2 Time symmetric quantum mechanics

Why is the fun­da­mental the­o­ret­ical frame­work of con­tem­po­rary physics a prob­a­bility cal­culus, and why are the events to which this assigns prob­a­bil­i­ties mea­sure­ment out­comes?

It seems to me that all pre­vious attempts to arrive at sat­is­fac­tory answers have foundered on two assump­tions that are at odds with the theory’s onto­log­ical impli­ca­tions. The first is the notion that physics can be neatly divided into kine­matics, which con­cerns the descrip­tion of a system at an instant of time, and dynamics, which con­cerns the evo­lu­tion of a system from ear­lier to later times. We may call this notion “the prin­ciple (or par­a­digm) of evolution.”

By itself, this prin­ciple implies that time is com­pletely dif­fer­en­ti­ated — or else that the world is com­pletely dif­fer­en­ti­ated with respect to time. It per­mits us to think of time as a set of dura­tion­less instants. When com­bined with the spe­cial theory of rel­a­tivity — specif­i­cally, its frame-​​dependent strat­i­fi­ca­tion of space­time into hyper­planes of con­stant time — it implies that space, too, is com­pletely dif­fer­en­ti­ated — or else that the world is com­pletely dif­fer­en­ti­ated with respect to space. This per­mits us to think of it as a set of exten­sion­less points. We may call these notions “the prin­ciple (or par­a­digm) of com­plete spa­tiotem­poral differentiation.”

In keeping with the prin­ciple of evo­lu­tion, the wave func­tion psi(x,t) is usu­ally awarded pri­mary status, while the prop­a­gator <B,tB|A,tA> is seen as playing a sec­ondary role, notwith­standing that both encap­su­late the same infor­ma­tion. At the bottom of this par­tiality lies the erro­neous notion that wave func­tions — and quantum states in gen­eral — are con­structs that are mean­ingful even in the absence of mea­sure­ments. If this were the case, mea­sure­ments would merely con­tribute (or even merely appear to con­tribute) to deter­mine quantum states, and if that were the case, quantum states would deter­mine prob­a­bil­i­ties that are absolute. In reality, the prob­a­bil­i­ties deter­mined by quantum states are always con­di­tional on mea­sure­ment outcomes.

The preva­lence of these mis­taken ideas can be traced back to two for­tu­itous cases of his­tor­ical prece­dence: that of Schrödinger’s “wave mechanics” over Feynman’s propagator-​​based approach, and that of Kolmogorov’s for­mu­la­tion of prob­a­bility theory[1] over an axiomatic alter­na­tive devel­oped by Rényi.[2,3] (Every result of Kolmogorov’s theory has a trans­la­tion into Rényi’s, but whereas in Kolmogorov’s theory absolute prob­a­bil­i­ties have pri­macy over con­di­tional ones, Rényi’s theory is based entirely on con­di­tional probabilities.)

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The time-​​symmetry of quantum mechanics

The prin­ciple of evo­lu­tion is more­over at odds with the time-​​symmetry of quantum mechanics. For Born’s rule can be used not only to pre­dict the prob­a­bil­i­ties of the pos­sible out­comes of a later mea­sure­ment on the basis of the actual out­come of an ear­lier mea­sure­ment but also to retro­dict the prob­a­bil­i­ties of the pos­sible out­comes of an ear­lier mea­sure­ment on the basis of the actual out­come of a later mea­sure­ment. Quantum mechanics even allows us to assign prob­a­bil­i­ties sym­met­ri­cally with respect to time, on the basis of both ear­lier and later out­comes. Positing an inter­po­lating quantum state evolving from later to ear­lier times would there­fore be just as legit­i­mate — or, rather, ille­git­i­mate — as positing an inter­po­lating quantum state evolving from ear­lier to later times.

The dif­fer­ence between the two uses of Born’s rule dimin­ishes if we think in terms of the ensem­bles needed to (approx­i­mately) mea­sure Born prob­a­bil­i­ties. Such ensem­bles can be post­s­e­lected as well as pre­s­e­lected. To pre­s­e­lect an ensemble is to take into account only those instances of a mea­sure­ment per­formed at the ear­lier time t1 that yield a par­tic­ular out­come v. The pre­s­e­lected ensemble — an ensemble of iden­ti­cally pre­pared sys­tems — serves to mea­sure, as rel­a­tive fre­quen­cies, the prob­a­bil­i­ties of the pos­sible out­comes of a mea­sure­ment per­formed at the later time t2. To post­s­e­lect an ensemble is to take into account only those instances of a mea­sure­ment per­formed at t2 that yield a par­tic­ular out­come w. The post­s­e­lected ensemble — an ensemble of iden­ti­cally retro­pared sys­tems — serves to mea­sure, as rel­a­tive fre­quen­cies, the prob­a­bil­i­ties of the pos­sible out­comes of a mea­sure­ment per­formed at t1.

To obtain the rule to be used for assigning prob­a­bil­i­ties on the basis of both ear­lier and later out­comes, we assume that three mea­sure­ments are per­formed at the respec­tive times t1<t2<t3, that the mea­sure­ment at t1 yields the out­come u, and that the mea­sure­ment at t3 yields the out­come w. We then cal­cu­late the prob­a­bility p(w,v|u) of obtaining v at t2 and w at t3, given that u has been obtained at t1. This is the product of two prob­a­bil­i­ties: the prob­a­bility |<v|u>|2 of obtaining v given u and the prob­a­bility |<w|v>|2 of obtaining w given v:

p(w,v|u) = |<w|v> <v|u>|2.

