2025 lines
66 KiB
Plaintext
2025 lines
66 KiB
Plaintext
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arXiv:quant-ph/9907009v2 10 Nov 1999
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The Importance of Quantum Decoherence in Brain Processes
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Max Tegmark
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Institute for Advanced Study, Olden Lane, Princeton, NJ 08540; max@ias.edu
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Dept. of Physics, Univ. of Pennsylvania, Philadelphia, PA 19104
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(Submitted to Phys. Rev. E July 2 1999, accepted October 25)
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Based on a calculation of neural decoherence rates, we ar-
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gue that that the degrees of freedom of the human brain that
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relate to cognitive processes should be thought of as a classical
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rather than quantum system, i.e., that there is nothing funda-
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mentally wrong with the current classical approach to neural
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network simulations. We find that the decoherence timescales
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(∼ 10−13 − 10−20 seconds) are typically much shorter than
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the relevant dynamical timescales (∼ 10−3 − 10−1 seconds),
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both for regular neuron firing and for kink-like polarization
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excitations in microtubules. This conclusion disagrees with
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suggestions by Penrose and others that the brain acts as a
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quantum computer, and that quantum coherence is related
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to consciousness in a fundamental way.
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I. INTRODUCTION
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In most current mainstream biophysics research on
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cognitive processes, the brain is modeled as a neural net-
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work obeying classical physics. In contrast, Penrose [1,2],
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and others have argued that quantum mechanics may
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play an essential role, and that successful brain simula-
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tions can only be performed with a quantum computer.
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The main purpose of this paper is to address this issue
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with quantitative decoherence calculations.
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The field of artificial neural networks (for an introduc-
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tion, see, e.g., [4–6]) is currently booming, driven by a
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broad range of applications and improved computing re-
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sources. Although the popular neurological models come
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in various levels of abstraction, none involve effects of
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quantum coherence in any fundamental way. Encouraged
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by successes in modeling memory, learning, visual pro-
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cessing, etc. [7,8], many workers in the field have boldly
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conjectured that a sufficiently complex neural network
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could in principle perform all cognitive processes that we
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associate with consciousness.
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On the other hand, many authors have argued that
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consciousness can only be understood as a quantum ef-
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fect. For instance, Wigner [9] suggested that conscious-
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ness was linked to the quantum measurement problem1,
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and this idea has been greatly elaborated by Stapp [3].
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There have been numerous suggestions that conscious-
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ness is a macroquantum effect, involving superconduc-
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1 Interestingly, Wigner changed his mind and gave up this
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idea [10] after he became aware in of the first paper on deco-
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herence in 1970 [11].
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tivity [12], superfluidity [13], electromagnetic fields [14],
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Bose condensation [15,16], superflourescence [17] or some
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other mechanism [18,19]. Perhaps the most concrete one
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is that of Penrose [2], proposing that this takes place
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in microtubules, the ubiquitous hollow cylinders that
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among other things help cells maintain their shapes. It
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has been argued that microtubules can process informa-
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tion like a cellular automaton [20], and Penrose suggests
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that they operate as a quantum computer. This idea has
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been further elaborated employing string theory methods
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[21–27].
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The make-or-break issue for all these quantum mod-
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els is whether the relevant degrees of freedom of the
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brain can be sufficiently isolated to retain their quan-
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tum coherence, and opinions are divided. For instance,
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Stapp has argued that interaction with the environment
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is probably small enough to be unimportant for cer-
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tain neural processes [28], whereas Zeh [29], Zurek [30],
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Scott [31], Hawking [32] and Hepp [33] have conjectured
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that environment-induced coherence will rapidly destroy
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macrosuperpositions in the brain. It is therefore timely
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to try to settle the issue with detailed calculations of the
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relevant decoherence rates. This is the purpose of the
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present work.
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The rest of this paper is organized as follows. In Sec-
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tion II, we briefly review the open system quantum me-
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chanics necessary for our calculations, and introduce a
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decomposition into three subsystems to place the prob-
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lem in its proper context.
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In Section III, we evaluate
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decoherence rates both for neuron firing and for the mi-
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crotubule processes proposed by Penrose et al., relegating
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some technical details to the Appendix. We conclude in
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Section IV by discussing the implications of our results,
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both for modeling cognitive brain processes and for in-
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corporating them into a quantum-mechanical treatment
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of the rest of the world.
