B.14 Chapters 29 to 31

B.14 Chapters 29 to 31

Problem 1:

Show that for vectors $\mathbf{v}\in {\mathbf{R}}^n$ and ${ϵ}_{jk}$ the basis element of $\mathfrak{so}(n)$ corresponding to an infinitesimal rotation in the jk plane, one has

$$e^{-\frac{\theta }{2}{\gamma }_j{\gamma }_k}\gamma (\mathbf{v})e^{\frac{\theta }{2}{\gamma }_j{\gamma }_k}=\gamma (e^{\theta {ϵ}_{jk}}\mathbf{v})$$

and

$$\left[-\frac{1}{2}{\gamma }_j{\gamma }_k,\gamma (\mathbf{v})\right]=\gamma ({ϵ}_{jk}\mathbf{v})$$

Problem 2:

Prove the following change of variables formula for the fermionic integral

$$\int F(\xi )d{\xi }_{1}d{\xi }_2\dots d{\xi }_n=\frac{1}{\det A}\int F(A{\xi }^{\prime })d{\xi }_1^{\prime }d{\xi }_2^{\prime }\dots d{\xi }_n^{\prime }$$

where $\xi =A{\xi }^{\prime },i.e.$

$${\xi }_j=\sum _{k=1}^n{A}_{jk}{\xi }_k^{\prime }$$

for any invertible matrix A with entries $A_{jk}$

For a skew-symmetric matrix A, and $n$ = 2d even, show that one can evaluate the fermionic version of the Gaussian integral as

$$\int e^{\frac{1}{2}{\sum }_{j,k=1}^n{A}_{jk}{\xi }_j{\xi }_k}d{\xi }_{1}d{\xi }_2\dots d{\xi }_n=Pf(A)$$

where

$$Pf(A)=\frac{1}{d!2^d}\sum _{\sigma }(-1{)}^{|\sigma |}{A}_{\sigma (1)\sigma (2)}{A}_{\sigma (3)\sigma (4)}\dots A_{\sigma (n-1)\sigma (n)}$$

Here the sum is over all permutations $\sigma$ of the n indices. $Pf(A)$ is called the Pfafian of the matrix A.

Problem 3:

For the fermionic oscillator construction of the spinor representation in dimension $n=2d.$ , with number operator $\begin{array}{r}{N}_F={\sum }_{j=1}^d{a_F^{†}}_j{a}_{Fj}\end{array}$ , define

$$\Gamma =e^{i\pi N_F}$$

Show that

$$\Gamma =\prod _{j=1}^d(1-2a_{Fj}^{†}{a}_{Fj})$$ $$\Gamma =c{\gamma }_1{\gamma }_2\dots {\gamma }_{2d}$$

for some constant c. Compute c.

for all $j$ .

$${\gamma }_j\Gamma +\Gamma {\gamma }_j=0$$ $${\Gamma }^2=1$$ $$P_{\pm }=\frac{1}{2}(1\pm \Gamma )$$

are projection operators onto subspaces ${\mathcal{H}}^{+}$ and ${\varkappa }^{-}$ of ${\mathcal{H}}_F$ .

• Show that ${\mathcal{H}}^{+}$ and ${\varkappa }^{-}$ are each separately representations of spin(n) (i.e., the representation operators commute with $P_{\pm })$ ).

Problem 4:

Using the fermionic analog of Bargmann-Fock to construct spinors, and the inner product 31.3, show that the operators ${\mathbf{a}}_{Fj}$ and $a_{F_j}†$ are adjoints with respect to this inner product.

Previous section: B.13 Chapters 27 and 28 · Next section: B.15 Chapters 33 and 34 · 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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