Mathematics Of Classical And Quantum Physics Solution Manual Pdf

1. Introduction

The foundations and interpretation of quantum mechanics can be investigated from various angles, depending on what one's deeper motivations are. In the early years of quantum mechanics, investigators were asking how this theory can relate to 'reality': what is it that really happens when a quantum process is observed? Then, when this question could not be answered in simple terms, the question became: how can this theory be used to predict the outcome of experiments?

When used correctly, it was found that the outcome of an experiment can be predicted precisely, but the answer often comes in a statistical form: upon repeating the experiment many times, statistical distributions will be found, and only those can be predicted by the theory, not the outcome of a single observation. Finally then, one could ask: what is the cause of all these observed statistical fluctuations? Can one sharpen the theory in this respect? At first sight, this seems not to be possible, but it is legitimate to ask whether the theory can be improved.

In the literature, one finds at least three kinds of approaches towards 'improving' the theory:

	(i) Assume that quantum mechanics is an approximation. One could regard the work of Nelson [1], Legget [2], Adler [3] and many others to be more or less in this class. There is an underlying theoretical, dynamical construction that is not quantum mechanical, while it does allow for a statistical approach, in the same sense as one may approach complex classical dynamical systems, by using procedures developed for statistical mechanics. If we are lucky, we might reproduce something that resembles a Schrödinger equation. It might be possible to predict deviations from the 'true' Schrödinger equation, which could be detected experimentally. An important issue will be Bell's theorem and the associated Clauser–Horne–Shimony–Holt (CHSH) inequality. It appears to tell us that classical logic cannot reproduce quantum mechanics exactly, without rather unwieldy non-local features in the dynamical laws.
	(ii) Alternatively, one could decide not to tamper with the equations of quantum mechanics, but ask the question: then what is the real world described by quantum mechanics? Does the theory refer to an infinite class of universes (the 'many world theory' [4])? What exactly is the role of DeBroglie's wave functions in this highly complex structure, and is it meaningful to replace our usual notion of reality by this much more complex one? This is Bohm's theory [5], which in our view is very similar to the Many World interpretation.
	(iii) One can however also ask: suppose we keep quantum mechanics exactly as it is, but identify another single real world behind it [6]. This other real world might not directly describe things we usually expect to be real, such as particles or fields. It will not be possible to describe such a world in general, but perhaps in some very special cases it is. Then the question becomes: which theories allow us to assume a single, real universe at their basis, and can situations arise that lead to violation of Bell's theorem? Even though the underlying real world might appear to be a rather cryptic one, we nevertheless have to cope with questions concerning 'superdeterminism' and 'conspiracy'.

Approaches along lines (i) and (ii) are well known and adhered to by many researchers. This report is about progress made along line (iii). Our motivation is not merely that, like Einstein, one might believe that 'God does not throw dice', or that anything else would be wrong with the logic of quantum mechanics as it is applied to theories such as the Standard Model of the subatomic particles. Rather, what we are searching for is efficiency in creating models that include the gravitational force and finiteness of the universe. What we do suspect is that a one-universe model may serve us more efficiently than today's more popular models that are juggling with infinitely many universes.

Moreover, we found that models with a single underlying reality, while fully admitting a quantum mechanical description, do exist. From discussions with colleagues, it became quite clear that such models are ill understood and confusing to many. We set as our aim to create a precise language for formulating them.

Our models are deterministic and appear to obey locality, so that they seem to be 'local hidden variable theories'. If they are to morph into quantum mechanics, they must be quite different from what we are used to. In most realistic examples, their elementary mass- and distance scales are assumed to be in the Planck domain, about which we have very little direct information. Nevertheless, we insist one should avoid any kind of 'magic'. The only accepted logic will be the standard, classical one.

The usual approaches towards understanding physics at the Planck scale are associated with difficulties; we have non-renormalizable divergences, the black hole information paradox and questions concerning non-trivial topologies in space–time. Many problems can be traced back to the group of Lorentz transformations being essentially non-compact, which creates problems when we wish to consider the possibility of a 'smallest distance' in space and/or time. Thus, our progress is slow, but our models may be pointing in interesting new directions of inquiry.

To set the scene, we have a number of very illuminating 'toy models', which display very well the promising nature of this approach. Three examples will be discussed here. All three are integrable, which makes their discussion very transparent, and all three models share the property that they are classical and quantum mechanical at the same time.

	(1) The quantum harmonic oscillator. It is mathematically equivalent to a classical point particle moving around the unit circle at constant speed. More generally, we claim that the quantum harmonic oscillator can be represented by any periodic classical system.
	(2) The chiral Dirac fermion. It is mathematically equivalent to an infinite plane sheet moving with the speed of light in one of the two possible transverse directions.
	(3) We briefly mention the bulk region of a fully quantized superstring, which we have discussed at full length in other publications. It is mathematically equivalent to a classical string moving in a target space that is a lattice. The lattice spacing a obeys , where α′ is the string slope parameter [7]. The interactions of strings with themselves and other strings still lead to problems that have not been solved.

