10 or 11 Dimensions? Phenomenological Conundrum. Drunken Risibility.

It is not the fact that we are living in a ten-dimensional world which forces string theory to a ten-dimensional description. It is that perturbative string theories are only anomaly-free in ten dimensions; and they contain gravitons only in a ten-dimensional formulation. The resulting question, how the four-dimensional spacetime of phenomenology comes off from ten-dimensional perturbative string theories (or its eleven-dimensional non-perturbative extension: the mysterious M theory), led to the compactification idea and to the braneworld scenarios.

It is not the fact that empirical indications for supersymmetry were found, that forces consistent string theories to include supersymmetry. Without supersymmetry, string theory has no fermions and no chirality, but there are tachyons which make the vacuum instable; and supersymmetry has certain conceptual advantages: it leads very probably to the finiteness of the perturbation series, thereby avoiding the problem of non-renormalizability which haunted all former attempts at a quantization of gravity; and there is a close relation between supersymmetry and Poincaré invariance which seems reasonable for quantum gravity. But it is clear that not all conceptual advantages are necessarily part of nature – as the example of the elegant, but unsuccessful Grand Unified Theories demonstrates.

Apart from its ten (or eleven) dimensions and the inclusion of supersymmetry – both have more or less the character of only conceptually, but not empirically motivated ad-hoc assumptions – string theory consists of a rather careful adaptation of the mathematical and model-theoretical apparatus of perturbative quantum field theory to the quantized, one-dimensionally extended, oscillating string (and, finally, of a minimal extension of its methods into the non-perturbative regime for which the declarations of intent exceed by far the conceptual successes). Without any empirical data transcending the context of our established theories, there remains for string theory only the minimal conceptual integration of basic parts of the phenomenology already reproduced by these established theories. And a significant component of this phenomenology, namely the phenomenology of gravitation, was already used up in the selection of string theory as an interesting approach to quantum gravity. Only, because string theory – containing gravitons as string states – reproduces in a certain way the phenomenology of gravitation, it is taken seriously.

But consistency requirements, the minimal inclusion of basic phenomenological constraints, and the careful extension of the model-theoretical basis of quantum field theory are not sufficient to establish an adequate theory of quantum gravity. Shouldn’t the landscape scenario of string theory be understood as a clear indication, not only of fundamental problems with the reproduction of the gauge invariances of the standard model of quantum field theory (and the corresponding phenomenology), but of much more severe conceptual problems? Almost all attempts at a solution of the immanent and transcendental problems of string theory seem to end in the ambiguity and contingency of the multitude of scenarios of the string landscape. That no physically motivated basic principle is known for string theory and its model-theoretical procedures might be seen as a problem which possibly could be overcome in future developments. But, what about the use of a static background spacetime in string theory which falls short of the fundamental insights of general relativity and which therefore seems to be completely unacceptable for a theory of quantum gravity?

At least since the change of context (and strategy) from hadron physics to quantum gravity, the development of string theory was dominated by immanent problems which led with their attempted solutions deeper. The result of this successively increasing self- referentiality is a more and more enhanced decoupling from phenomenological boundary conditions and necessities. The contact with the empirical does not increase, but gets weaker and weaker. The result of this process is a labyrinthic mathematical structure with a completely unclear physical relevance

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Superstrings as Grand Unifier. Thought of the Day 86.0

The first step of deriving General Relativity and particle physics from a common fundamental source may lie within the quantization of the classical string action. At a given momentum, quantized strings exist only at discrete energy levels, each level containing a finite number of string states, or particle types. There are huge energy gaps between each level, which means that the directly observable particles belong to a small subset of string vibrations. In principle, a string has harmonic frequency modes ad infinitum. However, the masses of the corresponding particles get larger, and decay to lighter particles all the quicker.

Most importantly, the ground energy state of the string contains a massless, spin-two particle. There are no higher spin particles, which is fortunate since their presence would ruin the consistency of the theory. The presence of a massless spin-two particle is undesirable if string theory has the limited goal of explaining hadronic interactions. This had been the initial intention. However, attempts at a quantum field theoretic description of gravity had shown that the force-carrier of gravity, known as the graviton, had to be a massless spin-two particle. Thus, in string theory’s comeback as a potential “theory of everything,” a curse turns into a blessing.

Once again, as with the case of supersymmetry and supergravity, we have the astonishing result that quantum considerations require the existence of gravity! From this vantage point, right from the start the quantum divergences of gravity are swept away by the extended string. Rather than being mutually exclusive, as it seems at first sight, quantum physics and gravitation have a symbiotic relationship. This reinforces the idea that quantum gravity may be a mandatory step towards the unification of all forces.

Unfortunately, the ground state energy level also includes negative-mass particles, known as tachyons. Such particles have light speed as their limiting minimum speed, thus violating causality. Tachyonic particles generally suggest an instability, or possibly even an inconsistency, in a theory. Since tachyons have negative mass, an interaction involving finite input energy could result in particles of arbitrarily high energies together with arbitrarily many tachyons. There is no limit to the number of such processes, thus preventing a perturbative understanding of the theory.

An additional problem is that the string states only include bosonic particles. However, it is known that nature certainly contains fermions, such as electrons and quarks. Since supersymmetry is the invariance of a theory under the interchange of bosons and fermions, it may come as no surprise, post priori, that this is the key to resolving the second issue. As it turns out, the bosonic sector of the theory corresponds to the spacetime coordinates of a string, from the point of view of the conformal field theory living on the string worldvolume. This means that the additional fields are fermionic, so that the particle spectrum can potentially include all observable particles. In addition, the lowest energy level of a supersymmetric string is naturally massless, which eliminates the unwanted tachyons from the theory.

The inclusion of supersymmetry has some additional bonuses. Firstly, supersymmetry enforces the cancellation of zero-point energies between the bosonic and fermionic sectors. Since gravity couples to all energy, if these zero-point energies were not canceled, as in the case of non-supersymmetric particle physics, then they would have an enormous contribution to the cosmological constant. This would disagree with the observed cosmological constant being very close to zero, on the positive side, relative to the energy scales of particle physics.

Also, the weak, strong and electromagnetic couplings of the Standard Model differ by several orders of magnitude at low energies. However, at high energies, the couplings take on almost the same value, almost but not quite. It turns out that a supersymmetric extension of the Standard Model appears to render the values of the couplings identical at approximately 10¹⁶ GeV. This may be the manifestation of the fundamental unity of forces. It would appear that the “bottom-up” approach to unification is winning. That is, gravitation arises from the quantization of strings. To put it another way, supergravity is the low-energy limit of string theory, and has General Relativity as its own low-energy limit.