?????????? ??????

×	1	i	j	k
1	1	i	j	k
i	i	?1	k	? j
j	j	? k	?1	i
k	k	j	? i	?1

?????????? ???? ???? ??? ???? ????? ????? ?? ?? ??????? ?????? ??? ????? ??? ?? ?? ??? ?????? ?????? ???? ??????? ?????? ???? ??????? ????? 1843/???? ???? ????? ?? ??? ??? ????-????? ???? ???? ??????? ???? ???? ???? ?? ??? ???????????? ?? ??? ??? ?? ?? ?? ?? ???????????? ?? ?????? ???-????????? ????? ??? ??????? ?? ??? ?????????? ??? ????-????? ???? ???? ?? ????????? ??????? ?? ?????? (????) ?? ??? ?? ????????? ???? ??? ?? ??????? ?? ?????? ?? ??? ?? ????? ????????? ?????

???????????? ?? ???? ??????? ??? ??????? ???? ????? ???? ????? ??, ??? ???? ????-????? ????????? ?????? ???????????? ?? ?????? ???? ???? ????? ??????? ????????, ??????? ????? ??? ????????????????? ?????? ????????? ?????? ??? ????????? ??????????? (???????? ???????) ????, ????? ??? ??? ??????? ?? ??? ??? ????? ?? ???? ??, ????? ???? ???? ??? ??????? ??????????, ??? ????? ?? ???? ????, ????? ?? ?????? ???? ???? ?? ???? ???

????? ?????? ????? ????, ?????????? ??????? ?????? ???? ??? ???-????? ??????? ????? ??????? ?????? ???? ??, ??? ?? ???? ????? ?? ??? ????? ????? ??? ?????, ?????????? ????? ????? ??? ???-????????? ??????? ?????? ??? ???????????? ?? ?????? ??? ???? H (??????? ???? ??) ??? ????? ????? ??, ??? ???????? ???? ???? ${\displaystyle \mathbb {H} }$ (Unicode U+210D, ? ) ????? ????? ????? ??? ?? ???????? ?????? ?????????? C ℓ _0,2 ( R ) ? C ℓ ⁰ _3,0 ( R ) ????? ?? ?????? ?? ????? ??? ?????????? ????? ???????, ?????? H ?????? ????? ???? ??? ??????? ???? ????? ??, ?? ??? ??? ??-???? ?? ??? ?? ??????? ???? ???? ???? ???? ?? ????-????? ??????? ?????? ?????? ??? ??, ???? ???? ??????? ???? ??? ????? ?????? ??? ??????? ???????? ?????? ?? ???? ????? ??, ?????? ?????? ?????????? ?? ??? ???? ????????? ?????? ???

???? ????? ???????????? ??? 3-????? S ³ ???? ???? ????? ???? ?? ??? ??? ?? ??? ???? ????? ?? ???? ?? ?? ????? Spin(3) ????? ??, ?? SU(2) ????? ?????????? ????? ?? ??? SU(3) ?? ????????? ??? ????? ?? ?????????? ????? ???

???????? [ ???? ]

??? ???? ?? ??? ????, ?????????? H, ??????? ?????? ???? ??? ???-????? ?????-???? R ⁴ ????? ??? H ?? ???? ??????? ??: ???, ????? ?????? , ??? ?????????? ??????? H ?? ?? ????????? ?? ???, R ⁴ ?? ????????? ?? ??? ?? ????? ?? ??? ?? ??? ???? ????????? ???? ????? ??? ??? ????, H ?? ???? ??????? ?? ???? ??????? ???? ??? ?????? R ⁴ ???? ???? ????? ??? ?????? ?? ???? ?? ????? ??? H ????? ?? ????????? ?? ?????? ??? ????????? ??? ?? R ⁴ ?? ????? ?? ??? ?? ??? ????? ??? ?? ????? ?? ????????? ??? ????? ?? ??? ?? 1, i, j, ??? k ????? ????? ??? H ?? ???? ??????? ????? ????? ????????? ?? ???? ?????? (?????) ??? (?????????) ?? ??? ???? ?????? ??? ?? ????? ?? ???? ??, ????? a1 + bi + cj + dk, ????? a, b, c, ??? d ??????? ???? ??? ????? ??????? 1 ??? H ?? ????????? ??????? ???? ?????, ????? ??? ?? ?? 1 ??? ?????? ??? ?? ???? ???? ?????, ??? ?? ???? ????, H ?? ????????? ??? ????? ?? ????? ??????? 1 ??? ???? ?? a + bi + cj + dk ????? ????? ??? ?? ????? ?? ????? ??? ??, ????????? ?????????? ?????? ???, ????? ????????? ?? ?????? ??? ?????? ????????? ???? ??? ??? ????????????? ???? ????? ???? ???? ?????? ????????? ????, ????????? ???? ????? ???

?????? [ ???? ]

?????????? ?????? 1843 ???? ??????? ????? ????? ??? ??? ?? ??? ??? ?????? ???? ??? ???? ????????? ????? ???? ???? ?? ???-??? ???? (???-?????? ?????????) (1748) ??? ?????? ?????????? ?? ??? ?????????? ???? ???? ????????? ?? ????????????????? (1840) ????? ??, ?? ????? ?????? ?????? ???? ?? ?? ??? ?????????? ????????? ??? ?????? ?? ??? ???? ???? ?????? ???? ????????? ????? ?? ?? 1819 ???? ?????????? ???? ???, ?? ???? ??? 1900 ??? ?? ?????

???????? ????? ??? ?? ??????? ?????? ?? ???? ???? ???? ?????? ??????? ?? ??? ???? ?? ???? ??, ??? ?? ????-????? ???? ???? ??????? ?? ??? ???? ??? ?? ????? ??? ???? ??? ???? ???? ??????? ??? ????? ?? ???????????? ?? ??? ???? ???????? ???? ?? ???? ???, ?? ?????? ?? ?????? (?????) ????? ??, ??? ?? ????? ??? ?? ??? ?????? ?? ?? ?????? ?? ??????? ??? ????? ????? ??? ?????? ??? ??? ??, ??????? ??? ???? ???? ??? ?????? (????????????) ??? ????? (???????) ?? ?????? ???? ???? ???? ??? ?? ?? ???? ???? ???? ?? ???? ???? ?? ??????? ?? ???????????? ?? ?????? ??? ????? ?????? ???

???????????? ???? ???? ??????? ?? ??? ??? ??? ?????? 16 ?????? 1843 ?? ???? ????? ???? ? ?? ???, ???? ??????? ???? ?????? ?????? ?? ???? ???? ?? ???? ??, ????? ?? ??? ????? ?????? ?? ???????? ??? ?? ???? ??? ???? ?? ?? ???? ???? ??? ???? ???? ?????? ???? ????? ?????, ???????????? ????? ?? ????? ???? ????? ???? ???? ?? ??? ??? ???? ???? ?? ??? ????? ?? ???, ??????? ???????????? ?? ?? ???????, ??? ?? ????? ????? ?? ???? ???? ?????? ??? ??? ?? ?? ????,

i ² = j ² = k ² = ijk = ?1,

???? ???, ??????? ?? ???? ???? ??? ???? ???? ??????? ???? ??. ?????? ??? ????? ????, ??? ???? ???? ??? ?? ????? ??? ???? ????? ?? ??? ????? ?? ??? ?? ????? ???? , ???????, ??? ????? ????????? ??????? ??? ???? ??? ?????, ?????? 15 (1844), ?? ?? 489-95 ???? ??? ???? ????? ???, ??????? ?? ???? ??? ???? ????? ?? ????? ????? ???? ?? ????? ???? ?? , ???? ??? ????, ??????? ??? ???????? ??? ?? ???? ?? ???? ?? ??? ???? ???? ??? ??? ???? ?????? ??…??? ????????? ???? ??? ????? ?????, ??? ??? ??????? ???? ?????

?????? ?? ????? ?????? ??? ??????? ?? ???????? ??? ??? ?????????? ?? ??? ?????, ??? ???? ???? ?????? ?? ???? ???? ??????? ????? ????? ??? ????? ??? ????? ??? ?? ?????? ????? ??????? ?? ????? ????? ??????????? ????? ????? ???? ?????? ??, ????? ???????????? ???? ??????? ???????????? ???? ??? ?????? ???? “???????????????” ?? ??? ???? ?? ???? ????, ??? ?? ??????? ???? ???? ???????????? ??? ??????? ??? ?? ??? ????? ?????? ??????? ?????? ????? ??? ?? ??? ????? ???? ?????, ???????? ?? ????????????, 800 ?????? ???? ????? ???; ?? ???? ??? ??? ???? ???? ??? ?? ???? ?? ???