In the following sections, we discuss these models. Although the results have been reported before [6,8], we present them here again, with some further editing. Most importantly, we add new discussions of these findings and their implications in connection with Bell's theorem and related considerations [9–12].

There is not enough space here to discuss in detail the cellular automaton models, which are much more general than the three examples mentioned above, but they are more difficult to handle with mathematical clarity [6]. We briefly indicate how cellular automaton models may lead to quantum theories of interacting particles, in §6.

2. The harmonic oscillator

Consider the spectrum of the harmonic oscillator. Its energy eigen states |n〉 _E have energies

2.1

First, consider a high-energy cut-off; the spectrum has an upper limit: n<N. Subsequently, we also remove the vacuum energy as being not very essential for the discussion that follows.¹ Our Hilbert space is N-dimensional, so a basis has always N elements |n〉 _E , n=0,…,N−1. The subscript E stands for 'energy'. Because of the equal spacings in the energy levels, we have indeed periodicity with period T:

2.2

This suggests that we look at a new basis of states, labelled with an integer k=0,…, N−1, defined by the finite Fourier transformation

2.3

where 'ont' stands for 'ontologic' ('ontic' for short). This turns our system into a model with classical periodicity; at time steps that are integer multiples of δt=2πT/N, the system evolves by permuting these 'ontological' states:

2.4

This is as classical as a model can be. Note that, restricting ourselves to integer multiples of δt as our time variable, is not a high price to pay as long as δt is kept reasonably small. If we really want to describe our universe at times in between the integer time steps, we can define the unitary transformations , that will bring us there, and as long as we limit ourselves to directly observable phenomena, these unitary transformations will stay close to the identity operator.

The operators and of the harmonic oscillator can be obtained from the creation and annihilation operators a, a ^†:

2.5

The operator can be retrieved in the basis of the harmonic operator states |n〉 _E as follows:

2.6

2.7

Note that the operator ordering was chosen here in such a way that we only encounter the square root of , which is always positive. The last, 'boundary', terms in equations (2.7) are important. This is because now the continuum limit, , can be taken, though it requires some caution.² We take the limiting case where the ontological states correspond to angles φ forming the unit circle,

2.8

where is the norm that normalizes the |φ〉 states such that 〈φ|φ′〉=δ(φ−φ′).

The normalized energy eigen states are

2.9

but, having in mind the limit of equation (2.3), we restrict ourselves to the non-negative values of n. It is very important to keep this potential threat to unitarity in mind.² Let us now compute the matrix elements of harmonic oscillator operators such as and in the ontological basis of the |φ〉 states. Given the explicit wave functions of the energy eigen states in the x and the p basis, we can calculate these operators.

For simplicity, set ω=1. The matrix transforming the |φ〉 states to the x eigen states and back is then found to be

2.10

where H _n (x) are the Hermite polynomials.

At all fixed values of x and φ, the sum (2.10) converges, but very slowly, because we need to include all energy eigen states that contribute at a given x, until the energies are so high that the bulk of the wave function stretches much further, away from x. Mathematically, working out the sum is laborious, and it does not lead to simple special functions because of the square root. We did find powerful approximation methods, and arrived at functions that can be plotted [6,8].

The most important conclusion of this article is that quantum harmonic oscillators are closely related to completely classical periodic features in a classical, deterministic description of Nature. In many branches of physics, one encounters oscillating modes, where the energy quanta play the role of fundamental particles. Now, if the periodic phenomena in the deterministic system would never deviate from their periodic pattern, they would not leave a trace elsewhere, and therefore play no significant role; this would imply that the quantum particles associated to them would not interact. Interacting particles are associated to not-exactly-periodic features of the automaton.

Operators such as have the property that they commute at all times. Indeed, in the case of the harmonic oscillator, our treatment leads to a basis transformation that allows us, starting from a system with non-commuting operators x and p (the harmonic oscillator), to re-describe it in terms of only commuting ones, |φ(t)〉. Such operators will be referred to as beables. Thus, beables are defined to be operators obeying

2.11

3. Massless chiral fermions

A more delicate system is that of massless, non-interacting, chiral fermions. It resembles something that does occur in the physical world: neutrinos. Here, we take the case that they are strictly massless and non-interacting. The particles are either always left handed or always right handed. Their antiparticles have this reversed, but this assumes second quantization; we will come to that. First, consider a single, first quantized, Dirac fermion. Its wave function is only a two-component spinor rather than the usual four component one. We call our particle a 'neutrino', between quotation marks to emphasize that it is the idealized version: no mass and no interaction. This first quantized 'neutrino' is described by the Hamiltonian³

3.1

where p is the momentum operator and σ are the Pauli matrices:

3.2

Note that, in equation (3.1), a sign choice was made: flipping the sign flips the handedness of the neutrinos.

We now claim that the following three types of operators form a set of beables:

3.3

The sign in the first equation is arbitrary; it could be defined such that . The equations of motion for these operators are derived as follows:

3.4

3.5

3.6

3.7

so that we have

3.8

Furthermore, we have

3.9

The first of these is obvious, and the last follows from the second. The fact that can be found by explicit calculation, but also understood as a consequence of the fact that, in momentum space, r is the dilation operator divided by |p|, while is invariant under dilations (its norm stays one).

