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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 ...
13
votes
2answers
64 views
Uniqueness of supersymmetric heterotic string theory
Usually we say there are two types of heterotic strings, namely $E_8\times E_8$ and $Spin(32)/\mathbb{Z}_2$. (Let's forget about non-supersymmetric heterotic strings for now.)
The standard argument ...
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 ...
10
votes
3answers
822 views
Adding 3 electron spins
I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers ...
9
votes
4answers
298 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
1answer
121 views
Why do we classify states under covering groups instead of the group itself?
Why do we always classify states under covering group representations instead of the group itself? For example see the following picture I lifted from 'Symmetry in physics' by Gross
So in the first ...
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 ...
7
votes
2answers
322 views
Groups acting on physics - a clarification on electrons and spin
My first question is fairly basic, but I would like to clarify my understanding. The second question is to turn this into something worth answering.
Consider a relativistic electron, described by a ...
7
votes
5answers
306 views
The role of representation theory in QM/QFT?
I need help understanding the role of representation theory in QM/QFT. My understanding of representation theory in this context is as follows: there are physical symmetries of the system we are ...
7
votes
1answer
557 views
Representation of the Galileo Group and Central Charges
I've arrived at this question because I've been reading Weinberg's Quantum Theory of Fields Volume I, and I'm in the second chapter about relativistic quantum mechanics. Weinberg discusses the ...
7
votes
1answer
332 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}^* / ...
7
votes
0answers
304 views
Wick rotation and spinors
I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; ...
6
votes
1answer
400 views
Modes of a QFT and irreducible representation of the gauge group
This is in reference to the calculation in section 3.3 starting page 20 of this paper.
I came across an argument which seems to say that the "constraint of Gauss's law" enforces gauge theory on ...
6
votes
1answer
128 views
Are group representations possible when the solution space is not a vector space?
As far as I understand, the motivation for using representation theory in high energy physics is as follows. Assume that a theory has some (internal or external) symmetry group which acts on a vector ...
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
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 ...
5
votes
1answer
216 views
Identification of the state of particle types with representations of Poincare group
In the second chapter of the first volume of his books on QFT, Weinberg writes in the last paragraph of page 63:
In general, it may be possible by using suitable linear combinations of the ...
5
votes
1answer
99 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 $$
And it is ...
5
votes
1answer
264 views
Wigner-Eckart theorem of SU(3)
I have just come across the Wigner-Eckart theorem and am not sure on how to apply it. How do I find the matrix elements of $\langle u|T_a|v\rangle$ in terms of tensor components and the Gell-Mann ...
5
votes
2answers
196 views
Are there irreducible tensors of half integral degree in quantum mechanics?
According to Ballentine, an irreducible tensor of degree k can be defined as a set of $2k + 1$ operators $\{T_q^{\;\;(k)}:(-k \le q \le k)\}$ satisfying the following commutation relations:
$$
...
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
2answers
373 views
Particle as a representation of the Lorentz group
In QFT one may refer to a particle as a representation of the Lorentz group (LG). More accurately - every particle is a quantum of some field $\phi(x)$ that belongs to some representation of the LG. I ...
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
2answers
185 views
If the S-matrix has symmetry group G, must the fields be representations of G?
If the fields in QFT are representations of the Poincare group (or generally speaking the symmetry group of interest), then I think it's a straight forward consequence that the matrix elements and ...
5
votes
0answers
122 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 ...
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
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 ...
4
votes
2answers
238 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 ...
4
votes
2answers
206 views
Calculating the commutator of Pauli-Lubanski operator and generators of Lorentz group
The Pauli-Lubanski operator is defined as
$${W^\alpha } = \frac{1}{2}{\varepsilon ^{\alpha \beta \mu \nu }}{P_\beta}{M_{\mu \nu }},\qquad ({\varepsilon ^{0123}} = + 1,\;{\varepsilon _{0123}} = - ...
4
votes
1answer
124 views
What is the definition of precession (in the context of Spinors)?
What is the definition of "precession"? How is it applicable to abstract objects such as Spinors? I understand the mathematics, but don't understand what one means by "precession angle" etc when it ...
3
votes
3answers
269 views
The Asymmetry between Real and Imaginary in the three Pauli Spin Matrices
The Pauli spin matrices
$$
\sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}),
\qquad\qquad
\sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 ...
3
votes
2answers
201 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 ...
3
votes
2answers
409 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
90 views
Supersymmetry and non-compact $R$-symmetry group?
The $R$-symmetry for $N$ supercharges is $U(N)$. Is it possible to generalize $R$-symmetry [let's take $U(4)$) to be something like $U(2,2)$ (maybe analogous to Wick rotation of $SO(3,1)$ to ...
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
1answer
710 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 ...
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
2answers
93 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 ...
3
votes
2answers
210 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
$$ \underbrace{\partial^\mu ...
3
votes
1answer
251 views
Angular Momentum Addition Theorem - Sanity Check
Looking back at my quantum mechanics notes, the angular momentum addition theorem is listed as:
$j=j_1+j_2,j_1+j_2-1, ..., |j_1-j_2| $ (Using conventional notation)
, but I'm a little unsure how to ...
3
votes
1answer
204 views
How do I find the tensor components of all weights of a representation of SU(3), e.g. the six dimensional representation (2,0)
How do I find the corresponding tensor component v^ij of the six dimensional representation of SU(3) with dynkin label (2,0).
3
votes
2answers
352 views
Lorentz transformations in Dirac equation
Let's denote a spinor $\xi$. If $(\theta ,\phi)$ are the parameters of a rotation and pure Lorentz transformation, then how $\xi$ could be written as
$$\xi ~\rightarrow~ \exp\left(\ i ...
3
votes
2answers
243 views
A question from Weinberg QFT text
In page 71 Weinberg's QFT,
$$A\Psi^{\theta }_{a,b}
~=~(a\cos{(\theta )}-b\sin{(\theta )})\Psi^{\theta }_{a,b}.$$
He says that massless particles represented by $\Psi ^{\theta }_{a,b}$ are not ...
3
votes
1answer
162 views
Why is there a phase factor when the two composite angular momentum is exchanged in Clebsch–Gordan coefficients
An identity exists for CG coefficients:
$$\langle j_1 m_1 j_2 m_2 |J M \rangle = (-1)^{j_1+j_2-J} \langle j_2 m_2 j_1 m_1|J M\rangle,$$
But why is there a phase factor $(-1)^{j_1+j_2-J}$?
It seems ...
3
votes
1answer
253 views
Does a spin-2 particle really return to its previous state after 180° rotation?
It is often claimed that spin-2 particles return to their previous state after $\pi$ rotation, just like spin-1/2 particles return after $4\pi$ rotation. But my calculation suggests otherwise.
Let z ...
3
votes
1answer
411 views
Introduction to Physical Content from Adjoint Representations
In particle Physics it's usual to write the physical content of a Theory in adjoint representations of the Gauge group. For example:
$24\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus ...
3
votes
1answer
38 views
A nice overview (and maybe derivation) of the Poincaré transformations of the Vector Spherical Harmonics
With $Y_{lm}(\vartheta,\varphi)$ being the Spherical Harmonics and $z_l^{(j)}(r)$ being the Spherical Bessel functions ($j=1$), Neumann functions ($j=2$) or Hankel functions ($j=3,4$) defining ...
2
votes
3answers
322 views
Spin decomposition in general
I can turn-the-crank and show that $\frac{1}{2}\otimes \frac{1}{2} = 1\oplus 0$ etc, but what would be a strategy to proving the general statement for spin representations that $j\otimes s ...
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 $\mathbf{SU}(N-1)$, ...
