ÆÓÒ¹ÒØÖÐ ËÒÐØ ÓÙÒÖÝ
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- Malcolm Noel Horn
- 6 years ago
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1 ÁÒØÖÐ ÓÙÒÖ Ò Ë»Ì Î ÊÐ ÔÖØÑÒØ Ó ÅØÑØ ÍÒÚÖ ØÝ Ó ÓÖ Á̳½½ ØÝ ÍÒÚÖ ØÝ ÄÓÒÓÒ ÔÖÐ ½ ¾¼½½
2 ÆÓÒ¹ÒØÖÐ ËÒÐØ ÓÙÒÖÝ
3 ÇÙØÐÒ ËÙÔÖ ØÖÒ Ò Ë»Ì Ì ØÙÔ ÏÓÖÐ Ø Ë¹ÑØÖÜ ÍÒÖÐÝÒ ÝÑÑØÖ ÁÒØÖÐ ÓÙÒÖ ÁÒØÖÐØÝ Ø Ø ÓÙÒÖÝ» ÖÒ Ò ØÛ Ø ÒÒ Ú»Ú ÖÒ»Ú ÖÒ Ò ÖÐ ØÛ Ø ÒÒ
4 ÇÙØÐÒ ËÙÔÖ ØÖÒ Ò Ë»Ì Ì ØÙÔ ÏÓÖÐ Ø Ë¹ÑØÖÜ ÍÒÖÐÝÒ ÝÑÑØÖ ÁÒØÖÐ ÓÙÒÖ ÁÒØÖÐØÝ Ø Ø ÓÙÒÖÝ» ÖÒ Ò ØÛ Ø ÒÒ Ú»Ú ÖÒ»Ú ÖÒ Ò ÖÐ ØÛ Ø ÒÒ
5 Ë»Ì ÓÖÖ ÔÓÒÒ Ì Ë/Ì ÓÖÖ ÔÓÒÒ ÓÖÒÐÐÝ ÓÒØÙÖ Ý ÂºÅÐÒ ØØ Ò ÕÙÚÐÒ ÓÖ ÙÐØݵ ØÛÒ ØÛÓ ÚÖÝ ÖÒØ ØÓÖ N = ÙÔÖ Ò¹ÅÐÐ ØÓÖÝ Ò ÛØ Ø Ù ÖÓÙÔ ËÍ(Æ) Ò ÓÙÔÐÒ ÓÒ ØÒØ Å Ò ÓÒÓÖÑÐ Ô ÌÝÔ ÁÁ ÙÔÖ ØÖÒ ØÓÖÝ ÓÒ Ë Ë ÛÖ ÓØ Ë Ò Ë Ú Ø Ñ ÖÙ Ò Ø ÓÙÔÐÒ ÓÒ ØÒØ Ë ÛØ Ø Ø ÒØØÓÒ Ë = ¾ Å º Ì Ë/Ì ÓÒØÙÖ ØØ ØØ Ø ØÓÖ ÒÐÙÒ ÓÔÖØÓÖ Ó ÖÚÐ ØØ ÓÖÖÐØÓÒ ÙÒØÓÒ Ò ÙÐÐ ÝÒÑ Ö ÕÙÚÐÒØ ØÓ ÓØÖº
6 ËÒÐ ÌÖ ÇÔÖØÓÖ ËÒÐ ØÖ ÓÔÖØÓÖ ÑÝ ÖÔÖ ÒØ ÔÒ Ò ÓÒ ØÖÙØ ÓÙØ Ó ÜØØÓÒ χ ={φ ½,φ ¾,ψ,ψ } Ò ÚÙÙÑ ÖÖÒ ØØ Z ÎÙÙÑ ÖÖÒ ØØ ÝÑÔØÓØ ØØ ÜØØÓÒ µ ¼=...Z Z...Z Z... χ ½...χ Ã = Ò ½... Ò Ã Ô ½Ò½ Ô Ã Ò Ã...Z χ ½ Z...Z χ Ã Z...
7 ÐÓ ËÔÒ Ò
8 ˹ÑØÖÜ
9 ˹ÑØÖÜ Ì Ë¹ÑØÖÜ Ò ÓÔÖØÓÖ ØÒ ÓÒ Ø ØÒ ÓÖ ÔÖÓÙØ Ó ÚØÓÖ Ô Ë(Ô ½, Ô ¾ ): Î(Ô ½,ζ) Î(Ô ¾,ζe Ô ½ ) Î(Ô ¾,ζ) Î(Ô ½,ζe Ô ¾ ). Ì ÙÒÑÒØР˹ÑØÖÜ Ü Ý Ø Ó ÓÒ ÝÑÑØÖ ØÓ Ó Ø ÓÖÑ Ë φ½ φ ¾ = φ { ¾ φ } ½ + φ [ ¾ φ ] ½ + ½ ψ ¾ ε α ε αβ ¾ ψ β ½, Ë ψ α ½ ψβ ¾ = ψ {α ¾ ψβ} ½ + ψ [α ¾ ψβ] ½ + ½ φ ¾ εαβ ε ¾ φ½, Ë φ½ ψβ ¾ = ψ β ¾ φ ½ + À φ¾ ψβ ½, Ë ψ α ½ φ ¾ = à ψ α ¾ φ ½ + Ä φ¾ ψα ½.
