From Soap Bubbles to String Theory-Yoni Kahn

From Soap Bubbles to From Soap Bubbles to String Theory: String Theory: Mathematical Mathematical

Reasoning in the Reasoning in the Physical WorldPhysical World

Yoni KahnYoni KahnNorthwestern UniversityNorthwestern UniversityUniversity of CambridgeUniversity of Cambridge

Massachusetts Institute of Massachusetts Institute of TechnologyTechnology

A little bit about myselfA little bit about myself Miller, 1997-2000Miller, 1997-2000

MATHCOUNTS, Mandelbrot, Math CircleMATHCOUNTS, Mandelbrot, Math Circle Math EnrichmentMath Enrichment Lots of self-study for contests!Lots of self-study for contests!

Lynbrook, 2000-2004Lynbrook, 2000-2004 Still did contests, but found I wasn’t as good Still did contests, but found I wasn’t as good

(qualified for USAMO freshman year, but never (qualified for USAMO freshman year, but never again!)again!)

Got interested in physics, took physics classes at SJ Got interested in physics, took physics classes at SJ State senior yearState senior year

Northwestern University, 2004-2009Northwestern University, 2004-2009 Majored in physics, but took so many math courses Majored in physics, but took so many math courses

I ended up with a math major as well!I ended up with a math major as well!

This talkThis talk What is physics?What is physics?

Math vs. physicsMath vs. physics Three mathematical principles in Three mathematical principles in

physicsphysics Lots of examples!Lots of examples!

Questions!Questions!

““God used beautiful mathematics in God used beautiful mathematics in creating the world.”creating the world.”- Paul Dirac- Paul Dirac

““The miracle of the appropriateness The miracle of the appropriateness of the language of mathematics for of the language of mathematics for the formulation of the laws of the formulation of the laws of physics is a wonderful gift which we physics is a wonderful gift which we neither understand nor deserve.”neither understand nor deserve.”- - Eugene WignerEugene Wigner

Quotes by famous Quotes by famous peoplepeople

(Image credit xkcd.com)

Math vs. PhysicsMath vs. Physics Perfect Perfect

idealizationsidealizations Constructing Constructing

objects to studyobjects to study Proof (deductive Proof (deductive

reasoning)reasoning) If a proof is If a proof is

correct, it’s correct correct, it’s correct foreverforever

Real world Real world (messy!)(messy!)

Describing objects Describing objects that existthat exist

Theory predicts, Theory predicts, experiment testsexperiment tests

Theories can Theories can alwaysalways be be disproved!disproved!However, many

similarities…

Shared concepts in math Shared concepts in math and physicsand physics

IntuitionIntuition Comes with time and Comes with time and lotslots of of practicepractice!!

Problem-solvingProblem-solving All the same strategies apply: what tools All the same strategies apply: what tools

do I need? What new data would be do I need? What new data would be helpful? How can I make an unusual helpful? How can I make an unusual problem look familiar?problem look familiar?

More than one solution to a problem: can More than one solution to a problem: can learn something unexpected by doing a learn something unexpected by doing a problem two different ways!problem two different ways!

Reasoning by analogyReasoning by analogy Absurdly powerfulAbsurdly powerful in physics in physics

Same math, different Same math, different problems!problems!

Soap bubble String theory

What could these possibly have in common? Stay tuned to find out…

SymmetrySymmetry Warm-upWarm-up: ABCDEFGH is a regular octagon, : ABCDEFGH is a regular octagon,

and O is the intersection of its long diagonals. and O is the intersection of its long diagonals. What is angle AOB? (No formulas, and no What is angle AOB? (No formulas, and no calculators!)calculators!)

ExerciseExercise: What kinds of polynomials : What kinds of polynomials f f ((xx) have ) have the property that their right half (the property that their right half (xx > 0) is the > 0) is the mirror image of their left half (mirror image of their left half (xx < 0)? < 0)?

