Lets suppose that neutrinos do travel faster than light.
This need not require any change to Relativity.
From the point of view of Relativity it would just mean the neutrinos need to be regarded as traveling backwards in time. Traveling backwards in time is not ruled by Relativity. Indeed Relativity requires that any conjectured FTL particles be deemed as traveling backwards in time.
There are two possible ways that FTL particles can do this without requiring any change to Relativity. However one way would violate intuitive causality rules. The other way wouldn't. The way which violates causality rules is if the neutrinos traveled back in time (by 60 nanoseconds) at the exact moment they were emitted. In this scenario, for the remainder of their flight, they would be traveling at the speed limit c. But if they had traveled back in time at the moment of emission this would result in violations of causality, be falsifiable, and/or violate the experimenter's sanity.
The other way is that FTL particles travel back in time, but over time. In this scenario, at the point of emission, the 'distance' traveled back in time is exactly zero, but over the duration of the flight, the 'distance' traveled back in time increases so that by the time it is observed it has traveled back in time by the observed 60 nanoseconds. This scenario doesn't violate causality rules.
Relativity doesn't rule out FTL particles. It is only non-time-travel rules that would rule such out, but there are no non-time-travel rules I've come across in theoretical physics. There are only informal conjectures at such.
Friday, November 25, 2011
Thursday, November 24, 2011
Mathematics, Statistics, Zero and Noise
The number zero plays a fundamental role in mathematics, and by extension, theoretical physics. But the language of physics, in particular quantum theory, is not just maths but statistics. A central player in statistics, I'd argue, is not zero as such, but noise, of the fundamental variety. There remains debate as to whether fundamental noise has any physical existence. It is something that can't be mathematically defined or proved (which is why it's existence is questioned). But we can nevertheless grasp the concept of such. Fundamental noise can be described (if not defined) as the selection of a number (or numbers) for which there is no reason for selecting that number (or numbers). From a statistical point of view the infinite set of all such selections will sum to zero, because for every selection that is positive, it can be coupled with a selection of equal magnitude but negative. The couples cancel each other out.
Statistics can be regarded as a way of reconciling observables with mathematics.
Quantum theory involves both a mathematical function (the wave function) and a random function. In computer simulations the random function has to be simulated (using a pseudo-random function) because random functions are mathematically impossible.
Why a random function?
It's required to account for the observables - the particle detections. The wavefunction describes the mathematical structure of the observables but can't describe the relationship between individual detections and that structure (the so called wave function collapse problem). This is because the relationship is (propositionally) random in one direction and undefined in the other. This doesn't mean there is no relationship, just that there is no mathematical relationship. While any single detection is random, the statistics of that randomness have a structure. In much they same way that one can say the statistics of a certain type of noise add up to zero one can say the statistics of particle detections add up to a wave function.
An easy way to understand how detections end up describing the wave function is to imagine each detection as a random sampling of the wave function. This doesn't help us to get any closer to a mathematical model of the relationship but it does help in computing simulations of such.
Another way to think of it, from a mathematical point of view, is to treat observables as literally errors for which there is no cause, ie. that the observable universe (whether we are here to see it or not) is just an error.
I prefer to think of the observable universe as just a different version of the mathematical one (and vice versa), that neither is a function of the other. The former is based on noise (presence) while the latter is based on zero (absence, the void). Each can be regarded as equally fundamental.
Statistics can be regarded as a way of reconciling observables with mathematics.
Quantum theory involves both a mathematical function (the wave function) and a random function. In computer simulations the random function has to be simulated (using a pseudo-random function) because random functions are mathematically impossible.
Why a random function?
It's required to account for the observables - the particle detections. The wavefunction describes the mathematical structure of the observables but can't describe the relationship between individual detections and that structure (the so called wave function collapse problem). This is because the relationship is (propositionally) random in one direction and undefined in the other. This doesn't mean there is no relationship, just that there is no mathematical relationship. While any single detection is random, the statistics of that randomness have a structure. In much they same way that one can say the statistics of a certain type of noise add up to zero one can say the statistics of particle detections add up to a wave function.
An easy way to understand how detections end up describing the wave function is to imagine each detection as a random sampling of the wave function. This doesn't help us to get any closer to a mathematical model of the relationship but it does help in computing simulations of such.
Another way to think of it, from a mathematical point of view, is to treat observables as literally errors for which there is no cause, ie. that the observable universe (whether we are here to see it or not) is just an error.
