we just discover what it's like. We don't get to pick whether it has a quantum configuration space or not, or whether it's a spacetime, or whether objects have position and momentum. So your questions seem meaningless to me.

In the string landscape or multiverse or others.. each universe has different laws of nature. I just want to differentiate or understand what it would be like to have spacetime only without configuration space.

Or I just want to understand how to relate spacetime and configuration space.

Is manifold used for configuration space? What is the relationship between manifold and configuration space?

 

[Azurite]

In the string landscape or multiverse or others.. each universe has different laws of nature. I just want to differentiate or understand what it would be like to have spacetime only without configuration space.

Or I just want to understand how to relate spacetime and configuration space.

Is manifold used for configuration space? What is the relationship between manifold and configuration space?

Configuration space is like the Hamiltonian phase space.. describing 2 particles with 6 dimensions.. and for more particles (infinite).. it is infinite dimensional...

A manifold is not a phase space but has a limit..

so I guess this is the main difference between the two. Right?

About the pilot wave.. it is vague as it is in configuration space... but can't you make the pilot wave like marbles in a box.. the marbles have multidimensional hamiltonian phase space but it doesn't mean the marbles don't exist in 3D.. likewise.. for those who propose pilot wave use some quantum force that has quanta or particles... can it occur just like the marbles? What manifold do you describe this real pilot wave? spacetime too? To distinguish. Others use the terms "Reciprocal space" but I think this is very vague terms hence mustn't be used. So what should the term/jargon for the manifold that house the real pilot waves?

 

[PeterDonis]

In the string landscape or multiverse or others.. each universe has different laws of nature.

No, each universe just has particular values for a bunch of physical constants that are free parameters in our current laws of nature. And anyway all that is speculative, there's no evidence for it. And in any case, the speculative models you mention don't look anything like what you are asking about.

I just want to differentiate or understand what it would be like to have spacetime only without configuration space.

I don't know of any such model.

I just want to understand how to relate spacetime and configuration space.

That's way too broad for a PF thread. You're asking for several undergraduate courses' worth of information.

Is manifConfiguration space is like the Hamiltonian phase spaceold used for configuration space? What is the relationship between manifold and configuration space?

Configuration space is an example of a manifold. If you want to know what a manifold is, again, you're basically asking for a course in mathematical physics. If you don't already have that background then you need to take the time to learn it. There are no shortcuts.

 

[PeterDonis]

Configuration space is like the Hamiltonian phase space

No, it isn't. Phase space contains "dimensions" for both position and momentum. Configuration space only contains "dimensions" for position.

A manifold is not a phase space but has a limit

Phase space is a manifold. I have no idea what you mean by "has a limit".

so I guess this is the main difference between the two. Right?

No. See above.

About the pilot wave

Again, this is much too broad for a PF discussion; you are basically asking for a course in QM. You need to take the time to learn it yourself.

 

[PeterDonis]

At this point I am closing the thread since no useful progress is being made. @Azurite , given your apparent lack of background in math and physics, the method you seem to be pursuing, of guessing what various terms mean and then asking if your guesses are right, is not going to work well. As I said in the last couple of posts, you need to take the time to actually work through several courses' worth of material, without trying to guess in advance what it is going to tell you. If during that process you have particular questions about particular items, you can post specific threads here about them. But you're basically trying to start from scratch and get all that several courses' worth of material from PF. That's not what PF is for (and, as noted above, the method you're using doesn't work well anyway).