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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Does the set of evolution operators in QM form any known algebraic structure (e.g., a monoid)?
The evolution operator, $U(t,t_0)$, in quantum mechanics is defined by
$$
|\Psi(t)\rangle = U(t,t_0)|\Psi(t_0)\rangle.
$$
with $t\geq t_0$. Is the set of such evolution operators, specified by two ...
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Projective representations and reduction of half-integer spin representations under $C_{\infty v}$
Suppose we have an orthonormal basis of states $|j,m,p\rangle$ where
$j=\frac{1}{2},\frac{3}{2},\ldots$ is the angular momentum quantum number associated with some angular momentum operator $\mathbf{...
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Vector/axial current and vector/axial transformation
As is well known, in particle physics, the vector current is defined as a current of the form
$$J_V^\mu = \bar{\psi} \gamma^\mu \psi $$
and the axial current as
$$J_A^\mu = \bar{\psi} \gamma^5 \gamma^\...
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On JD Jackson's derivation of Matrix Representations of Lorentz Tranformations
Jackson derives the ordinary rotation matrix for a rotation through angle $ \omega$ about the $z$ axis (eq.11.96) via the exponential map of the Lorentz group:
$$A=e^{-\vec{\omega}\cdot\vec S-\vec \...
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Group of time translations for finite-dimensional quantum systems and incommensurable eigenvalues
I am a bit confused by the group of the time translations in finite-dimensional quantum mechanics. Take a finite-dimensional Hamiltonian H. For instance, for a two-dimensional system
$$
H=\begin{...
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Inversion asymmetric little representation in a Inversion symmetric large representation in brillouin zone
I am not am having trouble understanding the little representation which characterize high symmetry points in the brillouin zone of a crystal.
I understand this essentially mean that for some high ...
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How can one says that a particle IS a representation of some group? [duplicate]
I suppose this question has been asked many times. I have been told that an elementary particle is a (moving) point, or (a section from) some field, or an excitation from some field. But now, I am ...
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Cartesian tensor transformation and representations of $SO(3)$
I have hard times understanding the transformation of tensors, that probably stems from my shaky understanding of representation theory.
A cartesian tensor can be decomposed in three terms:
$$
T_{ij}=...
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Can all solutions from any possible dynamical systems be formulated as combinations of harmonic oscillators?
In my understanding, all physical oscillators are unit vectors of a Hilbert space that is represented from the group $SU(3)\times SU(2)\times U(1)$. Every dynamical system operates under this group (...
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Recovering the electromagnetic field from the $(1,0) \oplus (0,1)$ representation
I'm confused as to how one recovers the electromagnetic field $E,B$ from the standard procedure of building the $(1,0) \oplus (0,1)$ representation of the Lorentz Lie algebra. The reason is the ...
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Why is the gauge field $A_\mu$ real for $\mathrm{U}(N)$ symmetry? [duplicate]
This question is a duplicate of this question asked on Maths S-E
From my notes I have that
The transformation law, $$A_\mu\to MA_\mu M+\frac{i}{g}\left(\partial_\mu M\right)M^\dagger\tag{1}$$ can be ...
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Group actions confusion
I have been using results from this paper in calculations. In sections 2.4 and 3.4 they perform a canonical transformation into new coordinates consisting of constants of motion. They then construct ...
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Group properties of density of state for i.i.d. particles microcanonical ensembles
For a $2N$ particle non-interacting gas, we could arbitrarily divide the gas into 2 sets of $N$ particles.
Then the density of states for each one is,
$$
D(N_A, V, U_A) = h^{-N} \int d^Nq~d^Np ~ \...
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Why is the gauge field $A_\mu$ real for $\mathrm{U}(N)$ symmetry?
From my notes I have that
The transformation law, $$A_\mu\to MA_\mu M+\frac{i}{g}\left(\partial_\mu M\right)M^\dagger\tag{1}$$ can be realised if $A_\mu$ is an element of the Lie algebra. It can ...
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Showing that a generator exponentiates to a $\mathbb{R}$ group
I have a generator $G$ that acts on the phase space of Schwarzschild and maps geodesics into each other.
In order to discuss the corresponding symmetry group, I need to exponentiate this generator and ...
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Goldstone Matrix for $SO(3) \longrightarrow SO(2)$ breaking
In these lecture notes (https://arxiv.org/abs/1506.01961) on composite Higgs models, on page 22, the authors calculate the Goldstone Matrix $U[\Pi]$ for an abelian composite Higgs scenario i.e. $SO(3)$...
