Understanding Quantum Theory

Introduction

It is now about 100 years ago that Erwin Schrödinger1 published his quantum theory, a few months after Heisenberg2 had published his. Very soon Schrödinger gave a proof that both formalisms are physically equivalent. Thus, for almost 100 years now the formalism of quantum mechanics seems clear. For the same time there has been a dispute, however, about the interpretation of that formalism, which is not settled yet even after 100 years. How is that possible, after those 100 years of extremely successfully applying quantum mechanics in physics as well as in technology?

Why is it that a physical formalism needs an interpretation at all? In classical mechanics it seems entirely clear how the results of the formalism have to be connected with empirical I think I can safely say that nobody understands quantum mechanics. Richard FeynmanI findings. Nobody ever called especially for an “interpretation” of classical mechanics. But for quantum mechanics that is different: The result of a quantum mechanical calculation is usually not the value of an observable but typically a probability or a probability density. So the question arises what that means from the viewpoint of physics—it calls for an interpretation.

What I attempt here is making understandable what quantum theory really says. It will be different from most well-known “popular” accounts of quantum theory, in that I do not want to present you that “miraculous” modern physics. Many popular descriptions of quantum theory concentrate on presenting a kind of curiosity cabinet with news like “a particle can be a wave as well” or “a particle might be at different places at the same time” or “quantum theory allows instant transfer of information over large distances”. Curios like that distort what quantum theory really says, and they reliably prevent any understanding of that theory. I will try to clarify, instead, what quantum theory is about and what it says—in rather sober words that give really a better “understanding”.

Thus, I am doing something similar to what Leonard Susskind and Art Friedman do in their ‘Quantum Mechanics. The Theoretical Minimum.’3 Their excellent account introduces into physical essentials of quantum theory whereas this text attempts to give an introduction on how to understand quantum theory, contrary to Feynman’s quote in the beginning. Whether Feynman is right depends, evidently, on what we understand by understanding. This is actually a philosophical question. The answer might become clearer in the course of this text.

Quantum vs. “Classical” Ontology

Classical ontology assumes that there is a nature “out there” we can watch and describe like one would describe e.g. the works of a clock. This view is described classically by P.S. Laplace4:

“Une intelligence qui, pour un instant donné, connaîtrait toutes les forces dont la nature est animée, et la situation respective des êtres qui la composent, si d’ailleurs elle était assez vaste pour soumettre ces données à l’analyse, embrasserait dans la même formule, les mouvements des plus grand corps de l’univers et ceux du plus léger atome : rien ne serait incertain pour elle, et l’avenir comme le passé, serait présent à ses yeux.”5 If you look close enough you can see that quantum theory is different from that picture. That is what I think leads many people to finding quantum theory so strange. What you can get from quantum theory is a system of predictions for possible measurements, and those predictions do not generally give certainty to an outcome but admit different possible outcomes with respective probabilities. It is true, probabilities occur in classical physics as well. But there one can always comfort oneself with the idea that “in reality” one outcome was certain, only the information the physicist had was not sufficient to obtain that “real” outcome. In quantum theory, this way out is barred. No quantum state that describes a system is such that it gives definite values to all observables. It rather gives for most observables

3 Leonard Susskind and Art Friedman, Quantum Mechanics. The Theoretical Minimum. London (penguin) 2015

4 Pierre Simon de Laplace, Essai philosophique sur les probabilités. Paris 1814

5 English version [MD]: “An intelligence which, in a given instant, knew all forces that animate nature and the correlations of the beings it is made up of; if it were, besides, huge enough to analyze these data, it would comprise in the same formula the movement of the largest bodies of the universe as well as those of the lightest atom: Nothing would be uncertain for it, and the future like the past were present before its eyes.” predictions and, in the optimal case, for every possible result a probability for finding that result. Thus, we have to admit that quantum theory is a fundamentally indeterministic theory. That insight forces us to abandon the “classical” picture of a world “out there” comparable to a clock work.

