NJV said:
What, physically, does a pilot wave in itself consist of?
It's elephants all the way down! ;)
You always run into this dilemma when looking for/at foundational substance. To "explain" electromagnetism you can propose an aether as a medium because we are used to thinking of waves as fluctuations of the medium. One must then ask about the components of this medium and mediators of its interaction and you are back where you started except you've moved beyond observable phenomena.
If one starts with quantum mechanics and one describes the behavior of quanta one will invoke a Hilbert space with "vectors" represented by wave-functions. At this level these wave-functions are not real but just mathematical constructs (like a force vector or real number line). The direct operational interpretation of the wave-function is a function yielding probability amplitudes for the observation of the quantum at each spatial point at a given time. It is thus effectively a probability distribution but encoding more information (the way a velocity vector encodes more information than just its magnitude=speed).
Understand this first and then consider the question of (ontological) interpretation of quantum theory. In trying to describe an objective reality underlying the phenomena we see in elementary quantum experiments and describe with these wave-functions you have some interesting historic proposals. Everett's MWI and Bohm's Pilot Wave being two such. The Bohmian's assert that the wave-function itself corresponds to some existent wave phenomenon guiding the classical^* particle and thus "explaining" quantum interference phenomena.
(* classical in the sense of having always an objective state of reality with position and momentum well defined but not known to the observer.)
As far as what underlies this wave the supposition stops there treating both pilot wave and elementary particles as foundational objects out of which other things are made.
For myself I am a Copenhagenist and deny the need to reify the wave-function. I see pilot-waves as analogous to the luminiferous aether invented by Newton.
Actually I just looked up Newton's aether on the web and it would seem he invented pilot waves before Bohm to explain the same phenomena of interference, diffraction and refraction along with his particulate light theory. I didn't realize the question went back that far!
So I guess the best answer to your question is...
Pilot waves are undulations of the aether.
