Questions tagged [group-theory]

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. Groups are used in physics to describe symmetry operations of physical systems.

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What is light cone? Explain to mathematicians who understand the Lorentz group but not light cone

Mathematically the Lorentz group is precisely the $O(1,3)$ is the 4-vector rotation preserving the inner product of 4-vector under this metric $$ \eta_{\mu \nu}=(+1,-1,-1,-1). $$ There are four ...
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Physically, what is a pseudoreal representation?

There are three kinds of representations: real, complex, and pseudoreal. A complex representation is not equivalent to its conjugate, and a real one is, which is pretty straightforward. A pseudoreal ...
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Parameters in Lie group theory

This is from P.Woit's "Quantum Theory, Groups and Representations", p. 151. If the Hilbert space is the space of complex-valued square-integrable functions on the circle, we want to find the ...
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$\rm SU(2)$ transformation of spinors

In the book QFT by Ryder on the topic $\rm SU(2)$ and the rotation group, it is stated that, The group $\rm SU(2)$ consists of $2\times2$ unitary matrices with unit determinant, $$UU^\dagger = 1, \...
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$U(\Lambda)$ that implements Lorentz transformations on the states of Hilbert space is unitary

In Peskin & Schroeder QFT book p.59, they said: The operator $U(\Lambda)$ that implements Lorentz transformations on the states of Hilbert space is unitary, even thought for boosts $\Lambda_{1/2}$...
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Orbit of the Little Group [migrated]

I have a problem with the comprehension of the definition of an orbit. Right now I'm writing my bachelor thesis about the analysis of solids with help of group theory and I need to understand this ...
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Mixing $SU(N)$ and $U(1)$ generators to form an unbroken $U(1)’$

I’m trying to understand some symmetry breaking patterns and have been reading David Tong’s Gauge Theory notes for an overview. I’m getting very confused about how one can mix $SU(N)$ and $U(1)$ ...
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Connected components of conformal group $ {\rm Conf}(p,q)$ containing $P$, $T$ and conformal inversion are same or different?

As we known (see this post), the global conformal group for $\mathbb{R}^{p,q}$ is $$ {\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}$$ The global conformal group ${\rm Conf}(p,q)$ has 4 ...
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What is the symmetry group of Mach's spacetime?

Newtonian spacetime can be modeled as a geometric object $M$ (affine space or manifold with connection with an absolute time function etc. etc.) that is symmetric under the action of the Galilean ...
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What is the relationship between the Galilean group and the Poincaré group?

What is the relationship between the Galilean group and the Poincaré group? Are they siblings within the Lie group? Or does the Poincaré group contain the Galilean group as a subgroup? I'm not so much ...
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Gauge Field Transformation Properties

I'm a bit confused about the gauge transformation properties of non-abelian gauge fields, and I just wanted some clarification. I keep seeing the statement that "gauge fields transform in the adjoint ...
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Witt algebra, $\mathfrak{sl}(2,\mathbb{R})$, $\mathfrak{sl}(2,\mathbb{C})$ and Bosonic String Theory

Suppose you know nothing about CFT, and suppose you have found in (closed bosonic) String Theory that \begin{equation} [L_n , L_m ]=(n-m) L_{n+m} \;\;\;\;(\mathrm{"right"\;Witt\;Algebra}\; \mathfrak{w}...
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Why should the infinite-dimensional representation of Poincare group induced by the unitary representation of little group be unitary?

In Weinberg's Quantum Field Theory (Vol. I, pages 64-67) it is stated that a unitary representation of little group induces a unitary representation of the Poincare group. But I don't understand how ...
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Different Casimirs and Casimirs of $E_6$

I am a bit confused by the notion of Casimirs (maybe it is related to terminology). In the simplest example of $su(2)$ with generators $L_i$, we get the Casimir operator $$ L^2=\sum_i L_i^2$$ ...
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Truncation of $D=5$, ${\cal N}=8$ Supergravity by $\mathbb Z_2^3$

The scalar manifold of $D=5, \mathcal N=8$ SUGRA is $$\mathcal M = \frac{E_{6(6)}}{Usp(8)}$$ where $USp(8)$ is a maximal compact subgroup of $E_{6(6)}$ and the 42 scalars of the theory correspond ...
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Connection between particle physics and weight diagrams

I have a hard time combining two topics that are often discussed in physics in a coherent way. In a lot of Introduction to particle physics-classes one will hear about "multiplets", which ...
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Constructing an operator from k.p hamiltonian

I have a question regarding to how to construct an operator from k.p hamiltonian. May be there are some problems in my understanding, I hope you can point me out and correct my description if I made ...
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Does a Super Noether Theorem exist?

