The lie-algebra tag has no wiki summary.
11
votes
1answer
214 views
Covariant derivatives
I need correctly define covariant derivatives on the coset space $G/H$, where a group $G \equiv \{X_i, Y_a\}$ ($X$ and $Y$ are generators) have a subrgroup $H \equiv \{X_i\}$
Lie algebra of $G$ has ...
10
votes
2answers
512 views
Is the G2 Lie algebra useful for anything?
Seems like all the simpler Lie algebras have a use in one or another branch of theoretical physics. Even the exceptional E8 comes up in string theory. But G2? I've always wondered about that one. ...
10
votes
2answers
85 views
When are there enough Casimirs?
I know that a Casimir for a Lie algebra $\mathfrak{g}$ is a central element of the universal enveloping algebra. For example in $\mathfrak{so}(3)$ the generators are the angular momentum operators ...
9
votes
4answers
297 views
Trace and adjoint representation of $SU(N)$
In the adjoint representation of $SU(N)$, the generators $t^a_G$ are chosen as
$$ (t^a_G)_{bc}=-if^{abc} $$
The following identity can be found in Taizo Muta's book "Foundations of Quantum ...
8
votes
2answers
345 views
Lie bracket for Lie algebra of $SO(n,m)$
How does one show that the bracket of elements in the Lie algebra of $SO(n,m)$ is given by
$$[J_{ab},J_{cd}]
~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$
...
8
votes
1answer
250 views
Why does a transformation to a rotating reference frame NOT break temporal scale invariance?
Naively, I thought that transforming a scale invariant equation (such as the Navier-Stokes equations for example) to a rotating reference frame (for example the rotating earth) would break the ...
7
votes
2answers
141 views
Embedding of $F(4)$ in $OSp(8|4)$?
Is the superconformal algebra in five dimensions, $F(4)$, a subalgebra of the (maximal) six-dimensional superconformal algebra $OSp(8|4)$?
7
votes
2answers
181 views
How to model a symmetry using Lie Groups?
I have been reading lately about Lie groups, and although all books keep listing the groups, and talk about Lie algebras and all that, one thing I still don't know how is it made, and I guess it's the ...
6
votes
2answers
158 views
Is this a simple Lie algebra?
This question comes from Georgi, Lie Alegbras in Particle Physics. Consider the algebra generated by $\sigma_a\otimes1$ and $\sigma_a\otimes \eta_1$ where $\sigma_a$ and $\eta_1$ are Pauli matrices ...
6
votes
1answer
144 views
Equivalent Representations of Clifford Algebra
I'm reviewing David Tong's excellent QFT lecture notes here and am a little confused by something he writes on page 94.
We've considered the standard chiral representation of the Clifford Algebra, ...
5
votes
1answer
264 views
Wigner-Eckart theorem of SU(3)
I have just come across the Wigner-Eckart theorem and am not sure on how to apply it. How do I find the matrix elements of $\langle u|T_a|v\rangle$ in terms of tensor components and the Gell-Mann ...
5
votes
1answer
343 views
Simultaneously commuting set
How does one determine the members of an simultaneously commuting set (of operators)? For example, I have read that for orbital angular momentum, the set is {$H,L^2,L_z$}. How does one know that these ...
5
votes
1answer
364 views
Could I see the quantum states as representations of the Galilei algebra (or Galilei group)?
In somes references of Relativistic Quantum Mechanics, the one-particle states are given by representation theory of Poincaré algebra.
Could I mimics this for the non-relativistic case? States in ...
5
votes
0answers
186 views
Coupling Coefficients in SO(4)
I have two equations (from two distinct authors) for the decomposition of a coupling coefficient of SO(4) (i.e. Wigner 3j-symbol for SO(4)). In the first:
...
4
votes
2answers
217 views
Definition of Casimir operator and its properties
I'm not sure which is the exact definition of a Casimir operator.
In some texts it is defined as the product of generators of the form:
$$X^2=\sum X_iX^i$$
But in other parts it is defined as an ...
4
votes
1answer
259 views
Why the Hamiltonian near degeneracies should be proportional to Pauli matrices?
When a quantum system have a double degenerescence at one point, the Hamiltonian should be proportional to Pauli matrices near this point (also known as diabolic point) [Ref.]. But, why the ...
