Part II Calculation Techniques
12.3 Automata that Are Continuous in Time
x=2 ∞ n=1
(−1)n−1sin(nx)
n . (12.8)
Next, writeHop=ω=x+π, to find that Eq. (12.8) gives ω=π−2
∞ n=1
sin(nω)
n . (12.9)
Consequently, as in Eq. (2.8), ω=π−∞
n−1
i n
U (nδt )−U (−nδt ) and
Hop=π−∞
n−1
i n
Uop(nδt )−Uop(−nδt ) .
(12.10)
Very often, we will not be content with this Hamiltonian, as it has no eigenvalues beyond the range(0,2π ). As soon as there are conserved quantities, one can add functions of these at will to the Hamiltonian, to be compared with what is often done with chemical potentials. Cellular automata in general will exhibit many such conservation laws. See Fig.2.3, where every closed orbit represents something that is conserved: the label of the orbit.
In Sect.13, we consider the other continuum limit, which is the limitδt→0 for the cogwheel model. First, we look at continuous theories more generally.
12.3 Automata that Are Continuous in Time
In the physical world, we have no direct indication that time is truly discrete. It is therefore tempting to consider the limitδt→0. At first sight, one might think that this limit should be the same as having a continuous degree of freedom obeying differential equations in time, but this is not quite so, as will be explained later in this chapter. First, in this section, we consider the strictly continuous deterministic systems. Then, we compare those with the continuum limit of discrete systems.
Consider an ontological theory described by having a continuous, multi- dimensional space of degrees of freedomq(t ), depending on one continuous time variablet, and its time evolution following classical differential equations:
d
dtqi(t )=fi(q), (12.11)
wherefi(q) may be almost any function of the variablesqj.
An example is the description of massive objects obeying classical mechanics in Ndimensions. Leta=1, . . . , N,andi=1, . . . ,2N:
{i} = {a} ⊕ {a+N}, qa(t )=xa(t ), qa+N(t )=pa(t ), fa(q) =∂Hclass(x, p)
∂pa , fa+N(q) = −∂Hclass(x, p)
∂xa , (12.12)
whereHclassis the classical Hamiltonian.
An other example is the quantum wave function of a particle in one dimension:
{i} = {x}, qi(t )=ψ (x, t ); fi(q) = −iHSψ (x, t ), (12.13) where nowHSis the Schrödinger Hamiltonian. Note, however, that, in this case, the functionψ (x, t )would be treated as an ontological object, so that the Schrödinger equation and the Hamiltonian eventually obtained will be quite different from the Schrödinger equation we start off with; actually it will look more like the corre- sponding second quantized system (see later).
We are now interested in turning Eq. (12.11) into a quantum system by changing the notation, not the physics. The ontological basis is then the set of states |q, obeying the orthogonality property
=δN q− q
, (12.14)
whereδ is now the Dirac delta distribution, and N is the dimensionality of the vectorsq.
If we wrote d
dtψ (q)= −? fi(q) ∂
∂qiψ (q) = −? iHopψ (q), (12.15) where the indexiis summed over, we would read off that
Hop ?
= −ifi(q) ∂
∂qi =fi(q)p i, pi= −i ∂
∂qi
. (12.16)
This, however, is not quite the right Hamiltonian because it may violate hermiticity:
Hop =Hop†. The correct Hamiltonian is obtained if we impose that probabilities are preserved, so that, in case the Jacobian of f (q) does not vanish, the integral dNqψ †(q)ψ ( q) is still conserved:
d
dtψ (q) = −fi(q) ∂
∂qiψ (q) −12
∂fi(q)
∂qi
ψ (q) = −iHopψ (q), (12.17) Hop= −ifi(q) ∂
∂qi −12i
∂fi(q)
∂qi
=12
fi(q)p i+pifi(q)
≡12
fi(q), p i
. (12.18)
The 1/2 in Eq. (12.17) ensures that the productψ†ψevolves with the right Jacobian.
