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ContentsContents

I.I. Introduction for Higher Spin theoryIntroduction for Higher Spin theory

II.II. Lagrangian formulation with constraintsLagrangian formulation with constraintsMassless case :Fang-Fronsdal formulationMassless case :Fang-Fronsdal formulation

III.III. Lagrangian formulation without constraintsLagrangian formulation without constraints

IV.IV.Lagrangian formulation by BRST-BFV; without constraintsLagrangian formulation by BRST-BFV; without constraintsLagrangian formulation:HS-algebra and BRST formulation Lagrangian formulation:HS-algebra and BRST formulation

V.V. A unified approach for integer and half-integer spinA unified approach for integer and half-integer spinReproduce Dirac-Fierz-Pauli, Reproduce Dirac-Fierz-Pauli, Bargmann-WingerBargmann-Winger

Free gauge fields of arbitrary spinFree gauge fields of arbitrary spin

高田浩行 , トムスク教育大学（ロシア） SSI, 2017.09.27-29

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I. I. Introduction for Higher Spin theoryIntroduction for Higher Spin theory

Goal of Higher spin theory Goal of Higher spin theory

● gravitational field 　　　 electromagnetic field 　　　 Yang-

Mills field 　　　　 Gauge field and matter fields will be unified with respect to their size of spin.

● Would like to spin independent theory. Spin s (s=0,1/2,1,3/2,2,5/2,3,....) is a parameter of the theory

● An approach to unified theory of particle including gravity. (relation to string theory is not assumed)

s=2 s=1

s=1

s=s

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I. I. Introduction for Higher Spin theoryIntroduction for Higher Spin theory

Why are we interested in Higher Spin theory?Why are we interested in Higher Spin theory?

● Similarity between EM field and gravity

force

Gauge symmetry

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I. I. Introduction for Higher Spin theoryIntroduction for Higher Spin theoryWhy are we interested in Higher Spin theory?Why are we interested in Higher Spin theory?

●Similarity exists to string theory ●Higher spin theory can be understood as that of rigid string that is given by "tension less limit in string theory

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I. I. Introduction for Higher Spin theoryIntroduction for Higher Spin theory

Arbitrary spin model or Higher spin theory - by naive extensionArbitrary spin model or Higher spin theory - by naive extension

We would like to treat model with any spin in the universal way.

● It has gauge symmetry with gauge parameter field of rank s-1 tensor

● Guess from electric theory and gravity “spin s model can be described by rank s tensor field”:

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I. I. Introduction for Higher Spin theoryIntroduction for Higher Spin theory

Initial work of Higher spin theory by Fierz and PauliInitial work of Higher spin theory by Fierz and Pauli

They considered field equation(EOM) of arbitrary spin field(1939).from condition(for massive case):●Lorenz inv●Positivity of energy, they found a set of equation, called Fierz-Pauli condition:

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I. I. Introduction for Higher Spin theoryIntroduction for Higher Spin theory

Spin as representation of Poincare groupSpin as representation of Poincare group

Condition of irreducibility of representationCondition of irreducibility of representation

Winger(1939), Bargeman and Wigner (1948)

Trace of rank s tensor = rank s-2 tensor(*)If we take trace of rank s tensor

So we need exclude this in order to consider irreducible representation for spin s. So we add traceless condition:

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I. I. Introduction for Higher Spin theoryIntroduction for Higher Spin theory

Summary of conditions:Summary of conditions:Our basic condition(constraints) for arbitrary spin model including irreducibility now has beenfound to be

E.g. massless spin 1

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II. II. Lagrangian formulation with constraintsLagrangian formulation with constraintsExamples: Lagrangian for spin 0,1,2Examples: Lagrangian for spin 0,1,2

There are traceless constraints by hand. It is complicated because of the constraint.

Massive

Massless

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Lagrangian for arbitrary integer spin (Fang-Fransdl type,Massless) Lagrangian for arbitrary integer spin (Fang-Fransdl type,Massless)

simplified notation

II. II. Lagrangian formulation with constraintsLagrangian formulation with constraints

Massless

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III. III. Lagrangian formulation Lagrangian formulation withoutwithout constraints constraintsLagrangian for arbitrary integer spinLagrangian for arbitrary integer spin

(Quartet (Quartet unconstrained formalism, IL Buchbinder, AV Galajinsky, VA Krykhtin,unconstrained formalism, IL Buchbinder, AV Galajinsky, VA Krykhtin,NPB779:155,2007)NPB779:155,2007)

This Lagrangian is given by gauge fixing from Lagrangian constructed by “BRST formalism”

