Povnoyu grupoyu Lorenca nazivayut mnozhinu peretvoren x m L m n x n 1 displaystyle x mu Lambda mu nu x nu qquad 1 yaki zalishayut kvadratichnu formu normu 4 vektora s 2 x m x m g m n x m x n x 0 2 x 2 2 displaystyle s 2 x mu x mu g mu nu x mu x nu x 0 2 mathbf x 2 qquad 2 invariantnoyu Nazvana na im ya Gendrika Antona Lorenca Zmist 1 Povna grupa Lorenca Komponenti zv yaznosti Diskretni peretvorennya grupi 2 Ortohronna grupa Lorenca Generatori ta algebra grupi 3 Rozsheplennya algebri grupi Operatori Kazimira grupi 4 Klasifikaciya poliv po vidnoshennyu do peretvoren za zagalnim nezvidnim predstavlennyam grupi Lorenca 5 Div takozhPovna grupa Lorenca Komponenti zv yaznosti Diskretni peretvorennya grupi RedaguvatiVirazi 1 2 displaystyle 1 2 nbsp dayut klasifikaciyu matric neperervnih peretvoren L displaystyle Lambda nbsp g m n x m x n g a b x a x b g a b L m a L n b x m x n g m n g a b L m a L n b g L T g L 3 displaystyle g mu nu x mu x nu g alpha beta x alpha x beta g alpha beta Lambda quad mu alpha Lambda quad nu beta x mu x nu Rightarrow g mu nu g alpha beta Lambda quad mu alpha Lambda quad nu beta Rightarrow hat mathbf g hat mathbf Lambda T hat mathbf g hat mathbf Lambda qquad 3 nbsp Zvidsi dlya viznachnikiv matric zliva ta zprava mozhna otrimati d e t g 1 r i g h t d e t L T g L d e t L 2 d e t L 1 displaystyle det left hat mathbf g right 1 right det left hat mathbf Lambda T hat mathbf g hat mathbf Lambda right det hat mathbf Lambda 2 Rightarrow det left hat mathbf Lambda right pm 1 nbsp Na vidminu vid zvichajnogo analizu matric peretvorennya Lorenca u ramkah STV treba zalishiti vipadok z vid yemnim viznachnikom Ci dva vipadki rozbivayut neperervni peretvorennya grupi na dvi pidmnozhini yaki ne mozhna otrimani odna z odnoyi shlyahom neperervnih peretvoren taki pidmnozhini nazivayutsya komponentami zv yaznosti Ci dvi pidmnozhini nazivayut L L displaystyle L L nbsp vidpovidno Persha z nih mistit odinichnij element v silu odinichnosti viznachnika a otzhe mozhe nazivatisya pidgrupoyu Druga odinichnij element ne mistit tomu ne ye pidgrupoyu Dali umova 3 displaystyle 3 nbsp mozhe rozbiti pidmnozhini she na dvi pidmnozhini Vikoristovuyuchi cej viraz dlya vipadku m n 0 displaystyle mu nu 0 nbsp mozhna otrimati g 00 1 r i g h t g a b L 0 a L 0 b L 0 0 2 i 1 3 L 0 i 2 L 0 0 2 1 i 1 3 L 0 i 2 1 displaystyle g 00 1 right g alpha beta Lambda quad 0 alpha Lambda quad 0 beta left Lambda quad 0 0 right 2 sum i 1 3 left Lambda quad 0 i right 2 Rightarrow left Lambda quad 0 0 right 2 1 sum i 1 3 left Lambda quad 0 i right 2 geqslant 1 nbsp Zvidsi mozhlivi dva vipadki L 0 0 1 L 0 0 1 displaystyle Lambda quad 0 0 geqslant 1 quad Lambda quad 0 0 leqslant 1 nbsp Znovu zh taki matrici dlya pershogo vipadku ne mozhut buti zvedeni do matric drugogo vipadku shlyahom neperervnih peretvoren Tomu kozhna z pidmnozhin L L displaystyle L L nbsp dodatkovo rozbivayetsya na dvi pidmnozhini L L displaystyle L uparrow L downarrow nbsp strilka vgoru vidpovidaye dodatnomu znachennyu nulovoyi komponenti matrici peretvorennya strilka vniz vid yemnomu Persha pidmnozhina mistit mozhe mistiti odinichnij element a druga ne mistit nulova komponenta mozhe buti lishe vid yemnoyu tomu odinichnij element ne mozhe buti predstavlenij Tomu persha pidmnozhina utvoryuye pidgrupu a druga ni Otzhe grupa Lorenca