1. 2 Time Physics and Field theory
Kuo, Yueh-Cheng

2. 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.

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

4. 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?

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

6. 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.

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

8. More examples of gauge fixing

9. 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.

10. 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

11. 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

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

13. 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.

14. 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) ( , ) similar to string field theory, Moyal star.
- BRST 2T Field theory
- Standard Model
5) - String/brane theory in 2T ( , )
-Twistor superstring in 2T ( , )
both 4 & 5 need more work

15. 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

16. 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

17. 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)

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

19. Twistor (group) gauge
Coupling of type-1

21. Twistors for d=4 superparticle with N supersymmetries

22. 2T field theory hep-th/xxxxxxx by I. Bars and Y. C. Kuo
BRST operator
Fix gauge:
Action

23. 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

24. 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.)