What is the meaning of non-Abelian in physics?

[KurtLudwig]

TL;DR Summary

I have read the paper arXiv:0901.4005v2 by A. Deur (and do not follow most of the mathematics). From my mathematical background, I do know that non-commutative means a x b is not equal to b x a. From Wikipedia, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them the other way round. At a physics undergraduate level, what is the meaning of non-Abelian in the context of this paper?

I have read the paper arXiv:0901.4005v2 by A. Deur (and do not follow most of the mathematics). From my mathematical background, I do know that non-commutative means a x b is not equal to b x a. From Wikipedia, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them the other way round. At a physics undergraduate level, what is the meaning of non-Abelian in the context of this paper?

 

[fresh_42]

https://arxiv.org/pdf/0901.4005.pdf

Implications of Graviton-Graviton Interaction to Dark Matter.
A. Deur, University of Virginia, Charlottesville, VA 22904 (Dated: October 24, 2018)

Our present understanding of the universe requires the existence of dark matter and dark energy. We describe here a natural mechanism that could make exotic dark matter and possibly dark energy unnecessary. Graviton-graviton interactions increase the gravitational binding of matter. This increase, for large massive systems such as galaxies, may be large enough to make exotic dark matter superfluous. Within a weak field approximation we compute the effect on the rotation curves of galaxies and find the correct magnitude and distribution without need for arbitrary parameters or additional exotic particles. The Tully-Fisher relation also emerges naturally from this framework. The computations are further applied to galaxy clusters.

Relevant quotation:

Cosmological observations appear to require ingredients beyond standard fundamental physics, such as exotic dark matter [1] and dark energy [2]. In this Letter, we discuss whether the observations suggesting the existence of dark matter and dark energy could stem from the fact that the carriers of gravity, the gravitons, interact with each others. In this Letter, we will call the effects of such interactions “non-Abelian”. The discussion parallels similar phenomena in particle physics and so we will use this terminology, rather than the one of General Relativity, although we believe it can be similarly discussed in the context of General Relativity. We will connect the two points of view wherever it is useful.

 

[KurtLudwig]

The next time I will follow your format: copy the summary of the paper followed by the relevant quotation.
Sorry for the omission.

 

[B.C.]

Fascinating paper. Can't believe no one ever considered self-interactions of gravitons before, since one would expect that to happen. Not to mention the example of the known behavior of gluons.

Why isn't this paper a bigger deal?

 

[haushofer]

The underlying gauge group of GR are, roughly speaking, general coordinate transformations (gct's). And gct's don't form an Abelian group; they don't commute.

 

[haushofer]

Fascinating paper. Can't believe no one ever considered self-interactions of gravitons before, since one would expect that to happen. Not to mention the example of the known behavior of gluons.

Why isn't this paper a bigger deal?

I haven't read the paper, but I guess the weakness of gravity is an issue; self interactions between gravitons are beyond perturbation theory, and the linear effects of gravity are already extraordinarily weak.

 

[KurtLudwig]

Is gravity as described by Newton considered Abelian? From the above answers: General relativity and graviton self-interactions are non-Abelian. The sequence of transformations make a difference in GR. Are these statements correct?

 

[haushofer]

Is gravity as described by Newton considered Abelian? From the above answers: General relativity and graviton self-interactions are non-Abelian. The sequence of transformations make a difference in GR. Are these statements correct?

Well,in Newtonian gravity the potential does not self-interact, so I'd say yes. On the other hand, it can be described as the gauge theory of a non-Abelean gauge group, just like GR. So then it becomes nomenclature.

Your statements are correct.

 

[ohwilleke]

The field of math known as "Abstract Algebra" is the study of the algebra you studied in high school (or earlier) but with some of the axioms (also sometimes called "properties") that apply in normal high school algebra removed or modified.

If you tweak certain properties of ordinary algebra you get a subfield in Abstract Algebra knowns as "group theory" (tweak different properties and you get different subfields like "rings" that have far fewer practical applications). And, within group theory you can have mathematical systems that are "Abelian groups" in which the commutative property applies, and "non-Abelian groups" in which it does not. Another subfield within group theory is the study of "Lie groups" which also have Abelian and non-Abelian versions. All defined at the end of this post.

Remarkably, the seemingly random intellectual exercise of tinkering with the properties of ordinarily algebra that you use and do not use, leads to mathematical structure like Abelian and non-Abelian Lie Groups that naturally provide a means to explain mathematically how the physical world behaves at the subatomic level.

As a practical matter, it is much easier to do calculations using Abelian groups (which are necessary for quantum electrodynamics (QED), the quantum mechanical version of electromagnetism) than it is to do so with non-Abelian groups (which are necessary, for example, for quantum chromodynamics (QCD), the Standard Model theory of the strong force that holds protons and neutrons and other particles made of quarks and gluons together and indirectly holds together atomic nuclei).

There is good reason to believe that a theory of quantum gravity should be non-Abelian, and one fruitful path to understanding it treats quantum gravity as "QCD squared". There is also good reason to think that quantum gravity should be a "chiral" theory (i.e. one in which right parity and left particle particles have different properties) even though neither Newtonian gravity, nor classical GR are chiral theories.