skladayetsya iz chotiroh komponent zv yaznosti L L L L L 4 displaystyle L L uparrow cup L uparrow cup L downarrow cup L downarrow qquad 4 nbsp Ne vrahovuyuchi ob yednan komponent yedinoyu pidgrupoyu u grupi ye komponenta L displaystyle L uparrow nbsp Cya komponenta nazivayetsya ortohronnoyu vlasnoyu grupoyu Lorenca Fizichno yij vidpovidayut peretvorennya Lorenca ta napryamlenist chasu u majbutnye Prote povna grupa Lorenca mozhe mistiti ne lishe neperervni peretvorennya a j diskretni Dijsno umova 3 displaystyle 3 nbsp dopuskaye takozh peretvorennya E x x 0 x T x x 0 x P x x 0 x P T x x 0 x displaystyle quad Ex x 0 mathbf x quad Tx x 0 mathbf x quad Px x 0 mathbf x quad PTx x 0 mathbf x nbsp Persha operaciya vidpovidaye odinichnomu elementu chisto formalno ce vidpovidaye diskretnomu peretvorennyu druga chasovij inversiyi tretya prostorovij inversiyi chetverta kombinaciyi chasovoyi ta prostorovoyi inversij Yavnij viglyad matric peretvoren vidpovidaye E d i a g 1 1 1 1 T d i a g 1 1 1 1 P d i a g 1 1 1 1 P T d i a g 1 1 1 1 displaystyle hat mathbf E diag 1 1 1 1 quad hat mathbf T diag 1 1 1 1 quad hat mathbf P diag 1 1 1 1 quad hat mathbf P mathbf T diag 1 1 1 1 nbsp Taki diskretni peretvorennya mozhut perevoditi odnu komponentu zv yaznosti u inshu Dijsno drugu komponentu zv yaznosti z 4 displaystyle 4 nbsp mozhna otrimati z pershoyi pri diyi na neyi peretvorennya T P displaystyle TP nbsp tretyu pri diyi T displaystyle T nbsp peretvorennya chetvertu pri diyi P displaystyle P nbsp peretvorennya Tomu maye sens analizuvati lishe ortohronnu grupu Lorenca yiyi poznachayut yak S O 3 1 displaystyle SO uparrow 3 1 nbsp Ortohronna grupa Lorenca Generatori ta algebra grupi RedaguvatiOrtohronna grupa Lorenca 6 parametrichna grupa S O 3 1 displaystyle SO 3 1 nbsp sho ob yednuye grupi obertan u 3 vimirnomu prostori ta lorencivskih bustiv u 4 vimirnomu prostori chasi Rozglyadayetsya prostir chas i dlya nogo obertannya navkolo fiksovanih ortogonalnih osej O x i displaystyle Ox i nbsp na kuti a i displaystyle alpha i nbsp ta peretvorennya Lorenca pri spivnapryamlenni vektora vidnosnoyi shvidkosti v i displaystyle v i nbsp ISV vzdovzh osej O x i displaystyle Ox i nbsp Vidpovidni peretvorennya mayut viglyad x i g x i v t x j x j t g t v x i c 2 displaystyle x i gamma x i vt quad x j x j quad t gamma t frac vx i c 2 nbsp x x c o s a y s i n a y x s i n a y c o s a displaystyle x xcos alpha ysin alpha y xsin alpha ycos alpha nbsp x x c o s a z s i n a z x s i n a z c o s a displaystyle x xcos alpha zsin alpha z xsin alpha zcos alpha nbsp y y c o s a z s i n a z y s i n a z c o s a displaystyle y ycos alpha zsin alpha z ysin alpha zcos alpha nbsp a matrici perehodu L x g 1 g 1 v 1 c 2 0 0 g 1 v 1 g 1 0 0 0 0 1 0 0 0 0 1 L y g 2 0 g 2 v 2 c 2 0 0 1 0 0 g 2 v 2 0 g 2 0 0 0 0 1 L z g 3 0 0 g 3 v 3 c 2 0 1 0 0 0 0 1 0 g 3 v 3 0 0 g 3 displaystyle hat L x begin pmatrix gamma 1 amp gamma 1 frac v 1 c 2 amp 0 amp 0 gamma 1 v 1 amp gamma 1 amp 0 amp 0 0 amp 0 amp 1 amp 0 0 amp 0 amp 0 amp 1 end pmatrix quad hat L y begin pmatrix gamma 2 amp 0 amp gamma 2 frac v 2 c 2 amp 0 0 amp 1 amp 0 amp 0 gamma 2 v 2 amp 0 amp gamma 2 amp 0 0 amp 0 amp 0 amp 1 end pmatrix quad hat L z begin pmatrix gamma 3 amp 0 amp 0 amp gamma 3 frac v 3 c 2 0 amp 1 amp 0 amp 0 0 amp 0 amp 1 amp 0 gamma 3 v 3 