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Arkady Bolotin

Publications and source records attributed to Arkady Bolotin.

50 records · Page 3Linked to original sources
arXiv2015

Constructibility of the universal wave function

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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arXiv2015

Examples of non-constructive proofs in quantum theory

Unlike mathematics, in which the notion of truth might be abstract, in physics, the emphasis must be placed on algorithmic procedures for obtaining numerical results subject to the experimental verifiability. For, a physical science is exactly that: algorithmic procedures (built on a certain mathematical formalism) for obtaining verifiable conclusions from a set of basic hypotheses. By admitting non-constructivist statements a physical theory loses its concrete applicability and thus verifiability of its predictions. Accordingly, the requirement of constructivism must be indispensable to any physical theory. Nevertheless, in at least some physical theories, and especially in quantum mechanics, one can find examples of non-constructive statements. The present paper demonstrates a couple of such examples dealing with macroscopic quantum states (i.e., with the applicability of the standard quantum formalism to macroscopic systems). As it is shown, in these examples the proofs of the existence of macroscopic quantum states are based on logical principles allowing one to decide the truth of predicates over an infinite number of things.

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arXiv2015

Correspondence Principle as Equivalence of Categories

If quantum mechanics were to be applicable to macroscopic objects, classical mechanics would have to be a limiting case of quantum mechanics. Then the category Set that packages classical mechanics has to be in some sense a 'limiting case' of the category Hilb packaging quantum mechanics. Following from this assumption, quantum-classical correspondence can be considered as a mapping of the category Hilb to the category Set, i.e., a functor from Hilb to Set, taking place in the macroscopic limit. As a procedure, which takes us from an object of the category Hilb (i.e., a Hilbert space) in the macroscopic limit to an object of the category Set (i.e., a set of values that describe the configuration of a system), this functor must take a finite number of steps in order to make the equivalence of Hilb and Set verifiable. However, as it is shown in the present paper, such a constructivist requirement cannot be met in at least one case of an Ising model of a spin glass. This could mean that it is impossible to demonstrate the emergence of classicality totally from the formalism of standard quantum mechanics.

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arXiv2015

Effectively calculable quantum mechanics

According to mathematical constructivism, a mathematical object can exist only if there is a way to compute (or "construct") it; so, what is non-computable is non-constructive. In the example of the quantum model, whose Fock states are associated with Fibonacci numbers, this paper shows that the mathematical formalism of quantum mechanics is non-constructive since it permits an undecidable (or effectively impossible) subset of Hilbert space. On the other hand, as it is argued in the paper, if one believes that testability of predictions is the most fundamental property of any physical theory, one need to accept that quantum mechanics must be an effectively calculable (and thus mathematically constructive) theory. With that, a way to reformulate quantum mechanics constructively, while keeping its mathematical foundation unchanged, leads to hypercomputation. In contrast, the proposed in the paper superselection rule, which acts by effectively forbidding a coherent superposition of quantum states corresponding to potential and actual infinity, can introduce computable constructivism in a quantum mechanical theory with no need for hypercomputation.

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arXiv2015

The Computational Limit to Quantum Determinism and the Black Hole Information Loss Paradox

The present paper scrutinizes the principle of quantum determinism, which maintains that the complete information about the initial quantum state of a physical system should determine the system's quantum state at any other time. As it shown in the paper, assuming the strong exponential time hypothesis, SETH, which conjectures that known algorithms for solving computational NP-complete problems (often brute-force algorithms) are optimal, the quantum deterministic principle cannot be used generally, i.e., for randomly selected physical systems, particularly macroscopic systems. In other words, even if the initial quantum state of an arbitrary system were precisely known, as long as SETH is true it might be impossible in the real world to predict the system's exact final quantum state. The paper suggests that the breakdown of quantum determinism in a process, in which a black hole forms and then completely evaporates, might actually be physical evidence supporting SETH.

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arXiv2015

Can category-theoretic semantics resolve the problem of the interpretation of the quantum state vector?

Do correctness and completeness of quantum mechanics jointly imply that quantum state vectors are necessarily in one-to-one correspondence with elements of the physical reality? In terms of category theory, such a correspondence would stand for an isomorphism, so the problem of the status of the quantum state vector could be turned into the question of whether state vectors are necessarily isomorphic to elements of the reality. As it is argued in the present paper, in order to tackle this question, one needs to complement the category-theoretic approach to quantum mechanics with the computational-complexity-theoretic considerations. Based on such considerations, it is demonstrated in the paper that the hypothesis of the isomorphism existing between state vectors and elements of the reality is expected to be unsuitable for a generic quantum system.

