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Une introduction aux liens entre Mécanique Statistique et Théorie des champs.

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(1)Une introduction aux liens entre Mécanique Statistique et Théorie des champs. Michel Bauer. To cite this version: Michel Bauer. Une introduction aux liens entre Mécanique Statistique et Théorie des champs.. université de Rabat - 1992, 2006. �cel-00092933�. HAL Id: cel-00092933 https://cel.archives-ouvertes.fr/cel-00092933 Submitted on 12 Sep 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(117) $D ( 4": ( )"Y3Q>V1l ( :!e•o+6)–# »zl–!+xF/3 ( 4I12  ( )+NJLe$/ ( /$+K$/$+NJL\à : ( )" 3Q>V1; ( \"O36+K:<+-)*1$//$01& 7^N)":s%nF/)L" ( 363-C)"A)S,.+=9:s> ( L/$71G+63636"$/P ( /$+-C)"': ( )"3 ( ;+-/ TUZ1O+ rP"$/$+6ÊLz:!1 ( g"Z:!3 ( )L] ( ].A445U ( HW43-A:<+67aJL  l x8 :cE‚:12 O +-/U)Ls  gF ( 7/P+KO3-A:!H4 (  $3 =?h!T \ œ F,.)L)"94 ( +6)W/O) ( )W/\( & 3 ( l+j])+6Ê" ( /$+j)p:![)L# ( 3KO3NTcqQ3A>V1$/9 (  !2 JCL3KJCLPDL#$:!e/OL/( & ^ ( +-/74 ( JC ( g3-T ‡ˆ)B4 (  ( )/x3 ( ^KO4e3-HDL +N$/$+KJCL\8 ™.E ( ,.13 ( ^N 4[3-Hg+-)S:Z2 Ê)+-\8;›WE %.+63UL/5F2 /A2 / )/ ( )/z:! :<+67MJCL G F2 / ( +j/x3 ( 4 (  $:!3 ( ( /$+KO3-T ‡ˆ)*^ ( ª +j/ G 1$/a3 ( 4 (  $94+KO# +KJCL%4 ( +K3 ( 4 ( O$4 ( O7#O+KJCLM( 1$ / ¼ K U ï ( a Šx`+x3Q>V)*c#$ ›.‰ƒëÄ Gi½¾ ¿s+63xRBl3jL12 $ ( )L4 ( O$a4+NO7# +NJLMO +j/$+KJCL Gi½¾ ë ŒQŽ a ¿n)Œ:! ._ ( & :!\3 ( JL363j3 ( ª /$DAF2  +-a1$/H4 ª ( 3X:2 Ê)+-T ª ª m ( /PDGF2  +jM)n> ( :<45F/a)L3=+64+j/ )L)£/$ +j,.+ ( 3-( & ] ( )":!5:<+KP/ ( )"JLl+Z3 ( 4 (  $\4+KO7! +KJCL / )":,.F3 ( 4 (  $zO +j/$+KJCLTW“a./ C)"'JCL%": ( )"F/ / a36+64+-/ A:!] ( )":!M:<+K$/ ( )"%#3j1A:ZF2 / ( +63KZ:<L12 $ (  )/':<+K; (  nTcZ1O+"1P/')"C7z,< ( +nl+c3Q>V)$D ( )L]<Y:!a/$R.cY:!aL12 $ ( nT.mna3j1/ A1$/x+6),<+-/M2 ( & 7^ ( +67G (  ( & ( Y3Q>VO+K

(118) L12 s12 :!)W/3-F JLe‰oëÁ›l3-€12 $ ( B/$ + ( )L]3 ( +=7LF2 ]36+-F/a3-

(119) L12 $ ( *DL ( ].) ( 3 LF2 ]36+-1T. mˆ>P12 JC ( /$+-C)B8;ÉWEu1P/A)"7, ( 3 ( g3-Tmn9 (   ( ].( & 3 ( 36+64+-/ a)W/P+6)#LMPa^ ( +j/G:!a4 ( )+;& 7 ( ) ( 3-.]LT m ( 7 1:Z2 Ê)+-/P+-):!_lž1$/x)ecL/A:<+6}Y2 7)/ A8Ù+63^ ( L/Z/ )+6'C4L/ A:<^ ( +-/ZJLG3-g1,.C3645iha:!H3 ( 4 ( +63=3- 2 312 4)W/ ( +  ( $D ( )L]x2 E % ( +6)"l+zJCL3 ( 73 ( /$+-)S3=+ ( )/}©€¿VjwF/F= ( ŠZ• ( +K9)L^N+Ke1O+^ ( +j/C) ( O +-,.S( & 3Q> 12 J ( /$+-)8$µ!E T  +6)"l+l%F4Œ 45Ul+.3-ˆ€12 $ ( g +K$ˆ3Q>{+6), ( O+ ( )"U ( n./ ( /$+j)n%1363-F”QO+1$/n1$/ ( LF2 G( & 3 ( 3=+64+j/ uC)W/$+=)#LT F/ / 

