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Constructibility of the Universal Wave Function

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Abstract

This paper focuses on a constructive treatment of the mathematical formalism of quantum theory and a possible role of constructivist philosophy in resolving the foundational problems of quantum mechanics, particularly, the controversy over the meaning of the wave function of the universe. As it is demonstrated in the paper, unless the number of the universe’s degrees of freedom is fundamentally upper bounded (owing to some unknown physical laws) or hypercomputation is physically realizable, the universal wave function is a non-constructive entity in the sense of constructive recursive mathematics. This means that even if such a function might exist, basic mathematical operations on it would be undefinable and subsequently the only content one would be able to deduce from this function would be pure symbolical.

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Notes

  1. Quantum formalism provides no indication as to how to exhibit, or calculate, macroscopic wave functions except the general form of the Schrödinger equation, which, to paraphrase Walter Huckel [2], has the good view but leads to misunderstanding, because it forces us to think that we have something that we do not have—i.e., a calculated wave function of any given physical system (i.e., of an arbitrary size and complexity).

  2. A constructive existence proof demonstrates the existence of a mathematical object by outlining a method (i.e., an algorithm) of “constructing” (that is, computing) this object [3].

  3. For example, most models in quantum cosmology—an application of quantum theory to the universe as a whole—are based on the approach of the Wheeler–DeWitt equation that combines mathematically the ideas of quantum mechanics and general relativity. This approach is first to restrict the full configuration space of three-geometries called “superspace” to a small number of variables (such as scale factor, inflaton field, etc.) and then to quantize them canonically; consequently, the resultant models are called “minisuperspace models”. More details can be found in [4] as well as in the review [5].

  4. However, while one can prove that any constructive function is a computable function (for instance, using Kleene’s realizability interpretation [9]), it is important to keep the distinction between the notions of constructivism and computability. As explained in [10], the notion of function in constructive mathematics is a primitive one, which cannot be explained in a satisfactory way in term of recursivity. Even so, unless it brings about a confusion, henceforward in this paper, we will use the adjectives “constructive”, “computable” and “recursive” interchangeably.

  5. If the spectrum of L is not bounded, then the domain of definition of L cannot be all of \(\mathcal {H}\), that is, \(\mathcal {D}(L) \ne \mathcal {H}\). Furthermore, if the spectrum of the operator L contains a continuous part, then the corresponding eigenvectors do not belong to \(\mathcal {H}\), but to a larger space.

  6. A general proof (not based on the computational representation of quantum mechanics), which demonstrates that the quantum world is in fact a “hostile environment” for many (if not all) varieties of constructivist mathematics, is given in the paper [19]. As it is argued there, from a thoroughgoing constructivist point of view, unbounded linear Hermitian operators in an infinite Hilbert space cannot be even recognizable as mathematical objects. Also, deserving of mentioning are the investigations [20, 21], which show that, due to the use of unbounded and thus discontinuous operators in an infinite Hilbert space, noncomputable and hence nonconstructive objects are unavoidable in the quantum theory. Even though it is still unclear whether or not at least some form(s) (albeit modified) of constructivist mathematics can be applied to develop a full counterpart of quantum formalism constructive in a sense, it is crucial to our investigation that such a counterpart cannot be recursive

  7. Suffice it to say that it is even unclear as to how to account for the probabilistic content of the wave function of the universe; see, for example, the discussion in [24].

  8. According to the commonly accepted paradigm, the observed universe evolved in a finite time from a dense singular state, before which classical space and time did not exist. However, as it is argued in the papers [25, 26], self-consistent, geodesically complete, and physically sensible steady-state (SS) eternally inflating universe, based on the flat slicing of de Sitter space, is also possible. In the SS, the universe always has and always will exist in a state statistically like its current one, and time has no beginning. Needless to say, the SS cosmology is appealing because it avoids an initial singularity, has no beginning of time, and does not require an initial condition for the universe.

  9. Putting it differently, because an essential feature of macroscopic assemblies of microscopic particles is that the state equations are size independent, one can naturally arrive at an idealization of an infinite universe as an infinite-volume limit of increasingly large finite systems with constant density.

  10. No ratio of positive integers (capable of being calculated in a finite number of steps) can be the exact value of \(\pi \); see the proof in [27].

  11. A typical example of such a ‘limited’ consideration is the Hartle–Hawking state (i.e., the wave function of the universe) satisfying the Wheeler–DeWitt equation defined in mini-superspaces [28].

  12. If not, then it would be hard for the proposition of explicit physical finitism to answer the charge of arbitrariness: Certainly, no matter where the limit for the particular parameter would be drawn, it would be always ad hoc and so perpetually subject to shifting.

  13. It is interesting to note that the existence of the upper bound on the complexity of the universal wave function could also result in constructiveness of this function. Indeed, according to [30], the complexity of a quantum state \({\vert {\Psi } \rangle }_N\) on N qubits is defined as the minimum number of gates (i.e., elementary unitaries) necessary to produce \({\vert {\Psi } \rangle }_N\) from a simple reference state \(\vert {\Psi }_0 \rangle \). The complexity of \({\vert {\Psi } \rangle }_N\) in principle depends on all the details of the construction, but more importantly, it is proportional to the number of active degrees of freedom of the system, that is, the number of qubits N. So, if the complexity of the state vector describing the universe \({\vert {\Psi } \rangle }_N\) were to be limited it would imply the limitation on N and thus the computability of this vector. The problem is that it is unclear why the complexity of the quantum state of the universe should be bounded (and not grow to infinity).

  14. For details see the critical analysis [35] that examines the possibility of computation-like processes transcending the limits imposed by the Church–Turing thesis.

  15. Besides, as expected by most logicians, one can construct an undecidable sentence (i.e., such that neither it itself nor its negation is provable) whose physical meaning seems to be hardly questionable. This happens both in classical and in quantum mechanics (see for example investigations by Pitowski [38], Mundici [39] and Svozil [40]). So, the wave function of the entire universe might be uncomputable and yet full of physical meaning.

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    Arkady Bolotin

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Bolotin, A. Constructibility of the Universal Wave Function. Found Phys 46, 1253–1268 (2016). https://doi.org/10.1007/s10701-016-0018-7

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