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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95 views

Permutations of two identical particles in two dimensions

In three spatial dimensions there are only two possible statistics: Bose-Einstein and Fermi-Dirac. This is the fact related with the statement that first homotopic group of 3-dimensional configuration ...
3
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1answer
129 views

Real representation is physically real?

In Peskin & Schroder, Introduction to Quantum Field Theory equation (15.82) states that $$ t^a_{\bar{r}} = -(t^a_{r})^* = (t^a_{r})^T $$ Why is the representation which satisfies $$ ...
3
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2answers
204 views

Why do the states of a spin multiplet have to have the same symmetry?

This was said in Prof. Balakrishnan lecture 19 on quantum mechanics for the case of exchange symmetry, but he showed no reason why. For example, the system corresponding to two spin $\frac{1}{2}$ ...
3
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1answer
334 views

Is the spin-rotation symmetry of Kitaev model $D_2$ or $Q_8$?

It is known that the Kitaev Hamiltonian and its spin-liquid ground state both break the $SU(2)$ spin-rotation symmetry. So what's the spin-rotation-symmetry group for the Kitaev model? It's obvious ...
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1answer
431 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 $(2,0)$?
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1answer
225 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
81 views

Why is the gauge potential $A_{\mu}$ in the Lie algebra of the gauge group $G$?

If we have a general gauge group whose action is $$ \Phi(x) \rightarrow g(x)\Phi(x), $$ with $g\in G$. Then introducing the gauge covariant derivative $$ D_{\mu}\Phi(x) = ...
3
votes
1answer
79 views

Is there a general theorem stating why the restricted Lorentz group's exponential map is surjective?

The exponential map for the restricted Lorentz group is surjective. An outline of why is shown on the wiki page Representation Theory of the Lorentz Group. Is there a more general theorem that states ...
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2answers
119 views

Identifying irreps of $SU(2)$

How does one verify that, the representations of $SU(2)$ corresponding to $j=1/2$ or $j=1$ is irreducible? I think showing the irreducibility (taking the representative matrices into a block-diagonal ...
3
votes
1answer
84 views

Two different transformation laws for Quantum Fields

I found a nice answer to a problem that was bothering me for quite a while in a lecture script (unfortunately in german). The first step of the answer, is what remains unclear to me. The script states ...
3
votes
1answer
151 views

Group theoretical reason that Gluons carry charge and anticharge

I was wondering how it is possible to see from the $SU(3)$ Gauge Theory alone that Gluons carry two charges colors: $g\overline{b}$ etc. Some background: The W-Bosons (pre-symmetry breaking) ...
3
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2answers
77 views

What is different in representation?

I'm sorry if this is somewhat a dumb question. First: "Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear ...
3
votes
1answer
168 views

How to construct an invariant Lagrangian under a Lie group $G$ generally?

How to construct an invariant Lagrangian under a Lie group $G$ generally? For example, if we have $SO(5)$'s generators which are constructed by some operators, then the question is that: is it ...
3
votes
1answer
104 views

Determining the group associated with a given potential?

I'm trying to understand how symmetry groups are related to potentials of the Schrodinger equation. In particular, I wish to know if it is possible to find the symmetry group of this potential $$V(x) ...
3
votes
1answer
151 views

Group analysis forbids band-crossing in 1D?

Group analysis forbids band-crossing in 1D in terms of conventional band theory. I read this in a good solid state physics book. But there's no explanation at all. Can anyone help on this?
3
votes
1answer
112 views

Is it possible to Vectorialize Quantum Field Theories?

If I take the rules for classical electrodynamics in the covariant formulation (the closest to QFT), I have a tensor that describes the field, $F_{\mu\nu}$. Now we know that we can take some of the ...
3
votes
1answer
73 views

Permutation symmetry - a continuous symmetry?

From quantum mechanics it is known that permutation between identical particles does not change the Hamiltonian. Assuming that the quantum system consists of a very high number of particles such that ...
3
votes
1answer
63 views

3D isotropic oscillator and angular momentum algebra

In our QM class, the prof said: "We are ready to begin constructing the individual states of the 3D isotropic harmonic oscillator system. The key property is that the states must organize ...
3
votes
1answer
128 views

Dirac group representation

I am currently taking a representation theory class (from a physicist), and I am very confused about the Dirac groups' irreducible representations. First of all, all the Dirac matrices in the ...
3
votes
1answer
93 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
389 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
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0answers
37 views

Symmetry breaking to a special subalgebra?

This is a follow-up to my question here. For regular subalgebras of some group's Lie algebra the root system of the subalgebra is a subset of the root system of the original's group algebra. In ...
3
votes
0answers
66 views

Is Witten's claim that gauge group representations get exchanged with its dual under parity correct?

I'm currently reading Physics and Geometry by Witten, which I really liked up to the point where he claimed that we exchange representations $R$ and $\tilde R$ under parity transformations, where $R$ ...
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74 views

Decomposing a representation under a subgroup [closed]

I am trying to understand what is the method for decomposing representations of a group under one of its subgroups. I already had a look in Slansky, but I could not extract a concrete set of ...
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votes
0answers
116 views

Why is only the third component of weak isospin used as a conserved quantity?

Using Noether's theorem \begin{equation} \partial_0 \int d^3x \left(\frac{\partial L}{\partial(\partial_0\Psi)} \delta \Psi \right) = 0 \end{equation} we get three conserved quantites $Q_i$ from ...
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0answers
56 views

Highest weight unitary representations of $psl(2|2)$

I'm having some trouble understanding how to extend representation theory from Lie algebras to super Lie algebras, in particular with $psl(2|2)$. Ultimately I'm interested in 2D quantum sigma models ...
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votes
0answers
87 views

Is general covariance a symmetry?

