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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. 1This 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 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 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. 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: H Symmetry XP, P-X More general: continuous GLOBAL symmetry: Sp(2,R) Invariant Even more general is GLOBAL canonical transformations Require local Sp(2,R) symmetry XP indistinguishable continuously at 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)=Q0ij(X)+QMij(X)PM+ QMNij(X)PMPN+… More generalizations: Particles with SPIN or SUSY; strings/branes (partial) 5 and finally Field Theory. 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). How does it work? Indistinguishable (Position & Time) (Momentum & Energy) at any instant Constraints: generators vanish !!! t2 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) t1,x,y,z, … w 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 6 3D room, and think of the shadows as different “beasts”, but can discover they are related. Xi+’ 7 Xi -’ Xi m Xi 0’ 8 Xi 0 Xi I 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 t reparametrization and one constraint remains. Gauge invariants xm=MN{LMN, xm}, pm=MN{LMN, pm}, SO(d,2) is hidden symmetry of the massive action. Looks like conformal tranformations deformed by mass. The symmetry in 1T theory is 9 a clue of extra 1+1 dimensions, including 2T. Note a=1 when m=0 Shadows from 2T-physics Massless Free or 10interacting systems, with or without mass, in flat or curved 3+1 spacetimes. Harmonic Analogy: multiple oscillator shadows on walls. 2 space dims 3rd mass = dim SO(2,2)xSO(2) 2T-physics predicts hidden symmetries and dualities (with parameters) among the “shadows”. 1T-physics misses these phenomena. hidden info in 1T-physics relativistic particle (pm)2=0 Hidden Symm. SO(d,2), d=4 Dirac singleton C2=1-d2/4= -3 conformal sy 2T-physics Sp(2,R) simplest example X2=P2=X∙P=0 . Background flat 6=4+2 dims SO(4,2) symmetry Massive relativistic (pm)2+m2=0 Non-relativistic H=p2/2m H-atom 3 space dims H=p2/2m -a/r SO(4)xSO(2) SO(3)xSO(1,2) These emerge in 2T-field theory as well Emergent parameters mass, couplings, curvature, etc. •Holography: These emergent holographic shadows are only some examples of much broader phenomena. 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 #1 kinematic #2 Kinematic eom´s say how to embed d dims in d+2 dims. dynamical eq. extended with interaction 11 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 gauge symmetry Physical part of field remainder 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 Can gauge fix to F0 which is independent of X2 12 kinematic #1,2 Action of the Standard Model in 4+2 dimensions Illustrates main features of 2T field theory Gauge fields SU(3) SU(2) U(1) 4*x4=adjoint quarks & leptons 3 families Yukawa couplings to Higgs Higgs and dilaton 13 4*x4*=6 vector quadratic mass terms not allowed 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 Result of gauge fixing and solving kinematic eoms is 14 fields only in 3+1 = k -1 Remainder is gauge freedom, remove it by fixing the 2Tgauge-symmetry at any l,k,x Dynamics only in 3+1 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. 15 Emergent gauge bosons in 3+1 dimensions start with YM axial gauge There is leftover YM gauge symm. Solution of X.A=0 kinematic equation simplifies homogeneous homogeneous L enough to gauge fix A+’=0 Only independent Use 2Tgauge symmetry to eliminate Vm gauge freedom proportional to X2 FMN is YM gauge invariant but 2Tgauge dependent 16 result is standard 3+1 YM Lagrangian 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 kinematic #1 dynamical eq. Minimizing the action gives two equations, so get all OSp(1|2) constraints as eom´s from the action kinematic #2 Although it looks like one equation one can show that each term vanishes separately due to X 2=0. 17 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. Emergent fermions in 3+1 dimensions choose X2x1 2Tgauge symm. Impose kinematical eom in extra dimension 2 component SL(2,C) chiral fermions choose x2 2Tgauge symm. 4+2 Lagrangian descends to 3+1 standard Lagrangian. No explicit X. 4 component SU(2,2) chiral fermions standard 3+1 kinetic term standard 3+1 Yukawa term Translation invariance in 3+1 comes from rotation invariance in 4+2 18 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) 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? 19 3. Can SO(4,2) help for mass hierarchy problem ? (need quantum 2T field theory, not ready yet) “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 20 Goldstone boson couples to everything the Higgs couples to, but with reduced amf/v 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? •Instead of choosing the flat 3+1 spacetime gauge, choose any conformally flat spacetime gauge. These include More complicated gauge choices with Robertson – Walker universe XM(xm,pm) mixed parametrization Cosmological constant (dS4 or AdS4) e.g. massive particle, etc. Any maximally flat spacetime These have non-local field interactions, AdS(4), AdS(3)xS(1), AdS(2)xS(2) but approximate local for small mass. 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. 21 Compare to flat case Sp(2,R) algebra puts kinematic costraints on fields 22 Solve kinematics, and impose Q11=Q12=0: One of the shadows 3 equations not just one Similarly for W, W 23 Self consistency of all equations, and consistency with Sp(2,R) make it a unique action. Solve Sp(2,R) kinematic constraints Conformal scalar, Weyl symmetry 24 All shadow scalars must be conformal scalars Effect on cosmology ? 25 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) 26 Summary of Current Status 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. 27 New technical tools There is more to space-time than can be garnered with 1T-physics. 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 …) New physical predictions and interpretations . It is more than a math trick. Hidden information is revealed by 2T-physics. 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. 28 In conclusion … 2T seems to be a promising idea on a new direction of higher dimensional unification. 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 30 •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). Field equations for fermions in 2T-physics Worldline gauge symmetry OSp(1|2) act like SO(d,2) gamma matrices on the two SO(d,2) Weyl spinors Vanishing constraints on physical states kinematic #1 kinematic #2 (homogeneous) Dynamic eq. of motion Notation: 31 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 32 must include dilaton factor F if d is not 4 due to 2Tgauge symmetry, or homogeneity 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 33 must include dilaton factor F if d is not 4 due to 2Tgauge symmetry, or homogeneity Gauge symmetries for the Model in 4+2 dimensions Standard Guiding principles : 2Tgauge symmetry, SU(3)xSU(2)xU(1) YM gauge symmetry, renormalizability Dilaton Higgs There is a separate 2Tgauge parameter for every field, so remainder of every field is gauge freedom. 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 34 emergent space-time emergent 1T dynamics LMN is Sp(2,R) gauge invariant 35 2 gauge choices made. t reparamet rization remains. Subtle effects in 3+1 dims: H-atom action is INVARIANT under SO(4,2). “Seeing” 4+2 dimensions through the H-atom 36 H-atom orbitals Energy (n) 0 1 2 3 Angular momentum (L) 4 states at the same energy. 1 at L=0 3 at L=1, etc. 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) Why L=0,1 same energy? Demo : the 4th dimension - shadow of rotating system. SO(4,2) > SO(4)xSO(2)