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II. SYSTEMS AND SUBSYSTEMS
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In this section, we review those aspects of quantum
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mechanics for open systems that are needed for our cal-
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culations, and introduce a classification scheme and a
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subsystem decomposition to place the problem at hand
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in its appropriate context.
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1
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A. Notation
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Let us first briefly review the quantum mechanics of
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subsystems. The state of an arbitrary quantum system
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is described by its density matrix ρ, which left in isolation
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will evolve in time according to the Schr¨odinger equation
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˙ρ = −i[H, ρ]/¯h.
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(1)
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It is often useful to view a system as composed of two
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subsystems, so that some of the degrees of freedom cor-
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respond to the 1st and the rest to the 2nd. The state of
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subsystem i is described by the reduced density matrix
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ρi obtained by tracing (marginalizing) over the degrees
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of freedom of the other: ρ1 ≡ tr 2ρ, ρ2 ≡ tr 1ρ. Let us
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decompose the Hamiltonian as
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H = H1 + H2 + Hint,
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(2)
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where the operator H1 affects only the 1st subsystem
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and H2 affects only the 2nd subsystem. The interaction
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Hamiltonian Hint is the remaining nonseparable part, de-
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fined as Hint ≡ H − H1 − H2, so such a decomposition
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is always possible, although it is generally only useful if
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Hint is in some sense small.
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If Hint = 0, i.e., if there is no interaction between
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the two subsystems, then it is easy to show that ˙ρi =
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−i[Hi, ρi]/¯h, i = 1, 2, that is, we can treat each subsys-
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tem as if the rest of the Universe did not exist, ignoring
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any correlations with the other subsystem that may have
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been present in the full non-separable density matrix ρ.
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It is of course this property that makes density matrices
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so useful in the first place, and that led von Neumann
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to invent them [34]: the full system is assumed to obey
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equation (1) simply because its interactions with the rest
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of the Universe are negligible.
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B. Fluctuation, dissipation, communication and
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decoherence
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In practice, the interaction Hint between subsystems
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is usually not zero. This has a number of qualitatively
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different effects:
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1. Fluctuation
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2. Dissipation
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3. Communication
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4. Decoherence
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The first two involve transfer of energy between the sub-
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systems, whereas the last two involve exchange of infor-
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mation. The first three occur in classical physics as well
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- only the last one is a purely quantum-mechanical phe-
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nomenon.
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For example, consider a tiny colloid grain (subsystem
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1) in a jar of water (subsystem 2). Collisions with water
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molecules will cause fluctuations in the center-of-mass
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position of the colloid (brownian motion). If its initial ve-
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locity is high, dissipation (friction) will slow it down to
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a mean speed corresponding to thermal equilibrium with
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the water. The dissipation timescale τdiss, defined as the
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time it would take to lose half of the initial excess energy,
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will in this case be of order τcoll × (M/m), where τcoll is
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the mean-free time between collisions, M the colloid mass
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M and m is the mass of a water molecule. We will define
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communication as exchange of information. The infor-
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mation that the two subsystems have about each other,
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measured in bits, is
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I12 ≡ S1 + S2 − S,
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(3)
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where Si ≡ −tr iρi log ρi is the entropy of the ith subsys-
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tem, S ≡ −tr ρ log ρ is the entropy of the total system,
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and the logarithms are base 2. If this mutual informa-
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tion is zero, then the states of the two systems are un-
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correlated and independent, with the density matrix of
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the separable form ρ = ρ1 ⊗ ρ2. If the subsystems start
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out independent, any interaction will at least initially
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increase the subsystem entropies Si, thereby increasing
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the mutual information, since the entropy S of the total
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system always remains constant.
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This apparent entropy increase of subsystems, which
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is related to the arrow of time and the 2nd law of of ther-
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modynamics [35], occurs also in classical physics. How-
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ever, quantum mechanics produces a qualitatively new
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effect as well, known as decoherence [11,36,37], sup-
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pressing off-diagonal elements in the reduced density ma-
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trices ρi. This effect destroys the ability to observe long-
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range quantum superpositions within the subsystems,
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and is now rather well-understood and uncontroversial
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[30,38–42] – the interested reader is referred to [43] and
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a recent book on decoherence [44] for details.
