The Contradiction between Two Versions of Quantum Theory Could Be Decided by Experiment

The incompatibility of Orthodox Quantum Mechanics with philosophical realism poses a serious challenge to scientists upholding such a philosophical doctrine. The desire to find a solution to this and other conceptual problems that quantum mechanics confronts has motivated many authors to propose alternative versions to Orthodox Quantum Mechanics. One of them is the Spontaneous Projection Approach, a theory grounded on philosophical realism. It has been introduced in previous papers and, with a few exceptions, it yields experimental predictions coincident with those of Orthodox Quantum Mechanics. One of these exceptions is analyzed in detail. The difference in predictions becomes apparent in a suggested experiment which could put both theories to the test.


Introduction and Outlook
Realism is a philosophical doctrine that revolves around two theses: the first (or ontological thesis) is that the world exists by itself as opposed to being the product of human mind; the second (or epistemological thesis) is that it can be known gradually and approximately [1]. Among authors adopting realism let us mention Albert Einstein and Mario Bunge. For Einstein, "The belief in an external world independent of the perceiving subject is the basis of all natural science" [2]. For Bunge, "Epistemological realism is the family of epistemologies which assume that a) the world exists independently of the knowing subject, and b) the task of science is to produce maximally true conceptual models of reality..." ([3] pp. [191][192].
In 1930, Paul Dirac published the first formulation of quantum mechanics [4].
Two years later John von Neumann published Mathematische Grundlagen der Quantenmechanik [5]. These first versions of quantum theory share two characteristics: 1) the state vector ψ (wave function ψ ) describes the state of an individual system, and 2) they involve two laws of change of the state of the system: spontaneous (natural) processes, governed by the Schrödinger equation; and measurement processes, ruled by the projection postulate. Many other versions of quantum theory followed. Those where ψ describes the state of an individual system and the projection postulate is included among its axioms are generally called standard, ordinary or Orthodox Quantum Mechanics (OQM), sometimes referred to as the Copenhagen Interpretation.
On the one hand, OQM is very successful a theory. "[It has provided] a strikingly successful recipe for doing calculations that accurately described the outcomes of experiments... [It has been] instrumental in predicting antimatter, understanding radioactivity (leading to nuclear power), accounting for the behavior of materials such as semiconductors, explaining superconductivity and describing interactions such as those between light and matter (leading to the invention of the laser) and of radio waves and nuclei (leading to magnetic resonance imaging). Many successes of quantum mechanics involve its extension, quantum field theory, which forms the foundations of elementary particle physics..." [6].
On the other hand, OQM contravenes philosophical realism. This is particularly clear as concerns its projection postulate. Scientists upholding philosophical realism have criticized this postulate because it introduces a subjective element into the theory: either it invokes an observer placed above the laws of nature ( [3], pp. 191-202) or it appeals to the interaction between the quantum system and a macroscopic measuring device (built, of course, by human beings). Referring to this issue Max Jammer points out: "As long as quantum mechanics one-body or many-body system does not interact with macroscopic bodies, as long as its motion is described by the deterministic Schrödinger time-dependent equation, no events could be considered to take place in the system... If the whole physical universe were composed only of microphysical entities, as it should be according to the atomic theory, it would be a universe of evolving potentialities (timedependent ψ functions) but not of real events" ( [7], p. 474).
It is worth noting that well-known scientists dealing with quantum mechanics renounce philosophical realism, either implicitly or explicitly. Let us mention two of them: Niels Bohr and Asher Peres. The last one declares: "Quantum theory is not an objective description of physical reality" ( [8], p. 423) and "any attempt to inject realism in physical theory is bound to lead to inconsistencies" [9]. In his analysis of Bohr's philosophy Aage Petersen asserts: "When asked whether the algorithm of quantum mechanics could be considered as somehow mirroring an underlying quantum world, Bohr would answer: 'There is no Journal of Modern Physics quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature'." [10].
Taking philosophical realism as a starting point, a Spontaneous Projection Approach (SPA) to quantum theory was formulated some years ago [11]. This approach was recently modified to account for quantum processes in the general case, including those where the Hamiltonian depends explicitly on time [12]. It is worth noting that SPA overcomes the main flaws of OQM [12] and in general yields experimental predictions coincident with those of OQM. There are, however, a few exceptions which could be exploited to confront both theories. This is the principal aim of our present study.
The contents of this paper are as follows: In Section 2, we reproduce the formulations of OQM and SPA, highlight their similarities and differences, mention some contributions aiming to solve the measurement problem and refer one of them in detail. In Section 3, we deal with a case where SPA and OQM yield different experimental predictions. We consider a silver atom in the ground state placed in a uniform constant magnetic field B. Starting with the atom in a given state we find 1) the evolution of its state assuming the validity of OQM; and 2) the possible changes of its state assuming the validity of SPA. The resulting contradiction could be decided by means of a suggested experiment. Section 4 is devoted to the discussion and conclusions.

