group_theory

# Group Theory

Basic idea:Numbers measure size, groups measure symmetry.” from Groups and Symmetry by Mark A. Armstrong

## Why do we care about Symmetries?

One way to understand the intention of designer of an universe is to find the symmetries. The search for symmetry can be substantially different mental activity from the usual explicit calculations. For example, consider the following simple arithmetic: $$34126 \times 12378 - 12378 \times 34126 = ?$$ You can compute the first term using calculator, and store in the memory. The second term is calculated, and subtracted from the previous result stored in memory. The final numeric result tells us that one should use brain instead of fingers in scientific problems. Of course, the commutative property of multiplication is enough to write down the answer. The morale of this example is: find the greatest possible symmetry whenever possible

From Magnetic Monopoles in Grand Unified Theories by I. G. Koh

Physicists are mostly agreed that the ultimate laws of Nature enjoy a high degree of symmetry. By this I mean that the formulation of these laws, be it in mathematical terms or perhaps in other accurate descriptions, is unchanged when various transformations are performed. Presence of symmetry implies absence of complicated and irrelevant structure, and our conviction that this is fundamentally true reflects an ancient aesthetic prejudice - physicists are happy in the belief that Nature in its fundamental workings is essentially simple. Moreover, there are practical consequences of the simplicity entailed by symmetry: it is easier to understand the predictions of physical laws. For example, working out the details of very-many-body motion is beyond the reach of actual calculations, even with the help of computers. But taking into account the symmetries that are present allows one to understand at least some aspects of the motion, and to chart regularities within it.

## Definition

A group $(G, \circ)$ is a set $G$, together with a binary operation $\circ$ defined on $G$, that satisfies the following axioms

• Closure: For all $g_1, g_2 \in G$, $g_1 \circ g_2 \in G$
• Identity: There exists an identity element $e \in G$ such that for all $g \in G$, $g \circ e = g = e \circ g$
• Inverses: For each $g \in G$, there exists an inverse element $g^{-1} \in G$ such that $g \circ g^{-1}=e = g^{-1} \circ g$.
• Associativity: For all $g_1, g_2, g_3 \in G$, $g_1 \circ (g_2 \circ g_3) = (g_1 \circ g_2) \circ g_3$.

## Quotes

We need a super-mathematics in which the operations are as unknown as the quantities they operate on, and a super-mathematician who does not know what he is doing when he performs these operations. Such a super-mathematics is the Theory of Groups.

Sir Arthur Stanley Eddington

Symmetry is the magic word that distinguishes theory from coincidence.

Seven Science Quests by Sudarshan

Almost anybody whose research requires sustained use of group theory (and it is hard to think of a physical or mathematical problem that is wholly devoid of symmetry) writes a book about it.

Rovelli: […] . I learned from your book that the world in which we happen to live is not Poincare invariant, and is not described by Poincare invariant theory. There is no sense in which general relativity is Poincare invariant. (If it were, what would be the Poincare transform of the closed Friedman-Robertson-Walker solution of the Einstein equation?) Thus, Poincare invariance is neither a symmetry of our universe, nor a symmetry of the laws governing our universe. Don't you find it a bit disturbing basing the foundation of our understanding of the world on a symmetry which is not a symmetry of the world nor of its laws?

Weinberg: Well, I think there's always been a distinction that we have to make between the symmetries of laws and the symmetries of things. You look at a chair; it's not rotationally invariant. Do you conclude that there's something wrong with rotation invariance? Actually, it's fairly subtle why the chair breaks rotational invariance: it's because the chair is big. In fact an isolated chair in its ground state in empty space, without any external perturbations, will not violate rotational invariance. It will be spinning in a state with zero rotational quantum numbers, and be rotationally invariant. But because it's big, the states of different angular momentum of the chair are incredibly close together (since the rotational energy differences go inversely with the moment of inertia), so that any tiny pertur- bation will make the chair line up in a certain direction. That's why chairs break rotational invariance. That's why the universe breaks symmetries like chiral invar- iance; it is very big, even bigger than a chair. This doesn't seem to me to be relevant to what we take as our fundamental principles. You can still talk about Lorentz invariance as a fundamental law of nature and live in a Lorentz non-invariant universe, and in fact sit on a Lorentz non-invariant chair, as you are doing. [Added note: Lorentz invariance is incorporated in general relativity, as the holonomy group, or in other words, the symmetry group in locally inertial frames.]

Conceptual Foundations of Quantum Field Theory edited by Cao

### Matrix Groups and Matrix Algebras

To help your intuition further: Lie groups are “almost always” matrix groups as follows. There is a corollary to a difficult theorem known as Ado's theorem that every Lie algebra can be realized as a Lie algebra of square matrices. The same is not true of Lie groups: not every Lie group can be represented as a group of matrices but it is almost true (a consequence of the Peter-Weyl theorem is that every compact group can be realized as a group of square matrices). Certainly, since we can find a square matrix realization for every Lie algebra, we can build a matrix Lie group with that algebra as its Lie algebra through the matrix exponential function; then we find that matrix group's universal cover and this is where we sometimes fail to get a matrix group. This is not typical and the first Lie groups that were not also matrix groups (so called metaplectic groups) weren't found until 1937. These oddballs are all covering groups of noncompact groups.

By exponentiating the Lie algebra elements, which can always can be written as matrices, we get representations of the corresponding universal covering group that belongs to this Lie algebra.

## Possible Points of Confusion?

Q: Why is the Jacobi identity important?

A: “[T]he real meaning of the Jacobi identity: it’s there so the adjoint representation of a Lie group, clearly a very basic and fundamental thing, induces a homomorphism in the corresponding Lie algebras that respects Lie brackets.” http://www.wetsavannaanimals.net/wordpress/the-adjoint-representation-of-the-lorentz-group/

Q: What does conjugation mean?

## Great Books

• Visual Group Theory by Nathan Carter. From the preface: “If you are interested in learning about group theory in a relaxed, intuitive way, then this book is for you.
• Naive Lie Theory by John Stillwell