02 May 2020

Perplexities of quantum mechanics

 The following is an excerpt from a work in progress.

3.4 Quantum-mechanical movement 

The advent of quantum mechanics as the successor to classical Newtonian-Galilean mechanics through the work of famous physicists such as Einstein, Heisenberg, Schrödinger, Dirac, Born and Bohr has been hailed by many physicists as an exquisite testament to the power of the human mind to unravel the mysteries of nature. It also gives rise to perplexing paradoxes. To quote just one of these enthusiastic quantum physicists, “Quantum mechanics is the greatest, the most profound of revolutions in our modern view of the physical world. Even for experts, acheiving a deep conceptual understanding of quantum mechanics can be an elusive goal.” (Bowman 2008 p.172)  Wherein lies the deeply paradoxical and downright confusing nature of quantum physics expressed in figures of thought such as quantum indeterminacy and the dual nature of quantum entities as both particle and wave? Are they ultimately paradoxes of the mathematized mind’s own making? Let us investigate the matter.
The basic entity of quantum physics is neither a particle, as in Newtonian physics, nor a force field, as in Maxwellian physics, but a dynamic quantum state that is expressed mathematically as a ket-vector in some linear vector (Hilbert) space of discrete finite, discrete, countably infinite or continuous, uncountably infinite dimensions with complex, rather than real scalars. These dimensions are purely mathematical, as a rule not (conceived as) physically spatial. The ‘dynamic’ in dynamic quantum state refers to the character of a quantum state as being in movement toward another quantum state under the impetus of a δύναμις, i.e. a power or force. A major consideration of quantum mechanics is thus what it calls the ‘time-evolution’ of a quantum state along linear real time, t, that can only be precalculated i) by conceiving an operator (the Hamiltonian) operating on the given quantum state and ii) by employing the infinitesimal differential calculus to solve (partial) differential equations in t that govern the movement. To this calculative end, the quantum state absolutely must be conceived (i.e. interpreted) as a real-infinitesimal state in a present instant of time along a real continuum of instants.
This present quantum state, in turn, is conceived as the prepared starting-point for an experiment set up to register the outcomes of movements of the quantum state to compare them with theoretical predictions. These possible outcomes are in the plural, that is, at least two and no longer one, as in classical physics. They may be discretely many, infinitely many but still discrete, or infinitely many in a continuum (such as a single, linear, spatial dimension). These multiple possible or potential outcomes are said to be, i.e. conceived as being, contained in the present quantum state which therefore must be in some sense a superposition of multiple potential quantum states, each of which could render a single observational outcome when measured in the future. Measurement itself is conceived mathematically as the operation of an observable-operator on the present quantum state, a multidimensional ket-vector. Whereas the present quantum state is conceivable mathematically by the physicist as a superposition of present alternative, as yet undecided, ambiguous, future physical states (say, of spatial position, linear momentum, angular momentum), the observable generated by the operation of the observable-operator on the quantum state is a certain, definite data-result recorded in a future present of the completed experiment.
Unlike the situation in classical mechanics, where experiments are repeated to ensure that the outcome of the experiment gives the same result within a margin for experimental error as quantified by mathematical statistics, thus allowing efficient causal inferences to be drawn within a statistical margin of error, the outcomes of quantum mechanical experiments repeated many times reveal a well-defined, distributed statistical pattern of multiple outcomes that can be plotted to show a regularity in their spread. The physicist tries to achieve a calculable grasp on this observed regularity by means of the mathematics of probability, which, having arisen in the 17th century with mathematicians such as Pascal, is today a well-established, highly sophisticated branch of mathematics. Like all mathematics, however, probability mathematics, too, can only operate on given mathematical entities in a mathematical ‘space’, mapping them to another single mathematical entity in another ‘space’, or to multiple entities forming a mathematically amenable probability distribution whose various parameters, such as expected value and standard deviation, can be calculated from the given data. Once these parameters have been experimentally ascertained and calculated, the road is clear for being able to reliably precalculate the outcomes of given quantum experiments with certain mathematically known probabilities and thus within statistical margins of error.