??????? ?? ?????? ??? ??? ??? ??? ???, ???? ???????? ???? ???? ?? ???????????? ?? ??????? ???? ????? ?? ??? ??, ????? ???? ?????????? ??? ????? ??????? ????? ??? ????? ?????? ??? ???????? (????????) ???? ????? ?? ??? ??????? ?? ???? ??? ????? ????? ?????, ????? ???? ???? ??????????? ??? ???????? ????????, ???? ?? ???? ?? ???? ???????????? ?? ?????? ???? ????? ????? ??? ????? ??? ?? ??? ?????????? ????? ????? “?????????? ???????” ?? ???, ?? ???????????? ??? ??? ?????- ??????? ???? ??????? ?? ????? ????? ?????? ???? ?? ???

???? 1880 ??? ????? ???, ???????????? ??? ????? ?????? ????? ?????? ??? ?????, ?? ?????? ??????? ??????, ????? ?????????, ??? ???? ??? ?????????? ????? ?????? ???? ?? ??? ????? ????????? ?? ??? ???? ??? ???????????? ?? ???? ??????, ???? ???? ???????????? ?? ????? ??? ???? ????? ??? ???????? ?????? ??? ????? ??? ??? ??, ????? ????????? ??????? ??? ?? ????? ??? ?????? ?? ??? ?? ??? ????? ??, ??? ??? ??? ?????????? ???? ??? ????? ?????? ???? ???? ???? ?????? ??? ???? ?? ??? ?? ????? ??? ?? ?????????? (??????????) ?? ????-?????? (????-???) ?? ?? ?? ??????? ?? ??? ?? ????? ?????? ?? ????? ???? ??? ??????? ???? ??? ??????????? ?????? ??????? ???? ?? ??? ???? ???? ?? ?????? ???? ????? ?????? ???? ?? ??? ???? ?? ???? ???

??? ??, ??? 20??? ??? ??? ???????????? ?? ??? ???????? ???, ???? ??? ?? ???????? (??????) ????????? ?????? ???? ????? ?? ???? ????? ???????????? ????? ????????? ???? ??????????, ?????????? ????? ?????????? ????? ????? ????? ??? ??? ??? ???? ??? ??, ???? ?????? ??? ???, ?? ????? ??? ?? ????? ???? ??? ?? ???? ????, ???????????? ??? ??????? ????????, ??????? ?????, ????????, ?????? ?????, ???? ???????????, ??????? ??????, ????? ?????? , ???????????????, ????????? ?????????, ??????? ????????????, ??? ?????? ??????? ???? ????? ????? ??? ?????? ?? ??? ??, ???????????? ?? ?????? ???? ?????????? ?? ??????? ?????? ??????? ??? ????????? ??? ?? ???? ?? ??? ??? ???????????? ?? ???? ????? ??? ??? ??? ???? ?????? ???? ?? ?????? ???? ?? ?????????? ?????? ??? ???? ????? ???

1989 ???, ?????? ????????? ?? ??????? , ????? ?? ???? ????? ?? ??? ????-????? ?????? ???? ??, ????? ??????? (??? ???? 2002 ???? ????? ??????? ?????? ???-???? , 2005 ???? ????? ?????? , ??? 2003 ???? ???? ??????? ??????? ?????? ) ??????? ?????????? ??? ???? ???? ????? ??? ????? ???? ??? ??????? ?? ?????? ??? ??????????? ???? ????

????? ?????? ???? ???????? ?????? [ ???? ]

??.?? ?????? ?? ??? “?? ?????????? ????? ??? ????? ????? ??????” ????? ?????? ???? ???????????? ???? ???? ????????? ?? ???? ???? ??? ?? ???????? ?? ?? ????? ?????? ?????? ????????? ?????: SO(3), ??????? ????? , ???? ?????????? ?????, ???????? ?????? SU(2), ??? ?????? ????? ????? ?????? ???? ?????????? ????? ??? ????? ???? ?? ???? ??? ?????? ?? ????? ?????????? ?? ???? ??? ???????????????? ?? ???? ???? ??????? ?? ???????? ??? ????? ????? ??? ?? ??? ?????? ?? ??? ?? ??????????? ??? ???? ????? ????? ??? ???? ?????? ?????????? ?? ??????? ????? ??? ??????? ??? ?? ??????? ???????????? (???????????????) ??? ?????, ??? ???? ???? ??????? ?????? ??? ???? ?????? ???????? ??? ????? ???? ???, ??? ???????? ?? ?????? ?????, ????? ??? ?????????? ???????? ?? ??? ???????? ?????? ??? ??????? ????????? ??? ??? ????? ???????????? ??????? ???? ?????? ?? ????? ??? ???? ?????????? ??? ?????, ???? ?????-?????? ????? ??? ??????? ???? ??????????????? (?????? ???????????????) ?? ??? ???????? ?????? ?? ?????? ???? ???? ??? ????, ??? ????????????? ?? ?????????? (???) ??? ???? ???? ???, ?????? ?? ???????? ???? ?????? ????? ??? ??????? ????? ??????? ????, ?????? ?? ?????????? ????????? ??? ???? "?????????? ???????" ?? ??????? ??? ?????? ????? 1999 ????, ???? ?????? ?? ????? ???????? ???? ???? ?????????? ???? ????????? ??? ???????? ?????? ???? ?????????? ?????? ?? ?????? ?? ?? ???????????? ??? ????? ????? ????? ???

???????????? ????? ??? ??? ????? ??????????? ????? ?????? ????? 1995 ???? ????? ??? ??? ???? ???? ??? “???? ??????? ?? ????? ?????? ???? ???????????? ??? ??????? ???? ?????’’ ???? ?????: “???? ???? ???? ??? ??????? ???????? ??? ?? ???? ???????? ?? ????? ????? ???? ??????” ???? ?? ?????? ??? ??? ?????? “??? ?????? ????? ??? ??????? ???? ??? ?? ?? ??? ??????? ????? ?????? ?? ???? ??????? ??? ???? ?? ?? ??? ???? ??????? ??????? ?? ????? ????? ???? ????????? ?? ?????????? ??? ?????-??? (???? ????) ?????? ??? ???? ???”

?????? [ ???? ]

?????? ??? ?? ?? ????? ????? ???? ?????? ??,

????? ????????? ?? ?????? [ ???? ]

????? ?????? ??????????? H ?? ????? ???????? i, j, ??? k ?? ???? ???? ?????? ???????? ?????? ???

${\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1}$,

?????? ?? ??? ??, ?1 = ijk ?? ???? ?????? ??? ???? ???? k ??? ???? ??? ?? ?? ????? ??;

${\displaystyle {\begin{aligned}-k&=ijkk=ij(k^{2})=ij(-1),\\k&=ij.\end{aligned}}}$

??? ????? ??? ???? ??? ???? ?????? ???????? ???? ?? ???? ??? ?? ?? ????? ????? ??;

${\displaystyle {\begin{alignedat}{2}ij&=k,&\qquad ji&=-k,\\jk&=i,&kj&=-i,\\ki&=j,&ik&=-j,\end{alignedat}}}$

?????? ??? ????? ????? ??? ?? ????? ?? ???? ?? ??????? ??????? ?????? (????) ?????? ?? ???? ????? (?????) ???????? ?????? ??? ??? ????? ????? ???? ???? ????? ???????? ???? ???, ????? ?? ?????? ?? ?????? ???? ?????? ??? ???

?????? ?? ???-???? ??????? ???? [ ???? ]

?????????? ?????? ?? ???-????????????

×	1	i	j	k
1	1	i	j	k
i	i	?1	k	? j
j	j	? k	?1	i
k	k	j	? i	?1

??????? ??? ??????? ?????? ?? ?????? ??? ???, ???????????? ?? ?????? ????????? ???? ?????? ?????? ?? ??? ??, ij = k ????? ?? ???? ?? ji = ?k ????? ??? ?????? ?? ???-???????????? ?? ???? ?? ???? ???? ??? ???? ????? ??? ?????? ?????? ??? ?? ?? ?? ???????????? ???? ?????????? ???????? ?????????? ?? ????? ??? ????? ????? ??? ??? ?????? ??? ?????? ?? ??? ??, ?????? z2 + 1=0 , ?? ???? ??? ?? ?? ??? z = bi + cj + dk ????? ?? ?????? ???? b2 + c2 + d2 = 1 ????, ??? ?? ?? ???, ????? ??????? ????? ???? ???????????? ?? ????-????? ?????? ?????? ????? ???? ???????? ???? ????? (????) ?? ??-????? ???? ???? ???? ???? ?? ????? ??????? ???? ??? ?? ??????? i ??? -i ???? ????? ???