Note that this Hamiltonian is Hermitian, and the evolution equation (12.11) follows immediately from the commutation rules
[qi, pj] =iδij; d
dtOop(t )= −i[Oop, Hop]. (12.19)
12.3 Automata that Are Continuous in Time 133 Now, however, we encounter a very important difficulty: this Hamiltonian has no lower bound. It therefore cannot be used to stabilize the wave functions. Without lower bound, one cannot do thermodynamics. This feature would turn our model into something very unlike quantum mechanics as we know it.
If we takeqspace either one-dimensional, or in some cases two-dimensional, we can make our system periodic. Then letT be the smallest positive number such that
q(T )= q(0). (12.20)
We have consequently
e−iH T|q(0) = |q(0), (12.21) and therefore, on these states,
Hop|q = ∞
n=−∞
2π n
T |nH Hn|q. (12.22) Thus, the spectrum of eigenvalues of the energy eigenstates|nH runs over all inte- gers from−∞to∞.
In the discrete case, the Hamiltonian has a finite number of eigenstates, with eigenvalues 2π k/(N δt )+δEwherek=0, . . . , N−1, which means that they lie in an interval[δE,2π/T +δE], whereT is the period, andδEcan be freely chosen.
So here, we always have a lower bound, and the state with that energy can be called
‘ground state’ or ‘vacuum’.
Depending on how the continuum limit is taken, we may or may not preserve this lower bound. The lower bound on the energy seems to be artificial, because all energy eigenstates look exactly alike. It is here that the SU(2)formulation for harmonic rotators, handled in Sect.12.1, may be more useful.
An other remedy against this problem could be that we demand analyticity when time is chosen to be complex, and boundedness of the wave functions in the lower half of the complex time frame. This would exclude the negative energy states, and still allow us to represent all probability distributions with wave functions. Equiv- alently, one could consider complex values for the variable(s)qand demand the absence of singularities in the complex plane below the real axis. Such analytic- ity constraints however seem to be rather arbitrary; they are difficult to maintain as soon as interactions are introduced, so they would certainly have to be handled with caution.
One very promising approach to solve the ground state problem is Dirac’s great idea of second quantization: take an indefinite number of objectsq, that is, a Hilbert space spanned by all states|q(1),q(2), . . .q(n), for all particle numbersn, and regard the negative energy configurations as ‘holes’ of antiparticles. This we propose to do in our ‘neutrino model’, Sect.15.2, and in later chapters.
Alternatively, we might consider the continuum limit of a discrete theory more carefully. This we try first in the next chapter. Let us emphasize again: in general, ex- cising the negative energy states just like that is not always a good idea, because any perturbation of the system might cause transitions to these negative energy states, and leaving these transitions out may violate unitarity.
The importance of the ground state of the Hamiltonian was discussed in Chap.9 of PartI. The Hamiltonian (12.18) is an important expression for fundamental dis- cussions on quantum mechanics.
As in the discrete case, also in the case of deterministic models with a continuous evolution law, one finds discrete and continuous eigenvalues, depending on whether or not a system is periodic. In the limitδt→0 of the discrete periodic ontological model, the eigenvalues are integer multiples of 2π/T, and this is also the spectrum of the harmonic oscillator with periodT, as explained in Chap.13. The harmonic oscillator may be regarded as a deterministic system in disguise.
The more general continuous model is then the system obtained first by having a (finite or infinite) number of harmonic oscillators, which means that our system consists of many periodic substructures, and secondly by admitting a (finite or infi- nite) number of conserved quantities on which the periods of the oscillators depend.
An example is the field of non-interacting particles; quantum field theory then corre- sponds to having an infinite number of oscillating modes of this field. The particles may be fermionic or bosonic; the fermionic case is also a set of oscillators if the fermions are put in a box with periodic boundary conditions. Interacting quantum particles will be encountered later (Chap.19and onwards).
