The paper-cutting experiment above illustrated Einstein's General Theory of Relativity. It showed that the motion of an object is determined by the structure of space-time: curved space gives curved orbits. The relationship between matter and space-time is mutual. On the one hand, the orbit of an object is due to the structure of space-time. On the other hand, that structure is determined by the arrangement of the massenergy- momentum of matter. We also saw that the state of motion of quantum particles is determined by the coupling at a Feynman vertex.

This raises the question: what determines this coupling? Or, in the language of our large-scale world: what determines the properties of forces?

The observation of reflection-and-transmission of light in a windowpane shows that quantum particles behave in ways that are radically different from the motions of big objects. Orbits in classical mechanics are fixed when all initial positions and velocities of the particles are prescribed together with a recipe for the accelerations. In quantum mechanics, we must determine the transition probability that connects a given initial state to a specified final state. These probabilities are computed by means of the Feynman diagrams introduced above.

The key to interaction in a Feynman diagram is the vertex, the point in space-time (called an event in relativity theory) at which particles are coupled. The interactions at a vertex are determined by symmetries. Before discussing how this works, let us look a little more closely at what is meant by ‘symmetry’.

A symmetry may be loosely described as ‘a change that leaves something unchanged’. Rotation is an example of a symmetry. A sphere is unchanged when it is rotated about its centre. Mathematicians say that a sphere is invariant under rotations. You might say that a sphere is in fact produced by a symmetry, because a sphere is the set of all points that have the same distance to a given point, and that ‘same distance’ is the ‘something that does not change’.

Galileo thought that the rotational symmetry of a circle implies that the ‘natural motion’ of planetary orbits is composed of circles, but Huygens proved that circular motion requires an acceleration. As we saw above, Huygens formulated a very different symmetry: the universe remains unchanged if everything in it were displaced over a fixed distance, and also if a fixed velocity were added to the velocities of all objects.