Abstract
This chapter introduces abstract groups and Lie groups, which are a formalization of the notion of a physical transformation. The chapter begins with a heuristic introduction that motivates the definition of a group and gives an intuitive sense for what an “infinitesimal generator” is. Then we give the definition of an abstract group along with examples, followed by a discussion of the groups that arise most often in physics, particularly the rotation group O(3) and the Lorentz group SO(3,1) o . These groups are discussed in coordinates and in great detail, so that the reader gets a sense of what they look like in action. Then we discuss homomorphisms of groups, which allows us to make precise the relationship between the rotation group O(3) and its quantum-mechanical “double-cover” SU(2). We then define matrix Lie groups and demonstrate how the so-called infinitesimal elements of the group give rise to a Lie algebra, whose properties we then explore. We discuss many examples of Lie algebras in physics, and then show how homomorphisms of matrix Lie groups induce homomorphisms of their associated Lie algebras.
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Notes
- 1.
This notation may be familiar if you did Problem 3-1 d). If not, it will be explained in the next section.
- 2.
The notation \(\mathfrak{s}\mathfrak{o}(3)\) may seem like it has a deeper meaning, and it does, but we can’t explain that until Sect. 4.6. Until then, just consider \(\mathfrak{s}\mathfrak{o}(3)\) as odd notation for the vector space of 3 × 3 antisymmetric matrices.
- 3.
You may recall having met \(GL(n, \mathbb{R})\) at the end of Sect. 2.1 There we asked why it isn’t a vector space, and now we know—it’s more properly thought of as a group!
- 4.
Hence the term “iso-metry” = “same length.”
- 5.
We are glossing over some subtleties with this claim. See Example 4.20 for the full story.
- 6.
In fact, the only reason for speaking of both “isometries” and “unitary operators” is that unitary operators act solely on inner product spaces, whereas isometries can act on spaces with non-degenerate Hermitian forms that are not necessarily positive-definite, such as \(\mathbb{R}^{4}\) with the Minkoswki metric.
- 7.
Meaning that there exists a continuous map \(\gamma: [0, 1] \rightarrow GL(n, \mathbb{R})\) such that γ(0) = I and γ(1) = R. In other words, there is a path of invertible matrices connecting R to I.
- 8.
Such as Goldstein [8].
- 9.
This fact can be understood geometrically: since orthogonal matrices preserve distances and angles, they should preserve volumes as well. As we learned in Example 3.28, the determinant measures how volume changes under the action of a linear operator, so any volume preserving operator should have determinant ± 1. The sign is determined by whether or not the orientation is reversed.
- 10.
One can actually think of the inversion as following or preceding the proper rotation, since − I commutes with all matrices.
- 11.
It’s worth noting that, in contrast to rotations, Lorentz transformations are pretty much always interpreted passively. A vector in \(\mathbb{R}^{4}\) is considered an event, and it doesn’t make much sense to start moving that event around in spacetime (the active interpretation), though it does make sense to ask what a different observer’s coordinates for that particular event would be (passive interpretation).
- 12.
Note that β is measured in units of the speed of light, hence the restriction − 1 < β < 1.
- 13.
See Griffiths [10], Ch. 12.2.
- 14.
See Problem 4-8.
- 15.
See Problem 4-6.
- 16.
We could also proceed by defining \(gK \equiv \{ gk\vert \,k \in K\}\), but this turns out to be the same as Kg. A subgroup K with this property, that \(Kg = gK\ \forall g \in G\), is called a normal subgroup. For more on this, see Herstein [12].
Fig. 4.6 
Schematic depiction of the subsets K and Kg of G. They are mapped respectively to e′ and h in H by the homomorphism \(\Phi \)
- 17.
See Problem 4-7 of this chapter, or consider the following rough (but correct) argument: ρ: SU(2) → O(3) as defined above is a continuous map, and so the composition
$$\displaystyle\begin{array}{rcl} \det \circ \,\rho: SU(2)& \rightarrow & \mathbb{R} {}\\ A& \mapsto & \det (\rho (A)){}\\ \end{array}$$is also continuous. Since SU(2) is connected any continuous function must itself vary continuously, so \(\det \circ \,\rho\) can’t jump between 1 and − 1, which are its only possible values. Since \(\det (\rho (I)) = 1\), we can then conclude that \(\det (\rho (A)) = 1\ \forall A \in SU(2)\).
- 18.
See Problem 4-7 again.
- 19.
See Problem 4-8 again.
- 20.
See Herstein [12] for a proof and nice discussion.
- 21.
- 22.
See Hall [11] for further information and references.
- 23.
This is actually only true when X and Y commute; more on this later.
- 24.
See any standard text on linear algebra and linear ODEs.
- 25.
- 26.
See Frankel [6] for a detailed discussion.
- 27.
