Mathematical Methods of Physics/Linear Algebra

The simplest structures on which we can study operations of both "algebra" and "calculus" is the Banach space. The crucial importance of Hilbert Spaces in Physics is due to the fact that the not only are Hilbert Spaces a special case of Banach space, but also because they contain the idea of inner product and the related conjugate-symmetry. (This chapter requires some familiarity with basic measure theory)

Inner Product

Let $V$ be a vector space over $F$ (here, $F$ stands either for $\mathbb{R}$ or $\mathbb{C}$). The binary operation $(\cdot):V\times V\to F$ is said to define an inner product if and only if,

For all $\mathbf{x},\mathbf{y}\in V$, $a,b\in F$

(i) (Conjugate Symmetry): $(\mathbf{x}\cdot \mathbf{y})=\overline{(\mathbf{y}\cdot \mathbf{x})}$

This implies that $(\mathbf{x}\cdot \mathbf{x})\in\mathbb{R}$ as $(\mathbf{x}\cdot \mathbf{x})=\overline{(\mathbf{x}\cdot \mathbf{x})}$

(ii) (Linearity in first variable): $(a\mathbf{x}\cdot \mathbf{y})=a(\mathbf{x}\cdot \mathbf{y})$

Conjugate symmetry implies that $(\mathbf{x}\cdot b\mathbf{y})=\overline{(b\mathbf{y}\cdot \mathbf{x})}=\overline{b(\mathbf{y}\cdot \mathbf{x})}=\overline{b}(x\cdot y)$

(iii) (Positivity): $(\mathbf{x}\cdot \mathbf{x})\geq 0$ for all $\mathbf{x}\in V$

(iv) (Definiteness): $(\mathbf{x}\cdot \mathbf{x})=0$ if and only if $\mathbf{x}=\mathbf{0}$

If an inner product is defined on $V$, we say that $V$ is an inner product space.

The complex-conjugate is sometimes denoted as $\overline{z}=z^*$

Hilbert Space

Observe that the positive-definite nature of the inner product implies that we can define a norm on $V$ as $\forall\mathbf{x}\in V$, $\|\mathbf{x}\|=(\mathbf{x}\cdot\mathbf{x})$

If $V$ is complete under this norm, we say that $V$ is a Hilbert Space.

Thus a Hilbert space is a complete inner product space.

Examples

In the examples of Hilbert spaces given below, the underlying field of scalars is the complex numbers C, although similar definitions apply to the case in which the underlying field of scalars is the real numbers R.

Euclidean spaces

Every finite-dimensional inner product space is also a Hilbert space. For example, Cⁿ with the inner product defined by

$\langle x, y \rangle = \sum_{k=1}^n x_k\overline{y_k}$

where the bar over a complex number denotes its complex conjugate.

Sequence spaces

Given a set B, the sequence space $\ell^2$ (commonly pronounced "little ell two") over B is defined by

$\ell^2(B) =\big\{ x : B \xrightarrow{x} \mathbb{C} \text{ and } \sum_{b \in B} \left|x \left(b\right)\right|^2 < \infty \big\}.$

This space becomes a Hilbert space with the inner product

$\langle x, y \rangle = \sum_{b \in B} x(b)\overline{y(b)}$

for all x and y in $\ell^2(B)$. B does not have to be a countable set in this definition, although if B is not countable, the resulting Hilbert space is not separable. Every Hilbert space is isomorphic to one of the form $\ell^2(B)$ for a suitable set B. If B=N, the natural numbers, this space is separable and is simply called $\ell^2$.

New Hilbert spaces from old

Two (or more) Hilbert spaces can be combined to produce another Hilbert space by taking either their direct sum or their tensor product.

Square-integrable functions

Among examples of Hilbert spaces, the one that holds the most interest for the physicists are the $\mathcal{L}^2$ spaces.

Consider $\mathcal{L}^2$ to be the set of all functions $f:[a,b]\to\mathbb{C}$ that are square integrable with respect to a real measure $\mu$, that is $\int_a^b \|f\|^2 d\mu$ is well-defined.

Define inner product on $\mathcal{L}^2$ as

$(f*g)=\int_a^b f(x)^*g(x)d\mu$

Provided the inner product exists for any pair of functions $f,g$, we can see that $\mathcal{L}^2$ is an inner product space.

The reader may notice an ambiguity here, as $(f*g)=(f*g')$ need not imply that $g=g'$. To resolve this, we use a different equivalence relation between functions, $f\sim f'\Leftrightarrow \int_a^b \|f-f'\|d\mu=0$, and hence, $f=f'$ at all points of $[a,b]$ except for a set of points of measure $0$.

The $\mathcal{L}^2$ space is an example of what are called the $\mathcal{L}^p$ spaces. It can be shown([1]) that all $\mathcal{L}^p$ spaces are complete, and hence, the Lebesgue space, $\mathcal{L}^2$ is also complete.

Thus, we have that $\mathcal{L}^2$ is a Hilbert space.

Let us identify $\psi\in\mathcal{L}^2$ as $\left|\psi \right\rangle$ and denote the inner product of $\psi,\phi\in\mathcal{L}^2$ as

$\left\langle\phi|\psi\right\rangle=\int_a^b \phi^*(x)\psi(x)d\mu$

The reader with previous experience in quantum mechanics will be able to recognise this as a formal justification for the dirac notation.