Formulation of OQM
The formulation of quantum mechanics due to von Neumann (OQM) ( [7], p. 5) ( [13], pp. 215-222) includes the primitive (undefined) notions of system, state and physical quantity (or observable). Its postulates are: I) To every system ζ corresponds a Hilbert space Hb whose vectors (state vectors, wave functions) ψ completely describe the states of the system. II) To every physical quantity  corresponds uniquely a self-adjoint operator A acting in Hb. It has associated the eigenvalue equations (ν is introduced in order to distinguish between the different eigenvectors that may correspond to one eigenvalue k α ), and the closure relation is fulfilled (here  is the identity operator). If either k or ν is continuous, the respective sum has to be replaced by an integral.  IV) For a system in the state ψ the probability that the result of a measurement of  lies between 1 α and 2 α is 2 η , where η is the norm of ( )  is the resolution of the identity belonging to A. V) Projection postulate. If a measurement of  yields a result between 1 α and 2 α , then the state of the system immediately after the measurement is an eigenfunction of ( )

Formulation of SPA
The formulation of SPA includes the primitive notions of system, state, physical quantity and probability [12].
is a constant of the motion.
Definition of preferential states: The system in the state ( ) t ψ has tendency to jump to the eigenstates of operators satisfying Equations (6) and (7) whether or not [12]. If there is a unique set of 2 N ≥ orthonormal vectors: we shall say that { } N ϕ is the preferential set of ζ in the state ( ) t ψ and the members of { } N ϕ will be called its preferential states.
Comment (iii): The validity of Equation (9) ensures the statistical sense of conservation laws [11]. Note that, by definition, the preferential set does not depend on ( ) it can either remain in the Schrödinger channel or jump to one of its preferential states k ϕ with probability Comment: The probability that the system will remain in the Schrödinger channel during the time interval ( ) see [12]. If the parameter τ defined by Equation (11) is constant, the state ( ) t ψ may be considered as an unstable state that can decay to one of its N preferential states [11] [15] [16]. In this case the probability that the system will remain in the Schrödinger channel during the interval ( ) with relaxation time τ .

OQM, SPA and Other Contributions Aiming to Solve the Measurement Problem
OQM assumes that there are two kinds of processes: spontaneous processes and is the Schrödinger evolution; see Equation (13). According to Equation (14) the Schrödinger evolution is also the dominant process for t τ .
Spontaneous projections seldom occur.
OQM measurements are somehow related to SPA projections. Let us highlight their similarities: SPA projections as well as OQM measurements 1) yield discontinuities of the state vector; 2) instantaneously break down certain superposition of different states; 3) violate conservations laws but respect their statistical sense [16] [17]; 4) imply a kind of action-at-a-distance [ OQM states that the probability that the measurement of  will yield the result n α and so (according to the projection postulate) According to SPA the probability that in the small time interval ( ) see Equation (10) The similarity of the mathematical expressions for the probabilities given by (16) and (19) is worth stressing. Even if it is not clear what a measurement is [20], the notion of measurement is included in two of the five postulates of OQM formulation. In any case, collapses or something similar to collapses are necessary to avoid paradoxes such as that of Schrödinger's cat ( [7], pp. 216-217). "Because of the linearity of the Schrödinger evolution there is no mechanism to stop the evolution and yield a single result for the measurement" ( [14], p. 264). However, "in common life as well as in laboratories, one never observe superposition of results; we observe that Nature seems to operate in such a way that a single result always emerges from a single experiment; this will never be explained by the Schrödinger equation, since all that it can do is to endlessly extend its ramifications into the environment, without ever selecting one of them only" [21]. The rule compelling to remain continuous as long as no measurements are performed poses a serious problem for OQM. By contrast, assuming that spontaneous projections are natural processes, SPA succeeds in stopping the endless ramifications resulting from Schrödinger evolution.

