The exponential map of GL(N)

Transcription

1 The exponental map of GLN arxv:hep-th/ v 9 Apr 996 Alexander Laufer Department of physcs Unversty of Konstanz P.O M KONSTANZ Aprl 9, 996 Abstract A fnte expanson of the exponental map for a N N matrx s presented. The method uses the Cayley-Hamlton theorem for wrtng the hgher matrx powers n terms of the frst N- ones. The resultng sums over the correspondng coeffcents are ratonal functons of the egenvalues of the matrx. e-mal Alexander.Laufer@un-konstanz.de 0

2 Introducton In the Le theory of groups and ther correspondng algebras the exponental map s a crucal tool because t gves the connecton between a Le algebra element H g and the correspondng Le group element g G g G exp : H g Fordetalssee[5and[,2andref.theren. Insomelowdmensonalcases, lkesu2andso3 the explct expanson of the exponental map s known. Some years ago the exponental map for the Lorentz group was gven by W. Rodrgues and J. Zen [7, 8. For the hgher dmensonal groups SU2,2 and O2,4 a method for the expanson was developed by A.O. Barut, J. Zen, and A.L. [, 2. The subject of the present paper s a generalzaton of the method developed n [, 2 to the general lnear groups GLN. Theresult wll be a method to calculate the exponental of a quadratc matrx H, where only ratonal functons of the egenvalues of H and the frst N- powers of H are nvolved. The key ponts are the Cayley-Hamlton theorem and the ntroducton of a multpler m. Theorganzaton of thspapers asfollows. Frstthemethodsshownnthelow dmensonalcase SU3 whch s a group occurrng qute often n physcs. Then the general case of the expanson of the exponental map for a N N matrx s presented. The method uses egnvalues The only real problem that remans s the determnaton of the egenvalues of the matrx H. Throughout the paper we assume that the groups GLN are represented as N-dmensonal matrces and that the egenvalues of H are all dfferent cf. remark n appendx B.3. Some possble applcatons of these results wll be presented n a future paper. 2 The exponental map for the group SU3 The group SU3 s used n several branches of physcs. The best known applcaton s the model of the strong nteracton see e.g. [3. For ths reason and because ts a good exercse to follow the steps of the general method we wll show the exponental mappng of SU3 n great detal. The calculatons depend n some ponts on the fact that we consder a specal group,.e., the sum over the egenvalues of the generator vanshes. There s no conceptonal problem to extend the method to U3. Lke n the other cases cf. [, 2 a typcal element U SU3 can be wrtten as exponental of the generator U e H H n wth H su3. The Cayley-Hamlton theorem and the terated form n ths case read H 3 b 0 H + c 0 and H 3+ a H 2 + b H + c where the coeffcents b 0 and c 0 are functons of the egenvalues of the egenvalues x,y,z of H. They satsfy the recurrence relatons a + b, b + a b 0 + c, c + a c 0. 2

3 Hence the coeffcents a satsfy a + a b 0 + a 2 c 0 3 wth the frst few values a 0 0, a b 0, a 2 c 0, a 3 b 0 2, a 4 b 0 c 0. The explct form of b 0 and c 0 can easly be derved from the secular equaton 0 λ x λ y λ z λ 3 x + y + zλ 2 + xy + xz + yzλ xyz }{{}}{{}}{{} a 0 0 b 0 The leadng coeffcent a 0 vanshes snce the generator H s traceless,.e., x +y + z 0. The second coeffcent can also be wrtten as b 0 2 x 2 + y 2 + z 2. There are also some nce relatons b 0 x + c 0 x 3, b 0 y + c 0 y 3, b 0 z + c 0 z 3. The dea now s to use the Cayley-Hamlton theorem for wrtng the sum lke U I 4 + H + 2 H2 + n+3! H n+3 I 4 + H + 2 H2 + n+3! an H 2 + b n H + c n Ths form contans only sum over ratonal functons, there are no hgher powers of the generator present anymore. The next step s now to fnd an analytc expresson for the sums over the coeffcents. A convenent form of the functons a n, b n, and c n can be obtaned f we ntroduce the multpler m x y x z y z x 2 y z + y 2 z x + z 2 x y. Then we get for the group element [ mu m I 4 + H + 2 H2 ma n + H 2 + n+3! [ mb n H + n+3! [ c 0. mc n n+3! It can easly be shown that the followng form for the coeffcents satsfy the recurrence relatons 2 and 3 The three sums are now [ ma n n+3! [ mb n n+3! [ mc n n+3! ma n y z x n+3 + z x y n+3 + x y z n+3 mb n y z x n+4 + z x y n+4 + x y z n+4 mc n yzy z x n+3 + xzz x y n+3 + xyx y z n+3 y z e x + z x e y + x y e z 2 m x y z e x + y z x e y + z x y e z m yz y z e x + xz z x e y + xy x y e z 2.

