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 ...
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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 ...
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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 ...
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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 ...
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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 $V = \lambda |{\xi}\rangle \langle{\xi}|$ ...
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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 ...
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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 ...
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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 ...
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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.
...
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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 \...
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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 ...
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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 ...
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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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What kind of a space is $\mathbb{C}_{-1}$?
In the first paragraph on page 24 of this paper: https://arxiv.org/abs/0904.1556, it's written
the left-handed leptons $\nu_L$ and $e^−_L$ both have hypercharge $Y=−1$, so each one spans a copy of $\...
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What are the infinite-dimensional unitary irreps of the little group for a massless particle, ISO(2)?
Wigner's classification implies that the internal state of a massless particle transforms under an irreducible unitary representation of ISO(2), the group of isometries on the plane. Since ISO(2) is a ...
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Confusion regarding Poincare group representations in Weinberg's Quantum Field Theory, Volume 1
I am a beginner to quantum field theory and my course is based on Weinberg's QFT (vol. 1; chapter 2) - I have quite a few confusions.
I have had an introductory group theory course before, so, I know ...
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Overlap between eigenstates of angular momentum operators
Consider the states $\left|j,m_x\right>_x$ and $\left|j,m_z\right>_z$ with total angular momentum $j$ and the angular momentum operators $\hat{S}_x$ and $\hat{S}_y$. In particular, assume that ...
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Irreps of symmetric group: Are projection operators always elements of the group algebra?
In the past, I studied the irreps of the symmetric group using chapter 5 and Appendix III of Wu-Ki Tung's book "Group Theory in Physics." I always thought that the reason that the theory ...
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More than one list of character for one IRREP in character tables
I would like to know why in the group $C_4$ (please see http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=204&option=4) there are two lists of characters for the irreducible ...
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Can someone identify this vector representation of $\rm SO(3)$ in terms of multi-variable polynomials?
I am trying to get some deeper intuition for the representations of $\rm SO(3)$ and how they combine with each other, and I ran into an odd object that I'm hoping that folks here might help me ...
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How do we relate vectors in spacetime to spin 1 representations of $su(2)$ algebra?
If we look at the representations of the $su(2)$ algebra we can construct a representation for every possible spin as follows
$$|0,0\rangle\,\,\, \text{(spin 0)}$$
$$|-1/2,1/2\rangle,|1/2,1/2\rangle\,\...
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Why intuitively do the eigenvalues of the Casimir operators classify irreps?
Distinguishing between distinct unitary irreducible representations is important from the point of view of distinguishing between different sorts of particles; the eigenvalues of the Casimir operators ...
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Proving multiplication with Dirac adjoint spinor is Lorentz scalar
I'm studying QFT with the book "Quantum Field Theory" by Dr. David Tong, and I have difficulties with understanding some used transformations.
On the page 88 the author calculates Hermitian ...
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Why, in QCD, are quarks in the fundamental representation of $SU(3)$?
QCD is built from the notion that Dirac's Lagrangian should be invariant under gauge colour transformations.
Here, quarks are elements of $\psi_{\alpha,f,c}(x)$, where $\alpha$, $f$ and $c$ stand for ...
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Selecting Kac determinant solutions for Yang-Lee Minimal Model
I'm looking into the non-unitary minimal model $\mathcal M_{5,2}$ associated with the Yang-Lee edge singularity. I'm trying to justify which conformal dimensions we expect to appear (easy enough) but ...
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Clebsch-Gordan decomposition for $SU(3)$ representations [duplicate]
I am currently working my way through Howard Georgei's text on Lie Algebras in Particle Physics. I am having some trouble understanding how to go about decomposing a general tensor product into ...
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Are the generators of the $SU(2)_L\times SU(2)_R$ group unitary?
In the chiral model, the quark field $q=\begin{pmatrix} u \\ d \end {pmatrix}$ transforms like $q\rightarrow\exp({i(\theta_a\tau^a+\gamma_5\beta_a\tau^a)})\;q$. Now, I understand that the ...
