*A short answer of mine.*

First *whats potential*, before *whats gauge potential*.

A potential is the variable: **energy**. ( — Energy in its various forms eg *per* unit charge or unit mass, known as; electric potential vs gravitational potential ).

*All energy are integration of vector-field quantity*, so

*Force is the vector field quantity whose integration is energy*.

We say; *Force is the potential gradient of energy or potential*. ( — and this potential gradient which is a *derivative* of a higher variable, it could be both *negative* and *positive*)

Then we say also, in Physics, *Force per unit mass or unit charge*? Like we did that for energy. So this time you say electric field vector or gravitational field vector. Like Force was *positive* or *negative* gradient (of energy), Field (a force or vector per unit charge or mass) is *negative* or *positive* potential gradient.

Potential was, as said above, *energy per unit mass or charge*. So, we see that, in defining the higher quantity *energy* or *potential* (higher therefore closer to action, hence more fundamental or unified) we have to INTEGRATE the *lower variable*, here, Force, Field or (Any ) 3-vector. This entails therefore arbitrariness into the Physical solution when we solve for these quantities. These physical problems, as they involve differentials or integration, leads to a differential equation. Under further suitable physical conditions called eg laws of nature or physics, become whats called a wave-equation or for *particles*, *equation of motion*. We can say equation of motion for particles or equation of motion for waves if they are separate.

Now that we understand *what are potential, field, vector and gradient and integral in relation to each other,* comes requirements called as *symmetry* or laws of nature or laws of physics or in simple, boundary conditions *to these* differential equations known as, wave equation or equation of motion of particles OR waves. ( — which are separate so far )

These equations constrained by the conditions or restrictions which are *attributes of physical observation*, must therefore unite these variables (*potential, field*) into one entity which would *satisfy* the* wave or particle* equations of motion, the differential equations of motion in PARTICULAR ways only, known as Laws of Nature or Physics. So they become, from their **3**–*vector* or *scalar* attributes, **4**–*vectors* (or still higher, *Tensors*).

One word of concept, the *potential* is also a *field* but a *scalar*. So instead of (*potential*, *field*) we can write (*scalar*, **3**–*vector*) components of the **field**, that is, (**scalar** *potential*, **vector** *potential*) which as you can see in a 4-dimensional world; 1+3 = 4 components.

Then comes the *conditions* or inter-relations between these vector and scalar and their differential and integrals. ( — again known as a **theory** or *equation of motion*)

These *conditions* have an *arbitrariness* built into them,** puchho kyon? ***Munna bhai* says: *because there are integration*. How did Munna bhai figure that out?

He was in a bar dancing with a lady Physicist from Munich, who had come to Mumbai for a conference in Particle Physics.

The method-of-integration has a constant of integration, any one?

In going from *lower variables* such as **velocity** and **momentum** we gradually integrate them to find the higher, closer-to-action or fundamental variables, and incurr *arbitrariness* into our understanding.

But not all of these solutions are *physical* in nature, they are merely *invalid* mathematical solutions even if *not necessarily* trivial solutions. To remove these arbitrary ness is whats called **Gauge Condition**. That is, to chose the *right* scalar and vector *potential* among all solutions, only that, which would be Physically *correct*.

Therefore a *gauge* or *gauge* *potential* ( — or a *gauge field scalar* or *gauge field vector*) are what’s *more* fundamental and closer to a **theory**. *Theory* is a jargon among Theoretical Messiahs for the longer version “equation of motion“. When *two kinds* of equation of motion, one for the **wave** and one for the **particle** are united into one, a feat called as **unification **we obtain equations whose specific example is **Schrodinger Equation** also called a **wave equation of motion**, with the implicit understanding that, such are *conditions for particles and waves* both, rather than *just one, either of them*.

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