Representations Definition and 201 Threads

Hi guys 1) We are looking at a Hamiltonian H. I make a rotation in Hilbert space by the transformation {\cal H} = \mathbf a^\dagger\mathsf H \mathbf a = \mathbf a^\dagger \mathsf U\mathsf U^\dagger\mathsf H \mathsf U\mathsf U^\dagger\mathbf a = \mathbf b^\dagger...

I have question, can someone please check whether my answer is correct or not: 1)Let \pi_i be representations of a group G on vector spaces Vi, i = 1, 2. Give a formula for the tensor product representation \pi_1 \otimes \pi_2 on V_1 \otimes V_2 Answer: \pi_1 V_1 \otimes \pi_2 V_2 2)Check...

These are probably a bit stupid, so I hope you don't mind me asking them... 1)what is a fundamental representation? 2)what is a unitary representation? (Is it just the identity matrix?) 3)What is meant by the 'orthogonal complement' in the following context? "If W\subset{V} is an...

I read somewhere that the irreducible representations of Lie groups were countable. But what about translations? Isn't each momentum value its own irreducible singlet, and there are a continuum of momentum values? For example take e^{ipx} . If you translate it, it doesn't mix with anything...

Hello everyone. I need someone to explain a concept to me. I'm confused about how a type of particle can be a representation of a lie group. For example, I read that particles with half-integer spin j are a representation of the group SU(2), or that particles with charge q are a...

The complexified Lie algebra of the Lorentz group can be written as a direct sum of two commuting complexified Lie algebras of SU(2). It is being said, that this enables us to classify the irreducible representations of the Lorentz algebra with two half-integers (m,n). But can someone...

Hi all. I have here a reference with a representation of the Lie algebra of my symmetry group in terms the fields in my Lagrangian. In order to calculate Noether currents, I would like to use this representation to derive formulae for the infinitesimal forms of the symmetry transformations...

I'm reading the wiki article on Representation theory of the Lorentz group and they seem to make a distinction between these two reps: (1/2,1/2) and (1/2,0) + (0,1/2) I did some checks and it seems that these two are the same. Am I wrong or is the wiki article wrong (won't be the...

Does anyone know how to classify the finite-dimensional irreducible representations of so(4,C)? Can they all be built from irreducible reps of sl(2,C) given the fact that so(4,C) \cong sl(2,C) \times sl(2,C). Thanks!

Never mind, I answered my own question...

This is not important, but it's been bugging me for a while. I'm struggling to see how the locally constant sheaves of vector spaces on X give rise to representations of the fundamental group of X. The approach I've been thinking of is the following. Given a locally constant sheaf F on X...

I have a few questions: 1) The tensor product of two matrices is define by A \otimes B =\left( {\begin{array}{cc} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \\ \end{array} } \right) for the 2x2 case with obvious generalisation to higher dimensions. The tensor product of two...

Howdy folks. The Gelfond-Schneider Constant 221/2 is transcendental. Of my understanding, transcendental numbers are a special case of irrational numbers in the sense of how they may or may not be derrived. This being the case, how is it that many infinite series are stated with such...

The fundamental representation of SU(N) has a basic form that allows you to deduce that there is a SU(N-1) subgroup. For example, in SU(3), the generators T_{1}, T_{2}, T_{3} form an SU(2) subgroup. I'm reading a book right now that goes into the adjoint representation of SU(N) to show that...

1. Can an algebra have an infinite number of non isomorphic representations? 2. Can an algebra have two different representations where one is irreducible and the other is reducible? 3. In general, is it easy to come up with a representation of an algebra? If so then is there a preference for...

Hi, I have a problem. Consider the representation of SU(2) which maps every U \in SU(2) into itself, i.e. U \mapsto U , and the vector space is given by \mathbb{C}^{2} with the basis vectors e_{1} = (1,0) and e_{2} = (0,1) How do I show that the tensor product (Kronecker) of the...

