KHOA HOC & PHAT TRIJN
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TOAN HOC VA VAT LY
Pierre Darriulat
Mersenne [so CO dang luy tbila dia 2 trd 1: 2n - 1 ) mang ten nha toan hoc ngifiS Phap Marin Mersenne.
1. Toan hoc dieu ky Nhiing n^fii khfing thuang xuydn tiep xiic vfii khoa hgc thuong ldn lgn giua vat ly vfii toan hgc, coi toan hgc la mgt cai gi dfi huydn bi, nhiing cflng thiic trim tugng khfing cfi lien he vfii thgc te. Nhung, nhu Feynman viet, "cdc nhd vgt ly su dung ly lugn do cdc nhd todn hgc tgo nen di md Id thi gi&i [...] cdc nhd todn hgc giup cde nhd vgt ly, nhung v&i vgt ly, bgn can phdi hieu mdl lien hi giira tir ngir vd thi gi&i ihuc."
Khfing nghi ngfi gi ca, toan hgc hdt sire cufin hiit. Ban bdt dau bang nhiing gia thuyet tufing chimg nhu don gian nhung lai kham phd ra mgt thd gifii tuyet didu den khfing ngfi. Ban ddm I, 2, 3... va rfii so nguyen to, hdm zeta Riemann, sfi Mersenne,
nhiing hinh vufing ky didu, nguyen ly Fermat, sfi hodn hao va vfi han ddy thach thiic ciia toan hgc xuat hidn. Gifing nhu each ma ngfln ngii eho phep ta choi voi ngfln tir vd mufing tugng ndn vii tru huydn difu, todn hgc cho phep ta choi vfii cdc tidn dd de tao ndn nhiing thii bgc ciia vfi han: dufing cong Peano, tinh khfing the quydt dmh va hang logt nhung ky quan khde.
Nhiing gia thuydt don gian ma dem lai su phong phu lam sao!
Cach xay dung cua todn hgc d?p de vd thanh tao dem lgi cho ta mgt cam giac thudn khidt! Khac vfii cac mfin khoa hgc phu thuge vao hien ttrgng vfii muc dich tim hieu va midu ta, toan hgc cfi thd tg trim tugng hfia tu nhiing hifn tugng nhu vdy, dung nen cac tidn dd ttr su ggi y cua ttr nhien dd cfi the khai ngufin cho nhiing cugc
kham pha. Trong khi nhimg gia thuyet ciia khoa hgc phdi dugc lidn tuc xem xet lai thi toan hgc la vTnh euu.
2. La ly thuyet hay quan sat thtfc tiln?
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Trufie khi minh hga vd viec iing dung todn hgc ttong vdt ly, cho ^ phep tfli chi ra mgt sfi sai ldm m^
nhidu nguoi thuang mac phai. .^;- Trufic het, khoang each giiia%- ly thuydt vd quan sdt khflng d&t^g nfii Ifin nhu nhidu nguoi ngjil ^ . Khi nfii: "Mat tr&i ldn &phia Tdy", thi dfi la ly thuydt hay quan sat? De ggi ten Mat ttfii, ta CA phai tufing tugng rdng nhung cai dia mau dfi bidn mat sau dirong ^ chdn ttoi vdo bufli ehidu ma ta J ^ nhin thdy cting Id cai ma ta shin thdy di Idn fi hufing bdn kia v i ^ ^ ; bufii sdng: dd ggi do Id M a t t i ^ dfing nghTa voi vide da cfi mgt IJ thuyet ve Mat ttfii. Ngay khi G6 gdng midu td cdi quan sat dugc, chuydn nfi thdnh nhung cau chii, con sfi, thfing tin md ndo bo lini trir va xir Iy, thi didu dfi ciing dflng nghia vfii vide chiing ta da tgo ndn mgt Iy thuyet. Hay nghi t
Trong khi nhQng giai cua khoa hpc phai d f tyc xem xet l^i thi toanp
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vd cdch mflt diia tte mfii sinh hgc each nhdn ra khufin mat va gigng nfii cua nguoi me. Chdng phdi dfi chinh la tgo ndn mgt ly thuydt vd ngufii me? Nhiing ly thuydt dugc hgc tu nhfi thi ta van ggi la le thuong tinh, cfin ly thuydt ma chua dugc nghidn ciiu rfi thi ta ggi la dieu bi dn.
