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the (472), that (104), and (103), this (95), for (89), with (58), are (47), one (44), have (40), from (39), can (39), theory (29), which (28), but (27), was (27), space (27), neutrino (26), there (24), their (23), miniboone (23), not (22), some (22), data (22), energy (21), tikz (21), they (21), #analysis (20), since (20), about (18), now (17), basis (17), anomaly (17), like (16), connection (16), 2009 (16), what (16), all (16), manifold (16), spacetime (16), excess (16), neutrinos (16), has (15), model (15), tangent (15), physics (14), also (14), then (14), out (14), will (14), only (14), into (14), these (14), paper (14), where (14), spin (13), new (13), been (13), how (13), action (13), group (12), more (12), first (12), region (12), index (12), gauge (12), background (12), still (12), comments (11), particle (11), posted (11), way (11), very (11), field (11), found (11), anti (11), effective (11), mev (11), you (10), sparticles (10), could (10), under (10), when (10), course (10), 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Text of the page (random words):
ndancy in how we describe our system it s a non physical degree of freedom that needs to be modded out when describing physical dynamics like any other gauge symmetry we are required to introduce a gauge field for the lorentz group which we shall call from the point of view of riemannian geometry this is just the connection so we can alternately call this creature the spin connection note that this is all different from the local diffeomorphism symmetry of general relativity for which we have the christoffel connection what do we know about the spin connection if we want to be consistent with general relativity while adding only minimal structure which gsw notes is not always the case we need to impose consistency when we take covariant derivatives in particular any vector field with manifold indices can now be recast as a vector field with tangent space indices by requiring that both objects have the same covariant derivative we get the constraint note that the covariant derivative is defined as usual for multi index objects a partial derivative followed by a connection term for each index for the manifold index there s a christoffel connection while for the tangent space index there s a spin connection this turns out to give just enough information to constrain the spin connection in terms of the vielbeins this is precisely equation 11 of hep ph 980547 eft for a 3 brane universe by sundrum and equation 4 28 of hep ph 0510275 tasi lectures on ewsb from xd csaki hubisz meade i recommend both references for rs model building but note that neither of them actually explain where this equation comes from well the latter cites the former so i thought it d be worth explaining this explicitly gsw makes a further note that the spin connection can be using the torsion since they are the only terms that survive the antisymmetry of the torsion tensor going back to our original goal of putting fermions on a curved spacetime in order to define a clifford algebra on such a spacetime it is now sufficient to consider objects where the right hand side contains a flat space constant gamma matrix with its index converted to a spacetime index via the position dependent vielbein resulting in a spacetime gamma matrix that is also position dependent left hand side one can see that indeed the spacetime gamma matrices satisfy the clifford algebra with the curved space metric there s one last elegant thought i wanted to convey from gsw in a previous post we mentioned the role of topology on the existence of the quantum mechanical spin representation of the lorentz group now once again topology becomes relevant when dealing with the spin connection when we wrote down our vielbeins we assumed that it was possible to form a basis of orthonormal vectors on our spacetime a sensible question to ask is whether this is actually valid globally rather than just locally the answer in general is no one simply has to consider the hairy ball theorem that states that one cannot have a continuous nowhere vanishing vector field on the 2 sphere thus one cannot always have a nowhere vanishing global vielbein topologies that can be covered by a single vielbein are actually comparatively scarce and are known as parallelizable manifolds for non parallelizable manifolds the best we can do is to define vielbeins on local regions and patch them together via lorentz transformations transition functions along their boundary consistency requires that in a region with three overlapping patches the transition from patch 1 to 2 2 to 3 and then from 3 to 1 is the identity this is indeed the case spinors must also be patched together along the manifold in a similar way but we run into problems the consistency condition on a triple overlap region is no longer always true since the double valuedness of the spinor transformation i e the spinor transformation has a sign ambiguity relative to the vector transformation if it is possible to choose signs on the spinor transformations such that the consistency condition always holds then the manifold is known as a spin manifold and is said to admit a spin structure in order to have a consistent theory with fermions it is necessary to restrict to a spin manifold january 2 2009 latex