B.3 Chapters 5 to 7

B.3 Chapters 5 to 7

Concept links · terms present in this machine draft; source roles are unverified: linear map · Lie algebra · adjoint representation

Problem 1:

On the Lie algebras $\mathfrak{g}=\mathfrak{su}(2)$ and $\mathfrak{g}=\mathfrak{so}(3)$ one can define the Killing form $K(\cdot ,\cdot )$ by

$$(X,Y)\in \mathfrak{g}\times \mathfrak{g}\to K(X,Y)=tr(XY)$$

1. For both Lie algebras, show that this gives a bilinear, symmetric form, negative definite, with the basis vectors $X_j$ in one case and $l_j$ in the other providing an orthogonal basis if one uses $K(\cdot ,\cdot )$ as an inner product.

2. Another possible way to define the Killing form is as

$$K^{\prime }(X,Y)=tr(ad(X)\circ ad(Y))$$

Here the Lie algebra adjoint representation $(ad,\mathfrak{g})$ gives for each $X\in \mathfrak{g}$ a linear map

$$ad(X):{\mathbf{R}}^3\to {\mathbf{R}}^3$$

and thus a 3 by 3 real matrix. This $K^{\prime }$ is determined by taking the trace of the product of two such matrices. How are K and $K^{\prime }$ related?

Problem 2:

Under the homomorphism

$$\Phi :Sp(1)\to SO(3)$$

of section 6.2.3, what elements of $SO(3)$ do the quaternions $i$, $j$, $k$ (unit length, so elements of $Sp(1))$ ) correspond to? Note that this is not the same question as that of evaluating ${\Phi }^{\prime }$ on $i$, $j$, $k$.

Problem 3:

In special relativity, we consider space and time together as ${\mathbf{R}}^4$ , with an inner product such that $(v,v)=-v_0^2+v_1^2+v_2^2+v_3^2$ , where $v=(v_0,v_1,v_2,v_3)\in {\mathbf{R}}^4$ The group of linear transformations of determinant one preserving this inner product is written $SO(3,1)$ and known as the Lorentz group. Show that, just as $SO(4)$ has a double cover $Spin(4)=Sp(1)\times Sp(1)$ , the Lorentz group has a double cover $SL(2,\mathbf{C})$ , with action on vectors given by identifying ${\mathbf{R}}^4$ with 2 by 2 Hermitian matrices according to

$$(v_0,v_1,v_2,v_3)\leftrightarrow \left(\begin{array}{cc}{v}_0+v_3& v_1-i{v}_2\\ v_1+i{v}_2& v_0-v_3\end{array}\right)=M$$

and using the action of $g\in SL(2,\mathbf{C})$ on these matrices by

$$M\to gM{g}^{†}$$

(hint: use determinants).

Note that the Lorentz group has a spinor representation, but it is not unitary.

Problem 4:

Consider a spin $\begin{array}{l}\frac{1}{2}\end{array}$ particle, with a state $|\psi (t)⟩$ evolving in time under the influence of a magnetic field of strength $B=|\mathbf{B}|$ in the 3-direction. If the state is an eigenvector for $S_1$ at $t=0$ , what are the expectation values

$$⟨\psi (t)|S_j|\psi (t)⟩$$

at later times for the observables $S_j$ (recall that $\begin{array}{r}{S}_j=\frac{{\sigma }_j}{2})?\end{array}$

Previous section: B.2 Chapters 3 and 4 · Next section: B.4 Chapter 8 · 中文版本

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 537、538、539、540、541、542、543、544、545、546、547、548、549、550、551、552、553、554、555、556

来源版本：2025-10-20

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