1

Field Theoretic Formulation of Two-

Time Physics including Gravity

Itzhak Bars, USC• String theory has encouraged us to study extra space dimensions.

• How about extra time dimensions? (easy to ask, not so easy to realize!! Issues!)

• 2T-physics solves the issues, agrees with 1T-physics, works GENERALLY.

• Unifies aspects of 1T-physics that 1T-physics on its own misses to predict.

• Key: Fundamental principles include momentum/position symmetry. This encodes fundamental laws in d+2 dimensions, with extra info for 1T.

Not much discussion of more time-like dimensions – why?

2

Because the 2T road is risky, dangerous and scary

1) Ghosts! = negative probability2) Causality violation (cause and effect

disconnect)

Illogical: You can kill your ancestors before your birth!

But in search of M-theory there has been some attraction to more T’s

Extended SUSY of M-theory is (10+2) SUSY (Bars -1995),F-theory (10+2 or 11+1), S-theory (11+2), U-theory (other

signatures), etc.2T or more used as math tricks, did not investigate full fledged

consequences.

Experience over ½ century taught us gauge symmetry overcomes inconsistencies for the first time dimension. More T’s would require a bigger gauge symmetry.A solution to inconsistencies: a new gauge symmetry (1998) 2T-

physics

Historical motivation for 2T-physics Extended SUSY of 11D M-theory is SUSY in (10+2) dims. (Bars -1995)

3

{Q32,Q32}=gmPm+gmnZ(2)mn+gmnpqrZ(5)

mnpqr in 11D= 10+1 32= real spinor of SO(10,1)

{Q32,Q32}=GMN Z(2)MN+GMNPQRS Z(6,+)

MNPQRS , in 12D =10+2 32= real Weyl spinor of SO(10,2) M=0’,m with m=0,1,2, … ,10 , 2 times!!

This led to S-theory (I.B. 1996), in 13D = 11+2 An algebraic BPS approach based on OSp(1|64) SUSY, Contains various corners of M-theory SUSYs: 11D SUSY, and 10D SUSYs type IIA, IIB, heterotic, type I

S-theory led to 2T-physics (1998). Taking 2 times seriously.

Clues for the fundamental principle:

Position Momentum GLOBAL symmetry

position/momentum are at same level of importance before a specific Hamiltonian is chosen in classical or quantum mechanics Boundary conditions, or any measurement. Poisson brackets or quantum commutators Any Lagrangian:

More general: continuous GLOBAL symmetry: Sp(2,R)

Even more general is GLOBAL canonical transformations

Invariant

H

Symmetry XP, P-X

5

New Gauge PrinciplesRequire local Sp(2,R) symmetry

XP indistinguishable continuouslyat EVERY INSTANT for ALL MOTION

Generalizes t reparametrization

Generalized Sp(2,R) generators Qij(X,P) instead of the simpler Qij(X,P)=Xi∙Xj=(X2,P2,X.P).

With these we can include ALL background fields. Gravity, E&M, high spin fields, etc.Qij(X,P)=Q0

ij(X)+QMij(X)PM+ QMN

ij(X)PMPN+…

More generalizations: Particles with SPIN or SUSY; strings/branes (partial)

and finally Field Theory.

6

1) New symmetry allows only highly symmetric motions (gauge invariants) little room to maneuver.

2) With only 1 time the highly symmetric motions impossible. Collapse to nothing.

3) Extra 1+1 dimensions necessary 4+2 !! No less and no more than 2T.

4) Straightjacket in 4+2 makes allowed motions effectively 3+1 motions (like shadows on walls).

Non-trivial: Gauge fix 4+2 to 3+1 (3 gauge parameters)many 3+1 shadows emerge for same 4+2 spacetime history. Each shadow contains only one timeline (a mixture of all 4+2)

How does it work?

t2

w

t1,x,y,z, …

(Position & Time) (Momentum & Energy)

Indistinguishable at any instant

Constraints: generators vanish !!!

Leftover Information: Observers stuck in 3+1 experience 4+2 dims through relations among “shadows”. Similar to observers stuck on 2D walls that see many shadows of the same object moving in a 3D room, and think of the shadows as different “beasts”, but can discover they are related.

7

Xi m

Xi+’ Xi

-’

8

Xi 0’

Xi 0 Xi

I

9

An example: Massive relativistic particle gauge

SO(d,2) is hidden symmetry of the massive action. Looks like conformal tranformations deformed by mass. The symmetry in 1T theory is a clue of extra 1+1 dimensions, including 2T.

embed phase space in 3+1 into phase space in 4+2

Make 2 gauge choices solve 2 constraints X2=X.P=0 t reparametrization and

one constraint remains.

x=MN{LMN, x}, p=MN{LMN, p},

Gauge invariants

Note a=1 when m=0

.

