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I will not get into theoretical details -- Luboš ad Marek did that better than I'm able to. Let me give an example instead: suppose that we need to calculate this integral: $\int d\Omega (Y_{3m_1})^*Y_{2m_2}Y_{1m_3}$ Here $Y_{lm}$ -- are spherical harmonics and we integrate over the sphere $d\Omega=\sin\theta d\theta d\phi$. This kind of integrals appear ...

20

Great, important question. Here's the basic logic: We start with Wigner's Theorem which tells us that a symmetry transformation on a quantum system can be written, up to phase, as either a unitary or anti-unitary operator on the Hilbert space $\mathcal H$ of the system. It follows that if we want to represent a Lie group $G$ of symmetries of a system via ...

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What you observe is the general phenomenon that in relativistic theories time translation is replaced by "affine-parameter-translation" or "wordline translation symmetry" and hence the corresponding Hamiltonian becomes a constraint, the constraint that states must be invariant under this symmetry. Yes, this works for the relativistic spinning particle and ...

19

By the "noncompact $U(1)$ group", we mean a group that is isomorphic to $({\mathbb R},+)$. In other words, the elements of $U(1)$ are formally $\exp(i\phi)$ but the identification $\phi\sim \phi+2\pi k$ isn't imposed. When it's not imposed, it also means that the dual variable ("momentum") to $\phi$, the charge, isn't quantized. One may allow fields with ...

16

Put into one sentence, Noether's first Theorem states that a continuous, global, off-shell symmetry of an action $S$ implies a local on-shell conservation law. By the words on-shell and off-shell are meant whether Euler-Lagrange equations of motion are satisfied or not. Now the question asks if continuous can be replace by discrete? It should immediately ...

15

No, the elements of the periodic table don't form any representation of a group or, more precisely, any irreducible representation. Even more precisely, the real insights by Mendeleev – that the reactivity etc. is a repeating function of the atomic number – doesn't follow from any property of a representation that could be derived by group theory. The ...

15

Here is a mathematical derivation. We use the sign convention $(+,-,-,-)$ for the Minkowski metric $\eta_{\mu\nu}$. I) First recall the fact that $SL(2,\mathbb{C})$ is (the double cover of) the restricted Lorentz group $SO^+(1,3;\mathbb{R})$. This follows partly because: There is a bijective isometry from the Minkowski space ...

14

$F_4$ is the centralizer of $G_2$ inside an $E_8$. In other words, $E_8$ contains an $F_4\times G_2$ maximal subgroup. That's why by embedding the spin connection into the $E_8\times E_8$ heterotic gauge connection on $G_2$ holonomy manifolds, one obtains an $F_4$ gauge symmetry. See, for example, http://arxiv.org/abs/hep-th/0108219 Gauge theories and ...

14

UPDATE - Answer edited to be consistent with the latest version of the question. The different definitions you mentioned are NOT definitions. In fact, what you are describing are different representations of the Lorentz Algebra. Representation theory plays a very important role in physics. As far as the Lie algebra are concerned, the generators ...

14

I don't know if these rise to the level of "useful," but: Yang-Mills theory with gauge group $G_2$ is interesting because $G_2$ has trivial center. So people simulate it on a lattice, try to understand in what sense it might be confining, how string tensions scale, if it has a deconfinement phase transition, and so on. The idea is that looking at a group ...

14

After the answers by joshphysics and user37496, it seems to me that a last remark remains. The quantum relevance of the universal covering Lie group in my opinion is (also) due to a fundamental theorem by Nelson. That theorem relates Lie algebras of symmetric operators with unitary representations of a certain Lie group generated by those operators. The ...

14

Two particles forming an $SU(2)$ doublet means that they transform into each other under $SU(2)$ transformation. In other words under $SU (2)$ transformations the proton and the neutron transform as, \left( \begin{array}{c} p \\ n \end{array} \right) \xrightarrow{SU(2)} \exp \left( - \frac{ i }{ 2} \theta_a \sigma_a \right) \left( ...

13

There is a book titled "Group theory and Physics" by Sternberg that covers the basics, including crystal groups, lie groups, representations. I think it's a good introduction to the topic. To quote a review on Amazon (albeit the only one) "This book is an excellent introduction to the use of group theory in physics, especially in crystallography, special ...

13

To develop physical intuition behind unitary groups, it's helpful to have intuition for the notion of unitary transformations in general since unitary groups are just groups consisting of unitary transformations. It helps to start with mathematical objects that are a bit more "familiar" in classical physics -- rotations, which are also known as orthogonal ...

12

Yes. G2 shows up often, starting with atomic physics (perhaps Racah is the first; see R. E. Behrends, J. Dreitlein, C. Fronsdal, and B. W. Lee, “Simple groups and strong interaction symmetries,” Rev. Mod. Phys. 34, 1 (1962).). You will find some refences in my 1976 Phys rev paper on cns.physics.gatech.edu/GroupTheory/refs . I have whole folder of physics ...

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I'll give you enough hints to complete the proof yourself. If you're desperate, I'm following the notes by Zuber, which are available online, IIRC. Let's start with some notation: pick some basis $\{t_a\}$ of your Lie algebra, then $$[t_a,t_b] = C_{ab}{}^c t_c$$ defines the structure constants. If you define $$g_{ab} = C_{ad}{}^e C_{be}{}^d,$$ then this ...