Like QED, and unlike QCD and quantum gravity designed to approximate GR, Newtonian gravity is also Abelian.

Take it as a given that non-Abelian gauge theories like QCD and quantum gravity mediated by a graviton, have carrier particles (called bosons because they have integer rather than half-integer spin) that interact with each other as well as with other particles that are charged under the relevant force (thus, quarks and gluons both carry color charge, and gravitons because they have mass-energy and couple to mass-energy, interact with each other). The logic behind why this is the case is rather obscure and technical.

In contrast, in QED, which is an Abelian gauge theory, the carrier particles (which are bosons) of the electromagnetic force, called photons, interact with particles that have electromagnetic charges, but do not interact with each other. The lack of self-interactions between photons is one of the things that makes QED calculations so much easier to do than QCD or quantum gravity calculations. This is basically because self-interactions add many more permutations of ways that a carrier boson can go from point A to point B, all of which much be considered to get a prediction (the path integral that determines the probability of this happening is called a propagator of the carrier boson).

Abelian gauge theories also have less elaborate emergent behavior from their seemingly simple
equations than non-Abelian gauge theories, and the self-interaction component is one of the important reasons that this is the case.

A related and important concept in quantum gravity is that the concept of a non-commutative geometry is important. To oversimplify, this is a geometry in which the path from A to B may have a different length than the path from B to A.

Relevant definitions are as follows:

In physics, a gauge theory is a type of field theory in which the Lagrangian does not change (is invariant) under local transformations from certain Lie groups.

The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory. . . .

Gauge theories are important as the successful field theories explaining the dynamics of elementary particles. Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and has one gauge field, the electromagnetic four-potential, with the photon being the gauge boson. The Standard Model is a non-abelian gauge theory with the symmetry group U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons.

From Wikipedia (underlining added).

In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a.[1][2] This class of groups contrasts with the abelian groups. (In an abelian group, all pairs of group elements commute).

Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them the other way round).

Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory.

From Wikipedia.

Group theory:

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. . . . A Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student https://en.wikipedia.org/w/index.php?title=Arthur_Tresse&action=edit&redlink=1, page 3.

Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.

Note also that Abelian v. non-Abelian is not the only important factor in the difficultly in doing calculations with different gauge theories.

Another reason that QED is easier to calculate with than QCD is that the dimensionless coupling constant of QED is much smaller relative to 1 than in QCD and that makes terms with higher powers of the coupling constant in the question contribute much less to the final result.

Another reason all Standard Model forces are easier to work with than a naive version of quantum gravity is that Standard Model gauge theories are renormalizable in perturbative mathematical methods, while quantum gravity with a spin-2 massless graviton is not, requiring non-perturbative mathematical methods instead.

Deur deals with some of the practical computational difficulties involved by using a scalar approximation of a naive full spin-2 massless graviton quantum gravity theory. This means that he uses real number values of quantities in the equation instead of tensors (i.e. four by four matrixes) that you would use if you did it without approximation. This is equivalent do a static, equilibrium approximation of the tensor valued variable case. Other scholars have shown that this is an acceptable approximation in the kinds of weak field deep space situations he is applying the theory to, because he is looking mostly at systems that are close to equilibrium and in which rest mass is the dominant mass-energy input to the stress-energy tensor of GR. (A scalar approximation might not be acceptable to approximate, for example, the behavior of fast moving particles near an electromagnetically charged black hole.)

 

[KurtLudwig]

Thank you so much for your detailed explanations.
Since I have time, I will now study up on group theory.
(I read in Wikipedia five or six times a day and contribute monthly.)
Currently, I am reading a textbook on linear algebra.
Also, I have read a book on Special Relativity and have started to read an introductory book on General Relativity, but I am having problems since I have not taken the required prerequisite courses in mathematics.
Physics requires so much math, one wonders is physics a branch of math? I do not think so, since mathematical models may not exist in the physical world.
To me, physics appears to be both beautiful and strange.
I have read the arXiv paper by Dr. Deur two times so far. It is amazing the connections he made between gluons and gravitons.

 

[ohwilleke]

Physics is much easier if you have the math first, even though most people learn them simultaneously.

Surprised anyone could write an entire book on special relativity. It's usually one chapter or part of a chapter in a larger textbook.

 

[PAllen]

Physics is much easier if you have the math first, even though most people learn them simultaneously.

Surprised anyone could write an entire book on special relativity. It's usually one chapter or part of a chapter in a larger textbook.

A rather ignorant comment, sorry for any offense. For example, the following 800 page book by a world renowned author:

https://www.amazon.com/dp/3642372759/?tag=pfamazon01-20

or how about the classic :

https://www.amazon.com/dp/B000GP7PX4/?tag=pfamazon01-20

apparently now available free: http://strangebeautiful.com/other-texts/synge-relativity-special.pdf

 

[KurtLudwig]

Just finished studying linear algebra and now I am studying "Elements of Group theory for Physicists".
Thanks for pointing me in the right direction.

 

[r731]

In algebra, non-commutativity is a property of an arbitrary operator ⊗ such that x ⊗ y ≠ y ⊗ x.