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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

arXiv:0704.0296v2 [hep-th] 12 Apr 2007

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 uni?es many types of spinning systems by mapping them to the same twistor ZA =

˙ ?α λα

, thus predicting an in?nite set of duality relations among spinning systems with

di?erent Hamiltonians. Usual 1T-physics is not equipped to explain the duality relationships and uni?cation 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 practical 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 backgrounds 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 x? , p? , s?ν to a twistor ZA =

1

˙ ?α λα

and reformulates the dynamics in terms of twistors instead of phase space. The

This work was partially supported by the US Department of Energy, grant number DE-FG03-84ER40168.

1

˙ twistor ZA is made up of a pair of SL(2, C ) spinors ?α , λα , α, α ˙ = 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]

˙ ˙ ? ˙ = p ˙, ?α = ?i (? x + iy ?)αβ λβ , λα λ β αβ ˙ where (? x + iy ?)αβ = 1 √ 2 ˙ (x? + iy ? ) (? σ? )αβ , and pαβ ˙ = 1 ? √ p 2

(1.1)

(σ? )αβ ?? = ˙ , while σ? = (1, σ ) , σ

h of the particle is determined by p · y = h. The spin tensor is given by s?ν = ε?νρσ yρ pσ , and it leads to 1 s?ν s?ν = h2 . The Pauli-Lubanski vector is proportional to the momentum 2 ε sνρ pσ = (y · p) p? ? p2 y ? = hp? , appropriate for a massless particle of helicity h. W? = 1 2 ?νρσ 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 p? of the ? ˙ automatically satis?es p? p? = 0. form p ˙ = λα λ

αβ β

(?1, σ ) are Pauli matrices. x? + iy ? is a complexi?cation of spacetime [2]. The helicity

However, recent work has shown that the same twistor ZA =

˙ ?α λα

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 p? p? = 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 ?at Minkowski space. The massive relativistic particle in d = 4 ?at 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 AdS2 ×S2 . The particle on AdS3 ×S1 or on R × S 3 .

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 modi?ed 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 = 0). It must be emphasized that while the phase spaces (and therefore dynamics, Hamiltonian, etc.) in these 2

˙ systems are di?erent, the twistors ?α , λα are the same. For example, the massive particle

phase space (x? , p? )massive and the one for the massless particle (x? , p? )massless are not the same (x? , p? ) , 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

˙ ?α , λα massive ˙ = ?α , λα massless ˙ = · · · = ?α , λα .

(1.2)

This uni?cation 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 uni?cation relies on an underlying spacetime in 4+2 dimensions. It was realized sometime ago that in 2Tphysics 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 ?xed 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 di?erent 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 ?eld 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 simpli?ed QCD computations [12][13], we expect that our twistor methods will ?nd useful applications.

3

II.

TWISTOR LAGRANGIAN

The Penrose twistor description of massless spinning particles requires that the pairs ˙ ? ?α , iλα ?α ) be canonical conjugates and satisfy the he˙ or their complex conjugates (λα , i? licity constraint given by

α ˙ ?α ? A ZA = λ Z ?α λα = 2h. ˙? + ?

(2.1)

Indeed, Eq.(1.1) satis?es this property provided y · p = h. Here we have de?ned the ? 4 of SU(2, 2) as the contravariant twistor ? A ≡ Z ? η2,2 Z

A

?α = λ ?α , η2,2 = ˙ ?

01 10

= SU (2, 2) metric.

(2.2)

? A ZA = 2h follows from the following The canonical structure, along with the constraint Z worldline action for twistors Sh =

A ?A Dτ Z A ? 2hA ? , Dτ Z A ≡ ?Z ? iAZ ? A. dτ iZ ?τ

(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 = 0 2T-physics action from which (2.3) is derived as a gauge choice. The derivative part of α ˙ ?α ?A ?τ Z A = i dτ λ this action gives the canonical structure S0 = dτ iZ ? α ?τ λα ˙ ?τ ? + ? ˙ ? that requires ?α , iλα or their complex conjugates (λα , i? ?α ) to be canonical conjugates. ˙ ? is a U(1) gauge ?eld on the worldline, Dτ Z A is the U(1) gauge covariant The 1-form Adτ ? = ?ε/?τ and δε Z A = iεZ A . The derivative that satis?es δε Dτ Z A = iε Dτ Z A for δε A ? is gauge invariant since it transforms as a total derivative under the in?nitesimal term 2hA ? was introduced in [6][7] as being an integral part of the twistor gauge transformation. 2hA 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 uni?cation 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 uni?cation is equivalent to the uni?cation achieved in 2T-physics.

4

III.

MASSLESS PARTICLE WITH ANY SPIN IN 3+1 DIMENSIONS

In our quest for the general twistor transform with spin, we ?rst discuss an alternative to the well known twistor transform of Eq.(1.1). Instead of the y ? (τ ) that appears in the

˙ complexi?ed spacetime x? + iy ? we introduce an SL(2, C ) bosonic 2 spinor v α (τ ) and its ˙ complex conjugate v ?α (τ ) , and write the general vector y ? in the matrix form as y αβ = ˙ β ˙ hv α v ? + ωpαβ , where ω (τ ) is an arbitrary gauge freedom that drops out. Then the helicity β condition y · p = h takes the form v ?pv = 1. Furthermore, we can write λα = pαβ ˙ v since this ? ˙ = p ˙ when p2 = (? automatically satis?es λα λ v pv ? 1) = 0 are true. With this choice of β αβ ˙

variables, the Penrose transform of Eq.(1.1) takes the new form

˙ ˙ λα = (pv )α , ?α = [(?ix ?p + h) v ]α , p2 = (? vpv ? 1) = 0, ˙ where the last equation is a set of constraints on the degrees of freedom x? , p? , v α ,v ?α .

(3.1)

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 x? , p? , v α ,v ?α

Sh =

e ? . v p) Dτ v ? Dτ v (pv ) ? 2hA dτ x ˙ ? p? ? p2 + ih (? 2

(3.2)

? is the U(1) gauge covariant derivative and we have included the where Dτ v = v ˙ ? iAv ? imposes the second constraint v motion for A ?pv ? 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 J ?ν = x? pν ? xν p? + s?ν , with s?ν =

2 3

Lagrange multiplier e to impose p2 = 0 when we don’t refer to twistors. The equation of

i hv ? (p σ ? ?ν 2

+ σ ?ν p) v. The helic-

This is similar to the fermionic case in [5]. The bosonic spinor v can describe any spin h. ? = 0 , then the excitations in the v sector describe If this action is taken without the U(1) constraint A √ an in?nite tower of massless states with all helicities from zero to in?nity (here we rescale 2hv → v ) Sall

spins

=

dτ

i e ˙ v ?pv ˙ ? vpv x ˙ ? p? ? p2 + 2 2

(3.3)

The spectrum coincides with the spectrum of the in?nite slope limit of string theory with all helicities 1 ?pv . This action has a hidden SU(2, 3) symmetry that includes SU(2, 2) conformal symmetry. This is 2v explained in the rest of the paper by the fact that this action is a gauge ?xed 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 uni?ed 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).