10 Ä ÐÖ Ì ÝÑÑØÖÝ ÐÖ Ó Ø ÐعÓÒ ÙÔÖ ØÖÒ ÓÒ Ë Ë Ò Ó Ø ÒÐ ØÖ ÓÔÖØÓÖ Ò Ø N = ËÅ psu(¾,¾ )º Ì ÛÓÖÐ Ø Ë¹ÑØÖÜ Ó Ø ØÓÖÝ ÑÒ ØÐÝ ÒÚÖÒØ ÙÒÖ Ö ÙÐ psu(¾ ¾) psu(¾ ¾) R ÛØ ÓÑÑÓÒ ÒØÖÐ Ö º [ L β α,j γ ]=δ β γ J α ½ ¾ δ β α J γ, [ R,J ] = δ J ½ ¾ δ J, { } α,qβ = ε ε αβ C, } Q { Q α,g β [ L β α,j γ] = δ γ αj β + ½ ¾ δ β α J γ, [ R,J] = δ J + ½ ¾ δ J, { },G β = ε αβ ε C, G α = δ L α β + δ β α R + δ δβ α H, Ö,, = ½,¾ Ò α,β,γ =,º ƺ ÖØ Ô¹Ø»¼½½¼¾
11 Ó¹ÔÖÓÙØ Ó psu(¾ ¾) R R L β α Q α G α =R ½+½ R, =Lα β ½+½ Lα β, =Q α ½+½ Q α, =G α C=C ½+½ C, ½+½ G α, C =C ½+½ C, H=H ½+½ H.
12 ÒÒ ÐÖ Ì ÒÒ (g) Ó Ä ÐÖ g ÓÖÑØÓÒ Ó Ø ÙÒÚÖ Ð ÒÚÐÓÔÒ ÐÖ U (g[ù]) Ó Ø ÔÓÐÝÒÓÑÐ ÐÖ g[ù]=g Ùg Ù ¾ g..., [Ù Ö g, Ù g] Ù Ö+ g. ÁØ ÒÖØ Ý Ö¹¼ g ÒÖØÓÖ J Ò Ö¹½ (g) ÒÖØÓÖ Ĵ º ÌÖ ÓÑÑÙØØÓÖ Ú Ø ÒÖ ÓÖÑ [J,J ]= J, [J,Ĵ ]= Ĵ, Ò ÑÙ Ø ÓÝ ÂÓ Ò ËÖÖ ÖÐØÓÒ [ J [, [ J,J ]]] [ = ¼, J [, [ J,Ĵ]]] = ¼, [Ĵ[, [ Ĵ,J ]]] = ½ À Ã ÀÃJ { J J }. Ì Ó¹ÔÖÓÙØ Ó Ø Ö¹¼ Ò Ö¹½ ÒÒ ÒÖØÓÖ Ö J =J ½+½ J, Ĵ =Ĵ ½+½ Ĵ + ½ ¾ J J.
13 ÒÒ Ó psu(¾ ¾) R ƺ ÖØ ÖÚ¼¼º¼¼¼ ˆR ˆL β α ˆQ α Ĝ α = ˆR = ˆL β α = ˆQ α =Ĝ α ½+½ ˆR + ½ δ G γ Q γ ½+½ ˆL β α ½ δ β αg γ Q γ ½+½ ˆQ α + ½ Q α + ½ ¾ R R ½ ¾ R + ½ δ Q γ G γ, ½ ¾ L γ α L β γ + ½ ¾ L β γ ½ δ αq β γ G γ, + ½ ¾ Q α H ½ H Q α ½+½ Ĝ α ½ G α ½ ¾ G α H+ ½ H G α R ½ ¾ R R L γ α Q α + ½ ¾ ε αγ ε C G γ R Ĉ=Ĉ ½+½ Ĉ ½ ¾ H C+ ½ ¾ C H, + ½ ¾ R Ĉ =Ĉ ½+½ Ĉ + ½ ¾ H C ½ ¾ C H, Ĥ=Ĥ ½+½ Ĥ+C C C C. G α ½ ¾ G γ + ½ ¾ G β + ½ ¾ Q γ Q γ Q α L γ α ½ ¾ ε αγ ε G γ C, ½ ¾ G γ L α γ ½ ¾ ε ε αγ C Qγ + ½ ¾ ε ε αγ Qγ C, ½ ¾ Q γ G γ + ½ ¾ Q α G β ½ ¾ L α γ Qγ + ½ ¾ L γ α G γ