Is general covariance a symmetry? If it is ,what is its symmetry group and corresponding generator?
3
votes
0answers
40 views

How does the choice of a particular vacuum in a field theory problem decide the number of Goldstone bosons?

How does the field expansion method (by this I mean expanding your fields about a chosen VEV and plugging into a given potential so that the masses of the fields are given by the coefficients in ...
3
votes
0answers
113 views

Show the Lie algebra is the same for $SU(2) \times SU(2)$ and Lorentz group [duplicate]

So I know: $$[\sigma_{I},\sigma{j}] = 2i \epsilon_{ijk} \sigma_{k}$$ So two products of this should give us the Lorentz group: $SO(4) = SU(2) \times SU(2)$ Where $SO(4)$ has 3 Lie algebra which can ...
3
votes
0answers
165 views

Group theory notation used in physics (AdS/CFT)

This in the context of the AdS/CFT correspondence. I am reading this review on AdS/CFT Aharony et. al. (The MAGOO review) The abstract can be found here Equation (2.50) of the above paper lists the ...
2
votes
3answers
300 views

Number of Parameters of Lorentz Group

We embed the rotation group, $SO(3)$ into the Lorentz group, $O(1,3)$ : $SO(3) \hookrightarrow O(1,3)$ and then determine the six generators of Lorentz group: $J_x, J_y, J_z, K_x, K_y, K_z$ from the ...
2
votes
3answers
171 views

What does “transform among themselves” mean?

I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context: An arbitrary rotation of the ...
2
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1answer
181 views

Group Theoretic definition of a particle

We intuitively have a sense of what a particle means in the conventional sense. But is it possible to have a group theoretical definition of a particle, I mean in terms of irreducible representations ...
2
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2answers
210 views

A whole lot of doubts on Lorentz representation

Can someone tell me in layman's language how the $(1/2,1/2)$ represents a vector field and $(0,1/2)$ or $(1/2,0)$ represents spinors and $(0,0)$ represents scalar field. Please don't be pedantic on ...
2
votes
1answer
83 views

Finding the stabilizer group given a state

Consider general pure state $|\psi\rangle$ in some hilbert space $\mathcal{H}$ (which could be a tensor product of other Hilbert spaces) I would like to know whether there is a way to ...
2
votes
2answers
458 views

Why does a Lorentz scalar field transform as $U^{-1}(\Lambda)\phi(x)U(\Lambda) = \phi(\Lambda^{-1}x)$?

This problem is from Srednicki page 19. Why $U^{-1}(\Lambda)\phi(x)U(\Lambda) = \phi(\Lambda^{-1}x)$? Can anyone derive this? $\phi$ is a scalar and $\Lambda$ Lorentz transformation.
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2answers
585 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
2answers
253 views

Why do we require the generators of $\mathrm{SU(N)}$ gauge theories to be $N \times N$ matrices?

I have often read that the generators for $\mathrm{SU(N)}$ gauge theories must be $N \times N$ matrices; see for instance these notes at the top of page 3: ...
2
votes
1answer
43 views

Why is the Mixed Faraday Tensor a matrix in the algebra so(1,3)?

The mixed Faraday tensor $F^\mu{}_\nu$ explicitly in natural units is: ...
2
votes
3answers
130 views

Generators of the Diffeomorphism Group

So what are the generators of a Diffeomorphism Group? For simplicity, let's consider $ Diff(R^2) $ (diffeomorphisms of the euclidean plane.) Diffeomorphisms are differentiable, invertible ...
2
votes
2answers
102 views

Constructing SUSY algebra via index structure

Often in literature the SUSY algebra is simply given, but various books, for example Bailin and Love, goes through the trouble of showing how the SUSY commutation relations are the only possible ones ...
2
votes
3answers
118 views

$SO(3)$ vs 3-Torus ${(S_1)}^3$

From rigid body rotations point of view, why are $SO(3)$ and 3-Torus not the same. Every rigid rotation is rotation about three axes. So how come $SO(3)$ is not ${(S_1)}^3$? It seems it should be. Is ...
2
votes
1answer
65 views

Unit determinant for relevant symmetry groups in QFT

When treating QFT we want our theory to be invariant under different symmetry groups, for example, the Standard Model is a non-abelian gauge theory with the symmetry group $U(1)×SU(2)×SU(3)$. ...
2
votes
1answer
139 views

Rotation of angular momentum eigenfunctions?

I am struggling to understand this apparently obvious example in my group theory notes: Where do the $e^{i\phi} $ and $e^{-i\phi} $ factors come from? I know that the $m_l$ = -1,0 and +1 angular ...
2
votes
1answer
410 views

Why do we use the complexification of the Lorentz group?

I do understand why we are using the double cover, but why exactly do we make the transition to complex Lorentz transformations? Where and why are they needed? To be precise: The double cover of ...
2
votes
1answer
284 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 ...
2
votes
3answers
145 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?
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1answer
105 views

Classical spin viewed as $SU(2)$

In which sense is the configuration variable of a classical spin $SU(2)$? I can view a classical spin as a unit vector in $\mathbb{S}^2$ (2-dim. sphere), but it seems it is really given by a matrix ...
2
votes
1answer
141 views

Is there a connection between Lie Groups and observable quantities in physics?

Good evening everybody. I have some questions about the relation between Lie groups and observables in physics. Indeed, taking the example of spin formalism of Quantum mechanics I know that Pauli's ...
2
votes
1answer
85 views

Symmetry transformation on Quantum Field

I stumbled upon this point several times, the latest beeing this question: Connection between conserved charge and the generator of a symmetry I want to understand, why Quantum fields transform under ...