Part II Calculation Techniques
15.3 The ‘Neutrino’ Vacuum Correlations
one, although we did the Fourier transform replacing the variablerby its momentum variablepr. The normalization is such that
α
d3pψ αstand(p) 2=
s
ˆ q3>0
d2qˆ π/δr
−π/δr
dprψsont(q, pˆ r)2, (15.91) see Eqs. (15.77)–(15.80).
In our case,ψhas only two spin modes, it is a Weyl field, but in all other respects it can be handled just as a massless Dirac field. Following Dirac, in momentum space, each momentumphas two energy eigenmodes (eigenvectors of the operator hβα in the Hamiltonian (15.87)), which we write, properly normalized, as
ustandα ±(p) = 1
√2|p|(|p| ±p3)
±|p| +p3
p1+ip2
; E= ±|p|. (15.92) Here, the spinor lists the values for the indexα=1,2. In the basis of the beables:
uonts ±(q, pˆ r)= 1
0
if ±pr >0, 0
1
if ±pr<0; (15.93)
E= ±|pr|. (15.94)
Here, the spinor lists the values for the indexs= +and−.
In both cases, we write
ψ (p) =u+a1(p) +u−a†2(− p); {a1, a2} = a1, a2†
=0, (15.95) a1(p), a †1
p
=
a2(p), a †2 p
=δ3 p− p
or δ
pr−pr δ2
ˆ q,qˆ
; (15.96) Hop= |p|
a1†a1+a2†a2−1
, (15.97)
wherea1 is the annihilation operator for a particle with momentump, and a†2 is the creation operator for an antiparticle with momentum− p. We drop the vacuum energy−1 . In case we have a lattice inr space, the momentum is limited to the values| p| = |pr|< π/δr.
15.3 The ‘Neutrino’ Vacuum Correlations
The vacuum state|∅is the state of lowest energy. This means that, at each momen- tum valuepor equivalently, at each(q, pˆ r), we have
ai|∅ =0, (15.98)
whereai is the annihilation operator for all states withH=σ· p=spr >0, and the creation operator ifH <0. The beable states are the states where, at each value of the set(q, r, s)ˆ the number of ‘particles’ is specified to be either 1 or 0. This means, of course, that the vacuum state (15.98) is not a beable state; it is a superposition of all beable states.
One may for instance compute the correlation functions of right- and left moving
‘particles’ (sheets, actually) in a given direction. In the beable (ontological) basis, one finds that left-movers are not correlated to right movers, but two left-movers are correlated as follows:
P (r1, r2)−P (r1)P (r2)
=δr2∅|ψs∗(r1)ψs(r1)ψs∗(r2)ψs(r2)|∅conn
= δr
2π π/δr
0
dp eip(r2−r1) 2=
δr2
π2|r1−r2|2 if r1δr−r2 =odd,
1
4δr1,r2 if r1δr−r2 =even,
(15.99) whereδr2, the unit of distance between two adjacent sheets squared, was added for normalization, and ‘conn’ stands for the connected diagram contribution only, that is, the particle and antiparticle created atr2 are both annihilated atr1. The same applies to two right movers. In the case of a lattice, whereδr is not yet tuned to zero, this calculation is still exact ifr1−r2is an integer multiple ofδr. Note that, for the vacuum,P (r)=P (r, r)=12.
An important point about the second quantized Hamiltonian (15.87), (15.88): on the lattice, we wish to keep the Hamiltonian (15.97) in momentum space. In position space, Eqs. (15.87) or (15.88) cannot be valid since one cannot differentiate in the space variabler. But we can have the induced evolution operator over finite integer time intervalsT =ntδr. This evolution operator then displaces left movers one step to the left and right movers one step to the right. The Hamiltonian (15.97) does exactly that, while it can be used also for infinitesimal times; it is however not quite local when re-expressed in terms of the fields on the lattice coordinates, since now momentum is limited to stay within the Brillouin zone|pr|< π/δr. This feature, which here does not lead to serious complications, is further explained, for the bosonic case, in Sect.17.1.1.
Correlations of data at two points that are separated in space but not in time, or not sufficiently far in the time-like direction to allow light signals to connect these two points, are called space-like correlations. The space-like correlations found in Eq. (15.99) are important. They probably play an important role in the mysterious behaviour of the beable models when Bell’s inequalities are considered, see PartI, Chap.3.6and beyond.
Note that we are dealing here with space-like correlations of the ontological de- grees of freedom. The correlations are a consequence of the fact that we are looking at a special state that is preserved in time, a state we call the vacuum. All physical states normally encountered are template states, deviating only very slightly from this vacuum state, so we will always have such correlations.
In the chapters about Bell inequalities and the Cellular Automaton Interpretation (Sect.5.2and Chap.3of PartI), it is argued that the ontological theories proposed in this book must feature strong, space-like correlations throughout the universe.
