B.4 Chapter 8

B.4 Chapter 8

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · unitary representation · irreducible representation

Problem 1:

Using the definition

$$⟨f,g⟩=\frac{1}{{\pi }^2}{\int }_{{\mathbf{C}}^2}\overline{f(z_1,z_2)}g(z_1,z_2)e^{-(|z_1{|}^2+|z_2{|}^2)}d{x}_{1}d{y}_{1}d{x}_{2}d{y}_2$$

for an inner product on polynomials on homogeneous polynomials on ${\mathbf{C}}^2$ • Show that the representation $\pi$ on such polynomials given in section 8.2 (induced from the $SU(2)$ representation on ${\mathbf{C}}^2)$ is a unitary representation with respect to this inner product.

• Show that the monomials

$$\frac{z_1^j{z}_2^k}{\sqrt{j!k!}}$$

are orthonormal with respect to this inner product (hint: break up the integrals into integrals over the two complex planes, use polar coordinates).

• Show that the diferential operator ${\pi }^{\prime }(S_3)$ is self-adjoint. Show that ${\pi }^{\prime }(S_{-})$ and ${\pi }^{\prime }(S_{+})$ are adjoints of each other.

Problem 2:

Using the formulas for the $Y_1^m(\theta ,\varphi )$ and the inner product of equation 8.3, show that

• The $Y_1^1,Y_1^0,Y_1^{-1}$ are orthonormal.

$Y_1^1$ is a highest weight vector.

$Y_1^0$ and $Y_1^{-1}$ can be found by repeatedly applying $L_{-}$ to a highest weight vector.

Problem 3:

Recall that the Casimir operator $L^2$ of so(3) is the operator that in any representation $\rho$ is given by

$$L^2=L_1^2+L_2^2+L_3^2$$

Show that this operator commutes with the ${\rho }^{\prime }(X)$ for all $X\in \mathfrak{so}(3)$ Use this to show that $L^2$ has the same eigenvalue on all vectors in an irreducible representation of so(3).

Problem 4:

For the case of the $SU(2)$ representation $\pi$ on polynomials on ${\mathbf{C}}^2$ given in the notes, find the Casimir operator

$$L^2={\pi }^{\prime }(S_1){\pi }^{\prime }(S_1)+{\pi }^{\prime }(S_2){\pi }^{\prime }(S_2)+{\pi }^{\prime }(S_3){\pi }^{\prime }(S_3)$$

as an explicit diferential operator. Show that homogeneous polynomials are eigenfunctions, and calculate the eigenvalues.

Previous section: B.3 Chapters 5 to 7 · Next section: B.5 Chapter 9 · 中文版本

来源与版本

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

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：7f317d1896fa748af4cb9570310ed7801a8bef7ef97dd5d1d2ea35f0099d5952

OCR 产物 SHA-256：7f317d1896fa748af4cb9570310ed7801a8bef7ef97dd5d1d2ea35f0099d5952