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Representation of permutation operators in quantum mechanics Seagraves, Paul Henry 1964

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REPRESENTATION OF PERMUTATION OPERATORS IN QUANTUM MECHANICS by PAUL HENRY SEAGRAVES B . S c . , New Mexico Ins t i tu te of Mining and Technology, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1964 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of • B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of / * y ? / <r ? The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date $*/>ite~l+y ? / ? / f Abstract A simple method i s presented for w r i t i n g the matrix elements of t ranspos i t i on operators for d i scre te sets of quantum numbers. A proper product of these leads to easy computation of general permutation operators . It i s shown how these operators may be constructed with operators defined i n angular momentum space. Results agree with Dirac for t ranspos i t ion of two p a r t i c l e s of spin and with Kaempffer for sp in 1. The c a l c u l a t i o n s are performed to extend the r e s u l t s to sp in 3/2 and 2 along with a l ternate representat ions . Spec ia l considerat ions are required for fermion crea t ion and a n n i h i l a t i o n operators . - i v -Acknowledgements I should l i k e to express my s incere thanks to Professor F . A . Kaempffer, for o r i g i n a l l y suggesting t h i s top ic and for providing valuable guidance at every stage of the work. I should a lso l i k e to thank the Un ivers i ty of B r i t i s h Columbia Physics Department for f i n a n c i a l a i d i n the form of a summer s t ipend . - i i i -Table of Contests Page Abstract > i i Acknowledgements iv Introduct ion . 1 I . General Permutations Operators for Discre te Sets of Quantum Numbers 5 I I . Transpos i t ion of Angular Momentum 11 I I I . Fermion and Boson Operators 18 IV. Determinants of Operators 23 Appendix A 1: Proof that (1.6) S a t i s f i e s (1.1) and (1.2) 28 Appendix A 2: Inversion of Vandermonde*s Matrix 31 Table I: Transpos i t ion Operators i n Angular Momentum Space 34 Table I I : Determinants of Operators 35 Bibl iography 36 Introduction Although permutation symmetry i s always invoked i n the quantum-mechanical d e s c r i p t i o n of systems of i d e n t i c a l p a r t i c l e s , l i t t l e i s found i n the l i t e r a t u r e on the ac tua l construct ion of permutation operators . A notable exception i s an a r t i c l e by Serber (1934), where permutation operators are constructed out of t ranspos i t ion operators for the s p e c i a l case of four objec t s . Two other types of symmetries are known. One type, the space time symmetries, can be grasped by in troduct ion of operators whose generators represent constants of the motion. For example, the generator of displacement operators i s the momentum of an objec t , the generator of r o t a t i o n i s the angular momentum and the generator of displacements i n time i s the energy (see, for example, Merzbacher 1961). The second type of symmetries deals with i n t e r n a l a t t r i b u t e s . The phys i ca l s i gn i f i cance of t h e i r generators i s not understood at a l l at present. Most attempts to understand these better use the existence of c e r t a i n gauge imparlances as a s t a r t i n g point (Yang and M i l l s 1954). Permutation symmetry i s unique i n that i t i s defined only for more than one objec t . However because of a r e l a t i o n between sp in and s t a t i s t i c s (Paul i 1940, Luders and Zumino 1958), even s ing le objects revea l through i n t e r n a l a t t r i b u t e s - 2 -faow c o l l e c t i o n s of them w