Hard Light Productions Forums
Off-Topic Discussion => General Discussion => Topic started by: Aardwolf on September 23, 2008, 03:39:14 pm
-
Tachyons are, as I understand it, hypothetical particles which travel faster than light.
Somehow, such a particle would have to have imaginary mass (I haven't find the source for that, but I saw it somewhere and I'm sure it should be fairly easy to locate).
So, what would something with imaginary mass do were it to collide with a particle with positive mass?
Feel free to discuss this and other things related to tachyons (in science, preferably, and not as used in science fiction).
-
I thought that they were originally theorized because there was loss of mass during an experiment. Basically since E=MC^2 wasn't preserved they knew there had to be a partial that disappeared somehow. FTL or time travel were possibilities.
-
I don't know about that really, but it sounds like it could have been explained by neutrinos instead.
-
I thought that they were originally theorized because there was loss of mass during an experiment. Basically since E=MC^2 wasn't preserved they knew there had to be a partial that disappeared somehow. FTL or time travel were possibilities.
Yeah that is how the existence of neutrinos was hypothetized and eventually proved to actually be the correct expectation.
Tachyons are a lot more hypothetical. Basically, they are based on the interpretation of relativity that essentially says that objects with a rest mass always travel at sub-light speeds in relation to others, objects with no rest mass will always be measured as traveling at light speed (which is the reason why light speed is constant and co-ordinate-independent), and objects iwht imaginary (or negative mass, depending on how one is to interpret mathematical strangeness like imaginary mass to reality) would always travel at superluminal speeds in relation to objects with rest mass.
Tachyons are the name given to these hypothetical particles. What makes them both interesting and dull is that if one is to assume that light speed is the speed of causality, tachyons cannot be used for information transmitting purposes because that would break the causality; you would have the effect before the cause.
Of course, one could argue that it wouldn't mean causality breaking, but instead it would just mean that light speed wouldn't be the highest information transmission speed in the universe.
If tachyons were to interact with particles that have rest mass, situation would be pretty interesting. Because tachyons basically would have negative mass (mathematically it would be imaginary mass, but the interpretation of equations to reality would pretty much mean that the mass is negative... at least to my understanding), their momentum vector would actually point to the direction they came from. Which is kinda interesting because if one were to assume that objects with mass would continuously emit tachyons, it would result in a force that would draw the objects with mass closer to each other... Of course we would notice if such a force were to exist in universe. Oh wait, there is such a force. Silly me... :drevil:
Note that this is just a thought experiment and is not a serious attempt to explain gravity as mass-tachyon-interactions. To do that, one would need to determine that the velocity of gravitational propagations is greater than light speed, AND find a mechanism that would explain why objects with mass would emit tachyons continuously, AND explain why and how exactly tachyons interact with particles that have mass.
Determining the velocity of gravitation has historically been quite difficult and results are as of yet inconclusive; in general relativity, the changes in the "tilt" of space-time fabric propagate at light speed as far as I know, but that's just one interpretation of theory and doesn't mean that reality is obliged to follow the model. Although the great accuracy of general realtivity gravitational model does suggest that either it's a very good approximation/abstraction of what's going on, or it actually corresponds to reality (within definite error bars).
Personally, I'm very eagerly expecting conclusive reports from LHC on whether the Higgs' boson exists or not...
-
There's been something bothering me about this whole can't go faster then the speed of light law. Isn't light made up of waves? If so aren't the photons or whatever the waves are composed of already exceeding the speed of light? If say a laser beam is going in a straight line the beam is moving at the speed of light. The waves that make up that been would have to be traveling faster since they have to cover a greater distance in the same time.
-
There's been something bothering me about this whole can't go faster then the speed of light law. Isn't light made up of waves? If so aren't the photons or whatever the waves are composed of already exceeding the speed of light? If say a laser beam is going in a straight line the beam is moving at the speed of light. The waves that make up that been would have to be traveling faster since they have to cover a greater distance in the same time.
