jordan-lie algebra of single-spin chiral fields

The Jordan-Lie structureof single-spin chiral fields

Selim Gómez-ÁvilaDivisión de Ciencia e IngenieríaUniversidad de Guanajuato

May 28, 2014

This work is about the description of high-spin fields

«The elementary particles known to present-day physics, the electron,positron, neutron, and proton, each have a spin of a half, and thus thework of the present paper will have no immediate physical application.All the same, it is desirable to have the equations ready for a possiblefuture discovery of an elementary particle with a spin greater than ahalf, or for approximate application to composite particles. Further, theunderlying theory is of considerable mathematical interest.»

P.A.M. Dirac

But why is it interesting?

We don’t know why the SM only includessome spins in some representations

(0, 0)(12 , 0

) (0, 12

)(1, 0) ( 12 ,

12 ) (0, 1)

( 32 , 0) (1, 12 ) ( 12 , 1) (0, 32 )

(2, 0) ( 32 ,12 ) (1, 1) ( 12 ,

32 ) (0, 2)

Higgs field

Quarks & Leptons

Gauge bosons

Graviton?

And these?

Are they allowed? If not, why are they forbidden?

We don’t know why the SM only includessome spins in some representations

(0, 0)(12 , 0

) (0, 12

)(1, 0) ( 12 ,

12 ) (0, 1)

( 32 , 0) (1, 12 ) ( 12 , 1) (0, 32 )

(2, 0) ( 32 ,12 ) (1, 1) ( 12 ,

32 ) (0, 2)

Higgs field

Quarks & Leptons

Gauge bosons

Graviton?

And these?

Are they allowed? If not, why are they forbidden?

We don’t know why the SM only includessome spins in some representations

(0, 0)(12 , 0

) (0, 12

)(1, 0) ( 12 ,

12 ) (0, 1)

( 32 , 0) (1, 12 ) ( 12 , 1) (0, 32 )

(2, 0) ( 32 ,12 ) (1, 1) ( 12 ,

32 ) (0, 2)

Higgs field

Quarks & Leptons

Gauge bosons

Graviton?

And these?

Are they allowed? If not, why are they forbidden?

We don’t know why the SM only includessome spins in some representations

(0, 0)(12 , 0

) (0, 12

)(1, 0) ( 12 ,

12 ) (0, 1)

( 32 , 0) (1, 12 ) ( 12 , 1) (0, 32 )

(2, 0) ( 32 ,12 ) (1, 1) ( 12 ,

32 ) (0, 2)

Higgs field

Quarks & Leptons

Gauge bosons

Graviton?

And these?

Are they allowed? If not, why are they forbidden?

We don’t know why the SM only includessome spins in some representations

(0, 0)(12 , 0

) (0, 12

)(1, 0) ( 12 ,

12 ) (0, 1)

( 32 , 0) (1, 12 ) ( 12 , 1) (0, 32 )

(2, 0) ( 32 ,12 ) (1, 1) ( 12 ,

32 ) (0, 2)

Higgs field

Quarks & Leptons

Gauge bosons

Graviton?

And these?

Are they allowed? If not, why are they forbidden?

We don’t know why the SM only includessome spins in some representations

(0, 0)(12 , 0

) (0, 12

)(1, 0) ( 12 ,

12 ) (0, 1)

( 32 , 0) (1, 12 ) ( 12 , 1) (0, 32 )

(2, 0) ( 32 ,12 ) (1, 1) ( 12 ,

32 ) (0, 2)

Higgs field

Quarks & Leptons

Gauge bosons

Graviton?

And these?

Are they allowed? If not, why are they forbidden?

Outline

▶ A survey of high spin fields▶ Jordan-Lie algebras and single-spin chiral fields▶ Multipolar moments and the parity-projection▶ Conclusions and perspectives

A survey of high spin fields

A theory of particle physics has three ingredients;to go beyond the SM we change one of them

A base spacetime fixes the asymptotically free states, a gauge group fixes thenumber and properties of gauge particles, and a matter spectrum includesmatter particles and their interactions.