This orthodox conception of the dynamic quantum state as a superposition of multiple possible quantum states upon measurement in a given instant must give rise to controversy because it is regarded as sacrosanct under the law of non-contradiction, that a present physical state cannot be in two or more states simultaneously. Simultaneity hence becomes a major stumbling block engendering paradoxes, confusion and denial among physicists. (Multiple simultaneous future potentials, however, provide the key to circumventing this apparent logical impasse.) To avoid this controversy over simultaneity, that is mostly regarded with disdain by physicists as ‘metaphysical’, some quantum physicists (e.g. Ballentine 1998) have proposed the statistical interpretation of quantum mechanics in which the focus is solely on the mathematically manageable statistical outcomes of quantum experiments in ‘real’ data, without bothering to concern themselves with the interpretation of the unseen ‘inner workings’ of the prepared quantum state. The quantum state is said to merely be a vector of probabilities for the observed ranges of experimental outcomes. The observable-operator can be represented as an operation on a complex linear combination of the set of the operator’s eigenstates spanning the ket-vector space that results in a weighted linear combination of the possible eigenvalue outcomes of the experiment. The weights are given by the modular squares of the expansion coefficients of a quantum state in the chosen eigenstate basis; they are interpreted as the mathematical probabilities in a probability distribution of observable outcomes.
For many a quantum physicist, this interpretation is highly appealing, since it avoids and apparently dissolves many of the perplexing conundrums of quantum indeterminacy and duality. However, does this statistical interpretation hold water phenomenologically? It proceeds from two implicit fundamental ontological-metaphysical pre-conceptions: i) that a dynamic physical state of any kind can be conceived as a state ‘frozen’ in a given now-instant and ii) that the physical state itself is objectively disjunct from any observer-subject who can only observe by registering physical data-signals received from any physical event in four-dimensional space-time (x, y, z, t). That is to say, any physical state is only instantaneous in time t, and any physical observation is likewise only a reception of physical data by the observer-subject’s senses, no matter whether they be assisted by sophisticated apparatuses such as various kinds of modern microscope or telescope or not.
These ontological-metaphysical preconceptions do not stand up to phenomenological scrutiny. Why not? Because they are based on misconceptions of the mind, time itself and therefore of physical movement, all of which are intimately interwoven. For modern mathematized physics there is no mind, nor can there be, because it operates from within an implicitly presupposed metaphysics of subject-object in which the observer-subject is endowed with an interior consciousness that is hooked up to the exterior world via sense-organs. These sense-organs are able to receive sense-data from the separated, external, objective world and process them internally through the intellect that itself is conceived as some kind of individual process ‘inside’ (mainly the brain) with many secrets yet to be deciphered by neuroscience. A dynamic physical state in the external world is conceived to ‘exist’ quite independently of any individual subject of consciousness who may or may not be observing it in any particular instant. Physical movement of any kind, quantized or not, is purportedly one ‘objective’ thing; its perception by an individual subjective consciousness quite another.
If there were no individual human consciousnesses around, this way of thinking prescribes, there still would be objective physical movement galore. But for whom? For what? For movement to ‘be’ it must present itself as such-and-such to and be understood as such-and-such by an understanding mind. This mind is not an internal individual consciousness, but a necessarily shared understanding in which all that occurs is understood or misunderstood in one way or another in a given historical time. Such a shared mind for understanding occurrences in the world as such-and-such is not merely the slave of the senses and the sense-data they deliver at the speed of light in the present. If it were, there would be no understanding of the world and its movement at all. Movement ‘exists’ only for a shared mind that understands it, conceives it as such-and-such, and this understanding must encompass, i.e. must mentally see, all three temporal dimensions at once to see that movement comes from a dimension of past and proceeds into a dimension of future. Movement is always underway toward, even when the final destination is seen mentally only indistinctly, without any reliable, definite prediction or is entirely unknown. The destination of a movement must be mentally in view for movement to be, i.e to present itself and thus presence in the mind, as movement at all. The destination may be completely obscure and unknown, but it is still a destination-in-mental-view as an εἰδος, an eidetic ‘look’ that is necessarily non-physical. There can never be a purely physical movement; it must be also ‘meta-physical’ in the sense of presenting itself to a mind that must be able to see temporally all at once in three dimensions.