Is that a drawback, compared to the “classical” ontology? We have to reflect on what we really expect of science: science should enable us to predict what will happen, using empirical data about the present state and theoretical calculations on the basis of a valid theory. In classical physics we had the singular situation that we could predict e.g. the positions of a planet, in principle, with certainty. This was, however, very principle-like; it applied mainly to the standard case of the movement of a planet. Even there, it is generally impossible to know the present situation and all influences on it exactly enough to predict with certainty. Thus, in the “classical” situation probabilities come in as well, not different from quantum theory. But still, in the foundations there is that decisive difference between a deterministic and an indeterministic theory: in a deterministic, “classical” theory we can always suppose that in fact every observable has an exact value.

When quantum theory was discovered, it was not clear from the beginning that indeterminism was its key feature. It was only after many futile attempts that Max Born published his seminal paper (1926)6. There he says that the square of the coefficient in the decomposition of the wave function is the probability of finding the corresponding result. Born continues his considerations with the suggestion that quantum theory is fundamentally indeterministic.

Among the “Copenhagen” experts, there had been speculations about indeterminism for some time before but most of them hesitated publishing their opinion because of the strong impact that such a change must have on our picture of the world. Thus Born was the first one who dared publishing that consequence of their discussions.

Indeterminism

In fact, indeterminism is the revolutionary new property of quantum theory, compared with all other physical theories. Indeterminism is the one property of quantum theory that is, to my mind, the real reason for the difficulties felt with understanding that theory.

Now, indeterminism does not mean that there is no way of predicting anything about the result of future measurements. Instead of predictions with certainty we have, in an

6 Max Born, ‘Zur Quantenmechanik der Stoßvorgänge.’ Zeitschrift f. Physik 37, 863–867. here p. 865

indeterministic theory, predictions with probability. As a consequence, many difficulties felt with quantum theory can be reduced to problems of probability. Thus, our first question is now, what is probability?

Probability7

The probability of a result, as used in empirical science, has to do with the relative frequency of that result in a series of measurements. Many physicists simply identify probability with relative frequency. But that does not work: Imagine throwing a coin, with probability of heads and tails both being ½. Now flip that coin 13 times. The relative frequency ½ would mean 6.5 results heads. That is not possible! – But there is still another objection that goes deeper: Using probability calculus we can calculate—we shall deal with that later—probabilities on a higher level. Consider, as an example, a series of 12 tosses of dice, and repeat that series many times. The probability, e.g., with a “good” die to get a ‘four’ is 1/6. From probability calculus we can calculate the probability that exactly 1/6 of the 12, i.e. 2 throws will show a ‘four’. The formula for the probability p(n) of a number of n results ‘four’ in the sequence of 12 throws (n = 0, …, 12) is:

12 12 1 5 6 6

n n p n n $$ ) = \left(\frac {1 2}{n}\right) \cdot \left(\frac {1}{6}\right) ^ {n} \cdot \left(\frac {5}{6}\right) ^ {1 2 - n} $$ ( ) The following table gives values p(n):

Click to enlarge

This gives p(2) = 29.6%, but the probability to get only one “four” is not much less, namely p(1) = 26.9%, and, just giving a few more examples, p(3) = 19.7%, p(0) = 11.2%, and p(12) = 5∙10-10. That means: All relative frequencies, not only 1/6, have positive probability, i.e. they are possible, and relative frequencies unequal but near the value of the probability are almost as probable as those exactly equal to the probability. Thus, probability theory itself excludes the

7 https://plato.stanford.edu/archives/sum2003/entries/probability- interpret/ identification of probability with relative frequency. We will come back later to such considerations.

The Necessity of Classical Concepts

There is a great problem with an indeterministic theory one would not think of in the beginning: How can we use such a theory for describing reality? – Quantum mechanics gives predictions of the form: “When quantity Q is measured then one will find the value q_1 with probability _p(q_1).” But that gives no possibility for describing _reality, quantum mechanics does not give a picture of a reality “out there”! This difference between quantum mechanics and classical physics seems to be the reason of the uneasiness some people felt immediately after the discovery of quantum mechanics, and some still feel uneasy today.

