2 Time Physics and Field theory Kuo, Yueh-Cheng

2T-physics 1T spacetimes & dynamics (time, Hamiltonian) are emergent concepts from 2T phase space The same 2T system in (d,2) has many 1T holographic images in (d-1,1), obey duality Each 1T image has hidden symmetries that reveal the hidden dimensions (d,2) 3) Duality • 1T solutions of Qij(X,P)=0 are dual to one another; duality group is gauge group Sp(2,R). • Simplest example (see figure): (d,2) to (d-1,1) holography gives many 1T systems with various 1T dynamics. These are images of the same “free particle” in 2T physics in flat 2T spacetime. 1) Gauge symmetry • Fundamental concept is Sp(2,R) gauge symmetry: Position and momentum (X,P) are indistinguishable at any instant. • This symmetry demands 2T signature (-,-,+,+,+,…,+) to have nontrivial gauge invariant subspace Qij(X,P)=0. • Unitarity and causality are satisfied thanks to symmetry. Sp(2,R) gauge choices. Some combination of XM,PM fixed as t,H 4) Hidden symmetry (for the example in figure) • The action of each 1T image has hidden SO(d,2) symmetry. • Quantum: SO(d,2) global sym realized in same representation for all images, C2=1-d2/4. 2) Holography • 1T-physics is derived from 2T physics by gauge fixing Sp(2,R) from (d,2) phase space to (d-1,1) phase space. Can fix 3 pairs of (X,P), fix 2 or 3. • The perspective of (d-1,1) in (d,2) determines “time” and H in the emergent spacetime. • The same (d,2) system has many 1T holographic images with various 1T perspectives. 5) Unification • Different observers can use different emergent (t,H) to describe the same 2T system. • This unifies many emergent 1T dynamical systems into a single class that represents the same 2T system with an action based on some Qij(X,P). 6) Generalizations found • Spinning particles: OSp(2|n); Spacetime SUSY • Interactions with all backgrounds (E&M, gravity, etc.) • 2T field theory (standard model) •2T string/p-brane • Twistor superstring 7) Generalizations in progress • New twistor superstrings in higher dimensions. • Higher unification, powerful guide toward M-theory • 13D for M-theory (10,1)+(1,1)=(11,2) suggests OSp(1|64) global SUSY.

Basic 2T physics Generalizations * Spinning particles Outlook * Supersymmetric particles and Twistors * 2T field theory Outlook

Heuristic Motivation for 2 Time 1985 Witten low energy, strong coupling limit of type 2A string  10+1 dim supergravity suggest a quantum theory (with N=1 supersymmetry) in 10+1 dim whose classical limit is supergravity The same supersymmetry can also be realized in 10+2 dim spacetime Note: 11+1 D spinor: chiral and complex 10+2 D spinor: chiral and real Self-dual But how about ghosts arising from one more time?

Spacetime signature determined by gauge symmetry EMERGENT DYNAMICS AND SPACE-TIMES return

Some examples of gauge fixing Covariant quantization: How could one obtain the three constraints from a Lagrangian of scalar field? 2 gauge choices made. t reparametrization remains.

Some examples of gauge fixing 3 gauge choices made. Including t reparametrization.

More examples of gauge fixing

Holography and emergent spacetime 1T-physics is derived from 2T physics by gauge fixing Sp(2,R) from (d,2) phase space to (d-1,1) phase space. SO(d,2) global symmetry (note: generators of SO(d,2) commute with those of Sp(2,R)) is realized for all images in the same unitary irreducible representation, with Casimir C2=1-d2/4. This is the singleton. Can fix 3 pairs of (X,P): 3 gauge parameters and 3 constraints. Fix 2 or 3. The perspective of (d-1,1) in (d,2) determines “time” and Hamiltonian in the emergent spacetime. Different observers can use different emergent (t,H) to describe the same 2T system. The same (d,2) system has many 1T holographic images with various 1T perspectives.

Many emergent spacetimes Duality 1T solutions of Qik(X,P)=0 (holographic images) are dual to one another. Duality group is gauge group Sp(2,R): Transform from one fixed gauge to another fixed gauge. Simplest example (figure): (d,2) to (d-1,1) holography gives many 1T systems with various 1T dynamics. These are images of the same “free particle” in 2T physics in flat 2T spacetime. Many emergent spacetimes

Generalizations obtained Spinning particles: use OSp(2|n) Spacetime SUSY: special supergroups Interactions with all backgrounds (E&M, gravity, etc.) 2T strings/branes (incomplete) 2T field theory (new progresses recently) Twistors in d=3,4,6,10,11 ; Twistor superstring in d=4

SO(d,2) unitary representation unique for a fixed spin=n/2.

Gauge Fixing: (for example, n=1) One could choose other gauges or do covaraint quantization. These on-shell condition will coincide with those constraints imposed from considering spinning 2T particle. Obtain E&M, gravity, etc. in d dims from background fields f(X,P, y) in d+2 dims. –> holographs.

5) - String/brane theory in 2T (9906223,. 0407239) Twistors emerge in this approach If D-branes admitted, then more general (super)groups can be used, in particular a toy M-model in (11,2)=(10,1)+(1,1) with Gd=OSp(1|64)13 4) – 2T Field Theory . - Non-commutative FT f(X,P) (0104135, 0106013) similar to string field theory, Moyal star. - BRST 2T Field theory - Standard Model 5) - String/brane theory in 2T (9906223,. 0407239) -Twistor superstring in 2T (0407239, 0502065 ) both 4 & 5 need more work

Spacetime SUSY: 2T-superparticle Supergroup Gd contains spin(d,2) and R-symmetry subgroups Local symmetries OSp(n|2)xGdleft including SO(d,2) and kappa Global symmetries: Gdright

Local symmetry embedded in Gleft local spin(d,2) x R acts on g from left as spinors acts on (X,P) as vectors Local kappa symmetry (off diagonal in G) acts on g from left acts also on sp(2,R) gauge field Aij

More dualities: 1T images of unique 2T-physics superparticle via gauge fixing 2T-parent theory has Y=(X,P,y) and g s-model gauge fix part of (X,P,y); LMN linear Integrate out remaining P e.g. AdS5xS5 sigma model SU(2,2|4)/SO(4,1)xSO(5)

Spacetime (or particle) gauge Residual local sym: reparametrization and K sym Global sym: superconformal

Twistor (group) gauge Coupling of type-1

Twistors for d=4 superparticle with N supersymmetries

2T field theory hep-th/xxxxxxx by I. Bars and Y. C. Kuo BRST operator Fix gauge: Action

2T field theory No Interaction terms maintaining the gauge sym of free field and without giving trivial physics ( ) can be written down. One could try to modify the BRST operators and hence the corresponding gauge sym

Outlook Standard Model as a 2T field theory 2T string/brane (New twistor superstrings in higher dimensions: d=3,4,6,10) Higher unification, powerful guide toward M-theory (hidden symmetries, dimensions). (13D for M-theory (10,1)+(1,1)=(11,2) suggests OSp(1|64) global SUSY.)