3 . Toan hpc la mai m a i va vat ly luon doi thay Ngufii ta thuong nfii rdng vai trfi eua ly thuydt vat ly la chi ra sg rang bufic giiia cae dgi lugng md chung ta diing dd mfi ta hifn tugng, vi du E=mc^ hay F=ma.
Tuy nhidn, didu ndy ngu y mgt gia dinh quan ttgng rang: cac dai lugng da dugc xdc dinh. Du nfii ve chat didm, Igc vecto, vecta bfln chidu khflng-thfii gian, trang thai Iugng tii khfing gian Hilbert hay sidu ddy thi nhirng khai nifm rdt khfing rfi rang, chung dd phdi dugc tu duy va nghidn ngdm tir trufie. Chiing ta cdn dd y rang nhiing khdi nifm nhu vdy dugc xdy dung mgt phan ndo dfi kha tiiy y va thidu chinh xac; chiing khfing phdi la nhung khdi nifm hodn hao md la kdt qua ciia su
Ta khdng the tuyen bo mot iv thuyet n^o 66 \k toi thugng va n^n tir bo tham vong CO duoc chSn ly tuy^t doi.
trim tugng hoa va cfi thd can dugc sira dfii lai. Su dep de va hodn hdo bd ngodi ciia mgt ly thuydt thufing lam cho chiing ta qudn di didu nay. Se la sai lam ndu coi mfit ly thuyet vat ly la chdn ly tuyft dfii. Chimg ta phai lufin san sang xem xet lai ca sfi nen mfing ciia cdc Iy thuyet de cfi thd tgo ra nhiing nen mong mfii gitip xay dung cac ly thuyet tfit hem, tfing quat hon va/hoac chinh xac hem. Ca hgc Newton khfing phai la mot ly thuyet tot nhu Thuyet tucmg dfii Dac biet nhung nfi Idifing hd kem phdn trim tugng. Ca hgc Newton dem lgi cam giac dd hieu vfii nhiing ngufii vfin da quen thuflc vfii nfi, ddn miic hg quen mat rang ly thuyet nay cung dua ttdn nhiing khai nidm trim tugng, cai ma hg da mac dinh la hidn nhien diing.
Tdt ca cac ly thuyet vdt ly deu
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trim tugng; vai ttfi ciia chiing Id tao ndn nhiing khai nifm trim tugng. Mfit Iy le tuong chirng hidn nhien thuc ra cung khfing cfi gi dgc bidt, nfi thuan tiiy chi Id mgt dieu ma chiing ta dd ITnh hgi duge, do thgc chdt vdn la mgt khai nidm chii quan, ngay cd khi nfi da trfi thanh quan diem chung cua sfl dfing. Khoa hgc lufin sin sang chap nhan mgt ly thuydt mfii, khong quan ttgng nfi trim tugng ddn miic nao, mien Id nfi tfit hon ly thuydt cu. Ta khfing the tuyen bfi mgt ly thuyet ndo dfi la tfii thugng va ndn tir bfi tham vong cfi dugc chdn ly tuyft dfii.
4 . ling diing toan hgc trong vat ly
Vo hgn vd viphdn trong vgt ly Bay gifi hay cho phep tfli dua ra mgt vai vi du ve each thirc cac nha vat Iy sir dung toan hgc.
