Phy622 Mathematical Methods 3

01-06

Special Rings

[Ring#Special Rings]

Division Rings

Multiplicative identity exists
But all x may not have inverses
- if x^- does not exist, x is singular
- Multiplicative inverse of 0 always does not exist ($00^- = 0 = 1$)

Field

A field is a ring with 0 as the only singular point.

Example -

Reals
Rationals

Zero divisors

For $r,s$ in a [Ring], if $rs=0$, then $r,s$ are called Zero divisors.

$Z_n$

${Z_n, +_n, *_n}$

0, 1 exist.
Group over addition
Monoid over multiplication
But elements apart from 0 may have a lack of inverses
- Except if n is prime
- $Z_5$ is a field but $Z_6$ is not a field

Exercises

Read pg 45-47 of P. Pal.
Ex 3.4, 3.9

Vector Spaces

$\mathbb{V}$ is defined as a set $V$ over Field $F$ and a $\oplus$ vector addition

$\mathbb{V} = {V, {F, +, *}, \oplus}$

Elements of $F$ are called scalars, of $V$ are called vectors

$\ket{x}, \ket{y}$

$\ket{x}\oplus \ket{y} \in V$

$a\ket{x} \in V$

Hence,

$\alpha \ket{x} + \beta \ket{y} \in V$

More formally,

Vector addition is commutative
$\oplus$ is associative
There is an unique identity for $\oplus$, called $\ket{0}$
Scalar multiplication defined, $\alpha\ket{x}\in V$
Multiplicative identity of F is also identity of Scalar multiplication
Additive inverse exists, denoted by $-\ket{x}$. Further, $-\ket{x} = -1 \ket{x}$
Scalar Vector multiplication is distributive over vector addition. 8.Scalar Vector multiplication is distributive over scalar addition. $(\alpha+\beta)\ket x = \alpha\ket {x}+\alpha\ket{x}$

Algebra

An algebra $\mathcal{A}$ is a [Ring] whose elements form a vector space over field F. The multiplication of this Ring is called a vector multiplication $*$.

$*: V\times V \rightarrow V$

Example

$N\times N$ matrices.
Lie Algebra: Lie Ring [Ring#Special Rings] whose elements form a vector space.
1. $\ket{x}*\ket{y} = -\ket{y}*\ket{x}$
2. Jacobi Identity
3. Eg. Set of antisymmetric matrices, with $* = [.,.]$ - Algebra of orthogonal group $O(N)$