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For in-
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stance, if our colloid was initially in a superposition of
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two locations separated by a centimeter, this macrosu-
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perposition would for all practical purposes be destroyed
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by the first collision with a water molecule, i.e., on a
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timescale τdec of order τcoll, with the quantum superpo-
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sition surviving only on scales below the de de Broigle
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wavelength of the water molecules [45,46].2 This means
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2Decoherence picks out a preferred basis in the quantum-
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mechanical Hilbert space, termed the “pointer basis” by
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Zurek [36], in which superpositions are rapidly destroyed and
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classical behavior is approached. This normally includes the
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position basis, which is why we never experience superposi-
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tions of objects in macroscopically different positions. Deco-
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herence is quite generic. Although it has been claimed that
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this preferred basis consists of the maximal set of commuting
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observables that also commute with Hint (the “microstable
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basis” of Omnes [43]), this is in fact merely a sufficient condi-
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tion, not a necessary one. If [Hint, x] = 0 for some observable
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x but [Hint, p] ̸= 0 for its conjugate p, then the interaction
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2
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that τdiss/τdec ∼ M/m in our example, i.e., that decoher-
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ence is much faster than dissipation for macroscopic ob-
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jects, and this qualitative result has been shown to hold
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quite generally as well (see [43] and references therein).
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Loosely speaking, this is because each microscopic par-
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ticle that scatters off of the subsystem carries away only
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a tiny fraction m/M of the total momentum, but essen-
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tially all of the necessary information.
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QUANTUM�
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SYSTEM
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NOT �
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INDEPENDENT�
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SYSTEM
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IMPOSSIBLE
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CLASSICAL�
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SYSTEM
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0.1
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1
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1
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0.1
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10
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100
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10
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100
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Dissipation time/Decoherence time
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Dynamical time/Decoherence time
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FIG. 1. The qualitative behavior of a subsystem depends on
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the timescales for dynamics, dissipation and decoherence.
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This
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classification is by necessity quite crude, so the boundaries should
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not be thought of as sharp.
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C. Classification of systems
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Let us define the dynamical timescale τdyn of a subsys-
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tem as that which is characteristic of its internal dynam-
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ics. For a planetary system or an atom, τdyn would be
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the orbital frequency.
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The qualitative behavior of a system depends on the
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ratio of these timescales, as illustrated in Figure 1. If
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τdyn ≪ τdec, we are are dealing with a true quantum sys-
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tem, since its superpositions can persist long enough to
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be dynamically important. If τdyn ≫ τdiss, it is hardly
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meaningful to view it as an independent system at all,
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since its internal forces are so week that they are dwarfed
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will indeed cause decoherence for x as advertised. But this
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will happen even if [Hint, x] ̸= 0 — all that matters is that
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[Hint, p] ̸= 0, i.e., that the interaction Hamiltonian contains
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(“measures”) x.
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by the effects of the surroundings. In the intermediate
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case where τdec ≪ τdyn <∼ τdiss, we have a familiar classi-
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cal system.
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The relation between τdec and τdiss depends only on
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the form of Hint, whereas the question of whether τdyn
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falls between these values depends on the normalization
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of Hint in equation (2). Since τdec ∼ τdiss for microscopic
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(atom-sized) systems and τdec ≪ τdiss for macroscopic
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ones, Figure 1 shows that whereas macroscopic systems
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can behave quantum-mechanically, microscopic ones can
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never behave classically.
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D. Three systems: subject, object and environment
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Most discussions of quantum statistical mechanics split
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the Universe into two subsystems [47]: the object under
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consideration and everything else (referred to as the en-
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vironment). Since our purpose is to model the observer,
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we need to include a third subsystem as well, the subject.
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As illustrated in Figure 2, we therefore decompose the
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total system into three subsystems:
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• The subject consists of the degrees of freedom as-
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sociated with the subjective perceptions of the ob-
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server. This does not include any other degrees of
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freedom associated with the brain or other parts of
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the body.
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• The object consists of the degrees of freedom that
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the observer is interested in studying, e.g., the
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pointer position on a measurement apparatus.
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• The environment consists of everything else, i.e.,
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all the degrees of freedom that the observer is not
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paying attention to. By definition, these are the
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degrees of freedom that we always perform a partial
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trace over.
