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towards tengen intermittent musings on physics philosophy linguistics life towards tengen intermittent musings on physics philosophy linguistics life skip to content home older posts muʿāwiyah ibn ʾabī sufyān posted on friday march 2 2012 1 comment i do not apply my sword where my whip is enough nor my whip where my tongue is enough i do not come between people and their tongues so long as they do not come between us and our rule on allowing people to mock him as the son of a jigar khor 1 comment posted in character history quotes trivia squirrel posted on wednesday june 18 2008 leave a comment leave a comment posted in uncategorized tagged painting aaaaaaaaaaaaaaaaaaaaaaaaarrrghhhhhhh posted on friday may 11 2007 1 comment computers will crash mistakes are unavoidable there lies great wisdom in regularly backing up data verily it is true 1 comment posted in random learning to paint posted on saturday may 5 2007 leave a comment well twiddling with gimp is pretty much fun here re some of my attempts at cleaning up scans and colouring them while i m nowhere near something like this the practice can only be good for me all this thanks to great artwork by aditi leave a comment posted in art random some books posted on saturday january 20 2007 leave a comment representation theory a first course fulton harris symmetry the monster mark ronan not yet read gödel escher bach an eternal golden braid douglas hofstadter a popular level book on gödel s theorem and ai and related stuff the first half is good but the book drags on too much with unfounded speculation towards the end surreal numbers donald e knuth aweird funny little book on set theoretic foundation of number theory one jump ahead jonathan schaeffer about chinook the checkers playing program sophie s world jostein gaarder a sort of informal fun introduction to western philosophy foucault s pendulum umberto eco the power of one courtenay bryce leave a comment posted in books literature random susy ii posted on friday july 14 2006 3 comments well i m done with the official presentation today so that leaves me a bit more free for some time considering that i ll be offline for a week or so once i go home i think i should probably try to spike my blogging density for some time now last time we d just set up the susy algebra for 0 1 d systems basically quantum mechanics that is actually enough for the proof of the atiyah singer index theorem the links at the end of the wikipedia are worth checking out and range from popular level to maths that is beyond me at present so for now let us take a slight detour and look at differential forms this may be redundant but let s just review differential forms horribly quickly in a way that might appear because it is pretty sloppy to a mathematician but it ll do for our purposes here so given a manifold m let s look at it s tangent space basically we can have vectors tensors in the tangent space now consider all the antisymmetric tensors in the tangent space now loosely speaking the p dimensional antisymmetric tensors is what we call p forms the p forms form an abelian group under ordinary addition now let us define an operator d as d a ijk d i a jkl where basically we get a p 1 form from a p form by differentiating it thus adding an extra index and then antisymmetrising so that the result is also a form antisymetric tensor this structure is what is called a cochain complex now we can also define a d operator which takes a p form to a p 1 form by contracting with an additional index ok that s enough of an interlude into this and it s getting to technical for a lazy guy like me to care to typeset in the horribleness that is html for mathematical typesetting that is so you might say all this is very interesting but why should i be interested in forms well there s tonnes of uses of this language in physics for one you can join the cool kids and say f d a d f 0 d f j instead of the boring old maxwell s equations but well it s just pretty neat stuff by itself but we ll see soon how all this can be useful in physics we ll consider a supersymmetric system which has as its hilbert space the space of all forms as i said on the top i ll be offline for a week at home so i will continue posting only after that if you re impatient then you can have a look at the report i wrote up about my summer work ciao 3 comments posted in physics susy posted on friday july 7 2006 3 comments it s been nearly three weeks since i promised to write about my activities here my apologies in case i actually have any readers i got swamped in last minute deadline beating and managed to sneak in my eport just at the last second as the clock was striking 4 i m writing this post thanks to some encouragement from tom so here goes day 1 symmetries have played a very important role as guiding principles in theoretical physics a given symmetry enables us to make some statement about the behaviour of a system namely the associated conserved charge thus one would naturally be interested in any possible new symmetry that physical models could have however there is a theorem due to coleman and mandula that forbids the possibility of any symmetries that transform as tensors under lorentz transformations other than p μ and m μν there is a very simple plausibility argument by witten as to why this should be so it basically says that the conservation of p μ and m μν leaves only the scattering angle unknown any new conservation law would leave us with only a discrete set of possible angles but since the scattering amplitude is an analytic function of angle it must be zero for all angles thus it would seem that the coleman mandula theorem rules out all possible symmetries other than the poincare group translations and lorentz transformations this seems to spoil all our fun by saying that no new symmetries lie in wait to be found by daring adventurous