▶ Supersymmetry and extra-dimensions modify the spacetime algebra.▶ Grand unification theories change the gauge group.▶ The addition of scalar fields or right neutrinos changes the matter

spectrum.

The first two posibilities have been systematically explored, but there are nofundamental principles to guide us in the third way.

At UG, we are currently studying the inclusion of higher spin matter fields.

For high spin matter, we consider the family of (j, 0)⊕ (0, j) representations.

We know three roads to high spin fields

▶ Dirac-Fierz-Pauli: Bosons and fermions obey equations that generalize theKlein-Gordon and Dirac formalisms, supplemented with constraints.

▶ Bhabha-Majorana: Boson and fermion fields are multiplets with severalmasses and spins, coupled by Dirac-like equations.

▶ Joos-Weinberg: Bosons and fermions with definity parity are described byfields with 2(2j + 1) component that obey no wave equations.

Wave equations mediate between Poincare and Lorentz

The Poincare algebra has two algebraic invariants

C2 = PµPµ, C4 =WµWµ with Wµ = 12εµστρM

στP ρ.

One-particle states satisfy

C2 |Ψ⟩ = m2 |Ψ⟩ , C4 |Ψ⟩ = −m2j(j + 1) |Ψ⟩

We callm the mass and j the spin ofΨ.

Quantum fields are built from operators that create or destroy these states:

Ψl(x) =

∫dΓ[κeip·xωl(Γ)a

†(Γ) + λe−ip·xωcl (Γ)a(Γ)].

Field coefficients like ωl(Γ) transform in representations of the Lorentzalgebra so(1, 3) ∼= su(2)A ⊕ su(2)B generated by the commuting sets

A =1

2(J − iK), B =

1

2(J + iK).

We label the Lorentz irreps with the su(2) numbers (a, b).

This mediation is at the origin of the high-spin problem

▶ To get the Poincaré states from the Lorentz representation we need ingeneral to include constraints.

▶ Minimally coupled Fierz-Pauli-Dirac theories suffer from the loss ofconstraints and propagate unwanted degrees of freedom1 .

▶ These can be wrong spin components or ghosts; both are undesirable.▶ Bhabha-Majorana type theories include by design spin and mass

multiplets not in correspondence with the SM fields.▶ The Joos-Weinberg theory has an unambiguos spin content.

1Velo & Zwanziger, Noncausality and Other Defects of Interaction Lagrangians for Particleswith Spin One and Higher, 10.1103/PhysRev.188.2218, 1969

Bhabha and DFP theories are too crowded

The representation (a, b) contains all spins between |a− b| and a+ b.

Formalism Representation Spin Content

Dirac-Fierz-Pauli Bosons:Äj2 ,

j2

ä0, . . . , j

Fermions:Ä2j+14 , 2j−1

4

ä⊕Ä2j−14 , 2j+1

4

ä12 , . . . , j

Bhabha Bosons and fermions:

(j, 0)⊕(j − 1

2 ,12

). . .

(12 , j −

12

)⊕ (0, j) 0(or 1

2 ) . . . j

Joos-Weinberg Bosons and fermions:

(j, 0)⊕ (0, j) j only

The presence of every spin between j = 0 or j = 1/2 up to j is the origin ofthe need for constrictions or auxiliary degrees of freedom.

There are two kinds of parity-invariant representations

Parity exchanges A and B:ΠAΠ−1 = B.

The parity-invariant Lorentz representations fall in two groups:▶ The (a, a) integer-spin Fierz-Pauli representations.▶ The reducible (a, b)⊕ (b, a) representations with a = b. These include

fermionic Fierz-Pauli representations as well as the Joos-Weinberg family.

We call these non-chiral and chiral representations, according with thevanishing of the chirality operator

χ =i Mµν M

µν

4a(a+ 1)− 4b(b+ 1).

We focus on representations with two spin sectors or less.

We can fully classify the few-spin representationswhere we can reconcile Lorentz with Poincaré

The parity-invariant Lorentz representations with at most two spin sectors are

1. The non-chiral representations (0, 0) and(12 ,

12

).