SPA Preferential Sets and Sudbery's Preferred Observables
Among the contributions appearing in the previous list the closest to SPA is textbooks do purport to derive decay rates for such events from the general principles of quantum mechanics. The derivation goes like this. To describe a decay A B C → + , we start at time 0 t = with a state A in which the unstable particle A is certain not to have decayed, and follows its time evolution, governed by the Hamiltonian H, to a superposition of the initial state and a state of the decay products B and C: The general principles of quantum mechanics are then supposed to yield the interpretation that ( ) is the probability that by time t there has been a transition from particle A to particles B C + . But where does this notion of a 'transition' come from? It appears nowhere in the general principles of the theory as usually stated. The literal application of these principles to the state (20) gives only the statement that if a measurement is made at time t (to determine, say, whether a particle of type B is present) then ( )  [36]. The problem Sudbery wants to discuss is that of formulating the theory so as to make this argument as sound as its conclusion. To do so, he claims, "transitions must be given a fundamental role in the theory; one of its basic postulates should be of the form 'If the system [however broadly conceived it] is in the state ψ at time t, there is a probability ( )d T t t ϕψ that it will take a transition to state ϕ between t and d t t + .' Such a postulate, if it is to be fundamental, would need to be accompanied by a clear statement of exactly what the eligible states ϕ , ψ are" [36].
In order to formulate a postulate concerning transitions, Sudbery adopts a suggestion made by Bell [34]. "The basic idea is that there is a set of special By adopting Bell's transition probabilities, Sudbery succeeds in giving a satisfactory description of the decay process. But, he points out, "Bell's formulation of quantum mechanics does not exists until one has specified the viable subspaces. In this respect, it is no improvement on the conventional formulation, which does not exist until one has specified precisely which physical arrangement constitutes 'measurement'… If one is aspiring to give an absolute description of the physical world, there seems to be no good empirical or theoretical reason why any particular set of physical quantities should have fundamental status" [36]. Nevertheless, he adds, "if arbitrary choices cannot be avoided, we must consider how to live with them" [36].
SPA fulfills all the requirements stated by Sudbery: According to Postulate E, if the system is in the state and k ϕ is one of its preferential states, the probability that it will take a transition from

A Case Where SPA and OQM Yield Different Experimental Predictions
In this section we shall deal with a case where SPA and OQM yield different    According to OQM the atom remains in the Schrödinger channel and at time L T recovers its initial state.

Possible Changes to the State Vector Assuming the Validity of SPA
In SPA spontaneous processes are not necessarily ruled by a deterministic equation. If the system has the preferential set { } N ϕ , it can either remain in the Schrödinger channel or jump to one of its preferential states. Thus, the same initial state According to SPA the probability that the atom will remain in the Schrödinger channel is a decreasing exponential with relaxation time τ . The probability that it will recover its initial state at time L T decreases when the polar angle ϑ increases.

Testing OQM vs. SPA Experimental Predictions
In the case of a silver atom in a uniform magnetic field, the experimental predictions of SPA are radically different from those yielded by OQM. This contradiction could be decided by means of the experiment described in ( Figure   2).  [37]. We assume, in addition, that at the exit of F the atoms travel with velocity υ . At the entrance as well as at the end of the magnet the magnetic field is strongly non uniform. Several authors have assumed that the evolution of a spin in a strongly non uniform magnetic field is ruled by the Schrödinger equation [37] ( [38], pp. 593-598). According to SPA this is so if 1) either the atom does not have a preferential set; or 2) the atom has a preferential set, but the time it spends in such a region is so short that it "has no time" to be projected. We shall assume one of these two conditions fulfilled in the experiment just described.

If OQM Is Valid
The atom enters the magnet in the state ( ) 0 u ψ = + . While flying through the magnet it precesses around the z-axis with the Larmor frequency ω , "it never aligns itself with the z-axis" ([14], p. 164). It will remain in the Schrödinger channel and at time T it will recover its initial state u + . Therefore, a measurement of u S yields the result 2 + with certitude whatever the angle ϑ may be (Figures 3(a)-(c)).
with obvious notation. The relation holds.
The following diagram illustrates the process just described. ϑ  , the same holds for the inverse of the relaxation time τ ; see Equation (41). For T τ , the atom "has no time" to decay to one of its preferential states during its passage through the magnet. The probability that it will remain in the Schrödinger channel is ( Figure 4(a)). In this case the difference between SPA and OQM predictions is irrelevant.
When ϑ increases, the relaxation time τ and the probability ( ) S P T decrease and the sum the intensity of the traces on plate P corresponding to both results will be very similar (Figure 4(c)).

