In trying to appreciate how the quarter-wave plate transforms the linearly polarized light, it is important to realize that the two components discussed are not entities in and of themselves but are merely mental constructs one uses to help appreciate what is happening. In the case of linearly and circularly polarized light, at each point in space, there is always a single electric field with a distinct vector direction, the quarter-wave plate merely has the effect of transforming this single electric field.