2. The single-spin chiral representations (j, 0)⊕ (0, j) with j ≥ 12 .

3. The double-spin chiral representations(j − 1

2 ,12

)⊕(12 , j −

12

)for

j > 1.

For these, the Poincare projection fixes the appropriate mass and spin. Thisproduces second order equations of motion

(Tµν PµP ν −m2)Ψ = 0,

for both fermions and bosons. 2

2Napsuciale et.al., Spin 3/2 beyond the Rarita-Schwinger framework,10.1140/epja/i2005-10315-8, 2006

Jordan-Lie algebrasand single-spin chiral fields

Symmetries obey Lie algebras,observables obey Jordan algebras

A Lie algebra, has a bilinear product ⋄ for which 3

x ⋄ y = −y ⋄ x,x ⋄ (y ⋄ z) = −y ⋄ (z ⋄ x)− z ⋄ (x ⋄ y).

Lie algebras are crucial in the description of symmetry and dynamics.A Jordan algebra is a non-associative algebra with bilinear product • satisfying 4

x • y = y • x(x • y) • (x • x) = x • [y • (x • x)].

Jordan algebras describe quantum observables.

If •, ⋄ are both defined in a consistent way, we have a Jordan-Lie algebra.

3Fuchs, Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists, 20034N. Jacobson. Structure and Representations of Jordan Algebras , 1968.

Each representation of a Lie algebrapossesses a different Jordan structure.

If we have two representation of a Lie algebra g

D1(x) ◦D1(y) = D1(x ⋄ y), D2(x) ◦D2(y) = D2(x ⋄ y).

then the tensor product

D1 ⊗D2(x) = D1(x)⊗ 12 + 11 ⊗D2(x).

is also a representation of g.

However, the tensor product of two representations of a Jordan algebra doesnot satisfy the original Jordan algebra.

Different representations of the same Lie algebra possess different Jordanalgebras.

The full Jordan-Lie algebras are properties of elementary systems, and relatedto their possible interactions.

A familiar example is the su(2) j = 1/2 algebra

J1 =1

2

Ñ0 1

1 0

éJ2 =

1

2

Ñ0 −i

i 0

éJ3 =

1

2

Ñ1 0

0 −1

é.

The Lie generators close under the Lie product ⋄

Jm ⋄ Jn = [Jm, Jn] = i εmnoJo,

but with the Jordan product • produce

Ji • Jj =1

4δije, e • Ji = J i, e • e = e,

with

e =1

j(j + 1)

3∑i=1

Ji • Ji =

Ñ1 0

0 1

é, e ⋄ Ji = [e, Ji] = 0.

An element of the j = 1/2 algebra has a natural decomposition in terms ofoperators transforming as j = 0, 1 objects.

so(1, 3) follows from su(2)

The su(2) irreps with Dynkin label [2j] are associated with the Young diagram

[2j] ⇔ · · ·︸ ︷︷ ︸2j

.

For j = 12 we obtain the operator descomposition:

j = 1/2 ⇔ [1] ⇔ ⊗2= ⊕ = 1⊕

Since the Lorentz algebra so(1, 3) is the simple sum so(1, 3) = su(2)⊕ su(2),irreps of so(1, 3) are built out of and :

[2a][2b] ⇔ · · ·︸ ︷︷ ︸2a

· · ·︸ ︷︷ ︸2b

.

The operator spaces contain only the products of the vectorial composite box5

= .

5Zhong-Qi Ma, Group Theory for Physicists, 2007

To find the Jordan structure we studythe single spin chiral representations

In the chiral basis the Lorentz generators take the form J = diag(τ, τ) andK = i diag(τ,−τ), where τ is the spin j angular momentum matrix.

The first Casimir operator is trivial

C2 =MµνMµν = 4(A2 + B2) = 4j(j + 1)diag(1,1)

but the chirality operator (proportional to the second Casimir C′2) is not

χ =A2 − B2

a(a+ 1)− b(b+ 1)= diag(1,−1)

For these representations, chirality connects the rotations and boosts through

K = iχJ or, covariantly, Mµν = −iχMµν .

We anticipate block diagonal operators built fromMµν and block antidiagonaloperators built fromMµν and parity.

