A minimalist's view of quantum mechanics

We analyse a proposition which considers quantum theory as a mere tool for calculating probabilities for sequences of outcomes of observations made by an Observer, who him/herself remains outside the scope of the theory. Predictions are possible, provided a sequence includes at least two such observations. Complex valued probability amplitudes, each defined for an entire sequence of outcomes, are attributed to Observer's reasoning, and the problem of wave function's collapse is dismissed as a purely semantic one. Our examples include quantum “weak values”, and a simplified version of the “delayed quantum eraser”.

this puzzle -that it is the way nature really' is.' In the above quote "we" clearly refers to conscious Observers. The probabilities, on the other hand, tend to be mentioned in literature in at lest three different contexts. Objective probabilities are related to frequencies, with which events occur [10], subjective probabilities describe degree of one's belief [11], and abstract probabilities, satisfying Kolmogorov's axioms [12], are a mathematical concept. In what follows, we will be interested mostly in the first option, thus assuming that the purpose of the theory is to predict relative frequencies of events, or of series of events, should the same experiment(s) be repeated a large number of times under the same conditions. What kind of events should then be considered? Physics is an empirical science, so the "events" must refer to objective experiences, i.e., accessible in principle to any number, or to all, conscious Observers. (Such is, for example, observation of the moon in the night sky, whereas one's dream about the moon must fall outside the remit of physical sciences.) At this point one faces a further choice to be made. Either quantum theory is so universal as to describe the nature of human consciousness, and of life in general, or it is a tool, specifically tailored to and constrained by the limitations of the Observer's perception. The Observer is either a subject of the theory, or its user, in which his/her place is outside the theory's scope. For an imperfect analogy, consider a community of mobile phone users whose ability, for reasons unknown, is limited to enacting applications on the set's screen. After some trying, the users will be able to compile a rule book, similar to a basic operation manual. But, unable to look inside the set, they will ultimately arrive at the level beyond which no further understanding of telephony's principles is possible. Conversely, although these basic rules will say something about how the users communicate, they will provide little insight into the origin of human consciousness. The analogy is imperfect since, unlike the physical world, a smartphone was made by user's peers, and more detailed descriptions of the set's design, Maxwell's theory, and the network's infrastructure are, in principle, available.
Thus, a choice needs to be made, and in the following, we will opt for the second proposition. An assumption that the make up of the world can be known in its entirety is a strong, and a relatively recent one. Arguably, the fact that a theory inevitably arrives at the level where no further explanation is possible, and the nature simply is as it is, may point towards the existence of phenomena, inaccessible to the Observer's experience (cf. the awkward analogy of the previous paragraph). There is little doubt that human observers have only limited perceptive powers, e.g., an ability of directly observing (leave other four senses aside) only surfaces of objects in three dimensional coordinate space. (Hence the need to equip a measuring device with a pointer, whose spatial displacement encodes the value of the measured variable.) Another reason for excluding the Observer from the remit of the theory is that no one has so far observed a state of human consciousness, while little is said to be known even about the consciousness of ants or trees [9]. (This is not to be confused with observation of physical or biochemical precesses in a live organism, accessible to direct or indirect measurements [13]). Even if the required observational technique could, at some stage, be found, the state of one's consciousness will be accessible to all except the conscious person, caught in a bad progression of being aware of being aware ... of being aware of his/hers own state. This, in turn, contradicts the earlier requirement that physics should deal only in phenomena, accessible to all in equal measure. The old view that "inner life of an individual is ... extra-observational by its very nature" [13], and that quantum mechanics should not try to describe the Observer entirely, is currently regaining its popularity. One example can be found in a recent paper by Frauchieger and Renner [14], although valid critique of the analysis was later given in [15].