There is one area where that lack is especially felt, namely in the description of measurements. Niels Bohr remarked that we have to be able to say “… what we have done and what we have learnt.” His conclusion for quantum mechanics is that classical concepts are necessary for a description of measurements. In classical theory it is presupposed that descriptions like “The present apparatus measures quantity Q”, or “The measured value is q_1” make sense. Since within the framework of quantum mechanics such descriptions of reality do not occur, we have to take the aid of classical physics in order to be able to give such descriptions. But there we have a problem: According to quantum mechanics classical physics is outdated; wherever quantum mechanics and classical physics give different results, quantum mechanics is right, classical physics is wrong. – This sounds awful, but actually there is no reason for worrying, because really in the usual cases quantum mechanics gives _almost the same results as classical physics. A working physicist, in any case, would not consider that a catastrophe since in physics, as mentioned above, we rarely have exact values at all. Still, many philosophers of science, who traditionally come from logic or mathematics, would insist that “almost correct” is, in fact, false. But in physics that argument is not valid. Approximation lies at the foundation of physics; without approximation physics is not possible at all. Maybe the most important argument is: Physics deals with objects which are defined by the observables that can be measured on those objects. Take, e.g., a classical point mass, which is defined as an object the state of which at a certain instant can be

8 Michael Drieschner, Found. Phys. 46(2016): 28-43

described completely by its position and momentum. There are no things in the world that are point masses. In order to describe anything as a point mass we use approximations: We discard every information except about position and momentum; we treat the mass of the body concerned as concentrated in one point in space; we give the results of our measurements real number values—all of which do not depict reality as it is but in a certain approximation. Without that we could not do physics.

It is actually quite normal to accept a proposition being ‘approximately true’ as a true proposition. Thus in physics there is no problem in accepting a “classical” proposition within the framework of quantum mechanics. And more than that: quantum mechanics is incomplete without those parts of classical physics; it could not be linked to reality without.

Interpretations9

In the course of almost 100 years of discussions about quantum theory, so many interpretations have been proposed that I will not endeavor giving a survey, and much less giving a systematic account of that abundance. My purpose here is rather to give some examples that may make my description clearer. Therefore, I will give short comments on three very different interpretations, namely

• The “Bohmian mechanics” introduced by David Bohm in 195210
• The “many-worlds” interpretation introduced by Everett, Wheeler, and DeWitt from 1957 on11
• The “Minimal Interpretation of Quantum Mechanics” I will present here.

I did not mention the “Copenhagen Interpretation”. Since it is universally acknowledged that it is difficult to get a clear cut definition of what that interpretation is, I rather do not attempt to give one. Possibly the “Minimal Interpretation” presented here comes close to what is usually called the Copenhagen Interpretation.

Mathematical Formalism of Quantum Mechanics

For the further discussion, let us start from the mathematical formalism: In quantum mechanics a physical object is described with the help of a complex vector space with an inner product (“Hilbert space”). How is physics connected with that space? – The observables of the object described are represented by self-adjoint operators of the Hilbert space. An eigenvalue $x_0$ of a self-adjoint operator $X$ represents a possible result of a measurements of the corresponding observable $X$. The present state of the object is described by a vector of norm 1 in the vector space (more abstractly, it is described by the eigenspace of the said eigenvalue; if the eigenspace is one-dimensional, it is conventionally represented by a vector in it of norm one. Still more generally, the state of the system is represented by a “statistical operator”). The eigenvector $x_0$ corresponding (in the special case mentioned) to the eigenvalue $x_0$ represents the state of the system after the eigenvalue $x_0$ has been measured. If the state of the system before the measurement is correctly described by the state-vector $\xi$ (of norm 1) then the probability of measuring value $x_0$ is the square of the absolute value of the inner product of the two vectors,

$$p(x_0) = \left| \xi, \xi_0 \right|^2$$

provided $(\xi_0, \xi_0) = (\xi, \xi) = 1$ (ignoring for the time being a more general description).