Wave motion doesn't really work like that. If you look at a photon as electromagnetic wave motion, you need to consider the elctric fields and the magnetic fields in vacuum kinda like the surface of water, where a wave propagates. The photon propagating through vacuum simply first changes the electric field to one direction and then to other while doign the same to the magnetic field, and then it's gone from that part of the vacuum. They themselves don't move anywhere, it's just the change in the zero potential of vacuum that travels along with the photon... or, depending of interpretation, forms the photon.
Also, phase velocity can exceed the speed of light in some conditions, but information velocity (distance from emission to absorption divided by the time used on it) of the photon (also sometimes called group velocity) itself cannot be greater than c.
Besides, the fact that behaviour of light can be modeled as electromagnetic wave motion doesn't mean that it is merely wave motion, because every particle can be modeled as wave motion; it's just that photons' deBroglie wavelength is the same as the wavelength of the electromagnetic wave. It's possible that it is merely electromagnetic wave motion, but just as well in some cases the behaviour of photons can be described with a particle model. The duality of photons is just much more pronounced than the duality of particles with rest mass.
For example, electron microscopes are based on the fact that electrons can have way shorter wavelength than photons - or rather, photons with low enough wavelength would just pass through or fry the sample. Shorter wavelength makes it possible to get better resolution, obviously, which is why electron microscopes are so much more useful than light microscopes when it comes to really small stuff.
-
There's been something bothering me about this whole can't go faster then the speed of light law. Isn't light made up of waves? If so aren't the photons or whatever the waves are composed of already exceeding the speed of light? If say a laser beam is going in a straight line the beam is moving at the speed of light. The waves that make up that been would have to be traveling faster since they have to cover a greater distance in the same time.
short answer,
no.
-
...one would need to determine that the velocity of gravitational propagations is greater than light speed...
Hasn't there already been work in that direction? I remember reading once (I forget where) that somebody posited that gravity should propagate at many times light speed. Otherwise, planetary orbits would experience lagged feedback and become chaotic.
-
No, gravity has been measured to propagate somewhere between .8 and 1.2 x c, (source is probably wikipedia).
I'd never thought about that, but it seems to work fine for us, and I'm pretty sure gravity propagation should be exactly c.
-
i can imagine you guys and girls chatting away to so many very intelligent scientists along with Samantha carter and then jack O'Neill comes in wonders what the hell is being said puts his hand up and says "AAAhh! oh no" looks around and walks out :P.
ftl hasn't been proven and words won't change that, making this whole discussion void. imo of course.
-
lolwut?
This is a discussion about the hypothetical particle called the tachyon. We're not trying to "prove ftl" (how do you prove "faster than light" anyway, it's not something that can be proven or disproven... it's just a term used to describe any hypothetical (keyword: hypothetical) means of travel or communication that reaches its destination before light would.) And your Stargate reference makes no sense and has no place here anyway.
Whatever.
-
hey, i have much as right as you do to ask and put my opinion across. i do not need you telling me how to run my life thank you. whatever, the same to you.
if you just explained the hypothetical thing a bit clearer i would of made a better remark, unless you think that is out of order as well?
lolwut???.. whatever you said... or whatever it means.
-
No, gravity has been measured to propagate somewhere between .8 and 1.2 x c, (source is probably wikipedia).
I'd never thought about that, but it seems to work fine for us, and I'm pretty sure gravity propagation should be exactly c.
Gravity should propagate at exactly C, I believe. Though I'd have to do some research to figure out what elements of the standard model support that (relativity seems pretty clear on the point.)
-
"Should" and "does" don't always coincide...
I would expect space-time changes to propagate at light speed, though. That is, assuming the model of general relativity actually is a description of reality. However measuring this is difficult because apart from tidal forces it's rather hard to detect gravitational waves; defining their direction and velocity would be even more difficult though not impossible. The gravitation waves inducing tidal forces have way too long a wavelength and amplitude (thankfully) for us to define what velocity it's propagating at.
The only reason I brought up gravity was because tachyon-mass interactions would look like gaining "negative" momentum, or in other words, particles would start moving towards the direction of tachyon source. That is, if mass and tachyons can interact, and if my interpretation of imaginary mass = negative mass is correct. Adding an assumption that every object with invariant mass does actualyl emit tachyons, and you get a rudimentary model of gravity.
