Gerard ’t Hooft, quant-ph/ Erice, September 6, 2006 Utrecht University 1

Motivations for believing in a deterministic theory: 0. Religion 1. “Quantum cosmology” 2. Locality in General Relativity and String Theory 3. Evidence from the Standard Model ( admittedly weak ). 4. It can be done. “Collapse of wave function” makes no sense QM for a closed, finite space-time makes no sense String theory only obeys locality principles in perturbation expansion. But only on-shell wave functions can be defined. In the S.M., local gauge invariance may point towards the existence of equivalence classes (see later in lecture) The math of QM appears to allow for a deterministic interpretation, if only some “small, technical” problems could be overcome. A few of my slides shown at the beginning of the lecture were accidentally erased. Here is a summary: 2

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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. 4

The use of Hilbert Space Techniques as technical devices for the treatment of the statistics of chaos... A “state” of the universe: A simple model universe: Diagonalize: 5

?? Emergent quantum mechanics in a deterministic system The POSITIVITY Problem 5a

In any periodic system, the Hamiltonian can be written as This is the spectrum of a harmonic oscillator !! 6

Lock-in mechanism In search for a 7

8 Lock-in mechanism

A key ingredient for an ontological theory: Information loss Introduce equivalence classes 9

With (virtual) black holes, information loss will be very large! → Large equivalence classes ! 10

bras ̶̶1 ̶̶2 ̶̶3 kets Consider a periodic variable: The quantum harmonic oscillator has only: Beable Changeable 1

Interactions can take place in two ways. Consider two (or more) periodic variables. 1: Do perturbation theory in the usual way by computing. This and the nect 2 slides were not shown during the lecture but may be instructive: 1212

2: Write for the hopping operator. Use to derive 1313

But this leads to decoherence ! However, in both cases, will take values over the entire range of values for n and m. Positive and negative values for n and m are mixed ! → negative energy states cannot be projected out ! But it can “nearly” be done! suppose we take many slits, and average: Then we can choose to have the desired Fourier coefficients for 1414

Consider two non - interacting systems: 1515

The allowed states have “kets ” with and “bras ” with Now, and So we also have: 1616

The combined system is expected again to behave as a periodic unit, so, its energy spectrum must be some combination of series of integers: For every that are odd and relative primes, we have a series: 1717

The case This is the periodicity of the equivalence class: 1818

A key ingredient for an ontological theory: Information loss Introduce equivalence classes 1919

With (virtual) black holes, information loss will be very large! → Large equivalence classes ! 2020

This is one equivalence class Note that is now equivalent with and are beables (ontological quantities), to be determined by interactions with other variables. Therefore, we should work with and instead of and In terms of these new variables, we have 2121

Some important conclusions: An isolated system will have equivalence classes that show periodic motion; the period ω will be a “beable”. The Hamiltonian will be H = (n +½) ω, where n is a changeable: it generates the evolution. However, in combination with other systems, n plays the role of identifying how many of the states are catapulted into one equivalence class: If all states are assumed to be non-equivalent, then n = 0. The Hamiltonian for the combined system 1 and 2 is then The ω are all beables, so there is no problem keeping them all positive. The only changeable in the Hamiltonian is the number n for the entire universe, which may be positive (kets) or negative (bras) This is how the Hamiltonian “locks in” with a beable ! Positivity problem solved. 2

1. In the energy eigenstates, the equivalence classes coincide with the points of constant phase of the wave function. Limit cycles We now turn this around. Assume that our deterministic system will eventually end in a limit cycle with period. Then is the energy (already long before the limit cycle was reached). If is indeed the period of the equivalence class, then this must also be the period of the limit cycle of the system ! The phase of the wave function tells us where in the limit cycle we will be. 2323

Since only the overall n variable is a changeable, whereas the rest of the Hamiltonian,, are beables, our theory will allow to couple the Hamiltonian to gravity such that the gravitational field is a beable. (Note, however, that momentum is still a changeable, and it couples to gravity at higher orders in 1/c ) Gauge theories Gravity The equivalence classes have so much in common with the gauge orbits in a local gauge theory, that one might suspect these actually to be the same, in many cases ( → Future speculation) 2424

General conclusions At the Planck scale, Quantum Mechanics is not wrong, but its interpretation may have to be revised, not only 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. We still have a problem: how to realize the information loss process before the interactions are switched on ! 2525

Claim: Quantum mechanics is exact, yet the Classical World exists !! The equations of motion of the (classical and [sub-]atomic) world are deterministic, but involve Planck-scale variables. The conventional classical e.o.m. are (of course) inaccurate. Quantum mechanics as we know it, only refers to the low- energy domain, where the fast-oscillating Planckian variables have been averaged over. Atoms and electrons are not “real” 26 This and the next slide were not shown during lecture

At the Planck scale, we presume that all laws of physics, describing the evolution. refer to beables only. Going from the Planck scale to the atomic scale, we mix the bables with changeables; beables and changeables become indistinguisable !! MACROSCOPIC variables, such as the positions of planets, are probably again beables. They are ontological ! Atomic observables are not, in general, beables. 2727

The End G. ’t Hooft A mathematical theory for deterministic quantum mechanics 2828