This has decisive consequences for the conception of a dynamic quantum state which cannot be properly conceived as an instantaneous state in a present now-instant. Why not? Because, as dynamic, the quantum state includes also its connection to a future state, or rather, multiple possible future states mathematized as probabilities that may be quantified and confirmed by frequency proportions of measured empirical data. This connection to the future lies in the very conception of a quantum state which is a conception for the encompassing, shared historical human mind (and not for the individual human consciousness of an observer-subject). At the very least, therefore, the conception of a quantum state must admit, upon unearthing its as yet hidden, suppressed, implicit presuppositions, the double temporal vision that can see conceptually a present state together with multiple possible future states. In other words, the very conception of a quantum state, adquately conceived, cannot be that of a purely present state at time t but must be temporally spread, which in itself poses a perplexing conundrum for any mathematization of the dynamic situation. Any possible future state, by definition, cannot be sensuously present for the senses, but this is no deficiency at all. Rather, the sensuous observations of experimental data are an impoverished act within an overall conception of what the quantum-physical experiment is supposed to predict or confirm, namely, a preconceived, mathematized theoretical result.
The perplexing paradox of Schrödinger’s cat is such only by virtue of conceiving the indeterminacy of a quantum state as a present, instantaneous state in which two or more physical states exist ‘simultaneously’ in the same instant; prior to physical-sensuous observation by an observer subject, the cat is said to be in a limbo of being simultaneously dead and alive, which is, within this construction, paradoxical, contradictory, impossible. The dead-or-alive status of the cat, however, refers only to possibilities that are set up experimentally so as to be mathematizable as 50:50 probabilities for the release of the poisonous gas that kills the cat. The release, in turn, depends on which of two equally likely paths a prepared photon in a strictly controlled experimental set-up will take. The ambiguity is said to ‘collapse’ on observation. This dynamic situation is on a par with leaving it to the toss of a coin whether a cat will be killed by some automated mechanism triggered by heads or stay alive if the outcome is tails. The tossed coin in its indefinite spinning movement contains both heads and tails as potential future outcomes of the toss, neither of which is present-as-actualized during the toss. Nevertheless the possible future outcomes are present (in the mind of anyone conceiving the experiment or watching it from within some preconception or other) as futurally absent. Futural absence, however, is a kind of presence, i.e. of existence.
Analogous considerations pertain also to the Stern-Gerlach experiment often employed to illustrate the indeterminacy of the finitely discrete spin ½ angular momentum in which a particle (a silver atom in the original 1922 experiment; cf. Bowman 2008 p. 102) with a magnetic moment proportional to the spin passes through a magnetic field that generates a force deflecting the particle in one of two directions (up or down), depending upon whether the quantum spin is positive or negative. Whereas the particle with its spin in flight cannot be physically observed, its deflection up or down, as a macro movement, can be, from which the particle’s spin, positive or negative, is inferred. The physically observed deflection registered by a detector makes sense within the framework of a precisely preconceived experiment in which the particle’s motion is (conceived as) influenced by the magnetic field due to the magnetic moment generated by the spin angular momentum. This spin must therefore be preconceived by the physicist’s mind as a quantum superposition of two eigenstates of the pertinent spin quantum state of the particle, the two eigenstates corresponding to two alternative eigenvalues, either of which can be experimentally observed. The experimental observation is thus strictly preconceived and preinterpreted in line with the mathematized theory which says that a moving particle of the kind experimented with can assume one of two opposite states that can be physically seen on coming to rest in a detector, just like a tossed coin is possibly in one of two equally likely states, either heads or tails, on coming to rest somewhere on the ground. While in motion, the spin state is as yet physically unobservable and insofar indeterminate for the observing mind. This does not entail, however, that, as long as the particle is in motion, it is in both spin states ‘simultaneously’ in the same instant. Rather, it has the potential to assume one of these states in the future and the particle’s spin-influenced motion has been precisely preconceived mathematically as indeterminate, wavering between two possible outcomes.

