Generalized Twistor Transform And Dualities, With A New Description of
  Particles With Spin, Beyond Free and Massless
USC-07/HEP-B3 hep-th/0704.0296
Generalized Twistor Transform And Dualities
With A New Description of Particles With Spin
Beyond Free and Massless1
Itzhak Bars and Bora Orcal
Department of Physics and Astronomy,
University of Southern California, Los Angeles, CA 90089-0484, USA.
Abstract
A generalized twistor transform for spinning particles in 3+1 dimensions is constructed
that beautifully unifies many types of spinning systems by mapping them to the same twistor
, thus predicting an infinite set of duality relations among spinning systems with
different Hamiltonians. Usual 1T-physics is not equipped to explain the duality relationships
and unification between these systems. We use 2T-physics in 4+2 dimensions to uncover
new properties of twistors, and expect that our approach will prove to be useful for practi-
cal applications as well as for a deeper understanding of fundamental physics. Unexpected
structures for a new description of spinning particles emerge. A unifying symmetry SU(2, 3)
that includes conformal symmetry SU(2, 2) =SO(4, 2) in the massless case, turns out to be
a fundamental property underlying the dualities of a large set of spinning systems, including
those that occur in high spin theories. This may lead to new forms of string theory back-
grounds as well as to new methods for studying various corners of M theory. In this paper
we present the main concepts, and in a companion paper we give other details [1].
I. SPINNING PARTICLES IN 3+1 - BEYOND FREE AND MASSLESS
The Penrose twistor transform [2]-[5] brings to the foreground the conformal symmetry
SO(4, 2) in the dynamics of massless relativistic particles of any spin in 3 + 1 dimensions.
The transform relates the phase space and spin degrees of freedom xµ, pµ, s
µν to a twistor
and reformulates the dynamics in terms of twistors instead of phase space. The
1 This work was partially supported by the US Department of Energy, grant number DE-FG03-84ER40168.
http://arxiv.org/abs/0704.0296v2
twistor ZA is made up of a pair of SL(2, C) spinors µ
α̇, λα, α, α̇ = 1, 2, and is regarded as
the 4 components A = 1, 2, 3, 4 of the Weyl spinor of SO(4, 2) =SU(2, 2).
The well known twistor transform for a spinning massless particle is [5]
µα̇ = −i (x̄+ iȳ)α̇β λβ, λαλ̄β̇ = pαβ̇, (1.1)
where (x̄+ iȳ)
(xµ + iyµ) (σ̄µ)
, and pαβ̇ =
pµ (σµ)αβ̇ , while σµ = (1, ~σ) , σ̄µ =
(−1, ~σ) are Pauli matrices. xµ + iyµ is a complexification of spacetime [2]. The helicity
h of the particle is determined by p · y = h. The spin tensor is given by sµν = εµνρσyρpσ,
and it leads to 1
sµνsµν = h
2. The Pauli-Lubanski vector is proportional to the momentum
εµνρσs
νρpσ = (y · p) pµ−p2yµ = hpµ, appropriate for a massless particle of helicity h.
The reformulation of the dynamics in terms of twistors is manifestly SU(2, 2) covariant.
It was believed that twistors and the SO(4, 2)=SU(2, 2) symmetry, interpreted as conformal
symmetry, govern the dynamics of massless particles only, since the momentum pµ of the
form pαβ̇ = λαλ̄β̇ automatically satisfies p
µpµ = 0.
However, recent work has shown that the same twistor ZA =
that describes massless
spinless particles (h = 0) also describes an assortment of other spinless particle dynamical
systems [6][7]. These include massive and interacting particles. The mechanism that avoids
pµpµ = 0 [6][7] is explained following Eq.(6.9) below. The list of systems includes the
following examples worked out explicitly in previous publications and in unpublished notes.
The massless relativistic particle in d = 4 flat Minkowski space.
The massive relativistic particle in d = 4 flat Minkowski space.
The nonrelativistic free massive particle in 3 space dimensions.
The nonrelativistic hydrogen atom (i.e. 1/r potential) in 3 space dimensions.
The harmonic oscillator in 2 space dimensions, with its mass ⇔ an extra dimension.
The particle on AdS4, or on dS4.
The particle on AdS3×S1 or on R× S3.
The particle on AdS2×S2.
The particle on the Robertson-Walker spacetime.
The particle on any maximally symmetric space of positive or negative curvature.
The particle on any of the above spaces modified by any conformal factor.
A related family of other particle systems, including some black hole backgrounds.
In this paper we will discuss these for the case of d = 4 with spin (h 6= 0). It must be
emphasized that while the phase spaces (and therefore dynamics, Hamiltonian, etc.) in these
systems are different, the twistors
µα̇, λα
are the same. For example, the massive particle
phase space (xµ, pµ)massive and the one for the massless particle (x
µ, pµ)massless are not the
same (xµ, pµ) , rather they can be obtained from one another by a non-linear transformation
for any value of the mass parameter m [6], and similarly, for all the other spaces mentioned
above. However, under such “duality” transformations from one system to another, the
twistors for all the cases are the same up to an overall phase transformation
µα̇, λα
massive
µα̇, λα
massless
= · · · =
µα̇, λα
. (1.2)
This unification also shows that all of these systems share the same SO(4, 2)=SU(2, 2)
global symmetry of the twistors. This SU(2, 2) is interpreted as conformal symmetry for the
massless particle phase space, but has other meanings as a hidden symmetry of all the other
systems in their own phase spaces. Furthermore, in the quantum physical Hilbert space, the
symmetry is realized in the same unitary representation of SU(2, 2) , with the same Casimir
eigenvalues (see (7.16,7.17) below), for all the systems listed above.
The underlying reason for such fantastic looking properties cannot be found in one-time
physics (1T-physics) in 3+1 dimensions, but is explained in two-time physics (2T-physics)
[8] as being due to a local Sp(2, R) symmetry. The Sp(2, R) symmetry which acts in phase
space makes position and momentum indistinguishable at any instant and requires one extra
space and one extra time dimensions to implement it, thus showing that the unification relies
on an underlying spacetime in 4+2 dimensions. It was realized sometime ago that in 2T-
physics twistors emerge as a gauge choice [9], while the other systems are also gauge choices
of the same theory in 4+2 dimensions. The 4+2 phase space can be gauge fixed to many
3+1 phase spaces that are distinguishable from the point of view of 1T-physics, without any
Kaluza-Klein remnants, and this accounts for the different Hamiltonians that have a duality
relationship with one another. We will take advantage of the properties of 2T-physics to
build the general twistor transform that relates these systems including spin.
Given that the field theoretic formulation of 2T-physics in 4+2 dimensions yields the
Standard Model of Particles and Forces in 3+1 dimensions as a gauge choice [10], including
spacetime supersymmetry [11], and given that twistors have simplified QCD computations
[12][13], we expect that our twistor methods will find useful applications.
II. TWISTOR LAGRANGIAN
The Penrose twistor description of massless spinning particles requires that the pairs
µα̇, iλ̄α̇
or their complex conjugates (λα, iµ̄
α) be canonical conjugates and satisfy the he-
licity constraint given by
Z̄AZA = λ̄α̇µ
α̇ + µ̄αλα = 2h. (2.1)
Indeed, Eq.(1.1) satisfies this property provided y · p = h. Here we have defined the 4̄ of
SU(2, 2) as the contravariant twistor
Z̄A ≡
Z†η2,2
λ̄α̇ µ̄
, η2,2 =
= SU (2, 2) metric. (2.2)
The canonical structure, along with the constraint Z̄AZA = 2h follows from the following
worldline action for twistors
− 2hÃ
, DτZ
− iÃZA. (2.3)
In the case of h = 0 it was shown that this action emerges as a gauge choice of a more
general action in 2T-physics [6][7]. Later in the paper, in Eq.(4.1) we give the h 6= 0
2T-physics action from which (2.3) is derived as a gauge choice. The derivative part of
this action gives the canonical structure S0 =
dτiZ̄A
λ̄α̇∂τµ
α̇ + µ̄α∂τλα
that requires
µα̇, iλ̄α̇
or their complex conjugates (λα, iµ̄
α) to be canonical conjugates.
The 1-form Ãdτ is a U(1) gauge field on the worldline, DτZ
A is the U(1) gauge covariant
derivative that satisfies δε
for δεÃ = ∂ε/∂τ and δεZ
A = iεZA. The
term 2hÃ is gauge invariant since it transforms as a total derivative under the infinitesimal
gauge transformation. 2hÃ was introduced in [6][7] as being an integral part of the twistor
formulation of the spinning particle action.
Our aim is to show that this action describes not only massless spinning particles, but also
all of the other particle systems listed above with spin. This will be done by constructing
the twistor transform from ZA to the phase space and spin degrees of freedom of these
systems, and claiming the unification of dynamics via the generalized twistor transform.
This generalizes the work of [6][7] which was done for the h = 0 case of the action in (2.3).
We will use 2T-physics as a tool to construct the general twistor transform, so this unification
is equivalent to the unification achieved in 2T-physics.