i l l behave under a permutation operator. Th i s i s not c l e a r l y understood, A system ^ of n ind i s t ingu i shab le p a r t i c l e s l a b e l l e d 1, 2, 3 , . n with complete sets of quantum numbers t\$ T2» • • • » T n i s wr i t ten ( o . i ) of = i t , . r ^ r , , : - . . J r „ > . "Indist ingulshabi l i ty** of p a r t i c l e s means that i t i s not poss ib le to a f f i x any other l a b e l to a p a r t i c l e beyond a complete set r , Expression (0.1) does not take care of t h i s requirement because there are a d d i t i o n a l p a r t i c l e l a b e l s l r 2, 3 , . . . , n beyond the sets r . Since the s tate (0.1) i s . i n d i s t i n g u i s h a b l e beyond the sets r , any reorder ing or "permutation** of the p a r t i c l e l a b e l s 1, 2, 3 , . . . , n would y i e l d a p h y s i c a l l y ind i s t ingu i shab le s t a t e . The operators that perform t h i s symmetry property are c a l l e d permutation operators . Since d i s s e c t i o n of any permutation i n t o a product of t ranspos i t ions i s always pos s ib l e , the primary task i s indeed the construct ion of t ranspos i t i on operators . One aims p a r t i c u l a r l y at a construct ion employing operators defined i n the space spanned by the a t t r i b u t e s whose p a r t i c l e l a b e l s are to be transposed. Dirac (1958) has done t h i s for t ranspos i t i on of p a r t i c l e l a b e l s attached - 3 -to two angular momenta of amount j = ^ 2 . Kaempffer (1964) has extended D i r a c ' s representat ion to the case of angular momenta of amount j » l , and has given representat ions for t ranspos i t i on of complete sets of a t t r i b u t e s i n terms of crea t ion and a n n i h i l a t i o n operators for both fermions and bosons. In the present work a simple method i s presented for w r i t i n g the matrix elements of t ranspos i t ion operators for a d i s cre t e set of quantum numbers. From t h i s representat ion , general permutation operators are e a s i l y computed. It i s then shown from another l i n e of reasoning how i n angular momentum space these operators may be constructed using operators defined i n angular momentum space. C a l c u l a t i o n s have been performed to extend work by Dlrac and Kaempffer mentioned above to include p a r t i c l e s of angular momenta 3/2 and 2 along with a l ternate representat ions i n each case. Spec ia l cons iderat ions are required for fermion creat ion and a n n i h i l a t i o n operators . Before one can discuss any poss ib le dynamical s i g n i f i c a n c e of permutation operators , one must possess the generators of the operators . Th i s remains an as yet un-solved problem. As a f i r s t attempt i n t h i s d i r e c t i o n , computation of the determinants of the operators was c a r r i e d out and shown to y i e l d +1 or - 1 . I f the determinant of an operator i s +1, i t should possess a connection with the - 4 -i d e n t i t y as a product of i n f i n i t e s i m a l operators . The group generated by these i n f i n i t e s i m a l operators should then contain the permutations with determinant +1 as a s p e c i a l subgroup. But t h i s group i s not known or not recognized. The fo l lowing convention about l a b e l l i n g matrix elements w i l l be used. Let p.