amp 0 amp 0 amp gamma 3 end pmatrix nbsp R x 1 0 0 0 0 1 0 0 0 0 c o s a 1 s i n a 1 0 0 s i n a 1 c o s a 1 R y 1 0 0 0 0 c o s a 2 0 s i n a 2 0 0 1 0 0 s i n a 2 0 c o s a 2 R z 1 0 0 0 0 c o s a 3 s i n a 3 0 0 s i n a 3 c o s a 3 0 0 0 0 1 displaystyle hat R x begin pmatrix 1 amp 0 amp 0 amp 0 0 amp 1 amp 0 amp 0 0 amp 0 amp cos alpha 1 amp sin alpha 1 0 amp 0 amp sin alpha 1 amp cos alpha 1 end pmatrix quad hat R y begin pmatrix 1 amp 0 amp 0 amp 0 0 amp cos alpha 2 amp 0 amp sin alpha 2 0 amp 0 amp 1 amp 0 0 amp sin alpha 2 amp 0 amp cos alpha 2 end pmatrix quad hat R z begin pmatrix 1 amp 0 amp 0 amp 0 0 amp cos alpha 3 amp sin alpha 3 amp 0 0 amp sin alpha 3 amp cos alpha 3 amp 0 0 amp 0 amp 0 amp 1 end pmatrix nbsp Shob otrimati generatori grupi treba matrici rozklasti v ryad po parametrah peretvorennya zalishivshi lishe linijni po parametrah matrici generatorom bude matricya pri parametri Dlya matric peretvorennya Lorenca u takomu vipadku g i 1 displaystyle gamma i approx 1 nbsp dlya matric povorotu c o s a i 1 s i n a i a i displaystyle cos alpha i approx 1 sin alpha i approx alpha i nbsp i todi generatori peretvoren mayut viglyad L 1 0 1 c 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 L 2 0 0 1 c 2 0 0 0 0 0 1 0 0 0 0 0 0 0 L 3 0 0 0 1 c 2 0 0 0 0 0 0 0 0 1 0 0 0 displaystyle hat mathbf L 1 begin pmatrix 0 amp frac 1 c 2 amp 0 amp 0 1 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 end pmatrix quad hat mathbf L 2 begin pmatrix 0 amp 0 amp frac 1 c 2 amp 0 0 amp 0 amp 0 amp 0 1 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 end pmatrix quad hat mathbf L 3 begin pmatrix 0 amp 0 amp 0 amp frac 1 c 2 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 1 amp 0 amp 0 amp 0 end pmatrix nbsp R 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 R 2 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 R 3 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 displaystyle hat mathbf R 1 begin pmatrix 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 1 0 amp 0 amp 1 amp 0 end pmatrix quad hat mathbf R 2 begin pmatrix 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 1 0 amp 0 amp 0 amp 0 0 amp 1 amp 0 amp 0 end pmatrix quad hat mathbf R 3 begin pmatrix 0 amp 0 amp 0 amp 0 0 amp 0 amp 1 amp 0 0 amp 1 amp 0 amp 0 0 amp 0 amp 0 amp 0 end pmatrix nbsp Algebru grupi viznachayut komutatori generatoriv Na prikladi komutatora generatoriv L 1 L 2 displaystyle hat mathbf L 1 hat mathbf L 2 nbsp peretvorennya Lorenca mozhna prodemonstruvati otrimannya komutacijnih spivvidnoshen L 1 L 2 0 1 c 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 c 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 c 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 c 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 c 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 c 2 0 0 0 0 0 0 displaystyle hat mathbf L 1 hat mathbf L 2 begin pmatrix 0 amp frac 1 c 2 amp 0 amp 0 1 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 end pmatrix begin pmatrix 0 amp 0 amp frac 1 c 2 amp 0 0 amp 0 amp 0 amp 0 1 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 end pmatrix begin pmatrix 0 amp 0 amp frac 1 c 2 amp 0 0 amp 0 amp 0 amp 0 1 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 end pmatrix begin pmatrix 0 amp frac 1 c 2 amp 0 amp 