10.1 The group R and its representations

10.1 The group R and its representations

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation · unitary representation · irreducible representation · Lie group · Lie bracket

Some of the most fundamental symmetries of nature are translational symmetries, and the basic example of these involves the Lie group $R$, with the group law given by addition. Note that R can be treated as a matrix group with a multiplicative group law by identifying it with the group of matrices of the form

for $a\in \mathbf{R}$ . Since

$$\begin{array}{c}\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right)\\ \left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& b\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}1& a+b\\ 0& 1\end{array}\right)\end{array}$$

multiplication of matrices corresponds to addition of elements of R. Using the matrix exponential one finds that

$$e^{\left(\begin{array}{cc}0& a\\ 0& 0\end{array}\right)}=\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right)$$

so the Lie algebra of the matrix group $R$ is matrices of the form

$$\left(\begin{array}{cc}0& a\\ 0& 0\end{array}\right)$$

with Lie bracket the matrix commutator (which is zero here). Such a Lie algebra can be identified with the Lie algebra $R$ (with trivial Lie bracket).

We will sometimes find this way of expressing elements of R as matrices useful, but will often instead label elements of the group by scalars ${}^{a,}$ and use the additive group law. The same scalars a are also used to label elements of the Lie algebra, with the exponential map from the Lie algebra to the Lie group now just the identity map. Recall that the Lie algebra of a Lie group can be thought of as the tangent space to the group at the identity. For examples of Lie groups like R that are linear spaces, the space and its tangent space can be identified, and this is what we are doing here.

The irreducible representations of the group $R$ are the following:

Theorem 10.1

Irreducible representations of R are labeled by $c\in \mathbf{C}$ and given by

$${\pi }_c(a)=e^{ca}$$

Such representations are unitary $(inU(1))$ when $c$ is purely imaginary.

The proof of this theorem is the same as for the $G=U(1)$ case (theorem 2.3), dropping the final part of the argument, which shows that periodicity $(U(1)$ is just R with a and $a+N2\pi$ identified) requires c to be i times an integer.

The representations of R that we are interested in are on spaces of wavefunctions, and thus infinite dimensional. The simplest case is the representation induced on functions on $R$ by the action of $R$ on itself by translation. Here $a\in \mathbf{R}$ acts on $q\in \mathbf{R}$ (where $q$ is a coordinate on $R$) by

$$q\to a\cdot q=q+a$$

and the induced representation $\pi$ on functions (see equation 1.3) is

$$\pi (g)f(q)=f(g^{-1}\cdot q)$$

which for this case will be

$$\pi (a)f(q)=f(q-a)\text{\hspace{1em}(10.1)}$$

To get the Lie algebra version of this representation, the above can be differentiated, finding

$${\pi }^{\prime }(a)=-a\frac{d}{dq}\text{\hspace{1em}(10.2)}$$

In the other direction, knowing the Lie algebra representation, exponentiation give

$$\pi (a)f=e^{{\pi }^{\prime }(a)}f=e^{-a\frac{d}{dq}}f(q)=f(q)-a\frac{df}{dq}+\frac{a^2}{2!}\frac{d^{2}f}{d{q}^2}+\cdots =f(q-a)$$

which is just Taylor’s formula.¹

In chapter 5, for finite dimensional unitary representations of a Lie group $G$ we found corresponding Lie algebra representations in terms of self-adjoint matrices. For the case of $G=\mathbf{R}$ , even for infinite dimensional representations on $\mathcal{H}=L^2({\mathbf{R}}^3)$ one gets an equivalence of unitary representations and selfadjoint operators², although now this is a non-trivial theorem in analysis, not just a fact about matrices.

Chapter contents · Next section: 10.2 Translations in time and space · 中文版本

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