What’s String Theory?

• I’m Yuji Tachikawa, a member of Department of Physics.

• I’m a string theorist.

• I got my PhD in 2006, here in U. Tokyo.

• I was a postdoc at the Institute for Advanced Study at Princeton till 2011.

• I’m working here since then.

• So I’m ~15 years older than you. Hmm.

String theory is ...

• A branch of theoretical physics / mathematical physics where we try to reconcile

• gravity and

• quantum mechanics.

• What’s gravity?

• What’s quantum mechanics?

• Why do we have to reconcile them?

• How do we reconcile them?

What’s Gravity?

Mostly described by Newtonian mechanics!

Motion of Mercury deviates significantly from it.

Einstein says it’s due to the warping of the spacetime itself.

Called General Relativity.

Attraction between apples can also bedescribed by General Relativity

if you want, but it’s probably overkill.

If you have a smartphone, it probably has GPS in it.

Due to general relativity effects, the time inside the satellite runs faster

(around 40 μs per day).

If uncorrected, this would totally ruin the accuracyof your satellite navigation system!

People who designed GPS knew this, and implementedprecaution against it.

Even without Einstein, we would have knownthe general relativity by now,

first as a mysterious source of error in the GPS system.

Gravity

Electromagnetism

F = GMm

4πϵ0

Light (electromagnetic wave) has two polarizations.

You rotate one 90 degrees, you get the other.

We can’t see polarization of light directly,but mantis shrimps can.

Some sunglasses have polarizers.

Gravitational wave also has two polarizations.

Called Spin 1.

Called Spin 2.

Called Spin 1.

Light (electromagnetism) is spin 1.

“Strong nuclear force” is also spin 1.“Weak nuclear force” is also spin 1.

Theoretically, you can have as many spin-1 forcesas you want.

Experimentally, there are three.

You rotate one 45 degrees, you get the other.Called Spin 2.

Gravity is spin 2.

Experimentally, there is only one spin 2 force.

Theoretically, physicists even don’t know how to write a theory with more than one spin-2 force.

It’s simply impossible.

• There are four forces in the world:

• Electromagnetism (light)

• “Weak nuclear force”

• “Strong nuclear force”

• Gravity

• Gravity is rather different !

spin 1

spin 2

What’s Quantum Mechanics?

ElectronSource

DoubleSlits

Screen

ElectronSource

DoubleSlits

Screen

ElectronSource

DoubleSlits

Screen

(c) Hitachi / Prof. Tonomura

One electron passes the two slitsat the same time...

Its “wavefunction” interfere,and causes

If nobody is watching the moon,

does the moon exist?

Is it even a physics question?

Yes it is a physics question.

If you always watch an electron,it goes through one particular path.

You won’t get this.

By not watching the electron during the way,it creates

• Many famous physicists didn’t like it.

• But nature doesn’t care if famous physicisits like it or not.

• For example, consider the gedanken experiment proposed by Schrödinger:

“Schrödinger’s me” experiment

In an hour, the Geiger counter“clicks” with 50% probability.

If it clicks, I drink and get drunk.

If it doesn’t click,I don’t drink and stay awake.

Suppose you keep me in a sealed boxand do this experiment.

Before you open the box after an hour,am I in a superposition?

I think I will be, from your point of viewfrom outside of the box.

Bigger and bigger things are put intoquantum mechanical superposition

experimentally.There doesn’t seem to be any upper bound

in principle.

structure and subsequently patterning the qubit. The fabrication pro-cess involved 13 layers of lithography, including metal and dielectricdeposition and etching steps (Supplementary Information). In the laststep, the device was exposed to xenon difluoride gas to release themechanical resonator. A photomicrograph of a completed device isshown in Fig. 2.