The obvious weakness in this model are plenty, mainly it's reliance to multiple assumptions that are pretty much unfalsifiable as of now. Like the fact that tachyons are completely hypothetical to begin with... Still, the idea of mass/tachyon interactions is pretty interesting.
-
About wave and particle-like duality: these are two things that should never be attempted to be used at the same time. It doesn't work that way. You either choose one to model that what you do.
Of course, QED states photons are particle -like - which poindexters with large particle accelerators have found to be a good model - and averaging masses photons yields electric field and wavelike behavior - which cannot explain what happens anything inside the atom or like - which in turn can be simplified to light rays - which cannot explain why rays don't always move in the direction stated by the local geometry - which can then be simplified to paraxial optics, where light ray angles are limited to a couple of degrees from local optical axis...
Mika
-
and if my interpretation of imaginary mass = negative mass is correct.
CP should be along to slap you shortly. :p Imaginary != negative.
However, the square of the mass would be negative. Though I can't offhand think of any equations with m2 in them.
-
and if my interpretation of imaginary mass = negative mass is correct.
CP should be along to slap you shortly. :p Imaginary != negative.
However, the square of the mass would be negative. Though I can't offhand think of any equations with m2 in them.
'tis a fair point, but I already explained my reasons for this interpretation earlier, and I'm the first to admit that it is a pretty big if there about the interpratation being correct:
If tachyons were to interact with particles that have rest mass, situation would be pretty interesting. Because tachyons basically would have negative mass (mathematically it would be imaginary mass, but the interpretation of equations to reality would pretty much mean that the mass is negative... at least to my understanding), their momentum vector would actually point to the direction they came from.
Since the observations made of an object hitting you at superluminal speeds would actually look like it was moving away from you after coming to contact with you, I find this actually the only sensible interpretation available, if one is to attach the term "imaginary mass" to reality in the context of our current knowledge of reality in the first place.
A tachyon passing you would be more interesting, since you would see two images of it, but if the tachyon interacts with you (ie. you stop it), you'll only see the image of the particle as if it was moving away from you after it hit you... :warp:
...In fact it would look like you actually emitted the tachyon in the first place. :nervous:
-
Something about the idea of imaginary mass just doesn't sit well with me. Even if you can come up with some sort of conceptual understanding for what "imaginary mass" would be (and I dispute this, heartily), you've just opened a huge can of worms. What happens when you have mass that falls somewhere in the complex realm m = a+(i*b)? What the gibbering balderdash would that be?
Negative mass (referred to in some circles as "exotic matter") makes some amount of conceptual sense as long as you don't get too bogged down in the particle physics of it. Imaginary mass is like adding a new axis, and dimension, to the previously scalar quantity "mass." The Standard Model would have to be wrong on so many levels its not even funny.
-
Ah, yes, tachyons. Quite interesting things, you can do a whole lot of stuff with those, I just recently stumbled upon a device that emits tachyons to "help you balance you inner energy patterns" or some crap like that.
But let's take a look at real world physics. What are tachyons really? Basically, until proven otherwise, they are one solution of Einsteins special theory of relativity. But does that mean they exist? In my opinion: NO.
I'll illustrate my point by an example of Pythagoras (yes, the triangle guy). As you all know, his famous formula for triangles is a² + b² = c², c being the hypotenuse. Now, if you solve this formula, you get not one but two solutions for the hypotenuse, c and -c. As far as math is involved, both solutions are correct. But if you try to take this solutions to real world physics, the solution -c disappears. A triangle with a negative length for the hypotenuse simply doesn't exist. It's basically the same with tachyons: they are the negative hypotenuse of your average particle.
-
While your argument does make sense and sounds compelling (for me at least), it has two weak points:
First, That works in Euclidian space Special relativity and general relativity are different, because you must add the time component in the world position vector [x y z t].
Second, Black hole is another curiosity that was completely predicted by the mathematics alone. Singularities tend to be a problem in the mathematical point of view, which is a reason a lot of Physicists didn't take that as a real phenomenom.