Having adopted a view that quantum theory is for rather than about conscious Observers, we can move on to more practical issues. We will do so by analysing the case where the events, perceived by an Observer, are the results of observations, made on a elementary quantum system, with which the theory associates a Hamiltonian operatorĤ, and a Hilbert space of a finite dimension N . An Observer may want to measure a variable C, represented by a hermitian operatorĈ, with eigenstates |c n , and eigenvalues C i , some of which can be degenerate. Quantum theory postulates that an accurate measurement of C must yield one of the discrete values C i . The outcome of a measurement cannot be predicted with certainty, but the probability of obtaining a C i is given by where |ψ(t) is the state in which the system is at the time of measurement,π(C i ) = N n=1 |c n ∆(C i − c n |Ĉ|c n ) c n | is the projector onto the state, or a subspace, corresponding to the value C i . [Above we have introduced ∆(X − Y ), which equals 1 if X = Y , and 0 otherwise.] A closer look at eq. (1), which descries a single measure-ment of C, shows that, in fact, it establishes a correlation between two Observer's experiences. An Observer must first determine that the system is indeed in |ψ(t) , prior to the measurement, and only then evaluate the odds on having the outcome C i . The first step can be made by preparing the system with the help of an apparatus, controlled by the Observer, or by measuring, at some t 0 < t, another variable, B with non-degenerate eigenvalues B i , so that obtaining a B j also helps establish that |ψ(t 0 ) = |b j . Either way, with |ψ(t) =Û (t, t 0 )|ψ(t 0 ) , whereÛ (t, t 0 ) = exp −i t t0Ĥ (t )dt , eq. (1) now yields a conditional probability for obtaining first B j and later C i , is a Feynman's transition amplitude [16] for a system which starts in |b j at t 0 and ends up in |c n at t. Importantly, the sequence C i ← B j cannot be reduced further, e.g., to predicting the statistics of measuring B i on its own. There are two compelling reasons why the concept of the state of a system, previously not a subject to an Observer's experience, can have little physical meaning. Suppose, Alice receives a spin-1/2 from a completely unknown source. One way to determine the state it is in would be produce a large number of its identical copies, and perform measurements on the ensemble, created in this manner. But this is forbidden by the no-cloning theorem [17], since the task cannot be performed by means of a unitary evolution, the only kind of evolution allowed by quantum theory. Alternatively, Alice could make a single measurement, but the result will depend on the choice of the measured operator, and cannot, therefore, reveal the true state of the system before Alice's intervention. (See also [18], for a proof that a measured value cannot pre-exist its measurement, and must be produced in the course of it.) We are, therefore, encouraged to shift the focus of our attention from the wave function |ψ(t) in eq. (1), to the transition amplitude A(c n ← b j ) in eq. (3), related the correlations between several events, experienced by the Observer. However, the two-measurements case (2)-(3) is not fully representative of the problem at hand, and we will turn to sequences in which three or more quantities Q , = 1, 2, ..., L, are measured times t , t +1 > t , with the possible outcomes Q 1 i1 , Q 2 i2 , ..., Q L i L (apologies for the cumbersome notations). To predict the probability of a given series of outcomes, P (Q L i L ... ← Q 1 i1 ), one must construct complex valued probability amplitudes for all possible scenarios, add them as appropriate, and take the absolute square of the result [9]. This rule needs to take into account the degeneracies of eigenvalues Q i of the op-eratorsQ , representing the quantities Q , and can be summarised as follows.
(I) Virtual (Feynman) paths. First, one needs to introduce L complete basis sets {|q n }, n = 1, 2, ..., N , in which the operatorsQ are diagonal. Connecting the states at different times t , yields N L virtual paths {q L n L ... ← q 2 n2 ← q 1 n1 }, each endowed with its own probability amplitude, These paths are the elementary building blocks, from which the observable probabilities will later be constructed.