Take-home challengeTake-home challenge: Given a point P at (: Given a point P at (xx,,yy), ), find the coordinates of P’, which is the image find the coordinates of P’, which is the image of P after rotating 90° counterclockwise about of P after rotating 90° counterclockwise about the origin. the origin. ProveProve that if P is on the circle that if P is on the circle then so is P’. then so is P’. 2 2 1,x y

Discrete symmetryDiscrete symmetry How many rotations or reflections How many rotations or reflections

leave this equilateral triangle the leave this equilateral triangle the same?same?

Discrete symmetryDiscrete symmetry How many rotations or reflections leave How many rotations or reflections leave

this equilateral triangle the same?this equilateral triangle the same?

AnswerAnswer: three rotations (0°, 120°, 240°), : three rotations (0°, 120°, 240°), three reflections (about each of the three reflections (about each of the dashed lines)dashed lines)

Discrete symmetryDiscrete symmetryHow do we know there aren’t any others? How do we know there aren’t any others? For example, what about a rotation, then a For example, what about a rotation, then a reflection?reflection?

rotate 120° clockwise

reflect about dashed line

A

BC AB

C

C

A

B

Discrete symmetryDiscrete symmetryHow do we know there aren’t any others? How do we know there aren’t any others? For example, what about a rotation, then a For example, what about a rotation, then a reflection?reflection?

rotate 120° clockwise

reflect about dashed line

A

BC AB

C

C

A

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.

Discrete symmetryDiscrete symmetry Let’s call the three different rotations R1 (0°), R2 Let’s call the three different rotations R1 (0°), R2

(120°), and R3 (240°), and the three reflections I1 (120°), and R3 (240°), and the three reflections I1 (about the line through A), I2 (line through B), and (about the line through A), I2 (line through B), and I3 (line through C)I3 (line through C)

We can make a “We can make a “multiplication tablemultiplication table:”:” R1 · R2 = R2 (R1 does nothing)R1 · R2 = R2 (R1 does nothing) R2 · R3 = R1 (rotation by 120°, then 240°, R2 · R3 = R1 (rotation by 120°, then 240°,

recovers the original triangle)recovers the original triangle) R2 · I3 = I1 (shown on last slide)R2 · I3 = I1 (shown on last slide)

The symmetries of an equilateral triangle have an The symmetries of an equilateral triangle have an interesting structure, which mathematicians call a interesting structure, which mathematicians call a groupgroup..

Exercise: finish the multiplication table!Exercise: finish the multiplication table!

Continuous symmetryContinuous symmetry How many rotations or reflections leave this How many rotations or reflections leave this

circle the same?circle the same?

Continuous symmetryContinuous symmetry How many rotations or reflections leave this circle the same?How many rotations or reflections leave this circle the same?

AnswerAnswer: infinitely many! Rotation by : infinitely many! Rotation by anyany angle, no matter angle, no matter how small, and reflection about how small, and reflection about anyany diameter diameter

The triangle had The triangle had discretediscrete symmetry (only certain angles), but symmetry (only certain angles), but the circle has the circle has continuous continuous symmetrysymmetry

Continuous symmetryContinuous symmetryCan 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 θ. For example:

• R(θ) is a rotation clockwise by the angle θ• I(θ) is a reflection about the diameter which makes an angle θ with the horizontal

Multiplication table: R(θ1) · R(θ2) = R(θ1 + θ1) (if you rotate twice, add the angles)

R(θ1) · I(0) = R(- θ1) (check this!)

OK, but what does this have OK, but what does this have to do with physics?to do with physics?

Put equal electric charges at each vertex of the triangle:

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!

+

++

P Q

R

OK, but what does this have OK, but what does this have to do with physics?to do with physics?

Or, consider the gravitational field of a sphere. Which direction does it point in?

OK, but what does this have OK, but what does this have to do with physics?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!

Real-world example: Real-world example: black holesblack holes

Einstein’s equations Einstein’s equations of General Relativity of General Relativity are are veryvery hard to solve hard to solve

But, if we assume But, if we assume spherical symmetry, spherical symmetry, the equations simplify the equations simplify dramatically: there is dramatically: there is essentially essentially one one unique solutionunique solution

This solution This solution predictspredicts the existence of black the existence of black holes!holes!

