In classical geometry, the distance between two points and is given by the length of any shortest path from to . However, this definition is not valid in quantum mechanics, where the concept of path between two points is not well defined. The idea introduced by Alain Connes in noncommutative geometry consists of defining a spectral distance from values taken by operator observables rather than from classical coordinates. In this way, the concept of geometrical point is not used, which allows the spectral distance to be applied to both classical and quantum spaces.
DISTANCE ON THE REAL LINE
In order to simplify the calculations, we choose an example of usual distance between two points and on the real line
In Riemannian geometry, the distance between two points and on a manifold can be formulated as an infimum
where the length of a path joining to is computed from the metric
In the case of the real line with unit metric
The distance on the real line can also be expressed as a supremum taken over continuously derivable functions on
At first sight, it can be strange to define a distance as a supremum whereas the usual definition uses an infimum. However, with this original approach, the primary objects are observables rather than points and , which is the inverse of the usual point of view. The advantage of this method is that it does not use the concept of path and thus can be generalized to quantum spaces, as explained further.
Let us consider the trivial function
We want to show that it corresponds to the supremum, that is to say that . To show this, we first note that and for all , thus is in the set defining the supremum. Suppose that is not the supremum. Then, there must exist such that for all and . This leads to
By the mean value theorem, there exists such that
From the two previous equations we deduce
which is in contradiction with for all
Although the above supremum formulation considers observables as primary objects, the expression is still incompatible with the operator framework of quantum theory. The approach suggested by noncommutative geometry is to replace points by states and functions by operators, which is exactly the way we do calculations in quantum mechanics.
First, a state is associated to each point as follows
Thus, a point is considered as an evaluator for observables, which reverses the classical point of view. This is called a state in accordance with the mathematical definition of positive linear functional on an algebra, which is in the above case . Note that complex numbers are preferred in quantum mechanics but they are not used here for simplification.
Next, the subexpression for all must be rewritten to be compatible with operator theory. In fact, each function can be considered as an operator acting by pointwise multiplication on the space of square integrable functions . That is to say for any vector . Then the above subexpression becomes equivalent to
where denotes the commutator and the operator norm.
Indeed, let us develop the expression for
Then we compute the operator norm
Therefore is equivalent to which in turn is equivalent to for all
Finally the supremum expression can be rewritten in the operator framework as
CONNES SPECTRAL TRIPLE
Intuitively, a spectral triple encodes geometrical information in an algebraic way.
Spectral triple : a spectral triple is composed of a unital C*-algebra acting on a Hilbert space and a self-adjoint operator with compact resolvent such that is bounded for all .
In this context, if and are two states, their distance is defined by
EXAMPLE : RIEMANNIAN MANIFOLD
Let be a compact Riemannian manifold. Although being a classical space, it is a good example to understand how noncommutative geometry works in the commutative case. A spectral triple is defined as follows.
We consider the algebra of continuous functions over . Then, we choose to be the Hilbert space of square integrable differentiable forms, the elements of acting by pointwise multiplication on . Finally, a first order differential operator on must be provided. A good choice is the Dirac operator, that is the square root of the Laplace operator
It is well known that the Laplace-de Rham operator satisfies
where is the exterior derivative and the exterior coderivative (obtained from and the Hodge star). Since we put
The distance between two points and considered as pure states and is thus
This is a generalization of the previous formula on the real line. Moreover, although the metric is not explicitly present in this expression, it is recovered by the distance. In other terms, encodes the metric of the Riemannian structure.