III. MASSLESS PARTICLE WITH ANY SPIN IN 3+1 DIMENSIONS
In our quest for the general twistor transform with spin, we first discuss an alternative
to the well known twistor transform of Eq.(1.1). Instead of the yµ (τ) that appears in the
complexified spacetime xµ + iyµ we introduce an SL(2, C) bosonic2 spinor vα̇ (τ) and its
complex conjugate v̄α (τ) , and write the general vector yµ in the matrix form as yα̇β =
hvα̇v̄β + ωpα̇β , where ω (τ) is an arbitrary gauge freedom that drops out. Then the helicity
condition y · p = h takes the form v̄pv = 1. Furthermore, we can write λα = pαβ̇vβ̇ since this
automatically satisfies λαλ̄β̇ = pαβ̇ when p
2 = (v̄pv − 1) = 0 are true. With this choice of
variables, the Penrose transform of Eq.(1.1) takes the new form
λα = (pv)α , µ
α̇ = [(−ix̄p+ h) v]α̇ , p2 = (v̄pv − 1) = 0, (3.1)
where the last equation is a set of constraints on the degrees of freedom xµ, pµ, v
α̇, v̄α.
If we insert the twistor transform (3.1) into the action (2.3), the twistor action turns into
the action for the phase space and spin degrees of freedom xµ, pµ, v
α̇, v̄α
ẋµpµ −
p2 + ih
(v̄p)Dτv −Dτv (pv)
− 2hÃ
. (3.2)
where Dτv = v̇ − iÃv is the U(1) gauge covariant derivative and we have included the
Lagrange multiplier e to impose p2 = 0 when we don’t refer to twistors. The equation of
motion for Ã imposes the second constraint v̄pv−1 = 0 that implies U(1) gauge invariance3.
From the global Lorentz symmetry of (3.2), the Lorentz generator is computed via
Noether’s theorem Jµν = xµpν − xνpµ + sµν , with sµν = i
hv̄ (pσ̄µν + σµνp) v. The helic-
2 This is similar to the fermionic case in [5]. The bosonic spinor v can describe any spin h.
3 If this action is taken without the U(1) constraint
Ã = 0
, then the excitations in the v sector describe
an infinite tower of massless states with all helicities from zero to infinity (here we rescale
2hv → v)
Sall spins =
ẋµpµ −
v̄pv̇ − v̇pv
(3.3)
The spectrum coincides with the spectrum of the infinite slope limit of string theory with all helicities
v̄pv. This action has a hidden SU(2, 3) symmetry that includes SU(2, 2) conformal symmetry. This is
explained in the rest of the paper by the fact that this action is a gauge fixed version of a 2T-physics
master action (4.1,5.4) in 4+2 dimensions with manifest SU(2, 3) symmetry. A related approach has been
pursued also in [15]-[18] in 3+1 dimensions in the context of only massless particles. Along with the
manifestly SU(2, 3) symmetric 2T-physics actions, we are proposing here a unified 2T-physics setting for
discussing high spin theories [14] including all the dual versions of the high spin theories related to the
spinning physical systems listed in section (I).
ity is determined by computing the Pauli-Lubanski vector W µ = 1
εµνλσsνλpσ = (hv̄pv) p
The helicity operator hv̄pv reduces to the constant h in the U(1) gauge invariant sector.
The action (3.2) gives a description of a massless particle with any helicity h in terms of
the SL(2, C) bosonic spinors v, v̄. We note its similarity to the standard superparticle action
[20][21] written in the first order formalism. The difference with the superparticle is that the
fermionic spacetime spinor θα̇ of the superparticle is replaced with the bosonic spacetime
spinor vα̇, and the gauge field Ã imposes the U(1) gauge symmetry constraint v̄pv − 1 = 0
that restricts the system to a single, but arbitrary helicity state given by h.
Just like the superparticle case, our action has a local kappa symmetry with a bosonic
local spinor parameter κα (τ), namely
α̇ = p̄α̇βκβ, δκxµ =
((δκv̄)σµv − v̄σµ (δκv)) , (3.4)
µ = 0, δκe = −ih
κ̄ (Dτv)−
, δκÃ = 0. (3.5)
These kappa transformations mix the phase space degrees of freedom (x, p) with the spin
degrees of freedom v, v̄. The transformations δκxµ, δκe are non-linear.
Let us count physical degrees of freedom. By using the kappa and the τ -reparametrization
symmetries one can choose the lightcone gauge. From phase space xµ, pµ there remains 3
positions and 3 momentum degrees of freedom. One of the two complex components of
vα̇ is set to zero by using the kappa symmetry, so vα̇ =
. The phase of the remaining
component is eliminated by choosing the U(1) gauge, and finally its magnitude is fixed by
solving the constraint v̄pv − 1 = 0 to obtain vα̇ = (p+)−1/2
. Therefore, there are no
independent physical degrees of freedom in v. The remaining degrees of freedom for the
particle of any spin are just the three positions and momenta, and the constant h that
appears in sµν . This is as it should be, as seen also by counting the physical degrees of
freedom from the twistor point of view. When we consider the other systems listed in the
first section, we should expect that they too are described by the same number of degrees
of freedom since they will be obtained from the same twistor, although they obey different
dynamics (different Hamiltonians) in their respective phase spaces.
The lightcone quantization of the the massless particle systems described by the actions
(3.2,3.3) is performed after identifying the physical degrees of freedom as discussed above.
The lightcone quantum spectrum and wavefunction are the expected ones for spinning mass-
less particles, and agree with their covariant quantization given in [15]-[19].
IV. 2T-PHYSICS WITH SP(2, R) , SU(2, 3) AND KAPPA SYMMETRIES
The similarity of (3.2) to the action of the superparticle provides the hint for how to
lift it to the 2T-physics formalism, as was done for the superparticle [22][9] and the twistor
superstring [23][24]. This requires lifting 3+1 phase space (xµ, pµ) to 4+2 phase space
XM , PM
and lifting the SL(2, C) spinors v, v̄ to the SU(2, 2) spinors VA, V̄
A. The larger
set of degrees of freedom XM , PM , VA, V̄
A that are covariant under the global symmetry
SU(2, 2) =SO(4, 2) , include gauge degrees of freedom, and are subject to gauge symmetries
and constraints that follow from them as described below.
The point is that the SU(2, 2) invariant constraints on XM , PM , VA, V̄
A have a wider set
of solutions than just the 3+1 system of Eq.(3.2) we started from. This is because 3+1 di-
mensional spin & phase space has many different embeddings in 4+2 dimensions, and those
are distinguishable from the point of view of 1T-physics because target space “time” and
corresponding “Hamiltonian” are different in different embeddings, thus producing the dif-
ferent dynamical systems listed in section (I). The various 1T-physics solutions are reached
by simply making gauge choices. One of the gauge choices for the action we give below in
Eq.(4.1) is the twistor action of Eq.(2.3). Another gauge choice is the 4+2 spin & phase
space action in terms of the lifted spin & phase space XM , PM , VA, V̄
A as given in Eq.(5.4).
The latter can be further gauge fixed to produce all of the systems listed in section (I)
including the action (3.2) for the massless spinning particle with any spin. All solutions still
remember that there is a hidden global symmetry SU(2, 2) =SO(4, 2) , so all systems listed
in section (I) are realizations of the same unitary representation of SU(2, 2) whose Casimir
eigenvalues will be given below.
For the 4 + 2 version of the superparticle [22] that is similar to the action in (5.4), this
program was taken to a higher level in [9] by embedding the fermionic supercoordinates in
the coset of the supergroup SU(2, 2|1) /SU(2, 2)×U(1). We will follow the same route here,
and embed the bosonic SU(2, 2) spinors VA, V̄
A in the left coset SU(2, 3) /SU(2, 2)×U(1) .
This coset will be regarded as the gauging of the group SU(2, 3) under the subgroup
[SU (2, 2)× U (1)]
from the left side. Thus the most powerful version of the action that
reveals the global and gauge symmetries is obtained when it is organized in terms of the
XMi (τ), g (τ) and Ã (τ) degrees of freedom described as
4+2 phase space
XM (τ)
PM (τ)
≡ XMi (τ) , i = 1, 2, doublets of Sp (2, R) gauge symmetry,
group element g (τ) ⊂ SU (2, 3) subject to [SU (2, 2)× U(1) ]L × U(1)L+R gauge symmetry.
We should mention that the h = 0 version of this theory, and the corresponding twistor prop-
erty, was discussed in [6], by taking g (τ) ⊂SU(2, 2) and dropping all of the U(1)’s. So, the
generalized theory that includes spin has the new features that involves SU(2, 2) →SU(2, 3)
and the U(1) structures. The action has the following form
XNj ηMN + Tr
(iDτg) g
− 2hÃ
, (4.1)
where εij =
is the antisymmetric Sp(2, R) metric, and DτX
i = ∂τX
the Sp(2, R) gauge covariant derivative, with the 3 gauge potentials Aij = εikA
For SU(2, 3) the group element is pseudo-unitary, g−1 = (η2,3) g
† (η2,3)
, where η2,3 is the
SU(2, 3) metric η2,3 =
. The covariant derivative Dτg is given by
Dτg = ∂τg − iÃ [q, g] , q =
14×4 0
 (4.2)
where the generator of U(1)L+R is proportional to the 5×5 traceless matrix
q ∈ u(1) ∈ su(2, 3)L+R . The last term of the action −2hÃ, which is also the last term
of the action (2.3), is invariant under the U(1)L+R since it transforms to a total derivative.