I am wondering if an extension of Noether theorem to supergroups exists. In particular the analogy with the usual case should be that supersymmmetries are in 1-to-1 correspondence to certain "...
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Why would a spinor transform under Lorentz transformations?

From my understanding of spinors, they arise as projective representations of $SO_0(1,3)$ that do not correspond to representations of $SO_0(1,3)$. But still one says here - and virtually everywhere - ...
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Idea of Covering Group

$SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence? I heard that this fact is related to the description of bosons and fermions. But how does it ...
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On the decomposition of rank two tensors

In the review of cosmological perturbations by Mukhanov et al linked: https://doi.org/10.1016/0370-1573(92)90044-Z on page 212 of the pdf, they introduce a symmetric, trace-free, rank-2 tensor ...
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Can the orbit of $|{↑↓}⟩$ under $\rm SU(2)$ be written as the span of some basis?

I am considering the action of $SU(2)$ on two spins. More precisely, I want to determine what states can be reached by acting with an element $\hat R \in SU(2)$ on $|\psi \rangle \in \text{span}\{|{\...
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Addition of Permutations in Second Quantisation of an Operator

Sorry, if the title doesn't provide any clarity, but I didn't really know how to call it. Anyways, I've been studying quantum field theory from Blundell's book and during the derivation of the formula ...
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Is $SU(2)\times U(1) = U(2)$?

In the many textbook of the Standard Model, I encounter the relation \begin{align} SU(2)_L \times U(1)_L = U(2)_L. \end{align} Here the subscript $L$ means the left-handness (i.e., the chirality of ...
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How do I decompose an $SO(10)$ antisymmetric tensor in $SO(4) \times SU(3) \times U(1)$?

My guess is that If I denote the $SO(4)$ indices $\mu, \nu = 1,...4$ and the $SU(3)$ indices by $I,J=1,2,3$, I think $N^{mn}$ should decompose as $N^{\mu \nu}, N^{IJ}, N^{I}_J, N_{IJ}$ plus other ...
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Why can all 4-momenta of fixed length square be constructed by applying a Lorentz transform on a "standard" 4-momentum?

In the subsection on 'One particle States' of Weinberg [1996], he says: Note that the only functions of $p^\mu$ that are left invariant by all proper orthochronous Lorentz transformations $\Lambda$, ...
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Spin Half in 4-Dimensions

The Stern Gerlach experiment in 3-Dimensions provides us with conditions on what properties Spin-Half vectors must satisfy, from which we can build our basis states in $x, y, \text{ and } z$. The ...
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Name for 16⊕1͞6 bosons of E6 & larger GUTs?

E6 GUTs put fermions into $27→16_1⊕10_{-2}⊕1_4$ and gauge bosons into $78→45_0⊕16_{-3}⊕\overline{16}_{3}⊕1_0$. And larger GUTs (e.g. E8 on 𝕋⁶/ℤ₆×ℤ₂, E8 on 𝕋⁶/ℤ₃×ℤ₃) usually incorporate these. The $...
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Trace of generators of Lie group

In most textbooks (Georgi, for example) a scalar product on the generators of a Lie Algebra is introduced (the Cartan-Killing form) as $$tr[T^{a}T^{b}]$$ which is promptly diagonalised (for compact ...
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What is the reason for $U(3)_{L} \times U(3)_{R} = U(1)_{V} \times U(1)_{A} \times SU(3)_{L} \times SU(3)_{R}$?

I am studying the QCD chiral symmetry, and by considering the $u$,$d$,$s$ quarks massless, the Lagrangian \begin{equation} \mathcal{L} = \sum_{i = u,d,s} \bar{q}_{k}i \gamma^{\mu}D_{\mu}q_{k} \end{...
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Meaning of the term 'states carry an irreducible representation of a group $G$'

If I have a group $G=SU(2)$, the $s=\frac{1}{2}$ irreducible representation is given by matrices $$ U(G)=\begin{pmatrix} \alpha & - \beta^* \\ \beta & \alpha^* \end{pmatrix}\;\;\;\; : \alpha \...
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How to determine which irrep the Hilbert space of states carry?

Is the following statement ([1]) correct? [1] If the universe has a symmetry under a group $G$, does this mean the Hilbert space carries a unitary representation formed by taking the direct product of ...
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Why is group theory important in General Relativity?

I came across the Poincaré group, and most importantly the Lorentz group while studying GR. What is the significance of these groups as well as any other groups used in GR? I mean, why should I care ...
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What does complexification mean for our particles in physics?