Parity is not a scalar

In the chiral basis, the parity operator that swaps (j, 0) and (0, j) is

Π =

Ñ0 1

1 0

é.

This operator satisfies the Lie rules

Π ⋄ J = 0 Π ⋄ K = 2ΠK,

Parity rotates as a scalar, but transforms under boosts into V = 2iΠK. In turn,

Ji ⋄ Vj = iεijkVk, Ki ⋄ Vj = −4ΠKi •Kj .

These rules involve the Jordan algebra of the Lorentz generators.

The covariant properties ofΠ will depend on the objects

Π{K,K • K, (K • K) • K, . . . },

that form a symmetric tensor Sµ1···µ2jwith time component S0···0 = Π.

The operator space of the Dirac representation is familiar

The first example is the (1/2, 0)⊕ (0, 1/2) space. The Lorentz decompositionof the operator space isïÅ

1

2, 0

ã⊕Å0,

1

2

ãò2= (0, 0)2 ⊕ (1, 0)⊕ (0, 1)⊕

Å1

2,1

2

ã2

.

This corresponds to a pair of scalars, an antisymmetric tensor, and a pair offour-vectors:(

⊕)⊗2

= ⊕ ⊕2⊕ ⊕ = 12 ⊕ 2

⊕ .

1. The scalars corresponds to the Lorentz invariants C2 and C′2, normalized

to the identity 1 and chirality χ.2. The asymmetrical (1, 0)⊕ (0, 1) tensor corresponds to the Lorentz

generatorsMµν .3. Parity transforms inside the vector object Sµ = η0µΠ− 2iΠM0µ . The

other independent vector is χSµ.No surprises here: {1, γ5, γµ, γ5γµ, σµν}

This is the j = 1 covariant structure

For the representation (1, 0)⊕ (0, 1), the Lorentz decomposition is

[(1, 0)⊕ (0, 1)]2= (0, 0)2 ⊕ (1, 0)⊕ (0, 1)⊕ (2, 0)⊕ (0, 2)⊕ (1, 1)2,

or in Young diagram form,

(⊕

)⊗2= ⊕ ⊕

2⊕ ⊕

⊕ ⊕

= 12 ⊕ 2⊕ ⊕ .

We identify the two (0, 0) operators with 1 and chirality; the (1, 0)⊕ (0, 1)operators are the Lorentz generatorsMµν .

We need to construct the symmetrical operators{Sµν , S

′µν

}and the

Weyl-like Cµναβ .

No problem in building the parity tensorand the Weyl-like tensor

The Lorentz generators mixes the set (Π,ΠM0i ,ΠM0i •M0j ). The covarianttensor containing parity is

Sµν = Πηµν − iΠ(η0µM0ν + η0νM0µ )−Π{M0µ ,M0ν }.

The other symmetrical tensor is χSµν .

The Cµναβ tensor transforms in the (2, 0)⊕ (0, 2) representation. Taking theproductMµν •Mαβ , removing all traces, and applying the Young projector

µ αν β

∝ SµαSνβAµνAαβ ,

we get

Cµναβ = 8Mµν •Mαβ+4Mµα •Mνβ−4Mµβ •Mνα−8(ηµαηνβ−ηναηµβ ).

This is a completely traceless tensor with the symmetries of the Weyl tensor.

The Lie part of the Jordan-Lie algebra

Lie ⋄ 1 χ Sµν χSµν Mµν Cµνρσ

1 0 0 0 0 0 0

χ 0 χSµν Sµν 0 0

Sµν Mµν χ,Cµνρσ Sµν χSµν

χSµν Mµν χSµν Sµν

Mµν Mµν Cµνρσ

Cµνρσ Mµν

Algebraic Lie structure of the (1, 0)⊕ (0, 1) basis.

The Jordan part of the Jordan Lie algebra

Jordan • 1 χ Sµν χSµν Mµν Cµνρσ

1 1 χ Sµν χSµν Mµν Cµνρσ

χ 1 0 0 Mµν Cµνρσ

Sµν 1, Cµνρσ Mµν χSµν Sµν

χSµν 1, Cµνρσ Sµν χSµν

Mµν 1, χ, Cµνρσ Mµν

Cµνρσ 1, χ, Cµνρσ

Algebraic Jordan structure of the (1, 0)⊕ (0, 1) basis.

