In Quantum Mechanics, by Wigner's theorem, a symmetry can be represented either by a unitary linear or antiunitary antilinear operator on the Hilbert space of states ##\cal H##. If ##G## is then a Lie group of symmetries, for each ##T\in G## we have some ##U(T)## acting on the Hilbert space and...
This is one problem from Robin Ticciati's Quantum Field Theory for Mathematicians essentially asking us to find Cartan subalgebras for the matrix algebras ##\mathfrak{u}(n), \mathfrak{su}(n),\mathfrak{so}(n)## and ##\mathfrak{so}(1,3)##. The only thing he gives is the definition of a Cartan...
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...
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.
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:
$$\Phi^\dagger...
Hey there,
I've suddenly found myself trying to learn about the Lorentz group and its representations, or really the representations of its double-cover. I have now got to the stage where the 'complexified' Lie algebra is being explored, linear combinations of the generators of the rotations...
90 years have gone by since P.A.M. Dirac published his equation in 1928. Some of its most basic consequences however are only discovered just now. (At least I have never encountered this before). We present the Covariant QED representation of the Electromagnetic field.
1 - Definition of the...
I'm having a bit of an issue wrapping my head around the adjoint representation in group theory. I thought I understood the principle but I've got a practice problem which I can't even really begin to attempt. The question is this:
My understanding of this question is that, given a...
Dear All
short explanation:
I am trying to leverage my limited understanding of representation theory to explain (to myself) how many non-vanshing components of, for example, nonlinear optical susceptibility tensor ##\chi^{(2)}_{\alpha\beta\gamma}## can one have in a crystal with known point...
Consider a three dimensional representation of ##U(1)\times SU(2)## with zero hypercharge ##Y=0##:
$$ 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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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}}...
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...
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...
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...
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...
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...