32 A Summary: Parallels Between Bosonic and Fermionic Quantization

To summarize much of the material we have covered, it may be useful to consider the following table, which explicitly gives the correspondence between the parallel constructions we have studied in the bosonic and fermionic cases.

Bosonic	Fermionic
Dual phase space $\mathcal{M}={\mathbf{R}}^{2d}$	Dual phase space $\mathcal{V}={\mathbf{R}}^n$
Non-degenerate antisymmetric bilinear form $\Omega (\cdot ,\cdot )$ on $\mathcal{M}$	Non-degenerate symmetric bilinear form $(\cdot ,\cdot )$ on $\mathcal{V}$
Poisson bracket $\{\cdot ,\cdot \}$ on functions on $M={\mathbf{R}}^{2d}$	Poisson bracket $\{\cdot ,\cdot {\}}_{+}$ on anticommuting functions on $V={\mathbf{R}}^n$
Lie algebra of polynomials of degree 0, 1, 2	Lie superalgebra of anticommuting polynomials of degree 0, 1, 2
Coordinates $q_j,p_j$ , basis of $\mathcal{M}$	Coordinates ${\xi }_j$ , basis of $\mathcal{V}$
Quadratics in $q_j,p_j$ , basis for $\mathfrak{sp}(2d,\mathbf{R})$	Quadratics in ${\xi }_j$ , basis for $\mathfrak{so}(n)$
$Sp(2d,\mathbf{R})$ preserves $\Omega (\cdot ,\cdot )$	$SO(n,\mathbf{R})$ preserves $(\cdot ,\cdot )$
Weyl algebra Weyl( $2d,\mathbf{C}$ )	Clifford algebra Cliff( $n,\mathbf{C}$ )
Momentum, position operators $P_j,Q_j$	Clifford algebra generators ${\gamma }_j$
Quadratics in $P_j,Q_j$ provide representation of $\mathfrak{sp}(2d,\mathbf{R})$	Quadratics in ${\gamma }_j$ provide representation of $\mathfrak{so}(2d)$
Metaplectic representation	Spinor representation

Stone-von Neumann,

Uniqueness of ${\mathfrak{h}}_{2d+1}$ representation

$\mathcal{M}p(2d,\mathbf{R})$ double cover of $Sp(2d,\mathbf{R})$

$$J:J^2=-1,\Omega (Ju,Jv)=\Omega (u,v)$$ $$\mathcal{M}\otimes \mathbf{C}={\mathcal{M}}_J^{+}\oplus {\mathcal{M}}_J^{-}$$

Coordinates $z_j\in {\mathcal{M}}_J^{+},{\overline{z}}_j\in {\mathcal{M}}_J^{-}$

$U(d)\subset Sp(2d,\mathbf{R})$ commutes with $J$

Compatible $J\in Sp(2d,\mathbf{R})/U(d)$

$a_j,a_j^{†}$ satisfying CCR

$a_j|0⟩=0,|0⟩$ depends on $J$

$$\mathcal{H}={\mathcal{F}}_d^{fin}=\mathbf{C}[z_1,\dots ,z_d]=S^{\ast }({\mathbf{C}}^d)$$ $$a_j^{†}=z_j,a_j=\frac{\partial }{\partial z_j}$$

Positivity conditions, leading to unitary state space:

$\Omega (v,Jv)>0$ for non-zero $v\in \mathcal{M}$

$⟨u,u⟩=i\Omega (\overline{u},u)>0$ for

non-zero u $\in {\mathcal{M}}_J^{+}$

Uniqueness of $\mathrm{Cliff}(2d,\mathbf{C})$ representation on spinors

$Spin(n)$ double cover of $SO(n)$

$$J:J^2=-1,(Ju,Jv)=(u,v)$$ $$\mathcal{V}\otimes \mathbf{C}={\mathcal{V}}_J^{+}\oplus {\mathcal{V}}_J^{-}$$

Coordinates ${\theta }_j\in {\mathcal{V}}_J^{+},{\overline{\theta }}_j\in {\mathcal{V}}_J^{-}$

$U(d)\subset SO(2d,\mathbf{R})$ commutes with $J$

Compatible $J\in O(2d)/U(d)$

$a_{Fj},a_{F_j}†$ satisfying CAR

$a_{Fj}|0⟩=0,|0⟩$ depends on $J$

$$\mathcal{H}={\mathcal{F}}_d^{+}={\Lambda }^{\ast }({\mathbf{C}}^d)$$ $$a_j^{†}={\theta }_j,a_j=\frac{\partial }{\partial {\theta }_j}$$

$(v,v)>0$ for non-zero $v\in \mathcal{V}$

$$⟨u,u⟩=(\overline{u},u)>0\mathrm{for}$$

non-zero u $\in {\mathcal{V}}_J^{-}$ or ${\mathcal{V}}_J^{+}$

中文目录

Chapter contents

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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 349、350

来源版本：2025-10-20

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