I prefer to think of the observable universe as just a different version of the mathematical one (and vice versa), that neither is a function of the other. The former is based on noise (presence) while the latter is based on zero (absence, the void). Each can be regarded as equally fundamental.
Saturday, November 19, 2011
Wave Function
The wave function of quantum mechanics can be regarded as a function of the experimental setup. The function represents the mathematical relationship between a setup described in terms of classical spacetime descriptors and what amounts to a Fourier transform of those descriptors, which we could call a "wave". Or a wave description of the setup.
If the following were the wave function
y = F(x)
then y would be the wave, and x the setup. The function is (with some caveats) invertible:
x = invF(y)
Now the wave function is defined mathematically, meaning that it does not depend on anything else to be correct. It is a definition. And it is global.
Now a mathematical function is very different from something that samples a mathematical function. For example a computer function can sample a mathematical function. Somewhat unfortunately the way you write a sampling function for a computer is often the same as the way you might write the mathematical function that was being sampled:
y = F(x)
The difference is that a single invocation of the sampling function results in just one of the infinite number of pairs identified by the mathematical function. The sampling function, in a sense, represents the relationship between one x and one y, whereas the mathematical function represents all of the pairs. Only if the sampling function was invoked for all possible x's, would the result become equivalent to a representation of the mathematical function. On the other hand the computer function's definition (it's code) could be regarded as equivalent to a representation of the mathematical function.
Now a sampling function is really interesting in relation to the problem of "wave function collapse". The function y = F(x) represents all x's and y's for which y = F(x) is true whereas a sampling version of the same function need only represent a single case (a state), as in the following computer program:
var x = 9
var y = F(x)
While the relationship between x and y is computed according to F(x) the resulting value of y could have been computed by any number of other functions. For example, if y turns out to be 3, then F could have been:
y = sqrt(x)
y = x/3
y =x - 6
It is only when one computes for all possible values of x, that the actual function used (F) becomes distinguishable from any other functions that might have also computed y. Of course it can also be indentified (distinguishable) by reading the code.
The point here is that while the rule can be regarded as determining the value of y (ie. mathematically - equivalent to reading the code), the relationship between any single value of y and it's corresponding x is not given by that rule (since any number of other rules could also express that relationship). Rules can be expressed mathematically but their indentity in the context of a sampling is statistical.
Here is a really rough sampling function that simulates/represents a particle detection in terms of a sampling function (one that samples the wave function).
x = Math.random() * detectorWidth;
y = Math.random() * detectorHeight;
r = Math.random();
a = WaveFunction(x,y);
b = WaveFunction(x,y);
if(r > (a * b))
displayParticleAt(x,y)
Apart from the QM formalism defining the probability of a detection as the square of the wave function the components a and b can be regarded as the two components of a joint probability, where one represents the particle while the other represents the detector. The joint probability represents the detector and particle being in the same place at the same time (ie. in the same frame of reference or the same classical universe).
...
If the following were the wave function
y = F(x)
then y would be the wave, and x the setup. The function is (with some caveats) invertible:
x = invF(y)
Now the wave function is defined mathematically, meaning that it does not depend on anything else to be correct. It is a definition. And it is global.
Now a mathematical function is very different from something that samples a mathematical function. For example a computer function can sample a mathematical function. Somewhat unfortunately the way you write a sampling function for a computer is often the same as the way you might write the mathematical function that was being sampled:
y = F(x)
The difference is that a single invocation of the sampling function results in just one of the infinite number of pairs identified by the mathematical function. The sampling function, in a sense, represents the relationship between one x and one y, whereas the mathematical function represents all of the pairs. Only if the sampling function was invoked for all possible x's, would the result become equivalent to a representation of the mathematical function. On the other hand the computer function's definition (it's code) could be regarded as equivalent to a representation of the mathematical function.
Now a sampling function is really interesting in relation to the problem of "wave function collapse". The function y = F(x) represents all x's and y's for which y = F(x) is true whereas a sampling version of the same function need only represent a single case (a state), as in the following computer program:
var x = 9
var y = F(x)
While the relationship between x and y is computed according to F(x) the resulting value of y could have been computed by any number of other functions. For example, if y turns out to be 3, then F could have been:
y = sqrt(x)
y = x/3
y =x - 6
It is only when one computes for all possible values of x, that the actual function used (F) becomes distinguishable from any other functions that might have also computed y. Of course it can also be indentified (distinguishable) by reading the code.
