[Mentor's note: forked from https://www.physicsforums.com/threads/conjugate-variable-clarification.878112/] [Broken](adsbygoogle = window.adsbygoogle || []).push({});

I think the list is very interesting. From https://en.wikipedia.org/wiki/Conjugate_variables Feeble Wonk said: ↑Just for curiosity sake... could you list a few other such pairs of operators?

By the way, the same wiki article offers this answer to the OPs question, and gives a hint on how to derive HUP.

- The energy of a particle at a certain event is the negative of the derivative of the action along a trajectory of that particle ending at that event with respect to the time of the event.
- The linear momentum of a particle is the derivative of its action with respect to its position.
- The angular momentum of a particle is the derivative of its action with respect to its orientation (angular position).
- The electric potential (φ, voltage) at an event is the negative of the derivative of the action of the electromagnetic field with respect to the density of (free)electric charge at that event.
- The magnetic potential (A) at an event is the derivative of the action of the electromagnetic field with respect to the density of (free) electric current at that event.
- The electric field (E) at an event is the derivative of the action of the electromagnetic field with respect to the electric polarization density at that event.
- The magnetic induction (B) at an event is the derivative of the action of the electromagnetic field with respect to the magnetization at that event.
- The Newtonian gravitational potential at an event is the negative of the derivative of the action of the Newtonian gravitation field with respect to the mass density at that event.

wiki said:Conjugate variablesare pairs of variables mathematically defined in such a way that they become Fourier transform duals of one another, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty in physics called the Heisenberg uncertainty principle relation between them.

That leads me to my own follow-up question [no hijack intended].What is the significance of the "at an event" qualifier in several of those pairs?I ask because @Dale once said one thing while Leonard Susskind said another that sounds contradictory.

Dale said: ↑The point is that there is not any energy just floating around as pure energy by itself. Energy is always a property of something, be it a system or a particle or a field.Susskind's statement sure sounds like electromagnetic energy does just float around by itself. If we added "at an event" to Susskind's statement, could that be the way to resolve those two statements? "At an event" can only occur in the presence of systems, particles, or fields, not in empty space. my paraphrase of Susskind said:In a cosmology course discussing the vacuum, Leonard Susskind said, "E and B can not both be zero at the same time and place, that would violate HUP."

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A Conjugate variable clarification

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**