BRST Quantization of String Theories

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BRST Quantization of String Theories BRST Quantization of String Theories Applying REDUCE to High Energy Physics y Werner M. Seiler Institut fur Theoretische Physik, Universitat Karlsruhe D-7500 Karlsruhe 1, West Germany BITNET: BE04@DKAUNI2 A REDUCE package for commutator calculations in sup ersymmetric theories including ordered pro ducts and for in nite sums is presented and applied to the computation of anomalies in string theory. 1. Intro duction While the use of computer algebra in general relativity has b ecome fairly common, most applications in high energy physics concern Feynman diagrams. Examples which tend more 2 to mo del calculations are rare: Castellani rep orts ab out a package for sup ergravity , other 6;8 authors tried REDUCE in sup erspace formalism . 7 In this pap er, we present a package written in REDUCE 3.3 for commutator calculations in sup ersymmetric theories, for the handling of ordered pro ducts and for the simpli cation of in nite sums. As an application, anomalies or Schwinger terms of constraint algebras in 3 string theory are computed. For commutator calculations Cecchini and Tarlani see also their contribution to this workshop recently presented a COMMON LISP program. But in our case, it is imp ortant, for the further pro cessing of the results, to integrate the package into a computer algebra system like REDUCE. We start with a very short description of the physical background. The following two sections describ e shortly the REDUCE package SUPERCALC implemented for the calculations. 9;10 A more explicit description can b e found elsewhere . First, we consider the computation of commutators and the handling of ordered expressions, then we regard the simpli cation of sums. The last section shows a concrete example and gives some conclusions. 2. String Theory In the last years, string theory has attracted a lot of interest as a p ossible candidate y Supp orted by Studienstiftung des deutschen Volkes New address: Institut fur Algorithmen und Kognitive Systeme, Universitat Karlsruhe for an uni ed theory. A striking feature of this theory is the existence of a so-called critical dimension: A consistent quantization is p ossible only for one sp ecial dimension of space-time. 5 The physical ideas of string theory can b e found in the \bible" . Here, we concentrate on the computational asp ects of quantization within the BRST formalism. Our basic ob jects are the Fourier mo des of the fundamental elds of the theory. They ob ey canonical commutation relations, for the b osonic string, for example, [ ; ]= m : 1 m n m+n;0 The string Lagrangian is degenerate and gives rise to constraints. Their Fourier mo des 1 X L = : : 2 m mn n 1 generate the famous Virasoro algebra with relations D 3 m m : 3 [L ;L ]=mnL + m+n;0 m n m+n 12 The colons denote normal ordering de ned by the pairings = m m : 4 m n m+n;0 These expressions already show all characteristics of the calculations: We always deal with bilinear currents like the L 's; b ecause of the ordering, central extensions or anomalies arise m in the algebras of constraints. In sup ersymmetric mo dels, the numb er of mo des and therefore the size of the algebras increases. Wemust work with fermionic mo des, to o then. Further mo des so-called \ghosts" are intro duced by the BRST theory. We call the quantization consistent, if the anomalies vanish. This condition determines the space-time dimension D . 1 To compute the commutator b etween ordered expressions, we need the theorem of Wick , expanding a pro duct in a sum of ordered pro ducts: A A ::: A = : A A :::A :+ 1 2 n 1 2 n X + : A :::A :::A :::A :+ 1 i j n 5 one pairing X + : A ::: A :::A :::A :::A :::A : + ::: 1 i k l j n two pairings A similiar formula exists for the pro duct of two ordered pro ducts. 3. Commutator Calculati ons For commutator calculations and the handling of ordered expressions, SUPERCALC pro- vides three pro cedures: bracket, wick and ordprod. bracket uses only the basic, algebraic prop erties of a graded commutator with the exception of ordered pro ducts, where the de - nition of a commutator is used: A B [A; B ]= [B; A] ; 6a [A + B; C ]=[A; C ]+[B; C ] ; 6b B C [AB ; C ]=A[B; C ]+ [A; C ] B 6c denotes the grading of the op erator A. Hence, it can also b e used for other structures A likePoisson or Jacoby brackets. The fundamental commutation relations are intro duced to SUPERCALC by means of LET rules. If the switch zerocomm is set, only the nonvanishing commutators must b e given. Any commutator without a de ning LET rule is then eliminated at once by the simpli er. A straightforward recursive implementation of the rules 6a-c leads to a fairly inecient algorithm. bracket tries to improve the eciency by rst computing the complexity of each argument, and starting with the more involved one. For the most common cases | sums, pro ducts and p owers | sp ecial pro cedures are called, working iteratively. wick expands a pro duct of op erators into a sum of ordered pro ducts following the theorem of Wick 5; ordprod do es the same for a pro duct of ordered pro ducts. A depth- rst approach is used to generate all terms. This algorithm p ossesses the highest complexity of the whole package. Foralownumber n of factors, the numb er of terms in the expansion is growing n approximately exp onentially, for a large numb er with n .For n =5 we get 25, for n =8 already 763 summands! 