The point here is that while the rule can be regarded as determining the value of y (ie. mathematically - equivalent to reading the code), the relationship between any single value of y and it's corresponding x is not given by that rule (since any number of other rules could also express that relationship). Rules can be expressed mathematically but their indentity in the context of a sampling is statistical.
Here is a really rough sampling function that simulates/represents a particle detection in terms of a sampling function (one that samples the wave function).
x = Math.random() * detectorWidth;
y = Math.random() * detectorHeight;
r = Math.random();
a = WaveFunction(x,y);
b = WaveFunction(x,y);
if(r > (a * b))
displayParticleAt(x,y)
Apart from the QM formalism defining the probability of a detection as the square of the wave function the components a and b can be regarded as the two components of a joint probability, where one represents the particle while the other represents the detector. The joint probability represents the detector and particle being in the same place at the same time (ie. in the same frame of reference or the same classical universe).
...
Faster Than A Speeding Photon
The headlines remain the same: "Faster Than Light Neutrinos".
The problem is not (or need not be) the experimental set up, or the data. A significant problem is that there is no theoretical basis for the headline. One can certainly say, in a relatively neutral way, that the neutrinos arrived 60 nanoseconds earlier than the photons. There is nothing wrong with translating the data in that way because, for example, the neutrinos might have departed 60 nanoseconds earlier than the photons. But as soon as you start saying the neutrinos are "traveling faster than light" (or even appearing to do so) you are automatically proposing that each and every other way of describing (or explaining) the data has been eliminated.
It's very much like a magician's trick where the magician performs a trick (faster than light particles) and challenges you to work out how they did it (the explanation), except that the magician, in this case, doesn't know how they did it (how to explain it). So instead they describe/explain it from the point of view of the audience - as a woman was being cut in half, or particles traveling faster than light.
Each and every other way of explaining the data must be eliminated before you can describe it in terms of faster than light particles - but even then, an FTL description (that the particles are traveling faster than light) will still be no more than a roundabout way of saying that one does not know how to describe the data.
The experiment is nevertheless interesting because the experimental equivalent of "exhausting alternative explanations" is what experimenters would call "calibration". And if they have correctly calibrated the set up, then they will have effectively eliminated all alternative explanations.
What would follow is then the much more daunting task - how to explain faster than light neutrinos. It's not just a case of saying - oh, Einstein is wrong and that's all there is to it. If particles can be observed to travel faster than light then there is an opportunity to "test" the theoretical frameworks in which particles do travel faster than light. Indeed, even Relativity Theory allow particles to travel faster than light (although neutrinos wouldn't be one of them).
Theoretical physics is about discovering the rules, not the exceptions. At the moment, FTL particles are the exception. Which means, from a theoretical physics point of view, there has not yet been any discovery.
The problem is not (or need not be) the experimental set up, or the data. A significant problem is that there is no theoretical basis for the headline. One can certainly say, in a relatively neutral way, that the neutrinos arrived 60 nanoseconds earlier than the photons. There is nothing wrong with translating the data in that way because, for example, the neutrinos might have departed 60 nanoseconds earlier than the photons. But as soon as you start saying the neutrinos are "traveling faster than light" (or even appearing to do so) you are automatically proposing that each and every other way of describing (or explaining) the data has been eliminated.
It's very much like a magician's trick where the magician performs a trick (faster than light particles) and challenges you to work out how they did it (the explanation), except that the magician, in this case, doesn't know how they did it (how to explain it). So instead they describe/explain it from the point of view of the audience - as a woman was being cut in half, or particles traveling faster than light.
Each and every other way of explaining the data must be eliminated before you can describe it in terms of faster than light particles - but even then, an FTL description (that the particles are traveling faster than light) will still be no more than a roundabout way of saying that one does not know how to describe the data.
The experiment is nevertheless interesting because the experimental equivalent of "exhausting alternative explanations" is what experimenters would call "calibration". And if they have correctly calibrated the set up, then they will have effectively eliminated all alternative explanations.
What would follow is then the much more daunting task - how to explain faster than light neutrinos. It's not just a case of saying - oh, Einstein is wrong and that's all there is to it. If particles can be observed to travel faster than light then there is an opportunity to "test" the theoretical frameworks in which particles do travel faster than light. Indeed, even Relativity Theory allow particles to travel faster than light (although neutrinos wouldn't be one of them).