Together, equations (3.8) and (3.9) thus obey equation (2.11). The best way to describe these observables physically is to say that they represent a sheet moving with the speed of light in one of the two possible orthogonal directions.

This is why we claim that the ontological theory behind a 'neutrino' is an infinite sheet moving with velocity v=c in one of its two possible orthogonal directions. If interactions or masses would be introduced, we have no obvious ontological theory anymore. See our remarks at the end of this section.

The mathematics of the transformation relating infinite sheets to neutrino states is delicate and interesting. The Hilbert space of 'neutrinos' can be assumed to be spanned by the states | p ,α〉_neutrino or the states | x ,α〉_neutrino, where α=±1 is the spin variable, or, the eigenvalue of the Pauli spin operator σ _z (see equation (3.2)). The sheet states will be denoted as . By first Fourier transforming the r variable to get the states , the orthogonal matrix relating these two sets of states can be calculated to be

3.10

and

3.11

where is the spin eigen state of the operator s with eigen value s, in the basis of σ _z . The factor p _r in equation (3.10) is necessary for normalization. It produces the unfamiliar factor δ′(z)≡(∂/∂z)δ(z) in equation (3.11).

To express the neutrino operators x and p in terms of sheet operators is not quite so easy. The phases of the sheet states still have to be defined. To do that right, we need an orthonormal set of two unit vectors and on the sheet. They have to be defined to be orthogonal to , so they will depend on :

3.12

We then define the rotation operators for the sheet as the generators that rotate the sheet. do nothing but rotate the unit vector . We have

3.13

but finding an operator x , with the proper commutation rules with p , is tedious. We first have to define spin flip operators s ₁ and s ₂, obeying, just like the Pauli matrices:

3.14

It was found (no derivation is given here [6]) that

3.15

Here, is the third component of in equation (3.12). Thus, we arrive at a picture where 'neutrinos' are identified with flat sheets spanning to infinity, able to move with light speed in one of the two directions orthogonal to the sheet, in a totally classical way.

Note that a single (first-quantized) neutrino can have positive or negative energy. This apparent disease can be cured by invoking Dirac's procedure of second quantization: if all negative energy levels are filled and all positive ones empty, we have the lowest energy state, the vacuum. Pauli's exclusion principle dictates that two particles cannot be in the same state. In the classical picture, this can be applied to the ontological basis, and we simply have to posit that there are infinitely many sheets, but two sheets never completely overlap. As in Dirac's picture, we must assume some cut-off. In the ontic basis, we may demand that the r variable is constrained to be an integer multiple of a small quantum δr. One may consider replacing also the variable by a dense lattice on the unit half-sphere.

We note the potential of our theory but also the difficulties. The potential is that systems normally considered as purely quantum mechanical may yet be completely classical, in disguise. One might object by suspecting that these sheets transfer signals with infinite speed inside a sheet, so that indeed the model seems to be non-local. This is not true; we do have local propagation rules for the sheet, but if we must observe that we have strong space-like correlations in our physical states: if we observe a sheet in the xy-direction at the origin, say, then correlation laws tell us that, at the same moment, there will be an xy sheet at all other points (x,y,0).

This apparent non-locality does yield difficulties when one wishes to consider mass terms and/or interactions. What can happen if two sheets interact at a point? First, one has to second-quantize the system: have many neutrinos described by many sheets. But even then, it is difficult to see how sheets can interact; a sheet cannot easily be made to change its direction of motion at one point by means of local interactions. One will have to consider the possibility of having imperfect sheets, or more realistically, modified correlation laws at space-like distances.

Indeed, an essential element in our work is the observation that strong non-local correlations may be the answer to objections often raised against our treatment of quantum mechanics in view of Bell's theorem [9–11]. There seems to be 'conspiracy' in our models. The conspiracy complained about could be nothing but unexpectedly strong space-like correlations. More about this important subject in §7.

4. Real numbers and integers

Imagine that, in contrast to appearances, the real world, at its most fundamental level, was not based on real numbers at all. Real numbers are a man-made invention in response to the apparent continuous nature of our world, such as the positions and sizes of things. Our impression of everything being continuous received its first blow when it became quite evident that matter is made by atoms, which are discrete. Subsequently, it was found that energy is quantized. Now the energy quantum is controlled by the frequency of a wave function, and since time still seems to be continuous as far as we know, frequencies can be varied continuously, and in spite of our discovery of quantum mechanics, we still use real numbers in most of our calculations.

On the other hand, however, there are indications that, eventually, real numbers might get to be gradually replaced as our understanding of space and time further improves. A curious indication that this might happen is the theory of the information content of black holes. The microscopic regions near the horizon of a black hole seem to harbour only discrete amounts of information, in the form of black hole microstates. Information is fundamentally discrete. It is generally believed that space and time cannot be represented as a continuum of real-valued coordinates at distance scales as tiny as the Planck scale.

