理论物理学要点及其发展（54）

（接（53））

53． 各类多线矢与1线矢叉乘形成的高次、线多线矢

     各多线矢中，各“元”不含有相同的“组成元”。

     例如：有如下各例：

1．1线矢叉乘1线矢

(s(X,x)[基矢(X,x)],x=0到3求和)(有4维)

     叉乘

(r(X,x)[基矢(X,x)],x=0到3求和)

=((s(X,0)r(X,j)-s(X,j)r(X,0)) [基矢(X,0j)]

+(s(X,k)r(X,l)-s(X,l)r(X,k))[基矢(X,kl)]

,jkl循环=123循环求和)

=(s(X,0j) [基矢(X,0j)]+s(X,kl)[基矢(X,kl)]

,jkl循环=123循环求和)

即：2线矢(有6维)，

2．2线矢叉乘1线矢

(s(X,0j)[基矢(X,0j)]+s(X,kl)[基矢(X,kl)]

,jkl循环=123循环求和) (有6维)

     叉乘

(r(X,x)[基矢(X,x)],x=0到3求和)

=(s(X,0kl) [基矢(X,kl)]+s(X,jkl)[基矢(X,jkl)]

,jkl循环=123循环求和)

即：3线矢=赝1线矢(有4维)，

实际上，以上这类都是由多个4维时空1线基矢组成的高(n)次、线基矢的个数(即该矢量的维数)=从4个基矢中取n个地组合数=C(4,n)=4(4-1)...(4+1-n)/(n(n-1)...1), n<,=4, 即：

n=2; C(4,2)=4x3/2=6,

n=3; C(4,3)=4x3x2/(3x2)=4,

n=4; C(4,4)=4x3x2/(4x3x2)=1,(标量)

3．22线矢叉乘1线矢

(s(X,(0k,0l))[基矢(X,(0k,0l))]+s(X,(0j,kl))[基矢(X,(0j,kl))]

 +s(X,(0k,kl))[基矢(X,(0k,kl))]+s(X,(0l,kl))[基矢(X,(0l,kl))]

+s(X,(kl,lj))[基矢(X,(kl,lj))]

,jkl循环=123循环求和) (有15维)

     叉乘

(r(X,x)[基矢(X,x)],x=0到3求和)

=(s(X,(0k,0l)j)[基矢(X,(0k,0l)j)]+s(X,(0k,kl))[基矢(X,(0k,kl))]

+s(X,(0l,kl)j)[基矢(X,(0l,kl)j)]+s(X,(kl,lj)0)[基矢(X,(kl,lj)0)]

,jkl循环=123循环求和)

即：22,1线矢(有12维)，

4．22,22线矢叉乘1线矢

(s(X,((0k,0l)(0l,0j)))[基矢(X,((0k,0l)(0l,0j)))]

+s(X,((0k,0l)(0j,kl)))[基矢(X,((0k,0l)(0j,kl)))]

+s(X,((0k,0l)(0k,kl)))[基矢(X,((0k,0l)(0k,kl)))]

+s(X,((0k,0l)(0l,kl)))[基矢(X,((0k,0l)(0l,kl)))]

+s(X,((0l,0j)(0j,kl)))[基矢(X,((0l,0j)(0j,kl)))]

+s(X,((0l,0j)(0k,kl)))[基矢(X,((0l,0j)(0k,kl)))]

+s(X,((0l,0j)(0l,kl)))[基矢(X,((0l,jl)(0l,kl)))]

+s(X,((0j,0k)(0j,kl)))[基矢(X,((0j,0k)(0j,kl)))]

+s(X,((0j,0k)(0k,kl)))[基矢(X,((0j,0k)(0k,kl)))]

+s(X,((0j,0k)(0l,kl)))[基矢(X,((0j,0k)(0l,kl)))]

+s(X,((0k,0l)(jk,kl)))[基矢(X,((0k,0l)(jk,kl)))]

+s(X,((0k,0l)(kl,lj)))[基矢(X,((0k,0l)(kl,lj)))]

+s(X,((0k,0l)(lj,jk)))[基矢(X,((0k,0l)(lj,jk)))]

+s(X,((0l,0j)(jk,kl)))[基矢(X,((0l,0j)(jk,kl)))]