5

1 ?νλσ ity is determined by computing the Pauli-Lubanski vector W ? = 2 ε sνλ pσ = (hv ?pv ) p? .

The helicity operator hv ?pv 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, v ?. We note its similarity to the standard superparticle action [20][21] written in the ?rst order formalism. The di?erence with the superparticle is that the

˙ fermionic spacetime spinor θα of the superparticle is replaced with the bosonic spacetime ˙ ? imposes the U(1) gauge symmetry constraint v spinor v α , and the gauge ?eld A ?pv ? 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 ih ˙ ˙ ?) σ? v ? v ?σ? (δκ v )) , δκ v α =p ?αβ κβ , δκ x? = √ ((δκ v 2 ? = 0. δκ p? = 0, δκ e = ?ih κ ? (Dτ v ) ? Dτ v κ , δκ A degrees of freedom v, v ?. The transformations δκ x? , δκ e 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 x? , p? there remains 3 positions and 3 momentum degrees of freedom. One of the two complex components of

˙ ˙ vα is set to zero by using the kappa symmetry, so v α = v 0

(3.4) (3.5)

These kappa transformations mix the phase space degrees of freedom (x, p) with the spin

. The phase of the remaining

1 0

component is eliminated by choosing the U(1) gauge, and ?nally its magnitude is ?xed by

˙ solving the constraint v ?pv ? 1 = 0 to obtain v α = (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 s?ν . 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 ?rst 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 di?erent dynamics (di?erent 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 massless particles, and agree with their covariant quantization given in [15]-[19]. 6

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 (x? , p? ) to 4+2 phase space ? A . The larger X M , PM and lifting the SL(2, C ) spinors v, v ? to the SU(2, 2) spinors VA , V ? A that are covariant under the global symmetry set of degrees of freedom X M , PM , VA , V 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. ? A have a wider set The point is that the SU(2, 2) invariant constraints on X M , PM , VA , V of solutions than just the 3+1 system of Eq.(3.2) we started from. This is because 3+1 dimensional spin & phase space has many di?erent 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 di?erent in di?erent embeddings, thus producing the different 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 ? A as given in Eq.(5.4). space action in terms of the lifted spin & phase space X M , PM , VA , V The latter can be further gauge ?xed 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, ? A in the left coset SU(2, 3) /SU(2, 2) ×U(1) . and embed the bosonic SU(2, 2) spinors VA , V

This coset will be regarded as the gauging of the group SU(2, 3) under the subgroup [SU (2, 2) × U (1)]L 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

7

? (τ ) degrees of freedom described as XiM (τ ), g (τ ) and A 4+2 phase space

X M (τ ) PM (τ )

≡ XiM (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 property, 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 Sh = where εij = dτ 1 ij N ε Dτ XiM Xj ηM N + T r (iDτ g ) g ?1 2 L0 00 ? , ? 2hA (4.1)

the Sp(2, R) gauge covariant derivative, with the 3 gauge potentials Aij = εik Akj = SU(2, 3) metric η2,3 = . The covariant derivative Dτ g is given by ? ? 1 0 ? [q, g ] , q = 1 ? 4×4 ? Dτ g = ?τ g ? iA 5 0 ?4

η2,2 0 0 ?1

0 1 ij ?1 0

M is the antisymmetric Sp(2, R) metric, and Dτ XiM = ?τ XiM ? Ai j Xj is AC CB

.

For SU(2, 3) the group element is pseudo-unitary, g ?1 = (η2,3 ) g ? (η2,3 )?1 , where η2,3 is the

(4.2)

where the generator of U(1)L+R is proportional to the 5×5 traceless matrix ? which is also the last term q ∈ u(1) ∈ su(2, 3)L+R . The last term of the action ?2hA, of the action (2.3), is invariant under the U(1)L+R since it transforms to a total derivative. g (or right side of g ?1 ) is (L)AB ≡ where ΓM N =

1 2 B Finally, the 4 × 4 traceless matrix (L)A ∈su(2, 2) ∈su(2, 3) that appears on the left side of

1 ΓM N 4i

B A N LM N , LM N = εij XiM Xj = XMP N ? XN P M.

(4.3)

? N ? ΓN Γ ? M are the 4×4 gamma-matrix representation of the 15 genΓM Γ

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 modi?cation of that discussion here is due to the inclusion of ? In the absence of the A ? coupling the global symmetry is given the U(1) gauge ?eld A. 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 . 8

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 ?1 L0 00 g=

1 J +4 J0 j ?? j ?J0 1 MN Γ JM N 4i

, J2,3 = η2,3 (J2,3 )? (η2,3 )?1 ,

(4.5)

B The traceless 4 × 4 matrix (J )A =

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 )AB = 0 and ?τ J0 = 0. The spinor charges jA , ? j A are not conserved4 due to the ? As we will ?nd out later in Eq.(6.8), jA is proportional to the twistor coupling of A. jA = J0 Z A , (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 classi?ed. The local symmetries of this action are summarized as ? ? 3 SU (2, 2) 4 kappa ? Sp (2, R) × ? 3 kappa U (1) 4

(4.7)

lef t

The Sp(2, R) is manifest in (4.1). The rest corresponds to making local SU(2, 3) transformations on g (τ ) from the left side g (τ ) → gL (τ ) g (τ ) , as well as transforming XiM = XM, P M as vectors with the local subgroup SU(2, 2)L =SO(4, 2) , and Aij 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] ?. The reason is that the U(1) despite the presence of A covariant derivative Dτ g in

L+R

? because Eq.(4.2) can be replaced by a purely U(1)R covariant derivative Dτ g = ?τ g + igq A the di?erence 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 A. We outline the roles of each of these local symmetries. The Sp(2, R) gauge symmetry can reduce X M , PM to any of the phase spaces in 3+1 dimensions listed in section (I). This

4

? = 0, the global symmetry is SU(2, 3) and jA , ? In the high spin version of (4.1) with A j A are conserved. R

9

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 ? A as shown in Eq.(5.3). The remaining 3/4 kappa by the SU(2, 2) ×U(1) spinors VA , V 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 ? ?xes the magnitude of V . in V . Finally the constraint due to the equation of motion of A 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 ?x 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 form