Finally, the 4× 4 traceless matrix (L) BA ∈su(2, 2) ∈su(2, 3) that appears on the left side of
g (or right side of g−1) is
(L) BA ≡
LMN , LMN = εijXMi X
j = X
MPN −XNPM . (4.3)
where ΓMN =
ΓM Γ̄N − ΓN Γ̄M
are the 4×4 gamma-matrix representation of the 15 gen-
erators of SU(2, 2). A detailed description of these gamma matrices is given in [11].
The symmetries of actions of this type for any group or supergroup g were discussed
in [9][23][24][7]. The only modification of that discussion here is due to the inclusion of
the U(1) gauge field Ã. In the absence of the Ã coupling the global symmetry is given
by the transformation of g (τ) from the right side g (τ) → g (τ) gR where gR ⊂SU(2, 3)R.
However, in our case, the presence of the coupling with the U(1)L+R charge q breaks the
global symmetry down to the (SU(2, 2)×U(1))R subgroup that acts on the right side of g.
So the global symmetry is given by
global: g (τ) → g (τ) hR, hR ∈ [SU(2, 2)× U(1)]R ⊂ SU(2, 3)R. (4.4)
Using Noether’s theorem we deduce the conserved global charges as the [SU(2, 2)×U(1)]R
components of the the following SU(2, 3)R Lie algebra valued matrix J(2,3)
J(2,3) = g
, J2,3 = η2,3 (J2,3)
(η2,3)
, (4.5)
The traceless 4× 4 matrix (J ) BA =
ΓMNJMN is the conserved SU(2, 2) =SO(4, 2) charge
and J0 is the conserved U(1) charge. Namely, by using the equations of motion one can
verify ∂τ (J ) BA = 0 and ∂τJ0 = 0. The spinor charges jA, j̄A are not conserved4 due to the
coupling of Ã. As we will find out later in Eq.(6.8), jA is proportional to the twistor
J0ZA, (4.6)
up to an irrelevant gauge transformation. It is important to note that J and J0 are invariant
on shell under the gauge symmetries discussed below. Therefore they generate physical
symmetries [SU(2, 2)×U(1)]R under which all gauge invariant physical states are classified.
The local symmetries of this action are summarized as
Sp (2, R)×
SU (2, 2) 3
kappa
kappa U (1)
(4.7)
The Sp(2, R) is manifest in (4.1). The rest corresponds to making local SU(2, 3) transfor-
mations on g (τ) from the left side g (τ) → gL (τ) g (τ) , as well as transforming XMi =
XM , PM
as vectors with the local subgroup SU(2, 2)L =SO(4, 2) , and A
ij under the
kappa. The 3/4 kappa symmetry which is harder to see will be discussed in more detail
below. These symmetries coincide with those given in previous discussions in [9][23][24][7]
despite the presence of Ã. The reason is that the U(1)L+R covariant derivative Dτg in
Eq.(4.2) can be replaced by a purely U(1)R covariant derivative Dτg = ∂τg + igqÃ because
the difference drops out in the trace in the action (4.1). Hence the symmetries on left side
of g (τ) → gL (τ) g (τ) remain the same despite the coupling of Ã.
We outline the roles of each of these local symmetries. The Sp(2, R) gauge symmetry
can reduce XM , PM to any of the phase spaces in 3+1 dimensions listed in section (I). This
4 In the high spin version of (4.1) with Ã = 0, the global symmetry is SU(2, 3)R and jA, j̄
A are conserved.
is the same as the h = 0 case discussed in [6]. The [SU(2, 2)×U(1)]L gauge symmetry can
reduce g (τ) ⊂SU(2, 3) to the coset g → t (V ) ∈SU(2, 3) /[SU(2, 2)×U(1)]L parameterized
by the SU(2, 2)×U(1) spinors
VA, V̄
as shown in Eq.(5.3). The remaining 3/4 kappa
symmetry, whose action is shown in Eq.(5.15), can remove up to 3 out of the 4 parameters
in the VA. The U(1)L+R symmetry can eliminate the phase of the remaining component
in V . Finally the constraint due to the equation of motion of Ã fixes the magnitude of V .
In terms of counting, there remains only 3 position and 3 momentum physical degrees of
freedom, plus the constant h, in agreement with the counting of physical degrees of freedom
of the twistors.
It is possible to gauge fix the symmetries (4.7) partially to exhibit some intermediate
covariant forms. For example, to reach the SL(2, C) covariant massless particle described by
the action (3.2) from the 2T-physics action above, we take the massless particle gauge by
using two out of the three Sp(2, R) gauge parameters to rotate the M = +′ doublet to the
(τ) =
, and solving explicitly two of the Sp(2, R) constraints X2 = X ·P = 0
XM = (
xµ (τ)), PM = (
x · p ,
pµ (τ)). (4.8)
This is the same as the h = 0 massless case in [6]. There is a tau reparametrization gauge
symmetry as a remnant of Sp(2, R) . Next, the [SU(2, 2)×U(1)]L gauge symmetry reduces
g (τ) → t (V ) written in terms of
VA, V̄
as given in Eq.(5.3), and the 3/4 kappa symmetry
reduces the SU(2, 2) spinor VA →
to the two components SL(2, C) doublet vα̇, with a
leftover kappa symmetry as discussed in Eqs.(3.4-3.5). The gauge fixed form of g is then
g = exp
2hvα̇
0 0 0
2hv̄α 0
1 hvα̇v̄β
2hvα̇
0 1 0
2hv̄α 1
∈ SU (2, 3) . (4.9)
The inverse g−1 = (η2,3) g
† (η2,3)
is given by replacing v, v̄ by (−v) , (−v̄) . Inserting the
gauge fixed forms of X,P, g (4.8,4.9) into the action (4.1) reduces it to the massless spin-
ning particle action (3.2). Furthermore, inserting these X,P, g into the expression for the
current in (4.5) gives the conserved SU(2, 2) charges J (see Eqs.(5.9,5.20)) which have the
significance of the hidden conformal symmetry of the gauge fixed action (3.2). This hidden
symmetry is far from obvious in the form (3.2), but it is straightforward to derive from the
2T-physics action as we have just outlined.
Partial or full gauge fixings of (4.1) similar to (4.8,4.9) produce the actions, the hidden
SU(2, 2) symmetry, and the twistor transforms with spin of all the systems listed in section
(I). These were discussed for h = 0 in [6], and we have now shown how they generalize
to any spin h 6= 0, with further details below. It is revealing, for example, to realize that
the massive spinning particle has a hidden SU(2, 2) “mass-deformed conformal symmetry”,
including spin, not known before, and that its action can be reached by gauge fixing the
action (4.1), or by a twistor transform from (2.3). The same remarks applied to all the other
systems listed in section (I) are equally revealing. For more information see our related paper
Through the gauge (4.8,4.9), the twistor transform (3.1), and the massless particle action
(3.2), we have constructed a bridge between the manifestly SU(2, 2) invariant twistor action
(2.3) for any spin and the 2T-physics action (4.1) for any spin. This bridge will be made
much more transparent in the following sections by building the general twistor transform.
V. 2T-PHYSICS ACTION WITH XM , PM , VA, V̄
A IN 4+2 DIMENSIONS
We have hinted above that there is an intimate relation between the 2T-physics action
(4.1) and the twistor action (2.3). In fact the twistor action is just a gauged fixed version
of the more general 2T-physics action (4.1). Using the local SU(2, 2) =SO(4, 2) and local
Sp(2, R) symmetries of the general action (4.1) we can rotateXM (τ) , PM (τ) to the following
form that also solves the Sp(2, R) constraints Xi ·Xj = X2 = P 2 = X · P = 0 [6][7]
XM = (
0), PM = (
0). (5.1)
This completely eliminates all phase space degrees of freedom. We are left with the gauge
fixed action Sh =
(Dτg) g
− 2hÃ
, where (iL) → 1
′−′, and
′−′ = 1. Due to the many zero entries in the 4×4 matrix Γ−′− [6], only one column
from g in the form
and one row from g−1 in the form
Z̄A,−Z̄5
can contribute in
the trace, and therefore the action becomes Sh =
iZ̄AŻA − iZ̄5Ż5 + Ã
Z̄5Z5 − 2h
Here Z̄5Ż5 drops out as a total derivative since the magnitude of the complex number Z5 is
a constant Z̄5Z5 = 2h. Furthermore, we must take into account Z̄
AZA − Z̄5Z5 = 0 which
is an off-diagonal entry in the matrix equation g−1g = 1. Then we see that the 2T-physics
action (4.1) reduces to the twistor action (2.3) with the gauge choice (5.1)5.