Mika
-
Ah, yes, tachyons. Quite interesting things, you can do a whole lot of stuff with those, I just recently stumbled upon a device that emits tachyons to "help you balance you inner energy patterns" or some crap like that.
But let's take a look at real world physics. What are tachyons really? Basically, until proven otherwise, they are one solution of Einsteins special theory of relativity. But does that mean they exist? In my opinion: NO.
I'll illustrate my point by an example of Pythagoras (yes, the triangle guy). As you all know, his famous formula for triangles is a² + b² = c², c being the hypotenuse. Now, if you solve this formula, you get not one but two solutions for the hypotenuse, c and -c. As far as math is involved, both solutions are correct. But if you try to take this solutions to real world physics, the solution -c disappears. A triangle with a negative length for the hypotenuse simply doesn't exist. It's basically the same with tachyons: they are the negative hypotenuse of your average particle.
Ah, but what if the sides of the triangle are vectors instead of just lines with some length? :drevil:
Vectors have two properties, length and direction. Scalar quantities only have one property, amount (or magnitude or whatever) and that can be anything depending on the quantity measured, though most scalar quantities do usually have positive values only.
However, one property of particles with negative (or imaginary, whichever interpretation you think better suits reality) mass is that they would always move at superluminal speeds, whereas particles with zero invariant mass would move exactly at c and particles with positive invariant mass would move at sub-light speeds. This means that the question of what a tachyon's rest mass (or invariant mass) actually would be is irrelevant since they can not be at rest any more than photons can. Which makes it much more interesting to just ignore the outlandishness of the concept of negative mass for a while and look at how such a thing would effect the properties of an object. Mainly in this case, momentum... or rather, four-vector, seeing how the origin of the .
When you consider the fact that the mass of moving objects always demonstrates as momentum more than anything else, negative mass becomes much less of a problem because momentum is a vector quantity - negative mass would just invert the vector (at least in the traditional sense of "momentum", I dunno how exactly the four-vectors would behave, but I suspect inversion of either time or the 3-vector would happen, both being completely in line with the tachyon hypothesis...).
Gravitational interactions between tachyons and bradyons would probably be a nightmare to define, though... :shaking:
Anyhow, inversion of momentum is exactly what you would expect from the interpretations of what tachyons would look like; After the tachyon hit you, you would essentially be able to see the tachyon gaining distance from yourself... in fact, tachyons cannot transmit information because their emission and absorption are indistinguishable from each other, which would mean that a tachyon detector would actually look like a tachyon emitter. ;7 Yes, they are hypothetical, and haven't been observed, but that doesn't make them any less possible. The black hole was completely hypothetical assumption based on some pretty exotic solutions of general relativity; similarly the neutrinos were predicted long before they were confirmed to exist. Right now the Higgs' boson and Higgs' field are under the scrutiny; quantum theories predict their existence, but they haven't yet been observed.
Tachyons are, obviously, a bit different; they aren't a direct prediction but rather a way of General Relativity to deal with all the possibilities - it just says that to exceed light speed, particles would need to have imaginary (or negative) mass; it doesn't really take any real opinion on their existence, for or against. But the black holes are a good example of how a mathematical strangeness can actually exist in real world... although I still don't think real singularities exist inside the event horizon. Singularities are even more worrisome than imaginary mass mathematically, since they imply infinities in a definite universe... :nervous:
-
Singularities are even more worrisome than imaginary mass mathematically, since they imply infinities in a definite universe...
Don't worry. I have never observed a singularity that is defined in the mathematical sense.
Mika
-
Don't worry. I have never observed a singularity that is defined in the mathematical sense.
Mika
:lol:
Well, that's re-assuring to know. :P
-
Complex mass >> Complex momentum >> Complex velocity >> Complex position >> BAD!
-
Herra, I still don't get your posit that the ideas of negative and imaginary mass are interchangeable. Unless the formulae you are using can be factored to m2 where ever mass is mentioned, there is a whooping big difference between -m and m*i. Negative mass would essentially curve space-time in the opposite fashion compared to normal mass. I can wrap my mind around that. I have doubts that such particles actually exist, but I can handle that conceptually. You start talking about imaginary mass and you suddenly start implying that mass has a phase component. It is no longer a scalar quantity. That is not compatible with the Standard Model.