X+ P +′

′

(τ ) =

1 0

, and solving explicitly two of the Sp(2, R) constraints X 2 = X · P = 0

+′ ? x , x? (τ )), 2 ?′ 2

XM = ( 1 ,

P M = ( 0 , x · p , p? (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 ? A as given in Eq.(5.3), and the 3/4 kappa symmetry g (τ ) → t (V ) written in terms of VA , V reduces the SU(2, 2) spinor VA →

˙ vα 0

˙ to the two components SL(2, C ) doublet v α , with a

The inverse g ?1 = (η2,3 ) g ? (η2,3 )?1 is given by replacing v, v ? by (?v ) , (?v ?) . Inserting the gauge ?xed forms of X, P, g (4.8,4.9) into the action (4.1) reduces it to the massless spin-

leftover kappa symmetry as discussed in Eqs.(3.4-3.5). The gauge ?xed form of g is then ? ? ? ? √ √ ˙ β ˙ ˙ 1 hv α v ? 0 0 2hv α 2hv α ? ? ? ? ? ? ? ? = (4.9) g = exp ? 0 0 ? ?0 0 ? ∈ SU (2, 3) . 0 1 ? ? √ ? ? √ 0 2hv ?α 0 0 2hv ?α 1

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 signi?cance of the hidden conformal symmetry of the gauge ?xed 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. 10

Partial or full gauge ?xings 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 = 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 ?xing 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 [1]. 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.

? A IN 4+2 DIMENSIONS 2T-PHYSICS ACTION WITH X M , P M , VA , V

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 ?xed 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 rotate X M (τ ) , P M (τ ) to the following form that also solves the Sp(2, R) constraints Xi · Xj = X 2 = P 2 = X · P = 0 [6][7] X M = ( 1 , 0 , 0 , 0 , 0),

+′ ?′ + ? i

P M = ( 0 , 0 , 1 , 0 , 0).

+′

?′

+

?

i

(5.1)

This completely eliminates all phase space degrees of freedom. We are left with the gauge ?′ ? ? , where (iL) → 1 Γ?′ ? L+′ ?′ , and ? 2hA ?xed action Sh = dτ T r 1 (Dτ g ) g ?1 Γ 0

2 0 0 2

= 1. Due to the many zero entries in the 4×4 matrix Γ [6], only one column A ? A , ?Z ?5 can contribute in and one row from g ?1 in the form Z from g in the form Z Z5

L

+′ ?′

?′ ?

?AZ ˙ A ? iZ ?5 Z ˙5 + A ? Z ?5 Z5 ? 2h . the trace, and therefore the action becomes Sh = dτ iZ ?5 Z ˙ 5 drops out as a total derivative since the magnitude of the complex number Z5 is Here Z ?5 Z5 = 2h. Furthermore, we must take into account Z ? A ZA ? Z ?5 Z5 = 0 which a constant Z is an o?-diagonal entry in the matrix equation g ?1g = 1. Then we see that the 2T-physics

11

action (4.1) reduces to the twistor action (2.3) with the gauge choice (5.1)5 . Next let us gauge ?x the 2T-physics action (4.1) to a manifestly SU(2, 2) =SO(4, 2) invariant version in ?at 4+2 dimensions, in terms of the phase space & spin degrees of ? A . For this we use the [SU(2, 2)×U(1)]lef t symmetry to gauge ?x g freedom X M , P M , VA , V gauge ?x: g → t (V ) ∈ SU(2, 3) [SU(2, 2) × U(1)]lef t (5.2)

? = The coset element t (V ) is parameterized by the SU(2, 2) spinor V and its conjugate V V ? η2,2 and given by the 5×5 SU(2, 3) matrix6 ? ? ?1/2 1 ? 2hV V 0 ? t (V ) = ?V 0 1 ? 2hV ?? √ ?

?1/2

The factor 2h is inserted for a convenient normalization of V. Note that the ?rst matrix

?? √

1 ? 2hV

2hV 1

?.

(5.3)

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 )?1 = t (?V ) , as can be checked explicitly t (V ) t (?V ) = 1. Inserting this gauge in (4.1) the action becomes Sh = = where dτ dτ ? LV V ? ˙ · P ? 1 Aij Xi · Xj ? 1 ?M N LM N ? 2hA X ?V ? 1 2 2 1 ? 2hV ? LV 1 ij ? M V N ? ε Dτ Xi Xj ηM N ? 2hA ?V ? 1 2 1 ? 2hV ? τ X M = ?τ X M ? A j X M ? ?M N XiN D i i j i composite SO(4, 2) connection ?M N (V (τ )) given conveniently in the following forms 1 MN ? ΓM N = (i?τ t) t?1 2

SU (2,2)

(5.4) (5.5)

(5.6)

is a covariant derivative for local Sp(2, R) as well as local SU(2, 2) =SO(4, 2) but with a 1 MN ? LM N = ?T r (i?τ t) t?1 2 L0 00

, or

.

(5.7)

Thus, ? is the SU(2, 2) projection of the SU(2, 3) Cartan connection and given explicitly as

˙V ?V ˙ ? ?V V ˙ ?V ?V ˙ ?V V V V ?V ?V 1 MN V V ? ΓM N = 2 h √ √ 2 ? V 1 + 1 ? 2hV ?V 1 ? 2hV

5

+h

? 1 VV ? ?V 4 V

˙V ?V ˙ ?V V ?V 1 ? 2hV

(5.8)

6

? (see footnote (3)), we replace Z5 = eiφ Z ? A ZA and after In the high spin version of (4.1) without A ? AZ ˙ A + ZZ ? φ ˙ . For a dropping a total derivative, the twistor equivalent becomes Sall spins = dτ iZ more covariant version that displays the SU(2, 3) global symmetry, we introduce a new U(1) gauge ?eld A ˙5 + B ? Z ? A ZA ? Z ?5 Z5 . ? AZ ˙ A ? iZ ?5 Z and write Sall spins = dτ iZ for the overall phase of Z Z5 ? are easily computed by expanding in a series and Arbitrary fractional powers of the matrix 1 ? 2hV V γ ? = 1+ V V ? 1 ? 2 hV ? V γ ? 1 /V ? V. then resuming to obtain 1 ? 2hV V

12

The action (5.4,5.5) is manifestly invariant under global SU(2, 2) =SO(4, 2) rotations, and ? A . The conserved global symmetry under local U (1) phase transformations applied on VA , V by inserting the gauge ?xed form of g → t (V ) into Eq.(4.5)7 J(2,3) = t?1 J =√ 1 L√ 1 currents J and J0 can be derived either directly from (5.4) by using Noether’s theorem, or

L0 00

t (5.9) (5.10)

