1 The standard axioms

Undoubt­edly the most effec­tive way of teaching the math­e­mat­ical for­malism of quantum mechanics is the axiomatic approach. Philo­soph­i­cally, how­ever, this has its dan­gers. Axioms are sup­posed to be clear and com­pelling. The stan­dard axioms of quantum mechanics are nei­ther. Because they lack a con­vincing phys­ical moti­va­tion, stu­dents — but not only stu­dents — tend to accept them as ulti­mate encap­su­la­tions of the way things are.

  • The first stan­dard axiom typ­i­cally tells us that the state of a system S is (or is rep­re­sented by) a nor­mal­ized ele­ment v of a Hilbert space H — in other words, a unit vector in a com­plex vector space that is com­plete (in a tech­nical sense that those who need to know it will know).
  • The next axiom usu­ally states that observ­ables — mea­sur­able quan­ti­ties — are rep­re­sented by self-​​adjoint linear oper­a­tors acting on the ele­ments of H, and that the pos­sible out­comes of a mea­sure­ment of an observ­able O are the eigen­values of O. (If Ov=cv, where c is a number, then v is an eigen­vector of O and c is the cor­re­sponding eigenvalue.)
  • Then comes an axiom (or a couple of axioms) con­cerning the (time) evo­lu­tion of states. Between mea­sure­ments (if not always), states are said to evolve according to uni­tary trans­for­ma­tions, whereas at the time of a mea­sure­ment, they are said to evolve (or appear to evolve) as stip­u­lated by the so-​​called pro­jec­tion pos­tu­late: if O is mea­sured, the sub­se­quent state of S is the eigen­vector cor­re­sponding to the out­come, regard­less of the pre­vious state of S.
  • A fur­ther axiom stip­u­lates that the state of a com­posite system is (or is rep­re­sented by) a vector in the direct product of the respec­tive Hilbert spaces of the com­po­nent systems.
  • Finally there are a couple of axioms con­cerning prob­a­bil­i­ties. According to the first, if S is “in” the state v, and if we do an exper­i­ment to see if it has the prop­erty rep­re­sented by the sub­space con­taining w, then the prob­a­bility of a pos­i­tive out­come is given by Born’s rule. According to the second, if S is “in” the state v, then the expected value of an observ­able O is <v|Ov>.

There is much here that is per­plexing if not simply wrong.

To begin with, what is the phys­ical meaning of saying that the state of a system is (or is rep­re­sented by) a nor­mal­ized vector in a Hilbert space? The reason why this ques­tion seems vir­tu­ally unan­swer­able is that prob­a­bil­i­ties are intro­duced almost as an after­thought. It ought to be stated at the outset that the math­e­mat­ical for­malism of quantum mechanics is a prob­a­bility cal­culus. It pro­vides us with algo­rithms for cal­cu­lating the prob­a­bil­i­ties of mea­sure­ment outcomes.

If the phase space for­malism of clas­sical physics and the Hilbert space for­malism of quantum physics are both under­stood as tools for cal­cu­lating the prob­a­bil­i­ties of mea­sure­ment out­comes, the tran­si­tion from a 0-​​dimensional point in a phase space to a 1-​​dimensional sub­space in a Hilbert space is readily under­stood as a straight­for­ward way of making room for the non­trivial prob­a­bil­i­ties that we need to deal with (and even to define) fuzzy phys­ical quan­ti­ties (which in turn is needed for the sta­bility of “ordi­nary” mate­rial objects).

Because the prob­a­bil­i­ties assigned by the points of a phase space are trivial, the clas­sical for­malism admits of an alter­na­tive inter­pre­ta­tion: we may think of (clas­sical) states as col­lec­tions of pos­sessed prop­er­ties. Because the prob­a­bil­i­ties assigned by the rays of a Hilbert space are non­trivial, the quantum for­malism does not admit of such an inter­pre­ta­tion: we may not think of (quantum) states as col­lec­tions of pos­sessed properties.

Saying that the state of a quantum system is (or is rep­re­sented by) a vector (in lieu of a 1-​​dimensional sub­space) in a Hilbert space, is there­fore seri­ously misleading.