This is a motivation for searching for procedures to replace real numbers by integers. As it turns out, standard mathematics used in quantum mechanics provides us with the tools to do this. Again we use mappings, where one basis of Hilbert space is replaced by another. Surprisingly, what we shall derive below is the following:

There is a natural way to map a Hilbert space where the basis elements are characterized by a set of two integers, which we shall call P and Q, onto a set of states where the basis elements are characterized by one real number, q.

The mapping is one-to-one, with only one single state forming an exception: it generates a singularity. The mapping can be trivially extended to the mapping of 2N integers, P _i and Q _i , i=1,2,…,N, onto basis elements of N real numbers, either in the form of positions q _i ,i=1,2,…,N or in the form of momenta p _i . The construction is such that q _i and p _i can be handled as positions and momenta in accordance with canonical theories of mechanics. Because , the | p  〉 states and the | q  〉 states are not independent.

The integers, which we shall always write as capitals Q _i and P _i , completely commute. For this reason, it will be relatively easy to formulate deterministic theories based on Q _i and P _i , as these can be the beables of our system.

In principle, the mapping is easy to formulate. Consider first just one set of states |Q〉, where Q is an integer. It may run from to . We can then introduce the unitary step operator U by

4.1

As U is unitary and is easily seen to be non-degenerate, we can write it as

4.2

for any integer N, where . The quantity η is an operator, and its relation to the Q operator is easy to derive: consider the function η in the domain . Let us write its Fourier transform,

4.3

and a ₀=0. Therefore, we can write

4.4

so that

4.5

The prefactor is defined to be such that the matrix element vanishes when Q ₁=Q ₂. It follows that

4.6

Here, the single state |ψ ₀〉 describes the one state for which the commutator is not the canonical one. We shall encounter such states more often; it is the exceptional state alluded to earlier, to be referred to as the edge state. We observe that this is also the state for which the operator η has the value , exactly the edge of the Fourier domain in equation (4.3).

The mapping that we wish to use is the mapping from the Q basis to the η basis and back. Just because we prefer to put η on an open interval rather than a closed circle, we encounter the 'problem' of the edge state; it will have to be taken proper care of. The transformation matrix is simply ϵ ^iQη . Because of our special choice of ϵ as the base of our exponentials, the normalization of the matrix became trivial.

Now, consider a real number q. Let us write

4.7

where Q is integer and η _P lies between and (as much as possible, we intend to write real numbers as lower case letters, integers as capitals, and numbers modulo 1, or fractional numbers between and , as Greek letters).⁴ As this splitting is unique, we can write basis elements |q〉 as

4.8

Next, we can Fourier transform

4.9

where P is integer. Any state |ψ〉 can thus be written in the real q basis or in the Q,P basis:

4.10

This is what we mean when we state that quantum mechanics on a basis described by a real number q can be transformed to quantum mechanics with a basis of a pair of integers, Q and P.

Subsequently, one may transform to momentum space, using the matrix

4.11

This procedure is not quite symmetric under the interchange , but this symmetry can be restored [6]. At the same time, one can reduce the effects of edge states. This we do by multiplying the coefficients of a wave function |ψ〉 in the space of the pairs η _Q , η _P , by a complex phase, ϵ ^{iφ(η _Q ,η _P )}. This phase can be chosen such that the coefficients become exactly periodic both in η _P and in η _Q . This would remove the edge state completely, except for the fact that the phase function ϵ ^iφ will feature a vortex singularity. This means that, at the very location of this vortex, we still will have an edge state.

The result of this procedure, requiring a calculation that has been published elsewhere [6], is that we obtain a transformation from the basis 〈q| or 〈p| to a basis of states 〈Q,P| that, apart from a few signs, is entirely symmetric. We write for the q and p operators

4.12

where the operators a _Q and a _P are not exactly restricted to the interval , but they will stay of order one:

4.13

where Q stands short for Q ₂−Q ₁, and P=P ₂−P ₁. From these, one derives that

4.14

In (η _Q ,η _P ) space, the edge state |ψ _edge〉 is the delta peak on the spot , exactly where we located the vortex of our phase function φ(η _Q ,η _P ). The eigen values of both operators q and p in equation (4.12) occupy the entire real line without overlappings, but we do have to restrict ourselves to the states that are orthogonal to the edge state |ψ _edge〉.

5. Free massless bosons in 1+1 dimensions and the Superstring

The field operators in a quantized field theory also have their eigenvalues on the real line. By splitting up these lines into integer and fractional parts, we would like to apply the results of the previous chapter to such theories, but in general this would not produce viable deterministic models. The reason is that the splitting (4.12) does not transform in a simple way when two real numbers are added or subtracted.

Nevertheless, the mapping can be significant for field theories, if we can arrange the dynamical variables in such a way that the classical field equations involve displacements without additions or subtractions.

The mathematical details of the 1+1 dimensional models in our formalism have also been presented before [6,8,13], so we would not repeat these again, but let us summarize them.

One can easily produce a cellular automaton, based on integers in every cell, such that it generates modes consisting of left-movers and right-movers. The integers cross past one another without interacting. The modes are characterized by integer-valued functions on the world-sheet lattice. As such, models of this sort can be used to describe discretized strings on a space–time lattice.

