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2 Time Physics and Field theoryKuo, Yueh-Cheng

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2T-physics2T-physics1T 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)

1) Gauge symmetry• Fundamental concept isSp(2,R) gauge symmetry: Position and momentum (X,P) are indistinguishableat any instant.• This symmetry demands 2T signature (-,-,+,+,+,…,+)to have nontrivial gaugeinvariant subspace Qij(X,P)=0.• Unitarity and causality aresatisfied thanks to symmetry.

2) Holography• 1T-physics is derived from 2T physics by gauge fixingSp(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 hasmany 1T holographic images with various 1T perspectives.

5) Unification• Different observers can usedifferent emergent (t,H) todescribe the same 2T system.• This unifies many emergent 1T dynamical systems into asingle class that represents the same 2T system with anaction based on some Qij(X,P).

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 givesmany 1T systems with various1T dynamics. These are imagesof the same “free particle” in 2T physics in flat 2T spacetime.

4) Hidden symmetry(for the example in figure)• The action of each 1T imagehas hidden SO(d,2) symmetry.• Quantum: SO(d,2) global symrealized in same representationfor all images, C2=1-d2/4.

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.

Sp(2,R) gauge choices. Some combination of XM,PM fixed as t,H

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• Basic 2T physics

• Generalizations * Spinning particles

* Supersymmetric particles and Twistors

* 2T field theory

• Outlook

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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

Q, Q p C

C

32332

528111021

55111098754321

462

Q, Q C C

Self-dual121121

6632332

528

121110987654321

12 462

But how about ghosts arising from one more time?

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Spacetime signature determined by gauge symmetry

EMERGENT DYNAMICS AND SPACE-TIMES

return

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Some examples of gauge fixing

2 gauge choices made. reparametrization remains.

Covariant quantization:

How could one obtain the three constraints from a Lagrangian of scalar field?

2 0 1XM M d 22 0 2

X2 0 33 X2F2 X2XM M d22F 0 41 'X2XM M

d 22F X2 2F 0 5

parametrize: XM ' '

x

x2

2XM M 4 FX20

d22 x,

x2

25

0

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Some examples of gauge fixing

3 gauge choices made. Including reparametrization.

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More examples of gauge fixing

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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.

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Duality • 1T solutions of Qik(X,P)=0

(holographic images) are dual to one another. Duality group isgauge 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

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Generalizations

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

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

13Obtain E&M, gravity, etc. in d dims from background fields (X,P, ) in d+2 dims. –> holographs.

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.

M XM 0, X2 0, XM PM 0, MPM 0, P2 0

XM ' '

1, x2

2,

x

PM 0, xp, p M 0, x , remainingconstraint: p2 0, p 0 ,

S XMPM

M M

MN

2AabYM

aYNb

S x p A222

p2C2 p

Dirac Equ.for masslessspin12 particle

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4) – 2T Field Theory

. - Non-commutative FT (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

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

Twistors emerge in this approach

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Spacetime SUSY: 2T-superparticle

Local symmetries OSp(n|2)xGd

left including SO(d,2)

and kappa Global

symmetries: Gdright

Supergroup Gd contains spin(d,2) and R-symmetry subgroups

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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

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More dualities: 1T images of unique 2T-physics superparticle via gauge fixing

Spacetime gauge-eliminate all bosons from g

keep only ½ fermi part: -fix Y=(X,P,) (d,2) to (d-1,1)

(x,p,)1T superparticle (& duals)

group/twistor gaugekill Y=(X,P,) completely

keep only g

constrained twistors/oscillators

2T-parent theory has Y=(X,P,) and g

-model gaugefix part of (X,P,); LMN linear

Integrate out remaining P

e.g. AdS5xS5 sigma modelSU(2,2|4)/SO(4,1)xSO(5)

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Spacetime (or particle) gauge

Residual local sym: reparametrization and K sym

Global sym: superconformal

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Twistor (group) gaugeCoupling of type-1

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Twistors for d=4 superparticle with N supersymmetries

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2T field theory hep-th/xxxxxxx by I. Bars and Y. C. Kuo

BRST operator

Action

gaugesym: X, c Q X, c S 0 QX, c 0

Q cmfm i2 m n kc

mcn bk

f1 14P2 X2

f2 14P2 X2

f3 14PMXM X

MPMX, c 0X cm mXccm mXccc 0X

Physicalstatesliesin Q cohomology

f1 0 0

f2 0 0

f3 0 0

m, m,

0 are puregauge

S Q dd2Xd3cQ

dd2X 0fm

m mfm 0 m n k m

fn k i m m

Fix gauge:

Sdd2X0f220dd2X0

P2

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addxdx'd 2 x' x2 d2 d0p2

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ad ddx0p2

20

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2T field theoryNo Interaction terms maintaining the gauge sym of free field and without giving trivial physics ( ) can be written down.

0 0

One could try to modify the BRST operators and hence the corresponding gauge sym

f11 X2

2

f22 P2

20

0f12

14PMXM X

MPM0

0 002d2

S constantddx 0p2

2 0

00dd2

d 4

Interaction 0 04

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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.)