Lecture 6: Quantum Geometry

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Lecture 6 Quantum Geometry Duality

• Recap higher dimensions from Lecture 5.
• Mathematics of Geometry
• Riemannian Geometry
• Cosmic bounce ?
• Wound up strings (R vs. 1/R)
• Coffee break!!
• Orbifolding and mirror symmetry
• Tearing space
• Flop transitions and mirror rephrasing
• Wittens stringy shield

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A 3D being might project a shadow of a cube on to
a flat plane to make it visible to a flatlander,
2D being.Picture from Hyperspace by MIchio Kaku
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Wormholes may lead to new Universes or other
dimensions. We may live in 10 or 11 dimensional
space.Picture left taken from Hyperspace by
Dr. Michio KakuPicture right from KITP talk by
Dr. Shing Tung Yau (Harvard Univ)
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Math LectureGeometry and Physics
UniteConveniently stolen (err borrowed) from
Dr.Yau Harvard University. From KITP talk by Dr.
Shing Tung Yauhttp//online.kitp.uscb.edu/online/
plectures/yau
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(No Transcript)
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
Euler Number
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From KITP talk by Dr. Shing Tung
Yauhttp//online.kitp.uscb.edu/online/plectures/y
au
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OH boy !!

• As fascinating as this is I think
• Thats enough textbook geometry.
• Lets move on...
• Read a good math book! Or look online.

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Riemann Geometry

• Einsteins genius lay in the bold statement that
  Riemanns geometry aligns perfectly with
  gravitational physics.
• Almost 100 yrs after Einstein, string theory
  gives a QM description of gravity which modifies
  Einsteins gravity at ultra-small distances
  (Planck length )
• On length scales around the Planck length we
  define a new geometrical framework called quantum
  geometry

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Ok I couldnt resist this was part of a public
lecture you know!
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Riemannian GeometryPicture from Hyperspace by
Michio Kaku
Angles inside triangle add up to 180 degrees
Angles inside this triangle add up to More than
180 degrees
Angles inside triangle add up to Less than 180
degrees.
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Cosmic bounce or Cosmic crunch ?

• If the average matter density exceeds a critical
  density or about 5 hydrogen atoms
  per cubic meter then there will be enough
  gravitational pull between matter in the universe
  to halt expansion and cause a collapse.
• If the universe began in a big bang it could end
  in a big crunch.

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Wound up stringsPicture from The Elegant
Universe by Brian Greene
(a) Unwrapped configuration
(b) Wrapped configuration
A string can move in UNIFORM motion and
OSCILLATORY (ordinary) motion. Ignoring ordinary
or oscillations of a stationary string for
now There are 2 ways a string can move
uniformly (ie. translational motion) on this 2
dim surface. They can slide on the surface (a)
or they can wrap around the circular dim. (b)
and circle around it.
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String Energy for Uniform motion

• There are two types of Energy
• Vibration energy (due to translational motion)
• Winding energy ( from sliding around the circular
  dimension)
• There are 2 frequencies, vibration No. v
• and winding No. w

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Energy of Vibration Winding

• x p h/2 p or m, is proportional to
  energy (Emc )
• Hence for radius R, energy 1/R for
  translational motion. (vibration)
• Strings have a minimum mass determined by their
  length. This (for a wound string) is determined
  by the radius R of the circular dimension and the
  No. of times the string is wrapped around it.
• Circumference is 2 R , hence energy R

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How to measure the SIZE of the Universe ?

• To us it looks like the Universe is HUGE!
• Our dimensions are gtgt Planck length
• That means R is really big
• The LIGHT weight strings are the unwound ones
  which have mass 1/R (recall mass energy)
• Wound strings are very heavy and not seen by
  experiment.
• We measure the universe by timing light weight
  particles (photons), which travel at a known
  speed, as they move from one place to another.
  This gives us a length scale.

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Physical Characteristics Energy of the Universe.

• Physical properties of the universe are sensitive
  to the total energy of the strings.
• The TOTAL energy is the sum of unwound and wound
  modes

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Cosmic Bounce ?

• Lets assume critical mass density has been
  reached and the universe starts to collapse.
• The unwound string modes (1/R) get heavier, and
  the wound modes (R) get lighter.
• When we reach the Planck length R1, both modes
  have the same mass-energy.
• The 2 approaches to measuring distance become
  equally difficult and yield the same answer.
• As the Universe shrinks ltR1, then the wound
  string modes become lighter and we should now use
  them to measure distance scales. The unwound
  modes get heavy.
• Now 1/R is gt Planck length since Rlt1. The
  universe appears to expand once again.

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Duality

• If the total energy of the universe is fixed
  then
• A small vibration energy for a large R universe
  must be equivalent to some small winding energy
  in a small R universe
• We say that size dimension R is dual to 1/R since
  they have the same physical characteristics in
  string theory by the interchange of vibration to
  winding modes of the string. The crunch only gets
  down to the Planck scale no further.
• Point particles do not have an equivalent wound
  mode and inevitably lead to a singularity in the
  cosmic crunch scenario.

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MORE after the break
10 min break !
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Welcome back !

• How general is the R 1/R duality ?
• What if space does not have a circular dim. do
  the conclusions about minimum size of universe
  hold?
• No-one knows for sure.
• Investigations show the answer depends on whether
  it is a full spatial dimension which is shrinking
  and not just an isolated part of space.

