11 The classical limit

In the clas­sical limit, the quantum-​​mechanical prob­a­bility cal­culus, which assigns non­trivial prob­a­bil­i­ties (prob­a­bil­i­ties between 0 and 1) to mea­sure­ment out­comes, degen­er­ates into a prob­a­bility cal­culus that assigns trivial prob­a­bil­i­ties (either 0 or 1) to mea­sure­ment out­comes. In the clas­sical theory, a prob­a­bility algo­rithm based on the max­imum of in prin­ciple avail­able infor­ma­tion is rep­re­sented by a point P in a phase space, and mea­sure­ment out­comes are rep­re­sented by sub­sets of this space. Because P assigns trivial prob­a­bil­i­ties, it can be thought of as a state in the clas­si­cal sense: a col­lec­tion of pos­sessed prop­er­ties. We are licensed to believe that if the prob­a­bil­ity of find­ing a given prop­erty is 1, it is because the sys­tem pos­sesses this prop­erty, inde­pen­dent of mea­sure­ments. In the quantum theory, a prob­a­bility algo­rithm based on the max­imum of in prin­ciple avail­able infor­ma­tion is rep­re­sented by a 1-​​dimensional sub­space L of a com­plex vector space, and mea­sure­ment out­comes are rep­re­sented by the sub­spaces of this space. Because L assigns non­trivial prob­a­bil­i­ties, it cannot be thought of as a state in the clas­si­cal sense.

Hence it may be said that the quantum laws, which cor­re­late the prob­a­bil­i­ties of mea­sure­ment out­comes prob­a­bilis­ti­cally, degen­erate into laws that cor­re­late pos­sessed prop­er­ties deter­min­is­ti­cally. And because deter­min­istic cor­re­la­tions between pos­sessed prop­er­ties lend them­selves to causal inter­pre­ta­tions, it may be said that the quantum-​​mechanical algo­rithms, which serve to com­pute prob­a­bil­i­ties of pos­sible mea­sure­ment out­comes on the basis of actual out­comes, degen­erate into algo­rithms that serve to com­pute effects that the prop­er­ties of mate­rial objects have on the prop­er­ties of mate­rial objects. What they don’t is degen­erate into (descrip­tions of) phys­ical mech­a­nisms or processes by which the prop­er­ties of mate­rial objects act on the prop­er­ties of mate­rial objects. As David Mermin[1] rem­i­nisced near the end of a dis­tin­guished career:

When I was an under­grad­uate learning clas­sical elec­tro­mag­netism, I was enchanted by the rev­e­la­tion that elec­tro­mag­netic fields were real. Far from being a clever cal­cu­la­tional device for how some charged par­ti­cles push around other charged par­ti­cles, they were just as real as the par­ti­cles them­selves, most dra­mat­i­cally in the form of elec­tro­mag­netic waves, which have energy and momentum of their own and can prop­a­gate long after the source that gave rise to them has vanished.

That lovely vision of the reality of the clas­sical elec­tro­mag­netic field ended when I learned as a grad­uate stu­dent that what Maxwell’s equa­tions actu­ally describe are fields of oper­a­tors on Hilbert space. Those oper­a­tors are quantum fields, which most people agree are not real but merely spec­tac­u­larly suc­cessful cal­cu­la­tional devices. So real clas­sical elec­tro­mag­netic fields are nothing more (or less) than a sim­pli­fi­ca­tion in a par­tic­ular asymp­totic regime (the clas­sical limit) of a clever cal­cu­la­tional device. In other words, clas­sical elec­tro­mag­netic fields are another clever cal­cu­la­tional device.

In other words, Feynman’s assess­ment of quantum physics[2] — “There are no ‘wheels and gears’ beneath this analysis of Nature” — applies equally to clas­sical physics. This should not come as a sur­prise. After all, in making the tran­si­tion from quantum to clas­sical we dis­card a con­sid­er­able amount of infor­ma­tion and, in so doing, sac­ri­fice a great deal of explana­tory power. What would be sur­prising is if clas­sical physics could explain what quantum physics can’t explain.

Having said this, it shouldn’t be nec­es­sary to give another thought to what the approx­i­mate laws of clas­sical physics tell us about the nature of Nature. Being approx­i­mate laws, they tell us nothing of the sort. As Feynman stressed at the begin­ning of his famous Cal­tech lec­tures,[3]philo­soph­i­cally we are com­pletely wrong with the approx­i­mate law” [orig­inal emphasis]. If it is nev­er­the­less nec­es­sary to dwell on the “world­view” of clas­sical physics, it is because the clas­sical laws are all but uni­ver­sally taught before the quantum laws, and because along with the clas­sical laws most stu­dent are force-​​fed a con­sid­er­able amount of meta­phys­ical embroi­dery that cannot but frus­trate their later efforts to make sense of the quantum laws. Mermin[1] was able to rid him­self of “our habit of inap­pro­pri­ately reifying our suc­cessful abstrac­tions,” but how many physi­cists are? In answer to this ques­tion, it may not be inap­pro­priate to quote philoso­pher of sci­ence Dennis Dieks:

As quantum theory is the reigning phys­ical par­a­digm, one would expect that physi­cists have a quantum mechan­ical world­view in the back of their minds when pur­suing their research. How­ever, it has become obvious by now that that is not the case. Still, the need for a phys­ical world­pic­ture makes itself felt. The reac­tion of most physi­cists is to sub­sti­tute a kind of common-​​sense, quasi-​​classical, pic­ture for the quantum mechan­ical one that really would be needed. Most physi­cists think of quantum objects as very small copies of everyday objects, and in effect use the con­cep­tual scheme of clas­sical physics. Of course, they know that some­thing is wrong here, and that a con­sis­tent use of clas­sical ideas will lead into trouble. But that problem rarely presents itself in an acute form. There is always the math­e­mat­ical for­malism of quantum mechanics with which cal­cu­la­tions that lead to obser­va­tional con­se­quences are made; the pic­tures asso­ci­ated with the theory do not really intrude into those cal­cu­la­tions. Because the math­e­mat­ical for­malism is con­sis­tent, no incon­sis­ten­cies will be encoun­tered when making pre­dic­tions. Para­doxes and bewil­der­ment only occur if one won­ders about how the cal­cu­lated and pre­dicted exper­i­mental out­comes can be real­ized by nat­ural processes.[4]

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1. [↑] [↑] Mermin, N.D. (2009). What’s bad about this habit? Physics Today 62 (5), 8–9.

2. [↑] Feynman, R.P. (1985). QED: The Strange Theory of Light and Matter, Princeton Uni­ver­sity Press, p. 78.

3. [↑] Feynman, R.P., Leighton, R.B., and Sands, M. (1963). The Feynman Lec­tures in Physics I, Addison–Wesley, pp. 1–2.

4. [↑] Dieks, D. (1996). The Quantum Mechan­ical World­pic­ture and Its Pop­u­lar­iza­tion. Com­mu­ni­ca­tion & Cog­ni­tion 29 (2), pp. 153–168.