figures with pgf and tikz posted by sparticles under tools 6 comments here s a new installment to the never ending debate about the best way to draw figures in latex a previous suggestion adobe illustrator stereographic projection image made using tikz by thomas trzeciak available at texample net a promising solution is the combination of pgf portable graphics format and tikz tikz ist kein zeichenprogramm or tikz is not a drawing program both developed by till tantau whom you may know better for creating beamer pgf is the base system that provides commands to draw vector images this layer of the graphics system is tedious to use directly since it only provides the most basic tools tikz is a frontend that provies a user friendly environment for writing commands to draw diagrams like other latex drawing packages the picture environment axodraw etc the tikz figures are a series of in line commands and so are extremely compact and easy to modify unlike other latex packages though tikz provides a powerful layer of abstraction that makes it relatively easy to make fairly complicated diagrams further the entire system is pdflatex friendly which is more than one can say for pstricks based drawing options the cost is that the system has a bit of a learning curve like latex itself there are many commands and techniques that one must gradually become familiar with in order to make figures fortunately there is a very pedagogical and comprehensive manual available unfortunately the manual is rather lengthy and many of the examples contain small errors that prevent the code from compiling texample net comes to the rescue however witha nice gallery of tikz examples with source code including those from the manual if you re thinking about learning tikz go ahead and browse some of the examples right now the range of possibilities is really impressive to properly learn how to use tikz i would suggest setting aside a day or two to go through the tutorials part i of the manual start from the beginning and work your way through one page at a time the manual was written in such a way that you can t just skip to a picture that you like and copy the source code you need to be sure to include all the libraries and define all the variables that are discussed over the course of each tutorial you can always consult the source code at the gallery of tikz examples in a pinch let me just mention two really neat things that tikz can do which sold me onto the system tikz feynman diagram by k fauske available at texample net the first example is of course drawing a feynman diagram the tikz code isn t necessarily any cleaner than what one would generate using jaxodraw but tikz offers much more control in changing the way things look for example one could turn all fermions blue without having to modify each line tikz arrows on a beamer presentation image by k fauske available at texample net the next example is a solution to one of the most difficult aspects of beamer drawing arrows between elements of a frame this is consistently the feature that powerpoint keynote and the ol chalkboard always do better than beamer no longer check out the source code for the example above adding arrows is as easy as defining some nodes and writing one line of code for each line the lines are curved naturally and the trick works with beamer s overlays beamer is also built on pgf anyway for those with the time to properly work through the tutorials tikz has the potential to be a very powerful tool to add to one s latex arsenal pros tools for creating high quality vector graphics node structure is very useful for drawing charts and feynman diagrams works with pdflatex images are drawn in line no need to attach extra files easy to insert tex into images cons a bit of a learning curve to overcome no standard gui interface download pgf tikz installation instructions place the files into your texmf tree for mac os x users this means putting everything into a subdirectory of library texmf tex latex january 2 2009 renormalization as dimensional analysis posted by sparticles under field theory 2 comments today i ll be reviewing p m stevenson dimensional analysis in field theory annals of physics 132 383 1981 it s a cute paper that helps provide some insight for the renormalization group a theory is a black box that we can shake to make predictions of physical observables we ve already said a few cursory words on dimensional analysis and renormalization it turns out that we can use simple dimensional analysis to yield some insight on the nature of the renormalization group without having to think about the technical heavy machinery required to do actual calculations first let us define a theory as a black box that is characterized by a lagrangian and its corresponding parameters coupling constants masses fields etc all these things however are contained within the black box and are in some sense abstract objects one can ask the black box to predict physical observables which can then be measured experimentally such observables could be cross sections ratios of cross sections or potentials as shown in the image above let s now restrict ourselves to the case of a naively scale invariant or naively dimensionless theory i e one where there are no couplings with mass dimensions for example theory or massless qcd we shall further restrict to