Particle in Roberstson-Walker

expanding

Universe

Particle in any

Maximally Symmetric Space, e.g.

AdS4-nxSn

Particle in any

Conformally flat Space

singular OK, some

Black Holes

Twistors SU(2,2)=SO(4,2

) twistor transform

for all these

2T-physics Sp(2,R) simplest example

X2=P2=X∙P=0 .

Background flat 6=4+2

dimsSO(4,2)

symmetry

Massless

relativistic particle (pm)2=0

conformal

sy Dirac

Massive relativistic

(pm)2+m2=0Non-

relativistic H=p2/2m

H-atom 3 space

dims H=p2/2m -a/r SO(4)xSO(2) SO(3)xSO(1,2

)

Harmonic oscillator

2 space dims mass

= 3rd dim SO(2,2)xSO(2

)

Shadows from 2T-physics hidden info in 1T-physics

10

2T-physics predicts hidden symmetries and dualities (with parameters) among the “shadows”. 1T-physics misses these phenomena.

• Holography: These emergent holographic shadows are only some examples of much broader phenomena.

Free or interacting systems, with or without mass, in flat or curved 3+1 spacetimes.Analogy: multiple shadows on walls.

Hidden Symm.SO(d,2),

d=4 C2=1-d2/4= -3 singletonEmergen

t paramet

ers mass,

couplings,

curvature, etc.

These emerge in 2T-field theory as well

11

Field equations in 2T-physicsConstraints = 0 on physical states i.e. Sp(2,R) gauge invariant state

Probability amplitude is the field

kinematic #1kinematic #2

dynamical eq. extended with interaction

gauge symmetry

Physical part of field

Can show: 3 eoms in d+2 1 eom in (d-1)+1. 1T interpretation of 1 eom depends on how the kinematic eoms are solved SHADOWS

remainder

Derived from Sp(2,R) in hep-th/0003100; also Dirac 1936 other approach

Kinematic eom´s say how to embed d dims in d+2 dims.

12

Action for scalar field in 2T-physicsObtain 3 equations not just one : 2 kinematic and 1 dynamic.

BRST approach for Sp(2,R). Like string field theory I.B.+Kuo hep-th/0605267

Gauge symmetry (a bit more complicated)

Works only for unique V(F)

Gauge fixed version is more familiar looking

Can gauge fix to F0 which is independent of X2

Minimizing the action gives two equations, so get all 3 Sp(2,R) constraints from the action

kinematic #1,2

dynamical eq.

After gauge fixing, eliminating redundant fields, and simplifications, boils down to a simplified partially gauge fixed form

Homogeneous F0 hence one less X, so alltogether two fewer dimensions

13

Action of the Standard Model in 4+2 dimensions Illustrates main features of 2T field theory

Yukawa couplings to Higgs

Higgs and dilaton

Gauge fields

quarks & leptons 3 families

quadratic mass terms not allowed

SU(3) SU(2) U(1)

4*x4=adjoint

4*x4*=6 vector

14

Emergent scalars in 3+1 dimensions

Embedding of 3+1 in 4+2 defines emergent spacetime xm. This is similar to Sp(2,R) gauge fixing (many shadows)xm and l are homogeneous

coordinates

Solve kinematic equations in extra dimensions

X2=0

Result of gauge fixing and solving kinematic eoms is fields only in 3+1

Remainder is gauge freedom, remove it by fixing the 2Tgauge-symmetry at any , ,l k xDynamics

only in 3+1

= k-1

15

Action for gauge fields in 2T-physicsObtain kinematic and dynamic equations from the

action

kinematic #1,2

dynamical eq.

remainder can be removed by gauge symm.

16

Emergent gauge bosons in 3+1 dimensions

start with YM axial gauge

kinematic equation simplifies homogeneous

There is leftover YM gauge symm.

homogeneous Lenough to gauge fix A+’=0Solution

of X.A=0Only independen

t Use 2Tgauge symmetry to eliminate Vm gauge freedom proportional to X2

FMN is YM gauge invariant but 2Tgauge dependentresult is

standard 3+1 YM Lagrangian

17

Action for fermion field in 2T-physicsObtain 3 equations not just one : 2 kinematic and 1

dynamic.

Minimizing the action gives two equations, so get all OSp(1|2) constraints as eom´s from the action

Any general spinor. Eliminates all spinor components proportional to X2=0.

Homogeneous spinor. Eliminates half of the leftover spinor to remain with spinor in d dimensions rather than spinor in d+2

These kinematic + dynamical equations for left/right spinors in d+2 dimensions descend to Dirac equations for left/right spinors in d dimensions. Extra components are eliminated because of kappa type fermionic symmetry.

kinematic #1 dynamical eq.

