Would you say that it takes energy for a dimension (of any kind) to exist?
I'd have to say no, because it's sort of like saying only the s-orbital in an atom exists if there's only one electron. There's still a p-orbital, even if it's empty. So I'd say there are N dimensions, however many necessary to resolve an event...
I guess I risk being called a fastidious nitpick, but I'm not sure the analogy holds. The orbitals are there because the nucleus is there, not because the electrons ;-)
Ah, but it takes two to tango, and in this analogy it takes three. The question would be phrased, "Does the p-orbital exist without the energy of a photon to excite the electron out of the s-orbital?" So the analogy is between the number of dimensions which exist without energy having to be there, and the number of energy levels in an atom... both infinite, correct? Does this sound better (as an analogy)?
-edit- Oops, well, how many energy levels for a bound electron? Maybe not infinite...
Ah, but it takes two to tango, and in this analogy it takes three. The question would be phrased, "Does the p-orbital exist without the energy of a photon to excite the electron out of the s-orbital?"
I can see that. So the energy levels are there even if no electron jumps and "probes" them. Got it.
Quote:
So the analogy is between the number of dimensions which exist without energy having to be there, and the number of energy levels in an atom... both infinite, correct? Does this sound better (as an analogy)?
-edit- Oops, well, how many energy levels for a bound electron? Maybe not infinite...
Well, you lost me again. What do you mean by the number of dimensions being infinite?
I'd have to say no, because it's sort of like saying only the s-orbital in an atom exists if there's only one electron. There's still a p-orbital, even if it's empty. So I'd say there are N dimensions, however many necessary to resolve an event...
Without bogging down in too much semantics, the potential exists for an electron to promote to a higher energy level by absorbing a photon, and the reverse process. We only know of orbitals as energy ( and angular momentum ) 'slots' in the EM interaction between the charged entities in an atom, and these were deduced by the reverse engineering of emission and absorbtion patterns/spectra. The quantisation of energy with bound states, and the continuous behaviour of the unbound ones, is a general qualitative result of wave models applied with boundary or 'edge' conditions. Bohr's 'old' quantum mechanics predicted quite well the hydrogen spectrum ( but little else ) essentially by assuming a fixed number of wavelengths fitting in the circumference of a circle! There's an infinite number of levels predicted here, but they are increasingly finely spaced as one approaches ionization level.
Aside: 'Dimension' can mean anything from 'another independent direction of spatial character' to 'another independent direction of timelike character' or simply 'another independent variable'. You can speak of a 'mass dimension' if you like, it's not directly 'visible' but it's effects sure are. EM 'charge', EW 'isospin', strong force 'colour charge', and 'gravitational charge' aka. mass etc....
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Well, you lost me again. What do you mean by the number of dimensions being infinite?
In an abstract sense, there are no rules prohibiting an infinite number of dimensions. So extended to quantum mechanics, there may be some other fundamental principle ('Least Action'?) that limits the number of utilized dimensions to a finite number, but otherwise there are as many dimensions as there types of energy. (I'm happy to be corrected, or otherwise nitpicked if I say anything wrong, so thanks Luis!)
In an abstract sense, there are no rules prohibiting an infinite number of dimensions. So extended to quantum mechanics, there may be some other fundamental principle ('Least Action'?) that limits the number of utilized dimensions to a finite number, but otherwise there are as many dimensions as there types of energy. (I'm happy to be corrected, or otherwise nitpicked if I say anything wrong, so thanks Luis!)
I noticed that you have a Penrose tiling as your picture (Sorry if that is actually your actual face ;-) I was reading a book by Penrose recently, "The road to reality". He does actually object to extra dimensions, even the few extra required by supersymetric theories, on the basis that it would cause instabilities in the classical limit. He also argues something else I couldn't quite understand about the functional degrees of freedom increasing in some exponential form with the dimension of the space in a catastrophic way. BTW, I'm not correcting anybody, I just like to argue.
I was reading a book by Penrose recently, "The road to reality". He does actually object to extra dimensions, even the few extra required by supersymetric theories, on the basis that it would cause instabilities in the classical limit. He also argues something else I couldn't quite understand about the functional degrees of freedom increasing in some exponential form with the dimension of the space in a catastrophic way.
Not quite sure, but it seems to me that each dimension roughly corresponds to a functional degree of freedom. So extra dimensions would add extra degrees of freedom, which would lead to null interactions instead of resolved ones. But is there a limit to energy density (gradient?) in a single dimension, that is, is there a limit whereupon excess energy in one dimension will suffuse into adjacent dimensions? How else could there be an equivalence (and transformations) between matter and energy?
In response to time being the fourth dimension, that was a big fad in the 80's and 90's. Physicists have been trying to tell people otherwise, but it's difficult to uproot such beliefs.
You're probably going to say "well, then what IS the fourth dimension?" The ebst way I can describe it is by saying try to describe a cube to a square. There is a book called "Flatland," which deals with the mathematical and physical difficulties of a two-dimensional world, when an ordinary square is told by a three-dimensional being that there is another land that exists in 3-dimensions. The church of Flatlanders tries to keep this advancement of thought back because their entire dogma is based on a two-dimensional God, etc. It's a very interesting book, and deals with social groups vs science, etc.