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3
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SUBJECT
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OBJECT
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ENVIRONMENT
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Hs
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Ho
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He
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Hso
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Hoe
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Hse
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Object �
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decoherence
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Subject�
|
|||
|
|
decoherence,�
|
|||
|
|
finalizing �
|
|||
|
|
decisions
|
|||
|
|
|
|||
|
|
Measurement,�
|
|||
|
|
observation,�
|
|||
|
|
"wavefuntion �
|
|||
|
|
collapse",�
|
|||
|
|
willful action
|
|||
|
|
|
|||
|
|
(Always traced over)
|
|||
|
|
|
|||
|
|
(Always zero entropy)
|
|||
|
|
|
|||
|
|
FIG. 2. An observer can always decompose the world into three
|
|||
|
|
subsystems: the degrees of freedom corresponding to her subjective
|
|||
|
|
perceptions (the subject), the degrees of freedom being studied (the
|
|||
|
|
object), and everything else (the environment). As indicated, the
|
|||
|
|
subsystem Hamiltonians Hs, Ho, He and the interaction Hamilto-
|
|||
|
|
nians Hso, Hoe, Hse can cause qualitatively very different effects,
|
|||
|
|
which is why it is often useful to study them separately. This paper
|
|||
|
|
focuses on the interaction Hse.
|
|||
|
|
|
|||
|
|
Note that the first two definitions are very restrictive.
|
|||
|
|
Whereas the subject would include the entire body of
|
|||
|
|
the observer in the common way of speaking, only very
|
|||
|
|
few degrees of freedom qualify as our subject or object.
|
|||
|
|
For instance, if a physicist is observing a Stern-Gerlach
|
|||
|
|
apparatus, the vast majority of the ∼ 1028 degrees of
|
|||
|
|
freedom in the the observer and apparatus are counted
|
|||
|
|
as environment, not as subject or object.
|
|||
|
|
The term “perception” is used in a broad sense in item
|
|||
|
|
1, including thoughts, emotions and any other attributes
|
|||
|
|
of the subjectively perceived state of the observer.
|
|||
|
|
The practical usefulness in this decomposition lies in
|
|||
|
|
that one can often neglect everything except the object
|
|||
|
|
and its internal dynamics (given by Ho) to first order,
|
|||
|
|
using simple prescriptions to correct for the interactions
|
|||
|
|
with the subject and the environment.
|
|||
|
|
The effects of
|
|||
|
|
both Hso and Hoe have been extensively studied in the
|
|||
|
|
literature. Hso involves quantum measurement, and gives
|
|||
|
|
rise to the usual interpretation of the diagonal elements of
|
|||
|
|
the object density matrix as probabilities. Hoe produces
|
|||
|
|
decoherence, selecting a preferred basis and making the
|
|||
|
|
object act classically if the conditions in Figure 1 are met.
|
|||
|
|
In contrast, Hse, which causes decoherence directly in
|
|||
|
|
the subject system, has received relatively little atten-
|
|||
|
|
|
|||
|
|
tion. It is the focus of the present paper, and the next
|
|||
|
|
section is devoted to quantitative calculations of decoher-
|
|||
|
|
ence in brain processes, aimed at determining whether
|
|||
|
|
the subject system should be classified as classical or
|
|||
|
|
quantum in the sense of Figure 1.
|
|||
|
|
We will return to
|
|||
|
|
Figure 2 and a more detailed discussion of its various
|
|||
|
|
subsystem interactions in Section IV.
|
|||
|
|
|
|||
|
|
III. DECOHERENCE RATES
|
|||
|
|
|
|||
|
|
In this section, we will make quantitative estimates
|
|||
|
|
of decoherence rates for neurological processes. We first
|
|||
|
|
analyze the process of neuron firing, widely assumed to be
|
|||
|
|
central to cognitive processes. We also analyze electrical
|
|||
|
|
excitations in microtubules, which Penrose and others
|
|||
|
|
have suggested may be relevant to conscious thought.