12

As you point out, the Minkowski metric $\eta = \mathrm{diag}(-1,+1, \dots, +1)$ in $d+1$ dimensions possesses a global Lorentz symmetry. A highbrow way of saying this is that the (global) isometry group of the metric is the Lorentz group. Well, translations are also isometries of Minkowski, so the full isometry group is the Poincare group. The general ...

11

I) Firstly, we are talking about the direct or Cartesian product $SU(2)\times SU(2)$ of groups, not the tensor product$^1$ $SU(2)\otimes SU(2)$ of groups. II) Secondly, $SU(2)\times SU(2)$ is not isomorphic to the Lorentz group $SO(3,1)$ but rather to a compact cousin $$[SU(2)\times SU(2)]/\mathbb{Z}_2~\cong~ SO(4).$$ In particular, a ...

11

Nice question! The short answer is that the group is not $SU(2)\times U(1)$, it is $SU(2)_L \times U(1)_{em}$. In other words the two groups act on different standard model particles differently. For example the left handed neutrino does interact weakly and so transforms under the $SU(2)_L$, but is electrically neutral so it doesn't transform under the ...

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Any good book in Semiconductor Physics will have a description of the k.p method. Try Fundamentals of Semiconductor Physics by Peter Yu and Manuel Cardona. Another reference for Kane Model and EFA are chapters 2 and 3 of "Wave Mechanics Applied to Semiconductor Heterostructures" by Gerald Bastard. If you want a more mathematically/group theory oriented ...

11

First of all, note that the real Abelian Lie group $U(1)$ comes in two (multiplicatively written) versions: Compact $U(1)~\cong~e^{i\mathbb{R}}~\cong~S^1$, and Non-compact $U(1)$ $~\cong~e^{\mathbb{R}}\cong~\mathbb{R}_+\backslash\{0\}$. Also note that in the physics literature, we often identifies charge operators with Lie algebra generators for a ...

11

Let the Minkowski metric $\eta_{\mu\nu}$ in $d+1$ space-time dimensions be $$\tag{1}\eta_{\mu\nu}~=~{\rm diag}(1, -1, \ldots,-1).$$ Let the Lie group of Lorentz transformations be denoted as $O(1,d;\mathbb{R})=O(d,1;\mathbb{R})$. A Lorentz matrix $\Lambda$ satisfies (in matrix notation) $$\tag{2} \Lambda^t \eta \Lambda~=~ \eta.$$ Here the superscript ...

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In this relatively simple example, one can observe that the subalgebras $\{\sigma_a \otimes \frac{1\mp\eta_1}{2}\}$ are the two commuting copies of $su(2)$. For more complicated situations, one actually has an algorithm to veify the simplicity of a Lie algebra. This is because (the root systems of) simple Lie algebras are classified by Cartan, thus one just ...

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It seems to me that the stabilizer formalism provides an answer to your question (see section 3.1 of quant-ph/0603226 for an introduction to the formalism). Given the two states, you simply take the stabilizer group for that 2-dimensional subspace of the total Hilbert space, and they will give you a such a set of invariants. However, this of course has ...

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I) The Casimir invariants of a Lie algebra $L$ over a field $\mathbb{F}$ are the central elements of the universal enveloping algebra $U(L)$. Example: The angular momentum square $\vec{J}^2$ is a quadratic Casimir invariant of the Lie algebra $L=sl(2,\mathbb{C})$. II) Given a bilinear associative/invariant form $B:L\times L\to \mathbb{F}$, one can create ...

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OP wrote (v1): What does "the ${\bf N}$ of a group" mean? 1) Physicists are referring to an irreducible representation (irrep) for whatever group $G$ we are talking about. The number ${\bf N}$ refers to the dimension of the irrep. The point is that irreps are so rare that irreps are often uniquely specified by their dimension (modulo isomorphisms). ...

10

By definition, the metric tensor $\eta_{ij}$ transforms trivially under the defining rep of $SO(n,m)$. $$\eta_{ij}=[D(g^{-T})]_{i}^{\ k}[D(g^{-T})]_{j}^{\ l}\eta_{kl} =[D(g^{-1})]^{k}_{\ i}[D(g^{-1})]^{l}_{\ j}\eta_{kl}$$ and this holds for all $g\in SO(n,m)$. Consider a one-parameter subgroup of the defining rep with matrices $D(g)=e^{tJ}$ where ...

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Here we will sketch a possible derivation. Let $\eta\in {\rm Mat}_{n\times n}(\mathbb{R})$ be a real symmetric matrix of signature $(p,q)$, where $n=p+q$. Define the Lie group $$O(p,q)~:=~ \{ \Lambda\in {\rm Mat}_{n\times n}(\mathbb{R}) \mid \Lambda^T\eta \Lambda = \eta \},$$ where $\Lambda^T$ denotes the transposed $\Lambda$ matrix. Prove for fun that ...

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This is a note on why angular velocities are vectors, to complement Matt and David's excellent explanations of why rotations are not. When we say something has a certain angular velocity $\vec{\omega_1}$, we mean that each part of the thing has a position-dependent velocity $\vec{v_1}(\vec{r}) = \vec{\omega_1} \times \vec{r}$. We might consider another ...

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