0 1 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 end pmatrix begin pmatrix 0 amp 0 amp 0 amp 0 0 amp 0 amp frac 1 c 2 amp 0 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 end pmatrix begin pmatrix 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 0 amp frac 1 c 2 amp 0 amp 0 0 amp 0 amp 0 amp 0 end pmatrix nbsp 1 c 2 R 3 displaystyle frac 1 c 2 hat mathbf R 3 nbsp Provodyachi analogichni rozrahunki mozhna otrimati komutacijni spivvidposhennya dlya usih matric R i R j e i j k R k L i R j e i j k L k L i L j 1 c 2 e i j k R k displaystyle hat mathbf R i hat mathbf R j varepsilon ijk hat mathbf R k quad hat mathbf L i hat mathbf R j varepsilon ijk hat mathbf L k quad hat mathbf L i hat mathbf L j frac 1 c 2 varepsilon ijk hat mathbf R k nbsp Vidno sho pri c gt displaystyle c gt infty nbsp ostannij komutator riven nulyu rezultat dlya generatora peretvorennya Galileya Nenulovi komutatori dlya generatoriv virazhayut toj fakt sho kompoziciya dvoh peretvoren u zagalnomu vipadku abo povoroti navkolo riznih visej abo peretvorennya Lorenca pri neparalelnih dvoh vidnosnih shvidkostyah ne daye znovu peretvorennya Lorenca Simvoli Levi Chivita u kozhnomu komutatori pov yazani z tim sho grupa Lorenca grupa obertannya u 4 vimirnomu prostori chasi Yak vidno komutatori dvoh generatoriv lorencevskih bustiv rivni z tochnistyu do koeficiyenta generatoru trivimirnih obertan Cej fakt pov yazanij iz tim sho grupa Lorenca ne ye unitarnoyu Dijsno perejshovshi vid antiermitovih kompleksne spryazhennya ta transponuvannya daye umovu na koeficiyenti a j i a i j displaystyle a ji a ij nbsp matric R i displaystyle hat mathbf R i nbsp do matric i R i displaystyle i hat mathbf R i nbsp yaki ne zminyuyutsya pri ermitovomu spryazhenni i vid matric L i displaystyle hat mathbf L i nbsp do matric i L i displaystyle i hat mathbf L i nbsp mozhna otrimati virazi komutatoriv R i R j i e i j k R k L i R j i e i j k L k L i L j i 1 c 2 e i j k R k displaystyle hat mathbf R i hat mathbf R j i varepsilon ijk hat mathbf R k quad hat mathbf L i hat mathbf R j i varepsilon ijk hat mathbf L k quad hat mathbf L i hat mathbf L j i frac 1 c 2 varepsilon ijk hat mathbf R k nbsp Iz nih vidno sho pri ermitovomu predstavlenni grupi obertan yaka ye pidgrupoyu Lorenca grupa lorencevskih bustiv ye antiermitovoyu a otzhe ne unitarnoyu i nekompaktnoyu komutator dvoh generatoriv bustiv daye generator obertan Umova unitarnosti operatora pov yazana iz zberezhennyam normi deyakoyi velichini nezalezhno vid sistemi vidliku Otzhe takim chinom predstavlennyam grupi Lorenca ne mozhna opisuvati chastinki voni mayut dodatno viznachenu lorenc invariantnu normu pov yazanu iz masoyu a u kvantovij mehanici gustina yih jmovirnosti opisuyetsya dodatno viznachenim kvadratom modulya amplitudi hvilovogo vektora Prote za dopomogoyu neyi mozhna klasifikuvati predstavlennya poliv po vidnoshennyu do peretvoren Lorenca oskilki velichinami sho sposterigayutsya ya bilinijni funkciyi poliv Rozsheplennya algebri grupi Operatori Kazimira grupi RedaguvatiMozhna rozshepiti algebru komutatoriv vvivshi ermitovi v silu ermitovosti generatora obertan ta antiermitovosti generatora bustiv operatori J k 1 2 R k i L k K k 1 2 R k i L k displaystyle hat mathbf J k frac 1 2 hat mathbf R k i hat mathbf L k quad hat mathbf K k frac 1 2 hat