The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
1answer
50 views

Representations of Lorentz algebra

It is well known that the Lorentz algebra can be written as two $SU(2)$ algebras. By defining $$N_i=\frac{1}{2}(J_i+iK_i), \qquad N^{\dagger}_i=\frac{1}{2}(J_i-iK_i)$$ we have ...
2
votes
1answer
115 views

Irreducible Representations from Cartan Matrix

I need some help understanding how you can construct the irreducible representations of an algebra from knowing its roots or its Cartan matrix, which I am being told you most certainly can. ...
1
vote
0answers
26 views

Decomposition of a tensor under transformations

To illustrate my question I'll take an example from theory of relativity: An arbitrary 4-tensor $A^{ik}$ changes under a general coordinate transformation: $$ A'^{ik} = C^{i}_mC^{k}_n A^{mn} $$ ...
2
votes
1answer
85 views

Angular momentum - proof for integer eigenvalues

I am confused about a proof my Quantum Mechanics textbook has left "as an exercise for the reader". So, we've got the angular momentum operator $\hat{L}$. We've also got the generalized angular ...
0
votes
1answer
174 views

General definition of vector, spinor, and spin

I am looking for basic and exact definitions of fundamental physical concepts in graduate level. I reach this following definitions. Could you please help to improve these definitions. Spin: ...
2
votes
1answer
73 views

Does scale invariance imply massless or continuous mass distribution?

$\newcommand{\ket}[1]{\lvert #1 \rangle}\newcommand{\bra}[1]{\langle #1 \rvert}\newcommand{\scp}[2]{\langle #1 \vert #2 \rangle}$ In his 2008 slides Unparticle Phenomenology (PDF), Tzu-Chiang Yuan ...
3
votes
1answer
272 views

What does Weinberg–Witten theorem want to express?

Weinberg-Witten theorem states that massless particles (either composite or elementary) with spin $j > 1/2$ cannot carry a Lorentz-covariant current, while massless particles with spin $j > 1$ ...
11
votes
2answers
379 views

Why are non-Abelian gauge theories Lorentz invariant quantum mechanically?

I seem to be missing something regarding why Yang-Mills theories are Lorentz invariant quantum mechanically. Start by considering QED. If we just study the physics of a massless $U(1)$ gauge field ...
8
votes
1answer
319 views

Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice

Happy holidays, everyone! The following is part question, part visual gallery, and part classical mechanics problem. Inspired by snow over the weekend I began simulating the vibrations of the ...
6
votes
1answer
241 views

How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group?

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...
1
vote
1answer
49 views

Lax representation of the harmonic oscillator

Peter Lax showed that the differential operators $$L=-6\partial_x^2-u,\quad B=-4\partial_x^3-u\partial_x-(1/2)u_x$$ fulfilling the Lax equation $$\dot{L}+[L,B]=0$$ is equivalent to the KdV equation ...
2
votes
2answers
49 views

Correct vector space of eigenkets of angular momentum

When we say an particle is in the state: \begin{equation} |l,m\rangle, \end{equation} what is the underlying state space, as a vector space? Is it a tensor product vector space, of dimension: ...
5
votes
1answer
597 views

What does it mean for a particle to have spin of 2? [duplicate]

When I first started to study quantum mechanics, my physics text book told that particles have spin of either 1/2 or -1/2. Then I recently read an article saying that gravitons are expected to be ...
2
votes
2answers
42 views

Why does the raising operator, when acting on a ket with a maximum second quantum number, yield zero?

I'm studying the general formalism of angular momentum in quantum mechanics from Zettili's "Quantum Mechanics: Concepts and Applications", and came across the following equation (labeled 5.37) on page ...
3
votes
2answers
54 views

Normalising Generators of a Lie Algebra

Ok, so I'm asking this in physics because I'm currently working through part of Srednicki's text on QFT, even though it's really a maths question. In Srednicki's chapter on non-Abelian gauge theory, ...
3
votes
1answer
33 views

Permissible combinations of colour states for gluons

My lecturer has said that there are 8 types of gluons (I'm assuming that the repetition of $r\bar{b}$ is a typo that is meant to be $r\bar{g}$) $$r\bar{b}, b\bar{r}, r\bar{g}, g\bar{r}, g\bar{b}, ...
1
vote
0answers
22 views

Recipe to compute dimension and decompose product of $SO(N)$ group representations [migrated]

As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts: There is a correspondence between irreps ...
1
vote
1answer
41 views