4. Sums Most op erators of string theory are de ned as in nite sums. Kronecker 's and step functions o ccur in the commutators and pairings of the mo des. To handle such expressions, SUPERCALC provides the op erator ssums,n,l,u and the pro cedure evalsum to simplify terms with ssum. Here s denotes the summand, n the index and l,u the lower resp. upp er b ound. The simpli cation of elementary sums at present: summands linear or quadratic in the index is integrated into the REDUCE simpli er and hence p erformed automatically.For more complex cases, evalsum must b e invoked. This pro cedure tries six simpli cation rules. The rst three consider single sums: The summand is checked for an even or o dd symmetry, sums over expressions containing a Kronecker are evaluated, and the b ounds are adjusted in sums with step functions. The other three rules work on linear expressions in ssum. First, sums over ranges of equal size are collected into a single sum to give REDUCE achance to simplify the summand. For sums over summands of the same typ e, two cases are considered: Either the sums cancel each other partially, or their ranges are adjacent. Both times a single sum is generated. All three rules search automatically for index shifts allowing their application. evalsum makes no use of summation theory. It is a completely heuristic approach which suces for the sums in our calculations. For multiple sums the evaluation can b e very time- consuming, b ecause every level must b e investigated sep erately. The elementary sums are computed by a kind of table lo ok-up. Hence, an extension of this list is easily p ossible. Besides the main pro cedures presented in this and the preceding section, SUPERCALC must tackle a lot of minor problems. In particular, a legiable output is imp ortant. Further p oints are e.g. e ective control of LET rules and the distinction b etween integer and half integer indices. 5. An example The following gure contains a complete session computing the algebra 3. To get the simplest form of the result, wemust give some hints with LET rules. A completely automatic calculations which also detects the necessary index shifts! would b e very dicult to program. Esp ecially the recognition of the Virasoro op erator in the nal expression, whichisvery easy in our example, requires nontrival pattern matching in the case of sup erstrings. setgreaterm,0; Time: 3 ms pp:=bracketnormord alpha m-j* alpha j, normordalphan-k* alpha k; PP := DELTA *DELTA *THETA-J+M*THETAJ*J *-J+ M -J-K+M+N,0 J+K,0 +DELTA *DELTA *THETA-K+N*THETAK* K*K- N -J-K+M+N,0 J+K,0 +DELTA *THETA-J+M*:ALPHA *ALPHA :*-J+M -J-K+M+N,0 J K +DELTA *THETA-K+N*:ALPHA *ALPHA :*K-N -J-K+M+N,0 J K +DELTA *DELTA *THETA-J+M*THETAJ* J*-J +M -J+K+M,0 J-K+N,0 +DELTA *DELTA *THETA-K+N*THETAK* K*K- N -J+K+M,0 J-K+N,0 +DELTA *THETA-J+M*:ALPHA *ALPHA :*-J+M -J+K+M,0 -K+N J -DELTA *THETAK*K*:ALPHA *ALPHA : -J+K+M,0 -K+N J +DELTA *THETA-K+N*:ALPHA *ALPHA :*K-N J-K+N,0 -J+M K +DELTA *THETAJ*J*:ALPHA *ALPHA : J-K+N,0 -J+M K +DELTA *THETAJ*J*:ALPHA *ALPHA : J+K,0 -J+M -K+N -DELTA *THETAK*K* :ALPHA *ALPHA : J+K,0 -J+M -K+N Time: 1378 ms for all x such that not freeofx,n let deltam+n,0*x=deltam+ n,0*s ubn=- m,x; Time: 15 ms qq:=evalsum ssumpp,j,-aleph,al eph; QQ := -2*DELTA *THETA-K*THETAK+M*K *K+M M+N,0 +2*DELTA *THETA-K-M*THETAK*K* K+M M+N,0 -THETA-K*K*:ALP HA *ALPHA : -K+N K+M -THETA-K*K*:ALP HA *ALPHA : K+M -K+N +2*THETA-K+N*:ALP HA *ALPHA :*K-N -K+M+N K +2*THETAK-N*:ALPH A *ALPHA :*K-N -K+M+N K -THETAK*K*:ALPH A *ALPHA : -K+N K+M -THETAK*K*:ALPH A *ALPHA : K+M -K+N Time: 3068 ms rr:=evalsum ssumqq,k,-aleph,al eph; 2 RR := DELTA *M*M -1 M+N,0 +6*SSUM:ALPHA *ALPHA :*I:1,I:1,-ALEPH,ALEPH M+N-I:1 I:1 -3*SSUM:ALPHA *ALPHA :*I:1,I:1,-ALEPH,ALEPH M+I:1 N-I:1 -3*SSUM:ALPHA *ALPHA :*I:1,I:1,-ALEPH,ALEPH N-I:1 M+I:1 -6*N*SSUM:ALPHA *ALPHA :,I:1,-ALEPH,ALEPH /3 M+N-I:1 I:1 Time: 2934 ms for all k let ssumnormordalpham+k *alpha n-k *k,k,- aleph, aleph = ssumnormordalpham+ n-k*a lphak *k- m,k,- aleph, aleph ; Time: 32 ms rr/4; 2 DELTA *M*M -1 M+N,0 +6*SSUM:ALPHA *ALPHA :,I:1,-ALEPH,ALEPH*M-N /12 M+N-I:1 I:1 Time: 275 ms A complete session computing the Virasoro algebra 3.
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