# Notions of Completeness in the EPR Discussion

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Classification Scheme

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Completeness in the EPR Debate and Beyond

#### 3.1. Theory Completeness

“A complete system of theoretical physics consists of concepts and basic laws to interrelate those concepts and of consequences to be derived by logical deduction.”

“One of the imperfections of the original relativistic theory of gravitation was that as a field theory it was not complete; it introduced the independent postulate that the law of motion of a particle is given by the equation of the geodesic. A complete field theory knows only fields and not the concepts of particle and motion. For these must not exist independently of the field but are to be treated as part of it.”

**Definition**

**5.**

#### 3.2. Bijective Completeness

“Whatever the meaning assigned to the term complete, the following requirement for a complete theory seems to be a necessary one: every element of the physical reality must have a counterpart in the physical theory”

“Since a complete description of a physical state must necessarily be an unambiguous description (apart from superficialities such as units, choice of the co-ordinates etc.), it is therefore not possible to regard the ψ-function as the complete description of the state of the system.”

“Man möchte nun gerne folgendes sagen: Ψ ist dem wirklichen Zustand des wirklichen Systems eindeutig zugeordnet. […] Wenn dies geht rede ich von einer vollständigen Beschreibung der Wirklichkeit durch die Theorie.”

“One would now very much like to say the following: Ψ stands in a one-to-one correspondence with the real state o f the real system. […] If this works, I talk about a complete description of reality by the theory.” (Translation by Arthur Fine [19] (p. 71))

**Definition**

**6.**

#### 3.3. Born Completeness

“According to the Bornian concept, a complete description is essentially nonprobabilistic; genuinely probabilistic assertions are necessarily incomplete.”

**Definition**

**7.**

#### 3.4. Schr ödinger Completeness

“By contrast the Schrödinger view is that probabilities can be fundamental, not to be reduced to something else. Thus the Schrödinger conception is that a complete description of a state of affairs can be a probabilistic assertion, with probability less than unity, which (somehow) tells the whole truth about that state of affairs. If there were some further truth to be told, then the probabilistic assertion would be an incomplete description.”

**Definition**

**8.**

“Nun beschreibe ich einen Zustand so: Die Wahrscheinlichkeit dafür, daß die Kugel in der ersten Schachtel ist, ist $\frac{1}{2}$. Ist dies eine vollständige Beschreibung?"

“I describe a state this way: The probability for a ball being in the first box is $\frac{1}{2}$. Is that a complete description?" (Translation by the author)

“Der Zustand vor dem Aufklappen ist durch die Zahl $\frac{1}{2}$ vollständig charakterisiert, deren Sinn sich bei Vornahme von Beobachtungen allerdings nur als statistischer Befund manifestiert."

“The state before opening the box is completely characterised by the number $\frac{1}{2}$, whose meaning becomes manifest as statistical account by applying measurements on the system." (Translation by the author)

#### 3.5. $\psi $-Completeness

“An ontological model is ψ-complete if the ontic state space Λ is isomorphic to the projective Hilbert space $\mathcal{PH}$ (the space of rays of Hilbert space) and if every preparation procedure ${P}_{\psi}$ associated in quantum theory with a given ray ψ is associated in the ontological model with a Dirac delta function centered at the ontic state ${\Lambda}_{\Psi}$ that is isomorphic to Ψ, $p\left(\lambda \right|{P}_{\psi})=\delta (\lambda -{\lambda}_{\psi})$.”

“It is quite clear that by “real state of the real system”, Einstein is referring to the ontic state pertaining to a system. Bearing this in mind, his definition of completeness can be identified as precisely our notion of ψ-completeness given in Definition 2”

“Man beschreibt in der Quantentheorie einen wirklichen Zustand eines Systems durch eine normierte Funktion ψ der Koordinaten (des Konfigurationsraumes). Die zeitliche Änderung ist durch die Schrödinger-Gleichung eindeutig gegeben. Man möchte nun gerne folgendes sagen: ψ ist dem wirklichen Zustand des wirklichen Systems eindeutig zugeordnet. Der statistische Charakter der Meßergebnisse fällt ausschließlich auf das Konto der Messapparate bzw. des Prozesses der Messung. Wenn dies geht rede ich von einer vollständigen Beschreibung der Wirklichkeit durch die Theorie. Wenn aber eine solche Interpretation nicht durchführbar ist, nenne ich die theoretische Beschreibung unvollständig.”