A final example: the j = 3/2 representation

The external product of states in the basis decompose as

(0, 0)2 ⊕Å3

2,3

2

ã2

⊕ (1, 0)⊕ (0, 1)⊕ (2, 0)⊕ (0, 2)⊕ (3, 0)⊕ (0, 3).

Here we have the decomposition

12 ⊕ 2⊕ ⊕ ⊕ .

{1, χ,Mµν , Sµνρ , χSµνρ , Cµνρσ , Dµνρσαβ}.

1. The symmetric tensors Sµνρ and µνρ are constructed along the lines ofthe j = 1 case.

2. The Weyl-like tensor Cµνρσ has the same form in terms of the Lorentzgenerators.

3. To build the sixth-order tensorDµνρσαβ , we apply the Young projector

µ α ρν β σ

,

to the productMµνMρσMαβ , and remove all traces.

We found the general covariant structureof (j, 0)⊕ (0, j) fields

For arbitrary j , the operator space will have the decomposition6

[(j, 0)⊕ (0, j)]2=

2j⊕i=0

[(i, 0)⊕ (0, i)]⊕ (j, j)2.

The covariant basis contains the scalars {1, χ}, the symmetrical (j, j) tensorsSµ1µ2...µ2j and χSµ1µ2...µ2j , and the series

2j⊕i=1

[(i, 0)⊕ (0, i)] = ⊕ ⊕ ⊕ . . . ,

{Mµν , Cµνρσ , Dµνρσαβ , Eµνρσαβτδ , . . . },

constructed by Young projecting the traceless product of 2j generators.

6S. Gómez-Ávila, M. Napsuciale, Covariant basis induced by parity for the (j, 0)⊕ (0, j)representation, 10.1103/PhysRevD.88.096012, 2013

And took a peek at the double spin chiral representations

For the double spin chiral representations the operator space is decomposed asïÅj − 1

2,1

2

ã⊕Å1

2, j − 1

2

ãò2=

2j−1⊕i=0

[(i, 0)⊕ (0, i)]

2j−1⊕i=0

[(i, 1)⊕ (1, i)]

⊕ (j − 1, j − 1)2 ⊕ (j, j)2

⊕ (j, j − 1)2 ⊕ (j − 1, j)2.

For example, in the-known Rarita-Schwinger j = 32 representation we haveïÅ

1,1

2

ã⊕Å1,

1

2

ãò⊗2

= 12 ⊕ 2⊕

2

⊕2⊕

2

⊕2

⊕ ⊕ .

This is complicated compared with the(32 , 0

)⊕

(0, 32

)case. There are

additional contributions to the spacetime tensor in the Poincare projection.

The study of this novel structures is an important perspective.

Multipolar moments and theparity-projection

Multipolar moments are sensitive to the Jordan-Lie structure

The multipolar moments are calculated from the q-derivatives of the electricand magnetic densities in the Breit frame

ϱE(q, j, λ) = e u0λ (ω − igK · n|q|)B(q)2u0λ

ϱM (q, j, λ) = i e g u0λ∇q ·ï1

ωB(q)

î(J · q) q − |q|2J

óB(q)

òu0λ.

For arbitrary j , only the charge and the magnetic moment are insensitive to theJordan-Lie structure in

B(q) =∞∑

n=0

(iφ)n

n!(K · n)n

= 1+ iφ(K · n)− φ2

2!

(12K · n ⋄ K · n + K · n • K · n

)+ . . .

For higher multipoles, the algebraic properties of K · n become relevant andmultipole moments are not universal.

The natural multipole moments depend on the representation

j “GE0“GM1

“GE2“GM3

“GE4“GM5

“GE6“GM7

“GE8

0 1

12 1 1

1 1 2 1

32 1 3 3 −3

2 1 4 6 −12 −3

52 1 5 10 −30 −15 5

3 1 6 15 −60 −45 30 5

72 1 7 21 −105 −105 105 35 −7

4 1 8 28 −168 −210 280 140 −56 −7

Generic high-spin waves propagate acausally,but the Poincare-projected fields are causal

For the (j, 0)⊕ (0, j) family, there are no constraints and the wave operator is

OABµν = 1

ABηµν − igMA

B µν .