4 Fnally, we get the expanson of a SU3 group element mu [yz y z e x + xz z x e y + xy x y e z I 3 + [x y z e x + y z x e y + z x y e z H + [y z e x + z x e y + x y e z H The exponental map of GLN As we have seen n the cases of the groups SU3 and SU2,2 [2 the exponental map can be wrtten as sum over the frst N- powers of the generator H sun. Where the coeffcents are functons of the egenvalues of H. In ths secton we generalze the results we have gotten on the low dmensonal examples. It seems that there s a relatvely easy concept of generalzaton. The desred result s an expanson of the exponental map of the form g e H H n k0 A k H k 5 where the coeffcents A k are ratonal functons of the egenvalues {λ ;,2,...,N} of the generator H. 3. The secular equaton The frst step wll be to take a look at the egenvalues, some auxlary functons, and ther nterrelatons. Let us consder the secular equaton of the matrx H 0 N N λ λ C k λ N k, 6 k0 where the coeffcents C k are functons of the egenvalues of H. In secton 3.4 some coeffcents are lsted n ther explct form. For later convenence we ntroduce also the truncated verson C k of the coeffcents C k, defned by λ λ j : j k0 C k λ N k 7 Essentally the C k contan all terms of C k wthoutλ. Theconnecton between these coeffcents can be seen easly va 3

5 N N λ λ λ λ λ λ 2 N λ λ λ C k λ N k λ N + λ C N! λ N k0 k N 2 k0 C k+ λ N k. [C k+ λ C k λ N k Snce the calculatons above can be generalzed to all egenvalues we get the relatons C k C k λ C k for k,2,...,n 8 C N λ C N. Let us defne the multpler m;.e., the dscrmnant of the secular equaton m : λ λ j 9 <j and the functons see also 30 m : m λ,..., λ, λ +,..., λ N j<k j,k λ j λ k. 0 In what follows we manly use the followng form of m whch can be obtaned by expandng the Slater determnant see 26 and 29 Ths formula can be generalzed to m + m λ N. m δ kl N l m C k λ For the proof see secton 3.4. for k,l,2,...,n Recurrence relatons The descrbed method reles on the Cayley-Hamlton theorem whch gves us the ablty to wrte all powers H N+n for n N n terms of the frst N- powers of H. The Cayley-Hamlton theorem for H gln reads H N C k H N k 3 k 4

6 The coeffcents C k are the same as those n the secular equaton 6 and satsfy the recurrence relatons 5 derved below. For the specal groups,.e., det g for g SLN the frst coeffcent vanshes snce the sum over the egenvalues s zero. Multplcaton of 3 wth H n and usng 3 agan gves the terated form Multplyng once more wth H gves H N+n+ H N+n Ck n HN k 4 k C n 2 + Cn C H N + C n 3 + Cn C 2H N C n n+ + C n C n H N n C n N + C n C N H + C n C N! k C n+ k and hence we get the recurrence relatons H N k C n+ C n 2 + Cn C C n+ 2 C n 3 + Cn C 2... C n+ k C n k+ + C n C k... 5 C n+ N Cn N + C n C N C n+ N C n C N. If we successvely plug n the C j k n the recurrence relaton of Cn we get a formula whch contans only terms wth C n and the coeffcents of the orgnal Cayley-Hamlton equaton 3 C n+ j C j for n N C n+ j 6 n C n j C +j + m C n+2 for n < N j0 For the other coeffcents Ck n+ k,2,...,n we get analogous formulae Ck n+ n j0 N k j0 C n j C k+j for n N k C n j C k+j + m C k+n+ for n < N k The coeffcents of the secular equaton have the explct form m C k + m λ N 8 C k + m λ C N k λ C k + m λ N C k + + m λ N C k }{{} 0 for k 0 7 5

