From Soap Bubbles to String Theory: Mathematical Reasoning in the Physical World

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From Soap Bubbles to String Theory Mathematical
Reasoning in the Physical World

• Yoni Kahn
• Northwestern University
• University of Cambridge
• Massachusetts Institute of Technology

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A little bit about myself

• Miller, 1997-2000
• MATHCOUNTS, Mandelbrot, Math Circle
• Math Enrichment
• Lots of self-study for contests!
• Lynbrook, 2000-2004
• Still did contests, but found I wasnt as good
  (qualified for USAMO freshman year, but never
  again!)
• Got interested in physics, took physics classes
  at SJ State senior year
• Northwestern University, 2004-2009
• Majored in physics, but took so many math courses
  I ended up with a math major as well!

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This talk

• What is physics?
• Math vs. physics
• Three mathematical principles in physics
• Lots of examples!
• Questions!

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God used beautiful mathematics in creating the
world.- Paul Dirac
Quotes by famous people
The miracle of the appropriateness of the
language of mathematics for the formulation of
the laws of physics is a wonderful gift which we
neither understand nor deserve. - Eugene Wigner
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(Image credit xkcd.com)
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Math vs. Physics

• Perfect idealizations
• Constructing objects to study
• Proof (deductive reasoning)
• If a proof is correct, its correct forever

• Real world (messy!)
• Describing objects that exist
• Theory predicts, experiment tests
• Theories can always be disproved!

However, many similarities
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Shared concepts in math and physics

• Intuition
• Comes with time and lots of practice!
• Problem-solving
• All the same strategies apply what tools do I
  need? What new data would be helpful? How can I
  make an unusual problem look familiar?
• More than one solution to a problem can learn
  something unexpected by doing a problem two
  different ways!
• Reasoning by analogy
• Absurdly powerful in physics

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Same math, different problems!
String theory
Soap bubble
What could these possibly have in common? Stay
tuned to find out
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Symmetry

• Warm-up ABCDEFGH is a regular octagon, and O is
  the intersection of its long diagonals. What is
  angle AOB? (No formulas, and no calculators!)
• Exercise What kinds of polynomials f (x) have
  the property that their right half (x gt 0) is the
  mirror image of their left half (x lt 0)?
• Take-home challenge Given a point P at (x,y),
  find the coordinates of P, which is the image of
  P after rotating 90 counterclockwise about the
  origin. Prove that if P is on the circle
  then so is P.

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Discrete symmetry

• How many rotations or reflections leave this
  equilateral triangle the same?

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Discrete symmetry

• How many rotations or reflections leave this
  equilateral triangle the same?
• Answer three rotations (0, 120, 240), three
  reflections (about each of the dashed lines)

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Discrete symmetry

• How do we know there arent any others? For
  example, what about a rotation, then a
  reflection?

A
C
A
rotate 120 clockwise
reflect about dashed line
B
C
A
C
B
B
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Discrete symmetry

• How do we know there arent any others? For
  example, what about a rotation, then a
  reflection?

A
C
A
rotate 120 clockwise
reflect about dashed line
B
C
A
C
B
B
This is the same as reflecting about the vertical
line through A! We have obtained one of our
original symmetries by composing two other
symmetries, one after the other.
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Discrete symmetry

• Lets call the three different rotations R1 (0),
  R2 (120), and R3 (240), and the three
  reflections I1 (about the line through A), I2
  (line through B), and I3 (line through C)
• We can make a multiplication table
• R1 R2 R2 (R1 does nothing)
• R2 R3 R1 (rotation by 120, then 240,
  recovers the original triangle)
• R2 I3 I1 (shown on last slide)
• The symmetries of an equilateral triangle have an
  interesting structure, which mathematicians call
  a group.
• Exercise finish the multiplication table!

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Continuous symmetry

• How many rotations or reflections leave this
  circle the same?

