A.1 Bilinear forms

对照

Parentheses

$$(\cdot ,\cdot )$$

will be used for a non-degenerate symmetric bilinear form (inner product) on a vector space ${\mathbf{R}}^n$ , with the same symbol also used for the complex linear extension to a bilinear form on ${\mathbf{C}}^n$ . The group of linear transformations preserving the inner product will be $O(n)$ or $O(n,\mathbf{C})$ respectively.

Angle brackets

$$⟨\cdot ,\cdot ⟩$$

will be used for non-degenerate symmetric sesquilinear forms (Hermitian inner products) on vector spaces ${\mathbf{C}}^n$ . These are antilinear in the first entry, linear in the second. When the inner product is positive, it will be preserved by the group $U(n)$ , but the indefinite case with $n=2d$ and group $U(d,d)$ will also occur. The quantum mechanical state space comes with such a positive Hermitian inner product, but may be infinite dimensional and a Hilbert space.

Non-degenerate antisymmetric forms (symplectic forms) will be denoted by

$$\omega (\cdot ,\cdot )\text{ or }\Omega (\cdot ,\cdot )$$

with the first used for the symplectic form on a real phase space $M$ of dimension $2d,$ and the second for the corresponding form on the dual phase space $\mathcal{M}$. The group preserving these bilinear forms is $Sp(2d,\mathbf{R})$ . The same symbol will also be used for the complex linear extension of these forms to ${\mathbf{C}}^{2d}$ , where the bilinear form is preserved by the group $Sp(2d,\mathbf{C})$ .

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上一节：Appendix A · Conventions · 下一节：A.2 Fourier transforms

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正文：英文 · 原著转录

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原书 PDF · 印刷页 533

来源版本：2025-10-20

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