Next let us gauge fix the 2T-physics action (4.1) to a manifestly SU(2, 2) =SO(4, 2)
invariant version in flat 4+2 dimensions, in terms of the phase space & spin degrees of
freedom XM , PM , VA, V̄
A. For this we use the [SU(2, 2)×U(1)]left symmetry to gauge fix g
gauge fix: g → t (V ) ∈ SU(2, 3)
[SU(2, 2)× U(1)]left
(5.2)
The coset element t (V ) is parameterized by the SU(2, 2) spinor V and its conjugate V̄ =
V †η2,2 and given by the 5×5 SU(2, 3) matrix6
t (V ) =
1− 2hV V̄
)−1/2
1− 2hV̄ V
)−1/2
2hV̄ 1
 . (5.3)
The factor 2h is inserted for a convenient normalization of V. Note that the first matrix
commutes with the second one, so it can be written in either order. The inverse of the group
element is t−1 (V ) = (η2,3) t
† (η2,3)
= t (−V ) , as can be checked explicitly t (V ) t (−V ) = 1.
Inserting this gauge in (4.1) the action becomes
Ẋ · P −
AijXi ·Xj −
ΩMNLMN − 2hÃ
V̄ LV
1− 2hV̄ V
(5.4)
XNj ηMN − 2hÃ
V̄ LV
1− 2hV̄ V
(5.5)
where
i = ∂τX
j − ΩMNXiN (5.6)
is a covariant derivative for local Sp(2, R) as well as local SU(2, 2) =SO(4, 2) but with a
composite SO(4, 2) connection ΩMN (V (τ)) given conveniently in the following forms
ΩMNΓMN =
(i∂τ t) t
SU(2,2)
ΩMNLMN = −Tr
(i∂τ t) t
. (5.7)
Thus, Ω is the SU(2, 2) projection of the SU(2, 3) Cartan connection and given explicitly as
ΩMNΓMN = 2h
V̇ − V V̄ V̇
V̄ − V
V̇ − V̄ V̇ V
1− 2hV̄ V
1− 2hV̄ V
) + h
V̄ V̇ − V̇ V
1− 2hV̄ V
) (5.8)
5 In the high spin version of (4.1) without Ã (see footnote (3)), we replace Z5 = e
Z̄AZA and after
dropping a total derivative, the twistor equivalent becomes Sall spins =
iZ̄AŻA + Z̄Zφ̇
. For a
more covariant version that displays the SU(2, 3) global symmetry, we introduce a new U(1) gauge field
for the overall phase of
and write Sall spins =
iZ̄AŻA − iZ̄5Ż5 + B̃
Z̄AZA − Z̄5Z5
6 Arbitrary fractional powers of the matrix
1− 2hV V̄
are easily computed by expanding in a series and
then resuming to obtain
1− 2hV V̄
= 1+ V V̄
1− 2hV̄ V
)γ − 1
/V̄ V.
The action (5.4,5.5) is manifestly invariant under global SU(2, 2) =SO(4, 2) rotations, and
under local U (1) phase transformations applied on VA, V̄
A. The conserved global symmetry
currents J and J0 can be derived either directly from (5.4) by using Noether’s theorem, or
by inserting the gauge fixed form of g → t (V ) into Eq.(4.5)7 J(2,3) = t−1
1− 2hV V̄
1− 2hV V̄
J0, J0 =
2hV̄ LV
1− 2hV̄ V
(5.9)
1− 2hV V̄
LV 1√
1− 2hV̄ V
(5.10)
According to the equation of motion for Ã that follows from the action (5.4) we must have
the following constraint (this means U(1) gauge invariant physical sector)
V̄ LV
1− 2hV̄ V
= 1. (5.11)
Therefore, in the physical sector the conserved [SU(2, 2)×U(1)]right charges take the form
physical sector: J0 = 2h, J =
1− 2hV V̄
1− 2hV V̄
. (5.12)
Let us now explain the local kappa symmetry of the action (5.4,5.5). The action (5.4) is
still invariant under the bosonic local 3/4 kappa symmetry inherited from the action (4.1).
The kappa transformations of g (τ) in the general action (5.4) correspond to local coset
elements exp
∈SU(2, 3)left/[SU(2, 2)×U(1)]left with a special form of the spinor KA
KA = Xi ·
Γκi (τ)
, (5.13)
with κiA (τ) two arbitrary local spinors8. Now that g has been gauge fixed g → t (V ), the
kappa transformation must be taken as the naive kappa transformation on g followed by a
[SU(2, 2)×U(1)]left gauge transformation which restores the gauge fixed form of t (V )
t (V ) → t (V ′) =
Tr (ω)
t (V ) (5.14)
The SU(2, 2) part of the restoring gauge transformation must also be applied on XM , PM .
Performing these steps we find the infinitesimal version of this transformation [22]
δκV =
1− 2hV V̄
1− 2hV̄ V
, δκX
i = ω
MNXiN , δκA
ij = see below, (5.15)
7 In the high spin version (Ã = 0) the conserved charges include jA as part of SU(2, 3)R global symmetry.
It is then also convenient to rescale
2hV → V in Eqs.(5.3-5.10) to eliminate an irrelevant constant.
8 In this special form only 3 out of the 4 components of KA are effectively independent gauge parameters.
This can be seen easily in the special frame for XM , PM given in Eq.(5.1).
where ωMN (K, V ) has the same form as ΩMN in Eq.(5.8) but with V̇ replaced by the
δκV given above. The covariant derivative D̂τX
i in Eq.(5.6) is covariant under the local
SU(2, 2) transformation with parameter ωMN (K, V ) (this is best seen from the projected
Cartan connection form Ω = [(i∂τ t) t
−1]SU(2,2)). Therefore, the kappa transformations (5.15)
inserted in (5.5) give
δκSh =
Xi ·Xj + iT r
(Dτ t) t
. (5.16)
In computing the second term the derivative terms that contain ∂τK have dropped out in
the trace. Using Eq.(5.13) we see that
LK = 1
εliXMl X
XLj ΓMNΓLκ
j (5.17)
εliXMl X
j (ΓMNL + ηNLΓM − ηMLΓN )κl (5.18)
εliXi ·Xj
Xl · Γκj
(5.19)
The completely antisymmetric XMi X
l ΓMNL term in the second line vanishes since i, j, l
can only take two values. The crucial observation is that the remaining term in LK is
proportional to the dot products Xi ·Xj. Therefore the second term in (5.16) is cancelled by
the first term by choosing the appropriate δκA
ij in Eq.(5.16), thus establishing the kappa
symmetry.
The local kappa transformations (5.15) are also a symmetry of the global SU(2, 3)R
charges δκJ = δκJ0 = δκjA = 0 provided the constraints Xi · Xj = 0 are used. Hence
these charges are kappa invariant in the physical sector.
We have established the global SO(4, 2) and local Sp(2, R)× (3/4 kappa)×U(1) symme-
tries of the phase space action (5.4) in 4+2 dimensions. From it we can derive all of the phase
space actions of the systems listed in section (I) by making various gauge choices for the
local Sp(2, R)× (3/4 kappa)×U(1) symmetries. This was demonstrated for the spinless case
h = 0 in [6]. The gauge choices for XM , PM discussed in [6] now need to be supplemented
with gauge choices for VA, V̄
A by using the kappa×U(1) local symmetries.
Here we demonstrate the gauge fixing described above for the massless particle of any spin
h. The kappa symmetry effectively has 3 complex gauge parameters as explained in footnote
(8). If the kappa gauge is fixed by using two of its parameters we reach the following forms
, V̄ A → (0 v̄α) , V̄ V → 0,
1− 2hV V̄
)−1/2 →
. (5.20)
By inserting this gauge fixed form of V, and the gauge fixed form of X,P given in Eq.(4.8),
into the action (5.4) we immediately recover the SL(2, C) covariant action of Eq.(3.2). The
U(1) gauge symmetry is intact. The kappa symmetry of the action of Eq.(3.2) discussed in
Eqs.(3.4,3.5) is the residual 1/4 kappa symmetry of the more general action ((5.4).
For other examples of gauge fixing that generates some of the systems in the list of section
(I) see our related paper [1].
