Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. ...
2
votes
2answers
58 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 ...
4
votes
2answers
216 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 ...
3
votes
1answer
87 views
Different representations of the Lorentz algebra
I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
1
vote
1answer
77 views
Spin(n) group SO(n) relation
Is it correct to state that the elements of Spin(n) fulfill a Clifford algebra and that the Lie group generators of Spin(n) is given by the commutator of the elements?
If not, then what is the ...
6
votes
1answer
158 views
A Game Of The Number Of Space-Time Dimensions
Holger Bech Nielsen, one of the founders of string theory, has apparently just played some sort of game between different potential dimensions for space-time and reached the conclusion that D4 wins in ...
4
votes
0answers
188 views
Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry
Background: Classical Mechanics is based on the Poincare-Cartan two-form
$$\omega_2=dx\wedge dp$$
where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, ...
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
0answers
147 views
What is the difference between the properties of Electron spin and Photon polarization/helicity?
What is the difference between a photon's polarization/helicity and an electrons spin half? I know that the photon is spin 1 but isn't its polarization analogous to spin half?
This question stems ...
5
votes
0answers
121 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 ...
4
votes
0answers
75 views
Is the search for a Simple-group-based Electro-Weak theory over?
Just wondering:
We know that, in its current form of the $SU(2)_L\times U(1)$, the electroweak theroy rides a wave of huge success. However, is it not possible that the correct simple group ...
2
votes
2answers
133 views
Two ways to form SU(2) singlets?
I am trying to reconcile the two ways of forming SU(2) singlets out of a pair of doublets.
Method (1):
If $v=\begin{pmatrix}v^1\\ v^2\end{pmatrix}$ and $w=\begin{pmatrix}w^1\\ w^2\end{pmatrix}$ are ...
2
votes
3answers
118 views
How to judge whether a symmetry will be spontaneously broken while only given a Hamiltonian preserving this symmety
As asked in the title, is Hamiltonian containing enough information to judge the existence of spontaneously symmetry breaking?
Any examples?
6
votes
1answer
102 views
What is the meaning of non-compactness in the context of $U(1)$ in gauge theories?
In John Preskill's review of monopoles he states
Nowadays, we have another way of understanding why electric charge is
quantized. Charge is quantized if the electromagnetic U(l)em gauge
group ...
3
votes
0answers
61 views
Finding symmetry of a part of an equation, given the group transformation property of another part
I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to ...
3
votes
1answer
57 views
Isometry group from information about the center of the group
I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to ...
15
votes
4answers
489 views
Elegant approaches to quantum field theory
I have been reading Quantum Mechanics: A Modern Development by L. Ballentine. I like the way everything is deduced starting from symmetry principles. I was wondering if anyone familiar with the book ...
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
$$ \underbrace{\partial^\mu ...
1
vote
0answers
69 views
Deriving term symbols from electron configuration using Young tableaux
Can somebody explain me how to derive all term symbols using Young tableaux? Our lecturer showed us but I couldn't quite understand it without any background on group theory. I have some vague ...
1
vote
1answer
114 views
Action of the Lorentz group on scalar fields
The Lorentz groups act on the scalar fields as:
$\phi'(x)=\phi(\Lambda^{-1} x)$
The conditions for an action of a group on a set are that the identity does nothing and that
$(g_1g_2)s=g_1(g_2s)$. ...
8
votes
1answer
279 views
Representations of Lorentz Group
I'd be grateful if someone could check that my exposition here is correct, and then venture an answer to the question at the end!
$SO(3)$ has a fundamental representation (spin-1), and tensor product ...
4
votes
2answers
238 views
Number of Components of a Spinor
I'm trying to develop my understanding of spinors. In quantum field theory I've learned that a spinor is a 4 component complex vector field on Minkowski space which transforms under the chiral ...
7
votes
2answers
181 views
How to model a symmetry using Lie Groups?
I have been reading lately about Lie groups, and although all books keep listing the groups, and talk about Lie algebras and all that, one thing I still don't know how is it made, and I guess it's the ...
1
vote
1answer
98 views
Commutation relation of $J^2$ and $R(\alpha,\beta,\gamma)$
If $R(\alpha,\beta,\gamma)$ is the Rotation operator and $\alpha,\beta,\gamma$ are Euler angles and $J$ is the total angular momentum then how to get to this:
$$[J^2,R]~=~0?$$
This is stated in ...
13
votes
3answers
338 views
Homotopy $\pi_4(SU(2))=\mathbb{Z}_2$
Recently I read a paper using $$\pi_4(SU(2))=\mathbb{Z}_2.$$ Do you have any visualization or explanation of this result?
More generally, how do physicists understand or calculate high dimension ...
