Tagged Questions
3
votes
2answers
104 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 ...
5
votes
1answer
102 views
Vector and Spinor Representation in Ramond-Neveu-Schwarz Superstring Theory
I am learning Ramnond-Neveu-Schwarz Superstring theory (RNS theory). I often find the following notation, especially in the closed string spectrum etc.:
$$\mathbf{8}_s,\mathbf{8}_v $$
And it is ...
5
votes
0answers
124 views
Decomposing a Tensor Product of $SU(3)$ Representations in Irreps
Can somebody explain in a simple way why, talking about representations, $3\otimes3=3\oplus6$, $3\otimes\bar{3}=1\oplus8$ and $3\otimes3\otimes3=1\oplus8\oplus8\oplus10$?
Here $3$ and $\bar{3}$ are ...
3
votes
2answers
211 views
Why does $\mathcal L = -\frac14 F^{\mu\nu} F_{\mu\nu}$ imply Photons are massless?
The Lagrangian $\mathcal L = -\frac14 F^{\mu\nu} F_{\mu\nu}$ with $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ results in the four-potential's equation of motion
$$ \underbrace{\partial^\mu ...
1
vote
1answer
114 views
Action of the Lorentz group on scalar fields
The Lorentz groups act on the scalar fields as:
$\phi'(x)=\phi(\Lambda^{-1} x)$
The conditions for an action of a group on a set are that the identity does nothing and that
$(g_1g_2)s=g_1(g_2s)$. ...
4
votes
2answers
239 views
Number of Components of a Spinor
I'm trying to develop my understanding of spinors. In quantum field theory I've learned that a spinor is a 4 component complex vector field on Minkowski space which transforms under the chiral ...
1
vote
2answers
165 views
high spin atoms SU(2) representation
I am very confused that some atoms called high spin or magnetic atoms have spin level more than $\frac{1}{2}$ but are still said to have $SU(2)$ symmetry.
Why not $SU(N)$?
4
votes
2answers
206 views
Calculating the commutator of Pauli-Lubanski operator and generators of Lorentz group
The Pauli-Lubanski operator is defined as
$${W^\alpha } = \frac{1}{2}{\varepsilon ^{\alpha \beta \mu \nu }}{P_\beta}{M_{\mu \nu }},\qquad ({\varepsilon ^{0123}} = + 1,\;{\varepsilon _{0123}} = - ...
8
votes
1answer
331 views
Schwinger representation of operators for n-particle 2-mode symmetric states
A bosonic (i.e. permutation-symmetric) state of $n$ particles in $2$ modes can be written as a homogenous polynomial in the creation operators, that is
$$\left(c_0 \hat{a}^{\dagger n} + c_1 ...
4
votes
2answers
161 views
Are there any known potentially useful nontrivial irreducible representations of the Lorentz Group $O(3,1)$ of dimension bigger than 4? Examples?
Are there any known potentially useful, nontrivial, irreducible representations of the Lorentz Group $O(3,1)$ of dimension more than $4$? Examples? A $5$-dimensional representation? EDIT: Is there ...
2
votes
1answer
117 views
How do representations of an isometry group correspond to degrees of freedom/entropy in a system?
To put the question into context: I am currently writing my bachelors thesis on de Sitter space, specifically, $dS_4$. I am trying to show that while the horizon entropy is finite, the isometry group ...
17
votes
4answers
1k views
Could the Periodic Table have been done using group theory?
These three questions are phrased as alternative-history questions, but my real intent is to understand better how well different modeling approaches fit the phenomena they are used to describe; see 1 ...
7
votes
1answer
334 views
Is this a quaternion representation of the equations of motion of General Relativity?
In The Quaternion Group and Modern Physics by P.R. Girard, the quaternion form of the general relativistic equation of motion is derived from
$du'/ds = (d a / d s ) u {a_c}^* + a u ( d {a_c}^* / ...
5
votes
2answers
473 views
Is this a quaternion Lorentz Boost?
The quaternion Lorentz boost $v'=hvh^*+ 1/2( (hhv)^*-(h^*h^*v)^*)$ where $h$ is $(\cosh(x),\sinh(x),0,0)$ was derived by substituting the hyperbolic sine and cosine for the sine and cosine in the ...
1
vote
0answers
40 views
How to obtain deconfined theory from an s-confined N=1 susy gauge theory?
Is there a systematic procedure for obtaining a deconfined theory from an s-confining theory (as defined in hep-th/9610139 for example)?
3
votes
1answer
714 views
How do I construct the $SU(2)$ representation of the Lorentz Group using $SU(2)\times SU(2)\sim SO(3,1)$ ?
This question is based on problem II.3.1 in Anthony Zee's book Quantum Field Theory in a Nutshell (I'm reading this for fun- it isn't a homework problem.)
Show, by explicit calculation, that ...
5
votes
1answer
266 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 ...
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).
1
vote
1answer
167 views
Irreducible tensor representations with “covariant” indices
As a follow-up of my question on the "most general" $\mathrm{SU}(2)$-symmetric interaction of two spin 1/2 particles, I ponder the following question:
Consider an operator acting just on one particle ...
1
vote
2answers
677 views
Irreducible representation in physics
I am confused about something.
Group theory books written for physicists say that any reducible representation can be decomposed in terms of irreducible representations (so correct me if I am wrong, ...
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)$)?
3
votes
2answers
410 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 ...
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:
...
1
vote
0answers
62 views
What is an isoscalar factor?
I need to find a definition for "the isoscalar factors of 3j-symbols for the restriction $SO(n)\supset SO(n-1)$...denoted by brackets with a composite subscript $(n: n-1)$..." They are given as:
$$
...
10
votes
2answers
515 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. ...