(II) Superposition principle. We will start with the case where first measured eigenvalue, Q 1 i1 , is nondegenerate, thus allowing for one to one correspondence Q 1 i1 ↔ |q 1 i1 , and return to a more general case in (IV) below. Other eigenvalues may, or may not, be degenerate, but different rules apply to the "present", at the last time t = t L , and the "past" at t = t , 1 < < L. If several eigenstates correspond to a "past" valueQ i , one must allow for the interference between the paths, not distinguished by the measurement. In this case, the amplitude for obtaining such a value, and ending up in a state |q L n L , is given by However, no interference is allowed for the paths, leading to different final states, even if the last observed ("present") value Q L i L is degenerate [9]. In this case we have which reduces to a simple Born rule for a non-degenerate The rule demonstrates, for example, that at present a particle cannot be at two different locations in space. Suppose that (we moved from finite-dimensional systems to point particles in one dimension), at t = t L , one measures a projector onto an interval [a, b],π [a,b] = b a |x x|dx. There is no amplitude for being inside [a, b], whose absolute square gives the probability to obtain an eigenvalue 1. Rather, there are probabilities for being at each location inside the interval, whose sum yields the odds on obtaining this eigenvalue. Not so, ifπ [a,b] is measured in the past, at some t < t L , where one has to define an amplitude for passing through the entire interval according to eq. (5), and take its absolute square, as prescribed by eq. (6). This is true in every representation, determined by the observations one wishes to make. We will return to the need for a distinction between the past and the present in the next paragraph, after noting that if Alice, wishes to test the theory, she can prepare a statistical ensemble by measuring aQ 1 , selecting those systems, for which the outcome is a non-degenerate eigenvalue Q 1 i1 , and proceeding to measure the values of the remaining Q . The gathered statistics will then be described by eqs. (6).
(III) Causality and consistency. Causality requires that the observations made in future may not affects the results already experienced. Indeed, it is easy to check that ignoring the outcomes, obtained at t = t L , restores the probabilities (6), for a shorter sequence The rule is also consistent, in the sense that to add one more measurement ofQ L+1 at t L+1 > t L , one should simply relegate the moment t L to the past, and consider ) Equation (9) helps to provide some insight into the form of eq. (6). Suppose, at t L one measures an operator Q L , whose eigenvalues are degenerate. It is then possible, without altering the probability of the previous sequence of outcomes, to measure an operatorQ L+1 , diagonal in one of the bases, in whichQ L is also diagonal. If the eigenvalues ofQ L+1 are all distinct, and the measurement is made immediately after t L , t L+1 → t L , the first factor in the r.h.s. of eq. (9) is a Kronecker delta, Inserting (9) into eq. (5) (with L replaced by L + 1), applying the Born rule (7), and using (8), yields eq. (6) for the probability of observing a sequence This illustrates Feynman's assertion [9] that scenarios, which can be distinguished in principle (in this case, by a future more detailed measurement), are always exclusive. In particular, there can be no interference between the paths leading to orthogonal final states.
(IV) Inconclusive preparation and consistency. Suppose next that the first measurement yields an 1 < m ≤ N -degenerate value Q 1 i1 , with which one associates an M -dimensional sub-space of the system's Hilbert space, spanned by a basis set |u m (Q 1 i1 ) , m = 1, 2, ..., M . This information is not sufficient for assigning to the system a particular initial state, and Alice, who still wishes to create a statistical ensemble (or asked to guess the next outcome given her incomplete knowledge [11]), must make an additional assumption about what that state should be. Consistent with the result Q 1 i1 , the system, prepared in any of the states |u m (Q 1 i1 ) . Assuming that m-th choice is made with a probability ω m ≥ 0, M m=1 ω m = 1, Alice obtains is the probability (6) for the system, which was prepared in a state |u m . With all possible choices of ω m , and of orthonormal bases spanning the M -dimensional subspace, Alice has many options. One does, however, stand out. With no other information available, she can decide to give all |u m (Q 1 i1 ) equal weights, thus choosing Now the probabilities (10) no longer depend on a particular choice of the basis (Note that Alice could as well consider all states in the subspace to be equally probable. A demonstration is straightforward for M = 2, where the states can be parametrised by the polar and