Potential has various meanings

[sirchasm]

The word potential has various meanings; it can mean "a possibility", or it can mean "a gradient".

Is there such a thing as an abstract potential? Is it "well-defined" in meaning, like an exact differential, or like the two exact values in base2?

 

[DeShark]

The word potential has various meanings; it can mean "a possibility", or it can mean "a gradient".

Is there such a thing as an abstract potential? Is it "well-defined" in meaning, like an exact differential, or like the two exact values in base2?

As far as the definition goes, my dictionary only has potential meaning a possibility or more accurately something having the inherent capacity for growth or development.

The potential from a physical point of view, the potential is a scalar field (each point in space is allocated a value). The gradient (or negative gradient) of this field is given the name "force" and indicates the amount of acceleration a unit mass would undergo via the effects of this field. If the field is time independent, then a particle, after following a closed path in the potential will end up with the same energy, independent of the path. This allows us to ascribe a set value to each point in the field. In electromagnetism, the potential at a point can be defined as the amount of work needed to move a test charge from infinity to the point. However, as only the relative values hold any meaning, we can arbitrarily define the field at one point and use that as out reference, if that would make solving the problem easier. It is an abstract concept which we concoct to explain observable phenomena, in this case acceleration. For a conservative field, the change in potential from point 1 to point 2 is given by the work done in moving from point 1 to point 2 and so the potential difference is well defined.

 

[confinement]

Yes, a given vector field in pure mathematics is said to be derivable from an (abstract) potential if there exists a scalar function (the potential) whose gradient is everywhere equal to the given vector field.

A sufficient condition for a vector field to be derivable from a potential is that the curl of the vector field vanishes everywhere.

Given a curl-free vector field the potential is unique up to a constant that is independent of space.

 

[sirchasm]

Ok thanks.

I'm trying to connect 'potential' in the physical, with information theory.
In IT, there are two dimensions (in binary logic) and neither are time or space. So is 'classical' binary as {0,1} equivalent to 'quantum' information, a Hilbert space? Are the 'abstract' potentials the same at some level.

I mean, eigenvalues are {0,1} in an orthonormal basis, or in the standard computational (physical) basis; can things be assumed at some abstraction? Classical binary is transformed into {-1,1} and back, say when the basis changes to another physical 'encoding', in communications.
QC does the same thing in a different 'quantum basis'.

Sorry if this is a little. er. abstracted.

 

[sirchasm]

Yep, the mathematical tour is probably worth taking; wikipedia explains potential theory and Laplace's eqn, which I've encountered in EE.
http://en.wikipedia.org/wiki/Potential_theory"

They appear to be saying an equation of state qualifies as a potential.
My own abstraction (which might be what they use in QIS) is that a Laplacian describes an initial state, so the goal is to derive this for the space you're looking at.

They discuss Markov chains and probability. This is interesting because I can remember a diagram of classical, probabilistic and quantum information; the first is two small spheres separated by a discontinuity; the second looks the same but has a cylindrical "probability space" between the two spheres; the last has a spherical space with a single central sphere 'inside'.
It's only a roughish sketch but it demonstrates, I suppose, the informational geometries in each space..?

 

[sirchasm]

Then, you see that, in the first case the space is disconnected, but each sphere might have another observable that 'colours' it - a 1 and a 0 and a way to determine which is whch - the domain of classical probability.
It implies a step between or a constant phase-separation - 90 Cartesian degees.

The second case is a cylindrical connection, with spheres 'stuck' in its ends. If you remove the connection you have the first case.

If you deform the cylinder into a sphere you have the third, or quantum space - the Bloch sphere, with a solid angle subtended by its surface, which are the transforms of the spheres in the first two. You can reflect the s-a and you have two - you're in electron spin space.

 

[sirchasm]

Then the reflection (of a single solid angle inside Bloch space) represents a rotation in Cartesian space and a twist in the Turing tape of electron and fermion spin.

A Mobius loop is a circle in some projection over Cartesian space.
The fiber of the space in the second case is over the spaces of the first and third, what's called a double cover. The fiber of electron precession is U(1).