VI. GENERAL TWISTOR TRANSFORM (CLASSICAL)
The various formulations of spinning particles described above all contain gauge degrees
of freedom of various kinds. However, they all have the global symmetry SU(2, 2)=SO(4, 2)
whose conserved charges J BA are gauge invariant in all the formulations. The most sym-
metric 2T-physics version gave the J BA as embedded in SU(2, 3)R in the SU(2, 2) projected
form in Eq.(4.5)
SU(2,2)
. (6.1)
Since this is gauge invariant, when gauge fixed, it must agree with the Noether charges
computed in any version of the theory. So we can equate the general phase space version of
Eq.(5.9) with the twistor version that follows from the Noether currents of (2.3) as follows
J = Z(h)Z̄(h) − 1
Z(h)Z̄(h)
1− 2hV V̄
1− 2hV V̄
J0 (6.2)
The trace corresponds to the U(1) charge J0 = Tr
Z(h)Z̄(h)
J0 = Z
(h)Z̄(h) =
1− 2hV V̄
1− 2hV V̄
. (6.3)
In the case of h = 0 this becomes
Z(0)Z̄(0) = L. (6.4)
Therefore the equality (6.3) is solved up to an irrelevant phase by
Z(h) =
1− 2hV V̄
Z(0). (6.5)
By inserting (6.4) into the constraint (5.11) we learn a new form of the constraints
V̄ Z(0) =
1− 2hV̄ V , V̄ Z(h) = 1. (6.6)
In turn, this implies
Z(0) =
1− 2hV̄ V
(6.7)
which is consistent9 with Z(0)Z̄(0) = L , and its vanishing trace Z̄(0)Z(0) = 0 since LL = 0
(due to X2 = P 2 = X · P = 0). Putting it all together we then have
Z(h) =
1− 2hV V̄
1− 2hV̄ V
V. (6.8)
We note that this Z(h) is proportional to the non-conserved coset part of the SU(2, 3) charges
J2,3, that is jA =
(h) given in Eqs.(4.5,4.6) or (5.10), when g and L are replaced by
their gauge fixed forms, and use the constraint10 J0 = 2h.
The key for the general twistor transform for any spin is Eq.(6.5), or equivalently (6.8).
The general twistor transform between Z(0) and XM , PM which satisfies Z(0)Z̄(0) = L is
already given in [6] as
Z(0) =
= −i X
, λ(0)α λ̄
X+P µ −XµP+
(σµ)αβ̇ . (6.9)
Note that (X+P µ −XµP+) is compatible with the requirement that any SL(2, C) vector
constructed as λ
must be lightlike. This property is satisfied thanks to the Sp(2, R)
constraints X2 = P 2 = X · P = 0 in 4+2 dimensions, thus allowing a particle of any
mass in the 3 + 1 subspace (since P µPµ is not restricted to be lightlike). Besides satisfying
Z(0)Z̄(0) = L, this Z(0) also satisfies Z̄(0)Z(0) = 0, as well as the canonical properties of
twistors. Namely, Z(0) has the property [6]
dτ Z̄(0)∂τZ
(0) =
dτ ẊMPM . (6.10)
From here, by gauge fixing the Sp(2, R) gauge symmetry, we obtain the twistor transforms
for all the systems listed in section (I) for h = 0 directly from Eq.(6.9), as demonstrated
in [6]. All of that is now generalized at once to any spin h through Eq.(6.5). Hence (6.5)
together with (6.9) tell us how to construct explicitly the general twistor Z
A in terms
9 To see this, we note that Eqs.(6.4,6.6) lead to LV V̄ L
1−2hV̄ V =
Z(0)Z̄(0)V V̄ Z(0)Z̄(0)
1−2hV̄ V = Z
(0)Z̄(0) = L.
10 For the high spin version (Ã = 0) we don’t use the constraint. Instead, we use Z(h) = 1√
1−2hV V̄
Z(0) only
in its form (6.5), and note that, after using Eq.(6.4), the jA in Eq.(5.10) takes the form jA =
1−2hV̄ V
, and it is possible to rescale h away everywhere
2hV → V.
of spin & phase space degrees of freedom XM , PM , VA, V̄
A. Then the Sp(2, R) and kappa
gauge symmetries that act on XM , PM , VA, V̄
A can be gauge fixed for any spin h, to give
the specific twistor transform for any of the systems under consideration.
We have already seen in Eq.(6.2) that the twistor transform (6.5) relates the conserved
SU(2, 2) charges in twistor and phase space versions. Let us now verify that (6.5) provides
the transformation between the twistor action (2.3) and the spin & phase space action (5.4).
We compute the canonical structure as follows
dτ Z̄(h)∂τZ
(h) =
dτ Z̄(0)
1− 2hV V̄
1− 2hV V̄
(6.11)
Z̄(0) 1√
1−2hV V̄
1−2hV V̄
+Z̄(0) 1
1−2hV V̄ ∂τZ
(6.12)
Ẋ · P + Tr
(i∂τ t) t
(6.13)
The last form is the canonical structure of spin & phase space as given in (5.4). To prove this
result we used Eq.(6.10), footnote (6), and the other properties of Z(0) including Eqs.(6.4-
6.7), as well as the constraints X2 = P 2 = X · P = 0, and dropped some total derivatives.
This proves that the canonical properties of Z(h) determine the canonical properties of spin
& phase space degrees of freedom and vice versa.
Then, including the terms that impose the constraints, the twistor action (2.3) and the
phase space action (5.4) are equivalent. Of course, this is expected since they are both gauge
fixed versions of the master action (4.1), but is useful to establish it also directly via the
general twistor transform given in Eq.(6.5).
VII. QUANTUM MASTER EQUATION, SPECTRUM, AND DUALITIES
In this section we derive the quantum algebra of the gauge invariant observables J BA
and J0 which are the conserved charges of [SU(2, 2)×U(1)]R. Since these are gauge invariant
symmetry currents they govern the system in any of its gauge fixed versions, including in
any of its versions listed in section (I). From the quantum algebra we deduce the constraints
among the physical observables J BA ,J0 and quantize the theory covariantly. Among other
things, we compute the Casimir eigenvalues of the unitary irreducible representation of
SU(2, 2) which classifies the physical states in any of the gauge fixed version of the theory
(with the different 1T-physics interpretations listed in section (I)).
The simplest way to quantize the theory is to use the twistor variables, and from them
compute the gauge invariant properties that apply in any gauge fixed version. We will apply
the covariant quantization approach, which means that the constraint due to the U(1) gauge
symmetry will be applied on states. Since the quantum variables will generally not satisfy
the constraints, we will call the quantum twistors in this section ZA, Z̄
A to distinguish them
from the classical Z
A , Z̄
(h)A of the previous sections that were constrained at the classical
level. So the formalism in this section can also be applied to the high spin theories (discussed
in several footnotes up to this point in the paper) by ignoring the constraint on the states.
According to the twistor action (2.3) ZA and iZ̄
A (or equivalently λα and iµ̄
α) are canon-
ical conjugates. Therefore the quantum rules (equivalent to spin & phase space quantum
rules) are
ZA, Z̄
= δ BA . (7.1)
These quantum rules, as well as the action, are manifestly invariant under SU(2, 2) . In
covariant SU(2, 2) quantization the Hilbert space contains states which do not obey the
U(1) constraint on the twistors. At the classical level the constraint was J0 = Z̄Z = 2h,
but in covariant quantization this is obeyed only by the U(1) gauge invariant subspace of
the Hilbert space which we call the physical states. The quantum version of the constraint
requires Ĵ0 as a Hermitian operator applied on states (we write it as Ĵ0 to distinguish it
from the classical version)
Ĵ0 =
A + Z̄AZA
, Ĵ0|phys〉 = 2h|phys〉. (7.2)
The operator Ĵ0 has non-trivial commutation relations with ZA, Z̄
A which follow from the
basic commutation rules above
Ĵ0, ZA
= −ZA,
Ĵ0, Z̄
= Z̄ A. (7.3)
By rearranging the orders of the quantum operators ZAZ̄
A = Z̄AZA+4 we can extract from
(7.2) the following relations
Z̄Z = Ĵ0 − 2, T r
= Ĵ0 + 2. (7.4)
Furthermore, by using Noether’s theorem for the twistor action (2.3) we can derive the 15
generators of SU(2, 2) in terms of the twistors and write them as a traceless 4 × 4 matrix
J BA at the quantum level as follows
J BA = ZAZ̄B −
δ BA =
ZZ̄ −
Ĵ0 + 2
. (7.5)
In this expression the order of the quantum operators matters and gives rise to the shift
J0 → Ĵ0 + 2 in contrast to the corresponding classical expression. The commutation rules
among the generators J BA and the ZA, Z̄A are computed from the basic commutators (7.1),
J BA , ZC
= −δ BC ZA +
J BA , Z̄D
= δ DA Z̄
B − 1
Z̄D δ BA (7.6)
J BA ,J DC
= δ DA J BC − δ BC J DA ,
Ĵ0,J BA
= 0. (7.7)
We see from these that the gauge invariant observables J BA satisfy the SU(2, 2) Lie algebra,
while the ZA, Z̄
A transform like the quartets 4, 4̄ of SU(2, 2) . Note that the operator Ĵ0
commutes with the generators J BA , therefore J BA is U(1) gauge invariant, and furthermore
Ĵ0 must be a function of the Casimir operators of SU(2, 2) . When Ĵ0 takes the value 2h
on physical states, then the Casimir operators also will have eigenvalues on physical states
which determine the SU(2, 2) representation in the physical sector.
From the quantum rules (7.3), it is evident that the U(1) generator Ĵ0 can only have
integer eigenvalues since it acts like a number of operator. More directly, through Eq.(7.4)
it is related to the number operator Z̄Z. Therefore the theory is consistent at the quantum
level (7.2) provided 2h is an integer.