|
|||
|
|
|
|||
|
|
A. Neuron firing
|
|||
|
|
|
|||
|
|
Neurons (see Figure 3) are one of the key building
|
|||
|
|
blocks of the brain’s information processing system. It is
|
|||
|
|
widely believed that the complex network of ∼ 1011 neu-
|
|||
|
|
rons with their nonlinear synaptic couplings is in some
|
|||
|
|
way linked to our subjective perceptions, i.e., to the sub-
|
|||
|
|
ject degrees of freedom. If this picture is correct, then if
|
|||
|
|
Hs or Hso puts the subject into a superposition of two
|
|||
|
|
distinct mental states, some neurons will be in a super-
|
|||
|
|
position of firing and not firing. How fast does such a
|
|||
|
|
superposition of a firing and non-firing neuron decohere?
|
|||
|
|
Let us consider this process in more detail.
|
|||
|
|
For in-
|
|||
|
|
troductory reviews of neuron dynamics, the reader is re-
|
|||
|
|
ferred to, e.g., [48–50].
|
|||
|
|
Like virtually all animal cells,
|
|||
|
|
neurons have ATP driven pumps in their membranes
|
|||
|
|
which push sodium ions out of the cell into the surround-
|
|||
|
|
ing fluids and potassium ions the other way. The former
|
|||
|
|
process is slightly more efficient, so the neuron contains a
|
|||
|
|
slight excess of negative charge in its “resting” state, cor-
|
|||
|
|
responding to a potential difference U0 ≈ −0.07 V across
|
|||
|
|
the axon membrane (“axolemma”). There is an inher-
|
|||
|
|
ent instability in the system, however. If the potential
|
|||
|
|
becomes substantially less negative, then voltage-gated
|
|||
|
|
sodium channels in the axon membrane open up, allow-
|
|||
|
|
ing Na+ ions to come gushing in. This makes the poten-
|
|||
|
|
tial still less negative, causes still more opening, etc. This
|
|||
|
|
chain reaction, “firing”, propagates down the axon at a
|
|||
|
|
speed of up to 100 m/s, changing the potential difference
|
|||
|
|
to a value U1 that is typically of order +0.03 V [49].
|
|||
|
|
The axon quickly recovers. After less than ∼ 1 ms, the
|
|||
|
|
sodium channels close regardless of the voltage, and large
|
|||
|
|
potassium channels (also voltage gated, but with a time
|
|||
|
|
delay) open up allowing K+ ions to flow out and restore
|
|||
|
|
the resting potential U0. The ATP driven pumps quickly
|
|||
|
|
restore the Na+ and K+ concentrations to their initial
|
|||
|
|
values, making the neuron ready to fire again if triggered.
|
|||
|
|
Fast neurons can fire over 103 times per second.
|
|||
|
|
|
|||
|
|
4
|
|||
|
|
|
|||
|
|
|
|||
|
|
Na+
|
|||
|
|
Na+
|
|||
|
|
|
|||
|
|
dendrites
|
|||
|
|
|
|||
|
|
axon
|
|||
|
|
|
|||
|
|
cell body
|
|||
|
|
|
|||
|
|
myelin�
|
|||
|
|
insulation
|
|||
|
|
|
|||
|
|
fraction f�
|
|||
|
|
not insulated
|
|||
|
|
|
|||
|
|
thickness h
|
|||
|
|
|
|||
|
|
Here�
|
|||
|
|
if�
|
|||
|
|
firing
|
|||
|
|
|
|||
|
|
Here�
|
|||
|
|
if not�
|
|||
|
|
firing
|
|||
|
|
|
|||
|
|
voltage�
|
|||
|
|
sensitive�
|
|||
|
|
gate
|
|||
|
|
|
|||
|
|
length�
|
|||
|
|
L
|
|||
|
|
|
|||
|
|
axon�
|
|||
|
|
membrane
|
|||
|
|
|
|||
|
|
pulse
|
|||
|
|
|
|||
|
|
di
|
|||
|
|
|
|||
|
|
re
|
|||
|
|
|
|||
|
|
ct
|
|||
|
|
|
|||
|
|
io
|
|||
|
|
|
|||
|
|
n
|
|||
|
|
|
|||
|
|
diameter d
|
|||
|
|
|
|||
|
|
FIG. 3. Schematic illustration of a neuron (left), a section of
|
|||
|
|
the myelinated axon (center) and and a piece of its axon membrane
|
|||
|
|
(right).