Khai niem ve vfi ban va vi phan lufin lufin kich thich cac nha toan hgc va ttiet hgc. Hay nghT den nghich Iy Hilbert ve mgt khach san vfii vfl sfl phfing cfi the chiia vfi han khach. Nhu la mfit vi du, tfii thich sir dung each chiing minh rang sfi nguyen ciing nhieu nhu sfl hiiu ti bdng each thay nhiing sfi hiiu ti ttong mgt bang cfi tir sfi la sfl cfit va mau sfi la sfi hang. Chi ddn dau the ky trufie, van dd nay mfii dugc lam rfi bfii Dedekind, Cantor, Frege va mgt sfl n^fii khac. Vat Iy sii dung rat nhieu khai niem vfl cung nhfi va vfi hgn nhung chi nhu la mgt cflng cu. De mfl ta the gifii, vat ly khfing can den vfi ban.
Theo ca hgc lugng tir, mgt chdt diem vfi ciing nhfi khfing cfi y nghla vat ly. Vu tru khfing vfi hgn ma nfi chiia 10*" hat ca ban. Nfii Vli tru vfi hgn chi don gian ngu y rang nhung kidn tbiic ta bidt vd vii try la nhirng thii nam trong
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dufing chdn trfii (khoang each chung ta cfi the nhan dugc anh sang tii vu nfi lon Big Bang,-14 ti nam anh sang), tuy nhien chang cfi ly do gi dd cho rang Vu tru khfing tidp tuc mfi rgng ra bdn ngodi bidn nay.
Ly thuyet nhom lrong vgt ly Ly thuyet nhfim cung cap mfit ndn tang dgc biet mau mo cho ly thuyet vdt ly. Cai dugc ggi la
"Mfi hinh chudn" mfi td vdt ly hgt ca ban dugc xay dung dua tren nhiing nhom Lie: nhfim Poincare (nhfim quay va tinh tien trong khflng thfii gian bao gflm ca nhiing phep bien doi Lorentz) va tieh ciia cac nhfim
S U ( 3 ) C X S U ( 2 ) L X U ( ] ) tao ra cac hgt fermion ca ban. Bdt bien chuan phdt ra nhiing hgt bonson chuan va pha vfi dfii xiing phat ra cac hat Higgs. Nhfim Poincare la nhfim Lie cua phep tinh tien vd phep quay trong khflng gian vfii chudn khoang each (metric) cua Thuyet tuong dfli Dac bift. Dieu nay ngu y tfin tai cac hat va phan hgt dugc mieu td bfii cac vecto bfin chieu nang-xung lugng vfii bat bien khfii lugng va spin. Tich cua cac nhflm
SU(3)cxSU(2)L>'U(l)mieuta dfii ximg dugc nhiing hat ca ban nay tuan theo, dieu dfi cfi nghia la cac hifn tugng vat ly bat bien khi hoan dfli hai hat cfi bieu didn tuang tu trong nhimg nhom nay, chang ban nhu hoan dfii proton thanh neutron. Den day, tdt ca cac hgt fermion (cac hat co spin Vi) tao ndn vat chat trong Vii tru va tinh chdt ciia chung da dugc mieu td mgt each diing ddn. Neu yeu cdu thdm bdt bi6n chuin (pha cua mgt trgng thai Iugng tu cfi thd duge cfi dinh mfit cdch tiiy y tai mgi diem trong khfing gian) se Iam xudt hifn vecta boson khflng khfii lugng, hic la nhiing
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hat gifing iinh sang: photon, boson truyen tuong tac yeu va gluon truyen tucmg tac manh.
Pha vfi dfii xirng tao nen khfii luang cho cac hat boson nay va phat them mot vo hufing (scalar) mfii la hat Higgs boson. Hat Higgs, mo! dirac phai hicn gan day tai Trung tam Hat nhan chau Au (CERN). la hat co ban cuoi cung con lai cua Vlu hmh chuan.
Nhinig duel! the Mgt vi du thii ba vc mgt khai niem toan hoc dugc ap dung thanh cong trong \ at ly la nhirnji chieu dtr (extra dimensions).