The time development of the state $\xi$ of the system is described by a unitary transformation depending on to the elapsed time $t$,

$$\xi(t) = e^{-iHt} \xi(0)$$

where $H$ is the Hamiltonian Operator, which represents the observable ‘energy’.

Lattice

The above presentation describes quantum mechanics as students learn it. It leads to some specific difficulties, though, for understanding quantum mechanics. One could, e.g., ask what the addition of vectors of Hilbert space means in reality—since it occurs in the mathematical description that is supposed to represent a corresponding structure of reality. But asking such a question does not make sense: ‘addition of vectors in Hilbert space’ does not have a meaning in a description of reality. This is important to know for an interpretation: The topic of admitting questions arises only from a formulation of quantum mechanics that does not easily lend itself for an understanding, namely in so far as it uses the vector space formalism.

This drawback might be cured by a more abstract but more comprehensible mathematical picture in using the concept of lattice:

A mathematical lattice is a partially ordered set (with an ordering we signify by ‘$\preceq$’) which is closed under meet $(\cap)$ and join $(\cup)$, where meet, in this general context, is the greatest lower bound according to the order relation ‘$\preceq$’, and join is the least upper bound. A lattice contains the elements $\square$ and $1$, the minimal and maximal elements of the whole structure, according to the order relation ‘$\preceq$’. A special example of a lattice is the set of all subsets of a given set, ordered by set inclusion $(\subseteq)$—isomorphic with the lattice of (classical) propositional logic. This type of lattice is called Boolean. But the concept of lattice is much more general.

• Let us be more specific for our case of quantum mechanics: A lattice is orthocomplemented iff for every element $E$ of the lattice there is an orthocomplement, namely an element $E^n$ such that
• $E \cap E^n = \emptyset$;
• $E \cup E^n = \mathbf{1}$;
• $E^n = E$;
• if $E \leqslant F$ then $F^n \leqslant E^n$.

Again, the set of subsets of a given set, ordered by set inclusion ($\subseteq$), is a special example, where the orthocomplement is the ordinary set complement. An example that is important for quantum mechanics is the set of all subspaces of Hilbert space. It is ordered by set inclusion, the orthocomplement of a subspace is the orthogonal subspace.

Let us now consider the structure of all properties of a given physical system or, maybe more realistic, the structure of all possible results of measurements of that system! The ordering relation in that case is implication ($\rightarrow$): $A \rightarrow B$ means: 'Whenever $A$ is necessary then $B$ is necessary as well'. The other properties of the orthocomplemented lattice are the following: the orthocomplement is negation ('not', ¬); the 'meet' operator represents conjunction ('and', ∧)—nothing but putting two predictions beside each other. From there, disjunction ('or', ∨) can be derived as:

$$A \vee B = (A^\perp \wedge B^\perp)^\perp$$

It can easily be seen from the definition of the orthocomplement that $A \wedge B$ is the greatest lower bound, and $A \vee B$ is the least upper bound of $A$ and $B$.

Again, classical propositional logic is an example, as well as the lattice of subspaces of Hilbert space. Because evidently there are similarities between the two, the lattice of the subspaces of a Hilbert space sometimes is called "Quantum Logic"$^{15}$.

Considering quantum mechanics from the lattice point of view gives us the means at hand that make it easier to understand quantum mechanics—without being obliged to dive into the depths of differential analysis or the like. All we have to do is fixing a few fundamentals of what we understand physics to be.