Massless

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spin2 example

IV. IV. Lagrangian formulation by BRST-BFV; Lagrangian formulation by BRST-BFV; without constraintswithout constraints

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How to construct Lagrangian by BRST method

By introducing oscillators satisfying

and defining state |Φ combining all fields over different spin:⟩

Basic condition for HS field can be rewritten as form of Operator × state=0: ex. integer spin Define

then our conditions for HS

Spin independent eq.'s

IV. IV. Lagrangian formulation by BRST-BFV; Lagrangian formulation by BRST-BFV; without constraintswithout constraints

are rewritten as

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lt may not commute each other. Example (massless unfixed integer spin)

They are the 1st class constraints.We may adopt sum of Lagrange multiplier(η's) terms as Lagrangian

We regard these η's are grassmann variable and introduce their conjugate momentum variables

Define operator Q asThis is nil-potent:

Compared with constraint system in Hamiltonian formalism of Higher spin

Study with Matsuo and Morozumi

c.f.

“Higher Spin Algebra”

IV. IV. Lagrangian formulation by BRST-BFV; Lagrangian formulation by BRST-BFV; without constraintswithout constraints

BRST-BFV construction

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We could replace constraints.Then we may wrote constraints and gauge transformations as,

Equation of motion, constraints and gauge symmetry

Then we may wrote constraints and gauge transformations as,

If |χ satisfy constraint then |χ ' also satisfy constraint.⟩ ⟩So we find gauge symmetry :

LagrangianDefine inner product appropriately, then Lagrangian is given. It is gauge invariant.

IV. IV. Lagrangian formulation by BRST-BFV; Lagrangian formulation by BRST-BFV; without constraintswithout constraints

Fix gauge

and use

Equation of Motion

Introduce auxiliary

fields and extend gauge

symmetry

BRST formalism

Nucl.Phys.B762:344-376,2007

Quartet formalism

Nucl.Phys.B779:155-177,2007

Fang-Fronsdal formalismPRD20,4:848,1979

Three formalism for HS gauge theory

III. III. Lagrangian formulation Lagrangian formulation withoutwithout constraints constraints

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V. V. A unified approach for integer and half-A unified approach for integer and half-integer spininteger spin

Higher Spin Algebra

, where

is Dirac operator:

C is a charge conjugation matrix

are oscillators.

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V. V. A unified approach for integer and half-A unified approach for integer and half-integer spininteger spin

Constraints for spin state

These include field equations both for integer and half integer field.

Examples

Ψ is written by SO(3,1) component, then Dirac equation is also written as

1. T1 constraint for spin 1/2

Dirac spinor

Dirac equation

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V. V. A unified approach for integer and half-A unified approach for integer and half-integer spininteger spin

Examples

Ψ is written by SO(3,1) component and omitting SL(2C) indices,

L1 corresponds divergence free condition

∴

Remember

2. L1 constraint for spin 1

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V. V. A unified approach for integer and half-A unified approach for integer and half-integer spininteger spin

3. T1 constraint for spin 1

We get

If we rewrite in 2 component ,

“Dirac-Fierz-Pauli” is reproduced

From these, KG equation is also reproduced

explicitly,

T1 corresponds field equation

Examples

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V. V. A unified approach for integer and half-A unified approach for integer and half-integer spininteger spin

4. T1 constraint for general spin : Bargmann-Winger equation

(V. Bargeman and E. Wigner, Proc. Nat. Acad. Sci. 34 (1948) 211.)

After changing representation of gamma matrix, we get

This is Bargmann-Winger type equation

(symmetric bracket is only for dotted indices)

This reduce to Rarita-Schwinger equation for spin 3/2 case.Note: another is gamma traceless condition in RS eq. is expected to be given from L

1 constraint).

Examples

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V. V. A unified approach for integer and half-A unified approach for integer and half-integer spininteger spin

Lagrangian preliminary result

BRST operator

Lagrangian is conveniently written in a matrix form as

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V. V. A unified approach for integer and half-A unified approach for integer and half-integer spininteger spin

Lagrangian for lower spin preliminary result

spin 0

s1 and s

5 are decoupled. So by using EOM of s

5:(□+m2)s

5=0, we get Lagrangian for one scalar s

1:

spin 1/2

similarly, Dirac Lagrangian

spin 1

Gauge transformation

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VI. SummaryVI. Summary

●Review of higher spin theory Fierz-Pauli+irreducible constraints determine HS field. Lagrangian with constraint field Lagrangian with un-constraint filed,

in particular BRST approach

●Unified approach for integer and half integer spin.● HS-algebra and HS constraints reproduce

div free and field equation( in examples)● Lagrangian (preliminary) is proposed

Lower spin examples