3.4.1 Non-commutability of operators and quantum indeterminacy

To recapitulate and extend the above discussion, I shall first give an orthodox interpretation of quantum indeterminacy in terms of the non-commutability of Hermitian operators on a complex Hilbert ket-vector space. A quantum state as a dynamic physical set-up or situation is a generic concept for everything physical in the world and, as such, supersedes the older basic physical concepts of particle and force-field. It is not localized in a particle, but is an entire physical-dynamic situation situated 3D-spatially, with this spatiality being captured by the notion of a spread-out wave with phases captured mathematically by complex-imaginary exponents of the Eulerian exponential number, e. (The conception of a 3D-temporal quantum situation would break with the strictures orthodoxly laid down by quantum physics as a predictive science, as discussed further below.) Mostly, however, a quantum state is carefully preconceived, designed and set up in line with a mathematized theory to experimentally interrogate the kinetic behaviour of (usually) sub-atomic (but also atomic or molecular) quantum entities conceived as quantum states to empirically validate or invalidate the theory. The aim is to gain a law-like, mathematically precalculative grasp on the movements of relevant quantum entities. What can be observed of quantum states is determined by operators on the ket-vector space operating on, i.e. transforming, such ket-vectors into the canonical state of a superposition of distinct eigenstates (represented mathematically as eigenvectors) associated with the eigenvalues of the given operator. Any state is then expressible as a complex-linear superposition of a chosen orthonormal basis of eigenvectors, usually for some observable. In line with a proposal by Max Born, the squared magnitudes of the complex coefficients in such an expansion are interpreted as the probabilities of obtaining each eigenvalue upon (future) observational measurement by the observable operator. Upon measurement, the quantum state must assume just one of its eigenstates (as prescribed by the mathematics of linear vector spaces employed) with its associated eigenvalue that is constricted to be a real number by stipulating that the observable operator be Hermitian. Any subsequent measurement of this eigenstate will reproduce only this one eigenvalue-measurement (a fundamental postulate of quantum mechanics).
To grasp the dynamics of the quantum state, however, observational measurements of more than one observable dynamic parameter are required. For instance, it is not sufficient to observe only the spatial location of a quantum state. Since physics is concerned with dynamics, i.e. with movement, the linear or angular momentum of the quantum state, for instance, must also be measured and accounted for theoretically. Momentum, be it linear or angular, as a dynamic parameter like spatial position, is also conceived as a superposition of multiple potential eigenstates of momentum. The same quantum state can therefore be represented also as a complex linear superposition of momentum eigenstates in a chosen orthonormal basis, say, of position. Once a position measurement has been carried out on the quantum state and it has irrevocably entered a single position eigenstate, its representation as a superposition of momentum eigenstates remains hovering in a multiplicity of potential momenta eigenvalues that may be actualized upon measurement, each weighted with the probabilities of the squared coefficients of the momentum expansion of the quantum state in the position basis. Hence repeated measurements of the momentum will result in a statistically manifested probability distribution, i.e. in an indeterminacy of measurement.
The repeated measurement of one observable, say, spatial position, of a carefully prepared quantum state will result in a spread of position measurements (real eigenvalues) whose spread, as calculated by standard deviation, can be reduced, thus increasing the precision of the position measurement in terms of probability by narrowing the spread of the distribution. For each of these repeated position-measurements, however, the quantum state is in different superpositions of another, interrelated dynamic parameter (standardly employing the example of the famous Schrödinger wave equation): linear momentum, which, upon its measurement, displays an ever greater spread as seen in the wider spread that is quantified by a larger standard deviation of the measurements. That is, the momentum measurement becomes less precise, the greater the precision (probability confidence) of the position measurement. This is the nub of Heisenbergian quantum indeterminacy: observable-operators do not commute; their order matters. The increased precision in measuring one observable parameter of a quantum state is paid for by less precision in measuring another. This is a general feature of non-commuting operators of a complex linear ket-vector space, not just of position and linear momentum nor of Hermitian observable-operators generating real measurements.
Position and (linear) momentum, as well as their commutator relationship to one another, are all quantized in quantum physics. They also remain conceived as continuous. How so? Both position and linear momentum are conceived as observable operators on a Hilbert vector space whose elements are ket-vectors of superposed quantized eigenstates of position or momentum. This Hilbert space, however, is conceived as infinitely dimensional with the infinity being an uncountable one of the real continuum. Hence both the operators, position and linear momentum, are continuous, represented by continuous functions. The relationship between position and momentum is also quantized by their non-zero commutator containing the Planck constant that keeps them apart. The probability distributions of both position and momentum on observation are again both continuous in vector spaces of continuous-infinite dimensions and are both continuously variable, but their joint spread cannot be made arbitarily small by repeated measurement, being held discretely apart as they are by the Planck constant. Hence a duality of discreteness and continuity holds sway in this quantized mathematization of a basic physical phenomenon, namely, motion itself in terms of position and momentum. It is one particular manifestation of the famous and notorious wave-particle duality of quantum physics.