1
vote
2answers
164 views
high spin atoms SU(2) representation
I am very confused that some atoms called high spin or magnetic atoms have spin level more than $\frac{1}{2}$ but are still said to have $SU(2)$ symmetry.
Why not $SU(N)$?
1
vote
0answers
46 views
Wigner $3j$ symbols
I am trying to determine the expansion that requires using $3j$ symbols; however, I am running into some conceptual snags. First, the expansion produces symbols that have m's that do not agree with ...
1
vote
2answers
177 views
What does “the N of a group” mean?
In the context of group theory (in my case, applications to physics), I frequently come across the phrase "the N" of a group, for example "a 24 of SU(5)" or "the 1" (the integer is usually typeset in ...
4
votes
2answers
205 views
Calculating the commutator of Pauli-Lubanski operator and generators of Lorentz group
The Pauli-Lubanski operator is defined as
$${W^\alpha } = \frac{1}{2}{\varepsilon ^{\alpha \beta \mu \nu }}{P_\beta}{M_{\mu \nu }},\qquad ({\varepsilon ^{0123}} = + 1,\;{\varepsilon _{0123}} = - ...
4
votes
4answers
866 views
How to prove that proper orthochronous Lorentz transformations form a group?
Proper orthochronous Loentz transform are Lorentz transforms that satisfy the conditions (sign convention of Minkowskian metric $+---$)
$$\det \Lambda=+1, \qquad \Lambda^0{}_0 \geq +1.$$
How to prove ...
1
vote
1answer
146 views
Proof of Pauli group preservation by Clifford group conjugation?
A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words:
$C(P_{1} \otimes P_{2})C^{\dagger} = P_{3} \otimes P_{4}$, with $C \in$ Clifford group and ...
1
vote
0answers
61 views
Matrix separability preservation under conjugation?
Someone know any paper about matrix separability preservation under conjugation? A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words:
$C(P_{1} ...
1
vote
3answers
176 views
Anybody have example of two-qubit non-Pauli and non-Clifford quantum gate?
A lot of known quantum gates are in the Pauli group (I,X,Z,Y) or in the Clifford group (H,P,Cnot). I need examples of the quantum gates that aren't in this groups. Also, are there are matlab functions ...
0
votes
1answer
99 views
Does anyone know the difference and relation between $k\cdot p$ method and tight binding (TB) method?
Among the methods of calculating energy bands for crystals, first-principles method is the most accurate. Besides first principles, two commonly used modeling methods are the $k\cdot p$ method and ...
4
votes
1answer
327 views
representation of SU(2)
The question is regarding SU(2) group and SU(2) algebra.
The SU(2) group can be generated by exponentiating the
generators of SU(2) algebra $X_a$ as $exp(i t_a X_a )$
with $t_a$ being three ...
0
votes
1answer
55 views
Proper notation when working with three Euclidean spatial coordinates in a setting with a time parameter
The How does the Euclidean metric is the symmetry group of Euclidean space. It includes rotations and translations.
Say I consider an Euclidean space and a time parameter. How does the Euclidean ...
4
votes
3answers
373 views
Must all symmetries have consequences?
Must all symmetries have consequences?
We know that transnational invariance, for example, leads to momentum conservation, etc, cf. Noether's Theorem.
Is it possible for a theory or a model to have ...
0
votes
0answers
30 views
How to learn relevant group theory for fundamental physics [duplicate]
Possible Duplicate:
Comprehensive book on group theory for physicists?
I don't know a lot about group theory, and that is preventing me from understanding a lot of physics.
Group theory is ...
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 ...
5
votes
3answers
245 views
Could general relativity and gauge theories in principle be covered in one course?
It's always nice to point out the structural similarieties between (semi-)Riemannian geometry and gauge field theories alla Classical yang Mills theories. Nevertheless, I feel the relation between the ...
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 ...
4
votes
1answer
362 views
Why does photon have only two possible eigenvalues of helicity
Photon is a spin-1 particle. Were it massive, its spin projected along some direction would be either 1, -1, or 0. But photons can only be in an eigenstate of $S_z$ with eigenvalue $\pm 1$ (z as 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}^* / ...
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 ...
6
votes
2answers
297 views
Do Lorentz Boosts in the same direction form a group?
I know that two consecutive Lorentz Boosts in different directions produce a rotation and therefore Lorentz Boosts don't form a group. But, my intuition tells me that, Lorentz Boosts in the same ...
4
votes
0answers
85 views
Calabi Yau compactification based on U(1) charges
In Green-Schwarz-Witten Volume 2, chapter 15, it is argued (roughly) that we need 6-dimensional manifolds of $SU(3)$ holonomy in order to receive 1 covariantly constant spinor field. And it turns out ...
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}),$$
...
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
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 ...