As gauge group let's consider the popular $SO(10)$ group. The fundamental representation $\pi$ of the corresponding Lie algebra $\mathfrak{so}(10)$ is $10$ dimensional $$ \pi: \mathfrak{so}(10) \...
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How to calculate the commutation relation for generator of $SO(3)$ with finite rotation in the group?

I am trying to show that $$e^{-iX_3\theta}X_+e^{iX_3\theta} = e^{-i\theta}X_+$$ where $X_1, X_2, X_3$ are the generators of $SO(3)$, which obey the usual commutation relations of their Lie algebra, ...
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What is the status of orbifolded 10D E8 theory?

A recent series of papers (1, 2) presents an $E_8$ GUT in 10 dimensions, where compactification on a $\mathbb T^6/(\mathbb Z_3\times\mathbb Z_3)$ orbifold ($\mathbb T^6/(\mathbb Z_6\times\mathbb Z_2)$ ...
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Representation of dilations

I am having some trouble getting some signs right on the representation of the dilation operator on a field. Let us follow the conventions of https://arxiv.org/abs/1511.04074. According to equation (2....
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How to physically interpret the symmetric tensor product of 2 solutions of the right-handed Weyl equations?

The Weyl equations are: $$ \sigma^\mu \partial_{\mu} \psi = 0,$$ where $\psi$ is a section of $M \times \mathbb{C}^2$ over Minkowski spacetime $M$. Let us say you have two solutions of the right-...
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Why are all transformations of quantum operators inner automorphisms?

Operators in quantum mechanics are basically related to each other through their Lie algebra i.e. the commutator $\times \frac{1}{i\hbar}$. This is then connected to the state space i.e. the Hilbert ...
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Error in Wikipedia on a flipped $SO(10)$ model

In the description of Wikipedia of flipped SO(10) model, it says that: In flipped $SO(10)$ models, however, the gauge group is $[SO(10)_F × U(1)_B]/Z_4$. If we suppose $[SU (5) × U(1)_χ ]/ Z_5$ ...
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What do these Casimir invariants of the Galilean group physically represent?

There exist Casimir invariants of the Galilean group which commute with all the generators of the group. They are, of course, Galilean scalars (i.e., scalars under space and time translations, ...
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Tachyons as vector representations on the surfaces transitivity of Lorentz group?

In Wikipedia's surfaces of transitivity (of Lorentz group $G$, it says "Standard vectors on the one-sheeted hyperbolas would correspond to tachyons. Particles on the light cone are photons, and ...
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Lie algebra generators as rank-16 matrix spinor representations of $𝑆𝑝𝑖𝑛(10)$

A simple Lie group $𝑆𝑝𝑖𝑛(10)$ has a spinor representations of 16 dimensions, which is distinct from the vector representation of 10 dimensions (coming from standard vector representation of SO(10))...
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Group Theory in a Nutshell for Physicists $SO(2)$ Irreducible Representations Dimensions

In Zee's book, in section 4.1 under the heading "The tensors of $SO(2)$," he states: "Hence the dimensions of the irreducible representations are $(j + 1) − (j − 2 + 1) = 2$. All of ...
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2answers
199 views

Rarita-Schwinger spin projection operators

Chapter 2 of the paper Symmetry of massive Rarita-Schwinger fields by T. Pilling mentions "the usual" spin projection operators. However, to me, they are not usual and I struggle with ...
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Vacuum manifold and moduli space

Vacuum manifold is just another name for the manifold spanned by the ground states of quantum field theory. It is also called moduli space. According to https://en.wikipedia.org/wiki/Vacuum_manifold, ...
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How to know the minimal coupling and how to find eigenvalues of arbitrary representations?

I'm dealing with the following problem. In a $ SU(3)_L\times U(1)_X$ model, the scalar representation content accomodates the following anti-sextet $$ S \equiv \begin{pmatrix} \sigma_1^0 & \frac{...
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1answer
75 views

Missing complex conjugate in (1/2,1/2) representation of Lorentz group Ticciati QFT

I've been working through some computations involving representations of the Lorentz group (now using the fantastic Ticciati QFT textbook). After some work, Ticciati gives the following formula $$D^{...
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1answer
944 views

Gauge invariance and the form of the Rarita-Schwinger action

in Weinberg Vol. I section 5.9 (in particular p. 251 and surrounding discussion), it is explained that the smallest-dimension field operator for a massless particle of spin-1 takes the form of a field ...
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2answers
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10-dimensional and 15-dimensional matrix representations of $SU(5)$: explicit 24 Lie algebra generators

There are some previous discussions in this post Representation of the $\rm SU(5)$ model in GUT which confused me. So I want to follow up with a new question. It is easy to write down the 5-...

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