7 where we appled Eq.34 to the second term n the last equaton. We wll need ths form as frst values n the proof of Eq.2 m C k m λ N C k for k,2,...,n 8 From the SU3 and SU2,2 cases one may assume that the recurrence relaton 6 has the soluton m C n + m λ N+n. 9 Proof: The proof of Eq.9 s done by nducton over n. The frst coeffcent n 0 s gven by m C + N m λ 20 whch s easy to prove f one wrtes C n the form C j λ j λ + j λ j λ + C. For the product m C we take m n the form of Eq.29 we get m C + m λ λ N + C + m λ N + + m C λ N } {{ } 0 The last equaton holds snce the exponent of λ should be N 2 n order to yeld a non-vanshng sum see Eq.34. Frst we treat the case of n N. The next step s to assume the valdty of 9 for n and to show that then t follows also for n+ mc n+ 6 j mc n+ j 9 C j j + m λ N+n+ j C j 8 j + N+n+ j m λ C j λ C j 6

8 + m C j λ N+n+ j N+n+2 j C j λ j j N+ + m C j λ N+n+2 j N+n+2 j C j λ j2 j + m C N λ n+ N+n+ C 0 λ }{{}}{{} 0 + N+n+ m λ mc n+. In the case of n < N there s an addtonal term mc n+ 8 n j0 mc n j C +j + mc n m C n+ λ N + m λ N+n+ + + m λ N+n+. C 0 λ N+n+ + mc n+2 + m C n+ λ N } {{ } mc n+2 + mc n+2 The coeffcents C n k can be wrtten n the form m C n k N+n m C k λ for k,...,n 2 Proof The proof s analogous to the one of m C n but uses the explct form 9 of these coeffcents. m C n+ k 8 N k j0 N k j0 m C n j 9 C k+j N k j0 + m λ N+n j + m C k+j λ C k+j λ N+n j C k+j 7

9 N k j0 + m C k+j λ N+n j N k j + m C k+j λ N++n j + m C k λ N+n+. If we now shft the summaton ndex j n the second sum most of the terms cancel wth those of the frst sum. The remanng term j N k n the frst sum contans C N 0, hence, vanshes also. Therefore, only the thrd sum remans what proofs the assumed form 2 of C n k. Agan there are the cases n < N k whch need to be treated separately m C n+ k N k j0 m C n j C k+j + m C k+n+... m C k λ N+n+ + m C k λ N+n+. + m C k+n λ N }{{} m C k+n+ 8 + m C k+n+ 3.3 The exponental map The expanson of a group element g G wth generator H gln can now be wrtten lke g e H Usng the multpler m we get mg m H n H n H n + + k H n N + [ We can now treat the sums for dfferent k separately + N k H N+n N + N + mcn k C n k HN k. 22 H N k. 23 N + mcn k N + m C k λ N+n 8

10 C k m C k m e λ C k m e λ + C k m e λ N + λn+n m N k! λ n + C k m λ n The last equaton reles on Eq.33 and 34. The terms m N k! cancel the frst sum n Eq.23. The fnal result turns out to be me H N N deth e λ N m I N + C λ N 2 m e λ H N C k m e λ H N k +... N N + m λ e λ H N m e λ H N or n closed form me H n N C n m e λ H N n 24 whch reads n terms of the adjonts of the Slater determnant me H N A n e λ H n The Slater determnant One crucal ngredent of the method s the usage of a multpler m, defned n Eq.9. From low dmensonal examples one may assume the forms and 2 of m. The general proofs can be done by wrtng m as Slater determnant. The Slater determnant s defned as cf. [4, 6... λ N λ N... λ λ 2 N λ 2 N... λ λ N N N... λ N λ N 9 <j λ λ j m. 26

11 We can now use the Laplacan method of expandng the Slater determnant deta n a j A j j n a j A j 27 where the so-called adjonts A j are the subdetermnants of a j multpled by the sgn factor +j. It s also well known that the Laplace expanson wth wrong adjonts gves zero 0 n a j A lj for l. 28 j The Laplacan method appled wth respect to the last row gves then the expanson m λ N A N+ N where m are the subdetermnants of m We get m m k< N+N+ m λ N + m λ N, 29 m λ k λ <j m j N 7 m k<j k,j λ λ j λ k λ j. 30 λ λ j for,2,...,n C n λ N n m C N n λ n 27 A n λ n 3 what proves Eq.33. Also Eq.34 s proven snce f the exponent of λ s not n the sum vanshes because t s an expanson wth the wrong adjonts A n. Hence we get an explct expresson for the adjonts A n m C N n. 32 We can also expand m wth respect to the n+-th lne and then use Eq.32 m A n λ n for n 0,,...,N m C N n λ n. 33 0