SPA Vs. OQM: Which One Is Valid (If Any)?
Taking into account the previous analysis we can assert that: 1) either all the atoms hit the same point on plate P (which depends on the polar angle ϑ ), as shown in Figure 3; or 2) the atoms make impact in one of the two opposite points on plate P as shown in Figure 4; here the positions of the opposite points depend on the polar angle and the rate between the number of impacts corresponding to the eigenvalues 2 − and 2 + of u S grows with ϑ .
In the first case we should conclude that when flying through the magnet

Discusion and Conclusions
John Bell points out: "In the beginning natural philosophers tried to understand the world around them. Trying to do that they hit upon the great idea of contriving artificially simple situations in which the number of factors involved is reduced to a minimum. Divide and conquer. Experimental science was born. But experiment is a tool. The aim remains: to understand the world. To restrict quantum mechanics to be exclusively about piddling laboratory operations is to betray the great enterprise. A serious formulation will not exclude the big world outside the laboratory" [20]. OQM is fine for all practical purposes. But besides excluding the big world outside the laboratory, its formulation lacks precision. The central notions of system, apparatus and measurement are neither included as primitive concepts nor defined in the theory. To circumvent this ambiguity-says Bell-discretion and good taste (born from experience) are needed ( [39], p. 160). It would be perhaps possible to improve OQM by replacing these ambiguous notions. "However, the idea that quantum mechanics, our most fundamental physical theory, is exclusively even about the results of experiments would remain disappointing" [20].
SPA is a version of quantum theory grounded on philosophical realism. It is not about the results of experiments; it talks about what happens [16]. It has been precisely formulated [12]. The notion of system is introduced as a primitive concept and those of apparatus and measurement are not included in its formulation. Spontaneous and measurement processes are treated on the same Journal of Modern Physics footing [16]. In SPA transitions to the continuum are spontaneous processes. The expressions of the probability density per unit interval of energy resulting from SPA and from OQM treatments coincide approximately and Fermi's golden rule is obtained [15].
In the framework of OQM, it is agreed that time-dependent perturbation theory (TDPT) must be used for solving all problems involving time, including spontaneous time-dependent processes. We have questioned TDPT on the following grounds: accounting for spontaneous time-dependent processes requires the application of a law (the projection postulate) which is not valid in such processes [40]. In the same direction, contradictions reminiscent of Zeno's paradoxes of motion have been pointed out [41]. By contrast, SPA has no relation with TDPT. Hence it does not confront these issues. According to SPA the Schrödinger equation not only rules processes where the system has no preferential set, but it also rules most processes where the system does have a preferential set. Spontaneous projections seldom occur. It is worth noting that spontaneous projections and projections resulting from OQM measurements share several traits. In particular, the mathematical expressions for the corresponding probabilities are quite similar; see Section 2.3.
The rule forcing the state vector to remain continuous as long as no measurements are performed poses a serious problem for OQM. By contrast, assuming that spontaneous projections are natural processes, SPA succeeds in stopping the endless ramifications resulting from Schrödinger evolution; see Section 2.4. Hence if SPA is valid there should be no paradoxes such as that of Schrödinger's cat.
SPA exhibits several advantages over OQM, but OQM enjoys the prestige of old age. In order to put both theories to the test, in this paper we suggest an experiment where at least one of them should fail. We study the behavior of a silver atom, initially in an eigenstate of the u-component of the spin, placed in a uniform magnetic field.
Assuming the validity of OQM, in Section 3.1 we find the state of the atom at a certain time L T . Assuming the validity of SPA, in Section 3.2 we find the possible states of the atom at the same time L T . In this particular case, the experimental predictions of SPA are radically different from those yielded by OQM. The resulting contradiction could be decided by means of the experiment described in Section 3.3.
Until today OQM experimental predictions seem to have no exceptions. Should the experiment mentioned yield the results predicted by SPA, it would be the first time that OQM experimental predictions fail. Journal of Modern Physics Mariana Delbue for her help with stylistic matters and Carlos Valero for his assistance with the figures and the transcription of the manuscript into LaTex.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.