Physical objects ("systems")

The first question: What is physics about? – One possible answer is: "Physics gives the opportunity for predictions, based on the present state and using established theories." – This answer is felt by some scholars as being too "operational". They are rather seeking an objective description of reality. But these two goals of physics are not as far apart as it might seem. What is an 'objective' description? It is a description that is valid independently of the person, time and place; it can be verified—in principle—by anyone anytime anywhere. 'Verifying' means, checking whether the respective proposition is true. This kind of checking is possible only if the proposition is a prediction which can come true or not. Thus, any objective description necessarily involves predictions.

Thus, we can record that physics deals with objective predictions. – How are such predictions connected among each other? – Actually, as mentioned above, the predictions do not concern the real world but an approximate, idealized "model", the physical system. It is defined by its observables. Here the fundamental role of predictions comes into play: A physical system (object) is composed of those observables that make, together, predictions about the same observables possible. Take, e.g., the classical object 'point mass'. Its defining observables are position and momentum. Momentum governs the change of position, and how momentum will change (i.e. the forces on the point mass) depends in many cases only on the position and, maybe in addition, on momentum (as for friction). Thus, for many cases position and momentum are a good choice of observables for a system to enable predictions about those same observables. Therefore, a point mass is a good object in such cases. – Let us have a look at a more complex example, the electromagnetic field, defined by its values at every point in space: The Maxwell equations show that the change of the magnetic field depends on spatial derivatives of the electric field, and the change of the electric field depends on spatial derivatives of the magnetic field; thus, again, the whole set enables predictions for the same whole set.

But what about quantum mechanics?

Quantum mechanical predictions are similar to classical ones: They predict outcomes of measurements. All possible outcomes of one measurement form a Boolean lattice, as in classical physics. But, other than in classical physics, in quantum mechanics there are incompatible observables. That means, if the state of the system can be described by the necessity of a certain outcome (i.e. in attributing the corresponding property—e.g. a certain position—to the system), then there are possible outcomes

15 The paper "The Logic of Quantum Mechanics" of 1936 by G. Birkhoff and J.v. Neuman, Annals of Mathematics, 37(1936)(4): 823–843. doi:10.2307/1968621

Drieschner M. Understanding Quantum Theory. Philos Int J 2021, 4(2): 000178.

Copyright © Drieschner M.

of other measurements—e.g. of momentum— that cannot be predicted with certainty, but only with certain probabilities: Such observables are called incompatible. “Almost all” pairs of observables of a system are incompatible.

So there we are, at the notorious indeterminism of quantum mechanics. Whereas in classical physics we could always suppose that there is one Boolean lattice that comprises all possible predictions about a system, in quantum mechanics there are several Boolean lattices for different incompatible observables. The question therefore is, how those Boolean lattices combine into a description of the respective system. In the completed quantum mechanics, to be sure, the combination is described as the orthocomplemented lattice of the closed subspaces of Hilbert space. That is known, in principle, empirically. But can we get at understanding that structure without jumping at once to the conclusion “Hilbert space”?

Are there Alternatives to Quantum Mechanics?

What do we know about that combination lattice, from general considerations?

1. There are Boolean sublattices, one for every set of compatible observables.
2. It is an orthocomplemented lattice.
3. There are probability functions defined for any Boolean sublattice of the whole lattice.
4. If there is a necessary prediction that defines the present state of the system, it defines probabilities for all predictions.
5. We can compose two independent systems abstractly into one system.

The probability of finding a joint result is the product of the two probabilities involved.

Can we conclude from general considerations more specifically what the quantum mechanical lattice is like? – In the completed quantum mechanics the lattice we use is, abstractly speaking, a complex projective geometry. This is, in more abstract terms, what is represented by the lattice of the closed subspaces of Hilbert space.