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Continuous symmetry

• How many rotations or reflections leave this
  circle the same?
• Answer infinitely many! Rotation by any angle,
  no matter how small, and reflection about any
  diameter
• The triangle had discrete symmetry (only certain
  angles), but the circle has continuous symmetry

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Continuous symmetry

• Can we make a multiplication table for the
  infinite number of symmetries of the circle?
• Answer yes! Instead of labeling the rotations
  and reflections with numbers (like R1, R2, and
  R3), we label them with an angle ?. For example
• R(?) is a rotation clockwise by the angle ?
• I(?) is a reflection about the diameter which
  makes an angle ? with the horizontal
• Multiplication table R(?1) R(?2) R(?1 ?1)
  (if you rotate twice, add the
  angles)
• R(?1) I(0) R(- ?1) (check this!)

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OK, but what does this have to do with physics?
Put equal electric charges at each vertex of the
triangle

Q
P


R
The electric fields at points P, Q, and R are
very simply related by the same symmetry
transformations we investigated earlier. - Three
solutions for the price of one!
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OK, but what does this have to do with physics?
Or, consider the gravitational field of a sphere.
Which direction does it point in?
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OK, but what does this have to do with physics?
Or, consider the gravitational field of a sphere.
Which direction does it point in?
Since a rotation by any angle is a symmetry of
the sphere, no direction along the sphere is
possible. Must be radial. - No calculations
needed!
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Real-world example black holes

• Einsteins equations of General Relativity are
  very hard to solve
• But, if we assume spherical symmetry, the
  equations simplify dramatically there is
  essentially one unique solution
• This solution predicts the existence of black
  holes!

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Probability

• Warm-up If I roll a six-sided die once, what is
  the probability that I roll either a 2 or a 3?
• Exercise What is the probability that I roll a 6
  two times in a row?
• Take-home challenge What is the probability that
  I roll a 6 twenty times in a row? How many zeroes
  after the decimal place will this number have? If
  I roll the die once a second, every second of
  every day, how many years (on average) will it
  take before I get twenty 6s in a row? (Does this
  answer surprise you?)

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Dealing with incomplete information

• How many molecules are in a glass of water?
• We have no hope of ever describing the exact
  location and speed of every single molecule in
  the glass
• However, we dont need to! Physics cares about
  bulk quantities
• Temperature
• Pressure
• Density

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Solution probability

• We can ask how do the internal configurations
  relate to the bulk quantities like temperature?
• Answer if each internal configuration is
  equally probable, then the most probable bulk
  quantity will correspond to the greatest number
  of internal configurations
• So maybe we cant predict temperature exactly,
  but we can predict it with 99.99999 certainty
  good enough!
• Turns a seemingly intractable problem into a
  problem of counting this branch of physics is
  statistical mechanics

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Example perfume in a room

• If I open a bottle of perfume at the far corner
  of the room, why can you smell it across the room
  almost immediately?

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Example perfume in a room

• If I open a bottle of perfume at the far corner
  of the room, why can you smell it across the room
  almost immediately?
• Answer there are a small number of ways the
  scent-carrying molecules can be concentrated in
  the perfume bottle, but there are a huge number
  of ways that the molecules could be randomly
  distributed across the room
• The perfume diffuses because of probability!

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Probability and quantum mechanics

• In the previous examples, we used probability
  because of ignorance we didnt know enough
  about the system being studied, so we had to
  approximate
• At the level of atoms and electrons, probability
  is fundamental
• An atom may not even have a definite position or
  speed until we measure it, so all we can predict
  is its probability (weird!)

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Mathematics of quantum mechanics

• Instead of an equation of motion (describing how
  a particle moves from its initial position), we
  have an equation of probability (describing how
  the probability of finding the particle at a
  certain place changes with time)
• Quantum mechanics is phrased in the language of
  probability expected value, standard deviation,
  etc.
• You will learn this in Algebra II! It really is
  useful!

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Symmetry in quantum mechanics

• Experiments show that all electrons are totally,
  completely, 100 identical
• This is a kind of symmetry if I switch two
  electrons while your back is turned, you cant
  tell the difference
• Same applies to protons, photons, etc. all
  quantum particles have this property (even whole
  atoms)

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Symmetry in quantum mechanics
Mathematically, this exchange symmetry leads to
some very cool consequences
Neutron stars
Lasers
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Optimization

• Warm-up What is the minimum value of the
  function
• Exercise What is the shortest path between any
  two points on a sphere? (Hint think about the
  North and South Poles.)
• Take-home challenge Use a graphing program (or
  not!) to graph the function
  What is the minimum value of f (x,y)?
  Is there more than one point (x,y) that gives
  this minimum value? Where is the axis of symmetry
  of this function? Investigate the tension between
  symmetry and optimization in this problem.

