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Yasuhiro Abe, Ph.D.

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Holonomy formalism in twistor space

Background   Looking back developments of theoretical physics in the last century, we find that the most important discovery has been that of the gauge principle. The most fundamental gauge theory is a theory of electromagnetic fields. As is well-known, this theory is invariant under local phase transformations. Namely, a Lagrangian which describes electromagnetic fields has a local symmetry under phase transformations. This symmetry is called gauge symmetry, which means that a coupling between an electromagnetic field and a charged particle is expressed by a principle of "minimal coupling." In effect, this principle corresponds to replacement of an ordinary derivative with a "covariant" derivative at the level of Lagrangian. Thus, the gauge principle naturally leads to incorporation of electromagnetic interactions into the Lagrangian. This also guarantees universality of the interaction, i.e., we find that the values of coupling constants (or elementary charges) are identical due to the gauge principle.

One can extend the electromagnetic gauge symmetry to non-abelian symmetries in defining a gauge theory. Such a theory is called Yang-Mills theory. A simple example of the Yang-Mills theories is given by a theory of weak interactions. Weak interactions appear in many physical phenomena; β-decay of neutrons is one of them. This refers to a physical process where a single neutron decays into a proton, an electron and an electric anti-neutrino. In order to explain such phenomena, it is not sufficient to resort to traditional frameworks, such as classical theories or statistical mechanics, where the total number of particles is preserved. Further, relativistic effects, i.e., conversions between mass and energy, require a framework that describes creation and annihilation of particles. It is quantum field theory that satisfies these requirements. In general, gauge theories are described by the quantum field theory.

Inside an atomic nucleus, there are protons besides neutrons. Despite repulsive electric forces among protons, the nucleons (neutrons and protons) are condensed in the nucleus of order 1×10-15m. This is due to nuclear forces, or strong interactions, which become dominant in such a short-distance range. The strong interactions are also described by the Yang-Mills theory. Therefore, three interactions (electromagnetic, weak and strong ones) in nature can be understood in a unified fashion by the Yang-Mills theory. At present, the so-called standard model of particle physics refers to this unified theoretical framework. In the study of particle physics, one tries to explain physical phenomena in terms of particles which represent both matter and interaction. As is well-known, electromagnetic interactions are described by photons. Since photons couple to a four-dimensional current, they are vector particles. Similarly, particles which represent weak or strong interactions are also vector particles. What had been confirmed towards the solid foundations of standard model is that the only consistent theory for interacting vector particles is the gauge theory. Here the consistency means the unitarity of the theory, i.e., the conservation of probabilities in a quantum theory.

The remaining interaction in nature, namely gravity, can be explained by the gauge theory as well. Einstein's general theory of relativity is derived from an "equivalent principle" which means that the theory is invariant under general coordinate transformations (or diffeomorphism). The general coordinate transformations correspond to Poincaré transformations that are a combination of spacetime translations and rotations. (The spacetime rotations are called Lorentz transformations.) Therefore, if we choose a gauge symmetry as a symmetry under Poincaré transformations, we can interpret a theory of gravity as a gauge theory. In other words, we can "in principle" understand all the interactions in nature by use of the gauge principle. Of course, there are cases where the gauge principle is not applicable. For example, we can not apply this principle to particles with interior structures, such as dipoles. In general, for those cases that the principle of "minimal coupling" is obviously inapplicable to, e.g., the cases of composite particles or many-body condensed systems, we can not use the gauge principle. For those particles that do not have interior structures, however, we can always use the principle of "minimal coupling" or the gauge principle.

As mentioned above, all interactions in nature can "in principle" be understood by the gauge principle. However, there are in fact two known problems in this interpretation. They are as follows.
  1. The standard model of particle physics consists of Yang-Mills theory and Higgs mechanism. In physics, there is an important theorem developed by Nambu and Goldstone in regard to spontaneous breaking of global symmetries. The Higgs mechanism arises from an application of the concept of spontaneous symmetry breaking to local gauge symmetries. In the standard model, masses are created by the Higgs mechanism. Hence, this is an indispensable element of the model which explains the origin of mass. But in this mechanism the gauge symmetry is broken, so is the gauge principle. In a language of Lagrangian, this corresponds to the fact that a mass term of vector particles (or gauge fields) can not be represented as a gauge invariant form.

  2. Construction of a gravitational theory in the framework of quantum field theory leads to a problem of divergence due to infinite degrees of freedom in the theory. As mentioned earlier, quantum field theory has infinite degrees of freedom in terms of creation and annihilation of particles. Thus, in order to obtain physical quantities from the theory, one needs some sort of regularization. Only when can we eliminate divergence in the calculations of physical quantities by regularization, quantum field theory makes sense as a (perturbative) theory of physics. Such a theory is called renormalizable. Standard model is one of the renormalizable theories. However, it is known that a similar regularization scheme does not work for a gravitational theory. In other words, if we describe gravity as a gauge theory in terms of perturbative quantum field theory, the theory is known to be non-renormalizable.
An influential candidate for a solution to the second problem is string theory. This is a theory that aims for a unified description of every matter and interaction (or everything) in terms of quantization of a string. It is known that this theory avoids non-renormalizability of gravity. There is, however, a stringent requirement that the dimension of spacetime is at least ten in order for the theory to be consistent. Owing to this, it seems extremely difficult to extract four-dimensional physics, e.g., the standard model, out of string theory. Further, string theory does not particularly give a solution to the first problem regarding the Higgs mechanism.

In the framework of quantum field theory, we can consider string theory as a conformal field theory in a two-dimensional space (or a worldsheet) where strings are moving. Conformal field theory is closely related to a three-dimensional gauge theory which is called Chern-Simons theory. The Chern-Simons theory is a theory that does not depend on spacetime metrics, and such a theory is generally called topological field theory. In 1989, before rapid progress in string theory, Witten has shown that the topological field theory has deep connections with the knot theory and the braid theory in mathematics. In order to tackle the above two problems, it is important to understand these connections between physics and mathematics.

Formalism   By use of an idea of twistor space developed by Penrose, one can obtain four-dimensional physical models from two-dimensional conformal field theory. In the context of scattering amplitudes in Yang-Mills theory, this point was first indicated by Nair. More explicitly, Nair has shown that a certain type of scattering amplitudes can be understood as a current correlator of a Wess-Zumino-Witten (WZW) model defined in an extended twistor space. A notion of holonomy operators for gauge fields in twistor space arises from an attempt to understand Nair's interpretation in a more universal point of view. For details of its formalism, see arXiv:0906.2524. For relations of holonomy operators to a gravitational theory, see arXiv:0906.2526.

Interests   Application of abelian holonomy formalism to elementary theory of numbers (see arXiv:1005.4299)
Predictions of physical quantities by use of holonomy formalism (under investigation)
Relation between holonomy formalism and physics on fuzzy S4 (plan)


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