Our quantum electrical circuit is a Josephson phase qubit23,24,30

comprising a Josephson junction shunted in parallel by a capacitorand an inductor. The qubit can be approximated as a two-levelquantum system with a ground state, jgæ, and an excited state, jeæ,separated in energy from jgæ by DE, whose transition frequency,fq 5DE/h, can be set between 5 and 10 GHz. The qubit frequency isprecisely controlled by a current bias, which is applied using anexternal magnetic flux coupled through the parallel inductor. The stateof the qubit is measured using a single-shot procedure23; accumulating,1,000 such measurements allows us to determine the excited-stateoccupation probability, Pe (Supplementary Information). We havepreviously used the phase qubit to perform one- and two-qubit gateoperations24, to measure and quantum-control photons in an electro-magnetic resonator27,28 and to demonstrate the violation of a Bellinequality31. Here the qubit and the mechanical resonator are coupledthrough an interdigitated capacitor of capacitance Cc < 0.5 pF, tomaximize the coupling strength between the qubit and resonator whilenot overloading the qubit. The coupled system can be modelled usingthe Jaynes–Cummings Hamiltonian32, allowing us to estimate thecoupling energy, g, between the mechanical resonator and the qubit.This energy involves the coupling capacitance as well as the electricaland mechanical properties of the mechanical resonator, as described inref. 5; the corresponding coupling frequency is designed to be V 5 2g/h < 110 MHz. The equivalent electrical circuit for the combined res-onator and qubit is shown in Fig. 2b.

Quantum ground state

The completed device was mounted on the mixing chamber of adilution refrigerator and cooled to T < 25 mK. At this temperature,both the qubit and the resonator should occupy their quantum

ground states. To study the cooled device, we performed microwavequbit spectroscopy23 to reveal the resonant frequencies of the com-bined system, using the pulse sequence shown in Fig. 2c. We mea-sured the excited-state probability, Pe, as a function of the qubitfrequency and the microwave excitation frequency, as shown inFig. 2d. The qubit frequency tunes as expected23,30 and displays thecharacteristic level avoidance of a coupled system as its frequencycrosses the fixed mechanical resonator frequency, fr. Similar observa-tions have been made using optomechanical systems33.

We note that the mechanical resonator produces two features in theclassical transmission measurement shown in Fig. 1d, generating amaximum (at fr) and a minimum (at fs) in the response. Whencoupled and measured using the qubit as in Fig. 2, the lower-frequencyresonance, at fs, does not produce a response, as this resonance doesnot correspond to a sustainable excitation of the complete circuit.However, the higher-frequency feature, at fr, does sustain such excita-tions and thus appears in the spectroscopic measurement.

To determine the coupling strength between the qubit and themechanical resonator, we fitted the detailed behaviour near the levelavoidance, as shown in Fig. 2e. The fitted qubit–resonator couplingstrength, V < 124 MHz, corresponds to an energy transfer (Rabi-swap) time of about 4.0 ns, and is in reasonable agreement withour design value.

We then performed a second spectroscopy measurement, similarto the qubit spectroscopy but coupling the microwaves to the mech-anical resonator through the capacitor of capacitance Cx shown inFig. 2b, rather than to the qubit. In this measurement, shown in Fig. 3,the mechanical resonator acts as a narrow band-pass filter, so signifi-cant qubit excitation (large Pe) should only occur near the mech-anical resonance frequency, fr, as observed. In general, the spectrumlooks very similar to that measured while exciting the qubit, provid-ing strong support that the fixed resonance is indeed due to themechanical resonator.

For higher-power microwave excitations, a new feature emerges inthe resonator spectroscopy, as shown in Fig. 3b. The qubit, althoughapproximated as a two-level system, actually has a double-well

Frequency (GHz)

60 μm

d0.025

5.4 5.6 5.8 6.0 6.2 6.4

Cm C0Lm

Figure 1 | Dilatational resonator. a, Scanning electron micrograph of asuspended film bulk acoustic resonator. Details on the fabrication of theresonator appear in Supplementary Information. The mechanical structurewas released from the substrate by exposing the device to xenon difluoride,which isotropically etches any exposed silicon; the suspended structurecomprises, from bottom to top, 150 nm SiO2, 130 nm Al, 330 nm AlN and130 nm Al. The dashed box indicates the mechanically active part ofstructure. b, Fundamental dilatational resonant mode for the mechanicallyactive part of the resonator. The thickness of the structure changes throughthe oscillation cycle. c, Equivalent lumped-element circuit representation ofthe mechanical resonator, based on a modified van Dyke–Butterworthmodel26,38. This circuit includes a series-connected equivalent mechanicalinductance Lm and capacitance Cm and a parallel geometric capacitance C0,with mechanical dissipation modelled as Rm and dielectric loss as R0.d, Measured classical transmission, | S21 | (blue), and fit (red) of a typicalmechanical resonance. The transmission has two features: one, at the

frequency fs < 1/2pffiffiffiffiffiffiffiffiffiffiffiffiLmCm

p< 6.07 GHz, due to the series resonance of the

equivalent mechanical components Lm and Cm, and one, at the slightlyhigher frequency fr < 1/2p