(120) +6)":2 c)": ( )":!\3 ( DWRl+KJCL( & ] ( )":! 12 PDL363-

(121) (  ( "/ ( pPD ( )L].4)W/z:<SL12 $ (  4+KO7# C+KJLH1$/ˆ)543-H/$01& ~ ( 7/$+KO36+j':!YJn>@) ( c363-H3Q>t)+-,. ( 36+-/2 T.A>V1$/U)L7O+¤F2 / 2 /$01& ˆL/$+63-%F/z$CL,.)W/A:s>{+=4c7/ ( )"O "O+ ( 3-Ty ( z43-%Llc#$)"aJCLa)L"H,.3=+-)"H:Z12 O +=7M:!1 "C3-R.4S& 1%0?>V1$/O”7( & ”Q:<+67A:!H3-C)L]L1ZPD ( ‹ )L1x:s> ( / 41T0“a"U,.C3-)"~3-H4:"& 3-U3-G36"xl+643-"#Ol+6g3-T ÃH)+=4 ( ]C+6)LH3-13=+ ( +K$C)"x)/$7 ( / C451H4451:!1gZ(  / C))LF/7%36+=g71)7./ ( /$+-)[3-1)"G ( H ( "C7/ (  ( L/$71T.mn1ˆ36+ ( +K$)"UC)W/ˆ)La3-)L]L€ ©Š!A)L:!1HJL1$/P+-)"U3j1x3="Zl+643-1U1$/3 ( ;+-, ( )/ 

(122) ²Wl+s3- "C3-R.4S&  ( bÅ4 ( +=363-)"%L( & JL3=3-M:<+K$/ ( )"Y$a/$7CL,.)W/H/$R<+KJL4)/U3-1W/$L2 4+-/12 x:!a3 ( $D ( ‹ )L¨  ¸ÃH) >{+=)W/2 71 $ ( S ( An£  & b 1$/a/$01& H] ( )":sTLyxa7)":<7

(123) 3-g3;& 436"z;+643j%WC)"L/a+=4c#$ (  ™0À.

