54
Great, important question. Here's the basic logic:
We start with Wigner's Theorem which tells us that a symmetry transformation on a quantum system can be written, up to phase, as either a unitary or anti-unitary operator on the Hilbert space $\mathcal H$ of the system.
It follows that if we want to represent a Lie group $G$ of symmetries of a system via ...
54
Yes. Both universal covers and central extensions incurred during quantization come from the same fundamental concept:
Projective representations
If $\mathcal{H}$ is our Hilbert space of states, then distinct physical states are not vectors $\psi\in\mathcal{H}$, but rays, since multiplication by a complex number does not change the expectation values given ...
31
UPDATE - Answer edited to be consistent with the latest version of the question.
The different definitions you mentioned are NOT definitions. In fact, what you are describing are different representations of the Lorentz Algebra. Representation theory plays a very important role in physics.
As far as the Lie algebra are concerned, the generators $L_{\mu\nu}$...
30
There are a number of imprecisions in your question, mostly having to do with confusing the Lie group and its Lie algebra. I suppose this will make it hard to read the mathematical literature. Having said that, the first volume of Kobayashi and Nomizu is probably the canonical reference.
Let me try to summarise. Let me assume that $H$ is connected.
The ...
27
It's an enormous subject, but briefly:
Lie groups are smooth groups. Technically, Lie groups are sets that are both a smooth manifold, like a sphere for instance, and also have a group structure (multiplication operator, inverses, and an identity). The group multiplication and inverse must be smooth (differentiable) functions on the manifold.
As you ...
26
Here's my two cents worth.
Why Lie Algebras?
First I'm just going to talk about Lie algebras. These capture almost all information about the underlying group. The only information omitted is the discrete symmetries of the theory. But in quantum mechanics we usually deal with these separately, so that's fine.
The Lorentz Lie Algebra
It turns out that the ...
25
After the answers by joshphysics and user37496, it seems to me that a last remark remains.
The quantum relevance of the universal covering Lie group in my opinion is (also) due to a fundamental theorem by Nelson. That theorem relates Lie algebras of symmetric operators with unitary representations of a certain Lie group generated by those operators. The ...
23
I'll give you enough hints to complete the proof yourself. If you're desperate, I'm following the notes by Zuber, which are available online, IIRC.
Let's start with some notation: pick some basis $\{t_a\}$ of your Lie algebra, then
$$ [t_a,t_b] = C_{ab}{}^c t_c$$
defines the structure constants. If you define
$$ g_{ab} = C_{ad}{}^e C_{be}{}^d,$$
then this ...
22
Short answer: complexifications facilitate representation theory.
In physics, we typically want to find representations of a Lie algebra $\mathfrak g$, and often times determining the representations of its complexification $\mathfrak g_\mathbb C$ is easier. Moreover, we have the following theorem (see ref 1. Proposition 4.6) which tells us that ...
22
Let me first remind you of (or perhaps introduce you to) a couple of aspects of quantum mechanics in general as a model for physical systems. It seems to me that many of your questions can be answered with a better understanding of these general aspects followed by an appeal to how spin systems emerge as a special case.
General remarks about quantum states ...
21
QuantumMechanic's links turn up a dizzying array of meanings for $SU(2)$ in physics, so your question probably turns out to be too broad for a simple answer. Nonetheless, I do like it, and similar questions that grope for pithy overviews of things, so I'll try to answer it with my non particle physicist's understanding.
Probably the "main" meaning of $SU(2)$...
20
The elements of the gauge transformations belong to a gauge group. In physics, it's most typically $SU(N)$ (both the electroweak theory, with its $SU(2)$, and the QCD for quarks, $SU(3)$, use these $SU(N)$ groups; $U(1)$ we first learn in electromagnetism – but we must reinterpret the charge as the "hypercharge" when we study the electroweak theory – is the ...
20
Ghostly Lie algebra cohomology
Let $\mathfrak{g}$ be our Lie algebra and $V_\rho$ a representation space with representation map $\rho : \mathfrak{g} \to \mathrm{End}(V_\rho)$. $V_\rho$ is, by the action through the representation, naturally a $\mathfrak{g}$-module (people missing the ring structure in $\mathfrak{g}$ - just embed it into the universal ...
20
Every quantum field theory has a symmetry group under which its Lagrangian is invariant. Like every group, it must be closed. The boosts are not closed under composition, so they cannot form a symmetry group by themselves.
Any rotation can be achieved by a composition of four boosts, so if each boost leaves the Lagrangian invariant then the resulting ...
19
Does it mean that a given rotation in 4-dimensional Euclidean space cannot be associated with a unique axis ($\hat{\textbf{n}}$) of rotation? If yes, why is that the case?
Yes, this is absolutely true. The notion of a one dimensional axis is an "accident" of three dimensions. Rotations transform planar (dimension 2) linear subspaces of Euclidean space and ...
18
I) The Casimir invariants of a Lie algebra $L$ over a field $\mathbb{F}$ are the central elements of the universal enveloping algebra $U(L)$.