Our observations are based on the simple notion that when the more standard quantized string theory is considered, based on a world sheet and a target space which are strictly continuous, then its quantum modes are also characterized by integers: the string excitation numbers, being the mass and spin eigen states of the world sheet energy operator [14–16]. All one has to do is make a mapping from one to the other, and that turns out not to be too hard. We do encounter, as usual, the notion of edge states, which are the exceptions to the rule, forming a subset of measure zero in the Hilbert space of all states. These we wish to ignore, and good arguments can be put forward to find permission to do that.

The result is very intriguing. It allows us to regard the bulk features of a string as being a classical cellular automaton in disguise. We then continue to add fermionic degrees of freedom. The bosonic equations, both for the string and for the automaton, take the form

5.1

where σ and τ are the world sheet coordinates. The solutions are then found to take the form

5.2

Fermionic fields are described by similar equations; their classical counterparts are Boolean degrees of freedom s ^μ (σ,τ), taking the classical values ±1, and obeying

5.3

which is solved by

5.4

where again the variables s _L,R take the values ±1.

In the quantized bosonic as well as the superstring, the independent variables have the index μ run from 1 to D−2, where D is the dimension of target space–time; the remaining values, μ=0 and μ=D−1, obey the same equations, but the variables are not independent, they are fixed by the constraints. We can now represent the D–2 transverse variables in the cellular automaton as classical degrees of freedom (both for the bosons and the fermions), while leave the remaining variables to be exclusive properties of the quantized (super)string only. They do not affect the dynamics, so our mapping remains valid.

This, however, leads to a problems concerning string interactions. In earlier publications, we proposed to admit classical interactions when strings meet in a point. The classical interaction could consist of an exchange of arms of the string at such meeting points. The problem with this proposal is that the question whether strings coincide at one point will involve the dependent degrees of freedom as well. If these are determined by quantum equations (the string- and superstring constraints in operator form), then it is difficult to see how these interactions can be seen as classical interactions in the automaton.⁵ The difficulty is further exacerbated by the fact that the procedure is not Lorentz invariant by construction; indeed, making the discretized cellular automaton Lorentz invariant is rather difficult, and this difficulty shows up when we try to include the string interactions.

This difficulty is also related to the question of background dependence. Our model, as yet, is assumed to have a flat background. It is this flat background that then takes the form of a rectangular lattice. Any interaction in the string would lead to gravitons interacting with it, and these in turn would generate curvature in the background. One may suspect what all this means physically: the lattice will feature local lattice defects of various forms; the strings may merge with defect lines in some delicate manner, but all this has not been sorted out.

What we did find is one rather astonishing fact, which seems to have been overlooked by string theoreticians: the lattice size parameter a is fixed in our theory. There is only one lattice size possible. It is exactly

5.5

where α′ is the string slope parameter.

Even if we do not know how exactly to deal with the string interactions, the above discussion appears to imply that their strength is also fixed. This fixes the strength of the gravitational interactions in terms of string parameters, and hence the Planck length is also determined by α′. Thus, we find that the lattice size parameter must be of the order of the Planck length itself.

6. On perturbations, interactions and cellular automata

In the other models discussed above, perturbations and interactions are not easy to introduce. In the harmonic oscillator, our mapping only works if the energy levels are all strictly equally spaced, which implies the absence of non-harmonic terms. The perturbed Zeeman atom will have to be treated in combination with many other degrees of freedom; one has to keep in mind that our models were chosen to demonstrate some basic principles, not as systems that can be subject to alterations so as to serve as more accurate descriptions of the world.

Thus, if we do want to know where perturbations and interactions come from, and find models that include those, we first have to understand the system we wish to study, at the quantum level, and also the set of classical models where we want to search for a system that can be used in the mapping. In this section, we briefly outline how this might be done.

The most important theories we eventually wish to understand are relativistic quantum particles that interact. Now these require a description in terms of a renormalized, relativistic quantum field theory, and we now know exactly how to classify all such systems: they consist of Yang–Mills gauge fields, Dirac fermion fields and scalars, all interacting as prescribed by a Lagrange density that takes the form of a four-dimensional polynomial in the fields, with all required gauge symmetry properties. This already indicates that the mathematical intricacies associated with such models will be considerably more complex that what we handled so-far, while we strongly suspect that that is not all: quite likely the gravitational force, requiring a dynamical, curved space–time, will have to be added as well.

Leaving out gravity as yet, we can envision a general strategy, although other options may also work. In general, the most promising classical models are cellular automata. These are systems of discrete variables, living on a space–time lattice, which evolve according to some local law. One may start with having non-interacting deterministic particles in the automaton. The best starting point can be reached if we have these particles move with the speed of light in a multitude of possible directions. That will not be exactly Poincaré invariant, but we suspect that, without gravity, this may be the best we can do.

Our second step is then to cast the evolution law of these particles in terms of a Schrödinger equation, by writing U ₁(δt)=e^{−iH ₁ δt} , where the index 1 refers to the fact that here we are dealing with only one particle. This Hamiltonian will have a spectrum ranging from −π/δt to πδt. Next, we subject this to second quantization, as follows.