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Identity for generators of $SU(N)$ in the adjoint representation
For the generators of $SU(N)$ in the fundamental representation, $T_{i j}^a$, the following identity holds
$$T_{i j}^a T_{k \ell}^a=\frac{1}{2}\left(\delta_{i \ell} \delta_{j k}-\frac{1}{N} \delta_{i ...
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Why is there no "internal" part of linear momentum? [closed]
One of the ways to derive the expression for how angular momentum operators act on fields from the corresponding action on coordinates is to define spin as $$M_{\mu \nu} \Phi(0) = S_{\mu \nu} \Phi(0) \...
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What is the displacement of E$_g$ mode of SrTiO$_3$ in tetragonal phase? [closed]
In the tetragonal phase, SrTiO$_3$ belongs to the D$_{4h}$ point group and possesses two Raman-active modes with the symmetry rep of A$_{1g}$ and E$_g$. However, in most books only 7 modes (normal ...
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Understanding the Role of $SO(3)$ in the Derivation of the Schwarzschild Solution
In MTW, the exact derivation of the Schwarzschild solution is based on the following assumptions:
Begin with a manifold $M^4$ on which a metric $ds^2$ of Lorentz signature is defined.
Assume $M^4$ to ...
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Why is it not possible to describe general Lorentz Boosts using Hyperbolic Quaternions?
The Lorentz Boosts (for 1+1D) can be described by the Split-Complex Numbers. A Lorentz Boost in the direction of $x$ with the rapidity $\alpha$ for a 1+1D-system can be calculated using
$$q \mapsto e^{...
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Why is it reasonable to use Biquaternions for Lorentz-Transformations?
I've read that biquaternions can be used for Lorentz-Transformations using the formula $$q \mapsto e^{\alpha h \mu/2}e^{\phi\epsilon/2} q \overline{e^{\alpha h \mu/2}e^{\phi\epsilon/2}}^{*},$$
$\alpha$...
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Group Representation and Particle
In quantum mechanics, a particle is an irreducible unitary Poincaré group representation on a Hilbert space $H$. This is because a particle exists objectively which does not depend on observers.
I ...
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Why is Dilatation subgroup $SO(1,1)$?
Why is dilatation subgroup of the conformal group $SO(1,1)$? I am not sure if this is a result for 2D theories or even higher dimensional ones too. (Parameter counting suggests there is one parameter ...
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Deducing the (special) relativistic Lagrangian only using Lorentz transform [duplicate]
I feel like there must be a way to deduce the (special) relativstic Lagrangian using only the Lorentz transform (without other knowledge of special relativity). So far, all the derivations I have seen ...
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$G$-invariant field equations
It is easily verifiable that if we are given a field theory with the Klein Gordon equation, it is Poincaré-invariant. So is the Dirac equation. More generally, for arbitrary spin, the Poincaré-...
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Confusion between trigonal and hexagonal systems
I'm studying space groups.
It's quite clear (I think) why trigonal and hexagonal systems collapse in the same primitive Bravais lattice, while are different when we introduce non-primitive unit cells, ...
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Orthogonality of the character of rotation group
I am currently reading the book 'Group Theory and Quantum Mechanics' by Michael Tinkham and there's something I don't quite get. To show the completeness of the irreducible representations of the ...
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Showing that the angular momentum operators and Laplace-Runge-Lenz operator together are generators of $SO(4)$
This is a question that popped up while reading Greiner's Quantum Mechanics Symmetries. For the sake of clarity I will omit the hat ($\hat{A}$) symbol on operators.
The quantum mechanical Laplace-...
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$(1/2,1/2)$ Representation transformation laws in Schwichtenberg has extra transpose
I'm reading Schwichtenberg's Physics from Symmetry. I have a question about the derivation of the $(1/2,1/2)$ representation of the lorentz group.
The issue is that Schwichtenberg adds an extra ...
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If a unitary operator is close to the identity, will it leave any state it acts on unchanged?
I have a question that feels obvious, but I have been having trouble proving it. In words, the question is: If a unitary is close to the identity, will it leave any state it acts on unchanged?
This, ...
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A reference for the fact that the second cohomology of the full Poincare algebra is zero
S. Weinberg in his book "The quantum theory of fields" vol. I says in page 86 that the full Poincare algebra is not semi-simple but its central charges can be eliminated (as he showed in the ...
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Given a representation $(n, m)$ of the Lorentz group, is the little group representation just the tensor product $n \otimes m$?
I've been reading Weinberg's QFT Vol 1. and more specifically section 5.6. I would like to know if my understanding is correct or if I missed something. He starts with the full Lorentz group $\mathrm{...