Theoretical physics is about discovering the rules, not the exceptions. At the moment, FTL particles are the exception. Which means, from a theoretical physics point of view, there has not yet been any discovery.
Friday, November 11, 2011
Woody Allen
In a Woody Allen film a protagonist could look closely at the world in which they were living and actually see the grain of the film in which they are being represented.
In a Woody Allen film people don't need time machines to travel into the past. They just walk out of shot and into another that is 90 years earlier. Or drive out of shot as the case may be.
Woody Allen is a master of simplicity.
Charlie Chaplin is his only peer.
In a Woody Allen film people don't need time machines to travel into the past. They just walk out of shot and into another that is 90 years earlier. Or drive out of shot as the case may be.
Woody Allen is a master of simplicity.
Charlie Chaplin is his only peer.
Weird Reality
It is a weird aspect of nature that the constituent elements, of which the physical universe is composed, (the atoms) are not there. In their place is either a representation of what is (otherwise) there, or there is nothing else there, other than the representation.
In the latter case the word "representation" is somewhat misplaced. For what is a representation representing if there isn't anything it is re-presenting?
There is a school of ancient thought (Stoicism) that does not distinguish between a representation and what it otherwise represents. Stoics treat these things as one and the same thing. A "presentation" perhaps. The universe is a surface (superficial) beyond which there is no deeper reality. The surface itself is the "reality". This is not the same thing as the universe proposed by The Matrix.
In the Matrix there is a deeper reality run by aliens.
But back to what is far more weirder: a weird aspect of nature.
We think of things being where we see them. We have an entire branch of mathematics (geometry) for defining where things are and how they relate to each other. We have theoretical physics (ie. mathematics) to formalise the forces between things in space and time. And Einstein to clarify the relations. But the "fact" is that when you look very closely at these things they do not behave according to the assumptions on which geometry (land measure), space and time are conceived. A branch of physics (quantum mechanics) is devoted to this. It is no less mathematical.
In quantum mechanics a thing (typically a particle) has no classical reality until it is observed. The observation (a particle detection), whether by man, machine or other entity, is the only thing conforming to classical reality.
From the observation (as in ordinary classical observation) one gains a sense of where things are supposedly in reality. But in quantum mechanics this sense (of where things are) gives rise to the thing having been in more than one place before we saw it.
Mathematically there is no way around this.
So either mathematics is not the right tool for analysing this, or things really are in multiple places at the same time. The latter is the position of most theoretical physicists.
While we observe things to be in one place only, when we use mathematics (and statistics) to determine where they are (or were) in "reality" (before we saw them) we end up with some weird conflated universe or multiverse of things. Yet we never see this. We never experience it. This multiverse of things is just what we mathematically infer from where we otherwise see things.
Now for the Stoics there is no deeper reality beyond what we observe (unless we're deranged, deluded etc). One of the original architects of quantum mechanics (Neils Bohr) held a similar position. He questioned the reality (classical reality) of what the mathematics was saying. He didn't question the mathematics. The mathematics, as far he was concerned, was not the issue. By definition mathematics is correct. If you've made a mistake in your mathematics it's not mathematics.
Einstein, on the other hand, treated the mathematics, not as incorrect (of course), but as incomplete (with respect to the apparent problem). Einstein believed there was a deeper reality, beyond the observation, and that a completed mathematical conception would reveal it.
A Theory of Everything is supposed to be the completion of this particular quest.
But what if the Stoics are right - that the universe, as we observe it, with our senses, or our machines, is all that is really there. A superficial reality. What if the mathematical/statistical inferences of a weird multiverse (and so on) is just a formalisation for otherwise managing, organising, manipulating what we otherwise sense (nothing more than a technological purpose), rather than necessarily exposing some deeper reality (scientific/theoretical/philosophical conclusion).
Interestingly, either way it's weird.
We're either living in some sort of weird image of nothing, or we're living in some sort of image of a weird something.
In the latter case the word "representation" is somewhat misplaced. For what is a representation representing if there isn't anything it is re-presenting?
There is a school of ancient thought (Stoicism) that does not distinguish between a representation and what it otherwise represents. Stoics treat these things as one and the same thing. A "presentation" perhaps. The universe is a surface (superficial) beyond which there is no deeper reality. The surface itself is the "reality". This is not the same thing as the universe proposed by The Matrix.
In the Matrix there is a deeper reality run by aliens.
But back to what is far more weirder: a weird aspect of nature.
