Does Newtonian Space Provide Identity to Quantum Systems?

We are intrinsically spatial and temporal beings. We can only observe particular entities, handle spatial things, and our thinking depends on the concept of individuals. How to reconcile nonspatial quantum characteristics with spatio- temporal individuation is perhaps the next great problem physics has to overcome.

(S. Auyang, How is quantum field possible?. Princeton Un. Press, 1995, p. 165)

In the microworld, we need uniformity of a strong kind: complete indistinguishability.

(F. Wilczek and B. Devine, Longing for Harmonies: Themes and Variations from Modern Physics. Penguin Books, 1987, p. 135)

Abstract

Physics is not just mathematics. This seems trivial, but poses difficult and interesting questions. In this paper we analyse a particular discrepancy between non-relativistic quantum mechanics (QM) and ‘classical’ (Newtonian) space and time (NST). We also suggest, but not discuss, the case of the relativistic QM. In this work, we are more concerned with the notion of space and its mathematical representation. The mathematics entails that any two spatially separated objects are necessarily different, which implies that they are discernible (in classical logic, identity is defined by means of indiscernibility)—we say that the space is T₂, or “Hausdorff”, or yet “separable”. But when enters QM, sometimes the systems need to be taken as completely indistinguishable, so that there is no way to tell which system is which, and this holds even in the case of fermions. But in the NST setting, it seems that we can always give an identity to them by means of their individuation, which seems to be contra the quantum physical situation, where individuation (isolation) does not entail identity (as we argue in this paper). Here we discuss this topic by considering a case study (that of two potentially infinite wells) and conclude that, taking into account the quantum case, that is, when physics enter the discussion, even NST cannot be used to say that the systems do have identity. This case study seems to be relevant for a more detailed discussion on the interplay between physical theories (such as quantum theory) and their underlying mathematics (and logic), in a simple way apparently never realized before.

Appendices

Appendix 1 : How Space and Time Enter the Quantum Schema

We shall use the Hilbert space formalism, common in most philosophical discussions. From a technical point of view, we can assume that we shall be working with the resources of ZFC. So, let us introduce the following definition [more details in Krause and Arenhart (2016)], which follows the style presented in Suppes (1957, Chap. 12):¹⁸

Definition 1.1

A non relativistic quantum mechanics is a 5-tuple of the form

$$ {\mathcal {Q}} = \langle S, \{{\mathcal {H}}_i\}, \{\hat{A}_{ij}\}, \{\hat{U}_{ik}\}, {\mathcal {B}}({\mathbb {R}})\rangle , \; \mathrm {with} \; i \in I, j \in J, k \in K $$

where:

• 1. S is a collection whose elements are called physical objects, or physical systems. Here is the novelty of our approach. The standard formalism doesn’t speak of the quantum systems directly, making reference only to their states and observables. The talk about the systems is made in the metalanguage. Here we are introducing the systems in the structure. Since they can be indiscernible in a strong sense, S would not be a set, but a quasi-set. But, if we aim at to consider this last option, we need to change ZFC by quasi-set theory.

• 2. $\{H_i\}$ is a collection of mathematical structures, namely, complex separable Hilbert spaces whose cardinality is defined by the particular application of the theory.

• 3. $\{\hat{A}_{ij}\}$ is a collection of self-adjunct (or Hermitian) operators over a particular Hilbert space $H_i$.

• 4. $\{U_{ik}\}$ is a collection of unitary operators over a particular Hilbert space $H_i$

• 5. $\mathcal {B}(\mathbb {R})$ is the collection of Borel sets over the set of real numbers.