|
|||
|
|
The axon is typically insulated (myelinated) with small
|
|||
|
|
bare patches every 0.5 mm or so (so-called Nodes of Ranvier) where
|
|||
|
|
the voltage-sensitive sodium and potassium gates are concentrated
|
|||
|
|
[51,52]. If the neuron is in a superposition of firing and not firing,
|
|||
|
|
then N ∼ 106 Na+ ions are in a superposition of being inside and
|
|||
|
|
outside the cell (right).
|
|||
|
|
|
|||
|
|
Consider a small patch of the membrane, assumed to
|
|||
|
|
be roughly flat with uniform thickness h as in Figure 3.
|
|||
|
|
If there is an excess surface density ±σ of charge near
|
|||
|
|
the inside/outside membrane surfaces, giving a voltage
|
|||
|
|
differential U across the membrane, then application of
|
|||
|
|
Gauss’ law tells us that σ = ǫ0E, where the electric field
|
|||
|
|
strength in the membrane is E = U/h and ǫ0 is the vac-
|
|||
|
|
uum permittivity.
|
|||
|
|
Consider an axon of length L and
|
|||
|
|
diameter d, with a fraction f of its surface area bare (not
|
|||
|
|
insulated with myelin). The total active surface area is
|
|||
|
|
thus A = πdLf, so the total number of Na+ ions that
|
|||
|
|
migrate in during firing is
|
|||
|
|
|
|||
|
|
N = Aσ
|
|||
|
|
|
|||
|
|
q
|
|||
|
|
= πdLfǫ0(U1 − U0)
|
|||
|
|
|
|||
|
|
qh
|
|||
|
|
,
|
|||
|
|
(4)
|
|||
|
|
|
|||
|
|
where q is the ionic charge (q = qe, the absolute value
|
|||
|
|
of the electron charge). Taking values typical for central
|
|||
|
|
nervous system axons [52,53], h = 8 nm, d = 10 µm,
|
|||
|
|
L = 10 cm, f = 10−3, U0 = −0.07 V and U1 = +0.03 V
|
|||
|
|
gives N ≈ 106 ions, and reasonable variations in our
|
|||
|
|
parameters can change this number by a few orders of
|
|||
|
|
magnitude.
|
|||
|
|
|
|||
|
|
B. Neuron decoherence mechanisms
|
|||
|
|
|
|||
|
|
Above we saw that a quantum superposition of the
|
|||
|
|
neuron states “resting” and “firing” involves of order a
|
|||
|
|
million ions being in a spatial superposition of inside and
|
|||
|
|
outside the axon membrane, separated by a distance of
|
|||
|
|
order h ∼ 10 nm. In this subsection, we will compute the
|
|||
|
|
timescale on which decoherence destroys such a superpo-
|
|||
|
|
sition.
|
|||
|
|
|
|||
|
|
In this analysis, the object is the neuron, and the su-
|
|||
|
|
perposition will be destroyed by any interaction with
|
|||
|
|
other (environment) degrees of freedom that is sensitive
|
|||
|
|
to where the ions are located. We will consider the fol-
|
|||
|
|
lowing three sources of decoherence for the ions:
|
|||
|
|
|
|||
|
|
1. Collisions with other ions
|
|||
|
|
|
|||
|
|
2. Collisions with water molecules
|
|||
|
|
|
|||
|
|
3. Coloumb interactions with more distant ions
|
|||
|
|
|
|||
|
|
There are many more decoherence mechanisms [44–46].
|
|||
|
|
Exotic candidates such as quantum gravity [54] and
|
|||
|
|
modified quantum mechanics [55,56] are generally much
|
|||
|
|
weaker [46]. A number of decoherence effects may be even
|
|||
|
|
stronger than those listed, e.g., interactions as the ions
|
|||
|
|
penetrate the membrane — the listed effects will turn out
|
|||
|
|
to be so strong that we can make our argument by sim-
|
|||
|
|
ply using them as lower limits on the actual decoherence
|
|||
|
|
rate.
|
|||
|
|
Let ρ denote the density matrix for the position r of a
|
|||
|
|
single Na+ ion. As reviewed in the Appendix, all three
|
|||
|
|
of the listed processes cause ρ to evolve as
|
|||
|
|
|
|||
|
|
ρ(x, x′, t0 + t) = ρ(x, x′, t0)f(x, x′, t)
|
|||
|
|
(5)
|
|||
|
|
|
|||
|
|
for some function f that is independent of the ion state
|
|||
|
|
ρ and depends only on the interaction Hamiltonian Hint.
