Mathematical Methods of Physics/Linear Algebra

< Mathematical Methods of Physics

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)

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Inner ProductEdit

Let   be a vector space over   (here,   stands either for   or  ). The binary operation   is said to define an inner product if and only if,

For all  ,  

(i) (Conjugate Symmetry):  

This implies that   as  

(ii) (Linearity in first variable):  

Conjugate symmetry implies that  

(iii) (Positivity):   for all  

(iv) (Definiteness):   if and only if  

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

The complex-conjugate is sometimes denoted as  

Hilbert SpaceEdit

Observe that the positive-definite nature of the inner product implies that we can define a norm on   as  ,  

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

Thus a Hilbert space is a complete inner product space.

ExamplesEdit

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 spacesEdit

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

 

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

Sequence spacesEdit

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

 

This space becomes a Hilbert space with the inner product

 

for all x and y in  . 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   for a suitable set B. If B=N, the natural numbers, this space is separable and is simply called  .

New Hilbert spaces from oldEdit

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 functionsEdit

Among examples of Hilbert spaces, the one that holds the most interest for the physicists are the   spaces.

Consider   to be the set of all functions   that are square integrable with respect to a real measure  , that is   is well-defined.

Define inner product on   as

 

Provided the inner product exists for any pair of functions  , we can see that   is an inner product space.

The reader may notice an ambiguity here, as   need not imply that  . To resolve this, we use a different equivalence relation between functions,  , and hence,   at all points of   except for a set of points of measure  .

The   space is an example of what are called the   spaces. It can be shown([1]) that all   spaces are complete, and hence, the Lebesgue space,   is also complete.

Thus, we have that   is a Hilbert space.

Let us identify   as   and denote the inner product of   as

 

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