Algebra

• Intuitive
• Concrete
• Abstract

Intuitive

Compare one thing, to another thing, using a mathematical formula.

Now take two formulas, and see if you can miss mash them together, in order to find out new things.

Repeat.

Algebra.

“Algebra is a merry science. We go hunting for a little animal whose name we don’t know, so we call it ‘x’. When we bag our game, we pounce on it and give it its right name.” ~ Albert Einstein

Concrete

Elementary algebra is the practice of calculating with symbols that stand for numbers. We write the relations we know as equations and transform them, using a small set of rules, until the quantity we want stands alone on one side. For real (or complex) numbers $a, b, c$ the rules are

$$\begin{aligned} a + b &= b + a, & ab &= ba, \\ (a + b) + c &= a + (b + c), & (ab)c &= a(bc), \\ a(b + c) &= ab + ac, & & \\ a + 0 &= a, & a \cdot 1 &= a, \\ a + (-a) &= 0, & a \cdot a^{-1} &= 1 \quad (a \neq 0). \end{aligned}$$

Every manipulation we do when "solving for $x$" is a chain of these steps, applied to both sides of an equation.

Example: eliminating a variable. For motion with constant acceleration $a$ we have $v = v_0 + a t$ and $x = x_0 + v_0 t + \tfrac{1}{2} a t^2$. If we do not care about the time, we solve the first equation for $t = (v - v_0)/a$ and substitute it into the second:

$$x - x_0 = \frac{v_0 (v - v_0)}{a} + \frac{(v - v_0)^2}{2a} = \frac{v^2 - v_0^2}{2a},$$

so $v^2 = v_0^2 + 2a(x - x_0)$, a relation between speed and position alone.

Example: completing the square. The equation $x^2 + px + q = 0$ can be written as $(x + p/2)^2 = p^2/4 - q$, which gives

$$x_{\pm} = -\frac{p}{2} \pm \sqrt{\frac{p^2}{4} - q}.$$

When the rules change. Physics uses objects that do not obey all of the rules above. Matrices and quantum mechanical operators do not commute in general, so $(A + B)^2 = A^2 + AB + BA + B^2$, and $AB - BA$ is a quantity of its own, the commutator $[A, B]$. The cross product of vectors in $\mathbb{R}^3$ is neither commutative nor associative: $\vec a \times \vec b = -\vec b \times \vec a$. Geometric algebra, the subject of the links below, defines a product of two vectors that combines the dot product and the wedge product, $ab = a \cdot b + a \wedge b$. This product is associative but not commutative, and every vector with nonzero square $a^2 = a \cdot a$ has an inverse $a/(a \cdot a)$. Before calculating with a new kind of object, one has to check which of the rules still hold.

• Algebra: Chapter 0 by Aluffi

• Learning about Geometric Algebra

• Primer on Geometric Algebra
• Geometric Algebra (University of Cambridge website)

Abstract

In modern mathematics "algebra" is the study of sets equipped with operations that satisfy a list of axioms, and of the structure-preserving maps between such sets. The structures that appear most often in physics are:

• A group $(G, \cdot)$: an associative product with a unit element and inverses. Symmetries of a physical system form a group; see group theory.
• A ring $(R, +, \cdot)$: an abelian group under $+$ with an associative product that distributes over $+$. Example: the $n \times n$ matrices with real entries, or the polynomials $\mathbb{R}[x]$.
• A field $K$: a commutative ring with $1 \neq 0$ in which every nonzero element has a multiplicative inverse. Examples: $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$. The rules listed in the Concrete layer, together with $1 \neq 0$, are the field axioms.
• A vector space $V$ over a field $K$: an abelian group with a scalar multiplication $K \times V \to V$; see linear algebra.
• An algebra over $K$: a vector space $A$ with a bilinear product $A \times A \to A$.

Two kinds of algebras over a field are central in physics. In an associative algebra the product satisfies $(ab)c = a(bc)$. The matrix algebra $M_n(K)$ and the algebra of bounded operators on a Hilbert space are examples. In a Lie algebra the product is a bracket $[\cdot,\cdot]$ that is antisymmetric and obeys the Jacobi identity

$$[a,[b,c]] + [b,[c,a]] + [c,[a,b]] = 0.$$

Every associative algebra becomes a Lie algebra with the commutator $[a, b] = ab - ba$. Lie algebras describe infinitesimal symmetries; see Lie algebras.

The geometric algebra of the links below is a Clifford algebra. Given a vector space $V$ over $\mathbb{R}$ with a quadratic form $q$, the Clifford algebra $\mathrm{Cl}(V, q)$ is the associative algebra with unit generated by $V$ subject to $v^2 = q(v)\,1$ for all $v \in V$. Polarizing this relation gives

$$uv + vu = 2B(u, v)\,1, \qquad B(u, v) = \tfrac{1}{2}\big(q(u + v) - q(u) - q(v)\big).$$

With $V = \mathbb{R}^{1,3}$ and $B$ the Minkowski metric, this is the relation $\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}$ obeyed by the Dirac matrices, which form a representation of this Clifford algebra (of its complexification) on $\mathbb{C}^4$.

The maps that matter are homomorphisms, maps that preserve the operations. A representation of an algebra $A$ on a vector space $W$ is an algebra homomorphism $A \to \mathrm{End}(W)$. Much of the use of algebra in physics consists of finding the representations of a given algebra, for example the angular momentum states as representations of the Lie algebra $\mathfrak{su}(2)$.