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Field Theoretic Formulation of Two-Time Physics including Gravity Itzhak Bars, USC

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!)

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Field Theoretic Formulation of Two-Time Physics including Gravity Itzhak Bars, USC

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  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.

  2. Not much discussion of more time-like dimensions – why? Because the 2T road is risky, dangerous and scary 1) Ghosts! = negative probability 2) 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

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

  4. 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 H Symmetry XP, P-X Invariant

  5. New Gauge Principles 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)=Q0ij(X)+QMij(X)PM+ QMNij(X)PMPN+… More generalizations: Particles with SPIN or SUSY; strings/branes (partial) and finally Field Theory. Require local Sp(2,R) symmetry XP indistinguishable continuously at EVERY INSTANT for ALL MOTION

  6. New symmetry allows only highly symmetric motions (gauge invariants)little room to maneuver. • With only 1 time the highly symmetric motions impossible. Collapse to nothing. • Extra 1+1 dimensions necessary  4+2 !! No less and no more than 2T. • Straightjacket in 4+2 makes allowed motions effectively 3+1 motions (like shadows on walls). How does it work? Indistinguishable at any instant t2 t1,x,y,z, … w Constraints: generators vanish !!! (Position & Time) (Momentum & Energy) 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) 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+’ Xi -’ Xi m

  8. Xi 0’ Xi 0 Xi I

  9. An example: Massive relativistic particle gauge • embed phase space in 3+1 into phase space in 4+2 • Make 2 gauge choices solve 2 constraints X2=X.P=0 • reparametrization and one constraint remains. Gauge invariants x=MN{LMN, x}, p=MN{LMN, p}, Note a=1 when m=0 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.

  10. Shadows from 2T-physics hidden info in 1T-physics Hidden Symm. SO(d,2), d=4 C2=1-d2/4= -3 singleton Free or interacting systems, with or without mass, in flat or curved 3+1 spacetimes. Analogy: multiple shadows on walls. Emergent parameters mass, couplings, curvature, etc. 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. These emerge in 2T-field theory as well

  11. Field equations in 2T-physics Derived from Sp(2,R) in hep-th/0003100; also Dirac 1936 other approach Constraints = 0 on physical states i.e. Sp(2,R) gauge invariant state Probability amplitude is the field kinematic #2 kinematic #1 Kinematic eom´s say how to embed d dims in d+2 dims. dynamical eq. extended with interaction gauge symmetry remainder 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

  12. Action for scalar field in 2T-physics Obtain 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 After gauge fixing, eliminating redundant fields, and simplifications, boils down to a simplified partially gauge fixed form Gauge fixed version is more familiar looking Gauge symmetry (a bit more complicated) Works only for unique V(F) dynamical eq. Minimizing the action gives two equations, so get all 3 Sp(2,R) constraints from the action kinematic #1,2 Homogeneous F0henceoneless X, so alltogethertwofewerdimensions Can gauge fix to F0 which is independent of X2

  13. Action of the Standard Model in 4+2 dimensions Illustrates main features of 2T field theory SU(3) SU(2) U(1) Gauge fields 4*x4=adjoint quarks & leptons 3 families Yukawa couplings to Higgs 4*x4*=6 vector Higgs and dilaton quadratic mass terms not allowed

  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 X2=0 Solve kinematic equations in extra dimensions = k -1 Remainder is gauge freedom, remove it by fixing the 2Tgauge-symmetry at any l,k,x Result of gauge fixing and solving kinematic eoms is fields only in 3+1 Dynamics only in 3+1

  15. Action for gauge fields in 2T-physics Obtain kinematic and dynamic equations from the action dynamical eq. kinematic #1,2 remainder can be removed by gauge symm.

  16. Emergent gauge bosons in 3+1 dimensions kinematic equation simplifies  homogeneous start with YM axial gauge homogeneous L enough to gauge fix A+’=0 There is leftover YM gauge symm. Solution of X.A=0 Only independent Use 2Tgauge symmetry to eliminate Vm gauge freedom proportional to X2 FMN is YM gauge invariant but 2Tgauge dependent result is standard 3+1 YM Lagrangian

  17. Action for fermion field in 2T-physics Obtain 3 equations not just one : 2 kinematic and 1 dynamic. 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 Minimizing the action gives two equations, so get all OSp(1|2) constraints as eom´s from the action kinematic #1 dynamical eq. kinematic #2 Although it looks like one equation one can show that each term vanishes separately due to X2=0. 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.