14 ÊÔÖ ÒØØÓÒ Ì psu(¾ ¾) R ÙÔÖÖ Ø ÓÒ Ø ØØ Qβ φ =δ ψ β, G β φ = ε β α ε ψ α, Q β ψ α= ε ε β α φ, G β ψ α=δ β α φ. Ì ÖÔÖ ÒØØÓÒ ÐÐ,,, Ö ( ζ Ü + ) = η, = ¾Ð ¾Ð η Ü ½, = ¾Ð ζ Ü η +, = ÛÖ ζ Ò ÓÚÖÐÐ Ô ØÓÖ η ÖØ Ø ÖÓÑ Ó Ø Ó Ó ÔØÖÐ ÔÖÑØÖ Ü ± ÓÝÒ e Ô = Ü+ Ü, Ü+ + ½ Ü + Ü ½ Ü = ¾Ð. Ì ÒÚÖÒ Ó Ø Ë¹ÑØÖÜ ÙÒÖ Ø ÝÑÑØÖÝ ÐÖ [ ] J, Ë(Ô ½, Ô ¾ ) = ¼, J =J ½+½ J, ÓÒ ØÖÒ ÐÐ ÙÒÑÒØР˹ÑØÖÜ ÓÒØ ÙÒÕÙÐÝ ÙÔ ØÓ Ò ÓÚÖÐÐ Ô ¾Ð Ü + ( Ü ) η Ü + ½,
15 ˹ÑØÖÜ ÓÒØ ( = ½, = ¾ Ü+ ¾ Ü ½ ܾ )( ½+Ü + ½ ܾ ) ܾ ( Ü ½ Ü ¾ + )( ½+Ü + ½ Ü ¾ + ) ½, = ( ܽ Ü ¾ ) η½ η ¾ ζ ( ܽ )( Ü+ ¾ ½+Ü + ½ Ü ¾ + ), ( Ü = ¾ Ü ½ + ) η½ η ¾ ( Ü ½ Ü ¾ + ), η½ η ¾ ( ܾ Ü+ ½ ( = ܽ + ¾ Ü+ ½ Ü ½ Ü )( ¾ ½+Ü ½ Ü + ) ¾ Ü+ ¾ ܽ ( Ü ½ Ü ¾ + )( ½+Ü + ½ Ü ¾ + ) = ζ ( Ü+ ½ Ü ½ ܾ )( Ü ½ Ü ½ + )( Ü ¾ Ü ¾ + ) ܽ ( Ü ¾ Ü ½ Ü ¾ + )( ½+Ü + ½ Ü ¾ + ), η½ η ¾ ) η½ η ¾ η ½ η ¾, = à = ( Ü + ½ Ü ¾ + ( ) η¾ Ü ( Ü ½ Ü ¾ + ), À = ½ ܾ ) η½ ( η½ Ü ½ Ü ¾ + ), η½ ( Ü ½ Ü ½ + ( ) η¾ Ü ( Ü ½ Ü ¾ + ), Ä= ¾ Ü ¾ + ) η½ ( η½ Ü ½ Ü ¾ + ). η¾
16 ÓÙÒ¹ ØØ Ë¹ÑØÖÜ su(¾ ¾) = = = ÄÓÒ =½ Ë psu(¾ ¾) = ÄÓÒ = ¾ Ë = ÄÓÒ ÄÓÒ =½ Ë Æº ÖØ Åº ÄÙÛ ºÌÓÖÖÐÐ ºÖÙØÝÙÒÓÚ Ò ËºÖÓÐÓÚ
17 Ò¹ÜØÖ ÕÙØÓÒ µ