??? ?? ?????????? ?????? ????????? ???? ?????, ???????????? ??? ???? ??? ????? ???? ?? ??? ???? ????? ????? ??? ???? ?????? ??? ????? ???

??????? ?????? [ ???? ]

?? ????????? a ₁ + b ₁ i + c ₁ j + d ₁ k ??? a ₂ + b ₂ i + c ₂ j + d ₂ k ??? ????? ?? ??????, ?????? ??????? ?????? ( a ₁ + b ₁ i + c ₁ j + d ₁ k ) ( a ₂ + b ₂ i + c ₂ j + d ₂ k ) ???? ????? ??, ????? ????????? ?? ?????? ??? ????????????? ???? ????? ???????? ????? ??? ?????????????? ???? ?????? ?? ?????? ???? ???? ??? ????? ?? ??? ?? ?? ????? ????????? ?? ?????? ?? ??? ????? ??? ?? ???? ?????? ??????? ????? ??;

${\displaystyle a_{1}a_{2}+a_{1}b_{2}i+a_{1}c_{2}j+a_{1}d_{2}k}$

${\displaystyle {}+b_{1}a_{2}i+b_{1}b_{2}i^{2}+b_{1}c_{2}ij+b_{1}d_{2}ik}$

${\displaystyle {}+c_{1}a_{2}j+c_{1}b_{2}ji+c_{1}c_{2}j^{2}+c_{1}d_{2}jk}$

${\displaystyle {}+d_{1}a_{2}k+d_{1}b_{2}ki+d_{1}c_{2}kj+d_{1}d_{2}k^{2}.}$

??? ????? ????????? ??? ???? ???? ?????? ??? ????? ??? ???? ???? ?? ???? ?? ??? ?? ?? ??? ???:

${\displaystyle a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2}}$

${\displaystyle {}+(a_{1}b_{2}+b_{1}a_{2}+c_{1}d_{2}-d_{1}c_{2})i}$

${\displaystyle {}+(a_{1}c_{2}-b_{1}d_{2}+c_{1}a_{2}+d_{1}b_{2})j}$

${\displaystyle {}+(a_{1}d_{2}+b_{1}c_{2}-c_{1}b_{2}+d_{1}a_{2})k.}$

?????? ???? ???? [ ???? ]

H ?? ????? 1, i, j, k ????? ???, H ??? ?????????? ?? ??? ???? ?? ??? ???? ????? ???? ?? ????? ??:

${\displaystyle \mathbf {H} =\{(a,b,c,d)\mid a,b,c,d\in \mathbf {R} \}.}$

??? ????? ??????? ?? ????? ??:

${\displaystyle {\begin{aligned}1&=(1,0,0,0),\\i&=(0,1,0,0),\\j&=(0,0,1,0),\\k&=(0,0,0,1),\end{aligned}}}$

??? ??? ??? ?????? ?? ??????? ?? ????? ??:

${\displaystyle (a_{1},\ b_{1},\ c_{1},\ d_{1})+(a_{2},\ b_{2},\ c_{2},\ d_{2})=(a_{1}+a_{2},\ b_{1}+b_{2},\ c_{1}+c_{2},\ d_{1}+d_{2}).}$

${\displaystyle {\begin{aligned}(a_{1},\ b_{1},\ c_{1},\ d_{1})&(a_{2},\ b_{2},\ c_{2},\ d_{2})=\\&=(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2},\\&{}\qquad a_{1}b_{2}+b_{1}a_{2}+c_{1}d_{2}-d_{1}c_{2},\\&{}\qquad a_{1}c_{2}-b_{1}d_{2}+c_{1}a_{2}+d_{1}b_{2},\\&{}\qquad a_{1}d_{2}+b_{1}c_{2}-c_{1}b_{2}+d_{1}a_{2}).\end{aligned}}}$

????? ??? ????? ????? [ ???? ]

a + 0i + 0j + 0k ?? ???? ?? ??? ????, ????? a ??? ??????? ???? ????? ??, ??? ??????? ???? ???? ????? ??, ??? ??? ?? ???? ?? 0 + bi + cj + dk ???? ?? ????, ????? b, c, ??? d ??????? ???? ???, ??? ???? ??? ????? ?????? b, c, ??? d ?????? ??? ?? ??? ???? ???-????? ????, ??? ????? ????? ??????? ???? ???? ????? ??? ???? a + bi + cj + dk ??? ?????????? ????, ??? a ??? ???? ????? ????? ???? ????? ?? ??? bi + cj + dk ??? ???? ????? ????? ???? ????? ??? ???? ?????????? ?? ????? ????? ??????? ?? ??????? ????? ??, ??? ????? ????? ????? ??????? ????? ??? ????? ???? ?????????? ??? ??? ???-????? ????? ???? ???? ???? ????? ?? ??? ?? ????? ?? ???? ??, ???? ???? ????? ??? ??? ????? ??????? ?????????? ?? ????? ???? ????????? ???? ?? ??? ??? ?? ??????? ??? ??? ?????, ????? ???? R ³ ?? ???? ??????? ????? ?? ????? ???

??? ??, ?? ??? ???? ????????? ?? ??, ???? ?????????? ?? ????? ?????, ?????, ??? “??????” (???? ????) ????? ??? “?????????” ????? ??, ??? ????? ??? “????” ????? ???? ?????, ????? ???.???. ??????? ????? ???? 1986 ??? ?? ????? “?????????,???????????? ??? ??? ??????” ?? ????? 12 ???? ?????? ???? ???? ??? ??? ??? ???? ????? ??? ??????? ???? ????, ??? ????? ?????????? ????? ???????????? (?????? ?? ??? ??, ???? ???? ?????????? ??? “???????? ?????????” ????? ?????? ????? ??) ???? ??????? ???? ?? ???? ??, ?? ????? ????? ?????? ???????? ???? ???? ???? ??????? ???? ?????? (180 ?????) ??????? ?????????? ?? ????, ????? ????? ????, ??????? ???? ?? ???? ????? ?? ????? ??? ????? ????? ???

??????? ?? ????? ??????? ???????????? ??? ???? ?????????? ???? ??? ??????? ?????? (?? ????? ????? ????? ???? ???????????? ?? ??? ?? ?? ????? ??) ??? ????? ?????????? ?????

???? ???? ?????????? ??? ??? ????? ????? ??? ??? ????? ????? ???? ????? ???? ????, ???? ??,

${\displaystyle q=(r,\ {\vec {v}}),\ q\in \mathbf {H} ,\ r\in \mathbf {R} ,\ {\vec {v}}\in \mathbf {R} ^{3}}$

??? ??? ??? ?????? ?? ??????? ?? ?? ????? ??:

${\displaystyle (r_{1},\ {\vec {v}}_{1})+(r_{2},\ {\vec {v}}_{2})=(r_{1}+r_{2},\ {\vec {v}}_{1}+{\vec {v}}_{2})}$

${\displaystyle (r_{1},\ {\vec {v}}_{1})(r_{2},\ {\vec {v}}_{2})=(r_{1}r_{2}-{\vec {v}}_{1}\cdot {\vec {v}}_{2},r_{1}{\vec {v}}_{2}+r_{2}{\vec {v}}_{1}+{\vec {v}}_{1}\times {\vec {v}}_{2})}$

????? "?" ??? ??????? ?? ??? "×" ???? ??????? ???

?????????, ???? ??? ?????????? [ ???? ]

???????????? ?? ????????? ??????? ?????? ?? ????????? ??? ???????? ?????? ?? ????????? ?? ???????????? (??? ??? ?????? ?? ???? ????? ??) ???? ??? ?? ??? ????????? ??? ?? ??? ?? ${\displaystyle q=a+bi+cj+dk}$ ??? ?????????? ??? q ?? ??????? ?????????? ${\displaystyle q^{*}=a-bi-cj-dk}$ ??????? ????? q ^? , q , q ^t , ??? ${\displaystyle {\tilde {q}}}$ ??? ????? ????? ??? ????????? ??? ????? ????? ??, ????? ??? ?? ?? ???? ??? ?? ??? (?????) ??, ???? ???? ??????? ??? ?? ??? ??????? ??? ??? ??? ??????? ??? ????? ? ????? ??? ?? ???????????? ?? ?????? ?? ??????? ???? ???? ???? ?? ????????? ?? ?????? ????? ????? ??? ???? ??, ???? p ??? q ?????????? ??? ??? (pq)? = q?p? ????? ??, ?? ?? p?q? ?????? ?

