Deterministic Quantum Mechanics

An outline is given of the interpretation of quantum mechanical expressions if one assumes that what is called Quantum Mechanics today, refers to an underlying reality as in Chap. 2, even if many authors claim to have evidence to the contrary, as is usually brought forward. If there exists a sub-microscopic classical world of real things, is that the same classical world as the world of large things such as planets, rocks and people? How does the notion of probability enter into quantum mechanics?

-At t = 0 postulate an initial state |ψ(0) , with ψ = 1, and a Schrödinger equation, H is the Hamilton operator (Hamiltonian) -Consider at t = T a possible final state ϕ(T )| with ϕ = 1 -The probability that the system will end in that mode will be P = ϕ(T )|ψ(T ) 2 Observables O are operators such asˆ x andˆ p , but their form depends on the basis chosen: Can we make good use of this, by choosing a special basis?
There is a special choice: the ontological basis. In this basis, U(T ) = P(T ), which is a permutator of basis elements |e i .
Can an ontological basis exist?
More complex, non integrable system: The classical, generic, cellular automaton, appears to be equivalent to a quantum field theory, however, it lacks rotation or Lorentz invariance.
And we could not prove this equivalence ... difficulty: the Hamiltonian of the field theory may be either non-local or unbounded below.

The cellular automaton interpretation
Theory: We assume that an ontological basis can be found for the complete Standard Model including the gravitational force.
The basis elements are "ontological states", called |ont i Often, we won't know which ontological state we are in. We can then describe the state we are in as a "physical state", As for the final state, see later! Ontological states evolve in accordance with the Schrödinger equ.
The universe is in a single ontological state.
Classical states (such as a planet moving in its orbit) can be distinguished by carefully analysing the statistics of the ontological data. Therefore: Classical states are ontological states (classical observables are diagonal in the ontological basis).
Therefore, the probability that a particular classical state, |ont(T ) c is seen at time t = T , is either 1 or 0.
Since in a physical state |ϕ , the probabilities are |λ i | 2 , these will also be the probabilities for obtaining the final class. states |ont c .
Thus, the |λ i | 2 , the probabilities obtained from a quantum calculation, actually reflect nothing else than the ordinary uncertainties we had in specifying the initial states! Born's rule.
Note that we are following the Copenhagen rules for quantum mechanics very strictly.
The rule that ψ|O|ψ = O = the expectation value for the observable O , only holds if -|ψ is the wave function of the universe (an ontological state), or -a template state (an Ansatz for the state of the universe), but -not if |ψ is a physical state.
Only modification: we do ask the question that should not be asked according to Copenhagen: What is going on in reality?
We claim to know the wave function of the universe: it is one of the ontological states |ont i .
From the one simple assumption that an ontological basis exists, it is easy to deduce that: -The Bell inequalities are violated (We have genuine QM) -The wave function always collapses automatically. This is because, in the real world, we always are in a single ontological state, therefore all observed phenomena end in single classical states.
-The Born rule follows: we get the Born probability distributions if we assume -for lack of more complete information -that the initial state was a physical state. Therefore we begin with probabilities |λ i | 2 , and we end with them.
We cannot write both the initial state and the final state as "physical" states, but they may be regarded as templates for the ontological states The mix-up of physical states as superpositions of ontological states begins at the Planck scale, so it will be complicated.
We have models, but these are simple. They enjoy locality We have superdeterminism (more about that later) The universe will refuse ever to go into a superposition of ontological states.
There is only one world; No many-world interretation there exists no ontological "pilot wave".
No Bohm pilot Note: This theory is primarily intended to describe the world at the Planck scale, and may be crucial for model building purposes. It is suspected that the gravitational force will play an important role in our understanding of quantum mechanics.
This is why the construction of explicit, realistic models is difficult; we often experience problems with locality and positivity of the hamiltonian (as in many other theories of Planck scale physics).
Some people think that they can "prove" that an ontological basis is impossible: The Bell -CHSH inequality Consider the production of an entangled state of two particles. In spin 1 / 2 notation: Atom → 1 √ 2 ( | ↑↓ − | ↓↑ ) (spin 1 2 notation)