^, ( i k , . . . be the number of d i scre te values the quantum numbers 1, k , . . . may take. For example, the eigenvalues of for given eigenvalue 2 j ( j + l ) of J has = 2j+l values . Now introduce running integers a±t «k» •••>. * ° order the poss ib le values of i , k , . . . i n a c e r t a i n sequence. The f i r s t value of i corresponds to a^l and so on by integer values to . A l l matrices w i l l be l a b e l l e d M e t c . In p a r t i c u l a r . aiak a matrix whose rows and columns are character ized by further sets of quantum numbers w i l l be wr i t ten ^axa^t^man ' With t h i s convention the matrix elements of the d i r e c t product G=A®B can be wri t ten as usual C * i « k , « m « n " Aa±am B f f l k « a < W i * n e r 1 9 5 9 » P- 1 7 > -I , General Permutation Operators for Discre te Sets of Quantum Numbers Needed i s an operator T which has the fo l lowing propert i e s : (a) T i s s e l f adjo int and un i tary (1.1) x = T t = T"1 so -tUt T1'-!. (b) T i s defined only i n a product space spanned by a d i r e c t product of two spaces of equal dimension. It should have the property of interchanging the order of the d i r e c t product of any two matrices i n the space. (1.2) T A ® 8 T" 1 - B ® A . Thus, i f the " f i r s t " subspace i s associated with the quantum numbers belonging to the " f i r s t " p a r t i c l e and the "second" subspace belongs to the "second" p a r t i c l e , the operator T has the e f fect of t ranspos i t ion of p a r t i c l e l a b e l s . Th i s may be expressed formal ly by denoting any operator belonging to the " f i r s t p a r t i c l e " (1 . 3 ) F(l] - F ®I and any operator belonging to the l*second" p a r t i c l e by - 6 -(1.4) F(2)--r®F remembering the a p p l i c a t i o n of (1.2) i s (1.5) T E f i ] F ( 2 ) T " ' - F ( D E ( 2 - J . These requirements are possessed by the fo l lowing representat ion (1.6) T<lXk)0Lm0L^ = (±) . The proof that (1.6) s a t i s f i e s (1.1) and (1.2), and unique i s s traightforward and i s relegated to Appendix A 1, It i s i n s t r u c t i v e to write T out e x p l i c i t l y for a given case. Take the product of two three-dimensional space, which may be i d e n t i f i e d with the product space spanned by two subspaces of J j each belong to angular momentum j = l . Then T has, with the l a b e l l i n g convention stated i n the i n t r o d u c t i o n , the foraj -7-(1.7) T - '(•+) 1 o o o o o o o o o o o 1 o o o o o 0 o o 0 o o 1 o o o 1 o o o o o o o o o o o 1 o o o o o o o o o o o 1 o o o 1 o o o o o o o o o o o 1 o o o o o o o o o o o 1 From the propert ies (1.1) i t fol lows that the eigenvalues of T are +,'1. Let an eigenstate of T be denoted by (1.8) \yy r | (p> ® cx> then (1.9) T l ^ > = f ' ^ > retires T2\V>-- 12\%>*\V> so that (1.10) f 2 = l f = * l . The proof leading to (1.6) for t ranspos i t ion operators allows an as yet a r b i t r a r y s ignj t h i s i s not a matter of convention however. Consider the eigenstate -8-|j,m)> belonging to J 2 = j ( j+1) and J3=m obtained by add i t ion of two angular momenta of amount j (say) . In p a r t i c u l a r P look at the state | 2 J p , 2 j p ^ . It i s wel l known that t h i s s tate i s symmetric and i n the usual column-vector representat ion has element 6 (with a r b i t r a r y a usua l ly chosen as a=0) i n the f i r s t p o s i t i o n , with a l l others 0 . That i s l2jpJ2jp> = lj>j'p>|®ljp,jp>2--( l . l l ) where 1 and 2 are the p a r t i c l e l a b e l s . The element T ^ ^ of T i s the only non-zero element i n the f i r s t column of T and hence i s the only element to a f fec t i 2 J 0 » 2 J « ^ and must be chosen as +1 to leave the s tate unchanged by e r o o 0 * — o * 0 e o o * • O transpos i t ion of p a r t i c l e l a b e l s . That i s — Tl2j P j2j p>=f+) | . . . o • * • e 0 e 0 o • • • o o (1.12) • • " « * * • • • • * o • . • • o O - — Hence the s ign i s p o s i t i v e throughout for angular momenta. The representat ion (1.6) can be extended to more than two p a r t i c l e s by in troduct ion of the i d e n t i t y for unaffected p a r t i c l e s . Thus the three poss ible t ranspos i t ion operators for three p a r t i c l e s are (1.13) . {Tl%)(j.ldi^c,^c^dp a ^ . ^ J ^ ^ ' J It i s important to keep track of the subscr ipts and the spaces to which they belong. Here " i ,m" belong to sub-space " i , a " , i . e . to the f i r s t p a r t i c l e and so on. By su i tab le products of t ranspos i t i ons , general per -mutations are found. For example, f or three objects the permutation T^23 a n d T132 a r e f o u n d a s fol lows (note that the operators are ef fected from the r i g h t ) (1.14) < T . Z 3 > W , , c ^ < , ^ ** s t (1.15) (T ?) - T " ( T 2 3 ) / T J ~ 2 — ^d./cf r » d > c f , 4/ ^ oL oL , S t -10-The remaining permutation of t h r e e - p a r t i c l e labe l s i s the i d e n t i t y , g iv ing s i x i n a l l , I , T 1 2 , T 1 3 » T 2 3 » T 1 2 3 » a n d T132< This process may proceed to any number of p a r t i c l e s without d i f f i c u l t y . For example the permutation T 1 4 g 3 2 can be wri t ten as T 1 3 2 T 1 4 5 » From (1.14) and (1.15) i t fol lows that (1.16) ( T j j g l ^ ^ ^ ^ ^ ^ a ' ^ j r f j = ^^^^JcC.df^^d^J (1.17) ( ^ 4 ^ ^ ^oT.dj^K^fiCp.oC^p z Sd.i*pJ«3*/*h*»(J4;*/*J'i:» so that r I J (1.18) ( T , 4 5 3 2 J ^ d i d ^ c ^ t f ^ ^ As a mnemonic a i d , i t may be noted the column l a b e l , aj for example, of any given symbol $ccady » appears always i n conjunction with a row l a b e l , a a i n t h i s example, which i n the l a b e l l i n g of the symbol T have the respect ive p o s i -t ions required by the permutation of p a r t i c l e s with quantum numbers P and a i n t h i s case. Thus i f the permutation i s (21) as i n (1.18) , i s found with a& among the subscr ipts of the corresponding J-symbol . -11-I I . Transpos i t ion of Angular Momentum Necessary propert ies for t ranspos i t ion operators i n angular-momentum space are found by examining the e igen-states for two p a r t i c l e s of given angular momentum. P a r t i -c u l a r l y needed are symmetry propert ies under exchange of p a r t i c l e l abe l s for i d e n t i c a l p a r t i c l e s . The elgenstate | J a > ^b» ™y 1 D e l o n g i n S t o J 2 = j ( j + D and J 3 - m obtained by add i t i on of two angular momenta j a and j b , have the we l l known symmetry property (Condon and Short ley , 1963) If the p a r t i c l e s have the same angular momentum j p the mul t ip l e t j = 2 j p i s symmetric, the mul t ip l e t j = 2 j p - l i s antisymmetric, and so on a l t e r n a t i n g l y to j=0. Thus the needed property of the t ranspos i t ion operator T i s (2.2) t / ; , » > > ( - i ; 2 j p " J l j > > . Now suppose there i s an operator F which i n the state j j , m ) has the eigenvalue f . . Further assume the f . are a l l d i f f e r e n t so that there are 2 j p + l values of f^. Then T can be wri t ten as a power ser i e s i n F , -12-(2.3) T = X0 + YIF +X2 FZ + X3F3 + - " + X2jpF with unknown c o e f f i c i e n t s X Q , X^, X 2 , . . . to be defined so that the eigenvalues t . of T must s a t i s f y (2.4) t j ^ x 0 + f ; + y t + m / • 2 i p i n accordance with (2 .2) . This system of 2j +1 equations form, (2.5) 2Ja ]2 *2» X 3' ... can be put ' o 4*2J> 11 x , • • • • * # 1 •C2if> \ -The c o e f f i c i e n t matrix i s a Vandermonde's matrix (see, for example, Browne, 1958)\ the inverse of which can be wri t ten as (see Appendix A 2) -13-where the fo l lowing notat ion has been used: t p _ sum of a l l d i f f e r e n t products formed with s values f^ (2.7) I {U-f.)(iri,)(ft-f2). • - Cf,-i2jp) and where the eigenvalue f t Is excluded i n the formation of products i n the numerator and £^.