Depending on whether our particles are bosons or fermions, the particle occupation number at each spot will have to be either an arbitrary, non-negative integer, or just ±1. This allows us to introduce (bosonic or fermionic) fields for these particles. These fields will not obey the usual Dirac or Klein Gordon equations, but something deviating from that due to lack of Poincaré invariance. The most important thing for the moment is that they are genuine quantum fields. The essential result of this operation is that only the positive part of the energy spectrum is handled as particles, but the negative energy part is interpreted as antiparticles, which flips their energies to become positive also.

Let the second-quantized evolution operator (describing infinitely many particles) be written as . As the evolution law for one particle was local, we can write , where a( x ) is some local operator, which can be directly obtained from H ₁. Now these fields do not interact, but now we can introduce interactions. In the classical system, we introduce a modification in the evolution law that makes particles feel each other's presence, but the modified evolution law only rarely deviates from the non-interacting one, say, it works when a number of particles occur at relative positions from one another that are fairly rare. We write the modification as , where b( x  ) is again local, but as it mostly does not act at all, the expectation value(s) of the operators b( x  ) are very small.

One can then use Baker–Campbell–Hausdorff to write

6.1

Now a( x  ) and b( x  ′) commute when x and x  ′ are far apart, so that H can again be written as a sum of local Hamilton density functions h( x  ), and furthermore, as b( x  ) was assumed to be small, we may limit ourselves to terms linear in b only. For this set of terms, the Baker–Campbell–Hausdorff series (6.1) can be summed. B will be a complicated expression in terms of the fields. It represents the interactions. The important reason to do things this way is that the resulting Hamiltonian will be bounded from below, as well as local. It is a genuine quantum Hamiltonian.

What remains to be done is scale over 20 orders of magnitude to reach the Standard Model domain. It is generally assumed that this scaling will filter away all non-canonical terms from the Hamiltonian, as these would not renormalize.

We do not claim to have herewith solved the issue of building a local, bounded Hamiltonian, because the explicit model calculations have not yet been done, but the above illustrates our general philosophy of how to link a quantum field theory language to a cellular automaton language. We have a limiting speed of information propagation, but not yet Lorentz invariance.

7. Bell inequalities and other fundamental quantum features

The idea that classical, non-quantum mechanical models should exist that can mimic quantum behaviour is met with much resistance by investigators of the foundations of quantum mechanics. Starting from some very reasonable looking assumptions, it was proved that such theories must be ruled out by Bell's theorem [9–11], as they would have to obey inequalities called the CHSH inequalities [12], which are strongly violated by quantum mechanics.

However, nothing seems to be wrong with the models presented here, accept that they all may well be far too simplistic to accommodate for the complex symmetry structures observed in the real world. The point we now bring forward is that the objections sprouting from Bell's gedanken experiments seem not to hinge on any symmetry arguments, so one should ask where the weakest link can be found in these various chains of our reasoning.

Knowing what the outcome should be, let us first construct a 'hidden variable model' much along the lines this is usually done; assuming that we can deal with classical statistics, we write some fancy model of a classical atom emitting classical photons. Later, we will replace this model with something much more in line with what we really think is going on, but first we need to identify the difficulty.

Momentarily then, we assume that we have a system illustrated in figure 1. A pair of entangled photons, α and β, is produced by a specially prepared unstable atom ε, at time t=t ₁. Alice analyses her photon by checking whether it passes a polarization filter rotated by an angle a, and Bob uses a filter at an angle b. In our naive deterministic model, the photons take the form of signals with some information c concerning their polarization. Ideally, the photons at α and β take the same⁶ polarization angle c.

The measurement A=±1 by Alice, becomes a classical bit of information at P, and B=±1 by Bob occurs at Q in the figure. They are predicted by quantum mechanics to be correlated as

7.1

whereas the CHSH inequality would dictate that the correlations cannot be that strong [12].

The 'natural' and 'fundamental' assumption made by Bell and by CHSH, was that the polarization angles a, b and c are uncorrelated. Alice and Bob were supposed to set their polarizers by 'free will' [13,17,18], so correlations are excluded there. However, the superdeterminism assumption states that Alice and Bob have no absolute free will, but instead, their decisions are dictated by laws of nature. We calculated in [6] which correlations would be required to reproduce the quantum result (7.1). In our naive model, we assume that Alice's result A=+1 if and A=−1 otherwise, and the same for Bob's measurement B. The answer became unique by assuming that the probabilities only depend on a–c and b–c:

If the choices made by Alice and Bob, and the angle c of the photons, are correlated as in the probability distribution

7.2

then equation (7.1) is obeyed.

The correlated probability function (7.2) seems to be difficult to defend: suppose Alice and Bob use tremendously complex algorithms to arrive at their decisions, or suppose that Alice lets her decision depend [19] on fluctuations of the light from distant Quasar Q _A and Bob chooses quasar Q _B , how could their choices still be strongly correlated with the atom prepared at S?

