What is a Gauge? Illustrative Explanation

  • Thread starter Winzer
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In summary, gauge transformations make calculations easier by allowing you to express potentials in simpler terms. They are most commonly used in electromagnetics, but can be applied to any potential.
  • #1
Winzer
598
0
I looked at the wiki article, and it's not clicking.
If someone could describe it through an illustration that would
be best.
 
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  • #2
Making an arbitrary (but valid) change in potentials is making a gauge transformation.

For example, in E&M, you have the scalar potential and vector potential that give you the electric and magnetic fields.

You can add the gradient of an arbitrary scalar to the vector potential if you make a corresponding change to the scalar potential, and the changes will not affect E or B at all.

The purpose is to make calculations easier.

Thats how I understand it...
 
  • #3
Hi there,

I thought of looking around the web to see what could be this "guage", for which I found many different meaning. I guess this the most stupid question to ask, but could you precise what type of "gauge" you are talking about.

By the way, I suppose that you mean "gAUge" and not "gUAge".

Cheers
 
  • #4
fatra2 said:
Hi there,

I thought of looking around the web to see what could be this "guage", for which I found many different meaning. I guess this the most stupid question to ask, but could you precise what type of "gauge" you are talking about.

By the way, I suppose that you mean "gAUge" and not "gUAge".

Cheers
Your absolutely correct, I should have been more specific.
I am talking about gauge transformations.
Nick R said:
Making an arbitrary (but valid) change in potentials is making a gauge transformation.

For example, in E&M, you have the scalar potential and vector potential that give you the electric and magnetic fields.

You can add the gradient of an arbitrary scalar to the vector potential if you make a corresponding change to the scalar potential, and the changes will not affect E or B at all.

The purpose is to make calculations easier.

Thats how I understand it...
How does it make calculations easier?
What else do we gain by doing a gauge transformation?
When are used?
 
  • #5
In electrodynamics, the underlying properties of the electric and magnetic fields are described by the scalar and vector potentials. While the fields are unique for a given problem, the potentials are not. There is a degree of freedom when choosing our potentials. This causes us to define additional constraints that allow us to find unique solutions to the potentials despite their invariance on the physical solution. This allows consistency in the results and can ease calculations. The shifting of the potentials within the degrees of freedom that still give rise to the same physical fields is a gauge transformation.

One set of constraints is called the Lorenz condition. The Lorenz condition does not fully constrain the potentials but it does give rise to a reduction in the mathematics. This is because it allows us to express the scalar and vector potentials as decoupled inhomogeneous wave equations. By decoupling the potentials, it allows us to use simpler mathematical methods to derive their solutions, like using a dyadic Green's function to relate the sources to the potentials.
 
  • #6
Born2bwire said:
In electrodynamics, the underlying properties of the electric and magnetic fields are described by the scalar and vector potentials. While the fields are unique for a given problem, the potentials are not. There is a degree of freedom when choosing our potentials. This causes us to define additional constraints that allow us to find unique solutions to the potentials despite their invariance on the physical solution. This allows consistency in the results and can ease calculations. The shifting of the potentials within the degrees of freedom that still give rise to the same physical fields is a gauge transformation.

One set of constraints is called the Lorenz condition. The Lorenz condition does not fully constrain the potentials but it does give rise to a reduction in the mathematics. This is because it allows us to express the scalar and vector potentials as decoupled inhomogeneous wave equations. By decoupling the potentials, it allows us to use simpler mathematical methods to derive their solutions, like using a dyadic Green's function to relate the sources to the potentials.
Thank you.
Are these transformations mostly applied to E&M? Or can we generalize for other kinds of potentials?
Could anyone give me a simple example calculation example or a link to one?
 
  • #8
Winzer said:
Thank you.
Are these transformations mostly applied to E&M? Or can we generalize for other kinds of potentials?
Could anyone give me a simple example calculation example or a link to one?

I have not really seen them outside of electromagnetics. They are much more prominent in quantum field theory but in a way that is still largely connected with electromagnetics. You could of course apply this to any general potential that would allow you to do so. However, I am at a lost of an example outside of electromagnetics where you could do so in a way that gives positive benefits. Heck, I'm sure I have seen an example in some of my mathematics courses, this would seem like something well suited for a lot of abstract problems.
 

Related to What is a Gauge? Illustrative Explanation

What is a Gauge?

A gauge is a scientific instrument used to measure and display a physical quantity, such as pressure, temperature, or force. It typically consists of a dial or digital display and a needle or pointer that indicates the measured value.

How does a gauge work?

A gauge works by converting a physical quantity, such as pressure or temperature, into a mechanical movement that is then displayed on a dial or digital screen. This conversion is typically achieved through the use of a sensor or transducer, which senses the physical quantity and produces an electrical signal that is then interpreted by the gauge.

What are the different types of gauges?

There are several different types of gauges, including analog gauges, which use a dial and pointer to display the measured value, and digital gauges, which use a digital screen. There are also specific types of gauges for measuring different physical quantities, such as pressure gauges, temperature gauges, and force gauges.

Why are gauges important in science?

Gauges are important in science because they allow scientists to accurately measure and monitor physical quantities, which is essential in conducting experiments and gathering data. They also play a crucial role in various industries, such as manufacturing and engineering, where precise measurements are necessary for quality control and safety.

How do I choose the right gauge for my needs?

Choosing the right gauge depends on the specific physical quantity you need to measure and the level of accuracy required. You should also consider the environmental conditions, such as temperature and pressure, in which the gauge will be used. Consulting with an expert or conducting research on the different types of gauges available can help you make an informed decision.

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