ProbabilityProbability Warm-upWarm-up: If I roll a six-sided die once, what : If I roll a six-sided die once, what

is the probability that I roll either a 2 or a is the probability that I roll either a 2 or a 3?3?

ExerciseExercise: What is the probability that I roll : What is the probability that I roll a 6 two times in a row?a 6 two times in a row?

Take-home challengeTake-home challenge: What is the : What is the probability that I roll a 6 twenty times in a probability that I roll a 6 twenty times in a row? How many zeroes after the decimal row? How many zeroes after the decimal place will this number have? If I roll the die place will this number have? If I roll the die once a second, every second of every day, once a second, every second of every day, how many years (on average) will it take how many years (on average) will it take before I get twenty 6’s in a row? (Does this before I get twenty 6’s in a row? (Does this answer surprise you?)answer surprise you?)

Dealing with incomplete Dealing with incomplete informationinformation

How many molecules are in a glass of How many molecules are in a glass of water?water?

We have no hope of ever describing the We have no hope of ever describing the exact location and speed of every single exact location and speed of every single molecule in the glassmolecule in the glass

However, we don’t need to! Physics cares However, we don’t need to! Physics cares about about bulk quantitiesbulk quantities:: TemperatureTemperature PressurePressure DensityDensity

Solution: probabilitySolution: probability We can ask: how do the internal configurations We can ask: how do the internal configurations

relate to the bulkrelate to the bulk quantities like temperature?quantities like temperature?AnswerAnswer: if each internal configuration is equally : if each internal configuration is equally probable, then the probable, then the most probablemost probable bulk quantity bulk quantity will correspond to the will correspond to the greatest numbergreatest number of of internal configurationsinternal configurations

So maybe we can’t predict temperature exactly, So maybe we can’t predict temperature exactly, but we can predict it with 99.99999% certainty but we can predict it with 99.99999% certainty – good enough!– good enough!

Turns a seemingly intractable problem into a Turns a seemingly intractable problem into a problem of problem of countingcounting – – this branch of physics is this branch of physics is statistical mechanicsstatistical mechanics

Example: perfume in a Example: perfume in a roomroom

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

Example: perfume in a Example: perfume in a roomroom

If I open a bottle of perfume at the far If I open a bottle of perfume at the far corner of the room, why can you smell it corner of the room, why can you smell it across the room almost immediately?across the room almost immediately?Answer:Answer: there are a there are a smallsmall number of ways number of ways the scent-carrying molecules can be the scent-carrying molecules can be concentrated in the perfume bottle, but concentrated in the perfume bottle, but there are a there are a hugehuge number of ways that the number of ways that the molecules could be randomly distributed molecules could be randomly distributed across the roomacross the room

The perfume diffuses because of The perfume diffuses because of probability!probability!

Probability and quantum Probability and quantum mechanicsmechanics

In the previous examples, we used In the previous examples, we used probability because of ignorance – we probability because of ignorance – we didn’t know enough about the system didn’t know enough about the system being studied, so we had to approximatebeing studied, so we had to approximate

At the level of atoms and electrons, At the level of atoms and electrons, probability is probability is fundamentalfundamental

An atom may not even An atom may not even havehave a definite a definite position or speed until we measure it, so position or speed until we measure it, so all we can predict is its probability all we can predict is its probability (weird!)(weird!)

Mathematics of quantum Mathematics of quantum mechanicsmechanics

Instead of an Instead of an equation of motionequation of motion (describing how a particle moves from its (describing how a particle moves from its initial position), we have an initial position), we have an equation of equation of probabilityprobability (describing how the probability (describing how the probability of finding the particle at a certain place of finding the particle at a certain place changes with time)changes with time)

Quantum mechanics is phrased in the Quantum mechanics is phrased in the language of probability: expected value, language of probability: expected value, standard deviation, etc.standard deviation, etc.

You will learn this in Algebra II! It really is You will learn this in Algebra II! It really is useful!useful!