|
|||
|
|
This assumes that we can neglect the motion of the ion
|
|||
|
|
itself on the decoherence timescale — we will see that
|
|||
|
|
this condition is met with a broad margin.
|
|||
|
|
|
|||
|
|
1. Ion–ion collisions
|
|||
|
|
|
|||
|
|
For scattering of environment particles (processes 1
|
|||
|
|
and 2) that have a typical de Broigle wavelength λ, we
|
|||
|
|
have [46]
|
|||
|
|
|
|||
|
|
f(x, x′, t) = e−Λt�
|
|||
|
|
1−e−|x′−x|2/2λ2�
|
|||
|
|
|
|||
|
|
≈
|
|||
|
|
|
|||
|
|
�
|
|||
|
|
e−|x′−x|2Λt/2λ2
|
|||
|
|
for |x′ − x| ≪ λ,
|
|||
|
|
|
|||
|
|
e−Λt
|
|||
|
|
for |x′ − x| ≫ λ.
|
|||
|
|
(6)
|
|||
|
|
|
|||
|
|
Here Λ is the scattering rate, given by Λ ≡ n⟨σv⟩, where
|
|||
|
|
n is the density of scatterers, σ is the scattering cross
|
|||
|
|
section and v is the velocity. The product σv is aver-
|
|||
|
|
aged over a the velocity distribution, which we take to
|
|||
|
|
be a thermal (Boltzmann) distribution for correspond-
|
|||
|
|
ing to T = 37◦C ≈ 310 K. The gist of equation (6) is
|
|||
|
|
that a single collision decoheres the ion down to the
|
|||
|
|
de Broigle wavelength of the scattering particle.
|
|||
|
|
The
|
|||
|
|
information I12 communicated during the scattering is
|
|||
|
|
I12 ∼ log2(∆x/λ) bits, where ∆x is the initial spread in
|
|||
|
|
the position of our particle.
|
|||
|
|
Since the typical de Broigle wavelength of a Na+ ion
|
|||
|
|
(mass m ≈ 23mp) or H2O molecule (m ≈ 18mp) is
|
|||
|
|
|
|||
|
|
5
|
|||
|
|
|
|||
|
|
|
|||
|
|
λ =
|
|||
|
|
2π¯h
|
|||
|
|
√
|
|||
|
|
|
|||
|
|
3mkT
|
|||
|
|
≈ 0.03 nm
|
|||
|
|
(7)
|
|||
|
|
|
|||
|
|
at 310K, way smaller than the the membrane thickness
|
|||
|
|
h ∼ 10 nm over which we need to maintain quantum
|
|||
|
|
coherence, we are clearly in the |x′ − x| ≫ λ limit of
|
|||
|
|
equation (6). This means that the spatial superposition
|
|||
|
|
of an ion decays exponentially Λ−1, of order its mean
|
|||
|
|
free time between collisions. Since the superposition of
|
|||
|
|
the neuron states “resting” and “firing” involves N such
|
|||
|
|
superposed ions, it thus gets destroyed on a timescale
|
|||
|
|
τ ≡ (NΛ)−1.
|
|||
|
|
Let us now evaluate τ. Coulomb scattering between
|
|||
|
|
two ions of unit charge gives substantial deflection angles
|
|||
|
|
(θ ∼ 1) with a cross section or order3
|
|||
|
|
|
|||
|
|
σ ∼
|
|||
|
|
� gq2
|
|||
|
|
|
|||
|
|
mv2
|
|||
|
|
|
|||
|
|
�2
|
|||
|
|
,
|
|||
|
|
(9)
|
|||
|
|
|
|||
|
|
where v is the relative velocity and g ≡ 1/4πǫ0 is the
|
|||
|
|
Coulomb constant. In thermal equilibrium, the kinetic
|
|||
|
|
energy mv2/2 is of order kT , so v ∼
|
|||
|
|
�
|
|||
|
|
|
|||
|
|
kT/m. For the
|
|||
|
|
ion density, let us write n = ηnH2O, where the density
|
|||
|
|
of water molecules nH2O is about (1 g/cm3)/(18mp) ∼
|
|||
|
|
1023/cm3 and η is the relative concentration of ions (pos-
|
|||
|
|
itive and negative combined). Typical ion concentrations
|
|||
|
|
during the resting state are [Na+] =9.2 (120) mmol/l and
|
|||
|
|
[K+] =140 (2.5) mmol/l inside (outside) the axon mem-
|
|||
|
|
brane [48], corresponding to total Na+ + K+ concentra-
|
|||
|
|
tions of η ≈ 0.00027 (0.00022) inside (outside). To be
|
|||
|
|
conservative, we will simply use η ≈ 0.0002 throughout.
