It's basically the most complex purely mathematical object ever. And certainly the most complex mathematical object to ever be described by humans. It also is showing up in certain theories about the structure of the universe, like String Theory.
I share your ignorance, too, and like Bob, I'm a mathematician. And being a discrete one, I could be expected to know my algebra. It happens all the time... I can read papers from specialized journals in several sciences, yet I get lost as soon as I walk a few steps into mathland beyond my usual haunts. I take confort from these words: "Most mathematics papers are incomprehensible to most mathematicians". Said by Tim Gowers, nothing less than a Fields medalist. So, philosophers of the world, don't despair. Or do despair, but know that you're not alone :-)
I wouldn't mind - everybody knows topologists are insane. But this is part of Group Theory and I have this nagging suspection[TM] that it has to do with classification, and so I ought to know about it.
Computer Scientists and Mathematicians will often reinvent stuff that was discussed at length in philosophy in some earlier period, and make it opaque (unlike philosophy!). I'm going to do a post on the Gene Ontology project sometime soon...
LOL!
Sorry, I just had to laugh aloud. Of course, you know that things look different from the other side (I may be extra-biased, since I suffered too much continental stuff before discovering the pristine analytical tradition).
Anyway, rest assured that whatever else it may be, the E8 work is not an opaque Manhattan-sized codification of any (previously) known philosophical system. ;-)
Ok, I'll bite and try. I will have to simplify beyond all real accuracy (you did ask). This will still take many words.
One big reason group theory is useful in physics is that current theories find that group operations are related to symmetries. That basically is what Lie groups are - groups whose group operations describe symmetries. Symmetries in physical theories (based on a wondrous theorem by Emmy Noether) imply conservation laws - and vice versa. Conservation laws are loved by physicists because they are small rules with huge testable consequences.
Several examples -
No matter when you do an experiment the laws of nature don't seem to change - that implies that energy is conserved. No matter where you do the experiment the laws seem the same - which implies conservation of momentum. No matter how you rotate the experiment the laws of nature seem the same - this implies conservation of angular momentum. These are the three great "continuous" symmetries in physics.
Discrete symmetries work somewhat the same. There is a symmetry in electric charge, like a group operation in a specific Lie group (U1 - I vaguely remember). This is the heart of Quantum Electrodynamics (QED), our theory of Electromagnetism that underlies all of Chemistry and biology. All electrons (or other charged particles of a given type) are the same - you can swap any one for another and no-one, even theoretically, could tell the difference. You could swap charges, and again, no-one could tell. All that is added in QED are rules for describing how these charged particles move, and (based on Lie group operations) how you can transform them into each other. The rules are complicated and all sorts of weird stuff falls out because of them, (e.g. Uncertainty relations, superposition rules, probabilistic natural laws...) but the structure is actually quite simple and a few Feynman diagrams show all the simple transformations.
There are another set of particles, quarks, with three "color" charges (just like positive and negative are really meaningless the colors of quarks are the same - there are three attributes that kind of follow color combining rules so that is the name they were given). If you could switch the colors of the quarks nothing would change. There are 3 colors in quarks, as well as 6 different kinds of quarks, so it was found that one uses SU(3), another Lie group to provide the rules for how quarks can change into each other and into other particles. The rules are much more complicated than in the QED case (vastly more, incredibly more), but the structure is the same - you use the Lie group operation rules to tell you what transformations are allowed, and you calculate probabilities based on sums of all the possible transformations based on some standard "Quantum Field Theory" rules.
Physicists found that with the aid of some different "bosons", force carrying particles, (instead of photons for QED) one could calculate even more transformations of particles - these were called "weak" interactions and they seemed to follow the SU(2) Lie group operations for what particles transformed into what.
In the 1970's people realized that the U1 and Su(2) groups could be combined into a Su(2)xU(1) Lie group that would combine all the transformations of Weak interactions and QED interactions. They "unified" the weak and the Electromagnetic (QED) theories.
This whetted the appetite of many people - perhaps one can find a Lie group that has all the transformations of Electroweak (Su(2)xU(1) group) and the Strong Interaction (SU(3)) group. "Grand Unified Theories" were proposed that used a bigger group S(5). They looked good but predicted that a proton (really one of the quarks inside it) could transform into an electron style particle - that is a proton could decay. Well we have never seen proton decay, so those theories were falsified and have been put aside or AH HAH! worked on by looking for other symmetry groups even bigger than S(5) that might have better rules. E8 is such a bigger group.