Symmetry in quantum Symmetry in quantum mechanicsmechanics

Experiments show that Experiments show that allall electrons electrons are totally, completely, 100% are totally, completely, 100% identicalidentical

This is a kind of symmetry: if I switch This is a kind of symmetry: if I switch two electrons while your back is two electrons while your back is turned, you can’t tell the differenceturned, you can’t tell the difference

Same applies to protons, photons, Same applies to protons, photons, etc. – all quantum particles have this etc. – all quantum particles have this property (even whole atoms)property (even whole atoms)

Symmetry in quantum Symmetry in quantum mechanicsmechanics

Mathematically, this exchange symmetry leads to some very cool

consequences:Neutron stars Lasers

OptimizationOptimization Warm-upWarm-up: What is the minimum value of the : What is the minimum value of the

function function ExerciseExercise: What is the shortest path between : What is the shortest path between

any two points on a sphere? (Hint: think about any two points on a sphere? (Hint: think about the North and South Poles.)the North and South Poles.)

Take-home challengeTake-home challenge: Use a graphing : Use a graphing program (or not!) to graph the function program (or not!) to graph the function What is the minimum value of What is the minimum value of f f ((xx,,yy)? Is there more than one point ()? Is there more than one point (xx,,yy) that ) that gives this minimum value? Where is the axis of gives this minimum value? Where is the axis of symmetry of this function? Investigate the symmetry of this function? Investigate the tension between tension between symmetrysymmetry and and optimizationoptimization in in this problem. this problem.

2( ) ( 4) 3f x x ?

2 2 2 2 2( , ) ( ) 2( ).f x y x y x y

Optimization: What’s the Optimization: What’s the best I can do?best I can do?

Many real-world problems involve finding Many real-world problems involve finding the best or optimal solution to some the best or optimal solution to some problemproblem What’s the largest amount of food I can buy for What’s the largest amount of food I can buy for

$10?$10? What’s the shortest route between home and What’s the shortest route between home and

school?school? Sometimes subject to Sometimes subject to constraintsconstraints::

I can drive at 60 mph, bike at 15 mph, and walk at 5 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 mph. What is the fastest way from point A to point B, given that I only have $15 to spend on gas, and I B, given that I only have $15 to spend on gas, and I don’t want to spend more than 10 minutes walking?don’t want to spend more than 10 minutes walking?

Optimizing a functionOptimizing a function Mathematically, the most interesting points Mathematically, the most interesting points

on a function are its on a function are its critical pointscritical points Something unusual happens there:Something unusual happens there:

Slope changes from negative to positive

Concave down becomes concave up

critical points

Optimizing a functionOptimizing a function How can we find the minima and How can we find the minima and

maxima of any general function?maxima of any general function? CalculusCalculus provides an answer: finding provides an answer: finding

critical points of critical points of f f ((xx) involves ) involves finding the zeros of a related finding the zeros of a related function function f ’ f ’ ((xx))

In other words, we can In other words, we can solve an solve an equationequation to find the minima and to find the minima and maxima – no guessing involved!maxima – no guessing involved!

Optimization and equations Optimization and equations of motionof motion

In physics, we solve equations to find In physics, we solve equations to find the behavior of a system (motion of a the behavior of a system (motion of a particle, expansion of a gas, speed of particle, expansion of a gas, speed of waves on water, etc.)waves on water, etc.)

What if we could reinterpret those What if we could reinterpret those equations as the equations for the equations as the equations for the critical points of some other function?critical points of some other function?

If the equations of motion are messy If the equations of motion are messy and complicated, maybe the function and complicated, maybe the function we’re trying to minimize will be simplerwe’re trying to minimize will be simpler

Example: General Example: General RelativityRelativity

General relativity (GR) tells us that space General relativity (GR) tells us that space must curve in the presence of mattermust curve in the presence of matter Gravity is really just the curvature of space!Gravity is really just the curvature of space!