ffiffiffiffiffiffiffiffiffiffiffiLmCsp

< 6.10 GHz, due to Lm and the equivalentcapacitance, Cs, of the capacitors Cm and C0 in series. These expressions areapproximate, as they do not take into account the effect of the dissipativeelements and external circuit loading. Inset, equivalent circuit for theresonator (Z, as shown in c) embedded in the measurement circuit,including two on-chip external coupling capacitors with Cx 5 37 fF and aninductive element with Ls < 1 nH that accounts for stray on-chip wiringinductance. Measurement is done using a calibrated network analyser thatmeasures the transmission from port 1 to port 2. We calculate C0 5 0.19 pFscaling from the geometry, and from the fit we obtain Cm 5 0.655 fF,Lm 5 1.043mH, Rm 5 146V and R0 5 8V. These values are compatible withthe geometry and measured properties of AlN29. We calculate a mechanicalquality factor of Q < 260 and a piezoelectric coupling coefficient ofk2

eff < 1.2% (ref. 38).

ARTICLES NATURE | Vol 464 | 1 April 2010

O’Connel et al., Nature 464 (2010) p. 697

Why do we have to reconcile them?

Reconcile?

• Electric field ~ 1/r2

• Energy per volume ~ | electric field |2

• Energy carried by the electric field ~

r2dr =

%r=∞

• An electron, if pointlike, always carry infinite amount of energy.

• Energy = mass × (speed of light)2

• Electron is infinitely massive!

• Of course the electron has finite mass.

• This puzzled physicists at the turn of 20th century greatly.

• One idea which didn’t work:

• Maybe all the mass is due to the electric field.

• Maybe electron has a finite radius.

r2dr =

%r=∞

• But it doesn’t work with relativity.

• Basically, relativity doesn’t like finite-sized rigid body.

• In relativity, nothing can exceed the speed of light.

a rigid body

one light yearPush!

• This can’t happen.

• Finite size = bad in relativity.

• In relativity, nothing can exceed the speed of light.

a rigid body

one light yearPush!

• This can’t happen.

• Finite size = bad in relativity.

• This problem was solved around in 1950.

• So it took almost half a century.

• I can’t easily say how it was resolved... as it is rather involved.

• It uses Quantum Mechanics in an essential way.

Real Electron Mass

Energy of the Electric Field

“Naked Electron Mass”

∞−

Naively, we expect something like

But you never see a naked electronwithout its electric field.

So you don’t and can’t think about them.

Real Electron Mass

Naively, we expect something like

But you never see a naked electronwithout its electric field.

So you don’t and can’t think about them.

• This theory is called the Quantum ElectroDynamics.

• It works extremely well.

• The most amazing example: the electron anomalous magnetic moment

Experiment: 0.001159652180...Theory: 0.001159652181...

They agree up to the expected experimental & theoretical errors.

Figure: 389 self-energy diagrams representing 6354 vertex diagrams of Set V.

T. Kinoshita () 34 / 57

• I took this slide from Prof. T. Kinoshita’s talk.

• He’s been doing this computation since 1971.

• He’s now 90, and can create slides in LaTeX himself.

The same problem of infinite energy arises for all four forces:

electromagnetism

“strong nuclear force”

“weak nuclear force”

gravity

electromagnetism

gravity

Problems solved.

Problems unsolved.

The same problem of infinite energy arises for all four forces:

electromagnetism

gravity

Called Spin 1.

Called Spin 2.

Problems solved.

Problem unsolved.

Gravity is different.

• Anyway, we learned by the late 70s how to treat quantum mechanically

• electromagnetism,

• “strong nuclear force,” and

• “weak nuclear force.”

• So it was unsatisfactory that we didn’t know how to treat gravity quantum mechanically.

• But, is it a must ?

• Maybe gravity is not quantum mechanical, after all.

• But trying to make gravity not quantum mechanical leads to many absurd things.

VOLUME 47, NUMBER 14 PHYSICAL REVIEW LETTERS 5 OCTOBER 1981

Indirect Evidence for Quantum GravityDon N. Page

Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802

C. D. GeilkerDePartment of Physics, Wit/iam Jetoell College, Liberty, Missouri 64068

(Received 9 June 1981)An experiment gave results inconsistent with the simplest alternative to quantum gra-

vity, the semiclassical Einstein equations. This evidence supports (but does not prove)the hypothesis that a consistent theory of gravity coupled to quantized matter should alsohave the gravitational field quantized.

