Gauge invariant Lagrangian for Massive bosonic higher spin field Hiroyuki Takata Tomsk state pedagogical university(ТГПУ) Tomsk, Russia Hep-th ＹＩＴＰ 2007 年８月７日 =Not always totally symmetric, but general tensor. Based on the work with I.L. Buchbinder and V.A. Krykhtin mixed symmetric

Motivations １ 1.Motivated by String/Brane theory. Point particlesting World lineWorld sheet brane World volume Higher dimensional extended object may naturally coupled with various tensor (gauge) field, those are mixed symmetric in general (ex. ) Various HS states are also in string theory. Similarity between Mixed sym. HS alg and Virasoro alg Note: we do not restrict ourselves from string/brane theory.

2.We are interested in investigating all irreducible tensor representation under Poincare group in arbitrary dimension. however Totally symmetric tensors do not cover all irreducible representation of Poincare group in case more than 4 space-time dimension. 3. Interacting HS theory is not considered here though, technique introduced here may be useful for that. Motivations 2, 3

For totally symmetric case, rank of tensor is its spin. For Mixed symmetric case, a set of number,(s1,s2,…) gives correspondent of “spin”. In other word, Young tableaux describe “spin” in this case.  ………… s1s1  …… s 2 …… symmetric for  ’s and ’s Each tensor symmetry is described by corresponding young tableaux … Introduction What is spin for mixed symmetric case?

What are conditions for irreducible representation under Poincare group? Essence of Mixed symmetiric case is in 2 rows case, so, Let us consider Massive arbitrary spin field like, which corresponds Young tableaux with 2 rows (s 1  s 2 ) 1. K-G equation, 2. Transversality condition 3. Traceless condition 4. One more condition for irreducibility under symmetric group of tenser indices

Procedure (plan of talk) Spin independent formulation introduce c-a op. Introduce and extending Fock space to formulate HS state with gauge sym. Treatment of arbitrary spin need more c-a op. Gauge inv. Formulatio n need ghosts Gauge fixing Irr.eq’s FROM eq. of irreducibility for tensor field TO Lagrangian. Constraint eq. With HS alg. BRST eq. with gauge sym. Lagrangian with gauge sym. Gauge fixing Idea for introduce gauge sym. : formula

Starting: Irreducibility condition Symmetric property of  (i) symmetric by permutation of  1  …  s 1 and 1  … s 2 We would like to find a Lagrangian that leads following conditions (i) - （ v) as its equations of motion (ii) after symmetrization of  1  …  s 1, 1, field vanishes Not exist for totally symmetric case

Following 3 conditions are the same as in the totally symmetric case (iii) Klein-Goldon equation (iv) Transversality condition (v) Traceless condition

Auxiliary Fock space representation When we consider this state, symmetry of indexes, i.e. condition (i) is automatically satisfied. Introduce auxiliary Fock space and creation-annihilation operators and rewrite above constraints. Unlike totally symmetric case, K kinds of c-a op. needed for K-row YT. Here we introduce 2 kinds of ones for 2-row YT. We define a basic state from the tensor field  Spin independent formulation introduce c-a op. Introduce and extending Fock space to formulate HS state with gauge sym. Treatment of arbitrary spin need more c-a op. Gauge inv. Formulatio n need ghosts Gauge fixing

To rewrite other constraints, define operators Not exist for totally symmetric case

Add Helmite conjugate of above operators (for Helmitisity of lagrangian) (These are not constraints for Ket state but for Bra.)

Higher Spin algebra for Mixed symmetry Independent generators: Essential for Irr. HS Virasoro algebra like Some sub-algebras

Since these m 2, G 11, G 22 are not regarded as 1 st class constraints for neither Ket or Bra, there appear 6 2 nd class constraints : In order to make these right hand sides 1 st class constraints, we can modify algebra and find new representation for that. Hint: If right hand sides of these commutators have some arbitrary constants, they may control model and make r.h.s constraints. 6 arbitrary parameters will be introduced Skip this slide

New representation New representation is sum of original and additional: To solve problem related mass, add we need to introduce 6 creation-annihilation operators corresponding above these 6 second class constraint, namely, Skip this slide Spin independent formulation introduce c-a op. Introduce and extending Fock space to formulate HS state with gauge sym. Treatment of arbitrary spin need more c-a op. Gauge inv. Formulatio n need ghosts Gauge fixing

To solve problem related spin, add There are two parameters in above expression, those determine value of spin in our model, by requiring remaining 4 class constraint should be the 1 st class. Skip this slide

It is easily seen that these “new” operators also satisfy the almost the same algebra to the original one difference: mass m does not cause 2 nd class constraint problem and it includes two parameters those make model consistent Note: modification of inner product is necessary because …Follow red colors Skip this slide

From algebra to BRST operator (General procedure) If constraint operators T a satisfy closed algebra: [T a, T b ] = f ab c T c Then, BRST operator is defined as Where  a, P b are canonically conjugate ghost variables. ….BRST equation ….Gauge transformation

BRST operator and Fock states BRST operator is calculated as, which is nilpotent. Here,

We define extended Fock state, which is independent of ghost momentum for G’s. -k is ghost number Spin independent formulation introduce c-a op. Introduce and extending Fock space to formulate HS state with gauge sym. Treatment of arbitrary spin need more c-a op. Gauge inv. Formulatio n need ghosts Gauge fixing

BRST equation and Reducible Gauge transformation We have equation of motion for physical state and gauge transformation for it This is reducible gauge theory because

Lagrangian from BRST operator Gauge invariant under reducible gauge transformation Massive Bosonic arbitrary 2row Mixed symmetric tensor Local theory without higher derivative or constraints Including auxiliary fields c.f. Generalization to k row case of mixed symmetry Now, we can construct Lagrangian for fixed spin K is a operator to define modified inner product.

Gauge fixing (with partial equations of motion) Gauge fixing conditions to reproduce starting equations Irreducible mixed spin state under Poincare group Gauge invariant mixed spin state BRST construction

Example the simplest mixed symmetric case: spin (1,1) State expansion

Lagrangian for spin (1,1) click to simplify

Gauge transformation where Tensor and Vector fields

Scalar fields where

Reducible Gauge transformation Gauge transformations of gauge parameters are where Return to Lagrangian

Generalization to multi row YT …… … … …… …… s 2 …  s1s1 ……  k rows Irr.condition & algebra, are the same form, but with i=1…k k(k+1) 2 nd Class constraints. k h’s are introduced. Gauge fixing cond. … … Gauge transformation is reducible, whose number of stage is k(k+1).

Summary Mixed symmetric Irreducible tensor field under Poincare group in arbitrary space- time dimension were studied. We found how to construct gauge invariant Lagrangian for the arbitrary mixed symmetric field by using BRST. Conversely, gauge fixing condition to reproduce irreducible field was found. Spin(1,1) example was explicitly given. Irr.eq’s FROM eq. of irreducibility for tensor field TO Lagrangian. Constraint eq. With HS alg. BRST eq. with gauge sym. Lagrangian with gauge sym. Gauge fixing