representation theory

  1. filip97

    A Gamma - traceless

    I read this question https://physics.stackexchange.com/questions/95970/under-what-conditions-is-a-vector-spinor-gamma-trace-free . Also I read Sexl and Urbantke book about groups. But I dont understand why spinors is irreducible if these are gamma-tracelees. Also I read many papers about higher...
  2. filip97

    A (A,A) representation of Lorentz group-why is it tensor?

    Why representation of Lorentz group of shape (A,A) corespond to totally symmetric traceless tensor of rank 2A? For example (5,5)=9+7+5+3+1 (where + is dirrect sum), but 1+5+3+9+7<>(5,5) implies that (5,5) isn't symmetric ? See Weinberg QFT Book Vol.1 page 231.
  3. Ramtin123

    A SU(2) Invariant Lagrangian

    Consider two arbitrary scalar multiplets ##\Phi## and ##\Psi## invariant under ##SU(2)\times U(1)##. When writing the potential for this model, in addition to the usual terms like ##\Phi^\dagger \Phi + (\Phi^\dagger \Phi)^2##, I often see in the literature, less usual terms like: $$\Phi^\dagger...
  4. tomdodd4598

    I Direct Sums of Lorentz Group Representations

    Hey there, I've suddenly found myself trying to learn about the Lorentz group and its representations, or really the representations of its double-cover. I have now got to the stage where the 'complexified' Lie algebra is being explored, linear combinations of the generators of the rotations...
  5. Hans de Vries

    A New Covariant QED representation of the E.M. field

    90 years have gone by since P.A.M. Dirac published his equation in 1928. Some of its most basic consequences however are only discovered just now. (At least I have never encountered this before). We present the Covariant QED representation of the Electromagnetic field. 1 - Definition of the...
  6. A

    I Adjoint Representation Confusion

    I'm having a bit of an issue wrapping my head around the adjoint representation in group theory. I thought I understood the principle but I've got a practice problem which I can't even really begin to attempt. The question is this: My understanding of this question is that, given a...
  7. Cryo

    A Nonlinear susceptibility and group reps

    Dear All short explanation: I am trying to leverage my limited understanding of representation theory to explain (to myself) how many non-vanshing components of, for example, nonlinear optical susceptibility tensor ##\chi^{(2)}_{\alpha\beta\gamma}## can one have in a crystal with known point...
  8. Ramtin123

    A Three dimensional representation of ##U(1)\times SU(2)##

    Consider a three dimensional representation of ##U(1)\times SU(2)## with zero hypercharge ##Y=0##: $$ L= \begin{pmatrix} L^+ \\ L^0 \\ L^- \end{pmatrix} $$ Then the mass term is given by [1]: $$ \mathcal{L} \supset -\frac m 2 \left( 2 L^+ L^- +L^0 L^0 \right) $$ I am wondering where the...
  9. C

    I Tensor representation of the Lorentz Group

    I've been trying to understand representations of the Lorentz group. So as far as I understand, when an object is in an (m,n) representation, then it has two indices (let's say the object is ##\phi^{ij}##), where one index ##i## transforms as ##\exp(i(\theta_k-i\beta_k)A_k)## and the other index...
  10. JuanC97

    I Minimum requisite to generalize Proca action

    Hello guys, In 90% of the papers I've read about diferent ways to achieve generalizations of the Proca action I've found there's a common condition that has to be satisfied, i.e: The number of degrees of freedom allowed to be propagated by the theory has to be three at most (two if the fields...
  11. JuanC97

    I ##A_\mu^a=0## in global gauge symmetries ?

    Hi, this question is related to global and local SU(n) gauge theories. First of all, some notation: ##A## will be the gauge field of the theory (i.e: the 'vector potential' in the case of electromagnetic interactions) also known as 'connection form'. In components: ##A_\mu## can be expanded in...
  12. Luck0

    A Diagonalizing Hermitian matrices with adjoint representation

    Suppose I have a hermitian ##N \times N## matrix ##M##. Let ##U \in SU(N)## be the matrix that diagonalizes ##M##: ##M = U\Lambda U^\dagger##, where ##\Lambda## is the matrix of eigenvalues of ##M##. This transformation can be considered as the adjoint action ##Ad## of ##SU(N)## over its...
  13. L

    A Tensor symmetries and the symmetric groups

    In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##) To discuss general properties of tensor symmetries, we shall use the representation theory of the...
  14. pellis

    A Who wrote "Ch 6 Groups & Representations in QM"?