PACS numbers: 04.60.+n

Quantum mechanics appears to govern all non-gravitational fields (here called matter), andmost people believe it also applies to the gravita-tional field. However, there has been no explicitexperimental test of this. Gravity is so weakthat Feynman' has questioned whether it mustbe quantized. As an alternative, Mgiller' andRosenfeld' have proposed a theory in which grav-ity is described by a classical field which obeysthe semiclassical Einstein equations

G~ —-8&(gl Tqv I g) ~

Here G„, is the Einstein tensor of the unquan-tized metric g~, . T„, is the stress-energy quan-tum operator, and p is the wave function or quan-tum state of the matter. (One could replace g bya density matrix or a C*-algebra state with noessential changes. ) In the Heisenberg picture,which we adopt, (1) is to be supplemented bythe appropriate covariant field equations and com-mutation relations for the quantized matter fieldoperators in the presence of the classical metric.The functional dependence of g„s upon g by (1)introduces a nonlinearity into the metric-depen-dent quantum evolution of the matter. " Thismakes it crucial to specify what happens duringa measurement. In the conventional view, thewave function collapses into an eigenstate of themeasured variable. ' This would change the right-hand side of (1) and produce objectionable conse-quences. "' For example, assuming that onecan make a measurement which collapses thewave function outside one's future light cone,Eppley and Hannah show' that one could use semi-classical gravity to transmit observable signalsfaster than light. Such consequences might welllead one to reject the conventional view in thecontext of the semiclassical theory of gravity,though one could argue that such unexpected ef-

fects have not been ruled out experimentally andthus should not yet be dismissed as unphysical.A more conclusive argument against the col-lapse of the wave function in the semiclassicaltheory is that if g collapses, in general the right-hand side of (1) will not be conserved, whereasthe left-hand side is automatically conserved.That is, if g =Q, c,. (x )g; with constant p;'s inthe Heisenberg picture but with c,.(x )'s whichchange during a measurement, then for almostall conceivable reductions of the wave packet,8.&yl T ""lip& .=8 P, ,(,*,)., (g, ~T""~j,)xO=G"".„. (2)

One might seek to avoid the inconsistency bysimply abandoning (1) during a measurement.However, one would need a replacement of (1)in order to determine the evolution of the gravi-tational field for any particular collapse of thewave function, and this would differ from thesemiclassical Einstein equations.Therefore, in order to retain (1) as the sim-

plest semiclassical theory of gravity, we mustassume that the universal matter wave functionj never collapses, as in the Everett formulationof quantum mechanics. ' One might think that theconventional collapse view is equivalent to this,as it is in practice for linear quantum theoriesin which one may ignore components of the wavefunction which have negligible interference withthe ones of interest. But in semiclassical grav-ity the metric depends upon all components of g,none of which can be ignored. Nevertheless,once the evolution of the gravitational field isdetermined by using the full wave function in (1),g may be decomposed into linear components onthat four-dimensional metric and any standardinterpretation may be applied to the components.One must simply remember that the individual

1981 The American Physical Society 979

Let me show how absurd it becomesif you try not to treat gravity quantum mechanically

following this paper:

“Schrödinger’s me” experiment

“Schrödinger’s me + gravity” experiment

In an hour, the Geiger counter“clicks” with 50% probability.

A 10g weight is always on a pan.

10g 20g

If it clicks, I put the 20g weight in the other pan.

If it doesn’t, I don’t put it there.

Now, let’s seal me inside the box.Then the weight is in the superposition.

Suppose gravity is not quantum mechanical.Then the nature needs to take the average

to decide what happens to the balance.

So the balance should be horizontal, no matter what.On the average, it weighs 10g on the right pan.

10g 20g

If it clicks, I put the 20g weight in the other pan.From inside,

10g 20g

If gravity is not quantum mechanical, the balance should stay horizontal.

This is clearly crazy.

If it doesn’t click, I don’t put the weight in the right pan.

From inside,

If gravity is not quantum mechanical, the balance should stay horizontal.

This is clearly crazy.

10g 20g

I’m in a superposition together with gravityIf gravity is quantum mechanical,

of this situation

and this situation.

10g 20g

I’m in a superposition together with gravityIf gravity is quantum mechanical,

• These arguments clearly shows that the world turns completely absurd if you don’t treat gravity quantum mechanically.

• But scientists should check it experimentally.

• It might be that the world is very absurd.