I'm not sure what you mean by "time being the fourth dimension" as a fad. General relativity is inextricably tied up with that notion, and I am not aware of any experiments or observations falsifying it.
Maybe you mean that string theorists are saying there are 10 or 11 dimensions, but most of them are spatial and are curved in on themselves very tightly. If that is true, time might be called the 10th or 11th dimension.
Holmes,
Are you asking if it takes extra energy to create a new dimension?
There is a subtle point here, that it's really changes in energy that are well defined rather than energy itself. When we talk about the energy in a bullet being so-and-so Joules, we are implicitly saying something like "the energy of this bullet moving this fast is so-and-so Joules greater than that of the bullet staying still." Or, even better, "it would take so-and-so Joules from some source (like gunpowder) to move the bullet from sitting still in the gun up to this fast moving out of it."
It's like with electric potential, where only differences are physically meaningful. Sure, we might talk about a power line as being at 50,000 volts, but that is implicitly with reference to the ground which we have chosen to call 0 volts. We could have just as well called the ground 1 volt and the power line 50,001 volts and it wouldn't make any physical difference.
In classical general relativity you can't really change the number of dimensions. But if the string theory version of quantum gravity is right and there are really 10 or 11 dimensions with all but 4 wrapped up tight at microscopic scales so we don't see them, you might be able to unwrap one to macroscopic scales. I have trouble keeping up with the different versions of string theory, but I believe the claim is that at the Big Bang all 10 or 11 were wrapped and the Bang was our favorite 4 unwrapping.
The existence of those extra dimensions could be tested by seeing how well the inverse square law (Newton's) works at small distances, as I believe Mike Hewson was saying. There has been an experimental push on this lately, but the answer so far is that Newton's law is observed to hold down to millimeters or less and it's experimentally very tricky to go much lower. Of course LIGO is very tricky too....
Then comes the question of does unwrapping one of those dimensions cost energy, and we get into the issue that the energy of a spacetime is not usually well defined. This post is already getting long enough, but remind me to come back to that later.
Hope this helps,
Ben
P.S. Abbott's "Flatland" has been a classic book for a century. It is very cheap and very short, and the mental tickle is well worth the small investment of time and money.
Last I knew, all dimensions denoted by a number n, were considered to be at right-angles to all other dimensions preceeding it. Granted, time may be thought of as "at right angles" to any preceeding dimension, but I feel there is a possibility that it may have a higher cardinality than "4", as you said with string theorists.
Last I knew, all dimensions denoted by a number n, were considered to be at right-angles to all other dimensions preceeding it. Granted, time may be thought of as "at right angles" to any preceeding dimension, but I feel there is a possibility that it may have a higher cardinality than "4", as you said with string theorists.
Hi A Brown - n dimensions all at right angles to each other might be the way to do it purely abstractly, but there are geometries (e.g., of particles) to consider. Lots of interpretations, but terms like embedded space and hyperspace are other common ways to consider how best to model things with more than 3 dimensions... Time plays a role in all the higher dimensions, so why not call it the first dimension, then? I think it's usually expressed something like "R3 + T" for regular flat space and time, and Dn for n additional "compact" or embedded dimensions, depending on which models/theories you're looking at...
RE: RE: RE: Would you
)
Ah, but it takes two to tango, and in this analogy it takes three. The question would be phrased, "Does the p-orbital exist without the energy of a photon to excite the electron out of the s-orbital?" So the analogy is between the number of dimensions which exist without energy having to be there, and the number of energy levels in an atom... both infinite, correct? Does this sound better (as an analogy)?
-edit- Oops, well, how many energy levels for a bound electron? Maybe not infinite...
RE: Ah, but it takes two
)
I can see that. So the energy levels are there even if no electron jumps and "probes" them. Got it.
Well, you lost me again. What do you mean by the number of dimensions being infinite?
RE: I'd have to say no,
)
Without bogging down in too much semantics, the potential exists for an electron to promote to a higher energy level by absorbing a photon, and the reverse process. We only know of orbitals as energy ( and angular momentum ) 'slots' in the EM interaction between the charged entities in an atom, and these were deduced by the reverse engineering of emission and absorbtion patterns/spectra. The quantisation of energy with bound states, and the continuous behaviour of the unbound ones, is a general qualitative result of wave models applied with boundary or 'edge' conditions. Bohr's 'old' quantum mechanics predicted quite well the hydrogen spectrum ( but little else ) essentially by assuming a fixed number of wavelengths fitting in the circumference of a circle! There's an infinite number of levels predicted here, but they are increasingly finely spaced as one approaches ionization level.
Aside: 'Dimension' can mean anything from 'another independent direction of spatial character' to 'another independent direction of timelike character' or simply 'another independent variable'. You can speak of a 'mass dimension' if you like, it's not directly 'visible' but it's effects sure are. EM 'charge', EW 'isospin', strong force 'colour charge', and 'gravitational charge' aka. mass etc....
