Are fields spatially quantised?

Electromagnetic waves are composed of disturbances in the electric and magnetic fields, which I have heard described thus: each point in the two fields is a vector and it is not that these points move but that the vectors change in direction and magnitude and these changes are passed from one point to the next in a manner that collectively forms a wave.

With or without this example, the concept of 'points' in a field or of there being one vector value in one area and another vector value in another area suggest that the field is spatially quantised, i.e. in order for there to be discrete points in a field, the field has to be divided up into discrete amounts.

Firstly, is this correct? I understand that the use of 'points' to explain concepts is semantic and not necessarily a true representation of reality but again how are we to assign different vectors to different place in a field if it is continuous?

Secondly, if fields are spatially quantised, is this because spacetime is quantised? The electromagnetic field is itself quantised in that photons exist, so would the division of the space that field exists in mean spacetime is the thing that is divided into discrete packets?

• An analagous sitution is a function considered as a graph over the real line: to every point in the number line a function assigns a value, but this does not imply the number line is discrete. Commented Nov 30, 2019 at 18:04

No, this isn’t correct. In conventional physics, both classical and quantum, spacetime is continuous and fields have a value at every point of this continuum. For example, classical electric and magnetic fields are just vector-valued functions of $$t,x,y,z$$. In quantum electrodynamics the quantized electromagnetic field is still defined on continuous spacetime. The same applies to all seventeen quantum fields in the Standard Model.