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Chieu du da timg bi cho la mot meo toan hoc ma khfing c6 ly giai vat ly cu the cho den khi Klein dua ra y tucmg rang chi^u du thir nam nay bi ufin cong din noi ta khong the nhin thay duac (ggi la bl xoan). Ly thuyet hien dai nhat ngay nay, thuyet tap M, yeu cau 10 chieu khfing gian va 1 chieu thai gian (trong do 7 chi^u khong gian bi xoan).
5. Toan hgc khong phai la chiec dua than Dirac luon luon qua quyet rang phuang trinh noi tieng cua ong
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Nghich ly Hilbert vekhaths;
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0 phong CO the chita vo han khach.
Nguoi ta da loai bo diarc dieu cam ky vc chieu du khi Thuyet tucmg doi Dac biet dira vao khai mem khong-thfii gian hay vecto bon chicu nang-xung luang. khai mem vc khoi lucmg, spin va phan hat. Them \ao do. Thuyet tuang doi Rgng mo ta hap dan bang each lien he tensor nen nang lugng vfii sir bicn dang khong gian do su nen nang lugng gay ra Sau do. Kaluza dira ra chieu du thir n;lni dc dua vao nhiing phuong tnnh Maxwell cua dien tir hoc (the \ecta va vo hufiniJ),
mo ta electron va positron duoc dan dat bai tham vgng xay dung nen mgt ly thuyet dep v e mat toan hgc. That khong may.
phuang trinh nay khong phai luc nao ciing co the ap dung dep de duac nhu vay. Trong nhiing nam 60. Chew, Mandelstam, Frautschi va Regge khai thac tinh kha tich cua ma tran S (lien he giiia ti-ang thai ban dau va trang thai cuoi ciia mot he) va khai niem doi xung cheo (hen he tinh chat ciia mgt tuong tac a+b^-*c+d vfii a^-c^-^b+d a do mgt vai hat
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chuydn thanh phdn hgt cita nfi) da de ra khai nifm dan chii hat nhan.
Y tufing cua khai nidm nay Id khfing cfi hgt nao "ca ban" hon nhiing hgt khac va tat cd cac hgt ddu dugc phdt boi co chd "tu nang gidy", lay y tufing tix hmh anh rngt ngufii ndng ca thd minh Idn bdng each tu keo gidy ciia minh. Y tuong dfi da rat phfi bidn, vd dfi cung Id Iy thuydt ma chiing tfii dugc hgc khi cfin la sinh vien, nhung nfi da khfing bao gia dugc ap dung mgt each cu thd cd vd, cufii cung, sau mgt the ky ly thuydt nay da bi loai bfi. Ngdy nay, Sidu dfli xung, ly thuydt ghep cap boson voi fermion, la y tuong vd mfit dfli ximg hodn my vd mat todn hge. Sg dep de ciia nfi cue k^ hdp ddn den nfii khfi CO the hinh dung dugc rdng Tu nhidn lgi lam nga vfii dfii xung nay. Tuy nhidn, cho den nay, chung ta van chua thanh cflng ttong vide tim kidm bdng chiing ve nfi ttong thgc
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dgi dd giai quydt van dd nay la thay thd cac hat co ban bdng cac vfing day vfii do nen ciia ddy xac dinh thang Planck. Vfii cdch lam nay, he thiic bdt dinh Heisenberg dugc sua dfii de co the hoat dfing fi thang Planck nhung nfi dfii hoi them 9 chidu khfing gian (trong dfi 6 chieu bi xoan). Bfin ly thuydt day da dugc tgo ra tuong ling vfii day dfing, day mfi, day cfi dinh hufing va khfing dinh huong. Tuy nhien, bfin ly thuydt nay hfia ra ciing chi la bg phgn cua mflt Iy thuyet khac dugc ggi
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Trong sufit 40 nam tiua, nhtmg nfi luc tim kiem sai Idch cua Mfi hinh chudn deu thdt bai. Trong khi dfi, nhung khdm pha lon gdn day ciia vat ly thidn vdn vd vdt chdt tfii va ndng lugng tfii lai ttfi thdnh nhiing vdn de thach thiic nhat dfli vfii vdt Iy Iy thuyet hifn dai. Van dd chinh cua vgt ly ngdy nay Id su thdt bai ciia chiing ta trong vifc mfi ta thang Planck, -10"^^
cm, tuong ung vfii ttgng thai ciia Vii tru tgi thfii didm Vu nfi Ifin.