18 ÉÙ ÙÑÑÖÝ Ó Ø ØØÖÒ ØÓÖÝ Ì ÙÒÑÒØÐ ÝÑÔØÓØ ØØ χ ={φ ½, φ ¾, ψ, ψ } Ö ÙÒØÓÒ Ó ÑÓÑÒØÙÑ Ô Ò Ô ζ Ò ÐÚ Ò (,,,) º Ì Ë¹ÑØÖÜ Ò ÓÔÖØÓÖ ØÒ ÓÒ Ø ØÒ ÓÖ ÔÖÓÙØ Ó ÚØÓÖ Ô Ë(Ô ½,Ô ¾ ): Î(Ô ½,ζ) Î(Ô ¾,ζe Ô ½ ) Î(Ô ¾,ζ) Î(Ô ½,ζe Ô ¾ ). Ì ÑØÖÜ ÐÑÒØ Ó Ø Ë¹ÑØÖÜ χ (Ô ¾,ζ ½ ) χ Ð (Ô ½,ζ ½ Ô ¾ ) Ë χ (Ô ½,ζ ½ ) χ (Ô ¾,ζ ½ Ô ½ ) = Ð, Ö ÓÒ ØÖÒ Ý Ø ÙÒÖÐÝÒ ÝÑÑØÖ χ (Ô ¾,ζ ½ ) χ Ð (Ô ½,ζ ½ Ô ¾ ) [Ë, J] χ (Ô ½,ζ ½ ) χ (Ô ¾,ζ ½ Ô ½ ) = ¼, ÙÔ ØÓ Ò ÓÚÖÐÐ Ö Ò ØÓÖº
19 ÇÙØÐÒ ËÙÔÖ ØÖÒ Ò Ë»Ì Ì ØÙÔ ÏÓÖÐ Ø Ë¹ÑØÖÜ ÍÒÖÐÝÒ ÝÑÑØÖ ÁÒØÖÐ ÓÙÒÖ ÁÒØÖÐØÝ Ø Ø ÓÙÒÖÝ» ÖÒ Ò ØÛ Ø ÒÒ Ú»Ú ÖÒ»Ú ÖÒ Ò ÖÐ ØÛ Ø ÒÒ
20 ÓÙÒÖÝ Ò¹ÜØÖ ÕÙØÓÒ µ
21 ÁÒØÖÐ ÓÙÒÖ ÓÖ Ø ÓÙÒÖÝ ØÓ ÒØÖÐ Ø ÑÙ Ø Ö ÔØ ÙµÐÖ Ó Ø ÙÐ ÐÖ h g Ò ØÙÖØ ÈË ÓÙÒº Ì ÓÙÒÖ Û ÐÐ ÓÒ Ö Ö ½º ÒØ ÖÚØÓÒ³ ÖÒ ÛÖÔÔÒ ÑÜÑÐ Ë Ë Ë ½º½ = ¼ ÒØ ÖÚØÓÒ³ ºÀÓÑÒ ² ºÅÐÒ ºÒ ² º ² ˺ÊÝ ÆºÅÃÝ ² κʺ ½º¾ = ¼ ÒØ ÖÚØÓÒ³ ºÀÓÑÒ ² ºÅÐÒ ºÒ ² ʺÆÔÓÑ ÆºÅÃÝ ² κʺ ĺÈÐÐ ¾º ÖÒ ÛÖÔÔÒ ÒØÖ Ë Ò ÑÜÑÐ Ë Ë ¾º½ = ¼ ÖÒ ºÓÖÖ ² ºÓÙÒ ÆºÅÃÝ ² κʺ ¾º¾ = ¼ ÖÒ ºÓÖÖ ² ºÓÙÒ ÆºÅÃÝ ² κʺ º ÖÒ ÛÖÔÔÒ Ë Ë ¾ º½ ÎÖØг ºÓÖÖ ² ºÓÙÒ ² κʺ º¾ ÀÓÖÞÓÒØг ºÓÖÖ ² ºÓÙÒ ² κʺ
22 ÁÒØÖÐ ÓÙÒÖ ¹ ËØØÖÒ ÌÓÖÝ Ì ÙÐ ÐÖ psu(¾ ¾) psu(¾ ¾) R º ÖÒ ÓÙÒÖÝ ÐÖ ÓÙÒÖÝ ÖÔ / psu(¾ ½) psu(¾ ½) ½ ½ Ú psu(¾ ¾) psu(¾ ¾) R ÚØÓÖ ÚØÓÖ Ú su(¾) su(¾) psu(¾ ¾) R ½ ÚØÓÖ psu(¾ ¾) + R ½ Ú psu(¾ ¾) + R ÚØÓÖ
23 ÁÒØÖÐ ÓÙÒÖ ¹ ËØØÖÒ ÌÓÖÝ Ì ÙÐ ÐÖ psu(¾ ¾) psu(¾ ¾) R º ÖÒ ÓÙÒÖÝ ÐÖ ÓÙÒÖÝ ÖÔ / psu(¾ ½) psu(¾ ½) ½ ½ Ú psu(¾ ¾) psu(¾ ¾) R ÚØÓÖ ÚØÓÖ Ú su(¾) su(¾) psu(¾ ¾) R ½ ÚØÓÖ psu(¾ ¾) + R ½ Ú psu(¾ ¾) + R ÚØÓÖ
24 » ÖÒ Ã(Ô): Î(Ô,ζ) Î (¼) Î( Ô,ζ) Î (¼)