4
votes
1answer
191 views
Holstein-Primakoff and Dyson-Maleev representation
In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In ...
4
votes
2answers
212 views
Equivalent Rotation using Baker-Campbell-Hausdorff relation
Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering ...
3
votes
3answers
269 views
The Asymmetry between Real and Imaginary in the three Pauli Spin Matrices
The Pauli spin matrices
$$
\sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}),
\qquad\qquad
\sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 ...
3
votes
2answers
408 views
How does non-Abelian gauge symmetry imply the quantization of the corresponding charges?
I read an unjustified treatment in a book, saying that in QED charge an not quantized by the gauge symmetry principle (which totally clear for me: Q the generator of $U(1)$ can be anything in ...
3
votes
1answer
87 views
Different representations of the Lorentz algebra
I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
3
votes
3answers
200 views
Quantum mechanical angular momentum and spin formalism/notation
I am currently stuck on the following notation:
$\frac{1}{2}\otimes\frac{1}{2} = 0 \text{ (antisym) } \oplus 1 \text{ (sym) }$
No matter what I tried, I couldn't derive the identity. I am sure that ...
3
votes
3answers
423 views
Building the meson octet and singlet
I am very lost in this topic. I understand that there are $3\times 3$ possible combinations of a quark and an anti-quark, but why should one decide arbitrarily (that's how it appears to me) that one ...
3
votes
1answer
204 views
How do I find the tensor components of all weights of a representation of SU(3), e.g. the six dimensional representation (2,0)
How do I find the corresponding tensor component v^ij of the six dimensional representation of SU(3) with dynkin label (2,0).
3
votes
1answer
90 views
Mathematically, how do we deduce that angular momentum is bounded?
So, how do we know $J_{+}|j,(m=j)\rangle =|0\rangle$?
I.e. that m is bounded by j.
We know that $J_{+}|j,(m=j)\rangle =C|j, j+1\rangle$, but how do I know that gives zero? Is it by looking at its ...
3
votes
1answer
331 views
Why is the value of spin +/- 1/2?
I understand how spin is defined in analogy with orbital angular momentum. But why must electron spin have magnetic quantum numbers $m_s=\pm \frac{1}{2}$ ? Sure, it has to have two values in ...
3
votes
1answer
524 views
Angular momentum coupling-calculation of Clebsch–Gordan coefficients
I am facing problem in calculating the value of given Clebsch–Gordan coefficients representing the coupled angular momenta of two-particle system. For example
$$\begin{pmatrix}2 & 1 & 2 \\ 1 ...
3
votes
0answers
116 views
Symmetries of separable potential
For separable potential, say $x^4+y^4$, its symmetry are degenerate.
Is that a generic case to every separable potential? I will explain my question:
The potential $x^4+y^4$ has $A_1, B_1, A_2, B_2, ...
3
votes
1answer
154 views
Question about the parity of the ghost number operator in BRST quantization
Given a Lie algebra $[K_i,K_j]=f_{ij}^k K_k$, and ghost fields satisfying the anticommutation relations $\{c^i,b_j\}=\delta_j^i$, the ghost number operator is then $U=c^ib_i$ (duplicate indices are ...
2
votes
2answers
105 views
Representations of Lie algebras in physics
Why is an invariant vector subspace sometimes called a representation? For example in Lie algebras, say su(3), the subspace characterized by the highest weight (1,0) is an irreducible representation ...
2
votes
2answers
144 views
In quantum mechanics(QM), can we define a high-dimensional “spin” angular momentum other than the ordinary 3D one?
Inspired by my previous question Questions about angular momentum and 3-dimensional(3D) space? and another relevant question How to define angular momentum in other than three dimensions? , now I get ...
2
votes
1answer
364 views
SU(N) symmetry and its representations
If a Lagrangian containing an N-multiplet of fields is invariant under global $\mathbf{SU}(N)$ transformations, does that necessarily imply it is invariant under $\mathbf{SU}(N-1)$, ...
2
votes
2answers
120 views
Quantization of orbital angular momentum
Probably a very simple question, but I can't find the answer on the Internet.
I know nearly to nothing about quantum mechanics, but in statistical physics I'm confronted with the idea that the orbital ...