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Chapter 13
The Continuum Limit of Cogwheels, Harmonic Rotators and Oscillators
In theN→ ∞limit, a cogwheel will have an infinite number of states. The Hamil- tonian will therefore also have infinitely many eigenstates. We have seen that there are two ways to take a continuum limit. AsN→ ∞, we can keep the quantized time step fixed, say it is 1. Then, in the Hamiltonian (2.24), we have to allow the quantum numberk to increase proportionally toN, keepingκ=k/N fixed. Since the time step is one, the Hamiltonian eigenvalues, 2π κ, now lie on a circle, or, we can say that the energy takes values in the continuous line segment[0,2π )(including the point 0 but excluding the point 2π). Again, one may add an arbitrary constantδEto this continuum of eigenvalues. What we have then, is a model of an object moving on a lattice in one direction. At the beat of a clock, a state moves one step at a time to the right. This is the rack, introduced in Sect.12.2. The second-quantized version is handled in Sect.17.1.
The other option for a continuum limit is to keep the periodT of the cogwheel constant, while the time quantumδttends to zero. This is also a cogwheel, now with infinitely many, microscopic teeth, but still circular. Since nowN δt=T is fixed, the ontological1states of the system can be described as an angle:
2π n/N→ϕ, d
dtϕ(t )=2π
T . (13.1)
The energy eigenvalues become
Ek=2π k/T +δE, k=0,1, . . . ,∞. (13.2) IfδEis chosen to beπ/T, we have the spectrumEk=(2π/T )(k+12). This is the spectrum of a harmonic oscillator. In fact, any periodic system with periodT, and a continuous time variable can be characterized by defining an angleϕobeying the evolution equation (13.1), and we can attempt to apply a mapping onto a harmonic oscillator with the same period.
1The words ‘ontological’ and ‘deterministic’ will be frequently used to indicate the same thing: a model without any non deterministic features, describing things that are really there.
© 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_13
135
Mappings of one model onto another one will be frequently considered in this book. It will be of importance to understand what we mean by this. If one does not put any constraint on the nature of the mapping, one would be able to map any model onto any other model; physically this might then be rather meaningless.
Generally, what we are looking for are mappings that can be prescribed in a time- independent manner. This means that, once we know how to solve the evolution law for one model, then after applying the mapping, we have also solved the evolution equations of the other model. The physical data of one model will reveal to us how the physical data of the other one evolve.
This requirement might not completely suffice in case a model is exactly inte- grable. In that case, every integrable model can be mapped onto any other one, just by considering the exact solutions. In practice, however, time independence of the prescription can be easily verified, and if we now also require that the mapping of the physical data of one model onto those of the other is one-to-one, we can be con- fident that we have a relation of the kind we are looking for. If now a deterministic model is mapped onto a quantum model, we may demand that the classical states of the deterministic model map onto an orthonormal basis of the quantum model.
Superpositions, which look natural in the quantum system, might look somewhat contrived and meaningless in the deterministic system, but they are certainly ac- ceptable as describing probabilistic distributions in the latter. This book is about these mappings.
We have already seen how a periodic deterministic system can produce a discrete spectrum of energy eigenstates. The continuous system described in this section generates energy eigenstates that are equally spaced, and range from a lowest state toE→ ∞. Mapping this onto the harmonic oscillator, seems to be straightforward;
all we have to do is map these energy eigenstates onto those of the oscillator, and since these are also equally spaced, both systems will evolve in the same way. Of course this is nothing to be surprized about: both systems are integrable and periodic with the same periodT.
For the rest of this section, we will putT =2π. The Hamiltonian of this harmonic oscillator can then be chosen to be2
Hop=12
p2+x2−1
; Hop|nH=n|nH, n=0,1, . . . ,∞. (13.3) The subscriptHreminds us that we are looking at the eigenstates of the Hamiltonian Hop. For later convenience, we subtracted the zero point energy.3
In previous versions of this book’s manuscript, we described a mapping that goes directly from a deterministic (but continuous) periodic system onto a harmonic os- cillator. Some difficulties were encountered with unitarity of the mapping. At first
2In these expressions,x, p, a, anda†are all operators, but we omitted the subscript ‘op’ to keep the expressions readable.
3Interestingly, this zero point energy would have the effect of flipping the sign of the amplitudes after exactly one period. Of course, this phase factor is not directly observable, but it may play some role in future considerations. In what we do now, it is better to avoid these complications.
13.1 The Operatorϕopin the Harmonic Rotator 137