(124) 4 ( 6+ 363-C)"x:s>KF /$Yl3-136+-)":s>{)L12 P ( nTcZ445€b 1$/H] ( )":s%L),.L/aJLa3-ac./ )/$+-3uPD+64+KJCL_j $+j/x7$DL:!M ( , ( 3-zO +-/$+KJCL€j He¿ Šmx>{)+j,. ( 3=+-/2 )L" ( Ol7JL3-1ˆL12 ;3-/ ( /7~)L:2 c)":<7)/ (  :<*L12 P ( B$DLC+Kl+QTL³94 ( )+l& 7+6)":<+61/ %)L")) ( +K $)"'3 ( L2 "C)"$TL‡x)*}nF/F%L3j36+-)*)W/P7^= F/_j ( & 3 ( 36+=4+-/Oz)/$+6)!L1$/ = ( ë  G ï Gi½( ¾ ™ Š ›<‰ © à `.+-/¤%€;>+ j*/ )":,.F3 j He¿ % j ( j He¿ À = ( Š ›.‰ à© m ( 4 (  $4+KO# +KJCLa1$€/ = ë G :s>VX &. = = eH ¿ à©. ë = .› ‰. ª(. )"7. =ÂÁÄà = à © = He¿. Š.  )"B:!aL2 iL!+-)e4)/$7zJCLM1O+n1$/H736+1 2 ( ^ ( +-/'JCLa3-1/$L2 4+-/12 x:s>t)LA4 ( PDL ( 3¤ 2 ( /O+67 :!^bÅ ( A$C)W/2 3-+-]C)GF2 1U)4R.))L:s>{)L:<+KP/ ( )"^Å bbŠ ³a+N$)"cY)"O36JCL)L./$a45:"& 3-:!c3-R<4p& 71x1P/Hg" ( "Cp/P7 +64+-/$+=^;%.)/$7 ( L/P71 ( MJCL3j1"3jR.4p& 71uLF2 3N'P)W/36"' +-]+K:!1GF/5F2 ,<+-/ )/x3-1 ( L/ .”j+6)/ P1/$+-)"T ) ( L/P7M].):!\7g3;& 41$/M:!l+64[3-M8N ( M43-M]!(  ( & )Œ:<+6) ( / 1E)L\4 ( $DL ( 31 2 ( / +=7: ( )"[3-ŒC)W/$+=)#nTH“z! ( 3KO3K[)L"545C)W/$)W/JCLŒ?>V1P/"# ;+6g3-TÃH)hcL/743 ( 3- )/$+6)!e ( )€12 $ ( o( & )":<+j/$+-)5:s> ( rP"$/ x3 ( 4 ( O$4+KO# +KJCLHF/3 ( 4 ( +6363j':<€12 $ ( 8N36"ˆ363- $ ( cF/$+-/ %~36"+63x^ ( ":< ( )Œ]7#aC:<+6) ( / ?EH"eJL\3 ( /$DGF2 O+-5( & 3-C)L]L:<+K$/ ( )" ( +-/3 ( 4 (  $ 4 ( O7# C+KJLa,.3=LT ql3nR ( :!)"a:!4 ( )+;& 71U43¤2 4)W/ ( +=71‚:s>t+6)W/O LF2 /Ox)L./$7z:2 4 ( $DLTql3"1$/H/$01& ~+64"/ ( )/ :!g+j) (  l+64+63-H/ "ˆ3-¶ 1A:F2 / ( +63NZ:!\ ( H/$01& Z;+643jT yxCM/  4+6)LF/ / :<+N · O" l+-C)n%c)L./ )"JCLe3 ( 73 ( /$+j)£8 ! EHÊ\3-1a +1F2 /W12 A:<+=45)"l+j))L363-1 ( :!  l x8 :XE Šn‡x) }XF^ (= ë \ K U ï [ ( a ¿"C)4/ )Wl /P7Y‚ :!V¿ Cj )"M1JC$L/Mw  ( l )"Mx8 :X:<+6V¿ 4jUeE = )"l( +-ïC)"az1%"$:!/ )"( F )"al :<+6x8 4:cq¿ )"j‚l E +-© a )"ï &T %U1$+6/) (  3-( )"45:<)+6€/ 4l )"x8 l:c+-q¿ )"j‚NE Tc= yx( #ï  a( )Wë / ª ª Æ e8 =?:XV¿ =?ª ©€E Š ª ª y (   ( )/M( & 3 ( 36+64+-/ _  ©[ëIÀ

(125) )*gL/$+-)/ l x8 :XExE ë = a ï ( Æ e8 =?:X¿À!EŠ. \. `+>=¬ë„À!¿l x8 :XEZ1$/H)La79+K  ( )"e8Q)lc#$‰|Ÿf›WE l x8 :cmE ÁÄÇÈ:>Ç ( ï a Š. ³ ( ") u3 ( l+j/ %.)L" ( 7)"H$CL,.)W/z( & L/$+=36+K$x3-G36+-))/$73 ( :12 O7+KO ( )"M( & 3Q>t+6)Ê)+X:s>{)7 ( ] ( /  F/3 ( $/$ "/$:!5 ( /$ ( )"l^N 4„F2 \:!F%LC +-e8Nc#l+-/$+j)Œ:!1ZC3-1E Tnyx3j$D ( 4*36+6g\36+-) 1$/ 2 312 4)W/ ( + T`+n)L"12 O +-,.)"3l )*/$ ( )"l^N 4„F2 A:!s%L +j1%!)L"HgL/ )LC)" \ l \ 8x:cExëÊÉ 8$:› Ëxa : E a ¾ (  ºè ÍÜ Ì ‚ = ( Š `+n3Q>@)+=)"& 7a3l>{+K:!)/$+-/ 2 ë É n  ï Î 8 Ì [ ƒ \ [ E 9: 8AŠ (¾ ºè ™ = ( Ê ™.™.

(126) 3Q>{+=)W/F2 ] ( /P+ C)[;'¾B:!F,<+-)/2 312 4)W/ ( +6'F/a:!))L yx# ( )W/3=ÇÑ:ZÇ<ëEÒ\F/€8=. l \ 8x:cExëÊÉ n .ïÎ \ [ ï T JÏ [Ð 8Q¶²Ë:98 8 E a ® (. (Ô ë ÓÒ9¤0›!¿)gL/P+-)W/ ' l \ 8 :cEUë « ›= Ò ( ­ ï ´[ É n.  ï [W 8 Õ. Š. ƒ ÖW Ep× 8Q¶Ë:9Ó~Ó E ( a®. Š. yx^Òe]C ( )":ƒ3 ( 4IF2 /$DL:!M:<k3H8;l3 ( JL363j)L"7F,<+-)":<7C)"( & 36"l+- +K$1: ( ") 3 ( l+-/ E :!))L ( l \ x8 :XEmØ ›™ 8;› =Ëxa E ï  W Ò ' ï [[ ï × [ ´ :!)"¨  = J+H1$/a3 ( "!l+-/$+-C)b:< Z3-%‚>{+6)/  0F& / D´ RW;+KJL4)/Y443Q>{+=)W,.F$e:!\3 ( 3- )L]CL:!  L2 3 ( /P+-)nTZaL12 l3-/ ( /Z1$/A)o^ ( +-/H/$01& ]x2 )G2  ( 3QT. ™0›.

