Potential has various meanings

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In summary, they discuss potential in terms of IT and classical binary, potential as an abstract concept, the potential difference between two points, the potential as a scalar field, the potential as a gradient, the potential as a force, the potential as a work required to move a charge, potential as an equation of state, potential as an initial state, potential as a connectable component, and the potential as a Mobius loop.
  • #1
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?
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  • #2

sirchasm said:
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.
  • #3

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.
  • #4

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.
  • #5

Yep, the mathematical tour is probably worth taking; wikipedia explains potential theory and Laplace's eqn, which I've encountered in EE.

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..?
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  • #6

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.
  • #7

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).

1. What is the definition of potential?

Potential can refer to a variety of concepts in science, but generally it is the possibility for something to occur or develop in the future. It can also refer to the energy or force that an object or system has due to its position, condition, or composition.

2. How is potential energy different from kinetic energy?

Potential energy is the energy that an object or system possesses due to its position or configuration, while kinetic energy is the energy that it possesses due to its motion. In other words, potential energy is stored energy, while kinetic energy is energy in motion.

3. What are some examples of potential energy?

Some common examples of potential energy include a stretched rubber band, a compressed spring, water at the top of a waterfall, and a book sitting on a shelf. In each of these cases, the object has the potential to release energy and do work when the potential energy is converted into kinetic energy.

4. How is gravitational potential energy calculated?

Gravitational potential energy is calculated by multiplying the mass of an object by the acceleration due to gravity (9.8 m/s2) and the height of the object above a reference point. The formula is PE = mgh, where PE is potential energy, m is mass, g is the acceleration due to gravity, and h is the height.

5. How is potential difference related to electric potential?

Potential difference, also known as voltage, is a measure of the difference in electric potential between two points. Electric potential, on the other hand, is a measure of the electric potential energy per unit charge at a point in an electric field. In other words, potential difference is the change in electric potential between two points, while electric potential is the potential energy per unit charge at a single point.

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