“In quantum theory a real state of a system is described by a normed function ψ of coordinates (of configuration space). The evolution in time is given by Schrödinger equation uniquely. One would like to say: ψ is corresponding to the real state of the real system uniquely. The statistical character of the measurement results is a consequence of the measurement apparatus respectively the measurement process. If this works, I speak about a complete description of reality by a theory. If such an interpretation is not possible, I call the theoretical description incomplete.” (Translation by the author)

#### 3.6. Remarks on Completeness and the EPR-ER Connection

“While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.”

“Es freut mich dass Du meinen kleinen Aufsatz gelesen hast. Hast Du auch gemerkt wie unlogisch Pauli darauf geantwortet hat? Er leugnet es, dass diese Art der Beschreibung unvollständig sei, sagt aber im selben Atemzuge, dass die ψ Funktion eine statistische Beschreibung des Systems sei, die Beschreibung einer System-Gesamtheit. Dies ist doch nur eine andere Form der Aussage. Die Beschreibung des (individuellen) Einzelsystems ist unvollständig!”

“I am pleased to hear that you read my little article. Have you recognised also how illogical Pauli responded? He denies that this description is incomplete, but states in the same breath that the ψ function is a statistical description of the system, the description of a system-totality. This is just a different formulation of the statement: The description of an (individual) single-system is incomplete!” (Translation by the author)

#### 3.7. Completeness and the Unified Field Theory Program

“I still believe in the possibility of giving a model of reality, a theory, that is to say, which shall represent events themselves and not merely the probability of their occurrence.”

“While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.”

“In favor of the theory (The Einstein-Rosen bridge model for elementary particles) one can say that it explains the atomistic character of matter as well as the circumstance that there exist no negative neutral masses, that it introduces no new variables other than the ${g}_{\mu \nu}$ and ${\varphi}_{\mu \nu}$, and that in principle it can claim to be complete (or closed). On the other hand one does not see a priori whether the theory contains the quantum phenomena.”(our underlining)

“Es wird aber gezeigt, dass die Überzeugung von der Unfähigkeit der Feldtheorie, diese Probleme mit ihren Methoden zu 1ösen, auf Vorurteilen beruht.”

“It will be shown that the conviction of the inability of field theory to solve these problems (The problems refer to the inability of field theory to give account for a molecular structure and to describe quantum phenomena with methods of field theory [5] (p. 347)) rests on a prejudice.”

“Angesichts dieser Sachlage erscheint es mir durchaus gerechtfertigt, die Frage ernsthaft zu erwägen, ob nicht doch die Grundlage der Feldphysik mit den Quanten-Tatsachen vereinbar ist.”

“In view of this situation it seems appropriate to me to reconsider the question if a unification of the foundations of field theory and quantum facts is possible.” (Translation by the author)

“Ich halte die statistische Physik trotz all ihrer Erfolge doch für eine vorübergehende Phase und habe Hoffnung, zu einer wirklich befriedigenden Theorie der Materie zu gelangen. Ich sende Dir gleichzeitig eine kurze Arbeit, die den ersten Schritt darstellt. Das neutrale und das elektrische Teilchen erscheinen gewissermassen als Loch im Raume, derart, dass das metrische Feld in sich selbst zurückkehrt. Der Raum wird als zweischalig dargestellt. In der Schwarzschild’schen strengen zentralsymmetrischen Lösung erscheint das Teilchen im gewöhnlichen Raume als Singularität vom Typus $1-\frac{2m}{r}$. Durch die Substitution $1-2m={u}^{2}$ wird das Feld regulär im r-Raume. Wandert u von $-\infty $ bis $+\infty $, so wandert r von $+\infty $ zu $r=2m$ und hierauf wieder zurück zu $r=+\infty $. So kommen beide "Blätter" im Riemann’schen Sinne zustande, die an der “Brücke” $r=2m$ bezw. $u=0$ stetig zusammenhängen. Aehnlich bei der Elektrizität. Die Aufgabe an der ich mit einem jungen Kollegen (russischer Jude) unablässig schwitze ist die Behandlung des Mehrkörperproblems auf dieser Basis. Wir haben aber die ernsthaften Schwierigkeiten des Problems bereits überwunden, sodass sich bald zeigen wird was daran ist. Jedenfalls ist es eine wundervolle mathematische Aufgabe.”

