The undecidable charge gap and the oil drop experiment

Abstract

Decision problems in physics have been an active field of research for quite a few decades resulting in some interesting findings in recent years. However, such research investigations are based on a priori knowledge of theoretical computer science and the technical jargon of set theory. Here, I discuss a particular, but a significant, instance of how decision problems in physics can be realised without such specific prerequisites. I expose a hitherto unnoticed contradiction, that can be posed as as decision problem, concerning the oil drop experiment, thereby resolve it by refining the notion of ‘existence’ in physics. This consequently leads to the undecidability of the charge spectral gap through the notion of ‘undecidable charges’ which is in tandem with the completeness condition of a theory as was stated by Einstein, Podolsky and Rosen in their seminal work. Decision problems can now be realised in connection with basic physics, in general, rather than quantum physics, in particular, as per some recent claims.

Appendix A: Exemplifying the ‘problem of universal validity’ with the continuity equation

Here I provide an example of how one can realise the decision problem as a ‘problem of universal validity’ in physics through self-inquiry (also see the Appendix B of ref. [84]) by studying the proof of the continuity equation that appears in standard textbooks and, otherwise, accepted beyond doubt from a logical point of view. This is actually an example of the second aspect of self-inquiry, i.e. extraction of the computational content.

The continuity equation, the so-called local conservation law, forms the basis of our general understanding of the flow of quantities like charge, mass, etc. in physics, e.g. see refs [97, 104,105,106,107]. Here, I point out how the continuity equation is founded on a decision problem that allows for making suitable choices concerning two apparently contradictory propositions. For the present discussion, it is convenient to consider the continuity equation involving charge and current [107]. To understand the issue at hand, it is sufficient and convenient to discuss the flow of charge along a wire that can be written as follows:

$$\begin{aligned} \mathbf {\nabla }\cdot \textbf{J}+\frac{\partial \lambda }{\partial t}=0, \end{aligned}$$

(A.1)

where $\textbf{J}=\lambda \textbf{v}$, $\textbf{v}=\textrm{d}\mathbf {\ell }/\textrm{d}t$, $\textrm{d}\mathbf {\ell }={\textrm{d}x}~{\hat{i}}+{\textrm{d}y}~{\hat{j}}+{\textrm{d}z}~{\hat{k}}$ and $\textbf{v}$ satisfies the condition $\mathbf {\nabla }\cdot \textbf{v}=0$ [104,105,106]; $\lambda [x(t), y(t), z(t), t]$ is the linear charge density (charge per unit length) at some instant t, $\textbf{J}[x(t), y(t), z(t), t]$ is the line current density and [x(t), y(t), z(t)] denote the spatial coordinate of any point on the line of flow at some time t. The construction of the relation $\textbf{J}=\lambda \textbf{v}$ is based on two scenarios of interpreting the charge flow as demonstrated in the figure shown here, which can be generalised for surface and volume flows (see ref. [107]).

• Scenario 1 (S1): Let a point charge dq is displaced through $\mathbf {\textrm{d}l}$ in time $\textrm{d}t$ such that we can write $\mathbf {\textrm{d}l}=\textbf{v}\textrm{d}t$. Then, we can write $|\mathbf {\textrm{d}l}|=|\textbf{v}|\textrm{d}t$ and hence, $\textrm{d}q=\lambda |\textbf{v}|\textrm{d}t$. In this case, the charge $\textrm{d}q$ is at the point A at some time t and then it is at the point B at some time $t+\textrm{d}t$, so that the displacement in time duration $\textrm{d}t$ is $\textbf{AB}=\mathbf {\textrm{d}l}$. In such a scenario, the following assertion holds true:

  $$\begin{aligned} M \equiv \text {d}q \text { is a point charge}. \end{aligned}$$

  (A.2)

• Scenario 2 (S2): At any instant t of the passage of a line current, there is a line charge density $\lambda$ (charge per unit length) such that the charge $\textrm{d}q$ distributed over the length $\textrm{d}\ell$ is written as $\textrm{d}q=\lambda \textrm{d}x$. In this case, the charge $\textrm{d}q$ is neither at point A, nor at point B, but is spread over the line AB. Therefore, the following proposition holds true for such explanation:

  $$\begin{aligned} \lnot M \equiv \text {d}q \text { is NOT a point charge}. \end{aligned}$$

  (A.3)

Therefore, the relation $\textbf{J}= \textrm{d}q/\textrm{d}t=\lambda \textbf{v}$ holds true if and only if M and $\lnot M$ both are true in one and the same process of reasoning that leads to such a construction. It may be written as

$$\begin{aligned} \textbf{J}=\lambda \textbf{v}~: M\wedge \lnot M. \end{aligned}$$

(A.4)

Now, it may be noted that, considering $\lnot \lnot M\equiv M$ (classical negation), $(M\wedge \lnot M)\equiv \lnot (\lnot (M\wedge \lnot M))$, which means there is a violation of the law of non-contradiction. Further, $M\wedge \lnot M\equiv \lnot (\lnot M\vee M)$ means the violation of the law of excluded middle. That is, (A.4) can be recast as

$$\begin{aligned} \textbf{J}=\lambda \textbf{v}~: \lnot (M\vee \lnot M). \end{aligned}$$

(A.5)

Hence, the relation $\textbf{J}= \textrm{d}q/\textrm{d}t=\lambda \textbf{v}$ holds true if and only if neither M nor $\lnot M$ (i.e. dq is neither a point charge nor not a point charge) in one and the same process of reasoning that leads to such a construction.

Therefore, the situation can be viewed as a decision problem where it can not be decided whether dq is a point charge or not i.e. this is a problem of universal validity of the formal statement M [4]. This, however, allows for suitable choices to be made by humans (us) according to context, i.e. $\textrm{d}q$ is considered as a point charge in S1 and it is not considered as a point charge in S2. To mention, such a choice cannot be made by a Turing machine which is, by definition, an automatic machine that excludes the scope of any choice [16].

Interestingly, this brings forward further questions regarding the notion of infinitesimal quantity. Dedekind would go as far as writing, on p. 1 of ref. [108], that the geometrically founded ‘introduction into the differential calculus can make no claim to being scientific’ and would decide to keep searching for an arithmetic foundation of the same. The proof of the continuity equation, as discussed above, justifies Dedekind’s criticism and reveals that the notion of infinitesimal quantity gives rise to undecidability when axiomatised in terms of geometry (axiom of point). So, naturally the question arises – Do we need new postulates or axioms to explain ‘infinitesimal quantity’? Do we need an arithmetic foundation of differential calculus? What are the meanings of the words ‘infinitesimal’ and ‘quantity’ in connection with operations? Does the definition of ‘quantity’ provided in quantity calculus be of any help regarding such issues?

Certainly, a few of these questions have been addressed in refs [6, 7, 83] and have indirect connection to ref. [85]. The present discussion regarding the oil drop experiment only showcases how subtle is the question regarding ‘quantity’ (its existence).