Foundations of Physics

, Volume 44, Issue 5, pp 463–471 | Cite as

Superstrings and the Foundations of Quantum Mechanics



It is put forward that modern elementary particle physics cannot be completely unified with the laws of gravity and general relativity without addressing the question of the ontological interpretation of quantum mechanics itself. The position of superstring theory in this general question is emphasized: superstrings may well form exactly the right mathematical system that can explain how quantum mechanics can be linked to a deterministic picture of our world. Deterministic interpretations of quantum mechanics are usually categorically rejected, because of Bell’s powerful observations, and indeed these apply here also, but we do emphasize that the models we arrive at are super-deterministic, which is exactly the case where Bell expressed his doubts. Strong correlations at space-like separations could explain the apparent contradictions.

1 Introduction

Roughly half a century was needed to reconcile the fundamental laws of quantum mechanics with Einstein’s theory of special relativity, in such a way the a vast body of accurate experimental observations could be incorporated at the same time. A picture emerged that became gradually more precise. The elementary particles must be identified with the energy quanta of a set of locally defined fields, which obey field equations such that the rules of quantum mechanics can be imposed on them, while these equations must at the same time be invariant under Lorentz transformations. When fields interact, the energy quanta interact, and it turned out to be crucial to require that these interactions can be kept under control mathematically. This is what renormalization is about, and it turned out that only a few different types of fields are allowed, and their interactions must be limited to some very basic, polynomial expressions.

The outcome of this research is now known as “the Standard Model” [1, 2]. It is fair to say that the Standard Model1 represents our present understanding of all subatomic particles that have been explicitly observed experimentally.

One of these basic sets of fields is the Dirac fields. Their energy quanta have spin 1/2, and can be looked upon as matter. According to the Standard Model, these matter fields come in three generations. Each generation has one doublet of particles called leptons—for the first generation these are the electron and the electron type neutrino. Then there is a doublet of quarks, which each form an \(SU(3)\) triplet, usually denoted as three colors: red, green and blue. For the first generation, these are the up-quarks and the down-quarks.

Dirac fields cannot interact directly with one another or with themselves; for interactions, other fields are needed: fields with spin zero (scalar fields) and/or fields with spin one (vector fields). Scalar fields seem to be the easiest, at first sight, but there are some problems with them, which may be the reason why their precise role, and their very existence, are only slowly becoming apparent today. Vector fields must be of the Yang-Mills type. The Standard Model makes use of three ingredients: A vector field based on the gauge group \(U(1)\), and, in a sense, responsible for the electro-magnetic force, a triplet of vector fields based on the group \(SU(2)\), responsible for the weak interactions, and an octet of vector fields based on \(SU(3)\), responsible for a very strong force holding the quarks together to form particles we call mesons and baryons.

According to the Standard Model, it is the virtual presence of a scalar field forming a doublet of the weak \(SU(2)\) group and carrying a \(U(1)\) charge as well, the Higgs field, that is responsible for the mixing between the \(SU(2)\) vector fields and the \(U(1)\) vector field to generate the observed features of electromagnetism and the weak force (electro-weak mixing), while the same scalar fields can also interact with the Dirac fields in such a way that the observed mass spectrum of the known particles is accounted for.

The Higgs field generates at least one energy quantum that is not virtual, but a real particle, the Higgs particle. This is the particle for which recent experiments at the Large Hadron Collider have given convincing evidence. Observations made at the Atlas detector [3] as well as the CMS detector [4] both are associated to a “Higgs-like” object with a mass of about 126 GeV.

There is a general consensus, however, that the Standard Model, as it is formulated today, cannot explain everything, even if we fill in all known details about quark- and neutrino mixing (see footnote 1). The question then rises how to obtain further information. Fortunately, we have the ability to do further experiments at higher energies, and the importance of the information gathered that way can hardly be over estimated. On the other hand it is also clear that further guidance from theoretical ideas is needed. 2

Quantum field theory has been used to identify the Standard Model, but it is not sufficiently restrictive to provide for reliable predictions as to what will come next. New input is needed.

The best way to proceed is to identify the weakest points of our present understanding. There are two. One weakness is the apparent “unnatural” nature of the postulated interaction scheme at distance scales much shorter than the LHC scale of around \(10^{-16}\,\)mm. It seems that naturalness can only be restored if a whole arena of new fields, particles and interaction types is assumed to exist there. Curiously, no sign of such schemes has yet been detected: no new particles or other deviations from the known Standard Model interaction mechanisms. Perhaps this has to be taken as a warning sign that new alleys have yet to be searched for.