-£^ i s excluded from the denominator. The s o l u t i o n can now be wri t ten (2.8) X k H - » l j r K T ( " f i r i J j • W* Z ' - « , V . * f ? . L = 0 -14-Suitable operators F remain to be found. C l e a r l y J with eigenvalues j ( j + l ) w i l l work. If J i s wri t ten as the sum of the sp in vectors J ( l ) and j (2 ) for the f i r s t and second p a r t i c l e s re spec t ive ly (2.9) J r J ( | ) 4 J (2 ) i t fol lows that (2.10) J 2 = j ' l D ' f W * •!<».• The eigenvalues gj of J ( 1 ) . J ( 2 ) s a t i s f y then the equation (2.11) j(j>;) = jP(jP^) i-Jp(jp-hl) + 2 jj^ A l t e r n a t i v e l y then, one may use a representat ion with J (1 ) . J (2 ) for F with the property (2.12) J(l)-I(?) ft/j,") w h e r e 'lj=-j,1fr0 + ij(j+l). 2 2 - 1 9. If J i s d iagonal ized such that VJ V A » J ^ f the eigenvalues 2 j ( j + l ) the appear along the diagonal and J~j can be factored as (2.13) j / = ^(Kj+I) 15 where K<j i s a diagonal matrix with eigenvalues j . Then by the operation V ~ 1 J 2 l V one has Thus i t i s always poss ib le to f i n d a matrix K such that (2.15) J< |j-,m> = j lj>> which i s another a l t e r n a t i v e for F . Results with J(1) .J(2) agree with Dirac (1958) for sp in V2 and Kaempffer (1964) for sp in 1. These and further r e s u l t s f or T are c o l l e c t e d i n Table I . An exponential representat ion of these operators should e x i s t . One would l i k e a matrix A such that T may be wri t ten (2,16) where the fac tor i T T i s introduced for convenience. The eigenvalues a . of A s a t i s f y (2.14) J2- K where itrA 1 - \ itr« i- \ (2.17) -16 In accordance with (2.2) one has (2.18) ( - / ) 2 J p " J = c ^ « j * * « ^ " « j * » ( - l ) a J because the eigenvalues of T are r e a l , r e q u i r i n g a to be in tegers . Indeed A can be found by much the same procedure jus t followed to f i n d T . That i s , A may be wri t ten as a power s er i e s (2.X9) A - Y0 + Y, F + Y, F 2 + Y3 F3+ • • • + Yl}f F2'f with the unknown c o e f f i c i e n t s Y Q , Y^, Y 2 , . . . d e f i n e d so that the eigenvalues a s a t i s f y (2.20) = y o + Y, fj f Y2 *J + Y3 *j + • • • * V2j.//J> where a^ i s a su i tab le even or odd integer to agree with (2.18) . This system of 2j^+l equations with 2 j p + l unknowns Yq» Y^, Y 2 , . . . can be solved i n the same manner as the system of equation (2 .4) . It i s seen that A i s not unique; the general form would be ;ir(A + B) lit-A (2.21) T = e - e where B i s any matrix with even integer eigenvalues, but -17-a l l the representat ions are equivalent and t h i s i s wri t ten i n (2.21). Rather than work out these power s e r i e s , which are not unique for a given s p i n , one would l i k e a c losed form which would always work. Using the operator K of (2.15), T may be wri t ten inr ( H(0+ he(2) - K) . p ' > ^ 2 J P " ^ ) (2.22) T = e - c for one has (2.23) i n accordance with (2 .2) . -18 -I I I . Fermion and Boson Operators The representat ion T proposed by (1,6) cannot be appl ied as i t stands to spaces spanned by fermion creat ion and a n n i h i l a t i o n operators , because fermion states require the s ign convention of Jordan and Wigner (1928) which has not been taken care of by (1 .6) . This s ign convention i s i n t r o -duced such that a + CfJ I . . . j . . . > = / . . . ; 1^} .. .