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arXiv2015

Concerning Infeasibility of the Wave Functions of the Universe

Difficulties with finding the general exact solutions to the Wheeler-DeWitt equation, i.e. the wave functions of the Universe, are known and well documented. However, the present paper draws attention to a completely different matter, which is rarely if ever discussed in relation to this equation, namely, the time complexity of the Wheeler-DeWitt equation, that is, the time required to exactly solve the equation for a given universe. As it is shown in the paper, whatever generic exact algorithm is used to solve the equation, most likely such an algorithm cannot be faster than brute force, which makes the wave functions of the Universe infeasible.

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arXiv2014

Justifying the Classical-Quantum Divide of the Copenhagen Interpretation

Perhaps the most significant drawback, which the Copenhagen interpretation (still the most popular interpretation of quantum theory) suffers from, is the classical-quantum divide between the large classical systems that carry out measurements and the small quantum systems that they measure. So, an "ideal" alternative interpretation of quantum theory would either eliminate this divide or justify it in some reasonable way. The present paper demonstrates that it is possible to justify the classical-quantum dualism of the Copenhagen interpretation by way of the analysis of the time complexity of Schrodinger's equation.

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arXiv2014

Computational solution to quantum foundational problems

This paper argues that the requirement of applicableness of quantum linearity to any physical level from molecules and atoms to the level of macroscopic extensional world, which leads to a main foundational problem in quantum theory referred to as the "measurement problem", actually has a computational character: It implies that there is a generic algorithm, which guarantees exact solutions to the Schrodinger equation for every physical system in a reasonable amount of time regardless of how many constituent microscopic particles it comprises. From the point of view of computational complexity theory, this requirement is equivalent to the assumption that the computational complexity classes P and NP are equal, which is widely believed to be very unlikely. As demonstrated in the paper, accepting the different computational assumption called the Exponential Time Hypothesis (that involves P!=NP) would justify the separation between a microscopic quantum system and a macroscopic apparatus (usually called the Heisenberg cut) since this hypothesis, if true, would imply that deterministic quantum and classical descriptions are impossible to overlap in order to obtain a rigorous derivation of complete properties of macroscopic objects from their microstates.

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arXiv2014

Any realistic model of a physical system must be computationally realistic

It is argued that any possible definition of a realistic physics theory -- i.e., a mathematical model representing the real world -- cannot be considered comprehensive unless it is supplemented with requirement of being computationally realistic. That is, the mathematical structure of a realistic model of a physical system must allow the collection of all the system's physical quantities to compute all possible measurement outcomes on some computational device not only in an unambiguous way but also in a reasonable amount of time. In the paper, it is shown that a deterministic quantum model of a microscopic system evolving in isolation should be regarded as realistic since the NP-hard problem of finding the exact solution to the Schrodinger equation for an arbitrary physical system can be surely solved in a reasonable amount of time in the case, in which the system has just a small number of degrees of freedom. In contrast to this, the deterministic quantum model of a truly macroscopic object ought to be considered as non-realistic since in a world of limited computational resources the intractable problem possessing that enormous amount of degrees of freedom would be the same as mere unsolvable.

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arXiv2013

Applying computational complexity to the emergence of classicality

Can the computational complexity theory of computer science and mathematics say something new about unresolved problems in quantum physics? Particularly, can the P versus NP question in the computational complexity theory be a factor in the elucidation of the emergency of classicality in quantum mechanics? The paper compares two different ways of deriving classicality from the quantum formalism resulted from two differing hypotheses regarding the P versus NP question -- the approach of the quantum decoherence theory implying that P = NP and the computational complexity approach which assumes that P is not equal to NP.

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arXiv2013

Computational Complexity and the Interpretation of a Quantum State Vector

The macro-objectivation problem derives from the fact that the Schrodinger equation is linear and thus requires that a macroscopic system interacting with an entangled state must be entangled as well. However, such a requirement entails that the Schrodinger equation must also be solvable in the macroscopic world in the same way as it is solvable in the microscopic world, which itself is an assumption. In this work, the alternative assumption that the Schrodinger equation is in general an intractable problem is considered.

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arXiv2012

Limits of reductionism and the measurement problem

In the paper, we tackle the following questions: Could the difficulty in solving the Schrodinger equation for an arbitrarily large system be a reflection of some nature intrinsic property? And if so, could this difficulty be a resolution to the measurement problem?

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