dimensionless observables such as the ratio of cross sections let s call a general observable where we have inserted a dependence on the energy with foresight that such things renormalize with energy scale dimensional analysis but one can immediately take a step back and realize that this is ridiculous how could a dimensionless observable from a dimensionless theory have a nontrivial dependence on a dimensionful quantity stevenson makes this more explicit by quoting a theorem of dimensional analysis thm a function which depends only on two massive variables and which is dimensionless uniquely defined defined without any dimensionful constants must then be a function of the ratio of only cor if is independent of then is constant then by the corollary must be constant this is a problem since our experiments show a dependence evading dimensional analysis the answer is that the theorem doesn t hold the theory inside the black box is not uniquely defined violating condition 2 this is what we meant by the stuff inside the black box being abstract the lagrangian is actually a one parameter family of theories with different bare couplings that is to say that the black box is defined up to a freedom in the renormalization conditions now that we see that it is possible to have dependence it s a bit of a curiosity how our dimensionless theory manages to define a dimensionful dependence of without any dimensonful quantities to draw upon the simplest way to do this is to have the theory define the first derivative where is the usual beta function calculated in perturbation theory it is dimensionless and is uniquely defined by the theory another way one can define dependence is to do so recursively one can read stevenson s paper to see that this is equivalent to defining the function one can integrate the equation for to write constant the constant of integration now characterizes the one parameter ambiguity of the theory the ambiguity can be mapped onto the lack of a boundary condition we may parameterize this ambiguity by writing constant for some arbitrary of mass dimension 1 this form is necessary to get a dimensionles logarithm on the left hand side the appearance of this massive constant is something of a virgin birth for the naively dimensionless theory and is called dimensional transmutation by setting we see that thus we see finally that the integral of the equation is all of the one parameter ambiguity of the theory is now packaged into the massive parameter is an integral that comes from the function which is in turn specified by the lagrangian of the theory on the left hand side we have quantities which depend on the arbitrary scale while the right hand side contains only quantities that depend on the energy scale if vanishes for some then we can write our observable in terms of this scale note that is arbitrary while is fixed for a particular theory this latter quantity is rather interesting because even though it is an intrinsic property of the black box it is not predicted by the black box it must be fixed by explicit measurement of an observable more january 1 2009 please cite me e mails posted by sparticles under phd life 1 comment i recently put up my first paper on the arxiv and have been dealing with the torrent of e mails asking for citations this is normal and part of the publication process though i ve been amused by some of the e mails i ve been getting one person decided to e mail my adviser even though it was my e mail address that was associated with the paper there is a reason why my e mail associated with the paper the senior collaborators don t want to have to deal your please cite me e mails don t worry i discuss everything with my collaborators but let s keep things organized yes e mails that start with i read your paper with great interest this is a very nice thing to say but of course when you send it just 30 minutes after the paper is made public then i know that you really mean i quickly searched your bibliography for my name with great interest all these e mails make me wonder if anything i ve done is original at all there is something to be said about being a competent writer when i skim some of these papers begging for citations it is clear why they re not part of the standard set of cited papers they re unreadable yes being a native english speaker is a huge advantage here and yes that s unfair for those who aren t native speakers but that s the way it is i ll cite papers even though they don t really have to be cited this is partly to avoid confrontation but also because i can sympathize with other grad students who keep an eye on their citations on spires december 22 2008 ssi 2009 posted by sparticles under announcements comments off on ssi 2009 since we appear to be in the business of summer school announcements it is perhaps worth passing along the date and title for next year s slac summer institute revolutions on the horizon 3 14 august 2009 the slac summer institutes are only half the length of tasi and tend to be at a more introductory level but they are great ways to dive into a research area one can get a sample by viewing some of their recorded lectures to the best of my knowledge they were the first summer school ot regularly do this the school website does not yet exist but the following blurb is posted the topic is a study of all the upcomin...