Finding that lattice from plausible assumptions would be the decisive step in the foundation of quantum mechanics. The ‘Plato’ database at Stanford17 says, in referring to this lattice as ‘L(H)’: “The point to bear in mind is that, once the quantum-logical skeleton  L(H)  is in place, the remaining statistical and dynamical apparatus of quantum mechanics is essentially fixed. In this sense, then, quantum mechanics— or, at any rate, its mathematical framework—reduces

17 https://plato.stanford.edu/entries/qt-quantlog/; chapter 1.4 to quantum logic and its attendant probability theory.” Varadarajan says in his seminal book,18 “For a long time it has been a desire within the community for finding rather comprehensible postulates that imply the structure of quantum mechanics.” Varadarajan was one of those scholars who contributed a lot to meet that desire. There has been, though, no mathematical proof so far that Hilbert space is the only possibility for a structure of quantum mechanics. But, up to now, no other lattice has been found that complies with the rules mentioned.

One property that makes this lattice special is its behavior when two systems are composed. When you take one observable of each of the systems, the compound observable is the direct product of the two separate ones. All those direct products form again a lattice of the same type as the component ones, in this case describing the compound system.—In quantum mechanics we describe the state space of the compound system as the tensor product of the two state spaces. This seems to be a very special combination, because it comprises the direct product of any two observables of the two component systems, and they again form an orthocomplemented lattice of the same type as the component ones, though, naturally, its dimension is the product of the dimensions of the components. I suspect that this special structure gives a handle for justifying the structure of quantum mechanics.

Starting from this structure (the orthocomplemented complex projective geometry), we can introduce the usual quantum mechanics of the ψ-function etc. (Hilbert space quantum mechanics) as a special representation of that structure, which facilitates calculating measurable results.

Taking quantum mechanics as a lattice of possible predictions, seems to make it easier to understand. Let me take up, in order to try that out, some of the much discussed stumbling blocks within quantum mechanics.

Conclusion

It is not so hard to understand quantum mechanics, once one has accepted its indeterministic character (and uses the lattice representation). Understanding the underlying concept of probability is much easier if one starts from the definition: probability is predicted relative frequency. Thus, Fuchs and Peres couldn’t be more right: Quantum Theory needs no ‘interpretation’!

Quotation from: Friebe et al 2018, p. 39

If one tries to proceed systematically, then it is expedient to begin with an interpretation upon which everyone can agree, that is with an instrumentalist minimal interpretation. In such an interpretation, Hermitian operators represent macroscopic measurement apparatus, and their eigenvalues indicate the measurement outcomes (pointer positions) which can be observed, while inner products give the probabilities of obtaining particular measured values. With such a formulation, quantum mechanics remains stuck in the macroscopic world and avoids any sort of ontological statement about the (microscopic) quantum-physical system itself. _(Friebe (2018), p.44)_33 Bryce: 1970: Quantum mechanics and Reality. Physics Today Sept. 1970: 30- 35.

33 Friebe (2018): Friebe, Cord; Meinard Kuhlmann; Holger Lyre; Paul M. Näger; Oliver Passon; Manfred Stöckler: The Philosophy of Quantum Physics. Cham (Springer)

The Ensemble Interpretation and the Copenhagen Interpretation The first stage of interpretation of the mathematical formalism establishes the connection to the empirical world as far as needed for everyday physics in the laboratory or at the particle collider. Born’s rule allows a precise prediction of the probabilities of observing particular outcomes in real, macroscopic measurements. The fact that this minimal interpretation makes statements only about macroscopic, empirically directly accessible entities such as measurement setups, particle tracks in detectors or pulses from a microchannel plate may be quite adequate for those who see the goal of the theory within an experimental science such as physics as being simply the ability to provide empirically testable predictions. For the metaphysics of science, this is not sufficient, and most physicists would also prefer to have some idea of what is behind those measurements and observational data, i.e. just how the microscopic world which produces such effects is really structured. In contrast to the instrumentalist minimal interpretation, however, every additional assumption which might lead to a further- reaching interpretation remains controversial.34

34 von Neumann (1932/1955/2018): Mathematical Foundations of Quantum Mechanics, by John von Neumann 1932, translated from the German 1955 by Robert T. Beyer, New Edition 2018 edited by Nicholas A. Wheeler. Princeton (UP) 2018