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: Well, you lost me
)
In an abstract sense, there are no rules prohibiting an infinite number of dimensions. So extended to quantum mechanics, there may be some other fundamental principle ('Least Action'?) that limits the number of utilized dimensions to a finite number, but otherwise there are as many dimensions as there types of energy. (I'm happy to be corrected, or otherwise nitpicked if I say anything wrong, so thanks Luis!)
RE: In an abstract sense,
)
I noticed that you have a Penrose tiling as your picture (Sorry if that is actually your actual face ;-) I was reading a book by Penrose recently, "The road to reality". He does actually object to extra dimensions, even the few extra required by supersymetric theories, on the basis that it would cause instabilities in the classical limit. He also argues something else I couldn't quite understand about the functional degrees of freedom increasing in some exponential form with the dimension of the space in a catastrophic way. BTW, I'm not correcting anybody, I just like to argue.
RE: I was reading a book by
)
Not quite sure, but it seems to me that each dimension roughly corresponds to a functional degree of freedom. So extra dimensions would add extra degrees of freedom, which would lead to null interactions instead of resolved ones. But is there a limit to energy density (gradient?) in a single dimension, that is, is there a limit whereupon excess energy in one dimension will suffuse into adjacent dimensions? How else could there be an equivalence (and transformations) between matter and energy?
In response to time being the
)
In response to time being the fourth dimension, that was a big fad in the 80's and 90's. Physicists have been trying to tell people otherwise, but it's difficult to uproot such beliefs.
You're probably going to say "well, then what IS the fourth dimension?" The ebst way I can describe it is by saying try to describe a cube to a square. There is a book called "Flatland," which deals with the mathematical and physical difficulties of a two-dimensional world, when an ordinary square is told by a three-dimensional being that there is another land that exists in 3-dimensions. The church of Flatlanders tries to keep this advancement of thought back because their entire dogma is based on a two-dimensional God, etc. It's a very interesting book, and deals with social groups vs science, etc.
A Brown, I'm not sure what
)
A Brown,
I'm not sure what you mean by "time being the fourth dimension" as a fad. General relativity is inextricably tied up with that notion, and I am not aware of any experiments or observations falsifying it.
Maybe you mean that string theorists are saying there are 10 or 11 dimensions, but most of them are spatial and are curved in on themselves very tightly. If that is true, time might be called the 10th or 11th dimension.
Holmes,
Are you asking if it takes extra energy to create a new dimension?
There is a subtle point here, that it's really changes in energy that are well defined rather than energy itself. When we talk about the energy in a bullet being so-and-so Joules, we are implicitly saying something like "the energy of this bullet moving this fast is so-and-so Joules greater than that of the bullet staying still." Or, even better, "it would take so-and-so Joules from some source (like gunpowder) to move the bullet from sitting still in the gun up to this fast moving out of it."
It's like with electric potential, where only differences are physically meaningful. Sure, we might talk about a power line as being at 50,000 volts, but that is implicitly with reference to the ground which we have chosen to call 0 volts. We could have just as well called the ground 1 volt and the power line 50,001 volts and it wouldn't make any physical difference.
In classical general relativity you can't really change the number of dimensions. But if the string theory version of quantum gravity is right and there are really 10 or 11 dimensions with all but 4 wrapped up tight at microscopic scales so we don't see them, you might be able to unwrap one to macroscopic scales. I have trouble keeping up with the different versions of string theory, but I believe the claim is that at the Big Bang all 10 or 11 were wrapped and the Bang was our favorite 4 unwrapping.
The existence of those extra dimensions could be tested by seeing how well the inverse square law (Newton's) works at small distances, as I believe Mike Hewson was saying. There has been an experimental push on this lately, but the answer so far is that Newton's law is observed to hold down to millimeters or less and it's experimentally very tricky to go much lower. Of course LIGO is very tricky too....
Then comes the question of does unwrapping one of those dimensions cost energy, and we get into the issue that the energy of a spacetime is not usually well defined. This post is already getting long enough, but remind me to come back to that later.
Hope this helps,
Ben
P.S. Abbott's "Flatland" has been a classic book for a century. It is very cheap and very short, and the mental tickle is well worth the small investment of time and money.
Last I knew, all dimensions
)
Last I knew, all dimensions denoted by a number n, were considered to be at right-angles to all other dimensions preceeding it. Granted, time may be thought of as "at right angles" to any preceeding dimension, but I feel there is a possibility that it may have a higher cardinality than "4", as you said with string theorists.
RE: Last I knew, all
)
Hi A Brown - n dimensions all at right angles to each other might be the way to do it purely abstractly, but there are geometries (e.g., of particles) to consider. Lots of interpretations, but terms like embedded space and hyperspace are other common ways to consider how best to model things with more than 3 dimensions... Time plays a role in all the higher dimensions, so why not call it the first dimension, then? I think it's usually expressed something like "R3 + T" for regular flat space and time, and Dn for n additional "compact" or embedded dimensions, depending on which models/theories you're looking at...