??????? ???? ???? ???????? ??? ???, ???? ?????????? ?? ????????? ??? ?????? ??? ??? ??? ???? ???? ?????? ?? ???? ??:

${\displaystyle q^{*}=-{\frac {1}{2}}(q+iqi+jqj+kqk).}$

????????? ?? ????? ???? ?????????? ?? ????? ??? ????? ??????? ??? ????? ???? ?? ???? ?? ???? ??? p ?? ????? ????? (p + p?) / 2 ????? ??, ??? p ?? ????? ????? (p ? p?) / 2 ????? ???

???? ?????????? ?? ???? ?? ??????? ??? ?????? ?? ?????? (?????? ???) ??? ???? ???? ???? ????? ?? ??? ????? ||q|| ????? ????? ?? (??????? ?? ?? ????? ??? q ?? ????? ???? ??, ?? ?? “?????” ?? ????? ??? ??? ???? ???? ??) ??????? ????, ????? ?? ???? ?????? ????? ??:

${\displaystyle \lVert q\rVert ={\sqrt {qq^{*}}}={\sqrt {q^{*}q}}={\sqrt {a^{2}+b^{2}+c^{2}+d^{2}}}}$

?? ??????? ?? ??? ???-??????? ??????? ???? ????? ??, ??? ?? ??? ????? ?? ?? ????? ???? R ⁴ ?? ??? ?? ??? ??? ???? ??????? ???? ????? ??? ???? ?????????? ??? ??? ??????? ???? ??? ???? ??? ?? ???? ???? ??? ???? ?? ????? ???? ??? ??? ????? ??? ???? ??, ???? α ??????? ???? ????, ???

${\displaystyle \lVert \alpha q\rVert =|\alpha |\lVert q\rVert .}$

?? ?? ??? ?? ??? ??? ????? ?? ?? ???? ???? ??? ???? (????????????) ????? ??, ????? ??? ?? ??; ???? ?? ?? p ??? q ??,

${\displaystyle \lVert pq\rVert =\lVert p\rVert \lVert q\rVert .}$

???? ??? ?? ????????? ???? ?????? ?? ??????? ?? ??????? ?? ????? ??? ???? ???? ???? ?? ????????? ??? ??? ???? ?? ??,

${\displaystyle \det {\Bigl (}{\begin{array}{cc}a+ib&id+c\\id-c&a-ib\end{array}}{\Bigr )}=a^{2}+b^{2}+c^{2}+d^{2},}$

(????? i ?? ??????? ????? ??? ???????? ??) ??? ?????? ?????? (???) ?????????? ?? ????????????? ?? ???? ??? ?? ????????? ??? ?? ??? ??? ???? ???

?? ???? p ??? q ?????? ??????? d(p, q) ??? ????? ?? ??? ?? ???? ???? ????????? ???? ?? ???? ???? ??:

${\displaystyle d(p,q)=\lVert p-q\rVert .}$

?? H ??? ??? ??????? ???? ??? ????? ??? ??????? ???????? ???? ??? ??? ???? ?????? ????? ??? ???????, ???? ?? ????? ???????? a ??, ?? ???? ????? ??,

${\displaystyle \lVert (p+ap_{1}+q+aq_{1})-(p+q)\rVert =a\lVert p_{1}+q_{1}\rVert .}$

A ??? ?????? ?? ???????? ???? ?????? ??? ??? ???? ?????? ?? ?? ????? ???

????? ?????????? [ ???? ]

??? ????? ?????????? ??? ???? 1 ???? ?????????? ????? ??? ???? ???-????? ?????????? q ??? ??? ?? ???? ??? ????? ??? ?? ??? ????? ?????????? Uq ????? ?? ?????? q ?? ???? ???? ????? ??:

${\displaystyle \mathbf {U} q={\frac {q}{\lVert q\rVert }}.}$

???? ?????????? ?? ??? ???? ???????????? q = ||q|| Uq ????? ???

????????? ??? ?????? ?? ????? ???? ???-????? ?????????? ?? ?????????? ??? ????????? ???? ???? ???? ??? ???? ?????????? ?? ??? ?? ?????????? ??? ?????? 1 ????? ???? ?????? ??, ??? ???? ??????? ???????? ??? ??? ?? ?? ${\displaystyle q}$ ??? ${\displaystyle q^{\ast }/\left\Vert q\right\|^{2}}$ (???? ?? ???? ????) ?? ?????? 1 ????? ??? ??????, q ?? ?????????? ??? ?????? ????????? ???? ????? ??:

${\displaystyle q^{-1}={\frac {q^{*}}{\lVert q\rVert ^{2}}}.}$

?? ?? ???????????? p ??? q ??? ?? ??? ????? ??????? (???? q ???-????? ????) ??? ????? ???? ???? ???? ??? ???? ??, ????? ?? ?????? p?q ^?1 ??? q ^?1 p ?????? ??? ?? ?? ???? ??? ????? p/q ?????? ?????? ?? ?????? ?? ??????? ??? ?? ???? ???? ?????? ?? q ???? ???? ????? ????? ?? ?? ???? ???? ????? ????? ???

??????? ???????????? [ ???? ]

???? ???????????? ?? ???? H ????????? 4 ???? ??????? ?????? ???? ??? ????? ???? ????? ??? (????? ?? ??? ??, ??????? ?????? ?? ????????? 1 ????? ??, ??????? ?????? ?? ????????? 2 ????? ??, ??? ?????????? ?? ????????? 8 ????? ??) ???????????? ?? ??????, ?????? ?? ??? ??, ????? ??? ???? ????????? ??? ????????????? ????? ??, ?? ????????? ???? ?????? ????, ?????????? H ??????? ?????? ???? ??? ???-????????? ????????? ?????? ????? ??? ????? H ???? ??????? ?????? ???? ????? ?????? ??????? ??, ??? ?? ?? ??????? ?????? ???? ????????? ?????? ???? ??????

?????? ???????????? ??? ????? ???? ???? ????? ??, ???? ?? ??? ??????? ?????? ???? ??? ?????? ?? ???-???????????? ??? ??? ?? ?? ???? ???? ???? ???? ????? ??? ??????? ?????? ???? ????-?????????? ????????? ??????? ?????? ???? ????? ?? ????? ??? ?????????? ????? ????? ?? ?? ????? ???? ????? ??: R,C, ??? H? ????, ???????????? ?? ??? ????? ?????? ??? ????? ??, ??? ??????? ?????? ???? ????? ??????? ?????? ???? ??? ????? ??: ???????? ????? ????? ?? ?? ????? ??? ????? ??: R,C,H ??? O (????????)? ?????????? ??? ?????????? ?????? ?? ?? ??? ?????? ????? ?? ??? ??????? ????? ?????? ?? ?? ??? ?????? ???

?????? ???? ?? ????? ??????? ?? ?????? ???? ??? ????? ????? ?? ??? ??? ????? ????? ??, ???? {±1, ±i, ±j, ±k} ?????? ???? ??? ????? ???? ??? ?? ????? ??? ?????????? ????? ???? ????? ?? ??? Q8 ????? ????? ??? ??????????, R[Q8] ?? ?????? ???? ????? ?? ?? ????????? 1 + (?1), i + (?i), j + (?j), ??? k + (?k) ????? ???? ????? ??? ???? ???? ?????? ????? ????? ??? ????? ????????? 1, i, j, ??? k ?????? ??? ????? ??, ??? ???? ??? ????? ????????? ?1, ?i, ?j, ??? ?k ?????? ??? ????? ??, ?? ?? 1, i, j, ??? k ?? ??? ??????

?????????? ??? R ³ ?? ???????? [ ???? ]

?????? ??? ?????????? ?? ????? ?????, R ³ ???? ??? ????? ????? ??, ???? R ³ ?? ???? ????, ???????????? ?? ??????? ???? ??? ??? ???? ??? ???????????? ?? ?????? ???? ??????? ???? ?? ??????? ????????? ???? ?? ???? ??, ??? ??? ??? ?????????? ??????? ??? ???? ???? ???? ???? ???? ?? ????? ???????? (???? ?????) ????? ???? ???? ?????? ?? ??? ??, ????????????????? ??? 3D ??????? ??????????? ???? ?? ??? ????? ???

?? ?????? ?? ???? ????? ??, i, j, ??? k , H ?? ??????? ????? ??????? ??? R3 ?? ??? ?????, ????? ??? ????????? ??? ??? ?? i ??? ?i ???, j ??? ?j ???, ??? k ??? ?k ??? ???? ?? ??? ????? ???? ???-??? (????? ????? ??? ???????) ?? ??? ?? ????? ??, ???? ???? ????? ?? ????? ?????, ???? ?????????? ?? ??? ?? ???? ??????? ?? ????? ??? ?? ???? ????, ????????? ??? ??? ??? ???????? ????? (?????? ?????) ?? ???? ????? ???