25 ÙÒÑÒØРùÑØÖÜ»µ Ì Ã ¹ÑØÖÜ ÓÖ Ø ÖØÓÒ Ó ÙÐ ÑÒÓÒ ÖÓÑ Ø ÓÙÒÖÝ ÚÙÙÑ ØØ Ò Ã : Î(Ô,ζ) Î (¼) Î( Ô,ζ) Î (¼), ÓÙÒÖÝ ÐÖ su(¾ ½)={Lα β,r½ ½,R ¾ ¾,Q α ½,G α,h}º ½ ÐÐ ÓÙÒÖÝ ÐÖ ÒÖØÓÖ ÒÒÐØ ÓÙÒÖÝ ÚÙÙÑ Øغ Ì ÙÒÑÒØÐ Ã ¹ÑØÖÜ ÑÝ ÖÔÖ ÒØ Ã φ ½ Ô = φ ½ Ô, à φ ¾ Ô = φ ¾ Ô, à ψ α Ô = ψ α Ô. Ì ÒÚÖÒ Ó Ø Ã¹ÑØÖÜ ÙÒÖ Ø ÝÑÑØÖÝ ÐÖ (J ½)Ã(Ô) Ã(Ô)(J ½)=¼, ÓÒ ØÖÒ ÐÐ Ã ¹ÑØÖÜ ÓÒØ ÙÒÕÙÐÝ ÙÔ ØÓ Ò ÓÚÖÐÐ Ô º ÀÓÑÒ ² ºÅÐÒ ÖÚ¼¼º¾¾¾
26 ùÑØÖÜ»µ ÙÒÑÒØÐ ØØ ÌÛÓ¹ÑÒÓÒ ÓÙÒ¹ ØØ psu(¾ ¾) R ½ ½ su(¾ ½) ½ ( ) ½
27 ÌÛ Ø ÒÒ Ó ¼ ÒØ ÖÚØÓÒ»µ ÄØ Ø ÓÙÒÖÝ ÐÖ ÙÐÖ Ó ÙÐ ÐÖ h g Ù ØØ Ø ÔÐØØÒ g=h m Ö ÔØ Ø ÝÑÑØÖ ÔÖ ÔÖÓÔÖØÝ [h,h] h, [h,m] m, [m,m] h, Ì ÖÙÐ Ò ÙÖÒØÒ Ø Ó¹Ð ÔÖÓÔÖØÝ Ĵ (g) (g,h). Ï ÒØÖÓÙ Ö ÒÚÓÐÙØÓÒ σ Ó g ÛØ Ø Ò Ô σ(h)=+½ Ò σ(m)= ½º ÌÒ ÓÙÒÖÝ ÒÒ (g,h) ÑÝ ØÓÙØ Ó ÓÖÑØÓÒ Ó Ø ÙÐÖ Ó U (g[ù]) Û ÒÚÖÒØ ÙÒÖ Ø ÜØÒ ÓÒ σ Ó σ Û Ò σ : Ù Ù h Ùm... g[ù]=(h m) Ù(h m)... ÀÒ Ø ÓÙÒÖÝ ÒÒ Ö ÑÙ Ø ÐÚ Ò Ø Ù Ô Ùmº
28 ÌÛ Ø ÒÒ Ó ¼ ÒØ ÖÚØÓÒ»µ σ ÐÚй¼ h m + ÐÚй½ + ÐÚй¾ +
29 ÌÛ Ø Ö ÀÓÛÚÖ ÛÐ Ø Ö¹¼ ÒÖØÓÖ Ó h ÐÖÐÝ Ö ÔØ Ø Ó¹Ð ÔÖÓÔÖØÝ Ø Ö¹½ ÒÖØÓÖ Ó Ùm Ó ÒÓØ Ó Ó Ĵ Ô =Ĵ Ô ½+½ Ĵ Ô + ½ ) (J ¾ Ô Õ J +J J Õ / (g) (g,h), Õ ÛÖ, ÖÙÒ ÓÚÖ Ø h¹ò Ò Ô, Õ ÓÚÖ Ø m¹ò º ÊØÖ Û Ò ÓÖÑØÓÒ Ó Ø Ö¹½ Ùm ÒÖØÓÖ Ò ØÖÓÖ Û Ò (g,h) ØÓ Ø ÐÖ ÒÖØ Ý {J, J Ô } ÛÖ JÔ :=ĴÔ + ½ ¾ Ô Õ JÕ J, Ö Ø ØÛ Ø ÓÙÒÖÝ ÒÒ ÒÖØÓÖ º
30 ÌÛ Ø Ö ÀÓÛÚÖ ÛÐ Ø Ö¹¼ ÒÖØÓÖ Ó h ÐÖÐÝ Ö ÔØ Ø Ó¹Ð ÔÖÓÔÖØÝ Ø Ö¹½ ÒÖØÓÖ Ó Ùm Ó ÒÓØ Ó Ó Ĵ Ô =Ĵ Ô ½+½ Ĵ Ô + ½ ) (J ¾ Ô Õ J +J J Õ / (g) (g,h), Õ ÛÖ, ÖÙÒ ÓÚÖ Ø h¹ò Ò Ô, Õ ÓÚÖ Ø m¹ò º ÊØÖ Û Ò ÓÖÑØÓÒ Ó Ø Ö¹½ Ùm ÒÖØÓÖ Ò ØÖÓÖ Û Ò (g,h) ØÓ Ø ÐÖ ÒÖØ Ý {J, J Ô } ÛÖ JÔ :=ĴÔ + ½ ¾ Ô Õ JÕ J, Ö Ø ØÛ Ø ÓÙÒÖÝ ÒÒ ÒÖØÓÖ º