? that follows from the action (5.4) we must have According to the equation of motion for A the following constraint (this means U(1) gauge invariant physical sector) ? LV V ? V = 1. 1 ? 2hV physical sector: J0 = 2h, J = √ 1 ? 1 ? 2hV V L√ 1 ? 1 ? 2hV V h ? . 2 (5.11)

? LV 2hV 1 ? J0 , J0 = ?V ? ? 4 1 ? 2hV 1 ? 2hV V 1 ? 2hV V √ 1 1 jA = 2h √ LV √ ? ?V 1 ? 2hV V 1 ? 2hV

Therefore, in the physical sector the conserved [SU(2, 2)×U(1)]right charges take the form (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

0 K ? 0 K

∈SU(2, 3)lef t /[SU(2, 2)×U(1)]lef t with a special form of the spinor KA KA = Xi · Γκi (τ )

A

= XM ΓM κ1

A

+ PM ΓM κ2

A

,

(5.13)

with κiA (τ ) two arbitrary local spinors8 . Now that g has been gauge ?xed g → t (V ), the kappa transformation must be taken as the naive kappa transformation on g followed by a [SU(2, 2) ×U(1)]lef t gauge transformation which restores the gauge ?xed form of t (V ) t (V ) → t (V ′ ) = exp ?ω 0 0 T r (ω ) exp 0 K ? 0 K t (V ) (5.14)

The SU(2, 2) part of the restoring gauge transformation must also be applied on X M , P M . Performing these steps we ?nd the in?nitesimal version of this transformation [22] δκ V = √

7

1 ? 1 ? 2hV V

K√

1 ?V 1 ? 2hV

, δκ XiM = ω M N XiN , δκ Aij = see below,

(5.15)

8

? = 0) the conserved charges include jA as part of SU(2, 3) global symmetry. In the high spin version (A R √ It is then also convenient to rescale 2hV → V in Eqs.(5.3-5.10) to eliminate an irrelevant constant. In this special form only 3 out of the 4 components of KA are e?ectively independent gauge parameters. This can be seen easily in the special frame for X M , P M given in Eq.(5.1).

13

˙ replaced by the where ω M N (K, V ) has the same form as ?M N in Eq.(5.8) but with V ? τ X M in Eq.(5.6) is covariant under the local δκ V given above. The covariant derivative D

i

SU(2, 2) transformation with parameter ω M N (K, V ) (this is best seen from the projected Cartan connection form ? = [(i?τ t) t?1 ]SU (2,2) ). Therefore, the kappa transformations (5.15) inserted in (5.5) give δκ Sh = dτ ? 1 δκ Aij Xi · Xj + iT r (Dτ t) t?1 2 0 LK ?L 0 ?K . (5.16)

In computing the second term the derivative terms that contain ?τ K have dropped out in the trace. Using Eq.(5.13) we see that LK = 1 li M N L ε Xl Xi X j ΓM N ΓL κj 4i 1 L = εli XlM XiN Xj (ΓM N L + ηN L ΓM ? ηM L ΓN ) κl 4i 1 = εli Xi · Xj Xl · Γκj 2i (5.17) (5.18) (5.19)

N The completely antisymmetric XiM Xj XlL ΓM N L 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 the ?rst term by choosing the appropriate δκ Aij 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 δκ J = δκ J0 = δκ jA = 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) symmetries 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 h = 0 in [6]. The gauge choices for X M , P M discussed in [6] now need to be supplemented ? A by using the kappa×U(1) local symmetries. with gauge choices for VA , V local Sp(2, R) × (3/4 kappa)×U(1) symmetries. This was demonstrated for the spinless case proportional to the dot products Xi · Xj . Therefore the second term in (5.16) is cancelled by

Here we demonstrate the gauge ?xing described above for the massless particle of any spin h. The kappa symmetry e?ectively has 3 complex gauge parameters as explained in footnote (8). If the kappa gauge is ?xed by using two of its parameters we reach the following forms VA →

˙ vα 0

? A → (0 v ? V → 0, 1 ? 2hV V ? , V ?α ) , V 14

?1/2

→

1 0

hv v ? . 1

(5.20)

By inserting this gauge ?xed form of V, and the gauge ?xed 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 ?xing 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) metric 2T-physics version gave the JAB as embedded in SU(2, 3)R in the SU(2, 2) projected form in Eq.(4.5) J = g ?1 L0 g 00 .

SU (2,2)

whose conserved charges JAB are gauge invariant in all the formulations. The most sym-

(6.1)

Since this is gauge invariant, when gauge ?xed, 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 1 1 1 ? (h) ? 1 T r Z (h) Z ? (h) = √ J = Z (h) Z L√ ? J0 ? ? 4 4 1 ? 2hV V 1 ? 2hV V ? (h) , so The trace corresponds to the U(1) charge J0 = T r Z (h) Z 1 1 1 ? ( h) = √ J + J0 = Z (h) Z L√ . ? ? 4 1 ? 2hV V 1 ? 2hV V In the case of h = 0 this becomes ? (0) = L. Z (0) Z Therefore the equality (6.3) is solved up to an irrelevant phase by Z (h) = √ 1 ? 1 ? 2hV V Z (0) . (6.5) (6.4) (6.3) (6.2)

By inserting (6.4) into the constraint (5.11) we learn a new form of the constraints ? Z (0) = V ?V , V ? Z ( h ) = 1. 1 ? 2hV 15 (6.6)

In turn, this implies Z (0) = √ ?V 1 ? 2hV LV (6.7)

? (0) = L , and its vanishing trace Z ? (0) Z (0) = 0 since LL = 0 which is consistent9 with Z (0) Z (due to X 2 = P 2 = X · P = 0). Putting it all together we then have Z ( h) = √ 1 ? 1 ? 2hV V LV √ 1 ?V 1 ? 2hV = 1 J + J0 V. 4

(6.8)

their gauge ?xed forms, and use the constraint10 J0 = 2h.