Causal propagation relies on the reality of the normals to the characteristicsurfaces.7 The characteristic determinant for this system is

det[O(n)] =(n2

)2(2j + 1) = 0.

The solutions for the time-like components of nµ are always real.

This problem was studied by Delgado-Acosta, et.al.(2013) for j = 1 underassumptions about the wave operator that are now proven to hold.

In this formalism all the components are dynamical. ⇒ Parity doublets.

7Courant and Hilbert, Methods of Mathematical Physics, 978-0471504399, 1989.

If we want well-defined paritywe need a constrained equation of motion

Well-defined parity states in the (j, 0)⊕ (0, j) satisfy the rest frame condition

P±(0)u±(0) =1

2(1±Π)u±(0) = u±(0).

This is boosted to a covariant wave equation involving Sµ1...µn ; for(12 , 0

)⊕

(0, 12

)it is Dirac’s equation:

1

2(1±Π)ψ(0) = ψ(0) ⇒ (SµP

µ ∓m)ψ(p) = 0.

For arbitrary j , parity eigenstates satisfy the (j, 0)⊕ (0, j) analogue of Dirac’sequation:

[B(p)2Π− 1]ψ±(p) = 0.

This was used by Weinberg to formulate perturbation theory for arbitrary j . 8

8Steven Weinberg. Feynman Rules for Any Spin, 10.1103/PhysRev.133.B1318, 1964.

We obtain two non-equivalent theories

With an on-shell projector

P±(p) =1

2

Å1± Sj(p)

m2j

ã,

we construct the following Lagrangian

L± = ψ±(x)(±Sj(i∂)−m2j

)ψ±(x).

with the adjoint field ψ = ψ†Π. We call this the type I theory.

Off-shell we need to use the projectorÅp2

m2

ãjP±(p) =

Åp2

m2

ãj1

2

Å1± Sj(p)

p2j

ã,

which corresponds to the Lagrangian

L± = ψ±(x)(−√∂4j ± Sj(i∂)− 2m2j)ψ±(x) = 0,

We call this a Type II theory. It is local only for integer j .

The j = 1 type I theory

For j = 1, the on-shell projector formalism leads to the Lagrangian

L0± = ±∂µψ(x)(Sµν − iκMµν )∂νψ(x)−m2ψ(x)ψ(x),

with κ a free parameter. Through minimal coupling we get the current

JIµ(p, p

′) = up′λ′[(p′ + p)µ + i(1 + κ)Mµν (p

′ − p)ν]upλ.

The particle has a gyromagnetic factor

g = 1 + κ,

where κ is an “anomalous” contribution to the magnetic moment.The multipoles are given by

q = e,

µ =e(1 + κ)

m,

Q =eκ

m2.

The j = 1 type II theory

In the off-shell projector formalism the Lagrangian is

L0± =

1

2∂µψ(x)(η

µν ± Sµν − 2iρMµν )∂νψ(x)−m2ψ(x)ψ(x),

with ρ a free parameter. Minimally coupling we get the current

JµII = u(p′λ′

[(p′ + p)

µ+ i

(12 + ρ

)Mµν (p′ − p)ν

]upλ.

Particles in this theory have a gyromagnetic factor

g = 12 + ρ,

with ρ also an “anomalous” magnetic moment. The natural value of theg-factor is g = 1/2, and the multipole moments are

q = e,

µ =e

m

Åρ+

1

2

ã,

Q =e

m2

Åρ− 1

2

ã.

The literature for spin 1 fits in our scheme

We can classify formalisms for j = 1 in the market by writting them in terms ofthe covariant basis, translated into tensor language through the identification

Ψ(x)A ↔ Fαβ(x).