> (3.1) where m-t (-*-/ <f Me No. .of occa^eeJ st«tc<, (3.2) Ct = T T C-2NtM\ it<**isev*« W < W L $ O J J and N_ stands for the occupation number of the quantum • i n -s tate Tm, and a + ( T m ) and a (T m ) are the we l l known creat ion and a n n i h i l a t i o n operators for these quantum s ta tes . The operators then s a t i s f y the anti-commutation r e l a t i o n s (3.3) -19-From i t fol lows that (3.5) Thus the t ranspos i t ion of p a r t i c l e l abe l s for fermions should have the propert ies (3.6) Tl2 a+tol 7,'1 = I3 7I2I2®*T,^ ae>T j etc. which i s not the property assumed, in. (1 .2) . The construct ion of the operator T 1 2 i n terms of crea t ion and a n n i h i l a t i o n operators for fermions i s r e l a t i v e l y easy, because there i s only a f i n i t e number of b i l i n e a r combinations containing the operators a ( l ) , a + ( l ) , a (2) , and a + (2 ) from which one can construct the most general un i tary operator invo lv ing a ( l ) , a + ( l ) , a (2) , and a + (2 ) of which T 1 2 , i f i t ex i s t s at a l l , must be a s p e c i a l case. One f inds (see Kaempffer, 1964) 0 . 7 ) T,t.-ie'<*>«"el<vs«. T l ; ' ^ ^ ' ( ? ) s ' 2 i ; ( ¥ ) ^ with -20-(3, 8) R , 2 - [ * + 0 ) « ( Z ) + At(2)A(0] (3 w i l l do the job required by (3 .6) , Indeed, s ince (3.10) [ S,Zf*(0] = "(I) ] ,*(2)]=A(2) one has *(» / 2 '* s a(i) + i * [Stt (iff ± [ ? l 2 p , 2 * 0 ) ] ] i (3.11) = = i*0) and s i m i l a r l y (3,12) c(2)e * 5 , 2 = d so that , because (3.13) [ R t l }A(I)] - -a(2) . L R n j A a ) J * - « a ) one has f i n a l l y J i t a ( 0 V = a ( i ) e " ' * R ' t = i { a ( 0 + (i$)[Rl2*aj]<»-] (3.14) / -21-and s i m i l a r l y (3.15) Ttl*<V l i = 1 { « (Z) ^ Z - *«(0^ T } a 0 ) -The representat ion (3.7) for the t ranspos i t ion operator turns out to be equal ly v a l i d for bosons, because the b i l i n e a r nature of the operators R^g a n d s^2 l e a d t o t h e commutation r e l a t i o n s (3.10) and (3.13) i f the fermion operators are replaced everywhere by the corresponding boson operators r^2 a n d S 12 ( s a v ) « The dec i s ive d i f ference between the two cases i s that n a s with fermion operators only the eigenvalue -1 and with boson operators i t has only the eigenvalue +1. For fermion operators S^ 2 a n d ?*i2 B&^s^7 the r e l a t i o n s (3.16) S,l - R , | - I • R „ Slt = O . = Rn . S,[ - Slt so that one can write s ^ s ^ = i + < sn + s,l £ < ~ f - ) ] = i+rsn - s* ( 3 - 1 7 ' i * R and thus, for fermions, the operator of t ranspos i t ion can be put i n the somewhat more transparent form -22-The deviat ion from (1.6) i s made more obvious by wr i t ing T 1 9 e x p l i c i t l y ; 1 o o o o o 1 o o 1 o o o o o -1 which d i f f e r s from (1.6) by the -1 instead of 1 i n the fourth row and column. Further one f inds (3.20) T , \ - S* + = I-(3.19) T 1 9 = -23-IV. Determinants of Operators The determinant of a matrix i s the product of i t s e igenvalues. The eigenvalues of a permutation P are +1; I f the determinant of an Operator i s +1 i t enjoys a connection with the i d e n t i t y . That i s , the e f fec t of the operator may be reached by successive a p p l i c a t i o n of i n f i n i t e -s imal operat ions . I f the determinant i s -1 such connection i s never pos s ib l e . One would l i k e to f i n d these i n f i n i t e -s imal operators , which generate some group that possesses the permutations of determinant +1 as a s p e c i a l subgroup. The present work points out , by computation of determinants, the operators where such a study might be f e a s i b l e . Some propert ies of determinants w i l l be needed. F i r s t , the determinant of a product of two matrices A and B i s the product of the determinants of A and B. therefore , the determinant P of P i s +1 or - 1 . (4.1) \ABh I / W I B I . Further the determinant of a d i r e c t product has values' (4,2) | C ® E l r \ C & \E\ -24-where d and d E are the dimensions of matrices C and E r e s p e c t i v e l y . This i s shown by d iagona l i z ing C#Dj so that the eigenvalues of C appear d times and those of E appear d^ times along the d iagonal . The determinants of t ranspos