29.1 Non-degenerate bilinear forms

Concept links · terms present in this machine draft; source roles are unverified: vector space · complex inner product · orthogonal group · complexification

In the case of $M = R^{2 d}$ , the dual phase space, the Poisson bracket determines an antisymmetric bilinear form on $M$ , which, for a basis $q_{j}, p_{j}$ and two vectors $u, u^{'} \in M$

u u^{'} = c_{q_{1}} q_{1} + c_{p_{1}} p_{1} + \dots + c_{q_{d}} q_{d} + c_{p_{d}} p_{d} \in M = c_{q_{1}}^{'} q_{1} + c_{p_{1}}^{'} p_{1} + \dots + c_{q_{d}}^{'} q_{d} + c_{p_{d}}^{'} p_{d} \in M

is given explicitly by

Ω (u, u^{'}) = c_{q_{1}} c_{p_{1}}^{'} - c_{p_{1}} c_{q_{1}}^{'} + \dots + c_{q_{d}} c_{p_{d}}^{'} - c_{p_{d}} c_{q_{d}}^{'} = (c_{q_{1}} c_{p_{1}} \dots c_{q_{d}} c_{p_{d}}) 0 - 1 ⋮ 00 10 ⋮ 00 \dots \dots \dots \dots 00 ⋮ 0 - 1 00 ⋮ 10 c_{q_{1}}^{'} c_{p_{1}}^{'} ⋮ c_{q_{d}}^{'} c_{p_{d}}^{'}

Matrices $g \in M (2 d, R)$ such that

g^{T} 0 - 1 ⋮ 00 10 ⋮ 00 \dots \dots \dots \dots 00 ⋮ 0 - 1 00 ⋮ 10 g = 0 - 1 ⋮ 00 10 ⋮ 00 \dots \dots \dots \dots 00 ⋮ 0 - 1 00 ⋮ 10

make up the group $S p (2 d, R)$ and preserve Ω, satisfying

Ω (g u, g u^{'}) = Ω (u, u^{'})

This choice of Ω is much less arbitrary than it looks. One can show that given any non-degenerate antisymmetric bilinear form on $R^{2 d}$ a basis can be found with respect to which it will be the Ω given here (for a proof, see [8]). This is also true if one complexifies $R^{2 d}$ , using the same formula for Ω, which is now a bilinear form on $\overset{ˉ}{C}^{2 d}$ . In the real case the group that preserves Ω is called $S p (2 d, R)$ , in the complex case $S p (2 d, C)$

To get a fermionic analog of this, all one needs to do is replace “nondegenerate antisymmetric bilinear form $Ω (\cdot, \cdot)^{†}$ with “non-degenerate symmetric bilinear form $(\cdot, \cdot)^{N}$ . Such a symmetric bilinear form is actually something much more familiar from geometry than the antisymmetric case analog: it is just a notion of inner product. Two things are diferent in the symmetric case:

• The underlying vector space does not have to be even dimensional, one can take $M = R^{n}$ for any $n,$ including n odd. To get a detailed analog of the bosonic case though, we will need to consider the even case $n = 2 d .$

• For a given dimension $n$ , there is not just one possible choice of $(\cdot, \cdot)$ up to change of basis, but one possible choice for each pair of non-negative integers $r$ , $s$ such that $r + s = n$ . Given $r, s .$ , any choice of $(\cdot, \cdot)$ can be put

in the form

(u, u^{'}) = u_{1} u_{1}^{'} + u_{2} u_{2}^{'} + \dots u_{r} u_{r}^{'} - u_{r + 1} u_{r + 1}^{'} - \dots - u_{n} u_{n}^{'} = (u_{1} \dots u_{n}) r + signs, s - signs 10 ⋮ 00 01 ⋮ 00 \dots \dots \dots \dots 00 ⋮ - 1 0 00 ⋮ 0 - 1 u_{1}^{'} u_{2}^{'} ⋮ u_{n - 1}^{'} u_{n}^{'}

For a proof by Gram-Schmidt orthogonalization, see [8].

We can thus extend our definition of the orthogonal group as the group of transformations g preserving an inner product

(g u, g u^{'}) = (u, u^{'})

to the case $r, s$ arbitrary by:

Definition (Orthogonal group $O ( r , s , { \bf R } )

) $. T h e g r o u p$ O ( r , s , \mathbf { R } ) $i s t h e g r o u p o f r e a l$ r + s $b y$ r + s $ma t r i ces$ g$ that satisfy
$g^{T} r + s i g n s, s - s i g n s 10 ⋮ 00 01 ⋮ 00 \dots \dots \dots \dots 00 ⋮ - 1 0 00 ⋮ 0 - 1 g = r + s i g n s, s - s i g n s 10 ⋮ 00 01 ⋮ 00 \dots \dots \dots \dots 00 ⋮ - 1 0 00 ⋮ 0 - 1$

$S O (r, s, R) \subset O (r, s, R)$ is the subgroup of matrices of determinant +1.

If one complexifies, taking components of vectors to be in $C^{n}$ , using the same formula for $(\cdot, \cdot)$ , one can change basis by multiplying the s basis elements by a factor of $i,$ and in this new basis all basis vectors $e_{j}$ satisfy $(e_{j}, e_{j}) = 1$ One thus sees that on $C^{n}$ , as in the symplectic case, up to change of basis there is only one non-degenerate symmetric bilinear form. The group preserving this is called $O (n, C)$ . Note that on $C^{n} (\cdot, \cdot)$ is not the Hermitian inner product (which is antilinear on the first variable), and it is not positive definite.

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 318、319、320、321、322、323、324、325

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：255e6de176b891a110e84138f84ce4224218312b2f275d3d598b59b1d077e160

OCR 产物 SHA-256：255e6de176b891a110e84138f84ce4224218312b2f275d3d598b59b1d077e160

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29.1 Non-degenerate bilinear forms

29.1 Non-degenerate bilinear forms

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