Gerard Gerard ’’t Hooftt Hooft

CERN Colloquium Genève,

January 11, 2007

Utrecht University

Will there be hints from LHC ?

Generation I Generation II

The Standard Model

Gauge Bosons

0Z W

W

g

c cc

Higgs

Generation III

b b

tt t

b

Leptons ee

Quarksu

d ssd

uu

sd

/ 7 / 8v c

Special Relativity

Niels Bohr

screen

holes

detector

Quantum Mechanics

Quantum Mechanics gives us statistical probabilities with fluctuations in the outcome:

String theory

Can one meaningfully talk about thequantum wave function of the universe,describing a big bang 13.7 ×10⁹ years ago,while also peaking at today’s observed phenomena such as the solar system and the distribution of the continents on earth?

Quantum mechanics was designed to be a theory to describe the outcome of anexperiment that will only be accurate ...

Thus, QM refers to events that happen intiny subsections of the universe, in space andin time.

But today, theories such as superstring theoryattempt to describe the entire universe

Can one meaningfully talk about “Quantum cosmology”?

if repeated many times!

More precise statements:

Quantum mechanics is a prescription to obtain thebest possible prediction for the future, given the past,in any given experimental setup.

In numerous experiments it has been verified that better predictions are not possible.

Quantum mechanics is not a description of theactual course of events between past and future.

One might imagine that there are equations of Nature that can only be solved in a statistical sense. Quantum Mechanics appears to be a magnificentmathematical scheme to do such calculations.

Example of such a system: the ISING MODEL

L. Onsager,B. Kaufman

1949

In short: QM appears to be the solution of amathematical problem.We know the solution, but what was the problem ?

Or, ...

1. Any live cell with fewer than two neighbours dies, as if by loneliness.

2. Any live cell with more than three neighbours dies, as if by overcrowding.

3. Any live cell with two or three neighbours lives, unchanged, to the next generation.

4. Any dead cell with exactly three neighbours comes to life.

iH

i

i e

e

eU

3/2

3/2

1

The use of Hilbert Space Techniques as technical

devices for the treatment of the statistics of chaos ...

, ... , , ..., , ..., , anything ... x p i

í ýA “state” of the universe:

A simple model universe: í 1ý í 2ý í 3ý í 1ý

Diagonalize:

0 0 1

1 0 0

0 1 0

U2 2 2

1 2 3, , P P P

;321

23

23

0

“Beable”

“Changeable”

ˆbut 0H ??

Emergent quantum mechanics in a deterministic system

†ˆ ˆ 2 Im( ) 0H H i f g

( ) ( )d

x t f xdt

ˆ

ˆˆ ( ) ( )

p ix

H p f x g x

ˆ( ) ( ) , ( )

dx t i x t H f x

dt

ThePOSITIVITY

Problem

In any periodic system, the Hamiltoniancan be written as

2; ;H p p i

T

21 ; 0, 1, 2, ...iHT n

e H n nT

0

54321

-1-2-3-4-5

This is the spectrum of aharmonic oscillator !!

φ

Interactions can take place in two ways.Consider two (or more) periodic variables.1:

int( );ii i i i i

dqq f q H f p

dt

Do perturbation theory in the usual way bycomputing .intn H m

1 1 12 2 2 ( 1)1; ; ;x xi i i

x x xe i e i e

intH

1q

2q2: Write for the

hopping operator.

0 1

1 0x

1 1 1(0)q q t

1 1 2 2 1 2( 1) ( ) ( )xH p p A q a q

Use

to derive 2

12

112

2

1

1exp exp ( )( 1)

if( ) 1

( )t

xt

x

i H dt i A q

A

q

a

But this leads to decoherence !

However, in both cases,

will take values over the entire range of valuesfor n and m .

intn H m

Positive and negative values for n and mare mixed !

→ negative energy states cannot be projected out !

intn H m

But it can “nearly” be done! suppose we take many slits, and average: Then we can choose to have the desired Fourier coefficientsfor

2 2( ) ( )q f q

Lock-in mechanism

In search for a

Lock-in mechanism

A key ingredient for an ontological theory: Information loss

Introduce equivalence classes

í 1ý,í 4ý í 2ý í 3ý

With (virtual) black holes, information loss will be very large! →

Large equivalence classes !

With (virtual) black holes, information loss will be very large! →

Large equivalence classes !

