Fourier Transform of Spacetime

[fanieh]

When you do a Fourier transform of spacetime.. what do you get? (or how does spacetime look in frequency domain? And what applications do this and what results are they looking or solving for?

 

[PeterDonis]

When you do a Fourier transform of spacetime

This concept doesn't even make sense. Spacetime isn't a function.

 

[fanieh]

This concept doesn't even make sense. Spacetime isn't a function.

Oh, I mean. If spacetime is of the distance-time variety.. is there an inverse version (a frequency domain)? If this frequency domain inverse version is not called spacetime.. then what is it called and what application use this frequency domain (or inverse of spacetime)? Thanks.

 

[Dale]

The Fourier domain of time is frequency, and the Fourier domain of space is k-space. So I guess it would be called k-omega space or something similar.

I don't think it is used physically much. Usually you would use the four momentum space instead.

 

[PeterDonis]

If spacetime is of the distance-time variety.. is there an inverse version (a frequency domain)?

The Fourier transform doesn't transform space and time. It transforms functions of space and time into functions of wave number and frequency (and the inverse transforms the other way). So talking about a Fourier transform of spacetime itself doesn't make sense.

The Fourier domain of time is frequency, and the Fourier domain of space is k-space.

I think this is misstated. A Fourier transform transforms functions, as above; it doesn't transform space or time themselves.

 

[Dale]

I think this is misstated. A Fourier transform transforms functions, as above; it doesn't transform space or time themselves.

it transforms a function in the time domain into an equivalent function in the frequency domain. The OP seems to be asking about the domains.

 

[fanieh]

The Fourier transform doesn't transform space and time. It transforms functions of space and time into functions of wave number and frequency (and the inverse transforms the other way). So talking about a Fourier transform of spacetime itself doesn't make sense.

Ok. The inverse of distance is number per unit distance which is is spatial frequency, and the inverse of time is number per unit time which is a temporal frequency?

So can you turn them into a four-dimensional subspace reference frame called wave number space and denoted by the vector Kx, Ky, Kz, Kt? What application need to plot it as 4 dimensional?

I think this is misstated. A Fourier transform transforms functions, as above; it doesn't transform space or time themselves.

 

[PeterDonis]

it transforms a function in the time domain into an equivalent function in the frequency domain. The OP seems to be asking about the domains.

Yes. I think part of the response needs to be to make that distinction clear.

The inverse of distance is number per unit distance which is is spatial frequency, and the inverse of time is number per unit time which is a temporal frequency?

Yes.

can you turn them into a four-dimensional subspace reference frame called wave number space and denoted by the vector Kx, Ky, Kz, Kt?

Yes. This is basically the same thing as the energy-momentum space that @Dale referred to, just in different units.

Note carefully, though, that, as @Dale pointed out, this is the domain of functions which have been Fourier transformed. It's important to keep clear the distinction between the domain--the argument to the functions--and the functions themselves. The functions are what get transformed, not the domains.

What application need to plot it as 4 dimensional?

To take just one example, particle physics experiments are analyzed in momentum space (which is the usual name for the energy-momentum space described above, mainly for brevity), because the energy and momentum of the incoming and outgoing particles are what the experiments actually measure.

More generally, energy and momentum, taken by themselves, are frame-dependent; the proper object for use in analysis is the energy-momentum 4-vector (often called "4-momentum", or just "momentum", again for brevity), just as the proper object for use in analysis in the space/time domain is the 4-position vector.

 

[fanieh]

Yes. I think part of the response needs to be to make that distinction clear.
Yes.
Yes. This is basically the same thing as the energy-momentum space that @Dale referred to, just in different units.

Note carefully, though, that, as @Dale pointed out, this is the domain of functions which have been Fourier transformed. It's important to keep clear the distinction between the domain--the argument to the functions--and the functions themselves. The functions are what get transformed, not the domains.
To take just one example, particle physics experiments are analyzed in momentum space (which is the usual name for the energy-momentum space described above, mainly for brevity), because the energy and momentum of the incoming and outgoing particles are what the experiments actually measure.