?
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Optimization Whats the best I can do?

• Many real-world problems involve finding the best
  or optimal solution to some problem
• Whats the largest amount of food I can buy for
  10?
• Whats the shortest route between home and
  school?
• Sometimes subject to constraints
• I can drive at 60 mph, bike at 15 mph, and walk
  at 5 mph. What is the fastest way from point A to
  point B, given that I only have 15 to spend on
  gas, and I dont want to spend more than 10
  minutes walking?

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Optimizing a function

• Mathematically, the most interesting points on a
  function are its critical points
• Something unusual happens there

Slope changes from negative to positive
Concave down becomes concave up
critical points
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Optimizing a function

• How can we find the minima and maxima of any
  general function?
• Calculus provides an answer finding critical
  points of f (x) involves finding the zeros of a
  related function f (x)
• In other words, we can solve an equation to find
  the minima and maxima no guessing involved!

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Optimization and equations of motion

• In physics, we solve equations to find the
  behavior of a system (motion of a particle,
  expansion of a gas, speed of waves on water,
  etc.)
• What if we could reinterpret those equations as
  the equations for the critical points of some
  other function?
• If the equations of motion are messy and
  complicated, maybe the function were trying to
  minimize will be simpler

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Example General Relativity

• General relativity (GR) tells us that space must
  curve in the presence of matter
• Gravity is really just the curvature of space!
• Einstein wrote down his equations after a long
  period of guess-and-check
• Final result was horribly messy could be solved
  case-by-case, but hard to see general properties
• Mathematician David Hilbert realized that
  Einsteins equations could be derived by
  minimizing a single function the total
  curvature!

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Optimization equations are everywhere!

• Classical mechanics motion of a particle
  minimizes a function called the Lagrangian
  (kinetic energy minus potential energy)
• Quantum mechanics equations of probability are
  also optimization equations
• In fact, all elementary equations in physics seem
  to be optimization equations!
• Why? Who knows

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Soap bubbles and string theory

• Because of surface tension, a soap bubble will
  find the shape of minimal area subject to
  constraints (pressure of air inside the bubble,
  shape of wire frame, etc.)
• Equations of motion in string theory are
  optimization equations resulting from minimizing
  the area of a 2-dimensional surface
• The mathematics of these two situations is
  identical! Same math, different problems.

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Symmetry, probability, and optimization in string
theory

• String theory is the leading candidate for a
  theory of all forces and matter, and relies
  heavily on all the math weve discussed
• Symmetry simplifies equations of motion
  enormously particles classified by group theory
• Probability quantum mechanics (the vibration of
  the strings gives rise to all quantum particles)
  and statistical mechanics (counting how many ways
  strings can fit inside a black hole!)
• Optimization equations of motion

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Some friendly advice

• If math and physics concepts like these look
  difficult, dont worry! Theyre difficult for
  everyone.
• A good foundation in math is essential.
  Everything you do in class this year will
  eventually find its way into your future work.
• Dont rush! Plenty of time to learn more advanced
  concepts (Ive been studying physics for 6 years,
  and Ive barely scratched the surface)
• Ask questions! Your teachers, friends, and fellow
  students, are your best resources.
• Parents support your kids in whatever they want
  to study. The best way to discover your passions
  is through exploration.

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My contact info

• Yoni Kahn y-kahn_at_u.northwestern.edu

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Backup slide Exchange Symmetry
Quantum mechanics says
For identical particles,
So
sign particles like to clump together (photons
in laser) - sign particles like to stay apart
(neutrons in neutron stars, electrons in atoms)
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Backup slide Temperature and Kinetic Energy

• Temperature is average kinetic energy (speed)
• Individual molecules dont matter, only average

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Backup slide GR