Abstract
In Sect. 2.4, I focused on the importance of solitons. The importance of course depends on what kind of field theory and solitons we consider. Solitons appear in various field theories, and an index characterizing those various solitons is the dimension of the solitons. The kinks, vortices, and monopoles having appeared in the preceding chapter are classified by their dimensions.
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Notes
- 1.
Though in the 1990s cosmic strings were considered to play an important role in density fluctuation structure formation of the universe, recently it is being denied by a precise observation of cosmological background radiation at 3 Kelvin. However, it is still possible that the cosmic string exists in the universe.
- 2.
This idea was proposed by K. Akama (1982), V. A. Rubakov and M. E. Shaposhnikov (1983).
- 3.
The spacetime dimensions are restricted in order to renormalize infinities in calculations of Feynman diagrams, but the constraint is not so strict as string theory. For example, even in 4-dimensional spacetime, one can write infinite kinds of field theories for particles, but we have limited kinds of string theories.
- 4.
Lorentz symmetry is a symmetry required by the special relativity. It is a generalization of a rotational symmetry of space to include also the time direction. Even for theories of particles, it is definitely required when we consider relativistic particles. The fact that the requirement of the Lorentz symmetry determine the dimensions of the spacetime in string theory is a result of a quantum mechanical treatment of string theory, and here I will not describe the mechanism.
- 5.
Strings oscillate in spacetime. In the case of the closed string theory, the oscillations propagating on the closed sting are either right-moving or left-moving. Now, there is a hetero-type string theory in which the left-movers are bosonic while the right-movers are of a superstring. This is called a heterotic superstring theory. Since it is useful to construct grand unified theories of elementary particles from the string theory, the heterotic string theory is enthusiastically studied as a candidate of the unified theory.
- 6.
The physics of each basic standing wave is equivalent to the quantum mechanics of a harmonic oscillator. For the harmonic oscillator, a quantum state is given by multiplying operators creating waves, on a vacuum state. In the preset case, since the number of species of the standing waves is as many as that of nodes, which is infinity, a string is the same as a system with infinite kinds of harmonic oscillators. Let us write the creation operator of the waves with the node number n as \({\alpha }_{-n}^{\mu }\). This creation / annihilation operator is given precisely as
$${X}^{\mu } = {x}^{\mu } + {l}_{\mathrm{s}}^{2}{k}^{\mu }\tau + i{l}_{\mathrm{s}}\!\!\!\!\!\!{ \sum \nolimits }_{n\in \mathbf{Z},\neq 0} \frac{1} {n}{\alpha }_{n}^{\mu }{e}^{-in\tau }\cos n\sigma,$$(3.5)where \({\alpha }_{n}^{\mu }\) is a Fourier coefficients of the solution of the equation of motion of the bosonic field \({X}^{\mu }\) on the worldsheet, \({\partial }_{\alpha }{\partial }^{\alpha }{X}^{\mu }(\tau,\sigma ) = 0\) with the free boundary condition. The generators have the index of the spacetime μ, because the position operator \({X}^{\mu }\) has the index of each coordinate of the spacetime, as in the quantum mechanics. Then, a general state of the string is written as
$$\vert 0; {k}^{\mu }\rangle,\quad {\alpha }_{ -1}^{\mu }\vert 0; {k}^{\mu }\rangle,\quad \cdots \,,{\alpha }_{ -1}^{\mu }{\alpha }_{ -1}^{\nu }{\alpha }_{ -3}^{\rho }\vert 0; {k}^{\mu }\rangle,\quad \cdots $$(3.6)Here \({k}^{\mu }\) is the center-of-mass momentum of the whole string. Although some creation operators α − n μ are there, those degrees of freedom can be seen as internal degrees of freedom, since they are all on the waves on the string. Then, one can regard this state (3.6) as a particle carrying the momentum \({k}^{\mu }\). In this way, string theory can accommodate infinite kinds of particles at once, by a single string.
- 7.
Quantum mechanically, N n counts the number of the creation operators α − n for each node number n contained in a state (3.6) of the string, and − 1 originates from the zero-point energy in quantum mechanics.
- 8.
In 2008, a particle accelerator called Large Hadron Collider (LHC) started to operate, and in 2009 it reached the energy which human being has never reached in history, and will reach higher energy in the near future.
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Hashimoto, K. (2012). Dimensions of Solitons, Dimensions of String Theory. In: D-Brane. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23574-0_3
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DOI: https://doi.org/10.1007/978-3-642-23574-0_3
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