Tai thang nay, hdp ddn manh den miic lam cho hf thuc bdt dinh Heisenberg, nen tang cua vat Iy Iugng tu, khfing cfin dp dung dugc nua. Mgt each tiep can hien
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Thuyet tap M yeu cau 10 chieu khSng gian va 1 chieu thcii gian, trong So 7 chieu khong gian bj xoSn.
la thuyet tap M vai 10 chieu khflng gian (7 chidu bi xoan).
Cac ly thuyet ddy nay dugc xay dung dd ap dung cho nhirng linh vuc rat khac nhau (khoang each nhfl vfii khoang each Ifin, tucmg tac manh vfii tuang tac ydu).
Chiing lien he vfii nhau bfii tinh dfii ngdu, mgt khai niem da trfi thanh trung tam ciia cufic chai vi nfi dua ra bang chung cho su tuang ducmg cua cac ly thuyet ddy. Cai ggi la dfii ngau AdS/CFT (Maldacena 1997) lien
he cac ly thuydt day trong khfing gian Anti-De Sitter vfii nhirng Iy thuydt trufing bdo giac Yang- Mills.
Mfi ta toan hgc khflng gian 10 chieu ro rang Id mgt su phiic tap cung cue vd mat hinh hgc vfii nhiing mang da chidu (ggi Id mang-p, tfing quat hfia ciia khai niem vong day). Chinh bfii vay, vdt Iy dang trong giai doan ma ly thuyet va thuc nghiem troi xa nhau tgi thang Planck, 6 do toan hgc thflng tri vat ly ly thuydt.
Mfit minh chirng ngoan muc cho su tien hfia nay la Ed Witten mgt trong nhihig ngufii cfi anh hufing Ifin tfii vat Iy Iy thuyet hien dai da dugc ttao giai thufing Fields nam 1990!
Ngdy nay, mgt sinh vidn vdt Iy Iy thuyet cao cap, dac biet chuyen nganh ve ly thuydt day, phai quen thufic vfii mgt danh sach dai cac chii de toan hgc; phan tich sfi thuc va sfi phiic, ly thuygt nhom, hinh hgc vi phan, nhfim Lie, cac dgng vi phan, dai so dfing deu, dfii dfing ddu, dfing luan, bfi sgi, Ifip ridng, diiih ly chi sfi, sieu dfii xiing va sidu hdp ddn.
Hien nay, nhting linh vuc hien dai nhat nghien cuu ve sieu day chinh la hai chuang cua toan hgc; Iy thuyet tap K (ly thuydt tfing quat hfia doi dfing ddu hi cac dang vi phdn tfii nhung bo vecta) va hinh hgc khfing giao hodn.
Ngufii dgc quen thuflc vfii nghien ciiu ciia Ngo Bao Chdu han nhan thdy rang nhirng dfing gfip ciia fing cho toan hgc, dac biet Ifii giai ciia bo de ca ban, cfi lien he vfii nhirng chii de toan hgc trong ly thuyet tap M. Rat thil vi khi dgc cai nhin ciia fing ve moi lien he giiia nhirng dfing gfip do vfii su phat ttien ciia vat Iy hien dai, tuy nhien didu nay
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Vide the gioi tu quan sat va tu ly giai ve ban than no, dua tren cAc kh^ nifm do chfnh no sang tao ra, chinh cai vonfi ludn quan nay cua khoa hoc ngan can n6 dua ra cau tra loi khich quan cho cau hoi co ban: Tai sao the gioi lai tOn tai ma khdng phai la khdng gi ca?
