Transcript Slide 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.
1This encodes fundamental laws in d+2 dimensions, with extra info for 1T.
Not much discussion of more time-like dimensions – why?
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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)
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

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{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
XP, P-X
More general: continuous GLOBAL symmetry: Sp(2,R)
Invariant

Even more general is GLOBAL canonical transformations
Require local Sp(2,R) symmetry
XP 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)
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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
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3D room, and think of the shadows as different “beasts”, but can discover they are related.
Xi+’
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Xi -’
Xi m
Xi 0’
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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
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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
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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
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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
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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
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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.
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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
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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.
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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
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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?
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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
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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.
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Compare to
flat case
Sp(2,R) algebra
puts kinematic
costraints on fields
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Solve kinematics, and
impose Q11=Q12=0:
One of the shadows
3 equations
not just one
Similarly for W, W
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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
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All shadow scalars must be conformal scalars
Effect on cosmology ?
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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)
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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.
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
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.
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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.
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•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:
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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
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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
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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
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emergent
space-time
emergent 1T
dynamics
LMN is Sp(2,R) gauge invariant
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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
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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)