Another motivation to look at other groups is that our methods for calculating how things move and transform assume low energies - if the energy of interaction is above a given level or the time considered is small enough our current Quantum Field theories cannot calculate any meaningful results. At a high enough energy even gravity matters which all current testable theories that calculate real world results ignore. Perhaps more encompassing transformation rules could allow us to calculate with these constraints - perhaps incuding gravity, or at least combining the transformation rules of the Su(3) and SU(2)xU(1) groups that our current standard model use.
I hope this give a very vague idea why Lie groups (symmetry groups) matter at all in Physics.
As an example (again simplified so don't jump all over nits) - the Lie group symmetry rules in QED allow an electron and a positron to come together and a photon to come out. One can flip the diagram 180 degrees and have a photon come in and an electron and a positron come out. They don't allow us to flip the diagram 90 degrees to have an electron and photon come in and a positron come out. Thus charge is conserved - say electrons have negative charge and positrons positive charge (or vice versa).
In mathematics, the circle group, denoted by T (or in blackboard bold by \mathbb T),
is the multiplicative group of all complex numbers with absolute value 1,
i.e. the unit circle in the complex plane.
T = \{ z \in C : |z| = 1 \}.
The circle group forms a subgroup of C*, the multiplicative group of all nonzero complex numbers. Since C* is abelian, it follows that T is as well.
Note: the set of all *nonzero* complex numbers. Zero messes up the inverse function of multiplication - it is no longer continuous let alone smooth.
and make it opaque (unlike philosophy!).
I'd like to see a mathematicization of Heidegger's stuff about "Nothing" that was less comprehensible than the original. (I once tried to read a short excerpt, and had to take two Advils and retire to bed).
No.
Mark does a decent job, but it does involve mathematical terms.
Nope, still don't get it. I'd rather talk shop with a molecular biologist...
Well, you get turned on by philosophy, so whatever rocks your boat.
It's basically the most complex purely mathematical object ever. And certainly the most complex mathematical object to ever be described by humans. It also is showing up in certain theories about the structure of the universe, like String Theory.
Stuff like that.
I think.
Don't believe any of them Mr Wilkins they're all just lying to you!
http://www.republicans.org/
John, I'm with you. And I work in a maths and stats department.
I was overjoyed a few months ago to discover that there is a Journal of Lie Theory. I'm guessing it's something to do with fuzzy logic.
Bob
Sorry to burst the jokes, but "Lie", as in "Lie Group", is pronounced like "Li" (or "Lee" with a shorter sound), not like the verb for lying.
"Sorry to burst the jokes, but "Lie", as in "Lie Group", is pronounced like "Li" (or "Lee" with a shorter sound), not like the verb for lying."
I actually knew that but I just couldn't resist a really, really bad joke. It comes of being English, anything for a terrible pun! Sorry.
Lie group? You mean, like the Discovery Institute?
I share your ignorance, too, and like Bob, I'm a mathematician. And being a discrete one, I could be expected to know my algebra. It happens all the time... I can read papers from specialized journals in several sciences, yet I get lost as soon as I walk a few steps into mathland beyond my usual haunts. I take confort from these words: "Most mathematics papers are incomprehensible to most mathematicians". Said by Tim Gowers, nothing less than a Fields medalist. So, philosophers of the world, don't despair. Or do despair, but know that you're not alone :-)
I wouldn't mind - everybody knows topologists are insane. But this is part of Group Theory and I have this nagging suspection[TM] that it has to do with classification, and so I ought to know about it.
Computer Scientists and Mathematicians will often reinvent stuff that was discussed at length in philosophy in some earlier period, and make it opaque (unlike philosophy!). I'm going to do a post on the Gene Ontology project sometime soon...
LOL!
Sorry, I just had to laugh aloud. Of course, you know that things look different from the other side (I may be extra-biased, since I suffered too much continental stuff before discovering the pristine analytical tradition).
Anyway, rest assured that whatever else it may be, the E8 work is not an opaque Manhattan-sized codification of any (previously) known philosophical system. ;-)
Ok, I'll bite and try. I will have to simplify beyond all real accuracy (you did ask). This will still take many words.
One big reason group theory is useful in physics is that current theories find that group operations are related to symmetries. That basically is what Lie groups are - groups whose group operations describe symmetries. Symmetries in physical theories (based on a wondrous theorem by Emmy Noether) imply conservation laws - and vice versa. Conservation laws are loved by physicists because they are small rules with huge testable consequences.
Several examples -
No matter when you do an experiment the laws of nature don't seem to change - that implies that energy is conserved. No matter where you do the experiment the laws seem the same - which implies conservation of momentum. No matter how you rotate the experiment the laws of nature seem the same - this implies conservation of angular momentum. These are the three great "continuous" symmetries in physics.