2
votes
2answers
87 views
Do generators belong to the Lie group or the Lie algebra?
In Physics papers, would it be correct to say that when there is mention of generators, they really mean the generators of the Lie algebra rather than generators of the Lie group? For example I've ...
2
votes
1answer
71 views
Charge of a field under the action of a group
What does it mean for a field (say, $\phi$) to have a charge (say, $Q$) under the action of a group (say, $U(1)$)?
2
votes
1answer
129 views
Similar masses and lifetimes of the $\Delta$ baryons
Why do the four spin 3/2 $\Delta$ baryons have nearly identical masses and lifetimes despite their very different $u$ and $d$ quark compositions?
2
votes
1answer
144 views
Dirac trace theorem
I am unable to prove exactly one trace identity that appears in the appendix of Peskin and Schroeder's QFT book. Can someone help me?
The theorem [Appendix A.4 eqn (A.28)] says that the order of ...
2
votes
1answer
157 views
What is the Lie algebra of the Galilean group and what is the structure of it?
I read Freeman Dyson's article Missed Opportunities, in which he talked about the mathematical attractiveness of the Lorenz group compared to the Galilean group. I am reading Florian Scheck's book on ...
2
votes
1answer
477 views
General procedure for Clebsch-Gordan expansions
I'm wondering if the Clebsch-Gordan series generalize to any orthonormal set of basis functions? If so, how would one go about deriving an expression for an arbitrary set of basis functions (perhaps ...
2
votes
1answer
98 views
Taylor series for unitary operator in Weinberg
On page 54 of Weinberg's QFT I, he says that an element $T(\theta)$ of a connected Lie group can be represented by a unitary operator $U(T(\theta))$ acting on the physical Hilbert space. Near the ...
2
votes
3answers
402 views
Hilbert space and Lie algebra in quantum mechanics
We are looking for a publication or website that explains the Standard Model in terms of Hilbert space and Lie algebra.
We are reading Debnath's Introduction to Hilbert Spaces and Applications and ...
2
votes
0answers
108 views
Derivation of the enhancement of U(1)$_L$ x U(1)$_R$ to SU(2)$_L$ x SU(2)$_R$ at the self-dual radius
Towards the end of the paragraph with the title String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract C with current algebras of this article, it is explained ...
1
vote
1answer
255 views
given the infinitesimal generator can we deduce the symmetry?
for example given the differential operator $ \partial _{x} $ we know that it belong to the translation $ y=x+a $ for some 'a'
given the differential operator $ x\partial _{x} $ we know that we are ...
1
vote
1answer
178 views
Commutators with a density matrix
The equation describing the evolution of our system is as follows:
$ \dot{\rho} = u_1(t)(a^\dagger a \rho - 2a\rho a^\dagger +\rho a^\dagger a) + u_2(t)(a a^\dagger \rho - 2a^\dagger\rho a +\rho a ...
1
vote
2answers
164 views
high spin atoms SU(2) representation
I am very confused that some atoms called high spin or magnetic atoms have spin level more than $\frac{1}{2}$ but are still said to have $SU(2)$ symmetry.
Why not $SU(N)$?
1
vote
1answer
246 views
Wigner-Eckart projection theorem
I'm following the proof of Wigner-Eckart projection theorem which states that:
$$\langle \bf{A} \rangle ~=~ \frac{\langle \bf{A} \cdot \bf{J} \rangle}{\langle {\bf{J}}^2 \rangle} \langle \bf{J} ...
1
vote
1answer
368 views
Weinberg's way of deriving Lie algebra related to a Lie group
I was reading the second chapter of the first volume of Weinberg's books on QFT. I am quite confused by the way he derives the Lie algebra of a connected Lie group.
He starts with a connected Lie ...
1
vote
1answer
41 views
Please provide the simplest example you can think of, of generators of time evolution and generalized coordinates
I was reading the Wikipedia article about Noether's theorem and this thing popped out:
Then the resultant perturbation can be written as a linear sum of the
individual types of perturbations
...
0
votes
1answer
629 views
Killing vector fields
I am facing some problems in understanding what is the importance of a Killing vector field? I will be grateful if anybody provides an answer, or, refer me to some review or books.