A more picturesque approach was brought forward by a participant in a blog discussion: imagine that both Alice and Bob carry with them a cage, and in it a mouse, and some grain to feed it. Just before making her decision about the angle a, Alice counts the number of droppings produced by the mouse in her cage. If it is even, she chooses the angle a to be 0, and if it is odd, a=45°. Bob does the same by inspecting his cage. Now, the function (7.2) tells us that the polarization angles of the two emitted photons have to be correlated with the way the bowels of these mice work, long before they produced their droppings! Would that not be a disgusting theory?

Let us first note that strong correlations are ubiquitous in our models [20]. For example, in our 'neutrino' model, §3, the dynamical variables take the form of flat sheets, which show strong space-like correlations even though their dynamical equations of motion are local, as was explained at the end of that section. And then, given the fact that we have these correlations, what exactly is the mathematical definition of disgusting?

More to the point, the correlation function (7.2) is still not quite in line with our theory of the Cellular Automaton Interpretation (CAI) of quantum mechanics. The correlation has to be complete! According to the principles of our ontological interpretation of quantum mechanics [6], the universe is always in a state where all beables at all times t, take precisely defined values. Therefore, the photons emitted by the source ε must have well-defined polarizations in exactly the same basis that is later employed by Alice and Bob, already when they are emitted, at time t=t ₂ in figure 1. This feature is an evident fact of all explicit models that we can produce, and is the central assumption of the CAI, but now it is also clear that it is exactly here that we have to face our 'conspiracy' problem. Did the photons commit 'conspiracy' by knowing in advance which basis to pick, the very same basis that Alice and Bob would decide to choose, by consulting the distant quasars, or the mice in their cages?

Quantitatively: we must explain how the photons could 'pick' the basis a for photon α and basis b for photon β, already at time t=t ₂. Here, a and b are described by the angles chosen by Alice and Bob, at time t=t ₃! As they both have the same polarization c, the atom ε gives them the same signs ±1 with probability cos ²(a−b) and opposite signs, ±1 and ∓1 with probability . One then obtains the quantum result (7.1), but how could this have happened?

According to the CAI, our universe started off being in exactly one ontological state (an ontic state, for short). From day 1 onwards, the universe evolves and continues to be in ontic states forever. It nevertheless obeys Schrödinger's equation, as our ontic states do, see the discussion of our harmonic oscillator and neutrino. The observations A by Alice and B by Bob are ontic, as soon as they become macroscopically visible. The universe can never be in a superposition of states. This indeed means is that the polarization c of the photons emitted by the decaying atom seen by Alice can only be parallel or orthogonal to the orientation of her filter. The photon 'knew' Alice's choice a ahead of time, or the photon 'knew' exactly about the quasars or the mouse droppings.

The explanation of this conspiracy phenomenon can be found by observing how the ontological interpretation of quantum mechanics arrives exactly at the Copenhagen prescriptions for deriving probabilities from the wave functions in quantum calculations. The wave functions normally employed in standard quantum mechanical calculations are regarded as templates, which are superpositions of our ontological states with arbitrary superposition coefficients. Both the ontological wave functions (where the beables have definite values) and the template wave functions obey Schrödinger's equation. Therefore, the superposition coefficients are constant in time. The templates are the functions we use in our models, such as the hydrogen atom or the Standard Model.

How are the superposition parameters chosen? Well, in practice, we pick these in such a way that the initial state is described as well as we can. What this really means is that, in the initial state, the superposition coefficients λ _i in

7.3

represent the probabilities P _i that we have the specific ontological state |ψ _i 〉 by writing P _i =|λ _i |².

Indeed, a central theme in our CAI of quantum mechanics is that the Born probabilities P _i , that usually are derived after the final state wave function is obtained, are seen to correspond to the probability distributions assumed for the initial states, as λ _i are time-independent. Thus, the probabilities, often thought to emerge from a separate axiom in the Copenhagen doctrine, are now reduced to our ubiquitous uncertainties concerning the initial state.

Even if we tried to describe the initial state as accurately as we can, the total number of ontological states participating in the sum (7.3) is huge, nearly comparable to the mathematical number 'googolplex' or 10¹⁰¹⁰⁰ . As soon as either Alice or Bob made anything like an infinitesimal modification of the settings of their polarization filters, all ontological states |ψ _i 〉 in equation (7.3) are replaced by others. This is because the classical description of their world is modified, and a crucial observation of our theory is that the classical states of the universe are ontological (a natural assumption which was found to serve as a perfect answer to what is known as the 'measurement problem' [21]). Any modification, no matter how small, in the classical parameters requires a completely different set of ontological states.

Now we can see what goes wrong in Bell's argument: it was assumed that Bob's template state is not affected by Alice's change of mind. By changing her mind, Alice had to replace all ontological states |ψ _i 〉 in her template state by others, and then Bell assumes that none of the ontological states in Bob's world are modified. If there is any kind of long-range correlations in the ontological variables—the beables—then this modified state will have zero probability. The template state describing the two entangled photons has to be repaired from scratch. Even Bob's world is replaced by a different world when Alice changes her mind, or the mouse in Alice's cage produced a different number of droppings.

This is the price one has to pay for a superdeterministic theory. Of course, we do not note the 'replacement of Bob's world by a different one', because we cannot modify the ontological states. Nature chose the state we are in when the Big Bang took place, and that is that.