Let us now compute the square of the matrix J BA . By using the form (7.5) we have
(JJ ) =
ZZ̄ − Ĵ0+2
ZZ̄ − Ĵ0+2
= ZZ̄ZZ̄ − 2 Ĵ0+2
ZZ̄ +
Ĵ0+2
where we have used
Ĵ0, ZAZ̄
= 0. Now we elaborate
ZZ̄ZZ̄
Ĵ0 − 2
Z̄B =
Ĵ0 − 1
B where
we first used (7.4) and then (7.3). Finally we note from (7.5) that ZAZ̄
B = J BA + Ĵ0+24 δ
Putting these observations together we can rewrite the right hand side of (JJ ) in terms of
J and Ĵ0 as follows11
(JJ ) =
Ĵ20 − 4
. (7.8)
11 A similar structure at the classical level can be easily computed by squaring the expression for J in Eq.(6.2)
and applying the classical constraint J0 = Z̄
AZA = 2h. This yields the classical version J CA J BC =
J BA + 316J
A = hJ BA + 34h
2δ BA , which is different than the quantum equation (7.8). Thus, the
quadratic Casimir at the classical level is computed as C2 =
J20 = 3h
2 which is different than the
quantum value in (7.16).
This equation is a constraint satisfied by the global [SU(2, 2)×U(1)]R charges J BA , Ĵ0 which
are gauge invariant physical observables. It is a correct equation for all the states in the
theory, including those that do not satisfy the U(1) constraint (7.2). We call this the
quantum master equation because it will determine completely all the SU(2, 2) properties of
the physical states for all the systems listed in section (I) for any spin.
By multiplying the master equation with J and using (7.8) again we can compute JJJ .
Using this process repeatedly we find all the powers of the matrix J
(J )n = αnJ + βn, (7.9)
where
αn(Ĵ0) =
Ĵ0 − 1
Ĵ0 − 2
Ĵ0 + 2
, (7.10)
βn(Ĵ0) =
Ĵ20 − 4
αn−1(Ĵ0). (7.11)
Remarkably, these formulae apply to all powers, including negative powers of the matrix J .
Using this result, any function of the matrix J constructed as a Taylor series takes the form
f (J ) = α
(7.12)
where
Ĵ0 − 1
Ĵ0 − 2
Ĵ0 + 2
, (7.13)
Ĵ0 − 1
Ĵ0 + 2
Ĵ0 − 2
Ĵ0 − 2
Ĵ0 + 2
 . (7.14)
We can compute all the Casimir operators by taking the trace of J n in Eq.(7.9), so we
find12
Cn(Ĵ0) ≡ Tr (J )n = 4βn(Ĵ0) =
Ĵ20 − 4
αn−1(Ĵ0). (7.15)
In particular the quadratic, cubic and quartic Casimir operators of SU(2, 2) =SO(6, 2) are
computed at the quantum level as
C2(Ĵ0) =
Ĵ20 − 4
, C3(Ĵ0) =
Ĵ20 − 4
Ĵ0 − 4
, (7.16)
C4(Ĵ0) =
Ĵ20 − 4
7Ĵ20 − 32Ĵ0 + 52
. (7.17)
12 Other definitions of Cncould differ from ours by normalization or linear combinations of the Tr (J n).
The eigenvalue of the operator Ĵ0 on physical states Ĵ0|phys〉 = 2h|phys〉 completely fixes
the unitary SU(2, 2) representation that classifies the physical states, since the most general
representation of SO(4, 2) is labeled by the three independent eigenvalues of C2, C3 and C4.
Obviously, this result is a special representation of SU(2, 2) since all the Casimir eigenvalues
are determined in terms of a single half integer number h. Therefore we conclude that all of
the systems listed in section (I) share the very same unitary representation of SU(2, 2) with
the same Casimir eigenvalues given above.
In particular, for spinless particles
Ĵ0 → h = 0
we obtain C2 = −3, C3 = 6, C4 = −394 ,
which is the unitary singleton representation of SO(4, 2) =SU(2, 2). This is in agreement
with previous covariant quantization of the spinless particle in any dimension directly in
phase space in d + 2 dimensions, which gave for the SO(d, 2) Casimir the eigenvalue as
MN → 1 − d2/4 on physical states that satisfy X2 = P 2 = X · P = 0 [8].
So, for d = 4 we get C2 = −3 in agreement with the quantum twistor computation above.
Note that the classical computation either in phase space or twistor space would give the
wrong answer C2 = 0 when orders of canonical conjugates are ignored and constraints used
classically.
Of course, having the same SU(2, 2) Casimir eigenvalue is one of the infinite number of
duality relations among these systems that follow from the more general twistor transform
or the master 2T-physics theory (4.1). All dualities of these systems amount to all quantum
functions of the gauge invariants J BA that take the same gauge invariant values in any of
the physical Hilbert spaces of the systems listed in section (I).
All the physical information on the relations among the physical observables is already
captured by the quantum master equation (7.8), so it is sufficient to concentrate on it. The
predicted duality, including these relations, can be tested at the quantum level by computing
and verifying the equality of an infinite number of matrix elements of the master equation
between the dually related quantum states for the systems listed in section (I). In the
case of the Casimir operators Cn the details of the individual states within a representation
is not relevant, so that computation whose result is given above is among the simplest
computations that can be performed on the systems listed in section (I) to test our duality
predictions. This test was performed successfully for h = 0 at the quantum level for some
of these systems directly in their own phase spaces [26], verifying for example, that the free
massless particle, the hydrogen atom, the harmonic oscillator, the particle on AdS spaces,
all have the same Casimir eigenvalues C2 = −3, C3 = 6, C4 = −394 at the quantum level.
Much more elaborate tests of the dualities can be performed both at the classical and
quantum levels by computing any function of the gauge invariant J BA and checking that it
has the same value when computed in terms of the spin & phase space of any of the sys-
tems listed in section (I). At the quantum level all of these systems have the same Casimir
eigenvalues of the Cn for a given h. So their spectra must correspond to the same unitary
irreducible representation of SU(2, 2) as seen above. But the rest of the labels of the repre-
sentation correspond to simultaneously commuting operators that include the Hamiltonian.
The Hamiltonian of each system is some operator constructed from the observables J BA ,
and so are the other simultaneously diagonalizable observables. Therefore, the different
systems are related to one another by unitary transformations that sends one Hamiltonian
to another, but staying within the same representation. These unitary transformations are
the quantum versions of the gauge transformations of Eq.(4.7), and so they are the duality
transformations at the quantum level. In particular the twistor transform applied to any of
the systems is one of those duality transformations. By applying the twistor transforms we
can map the Hilbert space of one system to another, and then compute any function of the
gauge invariant J BA between dually related states of different systems. The prediction is
that all such computations within different systems must give the same result.
Given that J BA is expressed in terms of rather different phase space and spin degrees
of freedom in each dynamical system with a different Hamiltonian, this predicted duality is
remarkable. 1T-physics simply is not equipped to explain why or for which systems there
are such dualities, although it can be used to check it. The origin as well as the proof of
the duality is the unification of the systems in the form of the 2T-physics master action
of Eq.(4.1) in 4+2 dimensions. The existence of the dualities, which can laboriously be
checked using 1T-physics, is the evidence that the underlying spacetime is more beneficially
understood as being a spacetime in 4+2 dimensions.
VIII. QUANTUM TWISTOR TRANSFORM
We have established a master equation for physical observables J at the quantum level.
Now, we also want to establish the twistor transform at the quantum level expressed as
much as possible in terms of the gauge invariant physical quantum observables J . To this
end we write the master equation (7.8) in the form
J − 3
Ĵ0 − 2
J + 1
Ĵ0 + 2
= 0. (8.1)
Recall the quantum equation (7.5) J + Ĵ0+2
= ZZ̄, so the equation above is equivalent to
J − 3
Ĵ0 − 2
Z = 0. (8.2)
This is a 4× 4 matrix eigenvalue equation with operator entries. The general solution is
Ĵ0 + 2
V̂ (8.3)
where V̂A is any spinor up to a normalization. This is verified by using the master equation
(8.1) which gives
J − 3
Ĵ0 − 2
J − 3
Ĵ0 − 2
J + 1
Ĵ0 + 2
V̂ = 0. Not-
ing that the solution (8.3) has the same form as the classical version of the twistor transform
in Eq.(6.8), except for the quantum shift J0 → Ĵ0 + 2, we conclude that the V̂A introduced
above is the quantum version of the VA discussed earlier (up to a possible renormalization
as belonging to the coset SU(2, 3) /[SU(2, 2)×U(1)].