-
Herra, I still don't get your posit that the ideas of negative and imaginary mass are interchangeable.
Not really interchangeable, I'm just trying to assign some kind of sensible (lol, I know) physical equivalent to imaginary mass that appears in the equations. Since negative mass offers a simpler way for physical interpretations of tachyons, one might as well use it.
Mathemathics is all fine and dandy, but in physics you need to have some kind of physical equivalent to what you're describing in the equations, otherwise it isn't physics. Imaginary mass is just as non-scalar quantity as negative mass, due to the non-scalable nature of imaginary numbers themselves. It is a very probably possibility that I'm wrong about it, but by Occam's razor I do find actual imaginary mass a lot less probable than interpreting it as negative mass.
Unless the formulae you are using can be factored to m2 where ever mass is mentioned, there is a whooping big difference between -m and m*i.
Yes. I know that. But the fact of the matter is that the imaginary mass still needs some kind of physical interpretation aside from mathematical means to take square roots of negative numbers.
Also, if you go by the numbers and calculate the time dilatation for objects traveling at velocities greater than c, using imaginary numbers to get some result from the equations, you would probably notice that the time for tachyons is flowing backwards... or the other interpretation is that their perception of direction of time is interchangeable; the end is the beginning of tachyon's journey from absorption to emission. This backwardness would also essentially invert the perceived vectors such as velocity and, indeed, momentum. And I can't really postulate any other interpretations for inverted momentum than negative mass.
Negative mass would essentially curve space-time in the opposite fashion compared to normal mass. I can wrap my mind around that.
You're assuming that tachyons have simialr qualities as bradyons as far as their interactions with space-time continuum are concerned. They might or they might not. And if the earlier wild hypothesis of tachyons being responsible of gravitational interactions would in fact be accurate, then they would actually be responsible for making up the curvature of space-time... or effects of it. After all, in General Relativity Einstein just gives us an energy tensor and says that mass and energy affect the local space-time continuum; explanations for it are less than vague. It works but we don't really know how or why mass/invariant energy curves space-time...
I have doubts that such particles actually exist, but I can handle that conceptually. You start talking about imaginary mass and you suddenly start implying that mass has a phase component. It is no longer a scalar quantity. That is not compatible with the Standard Model.
Then it isn't. :p It should be quite obvious by now that I can't actually provide any experimental data to support my hypotheses, and this is mostly just fun and games for me without the exactness of experimental science.
Also, imaginary mass automatically kinda adds a phase component to the concept of mass due to the fact that phase is an inherent part of imaginary numbers... or even better it makes mass a vector quantity on imaginary plane. What that amounts to in physical reality is, in my physical intuition, negative mass. Effectively, if not mathematically. You could go on calling it exotic mass or whatever, but I would still guess that it would behave as if the mass was negative.
Besides, for all we know, mass could actually have a phase component, it's just that we only observe the mass that has zero phase angle as "normal" mass. Perhaps we would perceive mass with 180 degree phase angle as negative mass. And what's in between would be something... else. :nervous: However I find this literary interpretation of imaginary mass somewhat more complex than simply saying that the imaginary mass in the equations corresponds to negative mass of particles. Because accurate mathemathics doesn't mean that all the results are physically sensible. It's obvious to use that triangles don't have sides with negative length.
-
However I find this literary interpretation of imaginary mass somewhat more complex than simply saying that the imaginary mass in the equations corresponds to negative mass of particles.
That makes even less sense than just talking about complex mass though. :p The equations simply aren't physically meaningful with complex quantities, so there isn't any point in trying to assign an interpretation to them.
Don't worry. I have never observed a singularity that is defined in the mathematical sense.
A singularity can be just about anything with locally non-smooth or non-analytic behavior. There are many situations where some basic physical quantity remains bounded but its derivative blows up, for example. Phase transition in materials or turbulence in fluids are types of singularities in some sense.
-
I had an idea just now, and although I have nothing to back it up, it sounds sort of interesting so I thought I'd bounce it off you guys.