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 = J0 Z (h) given in Eqs.(4.5,4.6) or (5.10), when g and L are replaced by

The key for the general twistor transform for any spin is Eq.(6.5), or equivalently (6.8). ? (0) = L is The general twistor transform between Z (0) and X M , P M which satis?es Z (0) Z already given in [6] as ? ? (0) ? ? , ?(0) Z (0) = ? (0) λ

α ˙

= ?i

X? σ ?? λ(0) X +′

α ˙

? = X + P ? ? X ? P + (σ? ) ˙ . (6.9) , λ(0) α λβ ˙ αβ

(0)

Note that (X + P ? ? X ? P + ) is compatible with the requirement that any SL(2, C ) vector (0) ? (0) constructed as λα λ ˙ must be lightlike. This property is satis?ed thanks to the Sp(2, R) β mass in the 3 + 1 subspace (since P ? P? is not restricted to be lightlike). Besides satisfying ? (0) = L, this Z (0) also satis?es Z ? (0) Z (0) = 0, as well as the canonical properties of Z (0) Z twistors. Namely, Z (0) has the property [6] constraints X 2 = P 2 = X · P = 0 in 4+2 dimensions, thus allowing a particle of any

? (0) ?τ Z (0) = dτ Z

˙ M PM . dτ X

(6.10)

From here, by gauge ?xing 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 ZA in terms

9 10 LV V L Z Z VVZ Z ? (0) = L. To see this, we note that Eqs.(6.4,6.6) lead to 1? = Z (0) Z ?V = ?V 2hV 1?2hV ? = 0) we don’t use the constraint. Instead, we use Z (h) = √ 1 For the high spin version (A ?

(0)

(h)

? (0)

?

(0)

? (0)

in its form (6.5),√and note that, after using Eq.(6.4), the jA in Eq.(5.10) takes the form jA √ √ ? 0 V 2h Z with J0 = √ , and it is possible to rescale h away everywhere 2hV → V. ?

1?2hV V

? 1?2hV V

Z (0) only √ = J0 Z (h)

16

? A . Then the Sp(2, R) and kappa of spin & phase space degrees of freedom X M , P M , VA , V ? A can be gauge ?xed for any spin h, to give gauge symmetries that act on X M , P M , VA , V the speci?c 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 ? (h) ?τ Z (h) = dτ Z 1 1 ? (0) √ dτ Z ?τ √ Z (0) ? ? 1 ? 2hV V 1 ? 2hV V ? ? ? ? ?Z ? (0) √ 1 ?τ √ 1 ? Z (0) ? ? 1?2hV V 1?2hV V dτ ? ? 1 (0) (0) ? ? ? +Z ? ?τ Z 1?2hV V ˙ · P + T r (i?τ t) t?1 X L0 00 (6.11)

=

(6.12)

=

dτ

(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.(6.4This 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 ?xed 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). 6.7), as well as the constraints X 2 = P 2 = X · P = 0, and dropped some total derivatives.

VII.

QUANTUM MASTER EQUATION, SPECTRUM, AND DUALITIES

In this section we derive the quantum algebra of the gauge invariant observables JAB 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 ?xed versions, including in any of its versions listed in section (I). From the quantum algebra we deduce the constraints among the physical observables JAB ,J0 and quantize the theory covariantly. Among other things, we compute the Casimir eigenvalues of the unitary irreducible representation of SU(2, 2) which classi?es the physical states in any of the gauge ?xed version of the theory 17

(with the di?erent 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 ?xed 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 ? A to distinguish them the constraints, we will call the quantum twistors in this section ZA , Z (h) ? (h)A from the classical Z , Z of the previous sections that were constrained at the classical

A

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. ? A (or equivalently λα and i? According to the twistor action (2.3) ZA and iZ ?α ) are canonical conjugates. Therefore the quantum rules (equivalent to spin & phase space quantum rules) are ?B = δ B . ZA , Z A (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 ? = 2h, U(1) constraint on the twistors. At the classical level the constraint was J0 = ZZ 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 ?0 as a Hermitian operator applied on states (we write it as J ?0 to distinguish it requires J from the classical version) ?0 = 1 ZA Z ?A + Z ? A ZA , J ?0 |phys = 2h|phys . J 2 ?0 has non-trivial commutation relations with ZA , Z ? The operator J basic commutation rules above ?0 , ZA = ?ZA , J ?0 , Z ? J

A A

(7.2)

which follow from the

? A. =Z

(7.3)

?A = Z ? A ZA + 4 we can extract from By rearranging the orders of the quantum operators ZA Z (7.2) the following relations ? =J ?0 ? 2, T r Z Z ? =J ?0 + 2. ZZ (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 18

JAB at the quantum level as follows JAB ?B ? 1 T r Z Z ? δB= = ZA Z A 4 ? ? ? J0 + 2 ZZ 4

B

.

A

(7.5)

In this expression the order of the quantum operators matters and gives rise to the shift ?0 + 2 in contrast to the corresponding classical expression. The commutation rules J0 → J ? A are computed from the basic commutators (7.1), among the generators J B and the ZA , Z

A

1 JAB , ZC = ?δCB ZA + ZC δAB , 4 JAB , JCD = δAD JCB ? δCB JAD ,

?D δ B ?D = δ DZ ?B ? 1 Z JAB , Z A A 4 ?0 , J B = 0. J

A

(7.6) (7.7)

We see from these that the gauge invariant observables JAB satisfy the SU(2, 2) Lie algebra, ? A transform like the quartets 4, ? ?0 while the ZA , Z 4 of SU(2, 2) . Note that the operator J commutes with the generators JAB , therefore JAB is U(1) gauge invariant, and furthermore ?0 must be a function of the Casimir operators of SU(2, 2) . When J ?0 takes the value 2h J 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. ?0 can only have From the quantum rules (7.3), it is evident that the U(1) generator J integer eigenvalues since it acts like a number of operator. More directly, through Eq.(7.4) ? . Therefore the theory is consistent at the quantum it is related to the number operator ZZ level (7.2) provided 2h is an integer. Let us now compute the square of the matrix JAB . By using the form (7.5) we have ?0 +2 ?0 +2 ?0 +2 2 ?0 +2 ??J ? Z ? ? 2J ?+ J ??J Z Z = Z ZZ Z Z where we have used (J J ) = Z Z 4 4 4 4 ? Z ? ?0 , ZA Z ? B = 0. Now we elaborate Z ZZ J

B A

?0 ? 2 Z ?B = J ?0 ? 1 ZA Z ? B where = ZA J ?0 +2 B ?B = J B + J δA . we ?rst used (7.4) and then (7.3). Finally we note from (7.5) that ZA Z A 4 Putting these observations together we can rewrite the right hand side of (J J ) in terms of ?0 as follows11 J and J ?0 3 ?2 J J ?4 . (7.8) ?2 J+ (J J ) = 2 16 0

11

A similar structure at the classical level can be easily computed by squaring the expression for J in Eq.(6.2) ? A ZA = 2h. This yields the classical version J C J B = and applying the classical constraint J0 = Z A C J0 3 2 B 3 2 B B B J + J δ = h J + h δ , which is di?erent than the quantum equation (7.8). Thus, the 0 A A A A 2 16 4 2 2 quadratic Casimir at the classical level is computed as C2 = 3 J = 3 h which is di?erent than the 4 0 quantum value in (7.16).

