8 From classical to quantum

Mate­rial objects occupy as much space as they do because atoms and mol­e­cules occupy as much space as they do. So how is it that a hydrogen atom in its ground state (its state of lowest energy) occu­pies a space roughly one tenth of a nanometer across? Thanks to quantum mechanics, we now under­stand that the sta­bility of “ordi­nary” mate­rial objects rests on the sta­bility of atoms and mol­e­cules, and that this rests on the fuzzi­ness of the rel­a­tive posi­tions and momenta of their con­stituents.

What, then, is the proper — that is, math­e­mat­i­cally rig­orous and philo­soph­i­cally sound — way to define and quan­tify a fuzzy prop­erty? It is to assign non­trivial prob­a­bil­i­ties — prob­a­bil­i­ties between 0 and 1 — to the pos­sible out­comes of a mea­sure­ment of this property.

To be pre­cise, the proper way of quan­ti­fying a fuzzy prop­erty or value is to make coun­ter­fac­tual prob­a­bility assign­ments. (To assign a prob­a­bility to a pos­sible mea­sure­ment out­come coun­ter­fac­tu­ally — con­trary to the facts — is to assign it to a pos­sible out­come of a mea­sure­ment that is not actu­ally made.) In order to quan­ti­ta­tively describe a fuzzy prop­erty or value, we must assume that a mea­sure­ment is made, and if we do not want our descrip­tion to change the prop­erty or value described, we must assume that no mea­sure­ment is made. We must assign prob­a­bil­i­ties to the pos­sible out­comes of unper­formed measurements.

Arguably the most straight­for­ward way to make room for non­trivial prob­a­bil­i­ties is to upgrade from a 0-​​dimensional point P to a 1-​​dimensional line L. Instead of rep­re­senting a prob­a­bility algo­rithm by a point in a phase space, we rep­re­sent it by a 1-​​dimensional sub­space of a vector space V. And instead of rep­re­senting ele­men­tary tests by sub­sets of a phase space S, we rep­re­sent them by the sub­spaces of V.

Exam­ples of vector spaces are lines, planes, and the familiar 3-​​dimensional space in which we appear to exist. Vector spaces con­tain vec­tors (no sur­prise there). Vec­tors can be added, and they can be mul­ti­plied with either real or com­plex num­bers. If only real num­bers are allowed, we can think of vec­tors as arrows, and we can visu­alize the addi­tion of two vec­tors as we did in Fig. 2.3.1. If com­plex num­bers are admitted, we have to go beyond visu­al­iza­tion. (Nobody should expect the inner work­ings of a prob­a­bility algo­rithm to be visualizable.)

What is char­ac­ter­istic of a vector space V is that if the two vec­tors a, b are con­tained in it, and if α and β are real or com­plex num­bers, then αa + βb is also con­tained in it. A vector space U is a sub­space of V if the vec­tors con­tained in U are also con­tained in V.

Math­e­mat­i­cally it is easy to intro­duce vector spaces of more then three (and even infi­nitely many) dimen­sions. The dimen­sion of a vector space is the largest number of mutu­ally orthog­onal (per­pen­dic­ular) vec­tors one can find in it. What makes it pos­sible to define orthog­o­nality for vec­tors is an inner product (a.k.a. scalar product). This is a machine that accepts two vec­tors a, b and returns a real or com­plex number <b|a>. Two vec­tors are orthog­onal if their scalar product van­ishes (equals zero).

A 1-​​dimensional sub­space L can be con­tained in a sub­space U, it can be orthog­onal to U, but it may be nei­ther con­tained in nor orthog­onal to U; there is a third pos­si­bility, and this is what makes room for non­trivial prob­a­bil­i­ties. As a prob­a­bility algo­rithm, L assigns prob­a­bility 1 to ele­men­tary tests rep­re­sented by sub­spaces con­taining L, it assigns prob­a­bility 0 to ele­men­tary tests rep­re­sented by sub­spaces orthog­onal to L, and it assigns prob­a­bil­i­ties greater than 0 and less than 1 to tests rep­re­sented by sub­spaces that nei­ther con­tain nor are orthog­onal to L.

The vir­tual inevitability of this “upgrade” has been demon­strated by J.M. Jauch.[1]

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1. Jauch, J.M. (1968). Foun­da­tions of Quantum Mechanics, Addison–Wesley.