We think of things being where we see them. We have an entire branch of mathematics (geometry) for defining where things are and how they relate to each other. We have theoretical physics (ie. mathematics) to formalise the forces between things in space and time. And Einstein to clarify the relations. But the "fact" is that when you look very closely at these things they do not behave according to the assumptions on which geometry (land measure), space and time are conceived. A branch of physics (quantum mechanics) is devoted to this. It is no less mathematical.
In quantum mechanics a thing (typically a particle) has no classical reality until it is observed. The observation (a particle detection), whether by man, machine or other entity, is the only thing conforming to classical reality.
From the observation (as in ordinary classical observation) one gains a sense of where things are supposedly in reality. But in quantum mechanics this sense (of where things are) gives rise to the thing having been in more than one place before we saw it.
Mathematically there is no way around this.
So either mathematics is not the right tool for analysing this, or things really are in multiple places at the same time. The latter is the position of most theoretical physicists.
While we observe things to be in one place only, when we use mathematics (and statistics) to determine where they are (or were) in "reality" (before we saw them) we end up with some weird conflated universe or multiverse of things. Yet we never see this. We never experience it. This multiverse of things is just what we mathematically infer from where we otherwise see things.
Now for the Stoics there is no deeper reality beyond what we observe (unless we're deranged, deluded etc). One of the original architects of quantum mechanics (Neils Bohr) held a similar position. He questioned the reality (classical reality) of what the mathematics was saying. He didn't question the mathematics. The mathematics, as far he was concerned, was not the issue. By definition mathematics is correct. If you've made a mistake in your mathematics it's not mathematics.
Einstein, on the other hand, treated the mathematics, not as incorrect (of course), but as incomplete (with respect to the apparent problem). Einstein believed there was a deeper reality, beyond the observation, and that a completed mathematical conception would reveal it.
A Theory of Everything is supposed to be the completion of this particular quest.
But what if the Stoics are right - that the universe, as we observe it, with our senses, or our machines, is all that is really there. A superficial reality. What if the mathematical/statistical inferences of a weird multiverse (and so on) is just a formalisation for otherwise managing, organising, manipulating what we otherwise sense (nothing more than a technological purpose), rather than necessarily exposing some deeper reality (scientific/theoretical/philosophical conclusion).
Interestingly, either way it's weird.
We're either living in some sort of weird image of nothing, or we're living in some sort of image of a weird something.
Monday, July 25, 2011
Writing a Flash Player in Javascript HTML5?
As the HTML5 standard consolidates into being across the various browsers, the state of JavaScript, across the browsers, could do with some more work.
Of particular concern is support (or lack thereof) for high performance byte array processing, especially when one would very much like to write efficient parsers (in JavaScript) for various binary data formats.
For example, I've been entertaining the idea of parsing a Flash swf file, through JavaScript, but I'm not filled with much confidence when looking at JavaScript and the various implementations of such in each of the available browsers.
Others have also been looking at JavaScript for similar requirements. See here for some tests on processing arrays in JavaScript:
http://blog.n01se.net/?p=248
Contrary to Steve Jobs 2010 claim that Flash is not an open standard, the format of Flash files is very much open and has been for a couple of years. Anyone can write a Flash player (if they are capable of doing so). Writing a player in Javascript would be great. But is JavaScript up to the job?
Many seem to think that HTML5 (+ Javascript) is a Flash killer. But in Flash I could, in principle, write a half decent Flash player. I'd very much like to think that was also possible in HTML5 (+ JavaScript) .
Of particular concern is support (or lack thereof) for high performance byte array processing, especially when one would very much like to write efficient parsers (in JavaScript) for various binary data formats.
For example, I've been entertaining the idea of parsing a Flash swf file, through JavaScript, but I'm not filled with much confidence when looking at JavaScript and the various implementations of such in each of the available browsers.
Others have also been looking at JavaScript for similar requirements. See here for some tests on processing arrays in JavaScript:
http://blog.n01se.net/?p=248
Contrary to Steve Jobs 2010 claim that Flash is not an open standard, the format of Flash files is very much open and has been for a couple of years. Anyone can write a Flash player (if they are capable of doing so). Writing a player in Javascript would be great. But is JavaScript up to the job?
Many seem to think that HTML5 (+ Javascript) is a Flash killer. But in Flash I could, in principle, write a half decent Flash player. I'd very much like to think that was also possible in HTML5 (+ JavaScript) .
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