# Representation Lost: The Case for a Relational Interpretation of Quantum Mechanics

## Abstract

**:**

## 1. Introduction

## 2. The Basic Set-Up

#### 2.1. Measurements

#### 2.2. States

#### 2.2.1. Betting-States

- Betting-states. The betting-state ascribed to a team is a triple $\langle \mathrm{w},\text{}\mathrm{d},\text{}\mathrm{l}\rangle $ where w, d, and l denote the number of wins, draws, and losses respectively.

#### 2.2.2. Odds Comparison

#### 2.2.3. Number-States

- Number-states. A number-state ascription is an ascription of a numerical value to a betting-state.
- Number-state functions. Number-state functions are linear functions from betting-states to the natural numbers.

- (1)
- The correspondence between betting-states and number-states is many-to-one. If, e.g., a team is assigned 15 points after 10 games (by the standard number-state function $\overline{\mathrm{s}}=\langle 3,\text{}1,\text{}0\rangle $), this is compatible with the team being in betting-states $\langle 5,\text{}0,\text{}5\rangle $ or $\langle 4,\text{}3,\text{}3\rangle $ or $\langle 3,\text{}6,\text{}1\rangle $. In general, therefore, knowledge of a team’s number-state only restricts, but does not determine, which betting-state the team can be said to be in. Although it is natural to say that number-states “encode information about the betting-states,” that information is not fully recoverable from the number-states. (There is common ground here between the football example and the toy-model developed by Spekkens in Ref. [27]: since the number-states put a limit on what can be known about the betting-states, they echo what Spekkens’ refers to as the “knowledge balance principle”, which he introduces as a postulate; cf. [27] (p. 3).) Introducing a further piece of terminology, I will say that a number-state “declares possible” all the betting-states that are compatible with it (so that, e.g., the number-state ascription “K has 15 points after 10 games” declares possible the betting-states $\langle 5,\text{}0,\text{}5\rangle $, $\langle 4,\text{}3,\text{}3\rangle $ and $\langle 3,\text{}6,\text{}1\rangle $ relative to the choice of number-state function $\overline{\mathrm{s}}=\langle 3,\text{}1,\text{}0\rangle $).
- (2)
- The specific number-state assigned to a betting-state has no objective significance, in the sense that (a) the choice of number-state function (from which it derives) is conventional, and (b) the relative ordering of the teams is not preserved under a general change of number-state function. In other words, since different choices of number-state functions (generically) produce different tables, the relevant relations— “…is better than…,” “…is worse than…,” and “…is equal to…”—are inherently relative to the choice of number-state function. Thus, these relations cannot be said to reflect (or represent) objective states of affairs.
- (3)
- Using a piece of terminology familiar from foundational studies on the reality of the quantum state, we would say that the number-states are “ontic” rather than “epistemic” [14,15]. Since this terminology will prove useful again below, it is worth outlining the main idea behind the ontological models framework, from which this terminology derives (cf. [15] esp. (pp. 82–88) for a comprehensive overview of the relevant issues). A model is called ontological, if each state in the model’s state-space, which will be denoted by $\mathsf{\Pi}$, corresponds to a classical probability distribution over some measurable space $\left(\mathsf{\Lambda},\mathsf{\Sigma}\right)$ (where $\mathsf{\Lambda}$ is called the ontic state-space, and $\mathsf{\Sigma}$ is a Borel ($\mathsf{\sigma}$-) algebra on $\mathsf{\Lambda}$). An ontological model is called ontic if, for any two states $\mathrm{n}$ and $\mathrm{m}$ in the state-space $\mathsf{\Pi}$, every element in the ontic state-space $\mathsf{\Lambda}$ which $\mathrm{n}$ declares possible, $\mathrm{m}$ declares impossible (this is a somewhat loose, though I hope appropriate, way of paraphrasing the definition given in Ref. [15] for the case in which the elements of the state-space $\mathsf{\Pi}$ don’t ascribe concrete probabilities to the elements of the ontic state-space $\mathsf{\Lambda}$ that they declare possible).

#### 2.3. Dynamics

#### 2.3.1. How the Situation Looks from A’s Perspective

#### 2.3.2. How the Situation Looks from B’s Perspective

## 3. A Classical Version of the Schrödinger Equation for Optimal Opinion Updating Relative to Non-Participating Observers