?? ?????????? p = b ₁ i + c ₁ j + d ₁ k ??? q = b ₂ i + c ₂ j + d ₂ k ????? ????? ?? ??? ??????? (??????? ??????) ?? ????? ??;

${\displaystyle p\cdot q=b_{1}b_{2}+c_{1}c_{2}+d_{1}d_{2}.}$

?? pq ^? , qp ^? , p ^? q , ??? q ^? p ?? ????? ??????? ???? ??? (??? ??? ?? ????? ???? ???????? ?? ????? ????? ????? ????? ??)? ???? ?? ??????? ?? ????? ??;

${\displaystyle p\cdot q=\textstyle {\frac {1}{2}}(p^{*}q+q^{*}p)=\textstyle {\frac {1}{2}}(pq^{*}+qp^{*}).}$

?????? ????? i, j, ??? k ????? ???????? ???? ????? ?? ??????? p ??? q ?? ???? ??????? (????? ??????) ?? ????? ??;

${\displaystyle p\times q=(c_{1}d_{2}-d_{1}c_{2})i+(d_{1}b_{2}-b_{1}d_{2})j+(b_{1}c_{2}-c_{1}b_{2})k.}$

(??? ??? ?? ???? ??? ???????? ??? ?? ????? ?????? ???? ??) ?? ?????? pq (???????????? ?? ??? ????) ?? ????? ????? ?? ???? ??, ??? ? q ^? p ^? ?? ????? ????? ?? ???? ?? ??? ???? ?? ??????? ?? ????? ??;

${\displaystyle p\times q=\textstyle {\frac {1}{2}}(pq-q^{*}p^{*}).}$

?? ??????? ???????????? ?? ???????? [p, q] = pq ? qp ??, ?? ?????? ????? ??;

${\displaystyle [p,q]=2p\times q.}$

??? ????? ??? ??, ??? ?? p ??? q ?????????? (???? ??? ?? ???-???????) ???, ??? ??? ???? ???

${\displaystyle p=p_{s}+{\vec {p}}_{v},}$

${\displaystyle q=q_{s}+{\vec {q}}_{v},}$

????? p ??? q ??, p _s ??? q _s ????? ????? ??, ???, ${\displaystyle {\vec {p}}_{v}}$ and ${\displaystyle {\vec {q}}_{v}}$ ????? ????? ??? ??? ???? ??? ?? ??????? ????? ??;

${\displaystyle pq=(pq)_{s}+({\vec {pq}})_{v}=(p_{s}q_{s}-{\vec {p}}_{v}\cdot {\vec {q}}_{v})+(p_{s}{\vec {q}}_{v}+{\vec {p}}_{v}q_{s}+{\vec {p}}_{v}\times {\vec {q}}_{v}).}$

?? ???? ???? ?? ?? ?????????? ?????? ?? ???-???????????? ????? ??????? ???????????? ?? ?????? ??? ????? ??? ?? ??? ?? ?? ???? ???? ?? ?? ?? ?????????? ???? ?? ???? ??? ?? ????? ???? ?? (???? ??????? ???? ????? ??) ???? ????? ?? ????? ????? ???????? (???? ???? ?? ??? ???????) ????

????-?????????? ??????? ?? ?????????? ???? ?? ??????? ???? ??? ??????????? ??????? ??, ???? "?????????? ??? ???????? ???????" ? ??? ???? ??????-?????? ??????? ??. ???? ????? ???? ???? ??? ??? ^[1]

???????? ???????? [ ???? ]

????? ??????? ?????? ??? ?????????? ?? ??? ???? ??????? ???? ?? ???? ??, ??? ???? ???????????? ??? ?? ??????? ???? ?? ???? ??? ???????????? ??? ?????????? ?? ???? ??????? ??? ?? ???? ??? ?? ????? ????? ?? ?? ?????????? ??? ??? ???? ???????? ??? ??? ???????? ?????? ??? ????? ????? ??? ??? ????? 2?×?2 ??????? ???????? ???? ?? ??, ??? ???? 4?×?4 ??????? ???????? ???? ?? ??? ???? ????? ????, ????? ??? ???? ???????? ????? ??? ?? ????? ?????????? ?? ??? ?????? ????? ??? ????? (?????????) ?????? ?? ???????? ????, ?? H ??? ???????? ?????? M(2, C) ??? M(4, R) ??? ??????? ???????? ?????????????? ??????? ???

2?×?2 ??????? ???????? ????? ???, ?????????? a + bi + cj + dk ??? ?????? ??????? ???? ?? ???? ??;

${\displaystyle {\begin{bmatrix}a+bi&c+di\\-c+di&a-bi\end{bmatrix}}.}$

?? ???????? ???? ????? ?????? ???????????? ??:

• b, c ??? d ?????? ???? ?? ??? ????? ??? ?? ????? ??? ?? ??????? ?????? ?? ??? ???????? ???? ????? ??? ?????? ?? ??? ??, c = d = 0 ???? ??? ?? ??????? ?????? ?? ??? ??? ??????? (?????) ??????? ???????? ???????? ???? ????? ??, ??? b = d = 0 ???? ??? ?? ??? ??????? ???????? ???????? ???? ????? ???

• ???? ?????????? ?? ???? (??????? ?????? ????, ???? ??????? ??? ?????? ?? ??????) ????? ???????? ?? ??????????? ?? ?????? ????? ???

• ???? ?????????? ?? ??????? , ???????? ?? ??????? ????????? ??? ????? ????? ???

• ?????? ????? ?? ???????? ????? ???????????? ??? ????? ?? ???? SU(2) ?? ??????? ?????? ??? ???????????? ???? ???? ??? ???????? ?? ??? ??, ????? ?????????? 3-????? ????? ??? ??? ?? ???? ??? SU(2) ?? ???? ?? ??? 3-????? ????? ??? ????? SU(2) ?????? ??????? ???? ????? ??? ?????? ?? ????????? ??; ???? ???? ???????? ?

4?×?4 ??????? ???????? ????? ???, ??? ?????????? ??? ?????? ????? ?? ???? ??;

${\displaystyle {\begin{bmatrix}a+bi&c+di\\-c+di&a-bi\end{bmatrix}}.}$

????? ?????-????? ???????? ?????? (?????) ???? ??? ?????, ????? ?????????? ?? ????? ????? 48 ?????? ???? ????? ??? ?? ???????? ????, ???? ?????????? ?? ??????? ???????? ?? ????????? ??? ????? ????? ??? ???? ?????????? ?? ???? ?? ??? ?? ??? (???? ????) ????? ???????? ?? ???????????? ????? ??? ????? ???? ????? 2?×?2 ??????? ???????? ???????? ???, ???????? ?? ???????? ????? ???? ??????? ?????? ??? ?????? ???? ???? ?? ???? ??; ?????? ?? ??? ??, ????? c = d = 0 ???? ????, ?? 2?×?2 ?????? ??? ???? ??????? ???????? ??? ????? ???

??? ????? ?? ??? [ ???? ]

???? ????? ????, ?????? ?? ???-?????? ????? (??? ??? ?????) ?? ?????? ?????? ??? ???? ?? ??? ?? ???????????? ??? ?? ????? ?? ???? ??, ?? ????? ?? ?? ???? ???-??????? ???? ??? ??? ???? ??? ????? ?? ??? ????? ??? ???? ?? ???? ??? ??????? ????? ??? ?? ????, ?????? ?? ??? ??? ????? ?? ???? ?? ?????? ???? ???? ????? ??? ???? ?? ??????? ????? ??, ????? ???????????? ??????? ?????? ?????????? ?????? ???? ???????? ???????????? ?? ???? ???? ??, ?? ???? ???????????? ?? ???? ?? ??? ??-???? ????? ?? ?????? ?? ???? ??? ??????? ???????? ????? ???

??????? ?????? ?? ?????? ?? ??? ???? ?????????? [ ???? ]

???????????? ??? ??????? ?????? ?? ?????? ?? ??? ???? ??????? ???? ?? ???? ??? ?? ??????????? ???, ?????????? , ??????? ?????? ???? ??????-?????? ???? ???? ??? ?? ????? ????? ??? ?? ??????? ?????? ?? ???? ?? [[??????? ????|??????? ??????}} ?? ?????? ?? ??? ???? ??? ????? ??? ???

