As said, the stability of “ordinary” material objects rests on the stability of atoms and molecules. This in turn rests on the fuzziness of the relative positions and momenta of their constituents, and it implies an inequality of the form (2.4.3): the product of the “uncertainties” associated with a relative position and the corresponding relative momentum must have a positive lower limit. This makes it impossible to simultaneously measure both quantities with arbitrary precision. In other words, the stability of matter requires, via the stability of atoms, that measurements of the two quantities be *incompatible*.

We now want to determine the conditions under which two elementary tests are compatible.

Let the two possible outcomes of an elementary test T_{1} be represented by the subspaces **M** and **M**, and let the two possible outcomes of another elementary test T_{2} be represented by the subspaces **N** and **N**. Suppose that whenever T_{1} yields **M**, a subsequent performance of T_{2} yields **N**. (This means that **M** is contained in **N**.) It follows that whenever T_{2} yields **N**, the outcome of the first test cannot have been **M**; it must have been **M**. (This means that **N** is contained in **M**.)

There are three possible combinations of outcomes: (i) **M** and **N**, (ii) **M** and **N**, and (iii) **M** and **N**. If the two tests are to be compatible, then for each of these combinations there must be an algorithm assigning to it a probability equal to 1. In other words, there must be

- a line (1-dimensional subspace)
**L**_{1}contained in both**M**and**N**, - a line
**L**_{2}contained in both**M**and**N**, - a line
**L**_{3}contained in both**M**and**N**.

It is easy to see that these requirements are satisfied: **M** is contained in **N**, so **L**_{1} exists. Because **M** is not equal to **N** (otherwise the two tests would be identical), **N** contains a line orthogonal **M**, so **L**_{2} exists. And **N** is contained in **M**, so **L**_{3} exists. This warrants the following conclusion: if either outcome of the first test implies either outcome of the second test, the two tests are compatible.

If neither outcome of the first test implies either outcome of the second test, then compatibility requires, in addition, the existence of

- a line
**L**_{4}contained in both**M**and**N**,

which assigns probability 1 to both **M** and **N**.

It is readily shown that these four requirements are satisfied if and only if

the entire vector space **V** is spanned by four intersections: that of **M** with **N**, that of **M** with **N**, that of **M** with **N**, and that of **M** with **N**.

The *span* of two subspaces **A**, **B** is the smallest subspace containing both **A** and **B**. The *intersection* of two subspaces **A**, **B** is the subspace that contains all vector that are contained in both **A** and **B**.

Our next order of business is to translate this result into the language of projectors.

Some definitions, to begin with:

- The
*norm*of a vector**a**is the positive square root of <**a**|**a**>. - The norm of a
*unit vector*equals unity. - A set of mutually orthogonal vectors
**a**_{1},**a**_{2},**a**_{3},… is*complete*if for every vector**b**in**V**there are real or complex numbers b_{1}, b_{2}, b_{3},… such that**b**= b_{1}**a**_{1}+ b_{2}**a**_{2}+ b_{3}**a**_{3}+ ··· - A
*basis*is a complete set of mutually orthogonal unit vectors. - The numbers b
_{1}, b_{2}, b_{3},… are the*components*of**b***with respect to*the basis**a**_{1},**a**_{2},**a**_{3},….

Next, a couple of facts:

For every subspace **A** of an *n*-dimensional vector space we can find a basis **a**_{1}, **a**_{2}, …, **a**_{n} such that **a**_{1}, …, **a**_{m} span **A**, while **a**_{m+1}, …, **a**_{n} span **A**.

For every *m*-dimensional subspace **A** of an *n*-dimensional vector space there is an operator P_{A} that projects vectors into **A**. Plug in the vector **b** = b_{1}**a**_{1} + ··· + b_{n}**a**_{n}, and out pops the vector P_{A}**b** = b_{1}**a**_{1} + ··· + b_{m}**a**_{m}. In other words, the projector P_{A} cuts off the terms orthogonal to **A**.

And finally, the translation of the above result into the language of projectors: T_{1} (with outcomes **M** and **M**) and T_{2} (with outcomes **N** and **N**) are compatible if and only if

P_{M} and P_{N} commute (that is, if for every vector **b**, P_{M}P_{N}**b** = P_{N}P_{M}**b**).

Be it mentioned in passing that if P_{M} and P_{N} commute, then so do P_{M} and P_{N}, P_{N} and P_{M}, and P_{M} and P_{N}.