3.4.2 Paradoxes of a quantized mathematization of the physical world

The complications of non-zero commutators for dynamic variables such as position and linear momentum for their precalculation is just one perplexing consequence of the quantized mathematization of the physical world. Another, intimately associated one is the observed phenomenon of wave-like interference of particle-like quantum entities, as in the well-known double-slit Gedankenexperiment, and the resultant ambiguity of particle-wave duality forced on quantum physicists attempting to theoretically precalculate the movements of quantized entities.
The introduction of physical quanta was first imposed on theoretical physics by the breakdown of a mathematized description of electromagnetic radiation in terms of continuous, real mathematical functions. Planck proposed the famous Planck constant, h, to quantize energy levels in an atom as discrete multiples of the finite frequency f of the radiation emitted: E = h.f. This theoretical move was made to avoid the consequence of infinite energy levels for continuously increasing, high-frequency (ultraviolet) radiation emitted from an idealized black body. From then on, the quantization of physical entities such as photons and electrons had to be taken into account mathematically. The first physical phenomenon to be quantized was light itself, the absolute physical motion of relativity theory. Electromagnetic light waves had to be conceived mathematically as quantized into photons, which are for physics tiny packets of pure motion, i.e. pure energy with massless momentum.
Hitherto, since the inauguration of mathematized physics in the 17th century with Galileo and Newton, physical entities and parameters had been uniformly treated as continuous real variables whose variability over linear time could be handled calculatively by the infinitesimal calculus. This required the mathematical physicist to imagine (conceive) that any physical entity, including space and time themselves, could continuously approach a zero-dimensional point-limit to make it amenable to the infinitesimal calculus of differentiation and calculation. Even after the quantization of mathematized physics, the predictive calculation of the movement of quantized entities (as quantum states) still had to employ differentials with respect to continuous, linear, real time, t, to maintain any sort of precalculative grip on physical movements of all kinds. Other classical variables, however, notably position and linear momentum, lost their imagined continuity to a discrete quantization. As we know from high school maths, classical mechanics operates with continuous time and continuous space to derive continuous velocity v at an instant in this continuous, linear time. A physical thing in this classical mechanics is conceived as composed of point-mass particles with linear momentum m.v in a continuum of matter whose motions can be precalculated by the application of three simple, Newtonian, mathematical laws of motion.
In quantum physics, these basic physical variables, position and linear momentum, lose (and also retain) their continuity with the consequence that they must be conceived mathematically as a complex-imaginary superposition of many different quantized states rather than the unique state calculated by differentiation and integration of continuous real position and continuous real momentum. Each possible physical state of position and momentum must then be quantized, and thus rendered precalculable, by the mathematical probabilities of an operatoe rather than unique solutions to differential equations. The calculative hold of mathematized physics on physical motion and movement is thus both weakened and maintained. Since quantum phenomena are endlessly repeated in nature, the stochastic quality of their movement in linear time can be easily coped with by means of probability calculus. This is sufficient for practical purposes, especially since all mathematized modern science, both natural and social, has to take into account experimental errors by employing quantitative stochastic-statistical methods. Hence, in a sense, all modern science has to calculatively cope with the indeterminacy of all movement.
Thus, despite the set-back of forced quantization, modern quantum physics remains true to the Pythagorean insight into numbers as a secret key to the world and also to the Galilean edict that the laws of nature are written in the language of mathematics. The wave-particle duality in ‘our’ (i.e. the mathematical physicists’) ‘picture’ of the physical world is ultimately a reflection of the duality in mathematics itself between (originally arithmetic) discreteness and (originally geometric) continuity, the countable infinity of the rational numbers and uncountable, irrational infinity of the continuum. Already the Greeks grappled with the irrational non-commensurability of certain real numbers of the continuum. The forced quantization of continuous, linear time itself, that is indispensable for calculating and thus predicting physical movement, would be a last bastion to fall in defending the mathematization of the world by the modern sciences. This last bastion could never fall due to any conceivable physical experiment, but only as a consequence of how time itself is conceived by our shared historical mind, that is, how time itself is hermontologically cast as what it is. This task of undermining the conception of linear, real time, however, is not and cannot be the concern of physics or any of the other modern sciences for the simple reason that the raison d’être of science ever since the Greeks has been to predict movements and thus enhance the hold of the unbridled will to power over movements of all kinds. 

Further reading: Ballentine, Leslie E. Quantum Mechanics: A Modern Development World Scientific Publishing Co. Pte. Ltd., Singapore 1998.
Bowman, Gary E. Essential Quantum Mechanics Oxford U.P., New York 2008.
Eldred, M. Movement and Time in the Cyberworld De Gruyter 2019

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