Bob Alice
Alice measures the spin in the direction a and Bob measures the spin in the direction b . Alice finds spin 1 2 A, A = ±1, Bob finds spin 1 2 B, B = ±1. The correlation they find, according to qm, will be But, according to Bell's theorem, this is impossible if this were a classical statistical system. Take the four cases: Bell's theorem proves that local counterfactual reality does not exist, not that you can't have a classical theory underlying qm We can still have superdeterminism: Theorem: there must be correlations between angles a , b and the atom ε. As implied by Bell's theorem: there are correlations between the choices of the angles a and b chosen by Alice and Bob, and the 'classical' angle c at which the atom emits its 'entangled' particles.
Only 3-point correlations; upon averaging over a , or over b , or over c , one finds the 2-point correlations to vanish.
If Bob's decision b , and the atom's emission angle c are given, Alice's decision a is not randomly distributed → no counterfactual realism, but "conspiracy": -Vacuum fluctuations indeed give spacelike correlations without violating causality -If indeed the ontological states contain such correlations, and superdeterminism demands that Alice and Bob are in an ontological state, then Alice and Bob will not be able to 'change their settings' in such a way that a non-ontological 'physical state' emerges; they have to pick an ontological state again. So the correlations persist.
-No counterfactual realism, but also no contradiction

Lecture II. Canonical Methods
The greatest challenges in finding deterministic versions of qm, or, finding a 'classical, 'ontological' theory generating all quantum features of the SM, are: -Include the symmetries of the SM: Lorentz, SU(3) × SU(2) × U(1) , . . . note, we most often need to restrict ourselves to discrete (lattice-like) models -Generate a Hamiltonian H that is bounded from below, H ≥ E 0 -Reproduce locality in this Hamiltonian: Therefore, we search for canonical models, as in classical mechanics and standard qm.

P Q theory:
skip Turning a pair of ontological integers into the quantum opsq andp : Make real number operators −∞ < q < ∞ as follows: q = Q + 1 2π η P There is a unitary transformation of states from one basis to another: Q, η P |ψ = q|ψ . Then transform Alternatively, find the p basis:

Extensive Hamiltonian
Let −π < x < π , then x = 2 ∞ n=1 (−1) n−1 sin(nx) n From that: But this is not extensive: 0 ≤ H < 2π An interesting extensive Hamiltonian is obtained when we use the classical Hamilton formalism for discrete systems.
Consider integer variables Q i and P i , depending on a discrete time variable T . Let there be given a discrete energy function H( Q, P). This function must vary sufficiently slowly as Q i and P i vary Can we define a unique evolution law by demanding H( Q, P) to be exactly conserved?
In the continuous case the answer is Hamilton's formalism: Wrong answer: replaceẋ by x(T + 1) − x(T ) and The path must have a sufficiently large number of points on it, and its orientation must be as in the continuum theory.
This can easily be made unique provided that H is sufficiently smooth. Borderline case: H is quadratic function of integers Q and P , if the coefficient for the squares is 1 2 , which is the smallest possible. In that case most paths have only 4 or so points on them.
More interesting cases have slower varying functions, with lower bounds, usually non-integrable.
With given Hamiltonian, and given cyclic order of indices, this is the most direct analogue of the continuous case.
With very smooth Hamiltonians, one finds the continuum limit to yield the standard Hamilton equations -by checking how fast a system runs along its path.
In the discrete lattice field theory, one can also formulate such Hamiltonians. The indices are then replaced by the discrete space coordinates x and the field indices.
We order the coordinates by first taking all even sites, then all odd sites. The action at the even sites all commute, same for odd sites.
In that case, this field evolution law obeys all locality conditions. Now define the evolution operator U E for a unit time step at a given classical Hamiltonian H class = E as after which H tot = H quant + 2πH class ≥ 0 H class is extensive, but H quant is not.
How to add multiples of 2π to make also H quant extensive?
The A B formalism for the cellular automaton The discrete lattice field theory just described is example of Cellular Automaton of A B type.