14 votes

Why does group representation theory look linear?

A group in itself is not necessarily related to linear spaces. Representation theory (which is a subfield of group theory) is the study of the way in which groups act linearly on vector spaces : a ...
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11 votes

Why does group representation theory look linear?

A group $G$ is an algebraic structure which encodes one pattern of composition of objects. This makes abstract the notion of transformations which may be composed to give new ones. The whole point, ...
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8 votes
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Basic Facts about Lie Algebras

Yes, OP is right: P&S do not distinguish between the abstract Lie group and its defining representation, which is usually the fundamental representation.
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5 votes

Help needed in understanding $2\otimes 2=1\oplus 1\oplus 2$ of $SO(2)$

You can parameterize a general $2\times 2$ matrix of real numbers in terms of 4 real numbers $t$, $a$, $s_1$, and $s_2$ as \begin{equation} A = \left( \begin{array} _ \frac{t}{2} + s_1 & s_2 + a \\...
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5 votes

Why does group representation theory look linear?

Concerning OP's title question Why does group representation theory look linear? it should be mentioned that there exist in physics (e.g. in the areas of spontaneous symmetry breaking and ...
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4 votes
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Invariants of inner product in pseudoreal representation of $SU(2)$

P&S are talking about the spinor/defining/fundamental representation $$\eta,\xi~\in~ V~\cong~ \mathbb{C}^2$$ of $SU(2)$. The expression $\epsilon^{\alpha\beta}\eta_{\alpha}\xi_{\beta}$ is $SU(2)$-...
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4 votes

Basic Facts about Lie Algebras

You learnt that in your group theory course, that sounds right. But you must consider the topology. Lie groups are manifolds, so they have tangent spaces (this is an additional structure to being a ...
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4 votes
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Help needed in understanding $2\otimes 2=1\oplus 1\oplus 2$ of $SO(2)$

The "2" is the space of all trace-free symmetric rank-2 tensors. So an arbitrary tensor product of two vectors can be decomposed as $$ A_i B_j = \frac{1}{2} \delta_{ij} A_k B_k + \frac{1}{2}...
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3 votes

Help needed in understanding $2\otimes 2=1\oplus 1\oplus 2$ of $SO(2)$

Any $2\times2$ matrix (4 dof) can be broken up into its trace part (1 dof), an antisymmetric part (1 dof) and a symmetric traceless part (2 dof). Hence, $2\times 2=4=1+1+2$.
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3 votes

Confusion in group theory at the example of the Lie algebra $su(2)$

Yes, the $T_i$ are supposed to be a basis of the algebra as a vector space. Therefore, choosing n-by-n matrices $\rho(T_i)$ that represent each of the $T_i$ defines a linear map $\rho : \mathfrak{su}(...
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2 votes
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Abstract definition of four-vector

95% correct, let me add some elements. We define the "mathematical Minkowski spacetime" as the vector space $\mathbb R^4$ endowded with a bilinear form we denote by $\eta$ ("mostly ...
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2 votes

Invariants of inner product in pseudoreal representation of $SU(2)$

I would like add to Qmechanic's excellent answer a bit of context from the practical point of view. Apparently the question arises in classical Quantum mechanics not accounting for relativity. So then ...
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2 votes

Importance of raising/lowering/ladder operators

That is just the thing about quantum mechanics. It is based on the fundamental principle that interactions involve the exchange of quanta. As a result of this principle each particle field taking part ...
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1 vote

Clebsch-Gordan Series and Rotation Matrices

The rotation operator \begin{align} D(R)=e^{i\vec \omega\cdot \vec L}& =e^{i\vec \omega \cdot(\vec L^{(1)}+\vec L^{(2)})} = e^{i\vec \omega \cdot\vec L^{(1)}}e^{i\vec \omega \cdot\vec L^{(2)}}\, ,\...
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1 vote

From a quant-info perspective why are the reals indexing irreps of the Lorentz group less suspect than continuousness in space-time and general QM?

The UV infinities in perturbative QFT is a mathematical artifact coming from the indiscriminate use of distributions. In causal perturbation theory, where more careful use of distributions is made, ...
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1 vote

A relation for adjoint representation of $U(N)$ acting on product of matrices and $SU(2)$ generators

Here is a partial remark on a special case that might help you get started on your problem, where everything is completely calculable directly. Take the case of N=2, where you might as well normalize ...
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1 vote

Importance of raising/lowering/ladder operators

The construction of raising and lowering operator is very natural when there is an underlying Lie algebra. Thus yes there's some systematic way of constructing them. In fact, if the Lie algebra is ...
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1 vote

Confusion in group theory at the example of the Lie algebra $su(2)$

...and wouldn't then linear combinations of them yield new representation? Not necessarily. Suppose you have a representation in matrix form like: $$ [M_i, M_j] = i\epsilon_{ijk}M_k $$ A linear ...
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1 vote
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Constructing gauge invariants

I am using this reference, Spin Multiplicities, T Curtright, T van Kortryk, and C Zachos, Phys Lett A381 (2017) 422-427. The character of a spin j, dimension 2j+1 irrep of SU(2) is $$ \chi_j (\theta)...
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1 vote

Why does group representation theory look linear?

Linearity is no restriction in the following sense: Every compact Lie group $G$ (in particular every finite group) has a faithful finite-dimensional linear representation, see the Peter-Weyl-theorem. ...
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