azimuthal angles, and the integration of the corresponding projectors over the entire Bloch sphere yields one half of the unity operator.) Furthermore, with the choice (11) made, the rule is consistent in the sense that if Q 1 is a constant quantity, Q 1 = λÎ, and the first measurement yields no information whatsoever, P (Q L i L ... ← Q i .... ← u m ) reduces to the probability of a shorter sequence, as if the first measurement, whose outcome is certain, were not made at all. (V) Composites and separability. With the help of the above, one can treat observations, made on a system of interest (labelled S, with a HamiltonianĤ S ), seen as a part of a larger composite system+environment (labelled E, with a HamiltonianĤ E ), whatever this environment might be. The full Hamiltonian is now given byĤ =Ĥ S +Ĥ E +Ĥ int , where the last term describes the interaction between the S and E. If, for example, the environment is a system in a Kdimensional Hilbert space, the eigenvalues Q i (S) of an operatorQ(S), representing a system's variable Q(S), are at least K-fold degenerate. The probability of a series of outcomes of observations made on the S are, therefore, given by eqs. (4)-(6), withÛ (t +1 , t ) = exp −i t +1 t Ĥ (t )dt . It is easy to check that if the system is completely isolated from the environment, so thatĤ Ĥ E (t )dt , after summing over the degeneracies one recovers eqs. (4)-(6) for the system only, here we have also assumed that at t = t 1 the result of the first measurement corresponds to a composite's product state |q 1 i1 (S) ⊗ |q 1 j1 (E) . A measurement of a more general collective quantity,Q(S + E), may yield Q 1 i1 (S +E), which would leave the composite in an entangled state |q 1 If no further interaction between S and E is possible, application of eqs. (4)-(6) to each term of the sum yields which simplifies to in a special case where φ j (E) are orthogonal, φ(E) j |φ(E) j = δ jj , and the system (S) can be said to start in a state |q 1 j (S) with a probability |β i1j | 2 . Equations (4)-(6) and (10) can be re-written in a compact and, perhaps, more familiar form where, the Heisenberg representation,π(Q i , t ) ≡ U −1 (t , i 1 )π(Q i )Û (t , i 1 ) is the projector onto the eigensubspace, associated with an outcome Q i and ρ(Q 1 i1 ) = M m=1 |u m ω m u m | is the system's density operator [13]. Similar strings of projectors appear, for example, in the consistent histories approach (CHA) [6], but there are important differences. Firstly, while the CHA aims to be a general theory, which includes observers, we put an Observer outside the theory's scope. Secondly, for us eq. (15) is a derived result, and the primary and most basic quantities are the probability amplitudes (4) and (5).
As an example where this difference is important, consider the case where the system is "pre-and postselected" in the states |q 1 i1 and |q 3 i3 at t 1 and t 3 , and in eq. (13) the role of environment is played by a von Neumann pointer [13], set up to measure someQ 2 at t 1 < t 2 < t 3 . The measurement can more accurate, or less accurate, yet any information about the system, gained from the pointer's final position, will have to be expressed in terms of an amplitude A(q 3 i3 ← Q 2 i2 ← q 1 i1 ) in eq. (5) [19]. IfQ 2 = |q 2 m q 2 m | is a projector onto a state |q 2 m , and the coupling to the pointer is small (the accuracy of the measurement is poor), the average shift of the pointer, f , turns out to be given by i1 . The last expression in eq. (16) was first obtained in [20], where the complex valued fraction in brackets was called "the weak value (WV) of the opera-torQ 2 ". Written in this way, a WV looks like a physical variable of a new kind [21], whose physical significance is still discussed in the literature (see, for example [22], [23]). However the first expression in the r.h.s. of eq. (16) identifies it as a previously known renormalised Feynman amplitude (or a weighted sum of such amplitudes if a more generalQ 2 is inaccurately measured) [24] - [29]. The problem is, little known about the probability amplitudes, apart from their relation to the observable frequencies, discussed above. Until, or, unless a deeper insight into the physical meaning of quantum amplitudes is gained, such an inaccurate "weak" measurement will remain merely an exercise in recovering the values of transition amplitudes from a response of a system to a small perturbation [19]. Our first simple example concerns a pair of entangled spins, travelling in opposite directions, and can be found in the Appendix A. As a further illustration of the use of the transition amplitudes, we revisit, in its simplest version, the "delayed choice quantum eraser experiment [31]. Figure 1a sketches a primitive doubleslit experiment, in which a two-level system (S) (a spin-1/2), which an observed outcome B 1 has left in a state |b 1 , is subjected to a later measurement of an operator C(S) = C 1 |c 1 c 1 | + C 2 |c 2 c 2 | at some t = t 2 . The final state |c 1 , playing the role of a point on the screen, can be reached via passing, at a t 1 < t 2 , through a pair of orthogonal states | ↑ and | ↓ , representing the two slits. The two virtual paths in the two-dimensional Hilbert space (Fig.1a, solid) interfere, and the probability to have C 1 (S) is given by (spin has no own dynamics) In Fig.2b, initial measurement of a collective variablê B(S + E) entangles the system with a two-level "environment" (E), whose orthogonal states are | + and | − . As before,Ĉ(S) is measured at t 2 , and then an environment's variableD(E) = D 1 |d 1 d 1 | + D 2 |d 2 d 2 | is measured at a t 3 > t 2 . Four relevant virtual paths in the now four-dimensional Hilbert space (solid lines in Fig.1b) are endowed with probability amplitudes Using the rules (I)-(VI), for the probabilities of the sequences of outcomes shown in Fig.1c, we easily find Much of the interest in the above scheme stems from the fact that while there is no interference term ∼ A * 1 A 2 in Eq. (20), this term reappears in P ( It is tempting to conclude that the coherence between the paths 1 and 2, that was lost after measuringĈ(S) at t 2 is somehow restored if the second system, (E), is found in |d 1 . All the more surprising is that this seems to happen after the outcome C 1 has already been observed. However, Fig.1b shows that here we are not comparing like with like. In Fig.1b, the interference term is controlled by the magnitudes of the amplitudes of two virtual paths, I and II, which connect states in a different four-dimensional Hilbert space of the composite system, and the argument cannot be reduced to to a discussion of the individual paths shown in Fig.1a. Quantum theory does its job of calculating probabilities for the outcomes in Fig.1c in an explicitly causal manner, and, we suspect, cannot be asked to do more than that. (For other recent attempts at "demystifying" the delayed eraser experiment we refer the reader to Refs. [32], [33]. ) We can now sum up the "minimalist view", advertised in the title. Quantum theory is a tool, allowing a conscious Observer to predict statistical correlations between the results of two or more of his/hers observations, first of which is needed to "prepare the system". With a particular series of results in mind, he/she may reason about its likelihood by associating with each outcome a state, or states, in a Hilbert space, including all systems which interact with each other during the time interval considered. A probability amplitude for the entire series is then constructed using the prescriptions (I)-(V), and taking its absolute square yields the required probability. Such attributes of the theory as amplitudes, Hilbert spaces, operators, and Hamiltonians are essentially Hertz's "symbols of external objects, formed by ourselves", "whose consequences are always the necessary consequences in the nature of the thing pictured" [1]. (A brief discussion, which we expect to be consistent with H. Hertz's original view is given in the Appendix B.) Observer's main effort then goes into identifying a system's HamiltonianĤ, and the operatorsQ 1 ,Q 2 ,...Q L , whose spectra contain all possible observational outcomes . As a tool, tailored to Observer's limited abilities, the theory is unable to progress beyond certain explanatory level, where it must admit, as in the opening quote, that nature simply is this way. For the same reason, Observer's conscience is not a valid subject of quantum theory, which must loose its pretence (if any) at explaining the world in its entirety, and give way to other complementary endeavours. Several other remarks can be in order. Firstly, the symbolic status of probability amplitudes does not preclude that their values can, under certain conditions, be deduced from the measured probabilities [19]. Indeed, this was done, for example, in the experiments reported in [34] and [35]. Secondly, different sets of measurements, e.g., of the quantities Q and Q , made on the same system, may produce essentially different statistical ensembles [30], [28]. For example, less detailed probabilities (some ofQ l 's eigenvalues are degenerate) cannot be obtained by adding the most detailed ones (obtained for aQ l , whose eigenvalues are all non-degenerate). This is, of course, a more elaborate version of a double slit experiment, where the price of knowing the way a particle has taken is the loss of the interference pattern on the screen. Finally, so far we made no mention of the collapse of the wave function. The possibility of avoiding this issue altogether, is precisely the point we intend to make here. An Observer, whose reasoning only requires him/her to evaluate certain matrix elements in an abstract space, may discard the "collapse problem" as a purely semantic one. It is, of course, possible to argue that in eq. (1) the evolution of the state |q L−1 n L−1 is mysteriously interrupted at t = t L , but it is equally possible not to enter into this argument at all. Conceptual economy from not having to worry about the fate of the wave function can be significant, as one avoids dealing with a universe which splits every time a measurement is made as it happens, for example, in Everett's many worlds (MW) picture [5]. Curiously, in 1995 M. Price [36] polled physicists in order to determine the level of support for the MW approach, and counted Feynman among the supporters. We note that Feynman's support must have been lukewarm at best. In [30] one reads: " Somebody mumbled something about a many-world picture, and that many-world picture says that the wave function ψ is what's real, and damn the torpedoes if there are so many variables, N R . All these different worlds and every arrangement of configurations are all there just like our arrangement of configurations, we just happen to be sitting in this one. It's possible, but I'm not very happy about it." To conclude, we note that the proposed viewpoint imposes strict limits and, if adopted, will have implications for such concepts as the "universal wave function" [37], for attempts to construct a quantum theory where no special role is given to an observer, [4]-[6], or for collapserelated theories of quantum mind [38]. None of these matters are trivial, and cannot be dismissed out of hand. The format of this Letter does not allow for detailed comparisons, so our purpose here was to articulate a maximally reduced view, which can later be extended, modified, or abandoned. For instance, it is possible that the simple model, used to illustrate it, will not suffice when dealing with extremely large or complex systems, or where the relativistic effects of various kinds need to be taken into consideration. It is also possible that, contrary to the Feynman's quote at the beginning of this article, quantum analysis has not yet arrived at its explanatory limit. If so, a more sophisticated theory will have to provide a further insight into the meaning of the transition probability amplitudes, which so far have played the role of basic elements of a quantum analysis.

I. APPENDIX A. A SIMPLE EXAMPLE: TWO SPINS 1/2 IN AN ENTANGLED STATE
Here we revisit a well known case of two spin-1/2 particles, 1 and 2, prepared in one dimensional wave packet states, |φ(±p 0 , 1, 2) , moving in opposite directions, and polarised up and down the z-axis, respectively. The spins do not interact with each other, there are no external fields, and the full Hamiltonian acts only on the spatial degrees of freedom,Ĥ =p 2 1 /2µ +p 2 2 /2µ. Reversing the spins' directions and adding up the two states one can entangle the spins in a state (for clarity we use the tensor product sign ⊗) where the "system" (S) consists of two spins, and the "environment" (E) includes the spatial degrees of freedom. By a time t 2 , after two wave packets have moved well away from each other, Alice measures the component the first spin along a direction, n = (θ, ϕ = 0),Q(S) 2 = σ n (1), Q(S) 2 i2 = ±1. At a later time t 3 ≥ t 2 Bob measures his spin along a different direction, n = (θ , ϕ = 0), Q(S) 3 =σ n (2), Q(S) 3 i3 = ±1, and we are interested in the four probabilities P (Q(S) 3 i3 ←Q(S) 2 i2 ← q(S +E) 1 i1 ). Since there is no spin-orbit interaction, no matter how far apart the wave packets are, we can use (V) to eliminate the environment, and consider only degrees of freedom of the two spins, prepared in a state, There are four basis states, and sixteen virtual paths {K ← J ← q 1 j1 }, J, K = I, II, III, IV shown in Fig.2. However, only four of them have non-zero amplitudes, These four paths lead to orthogonal filial states, and cannot interfere. A simple calculation yields the probabilities for Alice's and Bob's outcomes If Alice and Bob choose to measure along the same axis, θ = θ , they are guaranteed to find their respective spins pointing in the opposite directions, P (1 ← −1 ← q(S + E) 1 i1 ) = P (−1 ← 1 ← q(S + E) 1 i1 )) = 1/2. (This is true even if the distance between the wave packets is so large, that light cannot travel it in t 3 − t 2 seconds, and the information about Alice's result cannot reach Bob in time.) Needless to say, we haven't provided a new "explanation" for the entanglement phenomenon, since the probabilities in eqs. (25) can be obtained directly from (15). We did, however, provide a more detailed illustration of the fact that in elementary quantum mechanics correlations between parts of a system cannot depend on an environment, with which the system does not interact. There is simply no provision for such a dependence, even when the environment represents spatial positions of the system's parts.