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Why is a fixed $j$ eigenspace irreducible?
In his development of angular momentum, Ballentine writes:
The $(2j + 1)$-dimensional space spanned by the set of vectors $\{|j, m\rangle\}$, for fixed $j$ and all $m$ in the range $(−j \leq m \leq j)...
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How is the dimension of the vector space we represent $SO(3)$ on determined when discussing the spin of a particle?
Consider a single particle with Hilbert space $L^2(\mathbb{R}^3) \otimes V_\ell$ where $V$ is a vector space of dimension $2\ell + 1$ equipped with a projective unitary representation of $SO(3)$. ...
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Construction of $SU(2)_L$ invariant Lagrangian
I am reading this paper about "Scalar Electroweak Multiplet Dark Matter" written by Wei Chao et al. I am puzzled for their construction of $SU(2)_L$ invariant Lagrangian, Eq.(6).
$$
\begin{...
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Matrix representation of Lorentz boost acting on a scalar
For the unitary translation operator $\hat{X}_a$ in 1 dimension it is easy to show that its matrix elements can be expressed purely with a delta distribution or equivalently as a combination of a ...
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What is "spin" in spin representations?
When finding representations of Lorentz algebra let's say so(1,D-1), we can find some representations of this algebra using the Clifford algebra. I understand how to identify Weyl and Majorana ...
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Von Neumann Algebra decomposition
I am trying to understand the Von Neumann decomposition, according to which every Von Neumann Algebra can be uniquely decomposed as integral (or direct sum) of factors. More specifically, I am trying ...
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Tensor Product of Left Handed Spinor and Right Handed Spinor representations of the group $SU(2)_L \times SU(2)_R$ [duplicate]
From quantum mechanics, we know that the tensor product of two spin-half representations would give us spin-1, and spin-0 reps, i.e,
$$(1/2,0)\otimes(1/2,0)\cong(1,0)\oplus(0,0)$$
I want to know what ...
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Most general nonlinear Lorentz transformation law can be built from linear transformations?
Peskin and Schroeder give a Lorentz transformation law:
$$\Phi_a(x)\rightarrow M_{ab}(\Lambda)\Phi_b(\Lambda^{-1}x).\tag{3.8}$$
Then they say that
"the most general nonlinear [Lorentz] ...
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Why must boosts be non-compact?
It is a common argument in the theory of kinematic groups (the groups of motions for a spacetime) that the subgroups generated by boosts must be non-compact[1][2][3]. This is true of all commonly used ...
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Why on earth should we talk in terms of Lie brackets instead of commutators?
For a Lie group with structure constants $f_{abc}$, its Lie algebra is given by a relation between the generators in the form $$[T_a, T_b]=if_{abc}T_c\tag{1}$$ where the symbol $[T_a, T_b]$, is called ...
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Deriving the $(1/2,1/2) $ representation of the Lorentz Group as in Schwichtenberg
I'm currently studying Jakob Schwichtenberg's "Physics from Symmetry" where on subsection 3.7.8 he proves that the $(1/2,1/2)$ representation of the Lorentz Group corresponds to 4-vectors.
I ...
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Dimensionality of unitary representation of Lorentz group
I am reading the textbook "Lectures on Quantum Field Theory", second edition by Ashok Das. On page 137, he talks about the non-unitarity of the finite-dimensional representation of the ...
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Some intuitions for irreducible representations $SO(3)$ in classical physics
With a very naive and intuitive understanding of representation theory, I make my point below. Feel free to correct my intuition.
In order to find out the representations of $SO(3)$ on a vector space $...
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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.
...
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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 ...
2
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0
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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
$x,y,z$
...
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Action of Spin Operator on Dirac Spinor
This is my first post, although I often use physics stack exchange. Thanks.
I am solving P. Ramond’s text on QFT.
I am stuck with the action of the spin operator on the Dirac spinor. What confuses me ...
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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(\...
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