Conjugate variable clarification

[anorlunda]

[Mentor's note: forked from https://www.physicsforums.com/threads/conjugate-variable-clarification.878112/]

Just for curiosity sake... could you list a few other such pairs of operators?

I think the list is very interesting. From https://en.wikipedia.org/wiki/Conjugate_variables

1. 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.
2. The linear momentum of a particle is the derivative of its action with respect to its position.
3. The angular momentum of a particle is the derivative of its action with respect to its orientation (angular position).
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.

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

Conjugate variables are 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.

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.

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."

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.

 

[anorlunda]

I just realized that adding "At the event" to Susskind's statement about the vacuum would negate it being a vacuum. The rest of my question still stands; what is the significance of "at the event."

 

[Jilang]

Doesn't it just mean a point in space at a certain time?

 

[anorlunda]

Doesn't it just mean a point in space at a certain time?

I don't think so. An "event" implies change of quantum state of a particle, or interaction between particles. You can't have an event in empty space and time.

 

[Truecrimson]

The Wikipedia page the word "event" links to specifically says that it is a point in spacetime.

Susskind's statement sure sounds like electromagnetic energy does just float around by itself.

I don't see how the statement that an electric and magnetic fields can't be zero at the same spacetime point implies that electromagnetic energy can exists without these fields.

 

[anorlunda]

I don't see how the statement that an electric and magnetic fields can't be zero at the same spacetime point implies that electromagnetic energy can exists without these fields.

Not without the fields but it implies that there can not be a spacetime point in this universe with zero energy; that energy is everywhere.

 

[anorlunda]

The Wikipedia page the word "event" links to specifically says that it is a point in spacetime.

It does indeed say that ,but the complete sentence reads "In physics, and in particular relativity, an event is a point in spacetime (that is, a
specific place and time) and the physical situation or occurrence associated with it.". That definition leaves room to say it is a point with or without any occurrence.

But Merriam Webster also gives an alternate definition of event as a point without an occurrence.

So if event and point are synonyms, you are correct that at the event has no significance other than we don't compare conjugate variable values from different points. I guess my question is answered. Thank you.

 

[A. Neumaier]

Not without the fields but it implies that there can not be a spacetime point in this universe with zero energy; that energy is everywhere.

The electromagnetic and the gravitationl field are everywhere - since they are fields generated even by far away matter. This explains the way of talking.

if event and point are synonyms

There is the general meaning of event given in a standard disctionary. in addition, there is the specific meaning of event = space-time point used in relativity. One has to use the context to decide which meaning is intended. In your context clearly the second.