In order to connect the formalism with experience, we construct a mathematical framework for representing experience. Important to remark that this is another theoretical (abstract) construction: there is no connection, out of the informal one, of the formalism with reality per se. In order to do it, we need to elaborate reality, turning it a mathematical construct too. So, to each quantum system $s \in \mathcal {S}$ we associate a 4-tuple

$$ \sigma = \langle \mathbb {E}^4, \psi (\mathbf {x}, t), \Delta , P\rangle , $$

where $\mathbb {E}^4$ is the Newtonian spacetime (Penrose 2005, Chap. 17), where each point is denoted by a 4-tuple $\langle \mathbf {x}, t\rangle$ where $\mathbf {x} = \langle x, y, z \rangle$ denote the coordinates of the system and t is a parameter representing time, $\psi (\mathbf {x},t)$ is a function over $\mathbb {E}^4$ called the wave function of the system, $\Delta \in \mathcal {B}(\mathbb {R})$ is a Borelian, and P is a function defined, for some i (determined by the physical system s), in $\mathcal {H}_i \times \{\hat{A}_{ij}\} \times \mathcal {B}(\mathbb {R})$ and assuming values in [0, 1], so that the value $P(\psi , \hat{A}, \Delta ) \in [0,1]$ is the probability that the measurement of the observable A (represented by the self-adjunct operator $\hat{A}$) for the system in the state $\psi (\mathbf {x},t)$ lies in the Borelian set $\Delta$.¹⁹

Summing up: using standard mathematics, and classical space and time setting, this is all we can do, and it works! Physics can be made within such a framework, but philosophically we can raise questions such as those posed above. Summing up, I can give an answer to Omnès question about the consistence of the two languages: NO, in general they are not consistent one ano ther.

Appendix 2 : A Non Hausdorff Topology for QM?

In this Appendix, we advance an idea to be developed further, but which can complete the above discussion. The intention here is (I hope) to get some feedback. We shall be supposing the theory of quasi-sets (qsets) without further details (Arenhart and Krause 2014; French and Krause 2006; Krause and Arenhart 2018). Informally speaking, qsets may comprise absolutely indiscernible elements, entities to which the standard sense of identity ascribed by classical logic and set theory does not apply. In this theory, we can speak of indistinguishable but not identical objects. The qsets may have a cardinal, termed its quasi-cardinal even if its elements cannot be discerned form one another (this demands explanation, for it is usually—and wrongly, according to the mentioned works—supposed that once a collection has a cardinal greater than one, its elements are distinguishable). So, let us go.

Consider a qset A whose elements are all indiscernible one each other (we write $x \equiv y$ for all $x, y \in A$). Let B be a proper subqset of A. First, to avoid misunderstandings, let us explain why sentences such as “there is an element of A that belong to B” doesn’t imply that all elements of A belong to B, as it could be supposed since the elements of A are indistinguishable. It is in this sense that we can speak of a ‘proper’ subqset of A. The explanation is in term of cardinals, the quasi-cardinals of the qsets. When a qset has a quasi-cardinal, or just q-cardinal for short (not always a qset has a q-cardinal), then this q-cardinal is a cardinal in the standard sense, defined in the classical part of the theory (which is a model of ZFU, the Zermelo-Fraenkel set theory with Urelemente). Furthermore, it is postulated that for every qset A whose q-cardinal is $\lambda$, there are subqsets of A whose q-cardinals are $\eta$ for every $\eta \le \lambda$. So, in our hypothesis, once we have assumed that A has a not zero q-cardinal, then it makes sense in the theory to speak of a ‘proper’ subqset B, whose q-cardinal is less that the q-cardinal of A. So, although the elements of A are indiscernible, the theory is consistent with the affirmative that not every element of A is an element of B.

Now, let us define the cloud of B as being the qset of the elements of A that could be elements of B (we could call them the potential elements of B). The formal definition says that $Cl_A(B)$, the cloud of B relative to A, is the qset of all elements of A which are indistinguishable from some element of B.²⁰ In the supposed case, the cloud of B would be the whole A since its elements are indiscernible, so any element of Acould be in B. Now let us consider another subqset $C \subseteq A$disjoined from B. Depending on the q-cardinals of A, B and C, this is possible to suppose. For instance, suppose that the q-cardinal of A is 6 (take A as formed by the six electrons of the level 2p of a Sodium atom). Then the axioms of the theory are compatible with the intuitive idea (which in standard set theories is a theorem) that there are three disjoined subqsets of A with q-cardinals 2 each. The interesting thing is that we have no means to know which elements belong to each subqset, in the same vein that we can talk of two of the six electrons in the 2p shell but have no means to identify them. It is in this sense that we can say that the three subqsets are disjoined. But their clouds relative to the whole collection coincide with A.