Discrete symmetries work somewhat the same. There is a symmetry in electric charge, like a group operation in a specific Lie group (U1 - I vaguely remember). This is the heart of Quantum Electrodynamics (QED), our theory of Electromagnetism that underlies all of Chemistry and biology. All electrons (or other charged particles of a given type) are the same - you can swap any one for another and no-one, even theoretically, could tell the difference. You could swap charges, and again, no-one could tell. All that is added in QED are rules for describing how these charged particles move, and (based on Lie group operations) how you can transform them into each other. The rules are complicated and all sorts of weird stuff falls out because of them, (e.g. Uncertainty relations, superposition rules, probabilistic natural laws...) but the structure is actually quite simple and a few Feynman diagrams show all the simple transformations.
There are another set of particles, quarks, with three "color" charges (just like positive and negative are really meaningless the colors of quarks are the same - there are three attributes that kind of follow color combining rules so that is the name they were given). If you could switch the colors of the quarks nothing would change. There are 3 colors in quarks, as well as 6 different kinds of quarks, so it was found that one uses SU(3), another Lie group to provide the rules for how quarks can change into each other and into other particles. The rules are much more complicated than in the QED case (vastly more, incredibly more), but the structure is the same - you use the Lie group operation rules to tell you what transformations are allowed, and you calculate probabilities based on sums of all the possible transformations based on some standard "Quantum Field Theory" rules.
Physicists found that with the aid of some different "bosons", force carrying particles, (instead of photons for QED) one could calculate even more transformations of particles - these were called "weak" interactions and they seemed to follow the SU(2) Lie group operations for what particles transformed into what.
In the 1970's people realized that the U1 and Su(2) groups could be combined into a Su(2)xU(1) Lie group that would combine all the transformations of Weak interactions and QED interactions. They "unified" the weak and the Electromagnetic (QED) theories.
This whetted the appetite of many people - perhaps one can find a Lie group that has all the transformations of Electroweak (Su(2)xU(1) group) and the Strong Interaction (SU(3)) group. "Grand Unified Theories" were proposed that used a bigger group S(5). They looked good but predicted that a proton (really one of the quarks inside it) could transform into an electron style particle - that is a proton could decay. Well we have never seen proton decay, so those theories were falsified and have been put aside or AH HAH! worked on by looking for other symmetry groups even bigger than S(5) that might have better rules. E8 is such a bigger group.
Another motivation to look at other groups is that our methods for calculating how things move and transform assume low energies - if the energy of interaction is above a given level or the time considered is small enough our current Quantum Field theories cannot calculate any meaningful results. At a high enough energy even gravity matters which all current testable theories that calculate real world results ignore. Perhaps more encompassing transformation rules could allow us to calculate with these constraints - perhaps incuding gravity, or at least combining the transformation rules of the Su(3) and SU(2)xU(1) groups that our current standard model use.
I hope this give a very vague idea why Lie groups (symmetry groups) matter at all in Physics.
As an example (again simplified so don't jump all over nits) - the Lie group symmetry rules in QED allow an electron and a positron to come together and a photon to come out. One can flip the diagram 180 degrees and have a photon come in and an electron and a positron come out. They don't allow us to flip the diagram 90 degrees to have an electron and photon come in and a positron come out. Thus charge is conserved - say electrons have negative charge and positrons positive charge (or vice versa).
But to begin at the beginning, or at least a bit farther back:
What is a group?
http://en.wikipedia.org/wiki/Group_%28mathematics%29
On to the next needed definition:
http://en.wikipedia.org/wiki/Lie_group
http://en.wikipedia.org/wiki/Smooth_map
example:
The Lie group S1 is sometimes called the circle group.
http://en.wikipedia.org/wiki/Circle_group
Circle group
From Wikipedia, the free encyclopedia
In mathematics, the circle group, denoted by T (or in blackboard bold by \mathbb T),
is the multiplicative group of all complex numbers with absolute value 1,
i.e. the unit circle in the complex plane.
T = \{ z \in C : |z| = 1 \}.
The circle group forms a subgroup of C*, the multiplicative group of all nonzero complex numbers. Since C* is abelian, it follows that T is as well.
Note: the set of all *nonzero* complex numbers. Zero messes up the inverse function of multiplication - it is no longer continuous let alone smooth.
and make it opaque (unlike philosophy!).
I'd like to see a mathematicization of Heidegger's stuff about "Nothing" that was less comprehensible than the original. (I once tried to read a short excerpt, and had to take two Advils and retire to bed).
Markk, that was excellent. I understood enough to see the point, and insufficient to make a fooll of myself by repeating any of it.
Pete, I read those articles, and they always seemed to rely on something in a different one.
Eamon, I read Sein und Zeit in full. An entire year of my life is a blank to me...
That's the way it is until you get all the way back to the undefined terms, which make it all perfectly clear :)