Although it looks like one equation one can show that each term vanishes separately due to X2=0.

kinematic #2

18

Emergent fermions in 3+1 dimensions

choose X2x1 2Tgauge symm.

Impose kinematical eom in extra dimension

4 component SU(2,2) chiral fermions

choose x2 2Tgauge symm.

2 component SL(2,C) chiral fermions

4+2 Lagrangian descends to 3+1 standard Lagrangian. No explicit X. standard 3+1 kinetic

term

standard 3+1 Yukawa term

Translation invariance in 3+1 comes from rotation invariance in 4+2

19

Shadow Standard Model in 3+1 dimensions agrees with usual Standard Model, but

requires “dilaton” Every term in the 4+2 action is - proportional to k-4 after solving kinematic eoms - and is independent of l after 2Tgauge fixing, S

Normalize to 1Shadow Standard Model in 3+1 has “dilaton” in addition to usual matter.

1. Mass generation: a) electroweak phase transition is driven by dilaton (phenomenology?), b) new mechanisms of mass generation from extra dimension (massive shadow) ???

2. Duals of the Standard Model (other shadows) – new physics (e.g. mass, etc.), computational tools?

3. Can SO(4,2) help for mass hierarchy problem ? (need quantum 2T field theory, not ready yet)

What is new in 3+1 ? (semi-classical)

remainders proportional to X2

eliminated by 2Tgauge fix

Shadow SM is Poincare invariant. More, it has hidden SO(4,2) symmetry

20

“Dilaton” driven Electroweak phase transition Phenomenology of a well motivated SU(3)xSU(2)xU(1) singlet scalar The 4+2 Standard Model has 2Tgauge symmetry which forbids quadratic

mass terms in the scalar potential. Only quartic interactions are permitted Scale invariance in 3+1 ! Quantum effects break scale inv. (maybe in 2T?), give insufficient mass to the Higgs (10 GeV).

Goldstone boson couples to everything the Higgs couples to, but with reduced strength factor a. It is not expected to remain massless because of quantum anomalies that break scale symmetry. Can we see it ? LHC? Dark Matter? Inflaton?

amf/v

Goldstone boson due to spontaneous breaking of scale invariance

All space filled with vev. Makes sense to have dilaton & gravity & strings involved

More complicated gauge choices with XM(xm,pm) mixed

parametrization e.g. massive particle, etc.

These have non-local field interactions,

but approximate local for small mass.

21

Duals of the Standard ModelIB + S.H. Chen + G. Quelin; 0705.2834, 0802.1947

• Instead of choosing the flat 3+1 spacetime gauge, choose any conformally flat spacetime gauge. These include

Robertson – Walker universe Cosmological constant (dS4 or AdS4) Any maximally flat spacetime AdS(4), AdS(3)xS(1), AdS(2)xS(2) A spacetime with singularities (free function a(x)), etc.

All these have hidden SO(4,2) (note this is more than usual Killing vectors).

All are dual transforms of each other as field theories.Duality transformations: Weyl rescaling of background metric and general coordinate reparametrizations taking one field theory with backgrounds to another.The dualities can possibly be used for non-perturbative

computations.

22

Gravity in 2T-physics - 0804.1585 [hep-th]

Solve kinematics, and impose Q11=Q12=0: One of the shadows

Compare to flat case

Sp(2,R) algebra puts kinematic costraints on fields

23

Gravity in 2T-physics – Field Theory

Similarly for W, W

3 equations not just one

Self consistency of all equations, and consistency with Sp(2,R) make it

a unique action.

24

Solve Sp(2,R) kinematic constraints

Conformal scalar, Weyl symmetry

25

All shadow scalars must be conformal scalars

Effect on cosmology ?

26

The 2T field theory approach has been generalized to SUSY

4+2 dimensions, I.B. + Y-C.Kuo, and to appear soon:N=1 : HEP-TH 0702089, HEP-TH 0703002 N=2 : General, including coupling of hyper multipletsN=4 : Super Yang Mills.

N=1, SYM in 10+2 dimensions, to appear soon.10+2 Super Yang-Mills (gives N=1 SYM in 9+1, and N=4 SYM in 4+2) Towards a more covariant M(atrix) Theory?Preparing to develop 2T field theory for:

SUGRA (9+1)+(1,1)=10+2 (see also compactified, hep-th/0208012)SUGRA (11+2) , Particle limit of M-theory (10+1) + (1+1) = 11+2

Expect to produce a dynamical basis for earlier work on algebraic S-theory in 11+2 (hep-th/9607112, 9608061)M-theory type dualities, etc. (see hep-th/9904063)

Beyond the Standard Model or GUTs, and Gravity

27

Local Sp(2,R) 2T-physics seems to work generally! (X,P indistinguishable) is a fundamental principle that seems to agree with what we know about Nature, e.g. as embodied by the Standard Model, etc.

