On Group Theory: Symmetry Groups and Our Interest

Question

Over the past few years, I've been doing a lot of self education in the Quantum Mechanics and General Relativity, and of course, there are mathematical elements of both doctrines that are matrices. More specifically, linear operators and transformations. Most of my texts introduced these ideas as fundamental, and not part of some larger mathematical structure--Group Theory.

One thing throughout quantum mechanics that I've noticed is that there is a particular interest in matrices $ A $, such that $Ax = \lambda x$. The Eigenvalue Equation. This linear operator can be said to exhibit symmetry, since it leaves the operand the same up to a constant. For $ \lambda = 1 $, the transformation does indeed leave the operand the same. After some digging, this particular element of quantum mechanics that represents physical measurement, is an element of the Symmetry Group ( call it $ SG(n) $ where $n$ is the dimensionality).

Another example is for the Lorentz transformations in special relativity. Each transformation leaves the reference frame the same no matter where you are. I've usually seen it explained that the Lorentz transformation is just the result of a rotation, very similar to that of the Rotation Group $ O(3) $, but since we have a time dimension and we rotate spacetime and suppress the two other spatial variables, we get the Lorentz Group: $ SO(3,1) $, equipped with 3 space + 1 time, dimensions. Both of these groups have been shown to me to be elements of $ SG(n)$ as well.

At present it seems that Relativity and Quantum Mechanics are the two pillars that our ultimate goal of Cosmology rests upon. From the lens of Group Theory and by the relation of the elements in the theory aforementioned, I am compelled to conclude that the physical laws and principles that are being discovered present day forward are mathematical elements in this group theory that have meaningful physical and mathematical interpretation down at the physical doctrine level, ie the Energy Operator and the Hamiltonian which brought us to the Schrodinger equation, and the Lorentz Transformation that defined special relativity. This also compels me to believe that further advancement can be found if we have our mathematicians race to discover more and more elements in this larger Group Theory, and have Physicist determine and interpret any Physical meaning in the discovered constructs.

My question is then a question of how valid my conclusion is, and how sound does this in turn make my belief.

Answer 1

In a sense, quantum mechnaics is the applied theory of Lie groups and Lie algebras.

For a view of physics, and in particular of quantum mechanics, from the point of view of infinitesimal symmetries (i.e., in mathematical terms, Lie algebras), see my online book

Arnold Neumaier and Dennis Westra, Classical and Quantum Mechanics via Lie algebras, 2008, 2011. http://lanl.arxiv.org/abs/0810.1019

Answer 2

I like the way you see the connection between Group Theory and Physics, but your idea is not entirely new. In 2007, Garret Lisi propesed a theory of everything by using the $E_{8}$ lie group in order to account for the four forces present in nature. According to him, each kind of elementary particle can be associated with a point in a weight diagram. The coordinates of these points are the quantum numbers and the charges of elementary particles, which are conserved in interactions. To see an introduction of this idea consult New Scientist article :http://www.newscientist.com/article/dn12891-is-mathematical-pattern-the-theory-of-everything.html#.UyZphm8xohE, and for the original article in arXiv: http://www.newscientist.com/article/dn12891-is-mathematical-pattern-the-theory-of-everything.html#.UyZphm8xohE .

Enjoy it

Answer 3

Taken from this review of Conway and Smith's "On Quaternions and OCtonions:

It follows that the quaternions of norm 1 form a group under multiplication. This group is usually called ${\rm SU}(2)$, because people think of its elements as $2 \times 2$ unitary matrices with determinant 1. However, the quaternionic viewpoint is better adapted to seeing how this group describes rotations in 3 and 4 dimensions. The unit quaternions act via conjugation as rotations of the 3d space of 'pure imaginary' quaternions, namely those with ${\rm Re}(q) = 0$. This gives a homomorphism from ${\rm SU}(2)$ onto the 3d rotation group ${\rm SO}(3)$. The kernel of this homomorphism is $\{ \pm 1 \}$, so we see ${\rm SU}(2)$ is a double cover of ${\rm SO}(3)$. The unit quaternions also act via left and right multiplication as rotations of the 4d space of all quaternions. This gives a homomorphism from ${\rm > SU}(2) \times {\rm SU}(2)$ onto the 4d rotation group ${\rm SO}(4)$. The kernel of this homomorphism is $\{ \pm (1,1) \}$, so we see ${\rm SU}(2) \times {\rm SU}(2)$ is a double cover of ${\rm SO}(4)$.

These facts are incredibly important throughout mathematics and physics. With their help, Conway and Smith classify the finite subgroups of the 3d rotation group ${\rm SO}(3)$, its double cover ${\rm SU}(2)$, the 3d rotation/reflection group ${\rm O}(3)$, and the 4d rotation group ${\rm SO}(4)$. These classifications are all in principle `well known'. However, they seem hard to find in one place, so Conway and Smith's elegant treatment is very helpful.