This is a continuation of the previous post: Problems in Objectizing.
Wheeler's Second Question: "How come the quantum?”
Let me consider the second question, which seems to me to be less abstract. Indeed, we do have “the quantum” It is embodied in the Planck constant
It has the physical dimension of a “Joule” multiplied by a “second”, energy multiplied by time – physicists call it “action”. This constant seems to be numerically rather small in our accepted human-size units, but it enters into almost every formula of quantum theory.
Planck’s constant, by itself, indeed defines a “quantum”, but not a quantum of energy, as it sometimes written in popular books, but a quantum of action. Energy is not always quantized, there is no such thing as “quantum of energy”, except in some particular cases1, but there is a “quantum of action”. Therefore we should pose another question: “How come action?” Evidently it is a very important quantity, but not directly observed or measured. Why?
Energy and Momentum: Action
The physical dimension of action is that of energy multiplied by time, which happens to be the same as momentum multiplied by length. We all have a good general feeling or impression about what the time distance between, any two events is, and what the length distance between two points is. Energy and momentum seem to be more complicated and not so evident, and yet they happen to be fundamental in mechanics. Do they not deserve to have their own dimensions associated with them? Indeed they do, and they have.
Phase-space and Planck’s Constant
In mechanics, for quite some time now, we have been using not just space and time, and not just space-time, but also what is called “phase-space”. If space-time has four dimensions, three for space and one for time, then phase-space has these dimensions doubled. There are four dimensions added: three for momentum, and one for energy. Thus in relativistic mechanics, phase-space has eight dimensions.2
Things are somewhat more complicated when we are dealing with a system of particles rather than just one particle – we need a separate phase-space for every particle, but that can be thought of as a purely formal mathematical trick – we do it, because it makes calculation simple, and because “it works”.
Now, once we have phase-space, eight or six dimensional, depending on whether we respect the special theory of relativity or not, we can measure the volume of a region in this space. Then the Planck constant tells us that there is something like a “fundamental volume” – this phase-space has a kind of granular structure. That is to say, there is a discreteness at the fundamental level, though it has nothing to do with space or with time alone, but rather with a more complicated space that we are not immediately aware of – the phase-space. Energy and momentum, the additional coordinates of the phase-space have, intuitively, to do with frequencies and wave-lengths, not just time and position.
Are we somehow sensitive to frequencies and wavelengths, not just to positioning of objects in space and time? I think that in order to answer these questions the theory should be developed a little bit further. So, let me continue with my speculations about a possible shape of future physics.
To be continued...
1 An ideal harmonic oscillation, with a fixed frequency. is one such example.
2 U.S. Army seems to be interested in extending Einstein’s general theory of relativity to phase space: Howard E. Brandt, U.S. Army Research Laboratory, “Finslerian Quantum Field Theory”,
P.S.1. To my wife. She likes it.
P.S.2.
"If mankind could but see though in a glimpse of fleeting experience what infinite enjoyments, what perfect forces, what luminous reaches of spontaneous knowledge, what wide calms of our being lie waiting for us in the tracts which our animal evolution has not yet conquered, they would leave all & never rest till they had gained these treasures. But the way is narrow, the doors are hard to force, and fear, distrust & scepticism are there, sentinels of Nature, to forbid the turning away of our feet from her ordinary pastures."
Sri Aurobindo
P.S.3. The video below was taken 10 days ago in our yard.
P.S.4. I wish I would have such muscles:
