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.)