Theory Type Anomalous g-factor

Joos-Weinberg (1961) I 0

Shay-Good (1969) II 0

Tucker-Hammer (1971) II 0

Prabhakaran-Seetharaman (1973) II ρ = 1/2

Ecker(1989) II 0

Delgado-Acosta et.al. (2013) II & Poincare ρ = 1/2

Tomorrow, Rodolfo Ferro will talk in more detail about the canonicalquantizacion of type I and II theories for j = 1.

For j > 1 we have higher-derivatives,but we should not despair

The parity projection for higher spin (j > 1) leads to equations of motion withhigher than second derivatives of the fields.

By the Ostrogradski theorem, unconstrained higher-derivative theoriesdevelope unstabilities, either loss of unitarity or production of ghosts.

However:▶ The parity-projection equations are not unconstrained.▶ Some HD theories are known where unitarity is preserved. 9

▶ The Poincare and parity projectors commute; it would be surprising if thefree-field formalism propagated a ghost.

▶ Such a theory not be Poincare invariant while remaining Lorentz invariant.

9Grinstein, O’Connell and Wise, The Lee-Wick standard model, 10.1103/PhysRevD.77.025012,2007.

Conclusions and perspectives

Conclusions

▶ This work explored the Jordan-Lie algebraic structure for the(j, 0)⊕ (0, j) representations of the Lorentz algebra.

▶ The main result of this analysis is an algorithm for the calculation of acovariant basis for arbitrary j in a class of representations.

▶ An explicit construction of the covariant basis is given for j = 12 , 1,

32 .

▶ The propagation of spin j waves in an electromagnetic background iscausal in the Poincare formalism.

▶ Multipole moments are representation-dependant. Higher multipolemoments depend on the Jordan-Lie algebra.

Conclusions

▶ Two nonequivalent theories were found by using on-shell and off-shellprojectors for parity.

▶ The multipole moments of elementary spin j particles are dictated by asingle parameter g.

▶ For the j = 1 theories type I and II we studied the electrodynamics.Natural values of the gyromagnetic factor were obtained.

▶ Our formalism includes and clarifies the physical meaning of a number ofexisting descriptions.

▶ The type I theory carries a lineal realization of chiral symmetry.

Perspectives and questions

▶ Understanding the high derivatives in j > 1: is the theory unstable? doesit violate unitarity? Preliminary results suggest the answer is no.

▶ Understanding the role of discrete symmetries: do we have to identifythe parity spaces of boson particle-antiparticle?

▶ Phenomenology: can we describe dark matter as chiral spin 1 bosons?These would be SM singlets.

▶ What is the covariant basis for the Rarita-Schwinger type representations?We know there are two antisymmetrical structures: what is the multipolestructure?

Thanks!

This is the j = 1 Jordan-Lie algebra

Sµν ⋄ Sαβ = −i(ηµαMνβ + ηναMµβ + ηνβMµα + ηµβMνα

),

χSµν ⋄ χSρσ = i(ηµρMνσ + ηνρMµσ + ηµσMνρ + ηνσMµρ ),

χSµν ⋄ Sρσ =4

3

(ηµρηνσ + ηµσηνρ − 1

2ηµνηρσ

)− i

6

ÄCµρνσ + Cµσνρ

ä,

χ ⋄ Sµν = 2χSµν ,

χ ⋄ χSµν = 2Sµν ,

Mµν • Sαβ =1

2εµνσβχSασ +

1

2εµνσαχSβσ ,

Sµν • Sρσ =4

6

(ηµρηνσ + ηµσηνρ − 1

2ηµνηρσ

)− 1

12

(Cµρνσ + Cµσνρ

),

χSµν • χSµν = −4

6

(ηµρηνσ + ηµσηνρ − 1

2ηµνηρσ

)+

1

12

(Cµρνσ + Cµσνρ

),

χSµν • Sρσ =1

4

Äηµρ‹Mνσ + ηνσ ‹Mµρ + ηµσ

‹Mνρ + ηνρ ‹Mµσ

ä,

Mµν •Mρσ =4

6

(ηµρηνσ − ηµσηνρ

)− 4

6iεµνρσχ+

1

12Cµνρσ ,

χ • Sµν = 0,

χ • χSµν = 0.