Hat Higgiduac pti^higii taiTning tam Hat nhan Chau Au (CERN).
nam ngoai chii de ciia bai bao nay.
7. Chiia dang an noi dau?
Nhieu didu vfi ly da va vdn dang dugc nfii va vidt ve chu dd nay ttong hoan canh hien nay, ddc bidt vfii y dinh tao ra nhiing san phdm bdn chay nhat. Didn hinh ttong sfi do Id Bdn Thidt kd VT dgi ciia Steve Hawking hay Hgt cua chiia ciia Leon Lederman.
Dirac, Wigner, Einstein da timg tu hfii tgi sao Vii tnj lgi "eo the hidu dugc". Giong nhu Chan- drasekhar, hg hfii: "Tgi sao lgi cd the nhu vgy: tri ndo cua con ngu&i sdng tgo ra nhung khdi mem triru luang vd nhdn thdy chitng tuyit dep? Vd lgi sao nhirng khdi niim dd lgi lucmg img mgt cdch chinh xac din nhu
vgy trong tu nhien?"
Nhung, lam sao the gifii cfi the vira tu quan sat chinh no, dfing thfii lai dung mgt ngfii thii ba de nhdn xet ve sg quan sdt dy?
Nfi da tgo ra logic va toan hgc nham quan sat va mfi td ban than minh, vay tai sao nfi phdi ban khoan vd kha nang ki difu nay?
Dua vao ca sfi nao dd dfi dem mite do ki difu ay? Trong mgt khodng thfii gian ddi, mfit each vfl tinh, eac nha khoa hgc da hi danh bgi bfii do giac ciia chu nghia hidn thuc, dfi la quan nidm ve su ton tgi cua mfit hien thue dfic lap ben ngodi [chii the quan sat]. Tuy nhien, vide thd gifii tg quan sat vd tu ly giai vd ban thdn nfi, dua trdn cac khai nifm do chinh nfi sang tao ra, chinh cdi vfing Iuan qudn nay cua khoa hgc ngan can nfi dua ra cau ttd Ifii
khdch quan cho cau hfii cat Tgi sao the gioi lai ton tai mai khfing phdi Id khfing gi cd? Neu ngdy nay chung ta hieu ro hon v^
vdn dd nay ban truoc day, thi do Id nhfi cflng lao cdc nhd sinh h^c chii khfing phdi cdc nhd toan hpc hay vat ly, dac bift nhfi cac i than kinh hgc; "gia thuydt C kinh nggc" nam 1994 cua Franos Crick cho rdng chiing ta dugc tao ndn bfii khfing gi khac ngoai cac nguydn tii; rdng cac qua trinh vat ly va hfia hgc te bao dieu khien CO thd va tam tri ciia chiing ta.
Leibniz da timg cho rang phai c6 mgt ly do ndo do dd vii tru nay ton tai han Id ttfing rfing. Ngay nay chiing ta da tir bfi hy vgi^
rang cfi thd cfi mfit "ly do" nhu vdy, nhu Ludwig Wittgenstein ^' timg nfii; "Diiu bi dn khdngpfiai la vu tru nhu thi ndo. md ban thdn su tdn tgi ciia nd dd Id i bi dn. Phuangphdp hap iy t triel hgc Id nen giir yen i chi ndi khi ta da rd dieu can i tire Id nhirng phdt bieu dia I hge tu nhien khdng liin guan gi din triet hge. Vi nhihig dieu la khdng the ndi, tg nin giir im ldng".
Tfin tgi chii khfing phdi trfing rong, ngdy nay sinh hge hifn dai dd khidn hi dn ddng sau didunl^
ttfi ndn an sdu hon so vfii truoc ddy. Va ndu mufin thi ai cung co thd gan didu bi an ay la do su s^
ddt ciia Chiia ttfii. n
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