# Questions tagged [representation-theory]

The systematic study of group representations, which describe abstract groups in terms of linear transformations of vector spaces, such that group elements or their generators are represented as matrices, reducing group-theoretic problems to linear-algebraic ones.

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### Construction of an $N$ electron orbital and spin state

Consider a system of $N$ electrons. Their Hilbert space is the antisymmetric subspace of $\mathcal H = \mathcal H_e^{\otimes N}$, where $\mathcal H_e \cong L^2(\mathbb R^3)\otimes \mathbb C^2$ is the ...
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### Non-symmorphic space group representation

What is the representation of non-symmorphic group operation? I know that this results in phase factor multiplication in the original point group representation but do i need to multiply this phase ...
1 vote
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### Confused about two different definitions for the same object $\psi_i$ in the representation theory of $\mathrm{SU}(n)$

Let the entities $\psi^i$ transform as the fundamental representation of $\mathrm{SU}(n)$, denoted by ${\bf n}$: $$\psi^{\prime i}=U^{i}_{~j}\psi^j,$$ where, of course, $U$ represents $n\times n$ ...
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### How to derive the Lorentz Invariant bilinear form in the $(1/2, 1/2)$ representation?

We can represent the complexified proper Lorentz group Lie algebra as the direct sum $\mathfrak{sl}(2, \mathbb{C}) \oplus \mathfrak{sl}(2, \mathbb{C})$. The representation nomenclature is $(n,m)$, two ...
114 views

### Why is the Lorentz transformation of fields linear?

I know that the coordinate, $x^\mu = (t,\vec x)$ is a 4-vector and it transforms as $$x'^\mu={\Lambda^\mu}_\nu x^\nu.$$ The related (classical or quantized) field, $\phi_a(x)$, can be classified into ...
369 views

### Relation between Stone von Neumann Theorem and Bargmann's theorem

I am trying to understand a relation between the two theorems stated in the title. What I observed so far is that since $H^{2}(\mathbb{R},U(1))=\{e\}$, using Bargmann's theorem, we have that ...
1 vote
63 views

### What exactly is the definition of the representation of an operator in position or momentum space?

I apologize for this kind of silly question, I haven't brushed up on QM for a while. I was looking at a problem today, essentially I'm given some operator $V = \lambda |{\xi}\rangle \langle{\xi}|$ ...
79 views

### What is a particle in the context of QFT with interactions?

This is a crossposting of the same question from mathoverflow: https://mathoverflow.net/q/454768/ It seems that this question was not received well there, claiming that this question is not ...
29 views

### Parity symmetry complete/detailed definition and the group elements

I am trying to write down a complete/detailed definition for the parity symmetry. Symmetry as a concept is different in mathematics and in physics. There are also many other concepts which differ in ...
1 vote
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### Why the Double Covering?

It is known mathematically that given a bilinear form $Q$ with signature $(p,q)$ then the group $Spin(p,q)$ is the double cover of the group $SO(p,q)$ associated to $Q$, and that $Pin(p,q)$ is the ...
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### What is the difference between a ''unitary representation'' and a ''projective unitary representation''?

It is assumed that a valid QFT, must have states/observables in a Hilbert space, transform according to the projective unitary representations of the Poincare group. I can understand why the ...
60 views

### How do Poincare group act on Classical field, Quantum field operator, Field configuration states, Fock space states?

I will try to make each of my statement as clear as possible, if any of the statements are wrong prior to my question, please point them out:) For simplicity, we work in free QFT with scalar field. ...
52 views

### Algebra equation for rank-3 tensor

Suppose I work in $4$ dimensions. I have an algebraic equation in the following form, which contains a rank-3 tensor $X ^{\alpha \lambda \mu }$ \begin{equation} X ^{\alpha \lambda \mu }\eta ^{\beta \...
1 vote
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### How should we imbed massive spin-3 in a Lorentz covariant tensor? (degree of freedom-wise)

I'm not asking for Weinberg's systematic approach. Rather, I'm more concerned with how to get the correct degrees of freedom(dof) slickly for the moment. I believe it should be imbedded in the rank 3 ...
1 vote
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### Is the character table of $S_3$ unique? [migrated]

I'm trying to construct the character table of $S_3$ group. $n_c$ class $1$ $\bar{1}$ $2$ 1 $I$ 1 1 2 3 $(12),(23),(13)$ 1 a b 2 $(123),(132)$ 1 c d As a brute force method, I imposed every ...
1 vote
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### How is the Wigner little group representation of Poincaré group Unitary?

From Weinberg's QFT Vol.1, eq(2.5.11): $$U(\Lambda)\Psi_{p,\sigma}=({N(p)\over N(\Lambda p)})\sum_{\sigma'}D_{\sigma'\sigma}(W(\Lambda,p))\Psi_{\Lambda p ,\sigma '}.\tag{2.5.11}$$ However, this is not ...
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### Some references on Dh4 point group ans superconductors

Just wondering if anyone has any lecture notes or books/chapters which cover the representations of Dh4 CLEARLY. In particular, the form factors of superconductors are labeled with the representations ...
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### Coadjoint orbit method (geometric quantization) for second quantisation of free relativistic particles

The quantum theory for relativistic particles can be obtained by constructing the (projective) unitary irreducible representations associated to the integral coadjoint orbits of the Poincare group. ...
100 views

### Schur's Lemma in Zee's Group Theory book for reducible representations

Main question Schur's lemma says: $$D(g) A = A D(g) \Rightarrow A = \lambda I\tag{1}$$ if $D$ is irreducible. How can I use this to show that if $D$ is reducible and if $SDS^{-1}$ is a direct sum of ...
96 views

### Unitary representations of a Lorentz transformation

In QFT we have an action of the restricted Lorentz group which is implemented via a unitary transformation. In other words, if $\Lambda\in SO(1,3)^\uparrow$, then the corresponding unitary operator is ...
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### How to derive symmetry invariant terms in Hamiltonian from a space group?

If a space group of some crystal is know, how can we derive the Hamilton of it in spin form to do more theoretical calculations? For example, for a space group No.5 (C2), whose generators are $x,y,z$ ...
### Is $U(\Lambda,a)\hat\phi(x) U^{-1}(\Lambda,a) = R(\Lambda,a) \hat\phi(\Lambda^{-1}x + a)$ an axiom or can be it derived?
As stated in the question, and I have looked at other questions on this topic here, I am still confused whether the equation  U(\Lambda,a)\hat\phi(x) U^{-1}(\Lambda,a) = R(\Lambda,a) \hat\phi(\...