6 Spatial aspects of the quantum world

In a non-​​relativistic world, a max­i­mally fuzzy momentum and, con­se­quently, an infi­nite mean energy would be asso­ci­ated with a sharply local­ized par­ticle. In a rel­a­tivistic world, the attempt to pro­duce a strictly local­ized par­ticle results instead in the cre­ation of particle-​​antiparticle pairs. It is there­fore safe to say that no mate­rial object ever has a sharp posi­tion (rel­a­tive to any other object). This implies some­thing of para­mount impor­tance: the spa­tiotem­poral dif­fer­en­ti­a­tion of the phys­ical world is incom­plete — it does not go “all the way down.”

To see what exactly this means, let R3(O) be the (imag­i­nary) set of exact posi­tions (labeled by triplets of real num­bers) rel­a­tive to some object O. If no mate­rial object ever has a sharp posi­tion, we can con­ceive of a par­ti­tion of R3(O) into finite regions that are so small that none of them is the sen­si­tive region of an actu­ally existing detector. Hence we can con­ceive of a par­ti­tion of R3(O) into suf­fi­ciently small but finite regions Rk — k = 1,2,3,… — of which the fol­lowing is true: there is no object Q and no region Rk such that the propo­si­tion “Q is inside Rk” has a truth value. In other words, there is no object Q and no region Rk such that Rk exists for Q. But if a region of space does not exist for any mate­rial object, it does not exist at all. The regions Rk — or the dis­tinc­tions we make between them — cor­re­spond to nothing in the actual world. They exist solely in our minds.

What holds for the world’s spa­tial dif­fer­en­ti­a­tion holds for its tem­poral dif­fer­en­ti­a­tion as well. The times at which observ­ables pos­sess values, like the pos­sessed values them­selves, must be indi­cated in order to exist. Clocks are needed not only to indi­cate time but also, and in the first place, to make times avail­able for attri­bu­tion to indi­cated values. Since clocks indi­cate times by the posi­tions of their hands, the world’s incom­plete tem­poral dif­fer­en­ti­a­tion fol­lows from its incom­plete spa­tial dif­fer­en­ti­a­tion. (Dig­ital clocks indi­cate times by tran­si­tions from one reading to another, without hands. The uncer­tainty prin­ciple for energy and time, how­ever, implies that such a tran­si­tion cannot occur at an exact time, except in the unphys­ical limit of infi­nite mean energy.[1])


The shapes of things

There is one notion that is decid­edly at odds with the incom­plete spa­tial dif­fer­en­ti­a­tion of the phys­ical world. It is the notion that the “ulti­mate con­stituents” of matter are point­like (or, God help us, string­like[2]). A fun­da­mental par­ticle — in the con­text of the Stan­dard Model, a lepton or a quark — is a par­ticle that lacks internal struc­ture. It lacks internal rela­tions; equiv­a­lently, it lacks parts. This could mean that it is a point­like object, but it could also mean that it is form­less. (In order to leave a vis­ible trace — a string of bub­bles in a bubble chamber, a trail of droplets in a cloud chamber, or some­thing like that — an elec­tron does not need a shape; it only needs to be there. In fact, it was where it was only because its past where­abouts are indi­cated by bub­bles or droplets or some such thing.)

What does the theory have to say on this issue? It obvi­ously favors the latter pos­si­bility, inas­much as nothing in the for­malism of quantum mechanics refers to the shape of an object that lacks internal structure.

And exper­i­ments? While they can pro­vide evi­dence of internal struc­ture, they cannot pro­vide evi­dence of the absence of internal struc­ture. Hence they cannot pro­vide evi­dence of a point­like form.

The notion that an object without internal struc­ture has a point­like form — or any form, for that matter — is there­fore unwar­ranted on both the­o­ret­ical and exper­i­mental grounds.

In addi­tion, it explains nothing. Specif­i­cally, it does not explain why a com­posite object — be it a nucleon, a mol­e­cule, or a galaxy — has the shape that it does, inas­much as all empir­i­cally acces­sible forms are fully accounted for by the rel­a­tive posi­tions (and ori­en­ta­tions) of their mate­rial parts. All it does is encumber our efforts to make sense of the quantum world with a type of form whose exis­tence is com­pletely unver­i­fi­able, which is explana­to­rily com­pletely use­less, and which dif­fers rad­i­cally from all empir­i­cally acces­sible forms. If we reject this notion, we arrive at an appeal­ingly uni­form con­cept of form, since then all forms resolve them­selves into sets of spa­tial rela­tions — between parts whose forms are them­selves sets of spa­tial rela­tions, and ulti­mately between form­less parts.



Con­sider once more the fuzzy posi­tions in Figure 2.6.3.


hydrogen orbitals ray-traced

Figure 2.6.3 The posi­tion prob­a­bility dis­tri­b­u­tions asso­ci­ated with var­ious orbitals of atomic hydrogen.


Does the expanse over which these posi­tions are “smeared out” have parts? If it had, the posi­tions them­selves would have parts; they would be divided by the parts of space. But this makes no sense. One can divide an object, and thereby create as many posi­tions as there are parts (one for each part) or as many rel­a­tive posi­tions as there are pairs of parts, but one cannot divide a posi­tion. The expanse over which a fuzzy posi­tion is prob­a­bilis­ti­cally dis­trib­uted there­fore lacks parts. This con­firms an ear­lier con­clu­sion: if at all we think of space as an expanse, we must think of it as intrin­si­cally undivided.

Instead of attributing the prop­erty of spa­tiality — of being spa­tially extended — to an expanse to which all rel­a­tive posi­tions owe their spa­tial char­acter, we may attribute it to the rel­a­tive posi­tions them­selves. If we do so, there is no need of positing an inde­pen­dently existing expanse; space is nothing but the set of all rela­tions that share this prop­erty. If we think this through, we arrive at the fol­lowing con­clu­sions: Space con­tains the forms of all things that have forms, for the totality of spa­tial rela­tions con­tains — in the proper, set-​​theoretic sense of con­tain­ment — the spe­cific sets of spa­tial rela­tions that con­sti­tute mate­rial forms. What it does not con­tain is the cor­re­sponding relata — the form­less “ulti­mate con­stituents” of matter. And if we give the name of “matter” to these “ulti­mate con­stituents,” it does not con­tain matter either.


1. [↑] Hilgevoord, J. (1998). The uncer­tainty prin­ciple for energy and time. II. Amer­ican Journal of Physics 66 (5), 396–402.

2. [↑] Green, M.B., Schwarz, J.H., and Witten, E. (1988). Super­string Theory, Cam­bridge Uni­ver­sity Press.