19

?0 which This equation is a constraint satis?ed by the global [SU(2, 2) ×U(1)]R charges JAB , J 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 J J J . Using this process repeatedly we ?nd all the powers of the matrix J (J )n = αn J + βn , where ?0 ) = αn (J 1 ?0 ? 1 J ?0 ) = 3 J ?2 ? 4 βn (J 16 0 3 ? J0 ? 2 4

n n

(7.9)

?

?1 ? J0 + 2 4

,

(7.10) (7.11)

?0 ). αn?1 (J

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 ?0 J +β J ?0 f (J ) = α J where ?0 = α J ?0 β J 1 3 ? ?1 ? f J0 ? 2 ?f J0 + 2 , ? 4 4 J0 ? 1 ? ?0 ? 2 ? 3 J 1 ? J0 + 2 3 ? J0 ? 2 + = f f ?0 ? 1 4 4 4 J (7.13) ?1 ? J0 + 2 4 ? (7.12)

?nd12

We can compute all the Casimir operators by taking the trace of J n in Eq.(7.9), so we

?.

(7.14)

?0 ) ≡ T r (J )n = 4βn (J ?0 ) = 3 J ?2 ? 4 αn?1 (J ?0 ). C n (J (7.15) 4 0 In particular the quadratic, cubic and quartic Casimir operators of SU(2, 2) =SO(6, 2) are computed at the quantum level as ?2 ? 4 , C3 (J ?0 ) = 3 J ?2 ? 4 ?0 ) = 3 J C 2 (J 0 4 8 0 ?0 ) = 3 J ?2 ? 4 7J ?2 ? 32J ?0 + 52 . C 4 (J 0 64 0

12

?0 ? 4 , J

(7.16) (7.17)

Other de?nitions of Cn could di?er from ours by normalization or linear combinations of the T r (J n ).

20

?0 on physical states J ?0 |phys = 2h|phys completely ?xes The eigenvalue of the operator J the unitary SU(2, 2) representation that classi?es 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. ?0 → h = 0 we obtain C2 = ?3, C3 = 6, C4 = ? 39 , In particular, for spinless particles J 4 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

1 C2 = 2 LM N LM N → 1 ? d2 /4 on physical states that satisfy X 2 = 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 in?nite 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 JAB 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 su?cient 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 in?nite 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, 21

all have the same Casimir eigenvalues C2 = ?3, C3 = 6, C4 = ? 39 at the quantum level. 4 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 JAB and checking that it has the same value when computed in terms of the spin & phase space of any of the systems 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 representation correspond to simultaneously commuting operators that include the Hamiltonian. The Hamiltonian of each system is some operator constructed from the observables JAB , and so are the other simultaneously diagonalizable observables. Therefore, the di?erent 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 JAB between dually related states of di?erent systems. The prediction is that all such computations within di?erent systems must give the same result. Given that JAB is expressed in terms of rather di?erent phase space and spin degrees of freedom in each dynamical system with a di?erent 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 uni?cation 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 bene?cially 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 22

end we write the master equation (7.8) in the form J ? 3 ? J0 ? 2 4

?0 +2 J 4

J +

1 ? J0 + 2 4

= 0.

(8.1)

Recall the quantum equation (7.5) J + J ?

? so the equation above is equivalent to = Z Z, Z = 0. (8.2)

3 ? J0 ? 2 4

This is a 4 × 4 matrix eigenvalue equation with operator entries. The general solution is Z= J + 1 ? J0 + 2 4 ? V (8.3)

?A is any spinor up to a normalization. This is veri?ed by using the master equation where V ?0 ? 2 Z = J ? 3 J ?0 ? 2 ?0 + 2 V ? = 0. Not(8.1) which gives J ? 3 J J +1 J

4 4 4

ing that the solution (8.3) has the same form as the classical version of the twistor transform ?0 + 2, we conclude that the V ?A introduced in Eq.(6.8), except for the quantum shift J0 → J above is the quantum version of the VA discussed earlier (up to a possible renormalization13 ), as belonging to the coset SU(2, 3) /[SU(2, 2) ×U(1)]. ?A is a quantum operator whose commutation rules must be compatible with those Now V ? A, J ?0 and J B . Its commutation rules with J B , J ?0 are straightforward and ?xed of ZA , Z

A A

uniquely by the SU(2, 2) ×U(1) covariance ?0 , V ?A = ?V ?A , J ? ?0 , V J

A

? , =V ? JAB , V

D B D ? ? 1V ? δ B. = δAD V A 4

A

(8.4) (8.5)

?A + 1 V ?C = ?δ B V ?C δ B , JAB , V C A 4

?A follow from imposing the quantum property ZZ ? =J ?0 ? Other quantum properties of V 2 in (7.4). Inserting Z of the form (8.3), using the master equation, and observing the commutation rules (8.4), we obtain ? V J + ?0 + 2 J 4 ? = 1. V (8.6)

13

? is valid in the whole Hilbert space, not only in the subspace that satis?es the The quantum version of V ? → 2h. In particular, in the high spin version, already at the classical level we must U(1) constraint J √ √ 0√ ? take V = V ( 2h/ J0 ) and then rescale it V 2h → V as described in previous footnotes. So in the full √ √ ? = 2hV (J ?0 + γ )?1/2 (or the rescaled version V 2h → V ) with quantum Hilbert space we must take V ?0 + γ )?1/2 . the possibly quantum shifted operator (J

23

This is related to (5.11) if we take (5.9) into account by including the quantum shift J0 → ?0 + 2. Considering (8.3) this equation may also be written as J ?Z = Z ?V ? = 1. V

B

(8.7)

? ]. After some ? B = δ B to deduce the quantum rules for [V ?A , V Next we impose ZA , Z A ? B = δ B is algebra we learn that the most general form compatible with ZA , Z

A B

?A , V ? V

B

?V ? V δA B + =? ? J0 ? 1

?0 ? 2 ?0 ? 2 J J ? M (J ? 3 ) + (J ? 3 )M 4 4

,

A

(8.8)

? = (η2,2 ) M ? (η2,2 )?1 . The matrix M B could not where MAB is some complex matrix and M A ?A itself. be determined uniquely because of the 3/4 kappa gauge freedom in the choice of V ?A corresponds to eliminating 3 of its components A maximally gauge ?xed version of V 1,2,4 ?2,3,4 = 0 by using the 3/4 kappa symmetry, leaving only A ≡ V ?1 = 0. Then we ?nd V ? V =0

?1/2 ?iφ

? 3 = A? . Let us analyze the quantum properties of this gauge in the context of the and V formalism above. From Eq.(8.6) we determine A = (J3 1 ) then from Eq.(8.3) we ?nd ZA . ZA = JA 1 + ?0 + 2 J δA 1 4 J3 1

?1/2 ?iφ

e

, where φ is a phase, and

e

? A = eiφ J 1 , Z 3

?1/2

J3

A

+

?0 + 2 J δ3 4

A

. (8.9)

We see that, except for the overall phase, ZA is completely determined in terms of the gauge invariant JAB . We use a set of gamma matrices ΓM given in ([6],[11]) to write JAB =

1 MN J 4i B as an explicit matrix so that ZA can be written in terms of the (ΓM N )A

15 SO(4, 2) =SU(2, 2) generators J M N . We ?nd ?

1 12 J 2

+

1 +? J 2i i √ 2

+

′

1 +′ ?′ J 2i

+

?0 +2 J 4

? ? ? ZA = ? ? ? ? A = Z ? η2,2 and Z

1 √ 2 A

(J +1 + iJ +2 ) J+ + J + 1 + iJ + 2

′ ′

?

i √ 2

? ? e?iφ ? ?√ ′ , ? J+ + ?