We will freely use the identification between finite-dimensional linear operators and matrices here, so that we may think of Isom(V ) as a matrix Lie group and hence consider its Lie algebra. The exponential of linear operators is defined by (4.47), just as for matrices.
- 28.
\(\mathfrak{g}\mathfrak{l}(n, \mathbb{C})\) can also be considered a complex vector space, but we won’t consider it as such in this text.
- 29.
To be rigorous, we also need to note that \(\mathfrak{g}\) is closed in the topological sense, but this can be regarded as a technicality.
- 30.
The size of a matrix \(X \in M_{n}(\mathbb{C})\) is usually expressed by the Hilbert–Schmidt norm, defined as
$$\displaystyle{ \vert \vert X\vert \vert \equiv \sum _{i,j=1}^{n}\vert X_{ ij}\vert ^{2}. }$$(4.65)If we introduce the basis {E ij } of \(M_{n}(\mathbb{C})\) from Example 2.9, then we can identify \(M_{n}(\mathbb{C})\) with \(\mathbb{C}^{n^{2} }\), and then (4.65) is just the standard Hermitian inner product.
- 31.
- 32.
Which, of course, only holds for matrices when the matrices being added commute.
- 33.
The space of all traceless 2 × 2 complex matrices viewed as a complex vector space is denoted \(\mathfrak{s}\mathfrak{l}(2, \mathbb{C})\) without the \(\mathbb{R}\) subscript. In this case, the S i suffice to form a basis. In this text, however, we will usually take Lie algebras to be real vector spaces, even if they naturally form complex vector spaces as well. You should be aware, though, that \(\mathfrak{s}\mathfrak{l}(2, \mathbb{C})\) is fundamental in the theory of complex Lie algebras, and so in the math literature the Lie algebra of \(SL(2, \mathbb{C})\) is almost always considered to be the complex vector space \(\mathfrak{s}\mathfrak{l}(2, \mathbb{C})\) rather than the real Lie algebra \(\mathfrak{s}\mathfrak{l}(2, \mathbb{C})_{\mathbb{R}}\). We will have more to say about \(\mathfrak{s}\mathfrak{l}(2, \mathbb{C})\) in the appendix.
- 34.
There is a subtlety here: the vector space underlying \(\mathfrak{g}\mathfrak{l}(V )\) is of course just \(\mathcal{L}(V )\), so the difference between the two is just that one comes equipped with a Lie bracket, and the other is considered as a vector space with no additional structure.
- 35.
A function is “infinitely differentiable” if it can be differentiated an arbitrary number of times. Besides the step function and its derivative, the Dirac delta “function”, most functions that one meets in classical physics and quantum mechanics are infinitely differentiable. This includes the exponential and trigonometric functions, as well as any other function that permits a power series expansion.
- 36.
By this we mean infinite-dimensional, and usually requiring a basis that cannot be indexed by the integers but rather must be labeled by elements of \(\mathbb{R}\) or some other continuous set. You should recall from Sect. 3.7 that \(L^{2}(\mathbb{R})\) was another such “huge” vector space.
- 37.
That is, one-to-one and onto transformations which preserve the form of Hamilton’s equations. See Goldstein [8].
- 38.
Let \(\phi _{g}: P \rightarrow P\) be the transformation of P corresponding to the group element g ∈ G. Then H is invariant under G if \(H(\phi _{g}(p)) = H(p)\ \forall \,p \in P,\,g \in G\).
- 39.
- 40.
See Arnold [1] for a discussion of Noether’s theorem and a derivation of an equivalent formula.
- 41.
Of course, this identification depends on a choice of basis, which was made when we chose to work with \(\mathcal{B} =\{ S_{1},S_{2},S_{3}\}\).
References
V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd edn. (Springer, New York, 1989)
A.C. Da Silva, Lectures on Symplectic Geometry. Lecture Notes in Mathematics, vol. 1764 (Springer, Berlin, 2001)
T. Frankel, The Geometry of Physics, 1st edn. (Cambridge University Press, Cambridge, 1997)
H. Goldstein, Classical Mechanics, 2nd edn. (Addison-Wesley, Massachusetts, 1980)
D. Griffiths, Introduction to Electrodynamics, 3rd edn. (Prentice Hall, Upper Saddle River, 1999)
B. Hall, Lie Groups, Lie Algebras and Representations: An Elementary Introduction (Springer, New York, 2003)
I. Herstein, Topics in Algebra, 2nd edn. (Wiley, New York, 1975)
K. Hoffman, D. Kunze, Linear Algebra, 2nd edn. (Prentice Hall, Englewood Cliffs, 1971)
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Jeevanjee, N. (2015). Groups, Lie Groups, and Lie Algebras. In: An Introduction to Tensors and Group Theory for Physicists. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-14794-9_4
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