i t ion operators for a given set of p a r t i c l e l abe l s are e i ther a l l +1 or a l l - 1 . I t fol lows from property (4.1) that the determinant of an even permutation i s always +1, and that the determinant of an odd permutation i s the same as that of the t r a n s p o s i -t i o n operators . (An even permutation i s one formed by an even number of t ranspos i t i ons , and an odd permutation i s one formed by an odd number of t r a n s p o s i t i o n s ) . Angular-momentum operators w i l l now be used as an i l l u s t r a t i o n . The number of antisymmetric s tates determines the number of eigenvalues -1 of T . The product space of two p a r t i c l e s of angular momenta of amount j p , with 2 dimension d=2j p +l each, has dimension d . If j i s / h a l f - o d d integer \ , i t fol lows from (2.1) that the / s i n g l e t [ t r i p l e t i s antisymmetric, and each consecutive a n t i -has four more values of m. There are i n a l l / ~ 2 d-1 2 -2 5-m u l t i p l e t s . It fol lows from the formula for an ar i thmet ic sum that i n e i ther case ther are £{AT,?J. antisymmetric e igenstates . Thus the determinant of T may be wri t ten (4.3) \T\- tD (-ijf for even d , and hal f -odd <$(J-\) 1 integer sp in H) T(-i)'<lff or odd d , and integer spin . As an i l l u s t r a t i o n of these r e s u l t s the determinants of the s ix poss ible permutations of three objects w i l l be worked out . From (1.13) , (4.2) and (4.3) i t fol lows that 12 (4.4) j - y | i 0 T h i r / ' V / ' 7 = - < IT,,/-- Ir^J^l^l7,JIT„\ITnl J-/ I hie and thus (4.5) \l<u\-\T1%Tn\ 26-From (4.4) i t fol lows that the determinant of t ranspos i t ion operators for more than two p a r t i c l e s i s always +1 i n case of ha l f -odd integer s p i n , but formula (4.3) remains v a l i d i n case of integer s p i n . The representat ion (1.6) i s quite general except for s i g n . The d i scuss ion thus far has been with the plus s ign as demanded for angular momentum. Relations corresponding to (4 .3 ) , (4.4) and (4,5) for the minus s ign i n (1.6) can be found by rep lac ing T by - T i n the above d e r i v a t i o n . A f f i x i n g a (-) to T i s i n e f fec t m u l t i p l i c a t i o n by a diagonal matrix with a l l elements equal - 1 ; from (4.1) i t fol lows that | - T | = ( - l ) d 2 | T | = ( - l ) d | T | general r e s u l t s r evert d ( 4 - 6 > | T , t H One f inds the + I -far1 ir) at* t t^a. » (hi) remembering "d" i s the dimension of the subspace of a s ing le p a r t i c l e (d=2j p+l for s p i n ) , and the plus s ign must be chosen i n the representat ion (1.6) for sp in space. Results are c o l l e c t e d i n Table I I . -27-From (3.9) i t fol lows that the determinant of fermion t ranspos i t ion operators are +1. From (4.3) i t fol lows that for t w o » p a r t i c l e s ta tes , the t ranspos i t i on operators for p a r t i c l e s of sp in 1/2, 5/2, 9/2, . . . have | T | =» - 1 . Such p a r t i c l e s cannot be i d e n t i f i e d as fermions unless they have further a t t r i b u t e s so that the t o t a l t ranspos i t ion operator has determinant +1. 2 8-Appendix A 1: Proof that (1.6) s a t i s f i e s (1.1) and (1.2) By interchanging i n the row and column labe l s one has and a lso (A 1.2) ( T l , ^ ^ ^ - ! ! < k . « A « f V A « » : t.e., T=T-«. Thus r e l a t i o n s (1.1) are v e r i f i e d . Furhter i t fol lows that (A 1.3) tI*F)t{t ^ ' ^ F ^ so that (A 1.4) \Til*F%t.t J - ^ A * P ^ < - . ^ n = . ^ K « C m ^ < C ' » ocpocr and a l so o r . (A 1.6) T F ( 2 ) T " 1 n F ( l ) . -29-Then (A 1.7) TTF(2)TT * TF(1)T so that (A 1.8) T F d J T " 1 » F(2) For another operator E i t fol lows that (A 1.9) TECIJT"" 1, = E(2) so from (A 1.6) (A 1.10) TE(1)T" 1 TF(2)T"" 1 E(2)F(1) which gives (1.2) or (1 .5) , (A 1.11) TE(1)F(2)T~ 1 = F(1)E(2) as r e q u i r e d . Assume there i s another operator S with the propert ies (1.1) and (1 .2)j that i s , (A 1.12) S = S^-: S" 1 , SE(1)F(2)S" 1 F (1 )E(2) . It- .follows that (A 1.13) TSE(1)F(2)ST > TF(1)E(2)T - E(1)F(2) or (A 1.14) TSE(1)F(2) » E(1)F(2)TS. -30-That i s TS commutes with a l l E(1)F(2) and hence must be a mult ip le of the i d e n t i t y (Wigner 1959, p. 75). Hence (A 1,15) TS «* n l , n a constant and thus (A 1.16) T = nS, T 2 =n 2 S 2 =I=n 2 I , and n=+l. The uniqueness except for s ign i s thus proved. -31-Appendix A 2: Inversion of Vandermonde's Matrix Vandermonde's matrix V has the form (A 2.1) i o k 2 a, where a £ i s the number a^ to the power h . The notat ion *R i s given by (A 2 2) R = SU** °* a^ ^ l ^ e r e h * products wiH 5 oi tk< yuluta <z; (A 2.3) *R: and where the element a^ i s excluded i n formation of the products i n the numerator and a t ~ a t I s excluded from the denominator. With four numbers & Q , a^, a^, a^ as examples one has -32-(A 2 . 4 ) I R : The inverse V*** has the elements b where (A 2 . 5 ) ha =. J . R J That i s -I (A 2 . 6 ) V = (-1)" i R H)* Sf? • • • B f S*? • • • J ^ . . . , Ft 0 R • 0 * S R This i s r e a l i z e d by computing the product W \ An element c ^ of V V " 1 has the form (A 2 , 7 ) C , j = « * in - j / ? + « ? - 2 J > — J r , The denominator i s common to a l l ^R. I f i £ j then a ± i s used i n formation of the numerators of and the product s a n - s J R c a n b e wr i t ten i n the form i s -33-(A 2.8) • « ^ I R = j , * p f „ «-S j , ^ where ^ f i R means exc lus ion of both i and j information of s the numerator but s t i l l only a j"" a j i s l e f t out of the denominator. Now i t i s seen that C J J : » 0 when i ^ j . I f i = j , one should have c^j » 1. The denominator of C j j has the fac tors a j - a s with a^'-j* a g , so i t i s s u f f i -c i ent to show that the numerator fac tors the same way. Subs t i tu t ion of root a a g / a^ then should y i e l d zero as has already been shown above. Thus c i j = d~ij a n d the proposed invers ion i s v e r i f i e d . Table I . Transpos i t ion Operators i n Angular Momentum Space J p F T z. Sz K O, 2 2K-I 1 1(0-1(2) J 2 K ZK2-4K+I 3 Z J(0'J(2) t . K -li . 3 3 4 , 4 , 4 , 4 O, 2. ,6,12. 0, 1, 2, 3 Z J 2 K 0 , 2 , * , / 2 , 2 0 0, ' , 2 , 3 , 4 -35-Table I I . Determinants of Operators (a) A l l even permutations have determinant +1. (b) Determinant of odd permutations are number of p a r t i c l e s two many s ign of eq. (1.6) (+) (+) (-) dimension d d * 2 a p + l d=2j +1 P 2 -1 -1 +1 +1 3 -1 +1 -1 +1 4 +1 +1 +1 +1 5 +1 -1 +1 -1 6 -1 -1 +1 +1 7 -1 +1 -1 +1 8 +1 +1 +1 +1 9 +1 -1 +1 -1 10 -1 -1 +1 +1 Bibl iography Browne, E . T . Determinants and Matr ices , Univ . of North C a r o l i n a Press (1958), p . 78. Condon, E . U . and G . H . Shortley The Theory of Atomic Spectra, Cambridge Univ . Press , New York and London (1963), Chapter I I I . Dixac , P . A . M . C Quantum Mechanics, 4th e d . , ^58, Oxford Univ . Press , London and New York (1958). Jordan, P. and E . P . Wigner Z . Phys ik , 47, 631 (1928). Kaempffer, F . A . Concepts i n Quantum Mechanics, Section 27, Academic Press , New York (1964). Luders , G. and B . Zumino Phys .Rev. , 110. 1450 (1958). Merzbacher, E . Quantum Mechanics) John Wiley and Sons, Inc. New York and London (1961); pp. 363-365. Serber, R. Phys .Rev. , 45, 461 (1934). Wigner, E . P . Group Theory, Eng l i sh t r a n s . , Academic Press New York and London (1959), Yang, C . N . and R . L . M i l l s Phys .Rev. , 96, 191 (1954)* 

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