mathbf R k i hat mathbf L k nbsp Yih algebra zadovilnyaye nastupnim spivvidnoshennyam pereviryayetsya u matrichnomu viglyadi J i J j i e i j k J k K i K j i e i j k K k J i K j 0 5 displaystyle hat mathbf J i hat mathbf J j i varepsilon ijk hat mathbf J k hat mathbf K i hat mathbf K j i varepsilon ijk hat mathbf K k quad hat mathbf J i hat mathbf K j 0 qquad 5 nbsp Teper mozhna otrimati spektr vlasnih znachen dlya operatoriv J 3 K 3 displaystyle hat mathbf J 3 hat mathbf K 3 nbsp vin skladaye poslidovnist z 2 j 1 displaystyle 2j 1 nbsp chisel j j 1 j displaystyle j j 1 j nbsp de j displaystyle j nbsp dodatne cile chi napivcile chislo Analogichnij spektr mayut i operatori grup S O 3 displaystyle SO 3 nbsp ta S U 2 displaystyle SU 2 nbsp Otzhe za dopomogoyu vvedennya operatoriv J 3 K k displaystyle hat mathbf J 3 hat mathbf K k nbsp algebra grupi Lorenca rozshepilasya na dvi algebri vidu algebri grup S U 2 displaystyle SU 2 nbsp chi S O 3 displaystyle SO 3 nbsp Kozhna z algebr povnistyu harakterizuyetsya maksimalnim vlasnim chislom j i displaystyle j i nbsp Mozhna takozh vvesti matrici operatori yaki komutuyut iz usima operatorami vidpovidnoyi grupi Taki operatori nazivayutsya operatorami Kazimira Takimi matricyami u danomu vipadku ye kvadrati operatoriv vidpovidnoyi grupi J 2 J i J i J J J J J 3 2 K 2 K i K i K K K K K 3 2 displaystyle hat mathbf J 2 hat mathbf J i hat mathbf J i hat mathbf J hat mathbf J hat mathbf J hat mathbf J hat mathbf J 3 2 quad hat mathbf K 2 hat mathbf K i hat mathbf K i hat mathbf K hat mathbf K hat mathbf K hat mathbf K hat mathbf K 3 2 nbsp Neskladno pereviriti sho K i K 2 0 J i J 2 0 displaystyle hat mathbf K i hat mathbf K 2 0 quad hat mathbf J i hat mathbf J 2 0 nbsp tomu vektori ps k displaystyle psi k nbsp operatoriv vidpovidnoyi grupi ye vlasnimi vektorami operatoriv Kazimira i napriklad dlya vlasnogo vektora ps j displaystyle psi j nbsp J 2 ps j J J J J J 3 2 ps j J J 2 J J J 3 2 ps j J ps j 0 j 2 j ps j j j 1 ps j displaystyle hat mathbf J 2 psi j left hat mathbf J hat mathbf J hat mathbf J hat mathbf J hat mathbf J 3 2 right psi j left hat mathbf J hat mathbf J 2 hat mathbf J hat mathbf J hat mathbf J 3 2 right psi j hat mathbf J psi j 0 j 2 j psi j j j 1 psi j nbsp Klasifikaciya poliv po vidnoshennyu do peretvoren za zagalnim nezvidnim predstavlennyam grupi Lorenca RedaguvatiSimetriya u fiziciPeretvorennya Vidpovidnainvariantnist Vidpovidnijzakonzberezhennya Translyaciyi chasu Odnoridnistchasu energiyi C P CP i T simetriyi Izotropnistchasu parnosti Translyaciyi prostoru Odnoridnistprostoru impulsu Obertannya prostoru Izotropnistprostoru momentuimpulsu Grupa Lorenca busti VidnosnistLorenc kovariantnist ruhucentra mas Kalibruvalneperetvorennya Kalibruvalnainvariantnist zaryaduYak bulo napisano vishe predstavlennyami grupi Lorenca ne mozhna harakterizuvati chastinki Prote nimi mozhna a bilshe togo i treba opisuvati polya oskilki voni ne zobov yazani mati lorenc invariantnu normu Dijsno dovilne nezvidne predstavlennya grupi Lorenca mozhna podati cherez pryamij dobutok nezvidnih predstavlen grup S U 2 displaystyle SU 2 nbsp chi S O 3 displaystyle SO 3 nbsp Ce predstavlennya harakterizuyetsya naborom maksimalnih vlasnih znachen j 1 j 2 displaystyle j 1 j 2 nbsp vidpovidnih operatoriv cih grup S j 1 j 2 S j 1 S j 2 