About shift operators

The question is this: Does $$L_+ L_- Y_{lm} $$ ,where $Y_{lm}$ is a spherical harmonic function, equals to zero. If so, why? The two operators above are defined as $$L_+ ={L_x + iL_y } $$ $$L_-={L_x ...
0
votes
1answer
64 views

Indices of a Pauli matrix transformed in the Lorentz representation

When Peskin and Schroeder want to prove a Fierz identity on page 51, they make use of the identity $$(\sigma^{\mu})_{\alpha \beta} (\sigma_{\mu})_{\gamma\delta} = 2 \epsilon_{\alpha \gamma} ...
7
votes
4answers
348 views

Coadjoint orbits in physics

I am looking for some application of coadjoint orbits in physics. If you know some of them please let me know.
6
votes
2answers
75 views

Why is $\theta \over 2$ used for a Bloch sphere instead of $\theta$?

I'm a beginner in studying quantum info, and I'm a little confused about the representation of a qubit with a Bloch Sphere. Wikipedia says that we can use $$\lvert\Psi\rangle=\cos\frac{\theta}{2} ...
3
votes
1answer
93 views

What's the significance of the difference between the quantum numbers, $\ell$ and $m_{\ell}$?

I know that $m_{\ell}$ is associated with the projection of the angular momentum vector onto the $z$ axis and $\ell$ is associated with the length of the angular momentum vector. To me this implies ...
1
vote
1answer
43 views

Singleton representations of $SO(2, d)$?

What is meant by "singleton" representations of $SO(2, d)$ and 'small representations' in Witten's paper, Anti de Sitter Space and Holography?
0
votes
1answer
71 views

Why is orbital angular momentum quantized according to $I=\frac h{2 \pi} \sqrt{l(l+1)}$?

I simply have no idea how this result is found $$I=\frac h{2 \pi} \sqrt{l(l+1)}.$$ The result seems to just be dumped in textbooks rather than explained. I can get the result that ...
1
vote
1answer
46 views

Total angular momentum operator

How do the eigenfunctions of the total angular momentum operator analytically look like? I mean the operator is given by $J = L+S$ so the eigenfunctions have to be tensor-product states, right? Can ...
3
votes
3answers
271 views

Why does the electron spin with a particular tilt?

I found this image for the classical description of the electron spin at hyperphysics Can you explain why the axis of rotation makes an angle of 60° with the z-axis and how this particular ...
1
vote
1answer
40 views

Why do we restrict the maximal supercharge to 32?

Many supersymmetry textbook state that the maximal supersymmetry in any dimension has 32 hermitian supercharges. (Actually for lowest number of supersymmetry $N=1$ the highest dimension is $D=11$) I ...
0
votes
1answer
72 views

How do states in Hilbert Space act like irreducible representations?

I am reading Georgi's book on group theory and I came across this sentence..." Hilbert space of any parity invariant system can be decomposed into states that behave like irreducible representations". ...
1
vote
0answers
47 views

Interpretation of vector mesons in QCD

It is well-known that scalar mesons are interpreted as pseudogoldstone bosons which is connected with spontaneous broken $SU(3) \times SU(3)$ symmetry to $SU(3) \times SU(3) / SU(3)_{chiral}$. Is ...
2
votes
1answer
41 views

Decomposition of the gravitino into helicity $\pm \frac{3}{2}$ and $\pm \frac{1}{2}$ components

I'm reading this book on string theory. When they decompose two dimensional gravitino (formula 7.16) $$ \chi_\alpha = \frac{1}{2}\rho^\beta \rho_\alpha \chi_\beta + \frac{1}{2}\rho_\alpha \rho^\gamma ...
4
votes
1answer
145 views

Branching rules for $SU(3)$

How does one compute the branching rules for $SU(3)\to SU(2)\times U(1)$.? In particular, I do not know how to put the abelian charges. Take for example the adjoint $\mathbf{8}$ of $SU(3)$. I can ...
0
votes
0answers
26 views

Euclidean Continuation of spinor

I would like a clear and useful answer or explanation about these following argument or question that I'll place in logical order for my purpose that deal the Euclidean continuation of spinors in 5 ...
1
vote
0answers
74 views

Is an electron technically a set of two particles?

The electron - described as a four-spinor in the Dirac equation - transforms according to the $(1/2,0)\oplus(0,1/2)$ representation of the Lorentz group, so it is actually a direct sum of a left- and ...
0
votes
1answer
142 views

How do you add angular momentum of three or more particles in quantum mechanics?

