Gauge invariant Lagrangian for

Massive bosonic higher spin field

Hiroyuki Takata

Tomsk state pedagogical university(ТГПУ)

Tomsk, Russia

Hep-th 0707.218

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 particle sting

World line World 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 algNote: 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.

howeverTotally 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

… … s2

… …

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 (s1 s2)

1. K-G equation,2. Transversality condition3. 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. Formulation 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.

Idea for introduce gauge sym. : formula

Starting: Irreducibility condition

Symmetric property of (i) symmetric by permutation of 1…s1 and 1…s2

We would like to find a Lagrangian that leads following conditions (i) - （ v) as its equations of motion

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

•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.

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. Formulation 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 m2, G11, G22 are not regarded as 1st class constraints for neither Ket or Bra, there appear 6 2nd class constraints:

In order to make these right hand sides 1st

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

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New representation

New representation is sum of original and additional:

we need to introduce 6 creation-annihilation operators corresponding above these 6 second class constraint, namely,

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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. Formulation 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 1st class.

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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 2nd class constraint problem and it includes two parameters those make model consistent

Note: modification of inner product is necessary because

…Follow red colors

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From algebra to BRST operator (General procedure)

If constraint operators Ta satisfy closed algebra: 　　　　 [Ta, Tb ] = fab

cTc

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. Formulation need ghosts

Gauge fixing

BRST equation and Reducible Gauge transformation

We have equation of motion for physical state

Lagrangian from BRST operator

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

Reducible Gauge transformation

Gauge transformations of gauge parameters are

where

Return to Lagrangian

Generalization to multi row YT

………

…

… …

……

s2…

s1……

k rows•Irr.condition & algebra, are the same form, but with i=1…k

• k(k+1) 2nd 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