12 Wrtng the Laplacan expanson wth wrong adjonts leads to Ergo cf. Eq.2 0 A k λ n for k,n 0,,...,N k n m C N k λ n 34 m δ kl N l m C k λ for k,l,2,...,n 35 Appendx A Some detals Ths secton contans explct forms of some coeffcents and some proofs. Almost all of the equatons hold n the general case, but those whch hold only n the case of the specal groups, s.e., vanshng sum of egenvalues, are denoted by the sgn. For the coeffcents of the secular equaton we get, e.g., C N N+ N C N N N λ j λ j N+ deth N N deth λ 36 C 2 <j λ λ j s 2 k λ k 2 C Some truncated coeffcents C N N 2 j C N 2 N 3 k λ s 0, C0. λ j N deth λ j k, λ j C 2 j<k j,k λ j λ k, C j λ j s λ C 0, C N 0.

13 B Addtonal checks B. One-dmensonal subgroups One-dmensonal subgroups cf. [5 of GLN can be generated by { } e th ; H gln, t R. 37 From the known expanson of e H we can derve the expanson of e th by multplyng the occurrng expressons wth an approprate factor. Obvously, the egenvalues of the t-dependent generator th are tλ f the λ are the egenvalues of H. Therefore, we need to make the replacements λ tλ C k t k C k C k t k C k The expanson 25 reads now or t NN /2 m e th m t NN /2 m m t N N 2/2 m N t n C n t N N 2/2 m e th th N n N t NN /2 m e th N C n m e th H N n N C n m e th H N n. 38 Dfferentaton of the r.h.s of Eq.38 and settng t 0 gves the dervaton of the unt element N C n m λ H N n m H. } {{ } mδ n,n cf.2 Snce ths result concdes wth the one we get by dfferentatng the l.h.s t s an addtonal proof of the expanson 24. B.2 Egenvalues It s easy to demonstrate that Eq.25 gves also the rght connecton between the egenvalues of the generator H and the ones of the correspondng group element g e H. Let x be the egenvectors of H wth egenvalues λ For the powers of H we get H x λ x for,2,...,n. H n x λ n x for n N. 2

14 Pluggng ths n Eq.24 yelds m e H x j N Therefore, we get the desred result A n e λ A n λ n }{{} m δ j cf.2 H n x j x j m e λ j x j. A n e λ λ n x j g x j e λ j x j whch agan confrms the expanson 24. B.3 Remark In the cases where some of the egenvalues concde, the multpler m wll be zero. But n these cases m can be chosen n a smpler fashon so that there occur only non-vanshng factors. Essentally all factors whch wll become zero can be canceled out n Eq.24. Acknowledgement The author would lke to thank Dr. K-P. Marzln for the dscussons of some ponts. References [ A.O. Barut, J.R. Zen, and A. Laufer. The exponental map for the conformal group O2,4. J.Phys.A:Math.Gen., 27: , 994. [2 A.O. Barut, J.R. Zen, and A. Laufer. The exponental map for the untary group SU2,2. J.Phys.A:Math.Gen., 27: , 994. [3 Ta-Pe Cheng and Lng-Fong L. Gauge theory of elementary partcle physcs. Oxford Unversty Press, 984. [4 Sergo Fubn. Vertex operators and Quantum Hall Effect. Mod. Phys. Lett., A6: 347, 99. [5 Sgurdur Helgason. Dfferental Geometry, Le Groups, and Symmetrc Spaces. Academc Press, 978. [6 Claude Itzykson and Jean-Mchel Drouffe. Statstcal Feld Theory II. Cambrdge Unversty Press, 989. [7 J.Rcardo Zen and Waldyr A. Rodrgues. Hadronc Journal, 3:37, 990. [8 J.Rcardo Zen and Waldyr A. Rodrgues. A thoughtful study of Lorentz transformatons by Clfford algebras. Internatonal Journal of Modern Physcs A, 78:793 87,