There are two kinds of things that can be rep­re­sented by a vector (or a 1-​​dimensional sub­space) in a Hilbert space: pos­sible mea­sure­ment out­comes and actual mea­sure­ment out­comes. If a pos­sible mea­sure­ment out­come is thus rep­re­sented, it is for the pur­pose of cal­cu­lating its prob­a­bility. If an actual mea­sure­ment out­come is thus rep­re­sented, it is for the pur­pose of assigning prob­a­bil­i­ties to the pos­sible out­comes of whichever mea­sure­ment is made next. If v rep­re­sents the out­come of a max­imal test and if w rep­re­sents a pos­sible out­come of the mea­sure­ment that is made next, then the prob­a­bility of that out­come is |<w|v>|2. (If the Hamil­tonian is not zero, this prob­a­bility is |<w|Uv>|2, U being the uni­tary oper­ator that takes care of the time dif­fer­ence between the two measurements.)

It is essen­tial to under­stand that any state­ment about a quantum system between mea­sure­ments is “not even wrong” in Wolf­gang Pauli’s famous phrase, inas­much as such a state­ment is nei­ther ver­i­fi­able nor fal­si­fi­able. This bears on the third axiom (or couple of axioms), according to which quantum states evolve (or appear to evolve) uni­tarily between mea­sure­ments, which then implies that they “col­lapse” (or appear to do so) at the time of a measurement.

All that can safely be asserted about the time t on which a quantum state func­tion­ally depends is that it refers to the time of a mea­sure­ment — either the mea­sure­ment to the pos­sible out­comes prob­a­bil­i­ties are assigned, or the mea­sure­ment on the basis of whose out­come prob­a­bil­i­ties are assigned. What cannot be asserted without meta­phys­i­cally embroi­dering the axioms of quantum mechanics is that v(t) is (or rep­re­sents) an instan­ta­neous state of affairs of some kind, which evolves from ear­lier to later times. As Asher Peres point­edly observed, “there is no inter­po­lating wave func­tion giving the ‘state of the system’ between mea­sure­ments”.[1]

Again, what could be the phys­ical meaning of saying that observ­ables are (or are rep­re­sented by) self-​​adjoint oper­a­tors? We are left in the dark until we get to the last couple of axioms, at which point we learn that the expected value of an observ­able O “in” the state v is <v|Ov>. The expected value of a mea­sur­able quan­tity is defined as the sum of the pos­sible out­comes of a mea­sure­ment of this quan­tity each mul­ti­plied (“weighted”) by its (Born) prob­a­bility, and a self-​​adjoint oper­ator O can be defined so that this weighted sum takes the form <v|Ov>. That’s all there is to observ­ables “being” self-​​adjoint operators.

And finally, why would the state of a com­posite system be (rep­re­sented by) a vector in the direct product of the Hilbert spaces of the com­po­nent sys­tems? Once again the answer is self-​​evident if quantum states are seen for what they are — tools for assigning prob­a­bil­i­ties to the pos­sible out­comes of measurements.

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To be is to be measured

There is a widely held if not always explic­itly stated assump­tion, which for many has the status of an addi­tional axiom. This is the so-​​called eigenstate-​​eigenvalue link, according to which a system “in” an eigen­state of an observ­able O — that is, a system asso­ci­ated with an eigen­vector of O — pos­sesses the cor­re­sponding eigen­value even O is not, in fact, mea­sured. Because the time-​​dependence of a quantum state is not the con­tin­uous depen­dence on time ofbf an evolving state but a depen­dence on the time of a mea­sure­ment, we must reject this assump­tion. All that Ov(t) = ov(t) implies is that a (suc­cessful) mea­sure­ment of O made at the time t is cer­tain to yield the out­come o. In other words, prob­a­bility 1 is not suf­fi­cient for “is” or “has.”

If a system’s being in an eigen­state of an observ­able is not suf­fi­cient for the pos­ses­sion, by the system or the observ­able, of the cor­re­sponding eigen­value, then what is?

We came across sev­eral exper­i­mental arrange­ments that war­ranted the fol­lowing con­clu­sion: mea­sure­ments do not reveal pre-​​existent values; they create their out­comes. If so, the only suf­fi­cient con­di­tion for the exis­tence of a value o of an observ­able O is a mea­sure­ment of O. Observ­ables have values only if, only when, and only to the extent that they are mea­sured. In short, to be is to be mea­sured. This was the insight that Niels Bohr tried to convey when he kept insisting that, out of rela­tion to exper­i­mental arrange­ments, the prop­er­ties of quantum sys­tems are unde­fined.[2,3]

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1. [↑] Peres, A. (1984). What is a state vector? Amer­ican Journal of Physics 52, 644–650.

2. [↑] Jammer, M. (1974). The Phi­los­ophy of Quantum Mechanics, Wiley, pp. 68–69.

3. [↑] Petersen, A. (1968). Quantum Physics and the Philo­soph­ical Tra­di­tion, MIT Press.