**Box 1.**Kinematics of Interactional Probability Models.

**Kinematics of Interactional Probability Models**

**Measurement Context:**In the general case, our measurement context consists of n distinguishable outcomes of the interactions between the entities within the model’s scope.**Betting-states relative to B**: The betting-state relative to B is an n-tuple of numbers $\overline{\mathrm{x}}={\mathrm{x}}_{1},\text{}\dots ,{\text{}\mathrm{x}}_{\mathrm{n}}$, the components of which specify B’s best guess for how many times a particular outcome was observed (this best guess will be a true guess, if B happens to know the outcomes of the games). The set of betting-states has the following structure:- Betting-states are vectors. Betting-states can be added (component-wise) and multiplied by a scalar (component-wise).
- Odds-Comparison: The odds comparison of two betting-states $\overline{\mathrm{x}}=\langle {\mathrm{x}}_{1},\text{}\dots ,{\text{}\mathrm{x}}_{\mathrm{n}}\rangle $ and $\overline{\mathrm{y}}=\langle {\mathrm{y}}_{1},\text{}\dots ,{\text{}\mathrm{y}}_{\mathrm{n}}\rangle $ is defined (for $\sum {\mathrm{x}}_{\mathrm{i}}=\sum {\mathrm{y}}_{\mathrm{i}}$) as $\overline{\mathrm{x}}/\overline{\mathrm{y}}:=\langle {\mathrm{x}}_{1}/{\mathrm{y}}_{1},\text{}\dots ,{\text{}\mathrm{x}}_{\mathrm{n}}/{\mathrm{y}}_{\mathrm{n}}\rangle $(if well-defined).

**Number-states**: A number-state ascription is an ascription of a numerical value to a betting-state. All number-states that will be considered are required to arise from a choice of number-state function.- Number-state functions. Number-state functions are linear functions from betting-states to some choice of number field. Thus, number-state functions are characterized by n numbers, $\mathrm{s}=\langle {\mathrm{s}}_{1},\text{}\dots ,{\text{}\mathrm{s}}_{\mathrm{n}}\rangle $, such that: $\mathrm{s}\left(\overline{\mathrm{x}}\right)=\sum {\mathrm{s}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}$. Hence, they are vectors of the same mathematical type as the betting-states.
- Number-states are expectation values. From the point of view of interpretation, the equation $\langle \overline{\mathrm{x}}\rangle =\mathrm{s}\left(\overline{\mathrm{x}}\right)=\sum {\mathrm{s}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}$ yields an expectation value (for the number of points associated with a team after a certain number of games, for caveats, cf. Section 2.2.3).
- Number-state functions are coordinatizations. Since the primary role of number-state functions is that of generating an ordering of the teams, I will say that number-states are “coordinatizations” of the betting-states (cf. Section 4).

- Symmetry. A symmetry of the space of betting-states is a linear map U that maps the space onto itself such that the odds comparison of betting-states is preserved: $\mathrm{U}\left(\overline{\mathrm{x}}\right)/\mathrm{U}\left(\overline{\mathrm{y}}\right)=\overline{\mathrm{x}}/\overline{\mathrm{y}}$ (if well-defined). [24] (p. 857), [20] (p. 72).

**Theorem.**

**Proof.**

- Uniform motion. A uniform motion on the space of betting-states is an element of the set $\left\{\mathrm{U}\left(\mathrm{t}\right),\text{}\mathrm{t}\ge 0:\mathrm{U}\left({\mathrm{t}}_{1}\right)\circ \mathrm{U}\left({\mathrm{t}}_{2}\right)=\mathrm{U}\left({\mathrm{t}}_{1}+{\mathrm{t}}_{2}\right)\right\}$ of symmetries labelled by a continuous parameter t. [24] (p. 857), [20] (p. 72).

**Theorem.**

**Proof.**

**Box 2.**Non-participating observer B’s dynamical problem.

**Optimal Rational Opinion Updating Relative to Non-Participating Observers**

_{f}respectively) such that:

- U is a uniform motion: $\overline{\mathrm{x}}\left({\mathrm{t}}_{\mathrm{f}}\right)=\text{}\mathrm{U}\left({\mathrm{t}}_{\mathrm{f}}\right)\left(\overline{\mathrm{x}}\left(0\right)\right)$.

- 2
- Find real numbers ${\mathrm{k}}_{1},\text{}\dots ,{\text{}\mathrm{k}}_{\mathrm{n}}$ such that $\sum {\mathrm{k}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{f}}\right)\to \mathrm{min}$. This is subject to the constraints that:
- The final betting-state relative to B is normalized: $\sum {\mathrm{x}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{f}}\right)=\mathrm{n}\left({\mathrm{t}}_{\mathrm{f}}\right)$.
- The final number-state is agreed upon by both observers A and B: $\sum {\mathrm{s}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}\left({\mathrm{t}}_{\mathrm{f}}\right)=\mathrm{r}\left({\mathrm{t}}_{\mathrm{f}}\right)$.

**Theorem.**

**Proof.**

## 4. Preliminary Discussion—Some Advantages of the Example

#### 4.1. The Betting-States are not “Absolute Descriptions”

#### 4.1.1. Betting-States are Correlations

#### 4.1.2. “How Things Are” vs. “How Things Affect One Another”

#### 4.2. Completeness

#### 4.3. The Role of Number-States

#### 4.3.1. Number-States are Expectation Values

#### 4.3.2. Number-States are Coordinatizations

## 5. Quantum vs. Classical: The Similarities

- The goal of the discussion of the similarities between the quantum formalism and the classical example will be to provide a suggestive reason for why the above theorem can indeed be viewed as the counterpart of unitary evolution in quantum theory. This will rely on a result due to Lisi [25], who has proposed a heuristic derivation of the Schrödinger equations from similar assumptions as the ones that were required to prove the result of Section 3. Of course, and this is the important point here, such a discussion could only be suggestive (establishing too close a resemblance between the classical and the quantum case could only mean that we have made a mistake—the two cases are, after all, fundamentally different).
- The discussion of the differences between the two cases will be less suggestive. The argument will be structured around the thesis that unlike in the classical case, quantum theory is inconsistent with the assumption that each measurement has a determinate outcome. This argument will rely on a recent no-go theorem due to Frauchiger and Renner. [26], cf. [33] Since this no-go theorem is derived on the basis of the quantum formalism itself, this purports to show in what sense the formalism restricts the set of viable interpretations (this would not be visible if we stayed at the level of the reconstruction programs, for the reason articulated by Timpson).