Now V̂A is a quantum operator whose commutation rules must be compatible with those
of ZA, Z̄
A, Ĵ0 and J BA . Its commutation rules with J BA , Ĵ0 are straightforward and fixed
uniquely by the SU(2, 2)×U(1) covariance
Ĵ0, V̂A
= −V̂A,
Ĵ0, V̂
, (8.4)
J BA , V̂C
= −δ BC V̂A +
V̂C δ
J BA , V̂
= δ DA V̂
δ BA . (8.5)
Other quantum properties of V̂A follow from imposing the quantum property Z̄Z = Ĵ0 −
2 in (7.4). Inserting Z of the form (8.3), using the master equation, and observing the
commutation rules (8.4), we obtain
J + Ĵ0 + 2
V̂ = 1. (8.6)
13 The quantum version of V̂ is valid in the whole Hilbert space, not only in the subspace that satisfies the
U(1) constraint Ĵ0 → 2h. In particular, in the high spin version, already at the classical level we must
take V̂ = V (
J0) and then rescale it V
2h → V as described in previous footnotes. So in the full
quantum Hilbert space we must take V̂ =
2hV (Ĵ0 + γ)
−1/2 (or the rescaled version V
2h → V ) with
the possibly quantum shifted operator (Ĵ0 + γ)
−1/2.
This is related to (5.11) if we take (5.9) into account by including the quantum shift J0 →
Ĵ0 + 2. Considering (8.3) this equation may also be written as
V̂ Z = Z̄V̂ = 1. (8.7)
Next we impose
ZA, Z̄
= δ BA to deduce the quantum rules for [V̂A, V̂
]. After some
algebra we learn that the most general form compatible with
ZA, Z̄
= δ BA is
V̂A, V̂
= − V̂ V̂
Ĵ0 − 1
δ BA +
M(J − 3 Ĵ0 − 2
) + (J − 3 Ĵ0 − 2
, (8.8)
where M BA is some complex matrix and M̄ = (η2,2)M
† (η2,2)
. The matrix M BA could not
be determined uniquely because of the 3/4 kappa gauge freedom in the choice of V̂A itself.
A maximally gauge fixed version of V̂A corresponds to eliminating 3 of its components
V̂2,3,4 = 0 by using the 3/4 kappa symmetry, leaving only A ≡ V̂1 6= 0. Then we find V̂
1,2,4
and V̄ 3 = A†. Let us analyze the quantum properties of this gauge in the context of the
formalism above. From Eq.(8.6) we determine A = (J 13 )
e−iφ, where φ is a phase, and
then from Eq.(8.3) we find ZA.
J 1A +
Ĵ0 + 2
)−1/2
e−iφ, Z̄A = eiφ
)−1/2
J A3 +
Ĵ0 + 2
(8.9)
We see that, except for the overall phase, ZA is completely determined in terms of the
gauge invariant J BA . We use a set of gamma matrices ΓM given in ([6],[11]) to write
J BA = 14iJ
MN (ΓMN)
A as an explicit matrix so that ZA can be written in terms of the
15 SO(4, 2) =SU(2, 2) generators JMN . We find
J12 + 1
J+− + 1
′−′ + Ĵ0+2
(J+1 + iJ+2)
′1 + iJ+
e−iφ√
, (8.10)
and Z̄A =
Z†η2,2
. The orders of the operators here are important. The basisM = ±′,±, i
with i = 1, 2 corresponds to using the lightcone combinations X±
′ ±X1′
, X± =
(X0 ±X1).
From our setup above, the ZA, Z̄
A in (8.10) are guaranteed to satisfy the twistor commuta-
tion rules
ZA, Z̄
= δ BA provided we insure that the V̂A, V̂
have the quantum properties
given in Eqs.(8.4,8.5,8.8). These are satisfied provided we take the following non-trivial
commutation rules for φ
φ, Ĵ0
= i, [φ, J12] =
Ĵ0, e
= ±e±iφ, [J12, e±iφ] = ±1
e±iφ (8.11)
while all other commutators between φ and JMN vanish. Then (8.8) becomes [V̂A, V̂
] = 0,
so M BA vanishes in this gauge. Indeed one can check directly that only by using the Lie
algebra for the JMN , Ĵ0 and the commutation rules for φ in (8.11), we obtain
ZA, Z̄
= δ BA ,
which a remarkable form of the twistor transform at the quantum level.
The expression (8.10) for the twistor is not SU(2, 2) covariant. Of course, this is because
we chose a non-covariant gauge for V̂A. However, the global symmetry SU(2, 2) is still intact
since the correct commutation rules between the twistors and JMN or the J BA as given in
(7.6,7.7) are built in, and are automatically satisfied. Therefore, despite the lack of manifest
covariance, the expression for ZA in (8.10) transforms covariantly as the spinor of SU(2, 2) .
It is now evident that one has many choices of gauges for V̂A. Once a gauge is picked
the procedure outlined above will automatically produce the quantum twistor transform in
that gauge, and it will have the correct commutation rules and SU(2, 2) properties at the
quantum level. For example, in the SL(2, C) covariant gauge of Eq.(5.20), the quantum
twistor transform in terms of JMN is
µα̇ =
Jµν (σ̄
vβ̇ +
′−′vα̇, λα =
′µ (σµ)αβ̇ v
β̇. (8.12)
with the constraint
v̄σµvJ
+′µ = 1. (8.13)
This gauge for V̂M covers several of the systems listed in section (I). The spinless case was
discussed at the classical level in ([6]). The quantum properties of this gauge are discussed
in more detail in ([1]).
The result for ZA in (8.10) is a quantum twistor transform that relies only on the gauge
invariants J BA or equivalently JMN . It generalizes a similar result in [6] that was given at the
classical level. In the present case it is quantum and with spin. All the information on spin is
included in the generators JMN = LMN +SMN . There are other ways of describing spinning
particles. For example, one can start with a 2T-physics action that uses fermions ψM (τ)
[27] instead of our bosonic variables VA (τ) . Since we only use the gauge invariant J
MN , our
quantum twistor transform (8.3) applies to all such descriptions of spinning particles, with
an appropriate relation between V̂ and the new spin degrees of freedom. In particular in the
gauge fixed form of V̂ that yields (8.10) there is no need to seek a relation between V̂ and
the other spin degrees of freedom. Therefore, in the form (8.10), if the JMN are produced
with the correct quantum algebra SU(2, 2) =SO(4, 2) in any theory, (for example bosonic
spinors, or fermions ψM , or the list of systems in section (I), or any other) then our formula
(8.3) gives the twistor transform for the corresponding degrees of freedom of that theory.
Those degrees of freedom appear as the building blocks of JMN . So, the machinery proposed
in this section contains some very powerful tools.
IX. THE UNIFYING SU(2, 3) LIE ALGEBRA
The 2T-physics action (4.1) offered the group SU(2, 3) as the most symmetric unifying
property of the spinning particles for all the systems listed in section (I), including twistors.
Here we discuss how this fundamental underlying structure governs and simplifies the quan-
tum theory.
We examine the SU(2, 3) charges J BA , Ĵ0, jA, j̄A given in (4.5,5.9,5.10). Since these are
gauge invariant under all the gauge symmetries (4.7) they are physical quantities that should
have the properties of the Lie algebra14 of SU(2, 3) in all the systems listed in section (I).
Using covariant quantization we construct the quantum version of all these charges in terms
of twistors. By using the general quantum twistor transform of the previous section, these
charges can also be written in terms of the quantized spin and phase space degrees of freedom
of any of the relevant systems.
The twistor expressions for Ĵ0,J BA are already given in Eqs.(7.2,7.5)
Ĵ0 =
A + Z̄AZA
, J BA = ZAZ̄B −
Ĵ0 + 2
δ BA . (9.1)
We have seen that at the classical level (jA)classical =
J0ZA and now we must figure out
14 Even when jA is not a conserved charge when the U(1) constraint is imposed, its commutation rules are
still the same in the covariant quantization approach, independently than the constraint.
the quantum version jA =
Ĵ0 + αZA that gives the correct SU(2, 3) closure property
jA, j̄
= J BA +
A . (9.2)
The coefficient 5
is determined by consistency with the Jacobi identity
jA, j̄
j̄B, jC
[jC , jA, ] , j̄
= 0, and the requirement that the commutators of jA with
J BA , Ĵ0 be just like those of ZA given in Eqs.(7.6,7.7), as part of the SU(2, 3) Lie algebra.
So we carry out the computation in Eq.(9.2) as follows
jA, j̄
Ĵ0 + αZAZ̄
Ĵ0 + α− Z̄B
Ĵ0 + α
Ĵ0 + αZA (9.3)
Ĵ0 + α
Ĵ0 + α− 1
Z̄BZA (9.4)
Ĵ0 + α− 1
ZA, Z̄
+ ZAZ̄
B (9.5)
= δ BA
Ĵ0 + α− 1 +
Ĵ0 + 2
+ J BA (9.6)
To get (9.4) we have used the properties ZAf
Ĵ0 + 1
ZA and Z̄
Ĵ0 − 1
Z̄B for any function f
. These follow from the commutator
Ĵ0, ZA
= −ZA
written in the form ZAĴ0 =
Ĵ0 + 1
ZA which is used repeatedly, and similarly for Z̄
B. To
get (9.6) we have used
ZA, Z̄
= δ BA and then used the definitions (9.1). By comparing
(9.6) and (9.2) we fix α = 1/2. Hence the correct quantum version of jA is
Ĵ0 +
ZA = ZA
Ĵ0 −
. (9.7)
The second form is obtained by using ZAf
Ĵ0 + 1
Note the following properties of the jA, j̄
j̄AjA =
Ĵ0 −
Ĵ0 −
Ĵ0 −
Ĵ0 − 2
(9.8)
Ĵ0 +
Ĵ0 +
Ĵ0 +
Ĵ0 + 2
(9.9)
which will be used below.