The second glaring weakness in our present understanding is the gravitational force. Unfortunately, gravity seems to be a far way off: the distance scale at which gravity must become important may be another factor \(10^{-16}{\times }\) the LHC scale! these 16 orders of magnitude are a colossal gap to bridge, if intellectual logic is to be our only guide.3 Nevertheless, theoreticians are ready to make such jumps, and so, we are searching for difficulties and paradoxes that arise when a logically consistent scheme of quantum gravity is searched for. There are several difficulties.

2 Black Holes and Strings

An inevitable consequence of standard general relativity is that, given a sufficiently large concentration of mass, gravitational attraction causes this mass to collapse, to such an extent that a point of no return may be reached, and a “black hole” is formed. Thus, in large, massive systems, black holes are an inevitable form of matter, obeying their own laws of nature.

One of the laws of nature is that a black hole can have any size, between roughly the Planck size of the order of \(10^{-33}\,\)cm and billions of kilometers. The smallest black holes will have masses of the order of the Planck mass, some \(10^{16}\,\)TeV. This being the case, one might expect that the spectrum of particle-like objects near the Planck scale will vary from genuine elementary particles below \(10^{16}\,\)TeV, to black holes above that limit. There will be no obvious differences between these objects, so what is needed is a theory that unifies black holes with ordinary particles.

This requires a quantum description of black holes [5], just as we have for sub-atomic particles. Now, black holes obey the rules of general relativity, and so, different sets of space-time coordinates should be allowed to be used for their description, and here, something very special turns up. Black holes are equipped with a horizon, and what happens at this horizon can be described by two kinds of observers. An observer who falls into a black hole experiences a horizon that is transparent: (s)he can go straight through, apparently without being stopped, even without knowing exactly where the horizon is.

On the other had, we have observers who stay at a distance. Not only do these observers detect a steady stream of particles coming out of the black hole, exactly from where the ingoing observer would have been—while the ingoing observer would definitely deny the presence of such particles, which would have killed him/her instantly; the outside observer also deduces that information is distributed across the horizon in bits and bytes. To be precise, the outside observer would deduce the presence of one bit of information at exactly \(0.724\times 10^{-65}\,\hbox {cm}^2\). This seems to be much less than the information brought in by a steady stream of ingoing observers. Something seems to be wrong.

A very different approach to the questions concerning gravity at the Planck scale is represented by superstring theory. This theory resulted from investigations of what happens when the fundamental units of matter would no longer be represented as isolated points, as we have in quantum field theories, but rather as closed line like loops, or open line segments. The energy quanta of closed string loops were found to contain components behaving exactly as gravitons, the quanta of the gravitational field, and so this theory was further developed as a theory that should explain how gravity should be quantized.

Indeed, several obnoxious difficulties with the more standard picture of quantum gravity disappeared: ultraviolet infinities are no longer standing in the way of accurate higher order calculations, and furthermore, other superstring excitations appeared to be highly suitable to describe both the fermionic matter fields as the bosonic Yang-Mills and scalar fields. As the strings have to be subject to the very stringent rules of both quantum mechanics and Einstein’s general relativity, it was originally expected that their properties would be derived completely and the theory should come out as being unique.

The mathematics is very compelling. The theory indeed also includes structures that play the role of black holes, although strictly speaking they are black holes near the “extreme limit”, where we have two horizons rather than one, together behaving somewhat differently from that of a standard black hole. These string black holes do have a micro structure that correctly reproduces the information content of one bit per \(0.724\times 10^{-65}\,\hbox {cm}^2\), so the black hole “information paradox” is addressed. On the other hand, however, in terms of string theory, it is not yet understood how an in-falling observer should describe the metric.

This theory would really have been the ideal answer to the apparently contradictory questions we have about the gravitational force, except that its physical interpretation is confusing and troublesome. Some scientists tend to incline towards the mystic conclusion that, obviously, superstring theory will defy ordinary logic even more than quantum mechanics itself, and they embrace this as a gratifying feature. Without going into details (which will be published elsewhere [6]), we shall claim that the opposite should be true: string theory will be described with logic that is not only better than that of quantum mechanics itself, but also better than classical mechanics.

But, as it stands today in the literature, there are more unanswered questions. How do we describe the cosmos in terms of string theory? What does “quantum cosmology” mean? Is there a wave function in the “universe of universes”? Does this wave function collapse when we do a measurement?

The problem of understanding superstring theory is the difficulty of interpreting what it says. The foundations of this theory are amazingly weak. It is this author’s opinion that this difficulty does directly relate to the questions of the foundations of quantum mechanics itself.