31 ÌÛ Ø Ö ÆÓÛ Û Ò ÓÛ ØØ (g,h) ÐØ Ó¹Ð ÙÐÖ (g,h) (g) (g,h)º ÌÓ Ó Ø Û ÐÙÐØ ÜÔÐØÐÝ Ø Ó¹ÔÖÓÙØ Ó Ø ØÛ Ø ÒÒ ÒÖØÓÖ J Ô = ĴÔ + ½ ¾ Ô Õ JÕ J = Ĵ Ô ½+½ Ĵ Ô + ½ ¾ Ô Õ (J Õ J ½+½ J Õ J ) + ½ ¾ Ô Õ J J Õ + ½ ¾ Ô Õ JÕ J + ½ ¾ Ô Õ (J Õ J +J J Õ ) = J Ô ½+½ J Ô + Ô Õ JÕ J (g) (g,h), ÛÖ Û Ú Ù Ø ÝÑÑØÖ ÔÖ ÔÖÓÔÖØÝ ¹ Ø ÓÒÐÝ ÒÓÒ¹ÞÖÓ ØÖÙØÙÖ ÓÒ ØÒØ Ò Ø Ö Ô Õ Ò Ô Õ º
32 ÌÛ Ø ÒÒ (psu(¾ ¾),su(¾ ½)) R ¾ ½ = ( R ½ ¾ = ( ( Q α ¾ = ( G α ¾ = C= ˆR ¾ ½ ˆR ½ ¾ ˆQ ¾ α Ĝ α ¾ + ½ ¾ R ¾ ½ + ½ ¾ R ½ ¾ + ½ ¾ Q ¾ α ½ ¾ G α ¾ R ½ ½ R ½ ½ R ¾ ¾ R ¾ ¾ (Ĉ+ ½ ¾ CH ) ½, ( C = Ĉ ½ ) ¾ C H ½. ½ ¾ R ¾ ½ ½ ¾ R ½ ¾ ½ ¾ R ¾ ½ + ½ ¾ R ½ ¾ R ¾ ¾ R ¾ ¾ Q ½ α G α ½ ½ ) ¾ Q γ ¾ G γ ½ ½, ) ½ ¾ G γ ¾ Q γ ½ + ½ ¾ Q ¾ γ ½ ¾ G γ ¾ L γ α L α γ ½, + ½ Q α ¾ H ½ ) ¾ ε αγcg γ ½ ½, ) ½ G α ¾ H+ ½ ¾ εαγ C Qγ ½ ½, ƺÅÃÝ ² κʺ ÖÚ½¼½¼º ½
33 ÁÒØÖÐ ÓÙÒÖ ¹ ËØØÖÒ ÌÓÖÝ Ì ÙÐ ÐÖ psu(¾ ¾) psu(¾ ¾) R º ÖÒ ÓÙÒÖÝ ÐÖ ÓÙÒÖÝ ÖÔ / psu(¾ ½) psu(¾ ½) ½ ½ Ú psu(¾ ¾) psu(¾ ¾) R ÚØÓÖ ÚØÓÖ Ú su(¾) su(¾) psu(¾ ¾) R ½ ÚØÓÖ psu(¾ ¾) + R ½ Ú psu(¾ ¾) + R ÚØÓÖ
34 Ú»ÚÊ ÖÒ Ã(Ô, Õ) : Î(Ô,ζ) Î (Õ,ζ Ô ) Î( Ô,ζ) Î (Õ,ζ Ô )
35 ÙÒÑÒØРùÑØÖÜ Ú»Úʵ Ì Ã¹ÑØÖÜ ÓÖ Ø ÖØÓÒ Ó ÙÐ ÑÒÓÒ ÖÓÑ Ø ÓÙÒÖÝ ØØ Ò Ã(Ô, Õ): Î(Ô,ζ) Î (Õ,ζ Ô ) Î( Ô,ζ) Î (Õ,ζ Ô ), Ò Ü Ý Ø Ó ÓÒ ÝÑÑØÖ ØÓ Ó Ø ÓÖÑ Ã φ Ô φ Õ = φ { φ } Ô Õ + φ [ φ ] Ô Õ + ½ ψ ¾ ε α ε αβ Ô ψ β Õ, à ψ α Ô ψβ Õ + ½ φ ¾ εαβ ε Ô φ Õ, à à φ Ô ψ β Õ ψ α φ Ô Õ = ψ {α Ô ψβ} Õ = = à ψ β φ Ô Õ ψ α φ Ô Õ + ψ [α Ô ψβ] Õ + À + Ä φ Ôψ β Õ, φ Ô ψα Õ. Ì ÒÚÖÒ Ó Ø Ã¹ÑØÖÜ ÙÒÖ Ø ÝÑÑØÖÝ ÐÖ [ ] J, Ã(Ô, Õ) = ¼, J =J ½+½ J, ÓÒ ØÖÒ ÐÐ ÙÒÑÒØРùÑØÖÜ ÓÒØ ÙÒÕÙÐÝ ÙÔ ØÓ Ò ÓÚÖÐÐ Ô