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How to find Casimir operator eigenvalues of $SU(N)$? [closed]
The $[f1, f2, f3…fn]$ in the image represent the irreducible representations of $SU[n]$. How to find the irreducible representations of $SU[n]$ that conform to the form $[f1, f2...fn]$. Can you give ...
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Does all symmetry breaking have corresponding unitary group?
In high energy physics. Symmetry breaking like electroweak's has corresponding $SU(2)\times U(1)$ unitary gauge group broken down to $U(1)$. Does it mean all kinds of symmetry breaking (even low ...
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What is the importance of $SU(2)$ being the double cover of $SO(3)$?
To my understanding, it is important that $SU(2)$ is (isomorphic to) the universal cover of $SO(3)$. This is important because $SU(2)$ is then simply-connected and has a Lie algebra isomorphic to $\...
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$Ad\circ\exp=\exp\circ ad$ and $e^{i(\theta/2)\hat{n}\cdot\sigma}\sigma e^{-i(\theta/2)\hat{n}\cdot\sigma}=e^{\theta\hat{n}\cdot J}\sigma$
This question is inspired by my recent question How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$? with answer https://...
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How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?
Disclaimer: I'm sure this has been asked 100 times before, but I can't find the question asked or answered quite like this. If there are specific duplicates that could give me a simple satisfactory ...
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Help with Wigner-Eckert Theorem problem
Currently trying to solve the following problem:
Consider an operator $O_x$ for $x = 1$ to $2$, transforming according to the spin $1/2$ representation as follows:
$$ [J_a, O_x] = O_y[\sigma_a]_{yx} / ...
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Can you ever obtain a pure rotation from composing Lorentz transformations?
An exercise asks one to show that given $v, u$ speeds much smaller than $c$ and oriented orthagonally, the composition of the lorentz boosts $B(\mathbf{v})B(\mathbf{u})B(\mathbf{-v})B(\mathbf{-u})$ is ...
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Rotation and translation of a function of a 3D vector
I want to change the frame by doing translation and rotation.
$$f(\vec{v})=\sum_{n,l,m}R_{nl}(v)Y_{lm}(\hat{v})f_{nlm}^v.$$
Let, $\mathcal{R}$ be the rotation matrix and $\mathcal{T}$ be the ...
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From any element of $\mathrm{SO}(8)$, can we always find one corresponding $\mathrm{SU}(3)$ element?
I first recap the relation between $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$ and then raise my question concerning $\mathrm{SU}(3)$ and $\mathrm{SO}(8)$. Given any traceless hermitian matrix $H$, we can ...
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Is intrinsic spin a quantum or/and a relativistic phenomenon?
Ok, this is my reasoning. I am probably making some wrong assumptions here, pls tell me where I am going wrong.
Spin as a quantum phenomenon:
Quantum phenomena disappear as the Planck constant goes to ...
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Rotation of spherical harmonics
I have a question about the rotation of spherical harmonics. In Wikipedia it is mentioned that if we make a rotation in 3D space: $R\vec{r}=\vec{r}'$,then the Spherical Harmonics can be written as a ...
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What does the $N$ in $SU(N)$ mean?
So I know this is a very basic question, but I can't really wrap my head around it.
I was told $N$ is the number of dimensions in the rotations of the group theory that we are considering, so I ...
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Do GUT's really explain parity violation?
Every book on the Standard Model introduces early on the concept of left and right-handed quantum fields, defined as
\begin{align}
(\psi_L)_{\alpha} = \left(\frac{1-\gamma_5}{2}\right)_{\alpha \beta}\...
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Free fields in Weinberg QFT vol.1
Background:
In section 5.1 Weinberg discusses free fields. He had shown that for interaction of the form, $V(t) = \int{d^3x \mathscr{H}(\mathbf{x},t)}$ if $$U_0(\Lambda,a) \mathscr{H}(x) U_0^{-1}(\...
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Why are Lorentz transformations singular at $i^0$?
On pg. 16 of Strominger's lectures, it is said
Lorentz transformations themselves are not smooth at spatial infinity, because the vector
fields that generate them are singular at $i_0$. A boost ...
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Is the factorization method of Hamiltonian related to the theory of Lie groups?
I was learning about algebraic methods to solve the H atom, when I came across the factorization method. It is mentioned in various textbooks, notes and papers, like the one from Infeld and Hull.
I am ...
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One-Loop beta function for gauge couplings
I am currently doing my homework on Standard Model one-loop correction. When I am reading Quantum Field Theory by Mark Srednicki and Journeys Beyond the Standard Model by Pierre Ramond, I notice some ...
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