With the above arguments we have now constructed the quantum version of the SU(2, 3)
charges written as a 5× 5 traceless matrix
Ĵ2,3 =
quantum
(9.10)
B − 1
Ĵ0 +
Ĵ0 +
 , (9.11)
with Ĵ0,J given in Eq.(9.1).
At the classical level, the square of the matrix J2,3 vanishes since L2 = 0 as follows
(J2,3)
classical
= g−1
g = 0. (9.12)
At the quantum level we find the following non-zero result which is SU(2, 3) covariant
Ĵ2,3
ZZ̄ − 1
Ĵ0 +
Ĵ0 +
(9.13)
Ĵ2,3
− 1. (9.14)
By repeatedly using the same equation we can compute all powers
Ĵ2,3
, and by taking
traces we obtain the Casimir eigenvalues of the SU(2, 3) representation. For example the
quadratic Casimir is
Ĵ2,3
= −5. (9.15)
Written out in terms of the charges, Eq.(9.14) becomes
− 1. (9.16)
Collecting terms in each block we obtain the following relations among the gauge invariant
charges J , Ĵ0, j, j̄
− jj̄ + 5
+ 1 = 0, (9.17)
j − jĴ0 +
j = 0, (9.18)
−j̄j +
Ĵ0 + 1 = 0. (9.19)
Combined with the information in Eq.(9.9) the first equation is equivalent to the master
quantum equation (7.8). After using jĴ0 = Ĵ0j+ j, the second equation is equivalent to the
eigenvalue equation (8.2) whose solution is the quantum twistor transform (8.3). The third
equation is identical to (9.8).
Hence the SU(2, 3) quantum property
Ĵ2,3
Ĵ2,3
− 1, or equivalently
Ĵ2,3 + 2
Ĵ2,3 +
= 0, governs the quantum dynamics of all the sytems listed in sec-
tion (I) and captures all of the physical information, twistor transform, and dualities as a
property of a fixed SU(2, 3) representation whose generators satisfy the given constraint.
This is a remarkable simple unifying description of a diverse set of spinning systems, that
shows the existence of the sophisticated higher structure SU(2, 3) for which there was no
clue whatsoever from the point of view of 1T-physics.
X. FUTURE DIRECTIONS
One can consider several paths that generalizes our discussion, including the following.
• It is straightforward to generalize our theory by replacing SU(2, 3) with the super-
group SU(2, (2 + n) |N) . This generalizes the spinor VA to V aA where a labels the
fundamental representation of the supergroup SU(n|N) . The case of N = 0 and n = 1
is what we discussed in this paper. The case of n = 0 and any N relates to the
superparticle with N supersymmetries (and all its duals) discussed in [22] and in
[6][7]. The massless particle gauge is investigated in [17], but the other cases listed
in section (I) remain so far unexplored. The general model has global symmetry
SU(2, 2)×SU(n|N)×U(1) ⊂ [SU(2, (2 + n) |N)]R if a U(1) gauging is included, or the
full global symmetry [SU(2, (2 + n) |N)]R in its high spin version. It also has local
gauge symmetries that include bosonic & fermionic kappa symmetries embedded in
[SU(2, (2 + n) |N)]L as well as the basic Sp(2, R) gauge symmetry. The gauge sym-
metries insure that the theory has no negative norm states. In the massless particle
gauge, this model corresponds to supersymmetrizing spinning particles rather than
supersymmetrizing the zero spin particle. The usual R-symmetry group in SUSY is
replaced here by SU(n|N)×U(1) . For all these cases with non-zero n,N , the 2T-
physics and twistor formalisms unify a large class of new 1T-physics systems and
establishes dualities among them.
• One can generalize our discussion in 4+2 dimensions, including the previous paragraph,
to higher dimensions. The starting point in 4+2 dimensions was SU(2, 2) =SO(4, 2)
embedded in g =SU(2, 3) . For higher dimensions we start from SO(d, 2) and seek a
group or supergroup that contains SO(d, 2) in the spinor representation. For example
for 6+2 dimensions, the starting point is the 8×8 spinor version of SO(8∗) =SO(6, 2)
embedded in g =SO(9∗) =SO(6, 3) or g =SO(10∗) =SO(6, 4) . The spinor variables
in 6+2 dimensions VA will then be the spinor of SO(8
∗) =SO(6, 2) parametrizing the
coset SO(9∗) /SO(8∗) (real spinor) or SO(10∗) /SO(8∗)×SO(2) (complex spinor). This
can be supersymmetrized. The pure superparticle version of this program for various
dimensions is discussed in [6][7], where all the relevant supergroups are classified.
That discussion can now be taken further by including bosonic variables embedded
in a supergroup as just outlined in the previous item. As explained before [6][7], it
must be mentioned that when d + 2 exceeds 6 + 2 it seems that we need to include
also brane degrees of freedom in addition to particle degrees of freedom. Also, even in
lower dimensions, if the group element g belongs to a group larger than the minimal
one [6][7], extra degrees of freedom will appear.
• The methods in this paper overlap with those in [28] where a similar master quantum
equation technique for the supergroup SU(2, 2|4) was used to describe the spectrum of
type-IIB supergravity compactified on AdS5×S5. So our methods have a direct bearing
onM theory. In the case of [28] the matrix insertion
in the 2T-physics action was
generalized to
L(4,2)
L(6,0)
to describe a theory in 10+2 dimensions. This approach to
higher dimensions can avoid the brane degrees of freedom and concentrate only on the
particle limit. Similar generalizations can be used with our present better develped
methods and richer set of groups mentioned above to explore various corners of M
theory.
• One of the projects in 2T-physics is to take advantage of its flexible gauge fixing
mechanisms in the context of 2T-physics field theory. Applying this concept to the
2T-physics version of the Standard Model [10] will generate duals to the Standard
Model in 3+1 dimensions. The study of the duals could provide some non-perturbative
or other physical information on the usual Standard Model. This program is about to
be launched in the near future [29]. Applying the twistor techniques developed here
to 2T-physics field theory should shed light on how to connect the Standard Model
with a twistor version. This could lead to further insight and to new computational
techniques for the types of twistor computations that proved to be useful in QCD
[12][13].
• Our new models and methods can also be applied to the study of high spin theories
by generalizing the techniques in [14] which are closely related to 2T-physics. The
high spin version of our model has been discussed in many of the footnotes, and
can be supersymmetrized and written in higher dimensions as outlined above in this
section. The new ingredient from the 2T point of view is the bosonic spinor VA and
the higher symmetry, such as SU(2, 3) and its generalizations in higher dimensions or
with supersymmetry. The massless particle gauge of our theory in 3+1 dimensions
coincides with the high spin studies in [15]-[18]. Our theory of course applies broadly to
all the spinning systems that emerge in the other gauges, not only to massless particles.
The last three sections on the quantum theory discussed in this paper would apply
also in the high spin version of our theory. The more direct 4+2 higher dimensional
quantization of high spin theories including the spinor VA (or its generalizations V
is obtained from our SU(2, 3) quantum formalism in the last section.
• One can consider applying the bosonic spinor that worked well in the particle case to
strings and branes. This may provide new string backgrounds with spin degrees of
freedom other than the familiar Neveu-Schwarz or Green-Schwarz formulations that
involve fermions.
More details and applications of our theory will be presented in a companion paper [1].
We gratefully acknowledge discussions with S-H. Chen, Y-C. Kuo, and G. Quelin.
[1] I. Bars and B. Orcal, in preparation.
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http://arxiv.org/abs/hep-th/0606053
http://arxiv.org/abs/hep-th/0701177
http://arxiv.org/abs/hep-th/9812161
http://arxiv.org/abs/hep-th/9904063
http://arxiv.org/abs/hep-th/0407239
http://arxiv.org/abs/hep-th/0502065
http://arxiv.org/abs/hep-th/0605267
http://arxiv.org/abs/hep-th/9804028
http://arxiv.org/abs/hep-th/9810025
http://arxiv.org/abs/hep-th/9806085
http://arxiv.org/abs/hep-th/0208012
http://arxiv.org/abs/hep-th/0205194
	Spinning Particles in 3+1 - Beyond Free and Massless 
	Twistor Lagrangian
	Massless Particle With Any Spin in 3+1 Dimensions
	2T-physics With Sp( 2,R) , SU(2,3) and Kappa Symmetries
	2T-physics Action with XM,PM,VA,A in 4+2 Dimensions
	General Twistor Transform (Classical)
	Quantum Master Equation, Spectrum, and Dualities
	Quantum Twistor Transform
	The Unifying SU( 2,3)  Lie algebra
	Future Directions
	References