I was thinking, since gravity is effectively equivalent to a curvature of space, but there has yet to be a generalization of this law to the other forces...
What happens if you look at charge as the imaginary part of a complex mass value? I haven't done the math, but it seems like it might explain how gravity always attracts, and electrons and protons attract or repel depending on what they're interacting with. Furthermore, both gravity and electric interactions have a falloff rate of 1/r2.
I have no idea how this and the tachyons thing would fit together though.
What do you think?
Edit: I was trying to do some maths on this, assuming that the force (times d2 of course) should be proportional to the product of the two "complex masses." However, this gave complex results in many situations. So I don't really know what the correct solution would be.
-
what if there exists a complex axis to the universe? what if that is why gravity is so weak, cause it's bleeding over into the complex end of the universe. of the complex interactions we only see the real component. the three spacial dimensions are not scalar but in fact planar. this is frik'n nuts **** to be thinking about.
-
what if there exists a complex axis to the universe? what if that is why gravity is so weak, cause it's bleeding over into the complex end of the universe. of the complex interactions we only see the real component. the three spacial dimensions are not scalar but in fact planar. this is frik'n nuts **** to be thinking about.
It is? Good thing to know. Gives me another reason not to waste my time thinking about it.
/me throws black holes into the equation, just to piss people off.
-
A singularity can be just about anything with locally non-smooth or non-analytic behavior. There are many situations where some basic physical quantity remains bounded but its derivative blows up, for example. Phase transition in materials or turbulence in fluids are types of singularities in some sense.
Then the derivative is incorrect and is neglected. As long as you can have two discrete measurement points, you can always approximate between them, and construct the real derivative from the measurement data. Those listed examples I consider as discrete valued functions which result from the fact your measurement setup is not quick / accurate enough.
Locally non-smooth functions have never been that much of a problem. Approximate the steps with spline and voila, you have continuous functions.
Yeah, I know these tricks are not mathematically valid, but unfortunately work all too well in reality.
Mika
-
Then the derivative is incorrect and is neglected. As long as you can have two discrete measurement points, you can always approximate between them, and construct the real derivative from the measurement data. Those listed examples I consider as discrete valued functions which result from the fact your measurement setup is not quick / accurate enough.
I am talking about the underlying continuous-domain processes (or one of their derivatives) having jumps, not just a sequence of measurements of them. A derivative doesn't even make sense for discrete functions.
Locally non-smooth functions have never been that much of a problem. Approximate the steps with spline and voila, you have continuous functions.
That won't ever make them analytic, so you still have a singularity in one sense of the term. :p
-
I am talking about the underlying continuous-domain processes (or one of their derivatives) having jumps, not just a sequence of measurements of them. A derivative doesn't even make sense for discrete functions.
And yet somehow I have still managed to optimize a discrete function with a gradient based method... and back then the computed gradient made perfect sense. And if I'm walking down the stairs, I tend to think that the direction of my trajectory is perfectly predictable, even though there are discrete steps.
Regarding the underlying processes, it could be argued that physically you can't even know if the processes are really discrete or continuous. Why? Because you can't measure them continously (shortest physically achievable sampling time at the moment is of order 10^-18 seconds), and theoretically shortest possible time interval is Planck time!
That won't ever make them analytic, so you still have a singularity in one sense of the term.
Analytic? I really had to check what that meant from Wiki. Seemed like stuff I normally ignore, so no problem with that in the functions either. Hasn't yet caused any kind of trouble. Sounds like those Mathematical proofs that it is possible to reconstruct a drum from the sound it is making. While Mathematically proven, most of the Physicists are probably going to say "not gonna happen" and the extremists would be like "cannot be done".
With the singularity I mean something like 1/r where r approaches zero. There function value approaches infinity as does it derivative. If that is the case, then even I believe that practical Math is useless there, so one has to perform experiments to find the value.
This actually reminded me of one electronics course I took four years ago. Mathematically speaking, the amplification response of the circuit had amplification approaching infinity at some point, but still we could only get a bounded number from that place with measurements that was well within the measurement range of the instrument - no matter how hard we tried.