3 The Cellular Automaton

A view of our universe, and an approach to the deeper meanings of quantum mechanics that may appear to be entirely incompatible with any of the above theories, is the idea of a cellular automaton [7, 8, 9]. A Cellular Automaton is an information processing machine. At the beat of a clock, data in its memory are updated by some local, completely deterministic algorithm. Usually, we also demand this algorithm to respect locality, which means that the data are arranged in ‘cells’, and the rule for updating a cell only uses information from the cell itself and its nearest neighbors. See Fig. 1 as an illustration in 2 dimensions.
Fig. 1

An example of an evolving cellular automaton in 2 space dimensions, following an evolution law where each cell only reacts according to the data in its immediate neighbor cells

In addition, we assume that the algorithm can be run forwards as well as backwards4 in time. This means that the evolution law may be regarded as a permutation \(P(t)=P^{t/\delta t}\) in the entire set of all possible states. Here, \(\delta {t}\) is the fundamental time unit.

One then takes a step that should have been considered completely natural. We identify the states of the automaton as quantum states. To be precise: every (classical) state the automaton can be in is regarded as an element of an orthonormal basis in Hilbert space. In terms of the vectors in this Hilbert space, the evolution operator is the matrix associated with the permutator, that is, a matrix containing one 1 in every row and in every column, and all other elements are zero. If time \(t\) is limited to integer multiples of the time unit \(\delta t\) of the clock, we have
$$\begin{aligned} U(\delta t)=P,\quad U(t)=e^{-(i/\hbar )Ht}, \end{aligned}$$
where \(H\) is a hermitean operator. \(H\) is defined in this equation, but the definition is not unique: one may add any multiple of \(2\pi \hbar /\delta t\) to each of its eigen values.

We emphasize that, whichever hamiltonian \(H\) we choose, if it obeys Eq. (1), it reproduces standard quantum mechanics. It is easy to include all standard rules of quantum mechanics, in particular the Born rule, which is to be used if we take the superposition of several of the basis elements. The only difference with usual quantum mechanics is the fact that such a system, at the same time, has an obvious interpretation as being completely classical. During most of the 20th century, fundamental quantum physics was dominated by the Bohr–Einstein discussion about the differences between the quantum world and the classical world. Clearly, the ‘naive’ attempt described above is nearly universally rejected. This must be wrong!

More to the point: those “quantum models” one gets out of automaton models cannot possibly be realistic ones. A simple reason for being suspicious about the “automaton interpretation” would be that, for an automaton, only the basis elements of Hilbert space are ‘ontological’. Only they describe what may ‘really’ be happening. Superpositions of states should never occur in practice.

This, however, would not suffice to reject the automaton interpretation. We can easily include superpositions of states by positing that superpositions, by definition, could represent probabilistic distributions. More often than not, observers would be unable to identify exactly which automaton state we are looking at; the superposition would be a best guess. Since all basis elements evolve with the same hamiltonian \(H\), the superpositions would evolve with the same \(H\) as well, and the probabilities deduced using Born’s rule would evolve correctly. So, there is nothing wrong with superpositions.

4 Bell’s Theorem

A more important reason for rejecting automaton models of quantum mechanics is the possibility of doing quantum experiments with quantum-entangled states [10, 11, 12]. Again we raise a point of warning. All superpositions of ontological states would behave as probabilistic ensembles in the automaton model, so, at first sight, there seems to be no reason whatsoever to reject the automaton interpretation of quantum mechanics on the basis of the existence of quantum-entangled states. However, a problem arises.

The situation was most clearly spelled out by J. Bell, in terms of a theorem [13]. He simply asks how, in a theory with classical evolution laws, information can flow in such a way that correlations are seen as predicted in quantum mechanics. An experiment can be imagined that produces two particles in an entangled state. For instance, two photons could emerge whose total spin adds up to zero. Far away from the device that produced the photons, an observer called Alice checks one photon to see whether it passes through a polarization filter oriented in the direction \(\varvec{a}\). At far, space-like separation from her, another observer, Bob, detects the other photon and checks whether it passes through a polarization filter oriented in the direction \(\varvec{b}\) (Fig. 2).
Fig. 2

A Bell type experiment with entangled photons

According to quantum mechanics, the polarizations of the two photons are completely correlated if \(\varvec{a}\) is parallel to \(\varvec{b}\), and anti-correlated if \(\varvec{a} \perp \varvec{b}\). If the relative angle is \(45^\circ \) the correlation is absent, while it is very high at a relative angle of \(22.5^\circ \). Bell proves that such a correlation pattern cannot be reproduced when information is transported classically.

Something is wrong. Usually, it is concluded that cellular automaton models of the type discussed above cannot reproduce the quantum nature of our world entirely. What would be wrong with the cellular automaton model? It has a hamiltonian. Its evolution law obeys locality. There may be quite non-trivial, non-linear interactions between operators that could describe propagating particles. Some of these models may look somewhat like the universe we are in. In principle, as stated, the model allows us to describe entangled particles, just as in ordinary quantum mechanics.

