Two-Time Physics
The Unified View From
Higher Dimensional Space and Time
Itzhak Bars, USC
•One of the most important questions today: what is space and time?
•Such questions fascinated every civilization, from ancient to modern.
Discussed in theology (the heavens & earth; beginning & end), and in
philosophy, mathematics, astronomy, physics and cosmology.
•Einstein unified the concepts of space and time in the 20th Century.
•String theory: consistent Quantum Mechanics + Gravity requires
extra space dimensions (Kaluza-Klein idea: small, curled up,… ).
•How about extra time dimensions? (easy to ask, not so easy to realize!!)
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Themes in this lecture:
Symmetry
Deeper order in space-time, leads to predictions of extra dims.
Perspectives in higher space-time
Same event in higher dims., different interpretations in lower dims.
Reconstructing the substance from its shadows.
Subtle phenomena in our space-time of 3+1 dims.
Hard to notice, surprises, tests to confirm or reject theories
Surprising and far reaching preliminary conclusion:
The ordinary formulation of Physics in 3+1 LARGE dimensions is incomplete.
One extra timelike plus one extra spacelike LARGE dimensions are needed
to provide a more complete view of Nature at ALL SCALES of physics.
Related work: One corner of 2T-physics makes contact with work by
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Dirac, Salam, Adler, Ferrara, Siegel, Marnelius, Vasiliev,… (conformal symmetry)
Students / Collaborators
Shih-Hung Chen
Yueh-Cheng Kuo
Bora Orcal
Guillaume Quelin
Costas Kounnas (CERN and Ecole Normale, Paris)
Cemsinan Deliduman (now Assoc. Prof., Istanbul Tech. Univ.)
Oleg Andreev (now postdoc U. Munich)
Sasha Vongehr
Djordje Minic (now Prof. Virginia Tech)
Moises Picon (now postdoc U. Padova)
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space-time
(where and when)
Perspective: static observer free of forces, and emphasizing POSITION space
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

z
What is position in ordinary space ?
–3 dimensions (location in space) = where
What is time ?
– time on your watch = when
x
y
many kinds of watches (atomic clock, sand clock, etc.)
t

What is a space-time event in space-time ?
– an event at a specific time and specific place
x,y,z
What is space-time history ?
t
– a collection of events, just like a movie !
– motion of a point is represented by
x,y,z
a curve
in space-time
= worldline
This point
of departure
is a perspective
not symmetric in momentum vs position

space-time perspectives (relativity)
Ship’s watch: 1hr,
but Earth’s: 320 yrs. ,
as observed by astronauts.
Time duration seen on a
watch depends on the
speed of the observer
relative to that watch.
Perfect
identical
watches may
not measure
identical
time intervals.
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Time Dilation
Prediction :
Special
Relativity
allows time
travel to the
future.
Subtle effect:
Too small to
notice with
common
speeds. But
experiments
with short-lived
particles in an
accelerator
confirms this.
More Perspectives: Space Contraction
But, particles in
accelerators move
at great speeds,
and behave
precisely as
predicted.
Subtle effects.
The effect is too
small to notice in
daily phenomena
because the speed
is too small.
e.g. sphere looks
like pancake.
AT
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Identical rulers may measure differently the same space interval.
You guessed it – it depends on the speed of the observer !
Space and Time are not absolute
they are relative to the observer
t
Even more weird things happen close to the extremes.
x,y,z
Such effects
arise because
observers are
in relative
motion,
or under the
influence of
gravitational
forces.
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Close to the speed of light,
or in very strong gravity,
a watch appears to
almost stop.
Inside a black hole,
time & space switch roles.
t
x
x,y,z
t,y,z
The observer’s
coordinate
systems get
distorted
relative to
each other.
The Equivalence
Principle shows
how to relate
them. Then
observers see
same Law.
Equivalence Principle !! (Einstein)
The laws of Physics are independent of the
observer’s space and time perspective
This is implemented as a symmetry of the equations:
Invariance under “general coordinate transformations”
Origin of Gravity
The force of gravity exists because of this symmetry.
All details of the gravitational force is controlled by this symmetry.
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But there is more to explain:
Properties of
matter other
than history in
space-time?
Electric charge, …
Non-gravitational
forces?
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Space-Time and Higher Dimensions
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