Mathematically speaking, it was a true singularity, but at that point reality came back and bit mathematics in the arse. As it usually tends to do in these cases. Of course, sound quality was horrible in that region, but that was not the point. Hence my earlier comment of never encountering a singularity.
Mika
-
And yet somehow I have still managed to optimize a discrete function with a gradient based method... and back then the computed gradient made perfect sense. And if I'm walking down the stairs, I tend to think that the direction of my trajectory is perfectly predictable, even though there are discrete steps.
Sure, but you haven't dealt with derivatives anywhere in doing that.
Regarding the underlying processes, it could be argued that physically you can't even know if the processes are really discrete or continuous. Why? Because you can't measure them continously (shortest physically achievable sampling time at the moment is of order 10^-18 seconds), and theoretically shortest possible time interval is Planck time!
True, but that only applies to functions of time.
With the singularity I mean something like 1/r where r approaches zero. There function value approaches infinity as does it derivative. If that is the case, then even I believe that practical Math is useless there, so one has to perform experiments to find the value.
That is one type of singularity, but you said "defined in the mathematical sense," which includes a lot more than just that. :p And of course, any non-smooth function can be reduced to this case if you just look at derivatives, which are often physically interesting quantities in themselves.
Density type functions can have even worse local behavior, and in fact distributions were originally created to deal with that situation.
This actually reminded me of one electronics course I took four years ago. Mathematically speaking, the amplification response of the circuit had amplification approaching infinity at some point, but still we could only get a bounded number from that place with measurements that was well within the measurement range of the instrument - no matter how hard we tried.
Mathematically speaking, it was a true singularity, but at that point reality came back and bit mathematics in the arse. As it usually tends to do in these cases. Of course, sound quality was horrible in that region, but that was not the point. Hence my earlier comment of never encountering a singularity.
I think you're confusing math with physical theory. If the result does not correspond to something in reality, that is because the model is incomplete, not because there is anything wrong with the math.
-
Regarding the underlying processes, it could be argued that physically you can't even know if the processes are really discrete or continuous. Why? Because you can't measure them continously (shortest physically achievable sampling time at the moment is of order 10^-18 seconds), and theoretically shortest possible time interval is Planck time!
True, but that only applies to functions of time.
Out of curiosity, is there anything that disproves discrete space?
-
Out of curiosity, is there anything that disproves discrete space?
No. That is why theories of quantum gravitation like "Loop Quantum Gravity" and "Causal Dynamical Triangulation" are so interesting. The latter is especially interesting to me because its approach makes minimal assumptions beyond the Standard Model, and yet it is able to predict space-time "evolving" into pretty much the way we have observed it thus far.
-
Sure, but you haven't dealt with derivatives anywhere in doing that.
Actually, it is a derivative, further proven by the fact it actually can decrease the error function. It is mostly a question how big deviations one is ready to take from the lim x->a definition of derivates.
And it is not only time of which it cannot be said if there is really a continuous process or discrete steps. The same applies to the position by Planck's length. Every law of Physics can only be measured in discrete fashion. Basically every measurement set is discrete.
The problems arise when both the function values and the derivatives blow out to infinity. Then there is pretty much nothing that can be said by theoretical work alone.
I think you're confusing math with physical theory. If the result does not correspond to something in reality, that is because the model is incomplete, not because there is anything wrong with the math.
Yes, I know. Unfortunately, you cannot make the model perfect. Or maybe it is just Physicists way of thinking that you really cannot trust everything that you write on the paper since there are always some factors you couldn't know.
Mika
-
Out of curiosity, is there anything that disproves discrete space?
I don't think it is fully understood whether this is true or not, but I'm not really a physicist. :p
Actually, it is a derivative, further proven by the fact it actually can decrease the error function. It is mostly a question how big deviations one is ready to take from the lim x->a definition of derivates.
There isn't anything special going on there. Chances are that the discrete function you're optimizing is an approximation of a continuous-domain function and so the finite differences are approximations to derivatives.
It is probably possible to prove completely discrete-domain versions of most of these methods, if you put bounds on the modulus of continuity of the function (which basically does what you're saying) without requiring actual continuity.