More generally, energy and momentum, taken by themselves, are frame-dependent; the proper object for use in analysis is the energy-momentum 4-vector (often called "4-momentum", or just "momentum", again for brevity), just as the proper object for use in analysis in the space/time domain is the 4-position vector.

In the double slit experiments.. can the screen or detector interference patterns be nothing but the momentum 4 vector?

see: http://www.users.csbsju.edu/~frioux/diffraction/s00897040748a1.pdf

"In Marcella’s quantum mechanical analysis of the double slit experiment, what is subsequently measured at the detection screen is actually the particle’s momentum. In other words, the well-known diffraction pattern created by the double-slit geometry is the particle’s momentum distribution in the plane of the detection screen. Therefore, to calculate the diffraction pattern one needs a momentum wave function, and this is obtained by a Fourier transform of eq 1 into momentum space"

 

[PeterDonis]

can the screen or detector interference patterns be nothing but the momentum 4 vector?

I'm not sure what you mean. The two descriptions that are linked by a Fourier transform--the "position space" description (i.e., the description in terms of functions on ordinary spacetime) and the momentum space description (in terms of functions on momentum space) are mathematically equivalent; you can always convert between them. So just describing an experiment in momentum space instead of position space doesn't change any of the actual physics.

Based on that, I think this statement in the paper is misleading:

In Marcella’s quantum mechanical analysis of the double slit experiment, what is subsequently measured at the detection screen is actually the particle’s momentum.

What is actually measured in an experiment depends on the physical properties of the detector, not on what mathematical description we use. The actual pattern of light and dark that is observed on the detector is a function of position on the detector, not momentum of anything.

 

[fanieh]

I'm not sure what you mean. The two descriptions that are linked by a Fourier transform--the "position space" description (i.e., the description in terms of functions on ordinary spacetime) and the momentum space description (in terms of functions on momentum space) are mathematically equivalent; you can always convert between them. So just describing an experiment in momentum space instead of position space doesn't change any of the actual physics.

Based on that, I think this statement in the paper is misleading:
What is actually measured in an experiment depends on the physical properties of the detector, not on what mathematical description we use. The actual pattern of light and dark that is observed on the detector is a function of position on the detector, not momentum of anything.

Spacetime can be many dimensional such as the 11 dimensions of string theory. According to pilot wave quantum interpretation where there are many variants. One variant is the concept of duplex Reference Frame where there are two subspaces one of which is spacetime. Thus, as an approximation, one might consider it as a member of the general, eight-dimensional space. In this concept, the thing that makes this duplex RF unique and specific is that the other subspace is a reciprocal subspace to spacetime. Thus we have two potential four-dimensional subspaces; one is spacetime, the other is a reciprocal spacetime... and the pilot wave variant suggests the wave part lives in the reciprocal space.

Now in the double slit experiment and using that model. Is it really possible that in this duplex-space perspective, the slit structure itself, without the light waves, already has a Reciprocal space substance interference pattern existing around the slit regions of the ordinary spacetime structure. The model is that it is this reciprocal space pattern that guides the light into its maxima and minima ordinary space intensity locations behind the slits?

Do you have any arguments that can totally refute this pilot wave variant so if it's not plausible.. I can forget about it because I've been thinking of this from time to time for the past 12 years. So please address this. Thank you.

 

[Dale]

the pilot wave variant

This is probably best in the QM forum, not in the relativity section.

Now in the double slit experiment and using that model. Is it really possible that in this duplex-space perspective, the slit structure itself, without the light waves, already has a Reciprocal space substance interference pattern existing around the slit regions of the ordinary spacetime structure. The model is that it is this reciprocal space pattern that guides the light into its maxima and minima ordinary space intensity locations behind the slits

Do you have a reference for this? Be sure to post that reference in your opening post in the QM forum.

 

[martinbn]

I am pretty sure what you are asking doesn't make sense. But you may find this interesting

https://www.newscientist.com/article/mg21128241-700-beyond-space-time-welcome-to-phase-space/

 

[PeterDonis]

Spacetime can be many dimensional such as the 11 dimensions of string theory.

String theory does not have any experimental support. Discussion of it belongs in the Beyond the Standard Model forum.

 

[PeterDonis]

The OP question has been answered, and the further topics being brought up belong in separate threads. This thread is closed.