This would be the only way to explain how the Bell, or CHSH inequalities can be so strongly violated in these models. Now since our ‘neutrinos’ are non interacting, one cannot really do EPR-like experiments with them, so already for that reason, there
15.3 The ‘Neutrino’ Vacuum Correlations 167 is no direct contradiction. However, we also see that strong space-like correlations are present in this model.
Indeed, one’s first impression might be that the ontological ‘neutrino sheet’
model of the previous section is entirely non local. The sheets stretch out infinitely far in two directions, and if a sheet moves here, we immediately have some informa- tion about what it does elsewhere. But on closer inspection one should concede that the equations of motion are entirely local. These equations tell us that if we have a sheet going through a space–time pointx, carrying a sign functions, and oriented in the directionq, then, at the pointˆ x, the sheet will move with the speed of light in the direction dictated byqˆ andσ. No information is needed from points elsewhere in the universe. This is locality.
The thing that is non local is the ubiquitous correlations in this model. If we have a sheet at(x, t ), oriented in a direction qˆ, we instantly know that the same sheet will occur at other points(y, t ), if qˆ·(y− x)=0, and it has the same values forqˆandσ. It will be explained in Chap.20, Sect.20.7, that space-like correlations are normal in physics, both in classical systems such as crystals or star clusters and in quantum mechanical ones such as quantized fields. In the neutrino sheets, the correlations are even stronger, so that assuming their absence is a big mistake when one tries to draw conclusions from Bell’s theorem.
Open Access This chapter is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, dupli- cation, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link is provided to the Creative Com- mons license and any changes made are indicated.
The images or other third party material in this chapter are included in the work’s Creative Commons license, unless indicated otherwise in the credit line; if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation, users will need to obtain permission from the license holder to duplicate, adapt or reproduce the material.
PQ Theory
Most quantum theories describe operators whose eigenvalues form continua of real numbers. Examples are one or more particles in some potential well, harmonic os- cillators, but also bosonic quantum fields and strings. If we want to relate these to deterministic systems we could consider ontological observables that take values in the real numbers as well. There is however an other option.
One important application of the transformations described in this book could be our attempts to produce fundamental theories about Nature at the Planck scale. Here, we have the holographic principle at work, and the Bekenstein bound [5]. What these tell us is that Hilbert space assigned to any small domain of space should be finite-dimensional. In contrast, real numbers are described by unlimited sequences of digits and therefore require infinite dimensional Hilbert spaces. That’s too large.
One may suspect that uncertainty relations, or non-commutativity, add some blur to these numbers. In this chapter, we outline a mathematical procedure for a systematic approach.
For PQ theory, as our approach will be called, we employed a new notation in earlier work [114,115], where not buthis normalized to one. Wave functions then take the forme2π ipx=ipx, where=e2π≈535.5. This notation was very useful to avoid factors 1/√
2πfor the normalization of wave functions on a circle.
Yet we decided not to use such notation in this book, so as to avoid clashes with other discussions in the standard notation in various other chapters. Therefore, we return to the normalization=1. Factors √
2π (for normalized states) will now occur more frequently, and hopefully they won’t deter the reader.
In this chapter, and part of the following ones, dynamical variables can be real numbers, indicated with lower case letters:p, q, r, x, . . ., they can be integers indi- cated by capitals:N, P , Q, X, . . ., or they are angles (numbers defined on a circle), indicated by Greek letters α, η, κ, , . . ., usually obeying−π < α≤π, or some- times defined merely modulo 2π.
A real numberr, for example the numberr=137.035999074· · ·, is composed of an integer, hereR=137, and an angle,/2π=0.035999074· · ·. In examples such as a quantum particle on a line, Hilbert space is spanned by a basis defined on the line:{|r}. In PQ theory, we regard such a Hilbert space as the product of Hilbert
© The Author(s) 2016
G. ’t Hooft, The Cellular Automaton Interpretation of Quantum Mechanics, Fundamental Theories of Physics 185, DOI10.1007/978-3-319-41285-6_16
169
170 16 PQ Theory space spanned by the integers|Rand Hilbert space spanned by the angles,|. So, we have
√1
2π|r = |R, = |R|. (16.1) Note that continuity of a wave function|ψimplies twisted boundary conditions:
R+1, |ψ = R, +2π|ψ. (16.2) The fractional part, or angle, is defined unambiguously, but the definition of the integral part depends on how the angle is projected on the segment[0,2π], or: how exactly do we round a real number to its integer value? We’ll take care of that when the question arises.
So far our introduction; it suggests that we can split up a coordinateq into an integer partQand a fractional partξ /2π and its momentum into an integer part 2π P and a fractional partκ. Now we claim that this can be done in such a way that both[P , Q] =0 and[ξ, κ] =0.
Let us set up our algebra as systematically as possible.