Use as a synonym to the representation-theory tag
Use as a synonym to the representation-theory tag.
Questions tagged [group-representations]
542 questions
113
votes
1 answer
Why exactly do sometimes universal covers, and sometimes central extensions feature in the application of a symmetry group to quantum physics?
There seem to be two different things one must consider when representing a symmetry group in quantum mechanics:
The universal cover: For instance, when representing the rotation group $\mathrm{SO}(3)$, it turns out that one must allow also…
ACuriousMind
- 132,081
82
votes
21 answers
Comprehensive book on group theory for physicists?
I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that covers "all" the basics, and then, in addition, talks…
Lagerbaer
- 15,136
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48
votes
7 answers
Adding 3 electron spins
I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers needed are total spin and total z-component.
But how…
Gere
- 1,527
32
votes
2 answers
Why do we say that irreducible representation of Poincare group represents the one-particle state?
Only because
Rep is unitary, so saves positive-definite norm (for possibility density),
Casimir operators of the group have eigenvalues $m^{2}$ and $m^2s(s + 1)$, so characterizes mass and spin, and
It is the representation of the global group of…
user8817
32
votes
6 answers
Tensor Operators
Motivation.
I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. I then skimmed the Wikipedia page on tensor…
joshphysics
- 58,991
31
votes
3 answers
Why are only linear representations of the Lorentz group considered as fundamental quantum fields?
As described in many Q&As around here, fundamental quantum fields are expressed as irreducible representations of the Lorentz group. This argument is entirely clear - we live in a Lorentz-invariant world and those elements of an observed system that…
Void
- 21,331
29
votes
4 answers
Could the Periodic Table have been done using group theory?
These three questions are phrased as alternative-history questions, but my real intent is to understand better how well different modeling approaches fit the phenomena they are used to describe; see 1 below for more discussion of that point.
Short…
Terry Bollinger
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28
votes
2 answers
Wick rotation and spinors
I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; when we perform this Wick rotation, in a way we…
toot
- 2,946
27
votes
2 answers
Introduction to spinors in physics, and their relation to representations
First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms "up to a sign" when acted on by the rotation group. I have even rotated a plate on my palm to explain this to my fiancee! I…
Harry Wilson
- 877
25
votes
1 answer
Why do we identify symmetric 2nd rank tensors with spin-2 particles in string theory?
I am going through Tong's lecture notes on String Theory and came across the following irrep decomposition (Chap 2, p.43) of the bosonic string first excited states:
$$\text{traceless symmetric} \oplus \text{anti-symmetric} \oplus…
physguy
- 679
25
votes
2 answers
Do generators belong to the Lie group or the Lie algebra?
In Physics papers, would it be correct to say that when there is mention of generators, they really mean the generators of the Lie algebra rather than generators of the Lie group? For example I've seen sources that say that the $SU(N)$ group has…
Siraj R Khan
- 2,028
24
votes
2 answers
When are there enough Casimirs?
I know that a Casimir for a Lie algebra $\mathfrak{g}$ is a central element of the universal enveloping algebra. For example in $\mathfrak{so}(3)$ the generators are the angular momentum operators $J_1,J_2,J_3$ and a quadratic Casimir is…
Edward Hughes
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24
votes
1 answer
Why Lorentz group for fields and Poincaré group for particles?
Wigner treatment associates to particles the irreps of the universal covering of the Poincaré group
$$\mathbb{R}(1,3)\rtimes SL(2,\mathbb{C}).$$
Why don't we consider finite dimensional representations of this group?
I understand we ask for…
Issam Ibnouhsein
- 859
23
votes
2 answers
What does a $\rm SU(2)$ isospin doublet really mean?
What do we really mean when we say that the neutron and proton wavefunctions together form an $\rm SU(2)$ isospin doublet? What is the significance of this? What does this transformation really doing to the wavefunctions (or fields)?
SRS
- 27,790
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23
votes
4 answers
Why do all fields in a QFT transform like *irreducible* representations of some group?
Emphasis is on the irreducible. I get what's special about them. But is there some principle that I'm missing, that says it can only be irreducible representations? Or is it just 'more beautiful' and usually the first thing people tried?
Whenever…
BeneIT
- 333