Rather than concluding that the cellular automaton model must be wrong, we can also focus on the assumptions that went into Bell’s theorem. One assumption was that Alice and Bob must have the “free will” [14] to modify the settings \(\varvec{a}\) and \(\varvec{b}\) of their experiment just before the measurements are made, and the decaying atom should not be allowed to anticipate Alice’s and/or Bob’s decisions. Now, obviously, there is no free will in a cellular automaton model; Alice and Bob are parts of the system as well. On the other hand, one can reformulate Bell’s observations without making reference to free will. Then, however, assumptions must be made concerning the absence of space-like correlations at the early phases of the experiment. But here, we can argue that space-like correlations neither vanish in a quantum field theory, nor in a cellular automaton. Conceivably, loopholes may be found here.

5 Quantized Fields

Whatever is the case, we decided to look at the cellular automaton models in more detail, to see how close we can proceed to describe the real world this way. Quantum field theories (QFT) have some salient features in common with cellular automata. Both describe discretized pieces of information that propagate like particles. However, it turns out to be extremely difficult to match given quantized fields with given automaton models. In particular, rotational symmetry and Lorentz invariance are difficult to reproduce in a cellular automaton.

An exception is massless fields in 1 space- and 1 time dimension [15]. In this case, objects move with a fixed velocity either to the left or to the right. We succeeded in describing a mapping between such a field theory and a cellular automaton that describes data in the form of integers moving to the left and to the right. The mapping is one-to-one. Physically, therefore, a massless QFT in 2 space–time dimensions is indistinguishable from a cellular automaton.

6 Superstrings

The prototype of a massless, two dimensional quantum field theory is the description of the world sheet of a string moving in \(D\) dimensional space–time. The observation described in Section 5 directly applies here as well. So we do have a relativistic model in higher-dimensional space–time that is equivalent to a cellular automaton: string theory. Moreover, since also massless fermions on a 1+1 dimensional world sheet can be mapped onto a cellular automaton—though now with boolean degrees of freedom moving to the left and to the right, also superstring theory may be equivalent to a cellular automaton.

We found that also the string theory boundary conditions (describing the propagation of finite open string segments or closed string loops) appear to fit in a description of a deterministic automaton.

In fact, this led us to make an other observation: the physical degrees of freedom of a cellular automaton are fundamentally discrete. Bosonic fields take integer values. In terms of string theory, this means that the string target space is discrete: the cellular automaton corresponding to strings and superstrings are defined on a target space where the variables live on a lattice. The lattice length \(a\) can be determined precisely:
$$\begin{aligned} a=2\pi \sqrt{a'}\ , \end{aligned}$$
where \(\alpha '\) is the fundamental string slope parameter, which corresponds to the inverse of the string’s tension strength.

Interactions among strings can also be cast in a deterministic form: when two pieces of string hit at a specific point on the \(D\) dimensional lattice, these two string pieces exchange arms. Some careful inspection tells us that this requires strings or superstrings to have an intrinsic orientation, so we are dealing with oriented strings.

7 Conclusions

Since superstring theory is regarded as a serious candidate for a theory describing our physical world at the Planck scale [16, 17, 18], and since we arrived at evidence that this theory may be equivalent to a cellular automaton, It appears that we are homing in to physically significant models that may feature quantum behavior compatible with induced quantum mechanics, the quantum theory obtained by simply describing a classical evolution law in a quantum language.

Clearly, this leaves Bell’s theorem as a puzzle. We know that both quantum field theories and cellular automata allow for non-trivial correlations at space-like separations in the vacuum state—all field propagator functions can be regarded as such correlation functions, and they indeed do not vanish outside the light cone. We now ask whether these correlations could be sufficiently significant to be held responsible for the observed strong violations of Bell’s inequalities in quantum mechanics. It is our proposal to conclude from the existence of cellular automaton quantum models that the actual implications of Bell’s theorem should be reconsidered in this light.

As long as we are not sure how exactly to reconcile apparently conflicting observations, our working philosophy is: just shut up and calculate. Let us see what these models will bring us, and what further modifications will be needed to make them represent our world as precisely as possible.


  1. 1.

    Some researchers attribute the details of the mass spectrum and the mixing phenomena to features that go beyond the Standard Model. Here, we consider the entire system including the mass generating and mass mixing parameters, as being the “Standard Model”, since, in principle, it could be handled as a closed mathematical system. There are many introductory texts on the Standard Model. See for instance [1]; for more advanced [2].

  2. 2.

    After all, without theoretical input, the Higgs particle would never have been discovered.

  3. 3.

    For comparison: most of the Solar System easily fits in a box of \(10^{16}\) mm on a side.

  4. 4.

    It is conceivable that this condition may be relaxed in more sophisticated versions of this theory; cellular automata that are not time reversible have special and novel properties, but the complications are severe and we shall usually disregard this option.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUtrecht University and Spinoza InstituteUtrechtThe Netherlands

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