36 ùÑØÖÜ Ú»Úʵ = ÄÓÒ =½ à = ÄÓÒ = ¾ à = ÄÓÒ = ¾ à = ÄÓÒ ÄÓÒ =½ Ã
37 ÁÒØÖÐ ÓÙÒÖ ¹ ËØØÖÒ ÌÓÖÝ Ì ÙÐ ÐÖ psu(¾ ¾) psu(¾ ¾) R º ÖÒ ÓÙÒÖÝ ÐÖ ÓÙÒÖÝ ÖÔ / psu(¾ ½) psu(¾ ½) ½ ½ Ú psu(¾ ¾) psu(¾ ¾) R ÚØÓÖ ÚØÓÖ Ú su(¾) su(¾) psu(¾ ¾) R ½ ÚØÓÖ psu(¾ ¾) + R ½ Ú psu(¾ ¾) + R ÚØÓÖ
38 ùÑØÖÜ Úĵ ÙÒÑÒØÐ ÓÙÒ¹ ØØ psu(¾ ¾) R ½ ½ su(¾) Ä su(¾) Ê R ( Ä Ê ) ½ ( Ê Ä Ä/Ê ) ½
39 ÁÒØÖÐ ÓÙÒÖ ¹ ËØØÖÒ ÌÓÖÝ Ì ÙÐ ÐÖ psu(¾ ¾) psu(¾ ¾) R º ÖÒ ÓÙÒÖÝ ÐÖ ÓÙÒÖÝ ÖÔ / psu(¾ ½) psu(¾ ½) ½ ½ Ú psu(¾ ¾) psu(¾ ¾) R ÚØÓÖ ÚØÓÖ Ú su(¾) su(¾) psu(¾ ¾) R ½ ÚØÓÖ psu(¾ ¾) + R ½ Ú psu(¾ ¾) + R ÚØÓÖ
40 ùÑØÖܻڵ Ú psu(¾ ¾) psu(¾ ¾) R ½ psu(¾ ¾) + R + + ½ ( + + )
41 ÖÐ ØÛ Ø ÒÒ Ó»Ú ÖÒ Ì ÐÖ Ò Ø ÙÐ psu(¾ ¾) psu(¾ ¾) R g Ä g Ê º Ì ÓÙÒÖÝ ÐÖ psu(¾ ¾) + R g + º Ï ÛÖØ Ø ÝÑÑØÖ ÔÖ ØÖÙØÙÖ g Ä g Ê =g + g º Ì Ö ÔÒÒ Ý J ± =J Ä ± α(j Ê )º ÌÒ Ø ÓÙÒÖÝ ÒÒ ÝÑÑØÖÝ (g Ä g Ê,g + ) ÒÖØ Ý Ø ÐÚй¼ Ò J + Ò ØÛ Ø ÐÚй½ Ò J J := Ĵ + ½ (J J + +J + J ) = Ĵ + ½ ¾ J Ä α(j Ê ),
42 ÖÐ ØÛ Ø ÒÒ Ó»Ú ÖÒ Ì Ó¹ÔÖÓÙØ Ó Û J = Ĵ + ½ ( J + J + J J +) = Ĵ ½+½ Ĵ + ½ (J +J +J J +) ½+ ½ ½ (J +J +J J +) + ½ ( ) J J + +J + J + ½ ( ) J + J J J + = J ½+½ J + ½ ¾ J J + (g Ä g Ê ) (g Ä g Ê,g + )
43 ÖÐ ØÛ Ø ÒÒ Ó»Ú ÖÒ Ì Ó¹ÔÖÓÙØ Ó ØÛ Ø (g g,g) Ö ÑÝ ÛÖØØÒ ( J = Ĵ ½ ½ α Ä (Ĵ Ê )+ ½ ) ¾ JÄ α(jê ) ½ ( +½ Ĵ + ½ ) (J J ++J +J ) ( ½ ( + ¾ (J Ä ½ ½ α JÊ)) ) J +. Ì Ó¹ÔÖÓÙØ Ó ÒÝ (g) Ö Ĵ =Ĵ ½+½ Ĵ + ½ ¾ J J.