experimentalists thanfully there s a catch we have only ruled out symmetries whose generators transform as tensors under the lorentz group what about spinors that is another representation of the lorentz group we can have spinorial conserved charges this is because spinorial conserved quantities do not give us any observables which impose constraints on scattering matrices for the more technically minded the coleman mandula theorem applies to lie algebras whereas supersymmetry charges obey a different structure they have anticommutators instead of the lie bracket what they do is to relate boson boson and fermion fermion scattering now the anticommutator of two supersymmetry generators is going to have two spinor indices this quantity is like a vector if it consists of a left a right handed spinor but this thing being a conserved vector has to be nothing other than p μ all other anticommutators have to be zero because such a quantity does not correspond to anything that we observe more importanty p 0 the hamiltonian can be written as the anticommutator of a q its hermitian conjugate q this is the simple case with just one supersymmetry what is called n 1 supersymmetry therefore we have h q q now this simple way of writing h as an anticommutator has several interesting properties next day i think i ll write a bit about differential forms and slowly build up to a proof of a special case of the atiyah singer index theorem for the de rham cohomology in case anybody is reading this i d really like to know that somebody actually reads the stuff i m writing and considering that my audience would probably be a singleton i could try to make the stuff i put here more less technical as you might like it it would really feel nice to know that the stuff i m writing here is hopefully useful enjoyable to someone 3 comments posted in physics sinking swimming in susy posted on thursday june 15 2006 2 comments for quite some time i ve been thinking of blogging along as i try to learn supersymmetry and related stuff but it s taken quite a few pushes from several friends to overcome the initial inertia barrier today is as auspicious a day as any so let me start after some thought i ve decided that i ll let my posts slant more to the side of popular science with lesser equationy stuff mainly for the following reasons anyway have to work out most of the equations as part of my project so posting at a more expository level here will complement that by forcing me to think up of why certain things work the way they do how to better look at things most importantly ensure that i actually understand what i m doing i m lazy and it s pain to typeset equations in html made more horrible by wordpress not supporting mathml so here s hoping i ll start soon tomorrow 2 comments posted in meta physics the gamma song posted on saturday april 29 2006 12 comments well the idle mind is the devil s workshop but i d say that a busy one is a storehouse of crazy arbit stupid meaningless unwanted creativity this time during the endsems shankar i came up with the gamma song it s another parody of the famous llama song 1 check it out if you like physics maths stupidity or just have a lot of free time the lyrics here s a gamma there s a gamma and another little gamma real gamma virtua gamma gamma gamma look gamma gamma riemann gamma tangent wick rotate a gamma gamma gamma squareroot gamma gamma gamma look i was once a tensor i lived in a space but i never saw the way the indices were raised i was just an a mu but that fixed a gauge and now listen little child to this verbiage did you ever see a gamma miss a gamma hit a gamma gamma s gamma curve of gamma gamma gamma look affine gamma dirac gamma chiral gamma basu gamma gamma with psi bar a drama gamma gamma look is it all in bold now is this all so weird is it made of tangent bundle h cross sigma beard now our song is losing spin we ve run out of look time for me to retire now and become a crook 12 comments posted in linkage poetry random qft or why you should be glad i m not a poet posted on tuesday april 18 2006 4 comments quantum field theory it s come a long way since oscar walter started out one day rewriting schrödinger s equation in a relativistic way but their effort was partly in vain for the electron did not see fit to deign being a spinor with spin h cross by two physicists they were left without a clue then along came paul stared into the fire pulled out an equation with anticommuting stuff alack alas the equation seemed wrong negative energy eigenvalues it did acquire there was a solution though it was rough fill the vacuum with an electron sea now if there is a hole where an electron should be why that is a positron can t you see those were the days of old qft skip forward in time two dozen more years and richard and julian and shinichiro san they smoothened the wrikles and layed rest to fears and things it seem d were falling in place till other new problems blew up in our face are neutrinos massless are quarks confined is gravity quantised howe er shall we find m theory and unification why it s all still a puzzle and science marches on from hurdle to hurdle from this eternal story we can surely surmise that theories are made and problems are solved these theories then fade as new problems arise 4 comments posted in poetry older posts search for old stuff march 2012 1 june 2008 1 may 2007 2 january 2007 1 july 2006 2 june 2006 1 april 2006 2 march 2006 1 september 2005 3 august 2005 4 july 2005 1 blog at wordpress com towards tengen create a free website or blog at wordpress com subscribe subscribed towards tengen sign me up already have a wordpress com account log in now privacy towards tengen subscribe subscribed sign up log in report this content view site in reader manage subscriptions collapse this bar loading comments write a comment email required name required website design a site like this with wordpress com get started