Stumbling Blocks Against Unification

The following sections are included:

• Dedication
• Preface
• Acknowledgement
• Contents

Dissemination of a truth, novel insight, or discovery is not always straight-forward or easy. Just the contrary. In this respect, human society does not behave much better than the groups of monkeys in numerous experiments. For instance, the experimenters trained a low-rank-status monkey in isolation in how to obtain food from a box or similar equipment. When later observed by other monkeys, they very likely did not imitate this individual. He could be retrieving bananas from the automaton in front of the eyes of other monkeys, but no one took notice and imitated him. However, when the experimenters taught a high-rank-status monkey to retrieve food from the machine, the other monkeys started to imitate him and obtain bananas. Many other experiments show that chimpanzees, when given the opportunity to watch alternative solutions to a foraging problem, preferably copy the method shown by the older, higher-ranking individual with a prior trackrecord of success [2]. Records in the wild showed that majority of chimpanzee innovations are performed by low-ranking individuals, and consequently, their spreading within the community is impeded, because not observed and imitated by others. Such studies suggest that most chimpanzee innovations spread with difficulty, if ever…

A great stumbling block in current theoretical physics is the concept of negative energies. When they first occurred in relativistic mechanics and subsequently in the Dirac equation, there was initially much puzzlement among physicists. Problematic was the issue of stability of such systems. So Fermi [8] wrote…

In the previous chapter we reviewed various approaches to the problem of negative energies that occur in higher derivative theories, a toy model for them being the Pais-Uhlenbeck (PU) oscillator. We have noted the well-known fact that negative energies themselves are not problematic, if there are no interactions between positive and negative energy degrees of freedom. Problems are expected to arise in the presence of interactions. Therefore, the approaches in which the authors show how the description of the PU oscillator in terms of an indefinite Hamiltonian can be replaced with a positive definite Hamiltonian, do not solve the problem [15–19]. That is, such a reformulation of a free PU oscillator does not work for an interacting PU oscillator. For instance, if one attempts to reformulate a self-interacting PU oscillator in terms of a positive definite Hamiltonian, the Lagrangian acquires additional nonlinear terms that are not present in the original Lagrangian [12]. Therefore, an interacting PU oscillator, as a higher derivative system, has to be described in terms of the Ostrogradski or an equivalent formalism that contains negative energies. In Refs. [12,25–28] it was shown that if the potential is unbounded either from below or from above, then such systems are not necessarily unstable; they can be either stable or unstable, depending on the values of the coupling constant and the initial velocity. However, upon quantization such system would become unstable because of the tunnelling effect…

According to the generally accepted view spinors are objects that transform under rotations differently than vectors. However, it is also known by not so small group of experts that spinors can be considered as particular elements of Clifford algebras. If so, they must have the same transformation properties as any other Clifford number. This implies that spinor and vectors can both be transformed by the same set of transformations. In general, a Clifford number Φ transforms into a new Clifford number Φ′ according to Φ → Φ′ = R Φ S, i.e., by the multiplication from the left and from the right by two Clifford numbers R and S. We study the case of Cl(1, 3), which is the Clifford algebra of the Minkowski spacetime. Depending on choice of R and S, there are various possibilities, including the transformations of vectors into 3-vectors, and the transformations of the spinors of one minimal left ideal of Cl(1, 3) into another minimal left ideal. This, among others, has implications for understanding the observed non-conservation of parity.

It is shown that the generators of Clifford algebras behave as creation and annihilation operators for fermions and bosons. They can create extended objects, such as strings and branes, and can induce curved metric of our spacetime. At a fixed point, we consider the Clifford algebra Cl(8) of the 8-dimensional phase space, and show that one quarter of the basis elements of Cl(8) can represent all known particles of the first generation of the Standard Model, whereas the other three quarters are invisible to us and can thus correspond to dark matter. This chapter is a shortened and modified version of Ref. [91].

In this chapter we will see how a Dirac-Nambu-Goto brane can be described as a point particle in an infinite dimensional space with a particular metric. This can be considered as a special case of a general theory in which branes are points in the brane space ℳ, whose metric is dynamical, just like in general relativity. Such a brane theory, amongst others, includes the flat brane space, whose metric is the infinite dimensional analog of the Minkowski space metric η_μν. A brane living in the latter space will be called “flat brane”; it is like a bunch of non interacting point particles. Quantization of the latter system leads to a system of non interacting quantum fields. Interactions can be included if we consider a non trivial metric in the space of fields. Then the effective classical brane is no longer a flat brane. For a particular choice of the metric in the field space we obtain the Dirac-Nambu-Goto brane. We also show how a Stueckelberg-like quantum field arises within the brane space formalism. With the Stueckelberg fields, we avoid certain well-known intricacies, especially those related to the position operator that is needed in our construction of effective classical branes from the systems of quantum fields.

In the previous two chapters we have seen that branes can be created by the action of field operators on a vacuum. In such a way one can, for instance, create a 3-brane sweeping a 4-dimensional surface that, according to the braneworld scenario, represents our spacetime. We have also seen that matter in such setup arises as holes within the brane. Successful quantization of a generic brane would be a major step towards quantum gravity. Unfortunately, this has turned out to be a very tough problem that has not yet been completely resolved so far. But, as we have seen, by using the concept of brane created by an infinite set of field operators, considered as creation operators that create particles at certain positions, quantizing a brane reduces to the problem of interacting quantum field theory. But according to the usual understanding, operators that create particles at definite positions make no sense in relativistic quantum field theories. We will now show how in Refs. [166, 167] the issue concerning position in QFT has been carefully examined and shown that the conventional arguments have to be revised.

We have seen how within the framework of relativistic quantum field theory the wave packet profiles of single particle states represented in position space make perfect sense, in spite of the prevailing opinion that such states lead to inconsistencies, and thus make no sense. Besides the confusion regarding the Lorentz invariance and covariance of such states, the main argument against relativistic localized states was that they violate causality, because they leak outside the light cone. Our calculations show that such leakage is very small, and cannot be used for superluminal transmission of information, and therefore no causality violation in the sense of the grandfather paradox can take place. In this chapter we will go a step further and consider the case in which superluminal signals are indeed theoretically possible by means of tachyons, the particles that travel faster than light. We are going to report about the findings of Ref. [109].

A big stumbling block in our attempts to obtain consistent quantized versions of classical theories is the problem of ordering of operators. In simple cases, for instance in flat spaces, the order of quantum operators does not matter. However, if a space is curved, then the momentum operator acts on the position dependent metric, and there occurs the ambiguity concerning the order of operators. Such ambiguity typically occurs in the expressions for a Hamilton operator in curved space. In this chapter we discuss and further develop the finding of Ref. [227].

We started with the social behavior in chimpanzee’s communities in which, as experiments show, inventions are mostly achieved by the individuals of low-rank status and thus usually do not spread within the community. Such studies aim at a better understanding of social behavior in humans by studying social interactions in the less-involved chimpanzee’s communities. We see that as far as inventions or new discoveries are concerned, people are often not much better than monkeys, and sometimes, people are worse. Typically, a new out-of-the-box discovery is ignored, ridiculed or strongly opposed by the rest of the community. In the mildest case, such a discovery is ignored, and this happens if the discoverer is young and unknown; otherwise, the person usually faces ridicule and violent opposition…

The following sections are included:

• Bibliography
• Index