So, let us go back to our example. We had two disjoined subqsets of A, but their clouds relative to A intercept (in the given example, they coincide with A), so they are not disjoined. If we refer to B and C as the core qsets, then there is a sense in saying that they do not have common elements but that their clouds do have and, depending on the characteristics of A, will always intercept. This exemplifies a situation which can be taken as similar to the spatial overlapping of the wave-functions of two independently separated “identical” particles, a situation in which they become entangled and cannot be discerned in any way (Lo Franco and Compagno 2017).

Let us consider now the algebra of clouds. By this we mean, as in standard mathematics, the consideration of the clouds of the subqsets of some given qset (not necessarily with all elements indiscernible) with the usual mathematical operations (which have their counterpart in the theory of qsets). If we were considering sets, the algebra, as is known, would be a Boolean algebra. But here, as shown in Nascimento et al. (2011), this is not the case. The algebraic counterpart of the clouds is a non distributive lattice we have termed the lattice of indiscernibles. It resembles in much an orthomodular lattice, typical of quantum mechanics. But this is not what we intend to do here.

What we wish to say is that by considering qsets and their clouds, we get a (quasi-)set theoretical framework where for any point (element of a qset) we can consider a core to which it belongs to which is disjoined from another core of some ‘other’ element of the qset but so that their clouds intercept.²¹ So, the cores can be though as the two wells of Sects. 4 and 6. Since their clouds intercept, we can say that the elements are never completely isolated. The talk of ‘other’ element is, again, just a way of speech. All can be expressed in terms of q-cardinals.

The advantage of such a qset-theoretical approach is that the elements can be considered as non-individuals, as entities without a Principle of Individuality, without identity, contrary to what occurs in a standard set theory, where of course we can do similar things, for instance by considering either fuzzy sets or quasets (Dalla Chiara and Toraldo di Francia 1993) (see also French and Krause 2006, Sect. 7.4). But, in these cases, the objects would have identity, are individuals, contrary to our preferred metaphysics of seeing then as entities devoid of identity conditions.

The next step is to define a topology in qset theoretical terms. The definition is standard. We say that a qset $\Xi = \langle X, \tau \rangle$ is a q-topological space if X is a qset and $\tau$ is a family of clouds of subqsets of X satisfying the following conditions:

• 1. $\emptyset$ and X belong to $\tau$

• 2. $A \cap B \in \tau$ for every clouds A and B in $\tau$

• 3. For any family of clouds of $\tau$, their union also belongs to $\tau$

The definition is standard, but the consequences in taking it within quasi-set theory are not. So, where is the difference to a standard topology? The answer is that the clouds may ever intercept (depending of the chosen qsets). In this sense, the space is not Hausdorff and, consequently, given that we represent two similar quantum objects within such a framework, we cannot say that there are disjoined open balls centered in these points which do not intercept, since the ‘balls’ are taken as clouds of some qset. It seems that this non-Hausdorff topology fits better what NQM dictates.

But it seems that it is not any non Hausdorff topology that could be useful. There are ‘standard’ non-Hausdorff topologies which enable points to be indiscernible (not distinguished by open sets). For instance, take a Zariski topology (Wikipedia 2018) (here roughly) defined by means of closed sets as follows: take $\mathbb {R}^2$ with the closed sets being the zeros of polinomial equations with integer coefficients. The open sets are their complements. Then, the points (e, 2e) and $(\pi , 2\pi )$ cannot be discerned by open sets, for they are not zeros of any such a polinomial equation, hence belong to any open set.

Thus, even if (e, 2e) and $(\pi , 2\pi )$ are indiscernible by the defined Zariski topology, the topological space admits some hidden property, expressed for instance by the straight line $x = \pi$ that distinguish them: only the second one has such a ‘property’ (of belonging to the line). Thus, this seems not suitable for expressing the quantum case, where the systems should be taken as indiscernible by any means, encoded in the topology or not. There should be not hidden variables involved. Thus, what we need is a different kind of non-Hausdorff topology, perhaps one developed within the framework of quasi-set theory. But such a qset topology is still something to be pursued.