′

(8.10)

with i = 1, 2 corresponds to using the lightcone combinations X ± = (X 0 ± X 1 ). 24

. The orders of the operators here are important. The basis M = ±′ , ±, i

1 √ 2

′ ′

X0 ± X1 , X± =

? A in (8.10) are guaranteed to satisfy the twistor commutaFrom our setup above, the ZA , Z B ? B = δ B provided we insure that the V ?A , V ? have the quantum properties tion rules ZA , Z

A

given in Eqs.(8.4,8.5,8.8). These are satis?ed provided we take the following non-trivial commutation rules for φ ?0 , e±iφ = ±e±iφ , [J 12 , e±iφ ] = ± 1 e±iφ ?0 = i, [φ, J12 ] = i ? J φ, J 2 2 (8.11)

B

?A , V ? ] = 0, while all other commutators between φ and J M N vanish. Then (8.8) becomes [V so MAB vanishes in this gauge. Indeed one can check directly that only by using the Lie ?0 and the commutation rules for φ in (8.11), we obtain ZA , Z ?B = δ B , algebra for the J M N , J A 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 ?A . However, the global symmetry SU(2, 2) is still intact we chose a non-covariant gauge for V since the correct commutation rules between the twistors and J M N or the JA

B

as given in

(7.6,7.7) are built in, and are automatically satis?ed. Therefore, despite the lack of manifest covariance, the expression for ZA in (8.10) transforms covariantly as the spinor of SU(2, 2) . ?A . Once a gauge is picked It is now evident that one has many choices of gauges for V 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 J M N is

˙ ?α =

1 +′ ?′ α 1 1 ′ ˙ ˙ ˙ β β J?ν (? σ ?ν )α J v ˙ , λα = √ J + ? (σ? )αβ ˙v + ˙v . β 4i 2i 2 1 ′ √ v ?σ? vJ + ? = 1. 2

(8.12)

with the constraint (8.13)

?M covers several of the systems listed in section (I). The spinless case was This gauge for V 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 JAB or equivalently J M N . 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 J M N = LM N + S M N . There are other ways of describing spinning 25

particles. For example, one can start with a 2T-physics action that uses fermions ψ M (τ ) [27] instead of our bosonic variables VA (τ ) . Since we only use the gauge invariant J M N , our quantum twistor transform (8.3) applies to all such descriptions of spinning particles, with ? and the new spin degrees of freedom. In particular in the an appropriate relation between V ? that yields (8.10) there is no need to seek a relation between V ? and gauge ?xed form of V the other spin degrees of freedom. Therefore, in the form (8.10), if the J M N are produced with the correct quantum algebra SU(2, 2) =SO(4, 2) in any theory, (for example bosonic spinors, or fermions ψ M , 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 J M N . 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) o?ered 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 simpli?es the quantum theory. ?0 , jA , ? We examine the SU(2, 3) charges JAB , J j A 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. ?0 , J B are already given in Eqs.(7.2,7.5) The twistor expressions for J A ? ?A + Z ? A ZA , J B = ZA Z ? B ? J0 + 2 δ B . ?0 = 1 ZA Z (9.1) J A A 2 4 √ We have seen that at the classical level (jA )classical = J0 ZA and now we must ?gure 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.

26

the quantum version jA =

?0 + αZA that gives the correct SU(2, 3) closure property J 5? B jA , ? j B = JAB + J 0δ . 4 A (9.2) jA , ? j B , jC +

The coe?cient

5 4

is determined by consistency with the Jacobi identity

? j B , jC , jA + [jC , jA , ] , ? j B = 0, and the requirement that the commutators of jA with ?0 be just like those of ZA given in Eqs.(7.6,7.7), as part of the SU(2, 3) Lie algebra. J B, J

A

So we carry out the computation in Eq.(9.2) as follows jA , ? jB = ?0 + αZA Z ?B J ?0 + α ? Z ?B J ?0 + α J ?0 + αZA J (9.3) (9.4) (9.5) (9.6) =

?0 + α ZA Z ?B ? J ?0 + α ? 1 Z ? B ZA = J ?0 + α ? 1 = J = δAB ? B + ZA Z ?B ZA , Z + JAB ? ?0 + α ? 1 + J0 + 2 J 4

?0 To get (9.4) we have used the properties ZA f J

?0 + 1 ZA and Z ?B f J ?0 = f J

?0 = J ?0 + 1 ZA which is used repeatedly, and similarly for Z ? B . To written in the form ZA J ? B = δ B and then used the de?nitions (9.1). By comparing get (9.6) we have used ZA , Z A (9.6) and (9.2) we ?x α = 1/2. Hence the correct quantum version of jA is jA = ?0 + 1 ZA = ZA J 2 ?0 ? 1 . J 2 (9.7)

?0 ? 1 Z ? B for any function f J ?0 . These follow from the commutator J ?0 , ZA = ?ZA f J

?0 + 1 ZA . ?0 = f J The second form is obtained by using ZA f J Note the following properties of the jA , ? jA ? j A jA = jA ? jB = ?0 ? 1 ZZ ? J 2 ?0 ? 1 = J 2 ?0 ? 1 J 2 ?0 + 1 J 2 ?0 ? 2 J J+ 1 ? J0 + 2 4 (9.8) (9.9)

?B ?0 + 1 ZA Z J 2

?0 + 1 = J 2

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 ?2,3 = J g ?

?1 1 ? J +4 J0 j ?0 ?? j ?J ?

L0 00

g

quantum

=

(9.10)

=?