+s(X,((0l,0j)(kl,lj)))[基矢(X,((0l,0j)(kl,lj)))]

+s(X,((0l,0j)(lj,jk)))[基矢(X,((0l,jl)(lj,jk)))]

+s(X,((0j,0k)(jk,kl)))[基矢(X,((0j,0k)(jk,kl)))]

+s(X,((0j,0k)(kl,lj)))[基矢(X,((0j,0k)(kl,lj)))]

+s(X,((0j,0k)(lj,jk)))[基矢(X,((0j,0k)(lj,jk)))]

+s(X,((0k,kl)(0j,kl)))[基矢(X,((0k,kl)(0j,kl)))]

+s(X,((0k,kl)(0l,kl)))[基矢(X,((0k,kl)(0l,kl)))]

+s(X,((0l,kl)(0j,kl)))[基矢(X,((0l,kl)(0j,kl)))]

+s(X,((0l,kl)(0k,kl)))[基矢(X,((0l,kl)(0k,kl)))]

+s(X,((0j,kl)(0k,kl)))[基矢(X,((0j,kl)(0k,kl)))]

+s(X,((0j,kl)(0l,kl)))[基矢(X,((0j,kl)(0l,kl)))]

+s(X,((0j,kl)(jk,kl)))[基矢(X,((0j,kl)(jk,kl)))]

+s(X,((0j,kl)(kl,lj)))[基矢(X,((0j,kl)(kl,lj)))]

+s(X,((0j,kl)(lj,jk)))[基矢(X,((0j,kl)(lj,jk)))]

+s(X,((0k,kl)(jk,kl)))[基矢(X,((0k,kl)(jk,kl)))]

+s(X,((0k,kl)(kl,lj)))[基矢(X,((0k,kl)(kl,lj)))]

+s(X,((0k,kl)(lj,jk)))[基矢(X,((0k,kl)(lj,jk)))]

+s(X,((0l,kl)(jk,kl)))[基矢(X,((0l,kl)(jk,kl)))]

+s(X,((0l,kl)(kl,lj)))[基矢(X,((0l,kl)(kl,lj)))]

+s(X,((0l,kl)(lj,jk)))[基矢(X,((0l,kl)(lj,jk)))]

+s(X,((kl,lj)(lj,jk)))[基矢(X,((kl,lj)(lj,jk)))]

,jkl循环=123循环求和) (有105维)

     叉乘

(r(X,x)[基矢(X,x)],x=0到3求和)

=(s(X,((0k,0l)(0k,kl)j))[基矢(X,((0k,0l)(0k,kl)j))]

+s(X,((0k,0l)(0l,kl)j))[基矢(X,((0k,0l)(0l,kl)j))]

+s(X,((0l,kl)(0k,kl)j))[基矢(X,((0l,kl)(0k,kl)j))]

+s(X,((kl,lj)(lj,jk)0))[基矢(X,((kl,lj)(lj,jk)0))]

,jkl循环=123循环求和)

即：(22,22)1线矢(有12维)，

实际上，4维时空2线矢有6维，

两个2线矢组成的22线矢的维数=从6个2线基矢中取2个的组合数=C(6,2)=6(6-1)/2=15,

两个22线矢组成的22,22线矢的维数=从15个22线基矢中取2个的组合数=C(15,2)=15(15-1)/2=105,

对于多(n) 个22线矢和22,22线矢，基矢数分别为(n*)的维数=从其n*个基矢中取n个的组合数=C(n*,n)=n*(n*-1),,,(n*+1-n)/(n(n-1),,,1),

就会随n的增大而增大到成相应的赝矢量而减到1.

但是，4维时空22,1线矢的维数，就因其中1线基矢的各维不能与22,1各基矢的组成维相同，其维数=从4-1个基矢中取2个的组合数后的基矢数中再取两个的组合数，再乘以1线矢的4维数=C(C(3,2),2)x4=C(3,2)x4=3x4=12,

同理，

4维时空(22,22)1线矢的维数=C(C(C(3,2),2)x4=C(C(3,2),2)x4

=C(3,2)x4=3x4=12,

而且，4维时空((22,22)(22,22)1线矢,...,等的维数，就都=12。

    这些多线矢维数的变化规律，对以后研讨各类近程的多线矢力有重要意义与作用，须充分重视。

（未完待续）