I'm trying to find some information on how to add the angular momentum of three or more particles, but all the sources I look at deal with only two. In this case I understand that if the angular ...
2
votes
1answer
92 views

Embedding of particles into fields

For the classification of particles (Wigner 1939), we look for unitary representations of the Poincaré/Lorentz group. There are are only infinite-dimensional (non-trivial) unitary representations! To ...
1
vote
2answers
110 views

Explicit Spinor Representations in $SO(3)$ and $SO(4)$

The beautiful physical interpretation of the decomposition of the elasticity stress tensor into it's irreducible representations (mentioned in context below) has inspired the following question on ...
2
votes
0answers
57 views

Relation between representations/classifications

Generally a quantum system can be characterized in the following way: its states form a representation space for every symmetry group of that system. The representation has to be unitary (or ...
2
votes
0answers
33 views

manipulations with SU(N) Nekrasov partition function

Think of a Young tableau $R$ as collection of rows $y_1 \geq ... \geq y_d > y_{d+1}=0$ and all others zero, with $\ell(Y):= \sum_j y_j$ and for a box $s=(i,j)\in R$ we have $a_Y(s):=y_i-j$ and ...
0
votes
2answers
270 views

Tensor product of two different Pauli matrices $\sigma_2\otimes\eta_1 $

I'm solving problem 3.D in H. Georgi Lie Algebra etc for fun where one is to compute the matrix elements of the direct product $\sigma_2\otimes\eta_1$ where $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
0
votes
4answers
100 views

Conceptualization and modelling of spin

I'm trying to get a decent understanding of the Bell inequality, and so am trying to understand spin both conceptually and mathematically. When I picture spin, I imagine a sphere rotating about its ...
0
votes
1answer
70 views

Tensor operators and transformation of $O^s_{\ell}|j,m,\alpha\rangle$

In H. Georgi's Lie Algebras in Particle Physics one defines a tensor operator transforming under the spin-$s$ representation of $SU(2)$ as the set of operators $O^s_{\ell}$ (for $\ell=-s...s$) such ...
2
votes
1answer
60 views

Does $GL(N,\mathbb{R})$ own spinor representation? Which group is its covering group? (Kaku's QFT textbook)

In Kaku's QFT textbook page 54, there is a saying: $GL(N)$ does not have any finite-dimensional spinorial representation. This implicates that $GL(N)$ owns infinite-dimensional spinorial ...
8
votes
2answers
1k views

Irreducible Representations Of Lorentz Group

In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64. He define states as ...
8
votes
1answer
352 views

Vector spaces for the irreducible representations of the Lorentz Group

EDIT: The vector space for the $(\frac{1}{2},0)$ Representation is $\mathbb{C}^2$ as mentioned by Qmechanic in the comments to his answer below! The vector spaces for the other representations remain ...
3
votes
2answers
167 views

Why is the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group realized as the vector space of complex $2\times 2$ matrices?

Why can we write an arbitrary object $v_{a \dot{b} }$ our transformations in this basis act on as $$ v_{a \dot{b} } = v_{\nu} \sigma^{ \nu}_{a \dot{b} } = v^0 \begin{pmatrix} 1&0 \\ 0&1 ...
7
votes
1answer
113 views

How do you build a Lagrangian in particle/nuclear physics? (A specific example)

I know that the terms in the Lagrangian needs to be scalars (with respect to Lorentz symmetry etc.). Also I know that [see C. G. Tully (EPP) p. 85] in general, for $\psi$ in the fundamental ...
0
votes
1answer
111 views

Why do particles have spins such as $1/2$, $3/2$, $5/2$? [duplicate]

What does it mean to have 'half' spin? I have looked on Wikipedia and a few youtube videos on spin but they don't explain what it means to have $1/2$ spin. I am 18 and only starting to learning about ...
9
votes
1answer
109 views

Group notation $\otimes$ and $\oplus$ used for representations of quarks and mesons

I've been trying to figure out this statement from the PDG quark model summary (PDF). Following $\mathrm{SU}(3)$, the nine possible $q\bar{q}′$ combinations containing the light $u$, $d$, and $s$ ...
0
votes
0answers
19 views

How to know whether a representation is in $d$ or $\bar{d}$ using Young Tableaux?

For the sake of example, suppose we work with $SU(3)$ and we find that some product of reps decompose into a sum which contains a box consisting of four boxes: something like $$\tag{1}\Box\Box ...