I How to prove that some fields are different aspects of one general field?

[SEYED2001]

TL;DR Summary

I do not know how to show that some fields are just different aspects of another more general field. What does that more formally mean and how can it be proven?

Hi! I know some theorists believe all quantum fields and gravitational field are different aspects of one universal field. What does that formally (e.g. mathematically) mean "to be different aspects of" and how can one prove, let's say, fields A and B are different aspects of C?

By the way, I don't have a deep knowledge of QM or QFT and the math behind them. However, I still want to know a fairly accurate explanation. I would learn the math if required.

 

[PeroK]

Summary:: I do not know how to show that some fields are just different aspects of another more general field. What does that more formally mean and how can it be proven?

Hi! I know some theorists believe all quantum fields and gravitational field are different aspects of one universal field. What does that formally (e.g. mathematically) mean "to be different aspects of" and how can one prove, let's say, fields A and B are different aspects of C?

By the way, I don't have a deep knowledge of QM or QFT and the math behind them. However, I still want to know a fairly accurate explanation. I would learn the math if required.

In physics you don't prove things like that. What you are looking for is a mathematical model that predicts the range of phenomena you are interested in. For example, originally there were electric and magnetic fields. These were unified by showing that the model of a single electromagnetic field could encompass both electric and magnetic phenomena.

If you wanted to unify the strong, weak and EM forces, you would need a theory that somehow encompasses all aspects of all these phenomena.

 

[SEYED2001]

In physics you don't prove things like that. What you are looking for is a mathematical model that predicts the range of phenomena you are interested in. For example, originally there were electric and magnetic fields. These were unified by showing that the model of a single electromagnetic field could encompass both electric and magnetic phenomena.

If you wanted to unify the strong, weak and EM forces, you would need a theory that somehow encompasses all aspects of all these phenomena.

Thank you so much for your reply. Then, what is the general algorithm to do it? In other words, what do physicists do when they intend to theorize a unified field that encompasses all of the phenomena related to some other fields?

 

[hutchphd]

First one has to figure out what are the appropriate questions to ask. I know no algorithm for that..

 

[SEYED2001]

First one has to figure out what are the appropriate questions to ask. I know no algorithm for that..

I think you're right. I don't know how to ask it. But imagine you want to unify weak, strong and EM fields into a more general one. First, have they ever been unified? Second, how can one unify them? What are the steps to take?

 

[SEYED2001]

In physics you don't prove things like that. What you are looking for is a mathematical model that predicts the range of phenomena you are interested in. For example, originally there were electric and magnetic fields. These were unified by showing that the model of a single electromagnetic field could encompass both electric and magnetic phenomena.

If you wanted to unify the strong, weak and EM forces, you would need a theory that somehow encompasses all aspects of all these phenomena.

In addition, I wonder if there is a definite list of phenomena one's model need to encapsulate in his/her theory in order to unify weak, strong and EM fields.

 

[PeroK]

In addition, I wonder if there is a definite list of phenomena one's model need to encapsulate in his/her theory in order to unify weak, strong and EM fields.

A new theory must give the same results as all existing theories - as a first step. In essence this is everything from experiments involving elementary particles; plus gravity, dark matter and dark energy.

If it can also predict some new results then so much the better.

 

[SEYED2001]

A new theory must give the same results as all existing theories - as a first step. In essence this is everything from experiments involving elementary particles.

If it can also predict some new results then so much the better.

I can see that I am being given general answers. This maybe in part due to my inaccurate question. In fact I can make my question more accurate. Imagine you already have a field with some known dimensions and geometry as your candidate and want to check if this actually is a grand unified theory, i.e. it unifies EM, weak and strong interactions all into one single field. Now, how would you exactly test if your candidate field explain all known EM, weak and strong phenomena? Shall you derive every single field equation in each of these three fields or maybe something less mathematical is yet acceptable?

 

[PeroK]

Shall you derive every single field equation in each of these three fields or maybe something less mathematical is yet acceptable?

You would have to show, by whatever means, that your new theory recreates everything. For example, in QED if you take the non-relativistic limit you get Rutherford scattering. And, at high energies you get Mott scattering. Any new theory would have to show those results. Plus everything else that QED predicts and has been experimentally tested.

It's a lot of work.

 

[Stephen Tashi]

IImagine you already have a field with some known dimensions and geometry as your candidate

The field's dimensions and geometry, by themselves, don't necessarily produce a physical theory. For example, the physical theory called "the electromagnetic field" is more than a 10 dimensional vector space of vectors (ex,ey,ez,mx,my,mz,x,y,z,t). The theory is given by Maxwell's equations, which give descriptions of how properties of the field at different temporal and spatial locations must relate to each other.

A straightforward approach to unifying fields a,b,c is to think of a higher dimensional field of the form (a,b,c, p,q,r,...). If it is known that a,b are related by some equation like F(a,b) = 6 the equations of your new theory must imply this equation.

 

[Nugatory]

Shall you derive every single field equation in each of these three fields?

That's pretty much what it means to unify theories - you're showing that were previously thought to be two or more separate theories are actualy special cases of some more general theory. To do this, you have to be able to derive the older theories from the newer.

 

[Elroch]

A new theory must give the same results as all existing theories - as a first step. In essence this is everything from experiments involving elementary particles; plus gravity, dark matter and dark energy.

If it can also predict some new results then so much the better.

Not quite. A new theory must give predictions (what you meant by "results"?) that are compatible with the existing empirical data to within experimental uncertainty. These predictions way well be slightly different to those of an existing theory that is also compatible (to within experimental error) with the existing empirical data. For example, there may be additional processes in Feynman diagrams.
Of course, sometimes the experiments lead: an existing theory comes up against empirical data that shows it is not precisely correct and a better theory is needed to explain it.