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. ...
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191 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, ...
4
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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 ...
4
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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 ...
3
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2answers
411 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
2answers
130 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
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1answer
91 views
Different representations of the Lorentz algebra
I've found many definitions of Lorentz generators that satisfy the Lorentz algebra: ...
3
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1answer
715 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
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2answers
212 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
$$ \underbrace{\partial^\mu ...
3
votes
2answers
427 views
Is the Lorentz group compact (and if not, is U(1)?)
A common statement in any quantum field theory text is that only compact groups have finite-dimensional representations, and that the Lorentz group is not compact, since it is parameterised by $0\leq ...
3
votes
1answer
204 views
How do I find the tensor components of all weights of a representation of SU(3), e.g. the six dimensional representation (2,0)
How do I find the corresponding tensor component v^ij of the six dimensional representation of SU(3) with dynkin label (2,0).
3
votes
1answer
164 views
What's a pseudo-rotation?
I'm sorry for this lexical, probably extremely elementary, question. But what is a pseudo-rotation? I just read this term for the first time, in the beginning of the 4th chapter book of CFT by Di ...
3
votes
1answer
57 views
Isometry group from information about the center of the group
I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to ...
3
votes
1answer
146 views
An odd relation with the epsilon/delta invariant tensors of SO(3)
The rotation group SO(3) can be viewed as the group that preserves our old friends the delta tensor $\delta^{ab}$ and $\epsilon^{abc}$ (the totally antisymmetric tensor). In equations, this says:
...
3
votes
1answer
172 views
Linearizing Quantum Operators
I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator.
$$ \hat{A}={\langle}\hat{A}{\rangle}I+ \Delta\hat{a}$$
The right hand side ...
3
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0answers
62 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 ...
2
votes
2answers
134 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?
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
349 views
Generators of the lorentz group
I have the following minus sign problem:
Consider an infinitesimal Lorentz transformation for which $\Lambda^{\mu}_{\nu}=\delta^{\mu}_{\nu}+\lambda^{\mu}_{\nu}$, where $\lambda^{\mu}_{\nu}$ is ...
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 ...
1
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2answers
678 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, ...
1
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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
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2answers
141 views
Why is the crystallography restriction obeyed?
The crystallography restriction states that any 2-dimensional lattice can have rotational symmetry of degree 1, 2, 3, 4 or 6 - and that's it. A simple proof of I've heard of this is: the magnitude of ...
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)$. ...
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 ...
1
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2answers
180 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 ...
1
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1answer
80 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 ...
1
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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 ...
1
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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)$?
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
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0answers
70 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
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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
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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
1answer
105 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 ...
1
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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)?
1
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0answers
47 views
Books on representation theory [duplicate]
Possible Duplicate:
Best books for mathematical background?
I'm looking for a textbook on the group/representation theory for a student-physicist. The main questions of interest are ...
1
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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:
$$
...
0
votes
2answers
1k views
Wyckoff positions and lattice coordinations
Many papers use Wyckoff positions and Space Groups to report atom coordinates, making use of the structure's symmetries to save space in the notation (e.g. diamond = Fd-3m, only).
How can I ...
0
votes
3answers
376 views
Textbook on group theory to be able to start QFT
I am very enthusiastic about learning QFT. How much group theory would I need to master? Please could you recommend me a textbook on group theory, which would help me to start QFT?
0
votes
1answer
56 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 ...
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 ...
0
votes
0answers
175 views
What is the relationship between the structures of music and the universe? [closed]
Pitch sequences are classificed disregarding the effects of five musical transformations: octave shift, permutation, transposition, inversion, and cardinality change. These may be modelled according ...