II. APPENDIX B. OBSERVER'S ABILITIES AND LIMITATIONS
In Fig. 3 we tried to sketch a scheme, broadly consistent with the approach of [1]. In nature a phenomenon B always follows (is caused by) a phenomenon A. Neither of the two are fully accessible to the Observer (O), who, limited by his/her five senses, can only perceive some indications of A and B, namely A and B. In order to establish a connection between observations A and B, O associates A with a symbol a, A → a from his/her theoretical toolkit. He/she then uses the theory to reason about the consequences of a, the result being a new symbol b, a → b. Observer's theory is correct, if b corresponds to the observed result B, b ↔ B, and false otherwise. Observer's toolkit may include mathematical methods and concepts such as space, time, trajectories, forces, Lagrangians, Hilbert spaces, wave functions, etc. Next we must answer, at least to ourselves, the following questions. Do these items reflect the Observer's limited ability to reason about his/her limited experiences of an in principle intractable world? Or does his/her logical reasoning provide a deeper insight into the actual inner working of nature, hidden from sensual perception? Here we follow [1] in assuming the former. Classical mechanics is an elegant complete theory, yet it is no longer possible to insist that objects really obey Newton's laws, now that the theory has beed superseded by quantum mechanics. Quantum theory comes with its own conceptual baggage, and its even more difficult to attribute it solely to nature, leaving the human mind outside the picture. To use the well known quote by Asher Peres [39]: " Quantum phenomena do not occur in a Hilbert space. They occur in a laboratory". We conclude with a simple example. The laboratory setup consist of a copper coil, connected to an battery, a small object, S, (a spin-1/2 with associated magnetic moment), invisible to the naked eye, and meter (e.g., some sort of Stern-Gertach machine), M , which after a brief interaction with the spin gives one of two results, ±1. Alice switches the meter on, and something (A) happens between S and M . Alice can see (A) is that the meter's reading is +1. She begins to reason by associating condition of the spin (a), with a vector, | ↑ z in a twodimensional Hilbert space. The coil is represented by a magnetic field B, directed along the x-axis. Alice writes down an evolution operatorÛ (t) = cos(ωt) + i sin(ωt)σ x where ω ∼ B is the Larmor frequency, an amplitude for being able to attribute to the spin the same state | ↑ z A(↑ z ←↑ z ) = ↑ z |Û (t)|| ↑ z = cos(ωt), (26) and the probability to see the meter reading +1, if its is enacted after a time t = 2π/ω (B), P (1 ← 1) = |A(↑ z ←↑ z )| 2 = cos 2 (ωt) = 1.
Alice's prediction (b) that she will always see a result +1 can be compared with the actual meter's reading (B) turns out to be true -her theory is correct. The situation is a little more complicated if Alice wants top make predictions for an arbitrary time t. In general, she cannot say what the meter will read in each individual case. Alice can, however, prepare many copies oh the same setup, switch all meters at the same time and select N >> 1 cased where the result is +1. Now A in Fig.3 is a collective event, and Alice's outcome A is seeing N positive meters' reading. At t all meters are enacted again, and the outcome observed by Alice (B) is the ratio N + /N , where N + is the number of positive readings. For large N Alice has N + /N → cos 2 (ωt), as predicted by her theory. Note that neither Alice, nor other human observer, can know what "really" happens in the lab in full detail. But a causal link in the sense "if first A, then later B", has been established for certain phenomena, accessible to Alice's five senses. Collapse Theories, The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Editor E. N. Zalta,, URL =