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Mirror Symmetry

• By conventional geometry a circle of radius R is
  different from one whose radius is 1/R.
• String theory suggests they are physically
  indistinguishable.
• Might there be other geometrical forms of space
  which differ in more drastic ways than size,
  which nevertheless are physically
  indistinguishable in string theory?
• Could 2 different Calabi-Yau shapes give rise to
  the same physics?

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Orbifolding Picture from The Elegant Universe
by Brian Greene
Recall the number of holes in a Calabi-Yau 6D
surface determines the number of generations
(families) the strings excitations will arrange
themselves in. The string vibrations are
sensitive Only to the total number of holes Not
in which dimension they are in.
Orbifolding is a procedure in which a new
Calabi-Yau shape in made from Gluing together
various points of another Calabi-Yau shape.
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Mirror Manifolds by Ronen Plesser and Brian
Greene.See The Elegant Universe by Brian
Greene.

• Mirror manifolds describe physically equivalent
  yet geometrically distinct Calabi-Yau spaces.
• The individual spaces in a mirror pair of
  Calabi-Yau spaces are not literally mirror images
  of one another but they do give the same
  physical universe when used for the 6 extra
  spatial dimensions of string theory.
• The Calabi-Yau spaces differ by the interchange
  of the number of even and odd dimensional holes.
  A very hard calculation in one space can be easy
  in the mirror space, which has the same physical
  characteristics.

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Mathematics of Mirror Symmetry

• Einsteins rigid idea that the geometry of space
  and observed physics are intrinsically linked has
  been loosened up by String theory.
• Mirror symmetry is a powerful tool to unlock the
  physics of string theory and math of Calabi-Yau
  spaces.
• Mathematicians studying Algebraic Geometry have
  studied many Calabi-Yau spaces without any
  knowledge of their application to string theory.
  Mathematicians now use Mirror symmetry as a tool.

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Brian Greene in The Elegant Universe p260

• An example of how a calculation can be made
  simpler if we reorganize the problem a little.
• You are asked to count the number of apples in a
  storage bin 50x50 feet and 10 foot deep in size.
• You start to count apples one at a timetoo slow
• A friend comes by with a crate the apples
  originally came in. He tells you these crates
  were stacked 20 boxes long, 20 deep and 20 high.
• Now all you need to do is count the apples in one
  crate and multiply by 8000 for the total No. of
  apples.

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End of Riemannian Geometry

• Einsteins GR says that the fabric of space
  cannot tear.
• GR is rooted in Riemannian geometry. This is a
  geometrical framework that analyses distortions
  in the distance relations between nearby points.
  The math formalism requires a smooth fabric of
  space, no tears or creases.
• If tears exist then GR breaks down.

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Quantum Mechanics to the rescue

• QM leads to violent short-range undulations in
  space due to the Heisenberg uncertainty relation.
• Rips and tears are commonplace.
• Wormholes are a consequence (SG-1)
• Black-holes (experimentally established) are
  regions of immense curvature and the subject of
  our next lecture.

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Dr. Shing-Tung Yau 1987

• Dr. Yau with his student Gang Tian found that,
  using well-known mathematical techniques, certain
  Calabi-Yau shapes could be transformed into
  others by puncturing their surface and plugging
  up the resulting hole with a spherical surface
  (In 2D)
• This is called a flop transition, and results
  in a topologically distinct Calabi-Yau shape.

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Tearing Space Figures from The Elegant
Universe by Brian Greene
Fig 1.
Fig. 2.
A sphere in Fig 1a decreases until in Fig 1d it
comes to a point. In Fig 2a this is replaced by
a tear. You plug the tear with another sphere
which grows and replaces the original sphere in
Fig 2d. One says that the original sphere is
flopped to the new one. The flop transition is
a way of creating new Calabi-Yau spaces from old
ones.
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Could the flop transition occur in Nature?

• Andy Lutkin and Paul Aspinwall thought about what
  would happen in the perspective of the mirror
  Calabi-Yau space if a spacing tearing flop
  transition occurred in the original space.
• The motivation here is that the physics of the
  mirror space is identical to the original
• but the mathematical complexity of the
  calculations required to derive the physics from
  the space can be radically different.

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Physics of space tearing is really nasty!

• Plesser and Greene employed orbifolding to create
  mirror pairs of spaces. This technique is
  geometrically equivalent to the pinch and tear of
  flop transitions.
• The mirror space can have far less tricky math
  involved in the physics calculations.
• Question Does the mirror rephrasing have the
  same physics even after the tear?

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Flop Transitions Mirror Symmetry Picture from
The Elegant Universe by Brian Greene
The thing to do to prove this conjecture would be
to Calculate the physics for the tear and the
mirror space for the last diagram for each row
above. These were proved to be the same by
Greene, Aspinwall and Morrison.
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A String Shield- Space can rip! Picture from
The Elegant Universe by Brian Greene
Meanwhile Witten was working on a
similar Problem. He showed that microscopic
tears Can and do occur in space-time but are
not Catastrophic events. Papers by Witten and
Greene, Aspinwall Morrison were sent to the
e-print Archives simultaneously in Jan 1993.
Wittens approach Tears do not lead to
catastrophic events because Strings, unlike point
particles, sweep out tubes in space-time.
These 2D world sheets effectively encase the
tears in space rendering them harmless to the
surrounding universe. The tubes shield the rips.
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The End

• Lecture 7 Black holes revisited
• March 15th