??? ?? ??????? ?????? ???? ??? ??-????? ????? ???? ??? C ² ???? ????? ??? ?? ????????? 1 ??? j ??? ???? ??? ????? ????? ????? ????????? 1 ??? j ?? ?????? ???? C ² ???? ??? ????? ??? ?? ???? ????? ?? ???? ??:

${\displaystyle (a+bi)1+(c+di)j.\ }$

???? ???? j ² = ?1 ??? ij = ? ji ????????? ????, ??? ???? ????????????? ?????? ????? ??? ?? ??????? ??? ???? ?? ???? ???? ?????? ij ?? ??? k ????? ?? ???????????? ?? ?????? ?????? ???? ?????? ??? ??????? ??? ????, ??????? ?????? ?? ???? ?????? ?????, ?????????? a + bi + cj + dk ??? ????? ??? ???? ???? C ² ?? ????????? ??? ??????? ?????? ?? ??? ???? ????? ??? ???????????? ??? ?????????? ?? ??? ???? ?????, ??? ???? ?????? ???? ?? ???? ??;

${\displaystyle (a+bi,\ c+di)\leftrightarrow (a,b,c,d).}$

?1 ?? ?????? [ ???? ]

??????? ?????? ????, ???? ?? ???? ????? ??, i ??? ?i, ?????? ?? ?????? (???) -1 ????? ??? H ???? ????? ??? ?? ???? ??? ?? ???? ???? ?????? ??: -1 ?? ?????? ?? ?????????? ??? R ³ ???? ????? ????? ????? ??? ????? ???? ??, ??? ?? q = a + bi + cj + dk ??? ?????????? ??, ??? ??? ?? ?? ???? ??? -1 ??? a, b, c, ??? d ?? ?????? ???? , ???? ??? ?? ??

${\displaystyle a^{2}-b^{2}-c^{2}-d^{2}=-1,}$

${\displaystyle 2ab=0,}$

${\displaystyle 2ac=0,}$

${\displaystyle 2ad=0.}$

???? ???? ???????? ??? ??????? ??? ??, ??? ??? a=0 ???? ?????? ?? ??? b,c ??? d ???? ?? ???? ?? 0 ???? ?????? ??? ??? ???? ???? ????? ?? ?????? a ??? ??????? ???? ?? ??? ????? ?????? ??? ??? ?????? ?? a ² = ?1 ???? ?????? ??? ???? a = 0 ??? b ² + c ² + d ² = 1 ?? ???? ?????? ??? ???? ?????? ????, ??? ?????????? ?? ??? -1 ???? ?? ???? ??? ?? ????? ?? ???? ?? ??? ????? ????? ???? (???? ?? ????? ??????? ????) ????? ???? 1 ????? ???????? ???, ???? ????? ??????? ?? ???? ????? ????? ???? ???

???? ??????? ??????? ???????????? ?? ?? ???????? ?? ???? ????? ????? ??? ???? ?????? ???????????? ?? ???? ?? ?????? ?? ????? ?? (??? 0 ?? ????? ???? 1 ????? ??)?

H ???? ????? ??? ?? ???????? ?? ???? ??????? ????? ????? ?? ??, ?? ???? ??????? ???? ???? ???? ????? ????? ??? ??? 1971 ???? ??? ?????? ????? ????????? ?????? ?? ?????? (???? 39), ?? ?????? ?????? ?? ?????? ?? ?????????? ????? ???? ?? ???, ???? ????? ??? ???? ???? ?????? ?? ?????????? ???? ?????? ???? ??? ??? ????? ????? ????, ????? ??? ?? ??? ????? ?? ????? ??? ??? ??. ???????? ?? ????? ???????? ???????? ??? ?? ??????? ?????? (????????, 1995) ???? ???? 60 ???? ??? 8.13 ???? ?????? ??? ??? ?????? (2003) ???? ?? ???? 40 ???? ???????????? ??? ?????????? ???? ???? ???? ???: “???? ?? ??????? ????? ??? i ???? ?? ???? ??, ??? ????? ??? j , ??? ????? ?? ?????? ??? k ???? ?? ???? ??” , ?? ????? (????) ???? ??? ??? ??? ???

??????? ?????? ?? ??? ?? ??? ?? H [ ???? ]

-1 ?? ???????? ?? ???? ???? ???????????? ???? ??????? ?????? ?? ??? ????? ??? ???? ???? ??, ??? ??? ??? ?? ?????? ????? ???????? ???? ????? ??;

${\displaystyle a+b{\sqrt {-1}}\mapsto a+bq.}$

????????? (?????) ?????? ?? ????? ????, ???? ??, C ??? H ??? ??? ???????? ???? ???????????? ??? ???? (???????????) ???? ??????? ?? q ??? ?q ??? ????? ??????? ??, ???? ?????? ??????? ???

???? ???-??????? ?????????? , H ?? C ????? ??? ?????????? ?????? ???? ????? ????? ??? ???? ????? ????? ??? ????? ????? ?? ??? ???? q ??? ?????? ????:

${\displaystyle q=q_{s}+{\vec {q}}_{v}.}$

????? ????? ??? ??? ???? ???? ???? ??? ???? ???? ?? ??? ???? ??? ?? ??? ???:

${\displaystyle q=q_{s}+\lVert {\vec {q}}_{v}\rVert \cdot \mathbf {U} {\vec {q}}_{v}.}$

(??? ??? ?? ?? ${\displaystyle q_{s}+\lVert q\rVert \cdot \mathbf {U} q}$ ???? (?????) ???? ??)? q ?? ????? ????? ?? ???? ${\displaystyle \mathbf {U} {\vec {q}}_{v}}$, ????? ??????? ????? ?????????? ??, ???? ???? ??? -1 ??? ??????, ?? ?? ?????? ????? ??????? ?????? ?? ??? ??? ???????? ???? ???

${\displaystyle a+b{\sqrt {-1}}\mapsto a+b\mathbf {U} {\vec {q}}_{v}.}$

?? ?????? ????, q , ??????? ???? ${\displaystyle q_{s}+\lVert {\vec {q}}_{v}\rVert i}$ ?? ????? ????? ??? ??????, H, ???? ????? ??????? ???? ??? ???? ???? ??????? ?????? ?? ??? ??? (??????) ????? ??, ????? ?????? ??? ????? ??? ?? ???????? ???? ????? ???? ?? ???? ?? ????? ??? ??? ????? ?? ?? ??? ???? (????) ????? ?? ???????? (?????????, ???? ????) ??????? ??? ?? ????? ????? ???

????????? ??-???? [ ???? ]

??? ???? ??? ???????????? ?? ???? ???? H ?? ??????? ??-?????? ???? ????????? ??-?????? ?? ?????? ???? ?? ????? ??? ???? ?? ???? ??? ??? ?? ??, ?????? ?? ?????????? p ??? q ???? ?? ???? ??? ?? ????? (pq = qp) ???? ?? ?? ???? ?? ????? ?????????? , H ?? ???? ??????? ??-???? ???? ???? ??? ???, ??????? ?????? ?? ??? ??? (??????) ?? ??? ?? H ?? ??????? (????????) ???? ???? ????? ?? ???? ?????????? ???? ?? ???? ????????? ??-?????? ??? ???? ?? ???? ???? ????? ??? ????????? ??-?????? ???? ?? ???? ???????-???????????? ??? 2×2 ??????? ?????????? ?? ???????? (???????) ????? ?? ???? ????? ???

??? ?????????? ???????? ?? ?????? [ ???? ]

???? ??????? ???????? ?? ???????? ????, ???? ?????????? ???????? ?? ?????? ?? ???? ???? ??? ???? ?????? ???? ???????? ??? ?????? ?? ??? ??, "????????" ????? ??????? ??? ????????? ??? ???????? ?????? ???? ?????????? ???????? ?? ?????? ???

?????????????, ???????, ??? ???? [ ???? ]

???? ????? ??? ??????????,

${\displaystyle q=a+bi+cj+dk=a+\mathbf {v} }$

?? ???????????? ??? ?????? ????? ????? ??

${\displaystyle \exp(q)=\sum _{n=0}^{\infty }{\frac {q^{n}}{n!}}=e^{a}\left(\cos \|\mathbf {v} \|+{\frac {\mathbf {v} }{\|\mathbf {v} \|}}\sin \|\mathbf {v} \|\right)}$

???