Two (weakly) coupled degrees of freedom

bras

TT1TT2TT3

ketsConsider a periodic variable:

; [0, 2 ]d

dt

, 0, 1, 2,

zH p i L

m m

0123

The quantum harmonic oscillator hasonly:

, 0, 1, 2,H n m

Consider two non - interacting systems:

1E

2E 1 2E E

1 1 2 2H n n

The allowed states have “kets ” with

1 11 2 1 1 2 2 1,22 2( ) ( ) , 0H E E n n n

and “bras ” with

1 11 2 1 1 2 2 1,22 2( ) ( ) , 1H E E n n n

1 2 1 2 1 20 | | | |E E E E E E

12E t

11 1 2 2 1 2 1 2 1 2 1 22 ( )( ) ( )( )E t E t E E t t E E t t

Now, and

So we also have: 1 2 1 2( ) ( )t t t t

The combined system is expected again to behave asa periodic unit, so, its energy spectrum must be some combination of series of integers:

1 2E E

0

5

4

3

2

1

-1

-2

-3

-4

-5

11 1 2 22( )nE n p p

For everythat are odd andrelative primes, wehave a series:

1 2,p p

11 1 2 22( )nE n p p

11

2

T

22

2

T

1 25, 3p p The case

121 1 2 2

2T

p p

This is the periodicity of the equivalence class:

1 1 2 2 nst odC (M 2 )p x p x

1x

2x

22

Bell inequalities

And what about the

?

electron

Measuring device

vacuum

Free Will :

“Any observer can freely choose which feature of a system he/she wishes to measure or observe.”

Is that so, in a deterministic theory ?

In a deterministic theory, one cannot changethe present without also changing the past.

Changing the past might well affect the correlationfunctions of the physical degrees of freedom inthe present – if not the beables, then at least the phases of the wave functions, may well bemodified by the observer’s “change of mind”.

John S. Bell

R. Tumulka: we have to abandon one of [Conway’s] four incompatible premises. It seems to me that any theory violating the freedom assumption invokes a conspiracy and should be regarded as unsatisfactory ... We should require a physical theory to be non-conspirational, which means here that itcan cope with arbitrary choices of the experimenters, as if they had free will (no matter whether or not there exists ``genuine" free will). A theory seems unsatisfactory if somehow the initial conditions of the universe are so contrived that EPR pairs always know in advance which magnetic fields the experimenters will choose.

Do we have a FREE WILL , that does not even affect the phases?

Citations:

Using this concept, physicists “prove” that deterministic theories for QM are impossible.

The existence of this “free will” seems to be indisputable.

Bassi, Ghirardi: Needless to say, the [the free-will assumption] must be true, thus B is free to measure along any triple of directions. ...

Conway, Kochen: free will is just that the experimenter can freely choose to make any one of a small number of observations ... this failure [of QM] to predict is a merit rather than a defect, since these results involve free decisions thatthe universe has not yet made.

in his robā‘īyāt :

“And the first Morning of creation wrote /What the Last Dawn of Reckoning shall read.”

Determinism

Omar Khayyam (1048-1131)

The most questionable element inthe usual discussions concerning Bell’sinequalities, is the assumption of

It has to be replaced with

General conclusions

At the Planck scale, Quantum Mechanics is not wrong, but its interpretation may have to be revised, not for philosophical reasons, but to enable us to construct more concise theories, recovering e.g. locality (which appears to have been lost in string theory).

The “random numbers”, inherent in the usual statistical interpretation of the wave functions, may well find their origins at the Planck scale, so that, there, we have an ontological (deterministic) mechanics

For this to work, this deterministic system must feature information loss at a vast scale

Any isolated system, if left by itself for long enough time, will go into

a limit cycle, with a very short period. Energy is defined to be the inverse of that period: E = hν

Are there any consequences for particle physics?

Gauge theoriesThe equivalence classes have so much in common with thegauge orbits in a local gauge theory, that one might suspectthese actually to be the same, in many cases ( → Future speculation)

Understanding QM is essential for the constructionof the “Ultimate theory”, which must be more thanSuperstring theory.

the LANDSCAPE

12( ) iH n

Since only the overall n variable is a changeable, whereas therest of the Hamiltonian, , are beables, our theory willallow to couple the Hamiltonian to gravity such that thegravitational field is a beable.

i

( Indeed, total momentum can also be argued to be a beable ...)

Gravity 1:

Gravity 2:

General coordinate transformations might also connect membersof an equivalence class ... maybe the ultimate “ontological” theorydoes have a preferred ( rectangular ? ) coordinate frame !

The End

G. ’t Hooftdemystifying Quantum Mechanics

Gauge invariance(s) play an important role inthese theories .... → plenty of new gauge particles !

Predictions (which may well be wrong) :

No obvious role is found for super symmetry(a disappointment, in spite of attempts ...)