Example: The angular momentum square $\vec{J}^2$ is a quadratic Casimir invariant of the Lie algebra $L=sl(2,\mathbb{C})$.
II) Given a bilinear associative/invariant form $B:L\times L\to \mathbb{F}$, one can create ...
17
The general idea.
Let's restrict the discussion to matrix Lie Groups for simplicity. Determining the generators of a given Lie group $G$ simply means (by definition) determining a basis for its Lie algebra $\mathfrak g$. Here's a standard method for finding such a basis:
Recall that the Lie algebra $\mathfrak g$ of a matrix Lie group $G$ is defined as ...
17
Charge conjugation is extremely slippery because there are two different versions of it; there have been many questions on this site mixing them up (1, 2, 3, 4, 5, 6, 7, 8, 9), several asked by myself a few years ago. In particular there are a couple arguments in comments above where people are talking past each other for precisely this reason.
I believe ...
16
Here we will only discuss the case of finite-dimensional irreducible representations (irreps) of a complex semisimple Lie algebra $L$.
Recall that the set $Z$ of Casimir invariants is the center $Z(U(L))$ of the universal enveloping algebra $U(L)$, cf. e.g. this Phys.SE post.
OP's question is answered without proof on p. 253 in Ref. 1:
Theorem 2. For ...
16
The relevant Lie group isomorphism reads
$$ U(2)~\cong~[U(1)\times SU(2)]/\mathbb{Z}_2, \qquad Z(SU(2))~\cong~\mathbb{Z}_2.\tag{1a} $$
In detail, the Lie group isomorphism (1a) is given by
$$U(2)~\ni~ g\quad\mapsto\quad $$
$$ \left(\sqrt{\det g}, ~\frac{g}{\sqrt{\det g}}\right) ~\sim~ \left(-\sqrt{\det g}, ~-\frac{g}{\sqrt{\det g}}\right)$$
$$~\in ~[U(1)\...
16
The objects which matter in physics are Lie groups and not Lie algebras. Lie algebras approximate only infinitesimal group transformations and in quantum mechanics the finite and global properties of the transformations matter.
However, (considering quantum systems with a finite number of degrees of freedom), the spaces of quantum states are projective, as ...
15
In terms of classical general relativity: Einstein's equations
$$ G_{ab} = 8\pi T_{ab} $$
can be formulated, in local coordinates, as a system of second order partial differential equations for the metric unknown $g_{ab}$. The matter field equations further generate some family of partial differential equations.
Given a continuous symmetry (as guaranteed ...
15
I'd like to add to Josh's answer, because he didn't really explain what a universal covering group is. Essentially, a space $T$ is a covering space of another space $U$ if, for an open subset of $U$, there's a function $f$ that maps a union disjoint open subsets of $T$ to the subset of $U$. Or, more simply worded, pick a piece of your space $U$, and I'll ...
14
First of all, note that the real Abelian Lie group $U(1)$ comes in two (multiplicatively written) versions:
Compact $U(1)~\cong~e^{i\mathbb{R}}~\cong~S^1$, and
Non-compact $U(1)$ $~\cong~e^{\mathbb{R}}\cong~\mathbb{R}_+\backslash\{0\}$.
Also note that in the physics literature, we often identifies charge operators with Lie algebra generators for a Cartan ...
14
$F_4$ is the centralizer of $G_2$ inside an $E_8$. In other words, $E_8$ contains an $F_4\times G_2$ maximal subgroup. That's why by embedding the spin connection into the $E_8\times E_8$ heterotic gauge connection on $G_2$ holonomy manifolds, one obtains an $F_4$ gauge symmetry. See, for example,
http://arxiv.org/abs/hep-th/0108219
Gauge theories and ...
14
By definition, the metric tensor $\eta_{ij}$ transforms trivially under the defining rep of $SO(n,m)$.
$$
\eta_{ij}=[D(g^{-T})]_{i}^{\ k}[D(g^{-T})]_{j}^{\ l}\eta_{kl}
=[D(g^{-1})]^{k}_{\ i}[D(g^{-1})]^{l}_{\ j}\eta_{kl}
$$
and this holds for all $g\in SO(n,m)$. Consider a one-parameter subgroup of the defining rep with matrices $D(g)=e^{tJ}$ where $J^{i}...
14
Since I have gleaned some more idea of "where you are" in your learning and shown considerable enthusiasm for getting a thorough grasp of fundamentals, I'd like to add some more details (from a non-particle physicist, mind you, so there are many aspects of your question I must steer clear of) to Lubos's excellent answer. Also, in your question you spoke of ...
14
You may indeed identify the generators in the way you did. However, the Lie algebras and Lie groups are different because – as quickly said by Qmechanic – you must use different reality conditions for the coefficients.
A general matrix in the $SU(2)$ group is written as
$$ M = \exp[ i( \alpha J_+ + \bar\alpha J_- + \gamma J_0 )] $$
where $\alpha\in {\...
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