Tools for SU(n) 1.0 
Date Added: July 21, 2013  Visits: 195 


__________________________MATLAB filesdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcThese powerful functions study irreps of SU(n>1)... ddeD? suf.m  given an irrep, finds all weights and their individual dimensions ddeD? suw.m  using Weyl dimension formula, finds total dimension of all weights corresponding to an irrepThese very similar functions study specific SU(n) groups with the help of the above two functions... ddeD? su2.m  plots all weights of an SU(2) irrep ddeD? su3.m  plots all weights of an SU(3) irrep ddeD? su4.m  plots all weights of an SU(4) irrep ddeD? su5.m  plots all weights of an SU(5) irrep ddeD? su6.m  plots a 4dimensional slice of the weights of an SU(6) irrepThese functions study hadrons using all the above functions... ddeD? hadrons.m  creates flavorstate multiplets for hadrons! ddeD? hadr.m  used by hadrons.m to make a figure ddeD? had.m  used by hadrons.m to make an animationThe following function uses suw.m... ddeD? suy.m  finds all irreps within tensor products of SU(n>1) irrepsRun these in MATLAB making sure that the current folder contains these files. Use the help command to display help text (for example, type "help su3"). Octave, a free imitator of MATLAB, works very poorly with some of these functions due to its crappy graphics.__________________________my goaldldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcRepresentations of SU(n) interest me since they apply very much to the type of physics that is simple enough for me to presently understand. SU(2) describes spinx particles where x is any nonnegative halfinteger or integer. SU(3) describes color charge. There are many other physical uses for the SU(n) groups, and I have created general functions that help you study irreducible representations and their weights. I have also written some functions that draw and animate hadron multiplets. I am fascinated by the way that observed hadrons belong to the 4dimensional diagrams (which is a series of 3dimensional diagrams) created by the bizarre and abstract math of special unitary groups.Basically, I intended to write some code that very quickly does all the very long calculations required to study these very interesting groups while giving hadron flavorstates special attention.The big picture of my code is that you will use L to define an irrep. The length of L is r, which means that SU(r+1) is being studied. To find out what I mean by all these symbols and words, see my list of terminology below.I have done my best to make the code readable and expandable! Please conform it to your own needs and/or wants!__________________________terminology and other infodldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcdldcImportant vocab that I use... ddeD? SU(n) ddeOCL Lie group of all special unitary ndoOCon matrices (I usually assume n>1) ddeD? irrep ddeOCL irreducible matrix representation of SU(n) ddeD? weight ddeOCL an "eigenvalue" of the irrep's Cartan subalgebra ddeD? alpha ddeOCL simple roots of the Lie algebra for SU(n) that have been given the geometric interpretation as vectors so that weights can be plotted, where it is true that (1) all positive roots are sums of these, (2) the order of simple roots is chosen to be the standard choice that makes the Cartan matrix have zeros everywhere except the 3 central diagonals, and (3) the default basis used to write these vectors is the standard rectangular basis in n1 dimensional space ddeD? alpha basis ddeOCL I sometimes use alpha1=[1,0,0,...], alpha2=[0,1,0,...], etc. as a basis instead of the default rectangular basis ddeD? Dynkin coefficients of a weight ddeOCL every weight has integer Dynkin coefficients that uniquely label the weight, where these coefficients are the weight vector in a basis obtained by multiplying the inverse Cartan matrix with the column vector of alphas ddeD? highest weight ddeOCL this is the only weight of an irrep that can form all other weights by subtracting alphas (any list of nonnegative Dynkin coefficients is a valid highest weight) ddeD? dimension ddeOCL I sometimes refer to the number of spatial dimensions needed to draw something, and I even sometimes mean the dimension of the SU(n) manifold, but I usually am referring to the dimension of a weight space (i.e. the multiplicity of a weight) or the total dimension of the direct sum of all weight spaces (note that the total dimension of an ndoOCon irrep is ndimensional)Some variables I will use consistently... ddeD? r = rank = number of alphas of a group = the number of Dynkin coefficients per weight = the number of spatial dimensions a plot of weights will require = rank of SU(r+1) ddeD? L = Lambda = Dynkin coefficients of highest weight as a row vectorddeOCLddeOCLddeOCLddeOCLddeOCLddeOCLSee readme.htm for the rest of the description!ddeOCLddeOCLddeOCLddeOCLddeOCLddeOCL

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