${\displaystyle \ln(q)=\ln \|q\|+{\frac {\mathbf {v} }{\|\mathbf {v} \|}}\arccos {\frac {a}{\|q\|}}}$. ^[2]

????? ?? ??? ???? ?? ?? ???? ?????????? ?? ???? ???????????? ??? ?????? ????? ?? ???? ??;

${\displaystyle q=\|q\|e^{{\hat {n}}\theta }=\|q\|\left(\cos(\theta )+{\hat {n}}\sin(\theta )\right),}$

????? ???? ${\displaystyle \theta }$ ??? ????? ????? ${\displaystyle {\hat {n}}}$ ?????? ????????? ???? ????? ??:

${\displaystyle a=\|q\|\cos(\theta )}$

???

${\displaystyle \mathbf {v} ={\hat {n}}\|\mathbf {v} \|={\hat {n}}\|q\|\sin(\theta ).}$

??? ?? ????? ?????????? ???? ??? ???? ${\displaystyle e^{{\hat {n}}\theta }}$ ??? ?? ?????? ?? ???? ???

???? ?????????? ?? ???? ??? ???? ?????? (???????) ????????? ${\displaystyle \alpha }$ ??? ?? ???? ?????? ?? ???? ??:

${\displaystyle q^{\alpha }=\|q\|^{\alpha }e^{{\hat {n}}\alpha \theta }=\|q\|^{\alpha }\left(\cos(\alpha \theta )+{\hat {n}}\sin(\alpha \theta )\right).}$

???? ????? ??? ??? ????? ???????? ????? [ ???? ]

???? ????? ????? ?? ??????, ???? “?????????” ?? ??? ??? ?? ?? ?? ?? ???? ??????? ??? a ??? rar ^?1 ??? ???? ????? r ??? ???-????? ??????? (??????????) ??? ?? ???? ??????? ?? ???? ????? ??? ??????? ????? ??????? ??? (???? ??????? ?? ??? ??????), ?? ????? ????? ?? ??? ???? ??? ???? ??????? ????? ????? ?????? (?????? ??????? ?? ???? ??? ????????? ?????? ??? ??????? ???? ??)

??????, ???-????? ???????????? ?? ???? ??? ???? ?????, ????? ?? ????? ??????? ????? ???? ???????????? ??? ??? R ³ ?? ??? ???? ????????? ????? ????? ???? ??? ???? ????? ?????????? (???? 1 ???? ???? ??? ??????????) ????? ??????? ????? cos(θ) ????, ????? ????????? ?? ??? ?? ??? ???? 2θ ????? ???????, ??? ???? ??????? ?? ????? ??????? ????? ?? ????? ????? ??? ???????????? ?? ?? ????? ??:

1. ???-??????? ???????? (?????? ?? ??? ??, ???? ?????? ??? ????? ?? ??? ??)
2. ?????????? ????? ????? ????? (??? ????)?
3. ????? ?????????? ?? ???? 4D ???? ???? ??? ??????? ??????? ???? ?? (???? 4-?????????? ??????? ???? ???? ?????????: 4D ????????? ?? ??????)?

???? ????? ???????????? (??????) ?? ????, ???? ??? ??, ??? 3-????? S ³ ??? ??? ????? (??? ??? ?????) ???? ??, ??? ???? ?? ??????????? 1 ???? ??????? ??????? 3×3 ?????????? ?? ????? SO(3, R) ??? ?? ??? ??? ???? ?? ?????? ?? ????? ?????????? ???? ???? ???? ???? ????????? ???? ???? ??????? ??? ???? ????? ???

?? ????? ?? ??? ??????????? ??????? ?? ???? : ???? ?????? ???? ????? ??????

?????? ?? ???? ??-????? ?? ????? ??? ????? ????? ????? ??, ??? ????? ??? ????? ???, ???? ????? ????? ?? ????? ????? ?????? ?? ??? ??-????? ????? ??? ???? ???? ????? ????? ?? ??? ????? ??? ???? ?????? ??? ?? ??? ??? ??? ?????? ????? ??? ?????? ?? ??? ??, ?????????? ????? ?? ??? ????? "?????? ?????????? ?????" ????? ???

?????? ?? ?????, ???????????? 1 ???? ??????? ???????? 2×2 ?????????? ?? ????? SU(2) ????? ?????????? ????? ???

??? ?? A ??? ???????????? ?? ???? ?? ?? a + bi + cj + dk ??? ???? ?? ????? a, b, c, ??? d ??? ??? ???? ?? ???? ?? ???? ??? ?? ??? ?????? ???? ?? ?????? ?? ???????? ?? ??? ?????????? 2 ??? ???? A ??? ???? (??? ???? ??? ?????) ????? ?? ??? ??? ?????? ????? ??, ?????? ???????? ???????????? ?? ???? ???? ????? ??? ?? ???? ???? 24 ????? ?????????? ????? ??, ??? ?? ??????? ???? {3,4,3} ???? ??? 24-?????? ?? ?????? (??????) ??????? ?? ????? (????????) ????? ???

??? ???????? [ ???? ]

???? F ??? ????? ???? ???? ????? ????????? 2 ??? ????? ????, ??? a ??? b ???? ??????? ???, ??? F ???? ??? ????? ???-????? ???????? ????????? ?????? ????????? ???? ?? ???? ??, ????? ????? 1, i, j, ??? ij ???, ????? i ² = a , j ² = b ??? ij = ? ji (??? ?? (ij) ² = ? ab ) ????? ????? ???????? ??? ?????????? ?????? ???? ????? ?? ??? ?? F ???? ??????? ?????? ??? ??, a ??? b ?? ??? ?? ????? ???? ???, F ???? 2×2 ?????????? ?? ?????? ????? ????????? ????? ???

Cℓ3,0(R) ?? ??? ????? ?? ??? ?? ?????????? [ ???? ]

????????? ??????? ??????? ?? ???????????? ?? ??????? ????? ?? ???????? ?????? Cℓ _3,0 ( R ) ?? ??? ????? Cℓ ⁺ _3,0 ( R ) ?? ??? ???? ????? ?????, ??? ?????? ??? ??? ???????? ???? ?? ???? ??? ?? ????? ????? ????????? σ ₁ , σ ₂ , σ ₃ ??? ?????? ?????? ??? ??? ?? ???? ??? ????????? ????????? ?????? ????? ??;

${\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=1,}$

${\displaystyle \sigma _{i}\sigma _{j}=-\sigma _{j}\sigma _{i}\qquad (j\neq i).}$

???? ?? ????? ????? ??????? 3D ???? ???? ??????? ??? ??????? ??? ?? ?? ???, ??? ?? ????? ?????? ?? ?? ???? ????? ????? w ??? ????? ?? ???? ???? ???? ???? ????? r ?? ??????? (??????????) ?????? ????? ?? ???? ??:

${\displaystyle r^{\prime }=-w\,r\,w.}$

?? ????????????, ????? ?????????? ?????? ?????? ???? ??? ?????? ???? ????? ??? ??????? ????????? ??? ???? ,

${\displaystyle r^{\prime \prime }=\sigma _{2}\sigma _{1}\,r\,\sigma _{1}\sigma _{2}}$

?? ???? ???? ??? 180° ?? ??????? ??? ????? ??????? ?? ??? ???? ???? σ ₁ ??? σ ₂ ???? ????? ??? ?? ?? ????? ?????????? ??????? ??? ???? ????? ????? ????? ???? ??,

${\displaystyle r^{\prime \prime }=-\mathbf {k} \,r\,\mathbf {k} .}$

??? ????, ?? ????? ???? ???? ??, ???? ???? ???? ???? ???

${\displaystyle \mathbf {k} =\sigma _{2}\sigma _{1},\mathbf {i} =\sigma _{3}\sigma _{2},\mathbf {j} =\sigma _{1}\sigma _{3}}$

??? ?? ????? ?? ???? ?? ????? ?? ?? ?? ????? ??????? ?????? ??? ???????? ???? ??,

${\displaystyle \mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}=\mathbf {i} \mathbf {j} \mathbf {k} =-1.}$

?? ????? ????, ?????????? ??????? ??? ????? ???? ????? ???? ?????????? ??? ????? ????? ??- ?? ?????? ???????? ??????? ?? ?????? ?? ???? ??? ???????? ???? ??????? 2D ?????? ??? ?????? ????? ?? ?? ?? 1D ???????? ???? ??????? ?????? ??? 2D ???? ?? ???? ????? ?? ????? ??, ??? ???? ?? ????? ???????? σ1 ??? σ2 ??????? ??, ???? ??? ???????? ????? ??????? σ ₁ σ ₂ ????? ??, ???? ???? ??? ?? ??????? ????? ??? ??, ???? ????? ???????? ???? 3D ????, ???? ???????? ????? ??????? σ ₁ σ ₂ , σ ₂ σ ₃ , σ ₃ σ ₁ ????? ?? , ???? ???? ??????? ????? ???

