We study the classical equations of motion for scalar fields, Maxwell fields, and gravitational fields. We use the light-cone gauge to find plane-wave solutions to their equations of motion and the number of degrees of freedom that characterize them. We explain how the quantization of such classical field configurations gives rise to particle states – scalar particles, photons, and gravitons. In doing so we prepare the ground for the later identification of such states among the quantum states of relativistic strings.
Introduction
In our investigation of classical string motion we had a great deal of freedom in choosing the coordinates on the world-sheet. This freedom was a direct consequence of the reparameterization invariance of the action, and we exploited it to simplify the equations of motion tremendously. Reparameterization invariance is an example of a gauge invariance, and a choice of parameterization is an example of a choice of gauge. We saw that the light-cone gauge – a particular parameterization in which τ is related to the light-cone time X+ and σ is chosen so that the p+-density is constant – was useful to obtain a complete and explicit solution of the equations of motion.
Classical field theories sometimes have gauge invariances. Classical electrodynamics, for example, is described in terms of gauge potentials Aµ. The gauge invariance of this description is often used to great advantage. The classical theory of a scalar field is simpler than classical electromagnetism.
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