- (Equivalence of physical systems) “All systems are equivalent: Nothing distinguishes a priori macroscopic systems from quantum systems.” [11] (p. 4).
- (Relative facts postulate) Any system has a quantum state relative to other physical systems (which we, depending on context, consider as the “observing” or “reference” systems).

- (Empiricist facts postulate) “A quantum description of the state of a system S exists only if some system O (considered as an observer) is actually “describing” S, or, more precisely, has interacted with S.” [11] (p. 6).

- Optimization problem for the external observer in quantum mechanics. We require that the entropy—$\mathrm{H}=-\int \mathrm{Dq}\text{}\mathrm{p}\left[\mathrm{q}\right]\text{}\mathrm{log}\left(\mathrm{p}\left[\mathrm{q}\right]\right)$- is minimal, subject to the constraints that (1) the probabilities associated with each path sum up to 1: $\int \mathrm{Dq}\text{}\mathrm{p}\left[\mathrm{q}\right]=1$ (normalization), and (2) the expectation value of the action functional is fixed to be $\mathrm{S}=\int \mathrm{Dq}\text{}\mathrm{p}\left[\mathrm{q}\right]\text{}\mathrm{S}\left[\mathrm{q}\right]$.

“Main observation:In quantum mechanics different observers may give different accounts of the same sequence of events.”[11] (p. 4)

## 6. Quantum vs. Classical: The Differences

#### 6.1. Privileging Strategies

#### 6.1.1. Treating O’s State as Privileged

#### 6.1.2. Treating P’s State as Privileged

#### 6.1.2.1. Argument from Interference

#### 6.1.2.2. Argument from Scientific Realism: Explanatory Virtues

- (Non-interference) The different terms in the superposition are “causally shielded” from one another. They are eigenvectors of some observable (thus they are necessarily orthogonal), and hence no interference effects can be observed if we measure in that basis. [41] (pp. 60–63).
- (Functional instantiation of properties) Each of the non-interfering terms of the superposition represent (= is functionally/structurally equivalent to) a world that consists of objects which instantiate (at least some) determinate properties (such as a particle having the determinate properties of being either “spin-up” or “spin down”). (ibid.)

#### 6.2. (Hidden) Commonalities?

#### 6.2.1. The Quantum State Is Not Epistemic!

#### 6.2.2. Measurement Outcomes Are Not Objective!

## 7. Conclusions: Representation Lost

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs of the Theorems from Section 3

**Theorem.**

**Proof.**

**Theorem.**

**Proof.**

**Theorem.**

**Proof.**

## Appendix B. An Example for the Theorem in Section 3

## Appendix C. The Frauchiger and Renner 2016 Thought Experiment

_{0}and depending on the outcome of her measurement (heads or tails) at t

_{1}, she sends a spin-1/2 particle to the second friend F2. This spin particle is going to be prepared in either the z+ or the x+ direction, depending on whether F1 has observed outcome heads or tails respectively. F2 then conducts a measurement in the z-basis at t

_{2}on the particle she has received from F1. At t

_{3}the assistant A conducts a measurement on F1 in the basis ok/fail (which is an equally weighted superposition of the heads/tails basis—with a minus and a plus sign respectively—on the Hilbert space of F1). At t

_{4}Wigner conducts a measurement in the ok/fail basis on F2. The experiment is repeated many times, until Wigner and A have both received the outcome ok in their respective measurements on F2 and F1.

- By (B) and (Functional instantiation of properties) W concludes that, in one world, there is a version of F2 whose system instantiates spin-up. In another world, the particle that was measured by F2 instantiates spin-down. If W now wonders which version of F2 he shares a world with (what Section 6.1.2.2 called self-locating uncertainty), he will reason as follows.
- (Case 1) If the version of W (who is uncertain) is in the world in which F2’s system instantiates spin-up, then by (A) this is also the same world in which F1’s coin instantiates tails. But then, by (C), this is also a world in which A instantiates property fail.
- (Case 2) If the version of W (who is uncertain) is in the world in which F2’s system instantiates spin-down, then by (B) this is also a world in which F1 instantiates outcome fail. But then by (D) this is also a world in which A instantiates property fail.

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Representation Lost: The Case for a Relational Interpretation of Quantum Mechanics. *Entropy* **2018**, *20*, 975.
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Representation Lost: The Case for a Relational Interpretation of Quantum Mechanics. *Entropy*. 2018; 20(12):975.
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2018. "Representation Lost: The Case for a Relational Interpretation of Quantum Mechanics" *Entropy* 20, no. 12: 975.
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