???? ??? ???? ??? ????? ??? ???????? ?????? Cℓ _4,0 ( R ) ????, ?? ???????? ????? ??????? ????? ??, ?????? ??? ??????? ????? ????? ???????? ??????? ??? ?? ????? ???? ????? ?? ??? ?? ???? ?? ?? ????? ????? ??? ?? ?????? ?????? ??? ????????? ???? ?? ???? ??? ???????????? ???? ????? ?????????????? ??? ????? ??? ??????? ?????? ???? ?????????, ?????? ??? ???? (?????? ??????) ???? ????? ??, ?????????? ??????????? (?????? ????????????) ???? ??????? ?? ???? ??????? ?? ?????? ??? ?? ???? 3D ???? ?? ????? ?????????? ?? ?? ????? ????? ??????? ?? ????? ????? ????? ??, ??? ???? ???????? ??? ??? "?????????" ?? ??? ???? ?????? ?? ???? ???

??? ??????? ?? ??, ?? ??????? ???? ?????????? ??? ??????? ???? ????? ???? ????? ??? ?????:

• ?????????? ?????? ???? ?????? ?????? ??? ???-?????? ????? ?? ??? ??? ????? ?????????? ?? ?????-????? (?????????) ?? ??? ???? ????? ??? ??????? ????? ???
• ?????????? ?????? ????, ??? ???? ??? ??? ???? ?? ?? ????? ???? ??, ?? ????, ???? ??? ?????? ??? ?? ?????????? ???? ?? ????? ??? ????? ????? ?????? ??? ?????? ??? ??????? ??? (?????? ???) ?? ????? ??? ??? ?? ????? ?? (????? ???? ????? ????? ?? ???? ????? ?????? ??? ???????????? ??? ?????? ????? ??)?
• ?????? ?? ???? ?? ??????? ???? ??? ???? ??? ????????? ??? ?? ???? ???? ???? ??, ???? ??????? ??? ??? ???????????? ???? ?? ????, ???? ???????, ?????? (??????), ??????, ??????? (??????), ?????? (?????), ??? ??? ???? ???? ??? ???? ???? ?????? ?? ?? ???? ???? ?? ???? ???
• ??????? ???????? ?? ?????? ???? ????, ??????, ?????? ?? ???? ????? ??????? ???? ????? ???? ?? ???? ?????????, ????????? ??? ??????? ?? ????????? ?? ???? ????? ??? ??? ????, ???? ??? ?? ?? ?? ?????? ???? ?????? ???? ????? ????????? ??? ??????? ?? ???? ??, ?????? ?????????? ????? ????? ????? ??? ???? ???? ??? ???? ????? ???
• ????-???????? ??????????????? (???????) ?????? ??? ?? ????? ??? ???? ??????????? (????-????) ????????? ???

???????? ???????? ???? ?????????? ?????? ???? ??? ??????????? ??????? ??, ???? "?????????? ??????"

?????? ????? [ ???? ]

?????????? , ??????? ?????? ???? ???? ?? ?????? (???-????) ?????? ??? ?????? (?????? ????? ??????) (CSA) ?? ?????? ????????? ???? ???? CSA, ??? ??? ????????? ????? ??? ???????????? ????? ??? ?????? ????? ??? ????? ??? ????, ????????? ?? ?????? ????? ?? ????????? ??? ???? ????? ??, ?????? ??? ????????? ??? ???????????? ????? ??????? ???? ????? ??, ????? ?????? ????? ???? CSA?? ??, ??? ???????? ???? ???? CSA ?? ??? ??? ???? ??????? ?? ???? ??? ?? ???? ????? ??? ?????-???????? ????? (??????? ??? ??, ???????? ?????) ???????, CSA?? ??, ???? ??????? ?????? ???? ???? ???????? ??????, ??? ?????? ?????????? ??????? ????? ???? ?????? ???-???? ??????? ?????? ???

???? ???? ???? CSA?? ?? ????, ?? ????? ??? ?????? ????? ?? (?????? ?? ??? ?? ???-???? 2 ???? ????? ???? ?????, ????? ?????? ????) ?????? ?? ????? ??????? ???? ????? ??- ????????? ?????? ?? ???-????????? ???? ????? ??? ??? ????? ???? ???????? ??? ????? ???????? ???? ????? ??? ?? ??? ?? ?? ????????? ???? (??????? ???), ?????????? ?? ???? ?????? ???-???? CSA ????? ??, ?? ??? ??? ????? ???? ?? ???? ?? ?? ???? ??????? ???? ????????? ?? ?????? ?? ???-???? ???? ????? ????? ???

????? ????? ?????? [ ???? ]

• “??? ?? ??? ???????????? ???? ??? ????????, ??? ????? ??????, ??? ??? ????? ???? ????? ??? ??? ??????? ?? ?????? ??? ?? ???? ???? ?? x, y, z ??? ?? ????? ???? ????? ???? ??? ????? ????? ??”----?????? ???? ??????? (???? ??? ?????? ??? ?? ??? ????? ???? ????? ??? ????? ??? ???????)

• “???? ??? ???? ??? ???? ????? ???? ??? ??, ??? ???? ??? ???? ????? ???? ??? ??? ?????? ??? ?? ?????????? ????? ????? ????????? ??? ??????? ??? ?????? ????? ???? ????? ??: ?????? “????+????”, ??? “????+????” ???? ?? ???? ??: ??? ?? ??? ???????, ?????? ??? ???? ??? ?? ???? ??? ???? ?????? ??? ??? ????? ???? ?? ???, ???? ?? ???? ??? ?????? ?? ??? ???? ?????? ?? ???? ??” ----?????? ???? ??????? (??.??. ?????? ?? ????? “???? ?? ?? ?????? ???? ???????” ?????? ?????)

• “??????? ?? ??????? ???? ??? ??? ??? ??? ?????????? ??; ???, ????? ????? ?????, ????? ?? ??? ???????? ?????? ??? ?? ?????? ?? ????? ??? ???? ????? ?????, ?????? ???? ????? ???????? ?????? ??” ----???? ??????, 1892

• “??? ??? ???? ??? ???? ??, ????? ??? ???? ????? ?? ????? ????????? ?? ???? ??, ???????????? ?? ?? ???? ????? ?? ??, ???? ?? ???????????? ????? ?? ??? ???????? ?????? ??; ??? ????? ??? ??? ??? ????? ????????? ?? ?????? ???? ??? ???? ?? ?? ??? ???? ???? ??????? ?? ??? ???? ?? ??, ??? ????? ?? ????? ?????????? ??? ??? ??????????? ??? ???? ?? ???? ??” ----????? ????????, ???????????????? ?????, ?????? 1, ???? 134-135 (?? ????????????? ????????? ??? ????????? ?????, ????, 1893)

• “?? ?? ?????????? ?? ?? ?????????? ??? ????? ????? ?? ????? ?? (????) ??????? ??? ???? ???? ?? ??? ????? ????? ??????? ?? ???????? ???? ?? ??????, ???? ??? ??????, ?????? ?????????? ?? ??? ???? ??? ???? ???? ??????? ????? ?? ??????? ???? ??,?? ???????? ?????? ???????? ???? ???????? ??? ??? ??? ???, ?????? ???? ??? ??????? ?????? ???? ??, ??? ????? ??? ????? ???????? ?? ????? ????? ????? ??, ?????? ?? ???? ????????????? ??? ?????? ??????? ??, ?????? ??? ?????? ?????? ????? ????? ????? ?? ????????? (????? ????) ?????? ????? ??? ?????? ??? ?????? ?????? ????? ??” ---- ??????? ????????, ?? 1924 ???? ???? ?????????? ?? ????? ?? ???? ?????? ???? ?? ???? ???

• “…?????? ??????? ?? ??????-?-?????? ???? ??? ???? ??? ?? ??? ????, ??????????, ??????? ??? ????? ?? ??? ??? ?????? ???? ??? ???? ???????, ????? ?? ?? ?? ??? ?? ???? ????? ???? ???????????? ???? ???????? ??????? ???????????? ?? ??? ???????? ???? ??? ????? ??, ??, ????! ????? ???? ?? ???-????? ???? ?????? ??????? ?? ??? ?? ??? ?????”--- ????? ???. ???????, 1986?

????? [ ???? ]

1. ↑ "Visualizing Quaternions" . Morgan-Kaufmann/Elsevier. 2005.
2. ↑ "Lce.hut.fi" (PDF) . Archived from the original (PDF) on 2013-09-26 . Retrieved 2015-11-06 . {{ cite web }} : Unknown parameter |dead-url= ignored ( |url-status= suggested) ( help )