?B ?Z

?B ? 1 δ B ZA Z 2 A ?0 + J

?0 + 1 ZA J 2 ?0 ?J

1 2

?,

(9.11)

27

?0 , J given in Eq.(9.1). with J

At the classical level, the square of the matrix J2,3 vanishes since L2 = 0 as follows (J2,3 )2

classical

=

g ?1

L0 00

g

g ?1

L0 g 00

= g ?1

L2 0 0 0

g = 0.

(9.12)

At the quantum level we ?nd the following non-zero result which is SU(2, 3) covariant ?2,3 J

2

=? =?

?

5 ? J2,3 ? 1. 2

? ?Z

?? ZZ

1 2 1 2

?0 + 1 Z J 2 ?0 ?J

?0 + J

?2 ?

(9.13) (9.14)

n

?2,3 By repeatedly using the same equation we can compute all powers J quadratic Casimir is Tr ?2,3 J

2

, and by taking

traces we obtain the Casimir eigenvalues of the SU(2, 3) representation. For example the

= ?5.

(9.15)

Written out in terms of the charges, Eq.(9.14) becomes ? j J J +1 4 0 ?0 ?? j ?J

2

=?

5 2

? j J +1 J 4 0 ?0 ?? j ?J

? 1.

(9.16)

Collecting terms in each block we obtain the following relations among the gauge invariant ?0 , j, ? charges J , J j 1? J+ J 0 4

2

? j? j+

1? 5 J+ J 0 2 4

+ 1 = 0,

(9.17) (9.18) (9.19)

5 1? ? J+ J 0 j ? j J0 + j = 0 , 4 2 2 5 ?0 + 1 = 0. ?0 ? J ?? jj + J 2

Combined with the information in Eq.(9.9) the ?rst equation is equivalent to the master ?0 = J ?0 j + j, the second equation is equivalent to the quantum equation (7.8). After using j J 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 ?2,3 + 2 J ?2,3 + J

1 2

?2,3 J

2

5 ?2,3 J = ?2

? 1, or equivalently

= 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 28

property of a ?xed 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-

a group SU(2, (2 + n) |N ) . This generalizes the spinor VA to VA 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 symmetries 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 2Tphysics 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 embedded in g =SO(9? ) =SO(6, 3) or g =SO(10? ) =SO(6, 4) . The spinor variables 29 for 6+2 dimensions, the starting point is the 8×8 spinor version of SO(8? ) =SO(6, 2)

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 classi?ed. 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 compacti?ed on AdS5 ×S5 . So our methods have a direct bearing on M theory. In the case of [28] the matrix insertion generalized to

L(4,2) 0 0 L(6,0) L0 0 0

in the 2T-physics action was

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 ?exible gauge ?xing mechanisms in the context of 2T-physics ?eld 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 ?eld 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 30

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

a quantization of high spin theories including the spinor VA (or its generalizations VA )

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. [2] R. Penrose, “Twistor Algebra,” J. Math. Phys. 8 (1967) 345; “Twistor theory, its aims and achievements, in Quantum Gravity”, C.J. Isham et. al. (Eds.), Clarendon, Oxford 1975, p. 268-407; “The Nonlinear Graviton”, Gen. Rel. Grav. 7 (1976) 171; “The Twistor Program,” Rept. Math. Phys. 12 (1977) 65. [3] R. Penrose and M.A. MacCallum, “An approach to the quantization of ?elds and space-time”, Phys. Rept. C6 (1972) 241; R. Penrose and W. Rindler, Spinors and space-time II, Cambridge Univ. Press (1986). [4] A. Ferber, Nucl. Phys. B 132 (1977) 55. [5] T. Shirafuji, “Lagrangian Mechanics of Massless Particles with Spin,” Prog. Theor. Phys. 70 (1983) 18.

31

[6] I. Bars and M. Picon, “Single twistor description of massless, massive, AdS, and other interacting particles,” Phys. Rev. D73 (2006) 064002 [arXiv:hep-th/0512091]; “Twistor Transform in d Dimensions and a Unifying Role for Twistors,” Phys. Rev. D73 (2006) 064033, [arXiv:hep-th/0512348]. [7] I. Bars, “Lectures on twistors,” [arXiv:hep-th/0601091], appeared in Superstring Theory and M-theory, Ed. J.X. Lu, page ; and in Quantum Theory and Symmetries IV, Ed. V.K. Dobrev, Heron Press (2006), Vol.2, page 487 (Bulgarian Journal of Physics supplement, Vol. 33). [8] I. Bars, C. Deliduman and O. Andreev, “ Gauged Duality, Conformal Symmetry and Spacetime with Two Times” , Phys. Rev. D58 (1998) 066004 [arXiv:hep-th/9803188]. For reviews of subsequent work see: I. Bars, “ Two-Time Physics” , in the Proc. of the 22nd Intl. Colloq. on Group Theoretical Methods in Physics, Eds. S. Corney at. al., World Scienti?c 1999, [arXiv:hep-th/9809034]; “ Survey of two-time physics,” Class. Quant. Grav. 18, 3113 (2001) [arXiv:hep-th/0008164]; “ 2T-physics 2001,” AIP Conf. Proc. 589 (2001), pp.18-30; AIP Conf. Proc. 607 (2001), pp.17-29 [arXiv:hep-th/0106021]. [9] I. Bars, “ 2T physics formulation of superconformal dynamics relating to twistors and supertwistors,” Phys. Lett. B 483, 248 (2000) [arXiv:hep-th/0004090]. “Twistors and 2T-physics,” AIP Conf. Proc. 767 (2005) 3 [arXiv:hep-th/0502065]. [10] I. Bars, “The standard model of particles and forces in the framework of 2T-physics”, Phys. Rev. D74 (2006) 085019 [arXiv:hep-th/0606045]. For a summary see “The Standard Model as a 2T-physics theory,” arXiv:hep-th/0610187. [11] I. Bars, try”, Y-C. Kuo, “Field Theory in 2T-physics with N ibid. “Supersymmetric = 1 supersymme?eld theory”,

arXiv:hep-th/0702089;

2T-physics

arXiv:hep-th/0703002. [12] F. Cachazo, P. Svrcek and E. Witten, “ MHV vertices and tree amplitudes in gauge theory”, JHEP 0409 (2004) 006 [arXiv:hep-th/0403047]; “ Twistor space structure of oneloop amplitudes in gauge theory”, JHEP 0410 (2004) 074 [arXiv:hep-th/0406177]; “Gauge theory amplitudes in twistor space and holomorphic anomaly”, JHEP 0410 (2004) 077 [arXiv:hep-th/0409245]. [13] For a review of Super Yang-Mills computations and a complete set of references see: F.Cachazo and P.Svrcek, “Lectures on twistor strings and perturbative Yang-Mills theory,” PoS RTN2005 (2005) 004, [arXiv:hep-th/0504194].

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