Einstein wrote down his equations after a Einstein wrote down his equations after a longlong period of guess-and-check period of guess-and-check

Final result was horribly messy: could be Final result was horribly messy: could be solved case-by-case, but hard to see general solved case-by-case, but hard to see general propertiesproperties

Mathematician David Hilbert realized that Mathematician David Hilbert realized that Einstein’s equations could be derived by Einstein’s equations could be derived by minimizing a minimizing a single functionsingle function: the total : the total curvature!curvature!

Optimization equations are Optimization equations are everywhere!everywhere!

Classical mechanics: motion of a particle Classical mechanics: motion of a particle minimizes a function called the Lagrangian minimizes a function called the Lagrangian (kinetic energy minus potential energy)(kinetic energy minus potential energy)

Quantum mechanics: equations of Quantum mechanics: equations of probability are also optimization equationsprobability are also optimization equations

In fact, In fact, allall elementary equations in physics elementary equations in physics seem to be optimization equations!seem to be optimization equations! Why? Who knows…Why? Who knows…

Soap bubbles and string Soap bubbles and string theorytheory

Because of surface tension, a soap bubble Because of surface tension, a soap bubble will find the shape of will find the shape of minimal areaminimal area subject subject to constraints (pressure of air inside the to constraints (pressure of air inside the bubble, shape of wire frame, etc.)bubble, shape of wire frame, etc.)

Equations of motion in string theory are Equations of motion in string theory are optimization equations resulting from optimization equations resulting from minimizing the areaminimizing the area of a 2-dimensional of a 2-dimensional surfacesurface

The mathematics of these two situations is The mathematics of these two situations is identical! identical! Same math, different problems.Same math, different problems.

Symmetry, probability, and Symmetry, probability, and optimization in string optimization in string

theorytheory String theory is the leading candidate for a String theory is the leading candidate for a

theory of all forces and matter, and relies theory of all forces and matter, and relies heavily on all the math we’ve discussed:heavily on all the math we’ve discussed: SymmetrySymmetry: simplifies equations of motion : simplifies equations of motion

enormously; particles classified by group theoryenormously; particles classified by group theory ProbabilityProbability: quantum mechanics (the vibration : quantum mechanics (the vibration

of the strings gives rise to all quantum of the strings gives rise to all quantum particles) and statistical mechanics (counting particles) and statistical mechanics (counting how many ways strings can fit inside a black how many ways strings can fit inside a black hole!)hole!)

OptimizationOptimization: equations of motion: equations of motion

Some friendly adviceSome friendly advice If math and physics concepts like these look If math and physics concepts like these look

difficult, difficult, don’t worry!don’t worry! They’re difficult for everyone. They’re difficult for everyone. A A good foundationgood foundation in math is in math is essentialessential. Everything . Everything

you do in class this year will eventually find its way you do in class this year will eventually find its way into your future work.into your future work.

Don’t rush!Don’t rush! Plenty of time to learn more advanced Plenty of time to learn more advanced concepts (I’ve been studying physics for 6 years, concepts (I’ve been studying physics for 6 years, and I’ve barely scratched the surface…)and I’ve barely scratched the surface…)

Ask questions!Ask questions! Your teachers, friends, and fellow Your teachers, friends, and fellow students, are your best resources.students, are your best resources.

Parents: support your kids in Parents: support your kids in whateverwhatever they want to they want to study. The best way to discover your passions is study. The best way to discover your passions is through exploration.through exploration.

My contact infoMy contact infoYoni Kahn: Yoni Kahn:

[email protected]@u.northwestern.edu

Backup slide: Exchange Backup slide: Exchange SymmetrySymmetry

21 2 1 2( , ) ( , )P x x x x

1 2 2 1( , ) ( , )P x x P x xFor identical particles,

1 2 2 1( , ) ( , )x x x x So

+ sign: particles like to clump together (photons in laser)

- sign: particles like to stay apart (neutrons in neutron stars, electrons in atoms)

Quantum mechanics says

Backup slide: Temperature Backup slide: Temperature and Kinetic Energyand Kinetic Energy

Temperature is Temperature is average kinetic average kinetic energy (speed)energy (speed)

Individual Individual molecules don’t molecules don’t matter, only matter, only averageaverage

Backup slide: GRBackup slide: GR