(127) ¯e¦Z«Z¨»­© e§¨z«¨»§oÔ6¥"¥1£Ù†¨z §¥1­ ¨a*c¥  ÷ ‡ˆ) :<+64)"l+-)h›!%H3-*4:"& 3-*:s>{qPl+=)L]%z:!)/)L" ( ,.)"[ ( "31o 2 3Q>V +j]+6)L)ž+6)W/P7W:<"/P+-)n%H1$/  ( / 4)W/P36g3-TaÃH)Ä ( +-/ ( 3KO3-3 ( ^KC)"/$+-)h:! ( 7/$+-/$+j)n%x4 ( +K ( "Ol+9:!o)L4[g7"$1 ( L/P71 JC ( )W/P+-/12 a ( )"^ ( +6:s> ( 7?#+=4 ( /P+-?)"TL Èar ?%c36"l+-:<+j¢ ( +6)L1:!$C36L/$+-)"aC)W/F2 /2 g36+1F2 1T mx>{)LA:s>V3=3-1s4)W/P7U ( 7/$+NO36+;& 45)/g+-)

(128) 3-x3=+-) ( ,.1u3 ( /$DAF2  +-Z:!1U$D ( 4"%?F/n)L" ( 363j)"s3Q>V!"!$ g +;F& ,.4)W/FT ˆD ( JLl+-/Oz:s>{)€12 $ ( 5 (  LH2 "C7/ H)l+6? ) Úoëhö™.Šmn1Hl+6)"ˆ+6)/  ( ]+K $)/ ( ,.13-FxWPDL1 ,.+Nl+6)"F/H3Q> 2 )L7]+-a:s>{)L)ÊL]C ( /$+-C)51$/ ê ë & 3 ( )L./ ( /$+-) Û × Ý × ß l+j])+6ÊL Û × Ý × ß ÚUe8 Ò!xE Ú9e8 Ò c EZX $44l

(129) 3-1a ( +=71M:![7$DL1

(130) ,.+Nl+6)"%n?>V1$/ ”7( & àA”Q:<á+67õ $C4t 4l\3-1Y36+-)"Tn“a"\lõ c#$)"t 

(131) JL[3- L12 $ ( 1$/A] ( )":[4 ( +NUÊ)+ ( Ê)B:s> F2 ,<+-/ H3-1'7g3;& 451:!M)W,<7].)"T m~1')/$71:!1Z ( O€12 ~^N 4)W/

(132) F2 ] ( 3-4)W/~)eL12 $ ( n%#:<+-/ˆ€12 $ ( [:< ( 3X:<€12 s12 :!)W/FT“a" ( )" g"1P+6)I: ( )"93 ( l+-/O5:s>{)L^ ( 4+6363-e:!,.1/O9"eG27\3-19"C+6)W/:<£L12 $ (  F/:!P)I:< ( 3QT xDL+Nl+K $C)"x3 ( ^ ( 4+6363-al+-, ( )W/ \² e2. δ 1. δ 2. e1. e3 δ 3. δ 4 e4. xD ( CJ La36+-)B:<oL12 P ( ƒ:< ( 3u"9)36+j)B:<€12 $ ( +6)+-/$+ ( 3QTL`+'äÒ¿qÒ¯c çZ1$/H)*/ 3c36+-))L"G)L./ C)" r. r’. ó. ^N)"/$+-))L3=3-e ï ( ðÜÛÞÝò × Ý 8ß× t E ŠUql3G1P/O3 ( +6JL ( 3KO3j[:!1, ( 3-F

(133) 45R.))L1

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(135) r*. l. mc+j])LM:!:12 $:<7T 44e)"T!³ ( ") H ( %s),Z2  +6ÊLJL äpj\8J ' E Š=Š6ŠQj\8ãâ€E§Ú

(136) 8NÒ ' LE Š6Š6Š.Ú8JÒ. \. ì E çxë ™ Úî Ú\8JÒ ' E Š=Š6ŠäÚ

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