  18. Emergent fermions in 3+1 dimensions 4 component SU(2,2) chiral fermions choose X2x12Tgauge symm. Impose kinematical eom in extra dimension 2 component SL(2,C) chiral fermions choose x22Tgauge symm. 4+2 Lagrangian descends to 3+1 standard Lagrangian. No explicit X. standard 3+1 Yukawa term standard 3+1 kinetic 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, remainders proportional to X2 eliminated by 2Tgauge fix S Normalize to 1 Shadow Standard Model in 3+1 has “dilaton” in addition to usual matter. Shadow SM is Poincare invariant. More, it has hidden SO(4,2) symmetry What is new in 3+1 ? (semi-classical) • Mass generation: a) electroweak phase transition is driven by dilaton (phenomenology?), b) new mechanisms of mass generation from extra dimension (massive shadow) ??? • Duals of the Standard Model (other shadows) – new physics (e.g. mass, etc.), computational tools? • Can SO(4,2) help for mass hierarchy problem ? (need quantum 2T field theory, not ready yet)

  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). All space filled with vev. Makes sense to have dilaton & gravity & strings involved Goldstone boson due to spontaneous breaking of scale invariance 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

  21. Duals of the Standard Model IB + 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. More complicated gauge choices with XM(xm,pm) mixedparametrization e.g. massive particle, etc. These have non-local field interactions, but approximate local for small mass. 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] Compare to flat case Sp(2,R) algebra puts kinematic costraints on fields Solve kinematics, and impose Q11=Q12=0: One of the shadows

  23. Gravity in 2T-physics – Field Theory 3 equations not just one Similarly for W, W 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. Beyond the Standard Model or GUTs, and Gravity 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 multiplets N=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)

  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) Dilatondriven 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 toolsEmergent spacetimes and dynamics; unification; holography; duality; hidden symms.---Non-perturbativeanalysis 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-timethan 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 by 2T-physics. 2T seems to be a promising idea on a new direction of higher dimensional unification. 1T-physics on its own is not equipped to capture these hidden symmetries and dualities, which actually exist. 1T is OK only with additional guidance, so 1T-physics seems to be incomplete. extra 1+1 are LARGE, also not Kaluza-Klein. 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.

  29. 2T-physics works in the known world so far … and through work in progress we hope to the extend its domain of validity to solve the remaining mysteries!! The End

  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 gauges Coupling & mass in H-atom or Harmonic Oscillator Curvature 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) act like SO(d,2) gamma matrices Vanishing constraints on physical states on the two SO(d,2) Weyl spinors kinematic #1 kinematic #2 (homogeneous) Dynamic eq. of motion Notation:

  32. Yukawa interactions in 2T-physics d=4 SO(4,2) group theory explains why there should be XM 2Tgauge symmetry also explains why there should be XM vanishes against delta function must include dilaton factor F if d is not 4 due to 2Tgauge symmetry, or homogeneity

  33. Equations for gauge fields in 2T-physics 1) worldline gauge symmetry for spin 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 There is a separate 2Tgauge parameter for every field, so remainder of every field is gauge freedom. Dilaton Higgs remainders proportional to X2 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

  35. 2 gauge choices made. t reparametrization remains. emergent space-time emergent 1T dynamics LMN is Sp(2,R) gauge invariant

  36. Subtle effects in 3+1 dims: H-atom action is INVARIANT under SO(4,2). “Seeing” 4+2 dimensions through the H-atom Energy (n) H-atom orbitals Angular momentum (L) 0 1 2 3 … … … … 4 states at the same energy. 1 at L=0 3 at L=1, etc. Why L=0,1 same energy? 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 of space-time symmetry group SO(1,2) Demo : the 4th dimension - shadow of rotating system. SO(4,2) > SO(4)xSO(2)

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