Next we cal­cu­late the prob­a­bility pV(w|u) of obtaining w given u, and given that a mea­sure­ment M with the pos­sible out­come v is made at t2 (whose actual out­come is not taken into account). This prob­a­bility is obtained by adding the prob­a­bil­i­ties p(w,vk|u), k = 1,2,3…, for all pos­sible out­comes of M:

pV(w|u) = |<w|v1> <v1|u>|2 + |<w|v2> <v2|u>|2 + |<w|v3> <v3|u>|2 + ···
pV(w|u) = Σk |<w|vk> <vk|u>|2.

If we want to mea­sure the prob­a­bility p(v|w,u) of a par­tic­ular out­come v of M, given both the ini­tial out­come u and the final out­come w, we use an ensemble that is both pre– and post­s­e­lected — in other words, an ensemble of phys­ical sys­tems that are both iden­ti­cally pre­pared and iden­ti­cally retro­pared. To create this ensemble, we take into account only those instances in which the ini­tial mea­sure­ment yields u and the final mea­sure­ment yields w. We dis­card all runs in which either the ini­tial mea­sure­ment yields an out­come dif­ferent from u and/​or the final mea­sure­ment yields an out­come dif­ferent from w.

We gather from this that in order to cal­cu­late p(v|w,u), all we have to do is divide p(w,v|u) by pV(w|u). The result is known as the ABL rule:[4]

p(v|w,u) = |<w|v> <v|u>|2 /​ Σk |<w|vk> <vk|u>|2.

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An exper­i­ment with three holes

It will be instruc­tive to apply the ABL rule to the fol­lowing setup. Instead of a plate with two slits we have a plate we three holes, labeled A, B, and C. In front of the plate and equidis­tant from the holes there is a par­ticle source (say, an elec­tron gun G), and behind the plate, again equidis­tant from the holes, there is a par­ticle detector D. Finally, some­where between C and D there is a device that causes a phase shift by 180°, which means that the ampli­tude of the cor­re­sponding alter­na­tive gets mul­ti­plied by the com­plex number [1:180°] = −1.

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The experiment with three holes

The exper­i­ment with three holes, illus­trating the con­tex­tu­ality (in gen­eral) of time-​​symmetric prob­a­bil­i­ties cal­cu­lated with the help of the ABL rule.

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An elec­tron emerging from the holes is thus pre­pared “in” the state

u = (A + B + C)/√3

and retro­pared “in” the state

w = (A + BC)/√3.

We now intro­duce an appa­ratus MA that can indi­cate whether or not a par­ticle went through A. The pos­sible out­comes of this mea­sure­ment are rep­re­sented by two pro­jec­tors, one pro­jecting into the 1-​​dimensional sub­space con­taining the vector A, and one pro­jecting into the 2-​​dimensional sub­space con­taining the vec­tors B and C. If we now cal­cu­late the prob­a­bility p(A|w,u) of finding that the par­ticle has gone through A, given that it was pre­pared as described by u and retro­pared as described by w, we obtain the result p(A|w,u) = 1. Given the manner in which the par­ticle is pre– and retro­pared, we are sure to find that it went through A.

Instead of MA, we next intro­duce an appa­ratus MB that can indi­cate whether or not a par­ticle went through B. The pos­sible out­comes of this mea­sure­ment are rep­re­sented by two dif­ferent pro­jec­tors, one pro­jecting into the 1-​​dimensional sub­space con­taining B, and one pro­jecting into the 2-​​dimensional sub­space con­taining A and C. If we now cal­cu­late the prob­a­bility p(B|w,u) of finding that the par­ticle has gone through B, given that it was pre­pared as described by u and retro­pared as described by w, we obtain the result p(B|w,u) = 1. Given the manner in which the par­ticle is pre– and retro­pared, we are sure to find that it went through B.

Finally we use an appa­ratus that can indi­cate through which of the three holes a par­ticle went. The pos­sible out­comes of this mea­sure­ment are rep­re­sented by three pro­jec­tors, one pro­jecting into the 1-​​dimensional sub­space con­taining A, one pro­jecting into the 1-​​dimensional sub­space con­taining B, and one pro­jecting into the 1-​​dimensional sub­space con­taining C. If we now cal­cu­late the prob­a­bil­i­ties p(A|w,u), p(B|w,u), and p(C|w,u), we obtain the results

p(A|w,u) = p(B|w,u) = p(C|w,u) = 13.

Time-​​symmetric prob­a­bility assign­ments thus are (in gen­eral) con­tex­tual. If a par­ticle launched at (or by) G and detected at (or by) D, the prob­a­bility with which one finds that it went through A depends on the pos­sible out­comes of the inter­me­diate mea­sure­ment. If the only pos­sible out­comes are “through A” and “not through A,” then p(A|w,u) = 1. If the only pos­sible out­comes are “through B” and “not through B,” then p(B|w,u) = 1. And if the pos­sible out­comes are “through A,” “through B,” and “through C”, then the three out­comes are equally likely.

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1. [↑] Kol­mogorov, A.N. (1950). Foun­da­tions of the Theory of Prob­a­bility, Chelsea Pub­lishing Company.

2. [↑] Rényi, A. (1955). A new axiomatic theory of prob­a­bility. Acta Math­e­matica Acad­emia Sci­en­tiarum Hun­gar­icae 6, 285–335.

3. [↑] Rényi, A. (1970). Foun­da­tions of Prob­a­bility, Holden–Day.

4. [↑] Aharonov, Y. Bergmann, P.G., and Lebowitz, J.L. (1964). Time sym­metry in the quantum process of mea­sure­ment. Phys­ical Review B 134, 1410–1416.