Let H be a separable Hilbert space. What are the continuous linear representations of S^1 on H? I read in an article this is defined as in the finite-dim case. Why is this so? Thanks.

Homework Statement Given a field F, FS4 is a group algebra... we have a representation X that maps FS4 to 3x3 matrices over (presumably) F. Let V denote the FS4 module corresponding to X... do stuff. My question is, what the heck is V supposed to be? I assumed that V is F3, but that...

I'm very very very confused and extremely thick. If \Lambda_i is some element of the Lorentz group and \Lambda_j is another, different element of the group then under multiplication... \Lambda_i \Lambda_j is also an element of the Lorentz group, say \Lambda_i \Lambda_j...

Hey guys, I'm pretty new to group theory at the moment, what's the best way of understanding a 'representation' of a group? Thanks

In his book, "Representation and Reality" Hilary Putnam writes about Chomsky: So what do you think? Are there ‘semantic representations’ in the mind that are innate and universal? Or would you go along with Putnam? If they are not innate/universal, then how do you think meaning gets...

Hallo, I would like to know why physicists are always seeking for irreducible representation of a given group. I know that a reducible one is decomposable into irreducible representations (under special circumstances), but what is the physical motivation that irreducible reps are fundamental...

"Completeness" of irreducible representations Hi, For a finite group of order n each irreducible representation consists of n matrices [D(g)], one for each element in the group. For a given row and column (e.g. i,j) you can form an n-dimensional vector by taking the ij element of D(g) for each...

I was wondering if someone could give me an overview of what they are and how to get them. Thanks

Hi all, I asked this on the Quantum Physics board but didn't get a response. I'm reading Cahn's book on semi-simple lie algebras and their representations. http://www-physics.lbl.gov/~rncahn/book.html In chapter 1, he attempts to build a (2j+1)-dimensional representation T of the Lie...

I'm taking a course on Lie groups and am reading alongisde Cahn's semi-simple lie algebras and their representations. On page 4 he starts to construct a representation T of the Lie group corresponding to SU(2) acting on a linear space V, by defining the action of T_z and T_+ on a vector v_j...

[SOLVED] Lorentz Representations I am reading about the Lorentz group on Schweber. My problem is the following: I don't really understand how to make the direct product of Representations for this Group. I know that we need only 2 mubers since the invarints of the gropu are 2. I know the...

Homework Statement Plot the following for x = 0 to 8: This is part of a larger engineering problem, but I am stuck here, I have no idea how to plot a phasor. The original function is: y(x,\,t)\,=\,2\,cos\left(\frac{\pi}{6}\,t\,-\,\frac{\pi}{4}\,x\right) Homework Equations...

First question pertains to the Residue Theorem We are to use this theorem to evaluate the integral over the given path... There is one problem from this section that I am stuck on. An example in the book evaluates \int_{\Gamma} e^{1/z} dz for \Gamma any closed path not passing through...

Let q and q' be sufficiently close points on C^oo manifold M. Then is it true that any C^oo curve c:[a,b]-->M where c(a)=q, c(q)=q' can be represented as c(t)=exp_{q}(u(t)v(t)) where u:[a,b]-->R,v:[a,b]-->TM_{q} and ||v||=1? My question comes from Chapter 9 corollary 16 and 17 of Spivak vol1...

I'm trying to prove that any representation of a semisimple Lie algebra can be uniquely decomposed into irreducible representations. I have seen some sketches of proofs that show that any representation \phi of a semisimple Lie algebra which acts on a finite-dimensional complex vector space...

Sketch and represent parametrically the following: (a) \mid z+a+\iota b\mid =r \ \mbox { clockwise}\\ , (b) ellipse 4(x-1)^2 + 9(y+2)^2 =36 \ . Taking (a) first \mid z + a + \iota b \mid = r \mbox{- is the distance between the complex numbers }\ z=x+\iota y \ \mbox{ and } \ a + \iota b \...

