Chapter 4 – Forces

Gauge Invariance

The Standard Model gives the particles that mediate forces but where do the rules that govern the forces come from? Gauge invariance provides the rules but how it does it involves heavy mathematics that is out of scope for this presentation. Instead of getting deep into those details, we will give an example to show intuitively that it is possible but will skip the complexity.

What is a gauge

A gauge is a measurement scale. Deciding to measure weight in pounds or kilograms is selecting a gauge. Changing gauge alters values but not the underlying property. Even though the value changes when you switch between pounds and kilograms, the weight of the thing being measured remains the same.

Another example is measuring a direction. Some might measure it against true north, some against magnetic north and some against the grids for the local streets. For people to work together, they need to know how to convert between their own versions of the gauge. This is a gauge field.

The amazing fact is that requiring certain properties to be invariant produces the physics of the electromagnetic, strong and weak forces. How can keeping something constant produce rules for forces?

Gauge invariance example

Each country measures currency differently but exchange rates show how to convert between currencies – this is like a gauge field.

Gauge invariance looks at values that are expected to be the same everywhere – for example, the value of an ounce of gold should not matter where it is or what currency is used.

Take the price of an ounce of gold in dollars, convert to yen, then pounds then euros then back to dollars – it should be the value we started with.

If the loop ends with a different value, there is an inconsistency that gives someone the chance to make money

That inconsistency creates market drivers that will tend to force the movement of currency. This is similar to a force.

Rules can be developed to calculate the force pushing currency in a direction based on the change in value when an ounce of gold is moved around a loop.

Gauge invariance in Quantum Mechanics

In the same way a force on currency flow drops out of looking at invariants across the currency gauge field …

Start with the Dirac equation for wave functions. Then use an energy invariant to the electromagnetic field and evaluate how it changes under symmetries of quantum fields to get all of Maxwell’s rules for electrodynamics!

Gauge invariance similarly gives the rules for the strong and weak forces