    Who really wrote the best introductory account of representation theory in QM that I've seen so far ? [Likely mis-attribution discussed here below; prefixed "Advanced" to reach lecturers who are more likely to know the answer to this question.] It's available via...
  15. P

    Massive spin-s representations of the Poincare group

    Context The following is from the book "Ideas and methods in supersymmetry and supergravity" by I.L. Buchbinder and S.M Kuzenko, pg 56-60. It is about realizing the irreducible massive representations of the Poincare group as spin tensor fields which transform under certain representations of...
  16. arivero

    I Bootstraping a space from its tensor square

    By space, I mean a vector space which could be a representation of a group or even have some expanded algebraic structure. So I am not sure if this question goes here or in the Algebra subforum. Consider the tensor square r\otimes r of an irreducible group representation r with itself, and...
  17. V

    A How parity exchanges right handed and left handed spinors

    Reading through David Tong lecture notes on QFT. On pages 94, he shows the action of parity on spinors. See below link: [1]: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf In (4.75) he confirms that parity exchanges right handed and left handed spinors. Or for an arbitrary...
  18. B

    A Notation/Site for Representations of an Algebra

    I'm currently reading the paper "Higher Spin extension of cosmological spacetimes in 3d: asymptotically flat behaviour with chemical potentials in thermodynamics" I'm looking at equation (3) on page 4. I know that symmetrization brackets work like this A_(a b) = (A_ab + A_ba)/2. However I have...
  19. S

    I Representation of a vector

    Hello. If I represent a vector space using matrices, for example if a 3x1 vector, V, is represented by 3x3 matrix, A, and if this vector was the eigenvector of another matrix, M, with eigenvalue v, if I apply M to the matrix representation of this vector, does this holds: MA=vA? Also, if I...
  20. S

    I SU(2) representations

    Hello! I just started reading about SU(2) (the book is Lie Algebras in Particle Physics by Howard Georgi) and I am confused about something - I attached a screenshot of those parts. So, for what I understood by now, the SU(2) are 2x2 matrices whose generators are Pauli matrices and they act on a...
  21. G

    I About Lie group product ([itex]U(1)\times U(1)[/itex] ex.)

    I recently got confused about Lie group products. Say, I have a group U(1)\times U(1)'. Is this group reducible into two U(1)'s, i.e. possible to resepent with a matrix \rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}}...
  22. Giuseppe Lacagnina

    Lorentz transformations and vector fields

    Hi Everyone. There is an equation which I have known for a long time but quite never used really. Now I have doubts I really understand it. Consider the unitary operator implementing a Lorentz transformation. Many books show the following equation for vector fields: U(\Lambda)^{-1}A^\mu...
  23. JonnyMaddox

    Representations of spinors

    I'm currently reading a book on relativistic field theory and I'm trying to understand spinors. After the author introduces the four parts of the Lorentz group he talks about spinors and group representations: "....With this concept we see that the 2x2 unimodular matrices A discussed in the...
  24. Dilatino

    Introduction to Young-tableaux and weight diagrams?

    I am looking for a short pedagogical introduction to Young-tableaux and weight diagrams and the relationship between them, which contains many detailled and worked out examples of how these methods are applied in physics, such as in the context of the standard model and beyond for example. I am...
  25. terra

    2j+1 d representation for Poincaré group

    I want to learn how to write down a particle state in some inertial coordinate frame starting from the state ##| j m \rangle ##, in which the particle is in a rest frame. I know how to rotate this state in the rest frame, but how does one write down a Lorentz boost for it? Note that I am not...
  26. Primroses

    Why are invariant tensors also Clebsch-Gordan coefficients?

    On one hand, in reading Georgi's book in group theory, I comprehend the invariant tensor as a special "tensor", which is unchanged under the action of any generators. On the other hand, CG decomposition is to decompose the product of two irreps into different irreps. Now it is claimed that...
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