Hi. I am having trouble proving that the irreducible representations of SU(2) are equivalent to their dual representations. The reps I am looking at are the spaces of homogenous polynomials in 2 complex variables of degree 2j (where j is 0, 1/2, 1,...). If f is such a polynomial the action of...

Here's a question about inequivalent representations of the CCRs... For a given Hilbert space representation, what is it that determines which set of field operators \phi(x), or \phi(f) if we want to get rigorous a la Wightman, gives us THE field operators for that representation. For example...

I'm currently learning Python, but right now I'm interested in animating the motion of things like projecticles, bouncing balls, etc. What language is good for doing stuff like this? And are there any add ons for Python that allow this?

In characteristic zero any linear representation of a reductive group is semisimple. Also in characteristic zero any linear representation of a finite group is semisimple (Maschke's Thm). However is any linear representation of any group semisimple in characteristic zero?

Hello, perhaps this is the most dumb question ever, but I don't see why it holds. I'm looking at the irreducible representations of the Lie group U(1). To find them I considered the irreps of the lie algebra u(1). These irreps are obviously 1 dimensional and are given by f(a i ) = p a i for...

Lecture 1. Introduction and first examples.I thought that this forum was looking a little sparse recently and decided to try to write something interesting for people to read, think about, possibly do some work on. One aspect of algebra that is not taught anywhere at undergraduate level that I'm...

What are good resources on Young diagrams and tableaux for representations of the permutation groups Sn and the unitary groups U(n) of n x n unitary matrices?

Another problem I can't figure out how to start. Let S be the subspace of C[a,b] spanned by e^x , xe^x , (x^2)e^x . Let D be the differentiation operator of S. Find the matrix representing D with respect to [e^x, xe^x, (x^2)e^x ]

I've lost my Chemistry text :eek: and was wondering if anyone knew of any links that showed representations of what orbits may look like? In our halls I've seen wood models of 1p, 2p... but the text had MANY more and I can't remember the name even.

Hi, I keep having problems with a homework question regarding irreducible representations. For the C2 group,which has only two elements,say, e and a, Iwas able to find the regular representations for them, yet i don't know how to find an irreducible representation for them. I'm also supposed...

I'm a little bit confused about the difference between the spinor and vector representations of SU(N)--I guess I could start with asking how a spinor and a vector differ: is this only a matter of how they transform under Lorentz transformations? Following up, the covariant derivative for a...

I was wondering if anyone knew of any good books (or textbooks, or websites) which discuss finding series representations of integrals which exist, but don't have a closed form. I'm interested in the subject at the moment, but I haven't had much luck online. Furthermore, what branch of calculus...

Evaluate the indefinite integral as a power series and find radius of convergence. (i don't know how to type the integral and summations signs, sorry) (integral sign) (x-tan^-1x)/x^3 dx. ( if you write this out it makes more sense) i was able to find the power series of tan^-1x =...

Hello all Let us say we are given a sequence of order 2. By order 2 I mean that we have a sequence in which the differences between the terms forms a sequence of order 1, which has a constant difference between terms. How can I prove that the nth term of a sequence of order 2 can be...

Ok, I'm a little lost so if this subject is off topic for this forum then I appoligize. I am working on a modeling project that I'm trying to implement some code for and I don't even know where to start. I'm not that good of a coder nor am I that great at math but this project, in part, is to...

Does anyone know of any and can comment a good introduction to representaions of elementary particles?

I have to prove the following, and while I understand why the following is true, and I am not sure how to begin writing it out Let m.d1d2d3... and m'.d1'd2'd3' represent the same non-negative real number 1)If m<m', then I have to prove m'=m+1 and every di'=0 and di=0 2)If m=m' and there...

Any two noticeable R numbers a and b can be an open interval (a,b) of infinitely many R numbers, that cannot be separated form each other by any representation, and each R number can be represented only by aleph0 different representations. Is it right ? If what i wrote holds, than please...