VOLUME 47, NUMBER 14 PHYSICAL REVIEW LETTERS 5 OCTOBER 1981

Indirect Evidence for Quantum GravityDon N. Page

Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802

C. D. GeilkerDePartment of Physics, Wit/iam Jetoell College, Liberty, Missouri 64068

(Received 9 June 1981)An experiment gave results inconsistent with the simplest alternative to quantum gra-

vity, the semiclassical Einstein equations. This evidence supports (but does not prove)the hypothesis that a consistent theory of gravity coupled to quantized matter should alsohave the gravitational field quantized.

PACS numbers: 04.60.+n

Quantum mechanics appears to govern all non-gravitational fields (here called matter), andmost people believe it also applies to the gravita-tional field. However, there has been no explicitexperimental test of this. Gravity is so weakthat Feynman' has questioned whether it mustbe quantized. As an alternative, Mgiller' andRosenfeld' have proposed a theory in which grav-ity is described by a classical field which obeysthe semiclassical Einstein equations

G~ —-8&(gl Tqv I g) ~

Here G„, is the Einstein tensor of the unquan-tized metric g~, . T„, is the stress-energy quan-tum operator, and p is the wave function or quan-tum state of the matter. (One could replace g bya density matrix or a C*-algebra state with noessential changes. ) In the Heisenberg picture,which we adopt, (1) is to be supplemented bythe appropriate covariant field equations and com-mutation relations for the quantized matter fieldoperators in the presence of the classical metric.The functional dependence of g„s upon g by (1)introduces a nonlinearity into the metric-depen-dent quantum evolution of the matter. " Thismakes it crucial to specify what happens duringa measurement. In the conventional view, thewave function collapses into an eigenstate of themeasured variable. ' This would change the right-hand side of (1) and produce objectionable conse-quences. "' For example, assuming that onecan make a measurement which collapses thewave function outside one's future light cone,Eppley and Hannah show' that one could use semi-classical gravity to transmit observable signalsfaster than light. Such consequences might welllead one to reject the conventional view in thecontext of the semiclassical theory of gravity,though one could argue that such unexpected ef-

fects have not been ruled out experimentally andthus should not yet be dismissed as unphysical.A more conclusive argument against the col-lapse of the wave function in the semiclassicaltheory is that if g collapses, in general the right-hand side of (1) will not be conserved, whereasthe left-hand side is automatically conserved.That is, if g =Q, c,. (x )g; with constant p;'s inthe Heisenberg picture but with c,.(x )'s whichchange during a measurement, then for almostall conceivable reductions of the wave packet,8.&yl T ""lip& .=8 P, ,(,*,)., (g, ~T""~j,)xO=G"".„. (2)

One might seek to avoid the inconsistency bysimply abandoning (1) during a measurement.However, one would need a replacement of (1)in order to determine the evolution of the gravi-tational field for any particular collapse of thewave function, and this would differ from thesemiclassical Einstein equations.Therefore, in order to retain (1) as the sim-

plest semiclassical theory of gravity, we mustassume that the universal matter wave functionj never collapses, as in the Everett formulationof quantum mechanics. ' One might think that theconventional collapse view is equivalent to this,as it is in practice for linear quantum theoriesin which one may ignore components of the wavefunction which have negligible interference withthe ones of interest. But in semiclassical grav-ity the metric depends upon all components of g,none of which can be ignored. Nevertheless,once the evolution of the gravitational field isdetermined by using the full wave function in (1),g may be decomposed into linear components onthat four-dimensional metric and any standardinterpretation may be applied to the components.One must simply remember that the individual

1981 The American Physical Society 979

So the authors of this paper really did a variant of this thought experiment.

(They used a torsion balance.)

VOLUME 47, NUMBER 14 PHYSICAL REVIEW LETTERS 5 QQTQBER 1981

The procedure for each run was to generate aquantum decision with the Geiger tubes, positionthe macroscopic masses accordingly, and meas-ure the gravitational field by the torsion balance.If the quantum decision was o. , we set the massesin configuration AB, meaning the four-dimension-al configuration in which the large balls wereplaced in position A for 30 min of Cavendishbalance measurements and then in position B for30 min. If the Geiger counters gave P, we setconfiguration BA, meaning position B first andthen A. (Using a sequence of two positions rath-er than one increased the sensitivity and reducedthe effects of slowly varying nongravitational in-fluences on the torsion balance, but it is in prin-ciple unnecessary. ) In each experimental run,the appropriate configuration was started at apredetermined time, independent of the quantumdecision.Since the quantum process caused the wave

