Phy622 Mathematical Methods 3

01-07

Example 2 - Hermitian Matrices

$H_1*H_2 = \iota [H_1, H_2]$

$H_3 = H_1 * H_2$ is also hermitian

Example 3 - Symmetric Matrices

$S_1 * S_2 = {S_1, S_2}$

However this is not a [01-06#Algebra] as there is no Jacobi Identity.

Arithmetic

Set {S} with + and * defined.

1. $(S, +)$ is abelian group, with identity 0
2. $(S - {0}, *)$ is a group with identity 1
3. * is associative over S
4. * is distributive over +

Grassmann Arithmetic

Extension of c-no arithmetic.

$S = \mathbb C \cup G$

Where $\mathbb C$ is commuting, and $G$ are anticommuting, and c commutes with g. $g_1g_2 \in \mathbb{C}$. Hence, $g^2 = 0$, as $g^2 = -g^2$ and also $\in S$.

Hence, functions on $G$ are only linear.

Complex Grassman Numbers

$\zeta = \eta + i\chi$ and $\bar\zeta = \eta-i\chi$ Where $\eta, \chi$ are real Grassman numbers.

${\zeta}$ is also Grassman.

$\zeta \bar\zeta = -\bar\zeta \zeta$

Complex conjugation interchanges order of grassman #s

Exercise

1. $(\zeta_1\zeta_2)^* = \zeta_2^*\zeta_1^*$
2. $(F_1BF_2)^T = ?$

Matrices of Grassman Numbers

\((AB)^T = -B^TA^T\) \((AB)^\dagger = B^\dagger A^\dagger\)

Metric space

Non empty set S and $d:S\times S\rightarrow\mathbb{R}$

1. $d(x,y) \ge 0$
2. $d(x,y) = 0 \iff x=y$
3. $d(x,y) = d(y,x)$
4. $d(x,y) \le d(x,z) + d(z,y)$

Vector Spaces

Theorem 1:

For $0\in F$, $|0\rangle \in \mathbb V$, $0|v\rangle = |0\rangle$

Proof

$(0+1) |v\rangle = |v\rangle = 0|v\rangle +|v\rangle \implies 0|v\rangle = |0\rangle$

Linear Independance

Given a set of m non-null vectors, if we can find a set of m scalars ${\alpha}$ not all zero st $\sum_{i=1}^m \alpha_i|i\rangle = |0\rangle$, then the set is called linearly dependant. Otherwise, it is Linearly independant.

Dimension of Vector space

$\text{dim}(\mathbb V)$ is the max number of vectors which are simultaneously independant set of vectors.

If there is no such maximum, then the dimension of V is infinity.

Span

If $\forall\ |w\rangle \in \mathbb V \exists$ a set of scalars ${\alpha}$ st $|w\rangle = \sum_{i=1}^m \alpha_i |v_i\rangle$ is called a span of V

Linearly independant Span

It is a [01-07#Span|Span] which is also [01-07#Linear Independance|Linearly Independant] is called a basis set for V.

Dual Vector space

$|v\rangle \leftrightarrow^{\text{Duality}}_{\text{Hermitian Conjugation}} \langle w|$