44 ÖÒ Ì ÖØÓÒ ÑØÖÜ ÑÔ Ã : ½ ½, Ò ÑÝ ÒØÐÝ ÖÔÖ ÒØ ÓÒ ÙÔÖ Ô Ò ÓÔÖØÓÖ Ã : Î (Ô,ζ) Î( Ô,ζ Ô ) Î( Ô,ζ) Î(Ô,ζ Ô ) Ì ÒÓÒ¹ØÖÚÐ ÔÖØ Ó Ø Ó¹ÔÖÓÙØ Ó ØÛ Ø (g g,g) Ö ( J = Ĵ ½ ½ α Ä (Ĵ )+ ½ ) ¾ JÄ α(jê ) ½ Ò ØÚÐÝ Ö ÖÓÑ Ø Ø Ó¹ÔÖÓÙØ Ó (g) Ö Ý Ø ÑÒÙ Ò ÓÒÐÝ Ê Ĵ =Ĵ ½+½ Ĵ + ½ ¾ J J
45 ÖÒ ÓÐ Ò ÙÒÓÐ ÔØÙÖ Ó Ø ÖØÓÒ ( Ù, Ô,ζ) ( Ù, Ô,ζ Ô ) ( Ù, Ô,ζ) (Ù, Ô,ζ Ô ) Ë(Ô, Ô) κ κ Ë(Ô, Ô) κ κ (Ù, Ô,ζ) (Ù, Ô,ζ Ô ) (Ù, Ô,ζ) ( Ù, Ô,ζ Ô )
46 Ú ÖÒ Ì ÖØÓÒ ÑØÖÜ ÑÔ Ã Ú : ˇ ˇ, Û ØÓÖÞ ÓÑÔÓ ØÓÒ Ó ÙР˹ÑØÖÜ Ò ØÛÓ ÖÐ ÖØÓÒ ÑØÖ κ Ã Ú (Ô, Ô)=κ(Ô, Ü ) Ë(Ô, Ô)κ(Ô, Ü ). Ì ÒÓÒ¹ØÖÚÐ ÔÖØ Ó Ø Ó¹ÔÖÓÙØ Ó ØÛ Ø (g g,g) Ö ( J = Ĵ ½ ½ α Ä (Ĵ Ê )+ ½ ) ¾ JÄ α(jê ) ½ ( ½ ( + ¾ (J Ä ½ ½ α JÊ)) ) J +, Ò ØÚÐÝ ÓÑÔÓ Ó Ë¹ÑØÖÜ J = (Ĵ ½ ½ α Ä (Ĵ Ê Ò κ¹ñøöü J = κ ÔÖØ º Ë (Ĵ Ä ½ ½ α (Ĵ Ê )+ ½ ¾ JÄ α(jê ) ) ½, )) ½+ ½ ( )) ¾ (J Ä ½ ½ α J Ê J +
47 Ú ÖÒ ÓÐ Ò ÙÒÓÐ ÔØÙÖ Ó Ø ÖØÓÒ ( Ù, Ô,ζ)( Ù, Ô,ζ)(Ü,ζ Ô ) ( Ù, Ô,ζ) (Ü,ζ Ô ) (Ù, Ô,ζ) Ë(Ô, Ô) κ(ô, Ü ) κ(ô, Ü ) κ(ô, Ü ) Ë(Ô, Ô) κ(ô, Ü ) (Ù, Ô,ζ)(Ù, Ô,ζ)(Ü,ζ Ô ) (Ù, Ô,ζ) (Ü,ζ Ô ) ( Ù, Ô,ζ)
48 Ú ÖÒ ÄÄŹØÝÔ ÖÑ ζ Ô ζ Ô ζ Ô ζ Ô ζ Ô Ô ζ ζ Ô Ô ζ ζ Ô Ô ζ ζ Ô Ô ζ µ µ µ
49 ÓÐÒ¹ÙÒÓÐÒ Ì Ì ½ Î ½ Î ¾ ΠΠν Î Î ¾ Î
50 ËÙÑÑÖÝ Ï Ú ÓÒ ØÖÙØ ÙÒÑÒØÐ Ò ÓÙÒ¹ ØØ Ã¹ÑØÖ κ¹ñøö µ ÓÖ ÚÖÓÙ ÓÒÙÖØÓÒ Ó Ò ÖÒ Ò ÓÛ ØØ Ø ÓÙÒÖ Ö ÒØÖк Ï Ú ÓÒ ØÖÙØ Ó ØÛ Ø ÒÒ (g,h) Ò ÖÐ ØÛ Ø ÒÒ (g g,g) ØØ ÓÚÖÒ Ø ÖØÓÒ ÖÓÑ» Ò»Ú ÖÒ Ö ÔØÚÐݺ Ï ÓÛ ØØ ÖÐ ÖØÓÒ ÐÓ ÐÝ ÖÐØ ØÓ Ø ØØÖÒ Ò Ø Ùк ÇÙØÐÓÓ Ï Ò ØÓ ÙÒÖ ØÒ ÛØ Ö Ø ÙÒÖÐÝÒ ÒÒ ØÖÙØÙÖ ÓÖ Ú»Ú ÖÒ ÏØ Ø ÙÒÝÒ ÔØÙÖ
µ(, y) Computing the Möbius fun tion µ(x, x) = 1 The Möbius fun tion is de ned b y and X µ(x, t) = 0 x < y if x6t6y 3
ÈÖÑÙØØÓÒ ÔØØÖÒ Ò Ø ÅÙ ÙÒØÓÒ ÙÖ ØÒ ÎØ ÂÐÒ Ú ÂÐÒÓÚ Ò ÐÜ ËØÒÖÑ ÓÒ ÒÖ Ì ØÛÓµ 2314 ½¾ ½ ¾ ¾½ ¾ ½ ½¾ ¾½ ½¾ ¾½ ½ Ì ÔÓ Ø Ó ÔÖÑÙØØÓÒ ÛºÖºØº ÔØØÖÒ ÓÒØÒÑÒØ ½ 2314 ½¾ ½ ¾ ¾½ ¾ ½ ½¾ ¾½ ½¾ ¾½ Ì ÒØÖÚÐ [12,2314] ½ ¾ ÓÑÔÙØÒ
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