displaystyle hat mathbf S j 1 j 2 hat mathbf S j 1 times hat mathbf S j 2 nbsp de S j i displaystyle hat mathbf S j i nbsp matrici nezvidnih predstavlen grupi S U 2 displaystyle SU 2 nbsp chi S O 3 displaystyle SO 3 nbsp yaki mayut rozmirnist 2 j i 1 displaystyle 2j i 1 nbsp Vidpovidno do cogo yaksho ye deyaka bagatokomponentna velichina to vona peretvoryuyetsya po takomu zagalnomu predstavlennyu pri peretvorennyah Lorenca ta 3 povorotah yak ps a b S a m j 1 S b n j 2 ps m n displaystyle psi alpha beta S alpha mu j 1 S beta nu j 2 psi mu nu nbsp Rozmirnist bagatokomponentnoyi velichini skladaye 2 j 1 1 2 j 2 1 displaystyle 2j 1 1 times 2j 2 1 nbsp Teper mozhna klasifikuvati ci velichini u zalezhnosti vid znachen j 1 j 2 displaystyle j 1 j 2 nbsp Yaksho predstavlennya maye viglyad 0 0 displaystyle 0 0 nbsp to ce vidpovidaye skalyaru Dijsno nulovi znachennya maksimalnih vlasnih chisel dayut takozh vidsutnist matric nezvidnih predstavlen dlya peretvorennya danoyi velichini formalno zalishayutsya lishe odinichni matrici rozmirnosti 0 1 0 1 displaystyle 0 1 times 0 1 nbsp i sama velichina maye taku zh samu rozmirnist a otzhe velichina niyak ne zminyuyetsya pri 3 povorotah chi lorencivskih bustah Ce vidpovidaye skalyaru Vidpovidne pole sho pov yazane z nim mozhe buti suto dijsnim abo kompleksnim Yaksho predstavlennya maye viglyad 1 2 0 displaystyle left frac 1 2 0 right nbsp abo 0 1 2 displaystyle left 0 frac 1 2 right nbsp to rozmirnist bagatokomponentnoyi velichini skladaye 2 1 displaystyle 2 times 1 nbsp i vona nazivayetsya spinorom vidpovidno livim abo pravim Dlya predstavlennya 1 0 displaystyle 1 0 nbsp abo 0 1 displaystyle 0 1 nbsp peretvoryuvanoyu velichinoyu ye 3 vektor a i b displaystyle mathbf a i mathbf b nbsp okremi komponenti yakogo peretvoryuyutsya yak antisimetrichnij tenzor Nareshti predstavlennya 1 2 1 2 displaystyle left frac 1 2 frac 1 2 right nbsp harakterizuye 4 vektor Vin predstavlyayetsya matriceyu iz rozmirnistyu 2 2 displaystyle 2 times 2 nbsp za dopomogoyu spinornogo formalizmu Velichina j 1 j 2 displaystyle j 1 j 2 nbsp harakterizuye predstavlennya nastupnim chinom yaksho suma ye cilim chislom to predstavlennya ye odnoznachnim vektornim yaksho napivcilim to dvoznachnim spinornim Najprostishe ce prodemonstruvati na prikladi peretvorennya spinora pri povoroti na 2 p displaystyle 2 pi nbsp navkolo pevnoyi osi Jogo komponenti zminyuyut znak na protilezhnij A otzhe dlya danoyi oriyentaciyi komponenti spinora mozhut prijmati dva rizni znachennya Okrim togo velichina j 1 j 2 displaystyle j 1 j 2 nbsp ye maksimalnim vlasnim chislom nezvidnogo predstavlennya operatora J 3 K 3 R 3 displaystyle hat mathbf J 3 hat mathbf K 3 hat mathbf R 3 nbsp yakij ye generatorom obertan navkolo osi O z displaystyle Oz nbsp Take predstavlennya ye unitarnim tomu velichina j 1 j 2 displaystyle j 1 j 2 nbsp vidpovidaye sposterezhuvanij velichini Cya suma z tochnistyu do mnozhnika ℏ displaystyle hbar nbsp ye spinom Div takozh RedaguvatiGrupa Puankare Spin Kvantova teoriya polyaCya stattya ne mistit posilan na dzherela Vi mozhete dopomogti polipshiti cyu stattyu dodavshi posilannya na nadijni avtoritetni dzherela Material bez dzherel mozhe buti piddano sumnivu ta vilucheno serpen 2013 Otrimano z https uk wikipedia org w index php title Grupa Lorenca amp oldid 38626625