We do note that the existence of space-like correlations in phenomena such as vacuum correlations must be assumed. Glancing over our argument again, we do note that a modification of one single beable at one spot in Alice's world could in principle suffice to modify her settings. But then we must realize that even such a modification must have its traces back in the past, all the way back to the Big Bang. It cannot stay local when following the change in the past, so it certainly must affect Bob's world.

We could sharpen our argument as follows: even in a superdeterministic world, contradictions with Bell's theorem would ensue if it would be legal to consider a change of one or a few bits in the beables describing Alice's world, without making any modification in Bob's world. Now following what such a change would entail both for the distant past and for the distant future, it is easy to observe that, certainly in the distant past, the effects of such a modification would be enormous, and it may never be compatible with a simple low-entropy Big Bang beginning of the universe, which means that the change in question would be incompatible with the essential correlations that the Big Bang brought about in today's beables. Thus, we can demand in our theory that a modification of just a few beables in Alice's world without any change in Bob's world is fundamentally illegal. This is how an ontological, deterministic model can 'conspire' to violate Bell's theorem.

Perhaps, our argument here is not new, but it surely is not often raised, which is why in our opinion we must consider these observations very carefully: superdeterminism may evoke sarcasm among some investigators, but one should admit that the notion is a quite reasonable one to consider.

Certainly, saying that it is 'disgusting' should not suffice as a mathematically acceptable disproof. The elegance with which the theory presented here addresses the measurement problem, the collapse problem and Schrödinger's cat paradox, should remove some people's disgust.

Nevertheless, it may be wise to be prudent in drawing conclusions. The models produced so-far are still far from mimicking the Standard Model. We can have complicated interactions, such as in cellular automata that are more general than the ones that only produce non-interacting left-movers and right-movers, but mapping these into interesting quantum field theories turned out to be quite difficult or even impossible. We found that Hamiltonians can be produced that do the job correctly during short time intervals, but they seem to turn into non-local Hamiltonians (Hamiltonians that cannot be seen to be built up from local Hamiltonian densities) when we insist that also large time intervals are covered correctly.

Careful analysis of the models in question lead us to believe that the safest way to generate a local Hamiltonian density would be to consider models with a built-in invariance under local time diffeomorphisms. These are symmetries that are generated by a local Hamiltonian density. Now, if we combine this with the demand of Lorentz invariance, we invariably end up with systems that include a fully quantized gravitational force. Must gravity be included in quantum theory, before we can make a CA interpretation? This is only a suspicion, at this day. It could explain why, in an era that we are still totally mystified by the gravitational forces between quantum systems, anything resembling a CAI was as yet dismissed.

Our general conclusion is that theories including the gravitational force must be further studied and that classical interpretations of quantum mechanics may then surely be feasible. It is also a fact that standard arguments using Bell's theorem do not sufficiently take into account that correlation functions do not have to vanish at space-like distances. The 'mouse-dropping function' (7.2) displays correlations over space-like distances which are quite normal in physics. A liquid close to its critical point shows critical opalescence, which means that density fluctuations take place at distance scales in the order of microns, at which visible light is scattered, so that the correlations are optically visible. These correlation functions are also non-vanishing and large at space-like separations, so this is nothing to be concerned about.

A more complete description of the Cellular automaton interpretation of quantum mechanics will be published by the author [6].

Figure 1. A Bell-type experiment. Space runs horizontally, time vertically. Single lines denote single bits or qubits travelling; widened lines denote classical information carried by millions of bits. Light travels over 45°, as indicated by the light cone on the right. Meaning of the variables: see text. (Online version in colour.)

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Footnotes

1 The term makes the system antiperiodic with period T. Here, we consider a total shift of the energy that removes the as a harmless procedure, so that the minus sign over an odd number of periods is ignored.

2 In the case of finite N, the operator, derived from equation (2.7), connects the lowest energy state |0〉 _E to the highest energy state |N−1〉 _E ; one would be tempted to ignore this term in the limit , but then unitarity will be violated. The fact that there is a small subset of states, called 'edge states', for which the one-to-one mapping linking quantum mechanics to classical mechanics fails, may be an important feature to keep in mind; it happens in other examples as well; see our remarks about 'edge states' later.

3 In this section, the caret ( ) is reserved for vectors normalized to 1, not operators.

4 Alternatively, one may choose the boundaries to be 0 and 1.

5 We do note that the situation is not hopeless: the longitudinal string variables that need to coincide refer to points inside the string world sheet. Where exactly the strings meet inside their world sheet may perhaps be considered as 'unimportant'. Incorporating diffeomorphisms on the world sheet might conceivably resolve the difficulties.

6 In a total spin 0 state, two photons are polarized in the same direction; if we were dealing with spin electrons, their spins would be opposite, but such details are irrelevant for the general discussion.

One contribution of 13 to a theme issue 'New geometric concepts in the foundations of physics'.

© 2015 The Author(s) Published by the Royal Society. All rights reserved.

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Source: https://royalsocietypublishing.org/doi/10.1098/rsta.2014.0236

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