function to have amplitudes of comparable weightfor both decisions n and P, the correspondingpositioning of the masses led to simultaneouslyoccurring amplitudes for both mass configura-tions AB and BA. We of course assume that thefull wave function never collapses and that it in-cludes all aspects of the positioning (includingthe experimenter who recorded the Geiger tubecounts, calculated the ratio to classify the deci-sion, and then placed the masses in the corre-sponding positions), as is necessary even to dis-cuss the semiclassical Einstein equations con-sistently. We also assume that the positioningwas generally faithful to the quantum decisionrather than being determined by some systematiceffect. A refinement of the experiment mightemploy a completely inanimate positioning proc-ess, but this is not necessary so long as it isassumed that the experimenter did not put themasses in nearly the same configuration in near-ly all components of the wave function, disre-garding the quantum measurements. With theseassumptions we conclude that the wave functionreally did have a comparable measure of ampli-tudes for components with both mass configura-tions. We had no information about the compli-cated phase relations between these amplitudes,but that was not necessary since we were notdoing an interference experiment.Now in a quantum theory of gravity, we wouldpredict that the quantized gravitational fieldwould differ from component to component of thewave function and be highly correlated with themass configuration. Thus we would expect the

torsion balance to respond in each componentaccording to the mass configuration in that com-ponent. But in the semiclassical theory of grav-ity, we would predict a definite classical four-dimensional (i.e., not necessarily static) gravita-tional field that would correspond to the expecta-tion value of the stress-energy operator. Sincethe amplitudes for different components haverapidly varying relative phases, there would benegligible contributions from cross terms in theright-hand side of (1). For our nonrelativisticconfigurations it would essentially be a square-amplitude-weighted average over the mass dis-tributions of the different components of the wavefunction. Because the configurations AB and BAhave nearly equal weights, we would expect onlya small response by the torsion balance in thesemiclassical theory, and no correlations withthe particular mass configuration in our compo-nent of the wave function.The series of ten experimental runs gave 30-sec y-ray counts with means and standard devia-tions 1509.1+31.0 and 887.6+23.0 for the tworespective Geiger counters. The fluctuationsare consistent with Poisson statistics and thuswere attributed to the quantum mechanics of theradioactive decays and detections. There was anegligible background count rate when the cobalt-60 source was removed. The ratios of counts inthe two counters in our present component of thewave function gave the sequence of decisionsn, o., n, P, P, a, P, n, P, P, and the masses wereset in the appropriate configurations. Duringeach run the torsion balance responded to eachrepositioning of the masses and then underwentdamped oscillations with a mean period of 710sec. By fitting the extrema of the oscillationsto exponentially decaying sine waves during eachhalf-hour, the change in the equilibrium position(of the reflected light beam on the distant scale)as the large balls were moved from A to B or &to A was determined. The changes in equilibria(in cm) we measured were -61.3, -63.9,-36.0, +69.2, +36.1, -48.8, +46.4, -45.2,+51.3, and +59.6.Although the sensitive torsion balance was af-fected by temperature changes, vibrations, andother factors not under our control, so that thedata have a large scatter, they give a correla-tion coefficient with the quantum decisions of x=0.9788. If our data came randomly from an un-correlated population, as would be predicted bythe semiclassical Einstein equations, the corre-lation coefficient for X—2 =8 degrees of freedom

• So the world is not very absurd.

• We need to treat gravity quantum mechanically.

How do we reconcile them?

• So far I told you

• The world is quantum mechanical.

• There is gravity in the world.

• So we want to treat gravity quantum mechanically.

There are two known methods:

Loop Quantum Gravity String Theory

Which is correct? Nobody knows yet.

This is because the quantum mechanical effect ofgravity tends to be very, very tinyand very, very hard to observe.

• A theory is correct if it describes some aspect of this world we live in.

• Neither string theory nor loop quantum gravity is known to be correct in this sense.

• At least both are “logically consistent.”

• So, I’m studying a logically consistent entity, called String Theory.

• I’m doing it mostly disregarding whether it describes the world or not. The structure of the theory itself is interesting to me.

• This makes me a non-scientist.

• This makes me wonder why I’m speaking here in front of you.

• Mathematicians deal with “logically consistent idealized entities” rigorously.

• For example, ancient Greeks have found that there are five and only five regular polyhedra:

• This is the last proposition of Euclid’s Elements!

• Unfortunately, string theory is not quite rigorous yet.

• So mathematicians don’t consider string theorists mathematicians.