The Standard Model in 4+2 dimensions, Gravity, provide new guidance: a) Dilaton driven electroweak spontaneous breakdown. b) Cosmology Conceptually more appealing source for vev ; could relate to choice of vacuum in string theory Weakly coupled dilaton, possibly not very massive; LHC ? Dark Matter ? Inflaton? Can mass hierarchy problem be solved by conformal symmetry and/or 4+2 with remainders?

Beyond the Standard Model GUTS, SUSY, gravity; all have been elevated to 2T-physics in d+2 dimensions. Strings, branes; tensionless, and twistor superstring, 2T OK. Tensionful incomplete. M-theory; expect 11+2 dimensions OSp(1|64) global SUSY, S-theory.

New technical tools Emergent spacetimes and dynamics; unification; holography; duality; hidden symms. ---Non-perturbative analysis of field theory, including QCD? But wait until we develop quantum field theory directly in 4+2 dimensions. (analogs of AdS-CFT …)

There is more to space-time than can be garnered with 1T-physics. New physical predictions and interpretations . It is more than a math trick.

Summary of Current Status

28

In conclusion …Hidden information is revealed by2T-physics.

A lot more remains to be done with 2T-physics. New testable predictions at every scale of physics

are expected from the hidden dualities and symmetries.

1T-physics on its own is not equipped to capture these hidden symmetries and dualities, which actually exist. 1T isOK only with additional guidance, so 1T-physics seems to be incomplete.

2T seems to be a promising idea on a new direction of higher dimensional unification. extra 1+1 are

LARGE, also not Kaluza-Klein.

29

The End

… and through work in progress we hope to the extendits domain of validity

to solve the remaining mysteries!!

2T-physics works in the known world so far

30

Particle moving under the influence of forces due to any conformally flat space-time

This kind of space-time will always emerge if embedding is such that XM does not contain momentum p in lower spacetime.

Note moduli in the embedding of phase space in the higher dimensions:Mass in massive particle gaugesCoupling & mass in H-atom or Harmonic OscillatorCurvature parameters, Hubble constant, etc.

• The different “shadows” are predicted to be duals of each other. • They all have the same hidden SO(d,2) (if Sp(2,R) includes

backgrounds this changes)• Any function of the gauge invariant LMN has the same value in

different phase spaces• Example Casimir C2(SO(d,2)) =1-d2/4 (singleton).

31

Field equations for fermions in 2T-physics

Worldline gauge symmetry OSp(1|2)

Vanishing constraints on physical states

act like SO(d,2) gamma matrices on the two SO(d,2) Weyl

spinors

kinematic #2 (homogeneous)

kinematic #1

Dynamic eq. of motionNotatio

n:

32

Yukawa interactions in 2T-physics

d=4 SO(4,2) group theory explains why there should be XM

vanishes against delta function

2Tgauge symmetry also explains why there should be XM

must include dilaton factor F if d is not 4 due to 2Tgauge symmetry, or homogeneity

33

Equations for gauge fields in 2T-physics

worldline gauge symmetry for spin 1

1)

2) Spinless particle in gauge field background, and subject to Sp(2,R) gauge symmetry. Then the gauge field background must be kinematically constrained.

In the fixed ‘axial’ gauge it amounts to homogeneity

must include dilaton factor F if d is not 4 due to 2Tgauge symmetry, or homogeneity

34

Gauge symmetries for the Standard Model in 4+2 dimensions

Guiding principles : 2Tgauge symmetry, SU(3)xSU(2)xU(1) YM gauge symmetry, renormalizability

Dilaton Higgs

3 families of quarks and leptons but all are left/right quartet spinors of SU(2,2)=SO(6,2)

SU(3)xSU(2)xU(1) gauge bosons, but all are SO(6,2) vectors

There is a separate 2Tgauge parameter for every field, so remainder of every field is gauge freedom.

remainders proportional to X2

35

2 gauge choices made. t reparametrization remains.

emergent space-time

LMN is Sp(2,R) gauge invariant

emergent 1T dynamics

Energy (n) 0 1 2

3

Angular momentum (L)

… … …

…

“Seeing” 4+2 dimensions through the H-atom

36

H-atom orbitals

Demo : the 4th dimension - shadow of rotating system. SO(4,2) > SO(4)xSO(2)

Seeing 1 space (the 4th), and 2 times :SO(4,2) > SO(3) x SO(1,2)At fixed angular momentum (3-space)the energy towers are patterns ofspace-time symmetry group SO(1,2)

Subtle effects in 3+1 dims: H-atom action is INVARIANT under SO(4,2).

4 states at the same energy.

1 at L=03 at L=1, etc.

Why L=0,1 same energy?