“To me statistical physics despite its success is a transitory phase and I have hope that we arrive at a really satisfying theory of matter. I send you enclosed a short work, that represents a first step. The neutral and the electrical particle appear as a hole in space, of this kind that the metric field returns to itself. Space is represented as two sheets. In the strict spherical symmetric solution of Schwarzschild the particle appears in usual space as singularity of the kind $1-\frac{2m}{r}$. By substitution $1-2m={u}^{2}$ the field becomes regular in r-space. If u goes from $-\infty $ to $+\infty $, r is going from $+\infty $ to $r=2m$ and back again to $r=+\infty $. This represents both “sheets”, that are connected by the “bridge” at $r=2m$ respectively $u=0$. Likewise as it is in electricity. The challenge a young college (russian jew) and I labour away over is the many-body-problem on that basis. We have conquered the serious problems already, so it will show soon if there is something serious about it. Anyway it is a wonderful mathematical problem.” Translation by the author

“Erst die Untersuchung des Mehr-Brücken-Problems kann zeigen, ob diese theoretische Methode eine Erklärung für die empirisch erwiesene Massengleichheit der Teilchen in der Natur liefert, und ob sie den von der Quantenmechanik so wunderbar erfassten Tatsachen gerecht wird.”

“Only the examination of the many-bridge-problem can show if this theoretical model provides an explanation for the empirical proven equality of masses of particles in nature, and if it can reproduce the facts that are represented by Quantum Mechanics in such a delightful way.” (Translation by the author)

“General relativity contains solutions in which two distant black holes are connected through the interior via a wormhole, or Einstein-Rosen bridge. These solutions can be interpreted as maximally entangled states of two black holes that form a complex EPR pair. We suggest that similar bridges might be present for more general entangled states.”

“Now I feel that our current views of Quantum Mechanics are provisional; it’s the best we can do without a much deeper understanding of its connection with gravity, but it’s not final. The reason involves a very particular development, the so called ER=EPR principle. ER=EPR tells us that the immensely complicated network of entangled subsystems that comprises the universe is also an immensely complicated (and technically complex) network of Einstein-Rosen bridges.”

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Depiction of a theory fulfilling theory completeness: Laws (bold lines) relate mathematical entities (bold dots), which obtain physical meaning from the principles of the theory (areas).

**Figure 3.**Completeness —(Closedness) [14].

**Figure 4.**The completeness requirement of the EPR paper: An injective relation from the elements of reality (left side) to the elements of reality (right side).

**Figure 5.**Bijective completeness resulting from an unambiguousness relation between the elements of reality (left side) and the elements of the theory (right side).

**Table 1.**Classification scheme: We distinguish four levels of a theory to provide a categorization to analyze different notions of completeness.

1 | Mathematical laws between mathematical symbols | |

2 | Interpretation of physical quantities Relation of physical quantities and measurements Measurement laws | |

3 | Concepts and principles | |

4 | Ontology—beables— elements of reality |

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Krizek, G.C.; Mairhofer, L. Notions of Completeness in the EPR Discussion. *Entropy* **2023**, *25*, 585.
https://doi.org/10.3390/e25040585

**AMA Style**

Krizek GC, Mairhofer L. Notions of Completeness in the EPR Discussion. *Entropy*. 2023; 25(4):585.
https://doi.org/10.3390/e25040585

**Chicago/Turabian Style**

Krizek, Gerd Christian, and Lukas Mairhofer. 2023. "Notions of Completeness in the EPR Discussion" *Entropy* 25, no. 4: 585.
https://doi.org/10.3390/e25040585