Kaluza (1921) and Klein (1926) considered 1 extra
dimension. Explain charge and E&M (Gen Rel in 4+1)
String theory (so far unconfirmed) – requires 6 more
space dimensions in addition to the usual 3 space
and 1 time dimensions. This solves the puzzle of how
to marry Quantum Mechanics and Gravity.
“M-theory” , which includes string theory, adds one
more space dimension, for a total of
11 = 10 space + 1 time.
t
x,y,z, … 7 more
subtle effect
Where is the Extra Space?
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t
t
x,y,z, …. 7 more



x,y,z
7 more
Scenario: soon after the Big Bang the extra space
dimensions curled up into very tiny geometrical shapes,
and remained small. How small ???
They may exist at every space-time point of our own
3+1 space-time, and they are supposed to explain
charges, flavors, colors, families and more …
A tiny Christopher Columbus (e.g. electron) can explore
the small dimensions and bring the information to us.
Nobody discusses more time-like dimensions – why?
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
Because physicists are scared of them.

Two kinds of disasters:
1) Ghosts!
= negative probability
2) Causality (cause and effect) violation
 time machines:
Can kill your grandmother before
your mother or you are born. !

Both are nonsensical, and kept physicists away from
contemplating more time-like dimensions.
Easy!
A new symmetry
that unites
(Position & Time)
(Momentum & Energy)
Indistinguishable
at any instant
for any motion.
This wipes out the
source of disasters.
But requires
1 extra space and
1 extra time !

General setup of classical or quantum mechanics is
symmetric under XP, P-X. They are at same level
before a specific system or Hamiltonian is chosen.
H
Any Lagrangian:
 Poisson brackets or quantum commutators
 Boundary conditions


More general: GLOBAL symmetry, Sp(2,R) doublet
Invariant:

Even more general is symmetry under GLOBAL
canonical transformations
How? Here is the clue…
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2T-physics emerges from demanding that this global symmetry turn into a local symmetry
The new fundamental principle: A broader equivalence principle.
XM=(Position & Time)
PM= (Momentum & Energy)
indistinguishable
at any instant.
A new gauge symmetry of the Laws of Mechanics
Applied to all motion
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Gauge symmetries are so powerful that they highly constrain the
equations that can be written down, sometimes fully determining them.
All known forces in Nature are fully understood through some
remarkable gauge symmetries, and follow from them:
1) Symmetry under general coordinate transformations give
Einstein's General Relativity = Gravity
2) Yang-Mills gauge symmetry SU(3) x SU(2) x U(1) gives
The Standard Model of Particles and Forces
(strong, weak, and electro-magnetic)
To have the symmetry, the force must exist, its details fixed.
w
1) New symmetry allows only highly
symmetric motions.
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
t2 motions (like shadows on walls).
t1,x,y,z, …
How does it work?
(Position & Time)
(Momentum & Energy)
Indistinguishable
at any instant
Constraints: generators vanish !!!
Generalized Sp(2,R) generators Qij(X,P)
instead of the simpler Qij(X,P)=(X2,P2,X.P)
include ALL possible background fields.
Non-trivial: Gauge fix 4+2 to 3+1 (have 3 gauge parameters)
many 3+1 shadows emerge for same 4+2 spacetime history.
15Each shadow contains only one timeline (a mixture of all 4+2)
One higher dimensional reality gives rise to
many related lower dimensional “shadows”
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Demos: hand,
loop, sphere.
Shadows in 3+1
capture different
perspectives of 4+2
See ALL 4+2 !
Like many takes of a
movie scene from
different perspectives.
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Events
in
higher
dims.
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3
Physicists who
live only on the
“wall” in 3+1
dims. can
experiment only
on the “wall”.
The shadows
appear to be
independent
phenomena to such
observers.
From relations among many shadows could reconstruct full info in 4+2.
See similarities to Plato’s Allegory of the Cave. But Plato does not have many shadows or their
relations that are crucial in making the connections between 1T and 2T physics, & predictions.
Emergent spacetimes and dynamics, hidden symmetries & dualities
from gauge fixing the 2T theory Massless
Free or
17interacting
systems, with
or without
mass, in flat
or curved
3+1
spacetimes
relativistic
particle
(pm)2=0
Hidden Symm.
SO(4,2)
Dirac
singleton
C2=1-d2/4= -3
conformal sy
Harmonic
oscillator
2 space dims
mass = 3rd dim
SO(2,2)xSO(2)
2T-physics predicts
hidden symmetries
and dualities (with
parameters) among
the “shadows”.
1T-physics misses
these phenomena.
2T-physics
Sp(2,R)
Massive
X2=P2=X∙P=0 .
relativistic
(pm)2+m2=0
Non-relativistic
H=p2/2m
Background
flat 6=4+2 dims
SO(4,2)
symmetry
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
images are only
some examples
of much broader
phenomena.
Subtle effects:
“Seeing” 4+2 dimensions through the H-atom
Shadow of free motion in 4+2 subject to :
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Indistinguishable
(Position & Time)
(Momentum & Energy)
at any instant
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
Seeing1 space (the 4th), and 2 times :
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
Subtle effects:
“Seeing” the 4th dimension in celestial mechanics
One of the shadows in 3+1 of free motion in 4+2, subject to symmetry
(Position & Time)
(Momentum & Energy)
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Indistinguishable
at any instant.
Angular momentum (defines plane)
constant of motion (same at any point in orbit)
because of rotation symmetry in 3 dims.
Unchanging ellipses – No precession.
WHY?
An accidental symmetry – meaning?
Laplace-Runge-Lenz vector
(axis of ellipse // to A)
constant at any point in orbit.
It determines the eccentricity and the
orientation of the ellipse in the plane.
2T-physics says A is just the component
of angular momentum for rotating from
ordinary 3-space into the 4th dimension.
The unchanging L and A are due to
rotation symmetry in 4 dims.
Demo with shadows: eccentricity and orientation of ellipse
arises from orientation of orbit in 4 space dimensions.
Deeper into matter: 2T-physics and Field Theory
What are the true constituents of matter?
Macroscopic matter 
 molecules  atoms  nuclei (many isotopes)
 (protons + neutrons +…)  quarks + leptons
Matter particles
spin ½, fermions
6 quarks (in 3 colors)
6 leptons
(Including right-handed or
Majorana singlet neutrinos)
and their anti-particles
(anti-matter)
All known matter is made up of
quarks, leptons and “force
20 particles” that hold them together.
Currently we don’t know their sizes,
and what, if anything, is inside quarks,
leptons or force particles
(maybe “superstrings” ?).
The Standard Model of Particles and Forces
Four interactions or forces govern all known phenomena in the Universe.
Understood in the framework of Quantum Field Theory
(computational framework).
plus
Higgs
unconfirmed
Describes Nature down to
10-18 meters. Exquisite
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agreement with experiment.
This SM emerges as a “shadow” of a
4+2 field theory. With better features.
AND there are more shadows  duals
So far 2T-physics  known physics, but in new light with new predictions related to 1+1 dims.
Are there predictions of disagreements with 1T-Physics?? Yes, only in hitherto unknown realms.
Unsolved Mysteries:
•CP (time reversal) violation problem in the strong interactions (but axion not seen?)
 2T-physics solves this (a property of 4+2 dims.) – no need for axion!
•Why these degrees of freedom? And why in these patterns?
•25 parameters : masses, couplings, mixings – is there a theory that determines them
from first principles? Is Higgs the answer to the origin of mass ? LHC 2008 !!
 2T-physics modifies Higgs (adds dilaton). Also new possibilities for mass !
•Is there Supersymmetry (to resolve the mass hierarchy problem: stability) ?
a) If SUSY exists, there are new constraints from 2T-physics ! Tests !
b) 2T-physics may provide an alternative (conformal symmetry, 6 dims.!!) ?
•What is dark matter ? 25% of the matter in the Universe.
•What is dark energy? 70% of the energy in the Universe.
Possible candidates
for these exist
among the degrees
of freedom above.
•How do we solve the Quantum Gravity problem (strings 9+1 dims, M-theory 10+1
22 dims, curled up dimensions) ? This is a framework ! Not a theory yet !
 2T-physics requires extra 1-space + 1-time. M-theory in 11 space + 2 times !!
In conclusion …
These old Greeks
eventually
discovered the
concept of the
atom.
Similarly, you
can’t ignore
2T-physics.
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Hidden information
is predicted by
2T-physics. A new
idea on unification.
1T-physics on its own
cannot capture these
hidden symmetries
and dualities, which
actually exist. 1T is
OK only with
additional guidance.
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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