And it is not only time of which it cannot be said if there is really a continuous process or discrete steps. The same applies to the position by Planck's length. Every law of Physics can only be measured in discrete fashion. Basically every measurement set is discrete.
What about probability distributions? As far as I know, those are continuous in many contexts according to current understanding. It all depends on what you consider to be physically significant.
-
My comment about Planck's length was related to the question of discrete space. According to current understanding, it is impossible to measure distances shorter than Planck's length. This means you cannot really know if it the space is continuous or discrete. But as a side note, even if you knew the answer to this question, it would really not change your life in any way. [Comments like these get Physicists most likely ridiculed in the future. But I'm willing to take the risk.]
There isn't anything special going on there. Chances are that the discrete function you're optimizing is an approximation of a continuous-domain function and so the finite differences are approximations to derivatives.
It is probably possible to prove completely discrete-domain versions of most of these methods, if you put bounds on the modulus of continuity of the function (which basically does what you're saying) without requiring actual continuity.
Yes. This is most likely case in the large amount of physical situations that I have stumbled across - actually that was one of the first lessons in Physical Mathematics : "In Physics almost all functions are smooth enough" [And you can quote that to ridicule Physicists. Unfortunately, it appears to be true, so far in my cases at least.] It is here where physical insight comes to play. Is the phenomenom under research reasonably well sampled with respect to what I want to measure about it? This is the question what Physicists must answer. [And carry to grave if they made a mistake about it.]
For this reason I said that for example, phase transition is simply undersampled linear phenomenom. If you sample it accurately enough, there is no sudden phase transition.
What about probability distributions? As far as I know, those are continuous in many contexts according to current understanding. It all depends on what you consider to be physically significant.
Probability distributions with respect to what?
Mika
-
My comment about Planck's length was related to the question of discrete space. According to current understanding, it is impossible to measure distances shorter than Planck's length. This means you cannot really know if it the space is continuous or discrete.
I checked the wikipedia article on it and it looks like different models interpret this in different ways, especially its connection to gravity (for which the currently accepted models are fully continuous).
Yes. This is most likely case in the large amount of physical situations that I have stumbled across - actually that was one of the first lessons in Physical Mathematics : "In Physics almost all functions are smooth enough" [And you can quote that to ridicule Physicists. Unfortunately, it appears to be true, so far in my cases at least.
Well, how much is almost all and how much smoothness is enough? :D
As I said earlier, in some contexts a failure of analyticity is considered a singularity, and analyticity often fails somewhere with many physical situations (even say like, walking around a room).
For this reason I said that for example, phase transition is simply undersampled linear phenomenom. If you sample it accurately enough, there is no sudden phase transition.
I'm not sure this is correct, but in any case it's certainly not an analytic phenomenon and can be viewed as bad behavior in that sense.
Probability distributions with respect to what?
I was thinking along the lines of uncertainty principles, for which the discrete cases look a little different, but I guess that ties into the issue of discrete spaces.
-
Well, how much is almost all and how much smoothness is enough?
This is the question Physicist has to answer. As a result, he either becomes a hero or a laughingstock.
As I said earlier, in some contexts a failure of analyticity is considered a singularity, and analyticity often fails somewhere with many physical situations (even say like, walking around a room).
But that has never stopped Physicists from doing it anyway - or they never thought about it and then it never bothered them. Sometimes I think it is good not to know too much about Maths. Take a look at the history of the scalar diffraction theory by Kirchoff for example. In short, Kirchoff formulated the theory with a quite grave mathematical error, but the theory gave correct results anyhow. The modification by Sommerfeld (and Rayleigh?) and some other people put it on the sound mathematical basis, but generality was restricted.
Mika
-
But that has never stopped Physicists from doing it anyway - or they never thought about it and then it never bothered them.
What is "it?"
I'm just disputing your original statement that everything has perfect regularity and singularities never show up in real life. :p Whether that bad behavior actually poses a problem or not depends on the context. In many cases, the mathematical framework is set up to allow the types of irregularities you encounter and you don't need to do anything special about them.
Sometimes I think it is good not to know too much about Maths.
I think this Hilbert quote is oddly appropriate here: "Physics is too hard for physicists." :D