|
|||
|
|
Ion–ion collisions therefore destroy the superposition on
|
|||
|
|
a timescale
|
|||
|
|
|
|||
|
|
τ ∼
|
|||
|
|
1
|
|||
|
|
|
|||
|
|
Nnσv ∼
|
|||
|
|
|
|||
|
|
�
|
|||
|
|
|
|||
|
|
m(kT )3
|
|||
|
|
|
|||
|
|
Ng2q4en
|
|||
|
|
∼ 10−20 s.
|
|||
|
|
(10)
|
|||
|
|
|
|||
|
|
2. Ion–water collisions
|
|||
|
|
|
|||
|
|
Since H2O molecules are electrically neutral, the cross-
|
|||
|
|
section is dominated by their electric dipole moment
|
|||
|
|
p ≈ 1.85 Debye ≈ (0.0385 nm) × qe. We can model this
|
|||
|
|
|
|||
|
|
3 If the first ion starts at rest at r1 = (0, 0, 0) and the sec-
|
|||
|
|
ond is incident with r2 = (vt, b, 0), then a very weak scatter-
|
|||
|
|
ing with deflection angle θ ≪ 1 will leave these trajectories
|
|||
|
|
roughly unchanged, the radial force F = gq2/|r1 −r2|2 merely
|
|||
|
|
causing a net transverse acceleration [57]
|
|||
|
|
|
|||
|
|
∆vy =
|
|||
|
|
� ∞
|
|||
|
|
|
|||
|
|
−∞
|
|||
|
|
|
|||
|
|
�y · F
|
|||
|
|
|
|||
|
|
m dt =
|
|||
|
|
� ∞
|
|||
|
|
|
|||
|
|
−∞
|
|||
|
|
|
|||
|
|
gq2b dt
|
|||
|
|
|
|||
|
|
[b2 + (vt)2]3/2 = 2gq2
|
|||
|
|
|
|||
|
|
mvb .
|
|||
|
|
(8)
|
|||
|
|
|
|||
|
|
The approximation breaks down as the deflection angle θ ≈
|
|||
|
|
∆vy/v approaches unity. This occurs for b ∼ gq2/mv2, giving
|
|||
|
|
σ = πb2 as in equation (9).
|
|||
|
|
|
|||
|
|
dipole as two opposing unit charges separated by a dis-
|
|||
|
|
tance y ≡ p/qe ≪ b, so summing the two corresponding
|
|||
|
|
contributions from equation (8) gives a deflection angle
|
|||
|
|
|
|||
|
|
θ ≈ 2gqep
|
|||
|
|
|
|||
|
|
mv2b2 .
|
|||
|
|
(11)
|
|||
|
|
|
|||
|
|
This gives a cross section
|
|||
|
|
|
|||
|
|
σ = πb2 ∼ gqep
|
|||
|
|
|
|||
|
|
mv2 .
|
|||
|
|
(12)
|
|||
|
|
|
|||
|
|
for scattering with large (θ ∼ 1) deflections. Although σ
|
|||
|
|
is smaller than for the case of ion–ion collisions, n is larger
|
|||
|
|
because the concentration factor η drops out, giving a
|
|||
|
|
final result
|
|||
|
|
|
|||
|
|
τ ∼
|
|||
|
|
1
|
|||
|
|
|
|||
|
|
Nnσv ∼
|
|||
|
|
|
|||
|
|
√
|
|||
|
|
|
|||
|
|
mkT
|
|||
|
|
|
|||
|
|
Ngqepn ∼ 10−20 s
|
|||
|
|
(13)
|
|||
|
|
|
|||
|
|
3. Interactions with distant ions
|
|||
|
|
|
|||
|
|
As shown in the Appendix, long-range interaction with
|
|||
|
|
a distant (environment) particle gives
|
|||
|
|
|
|||
|
|
f(r, r′, t) = �p2 [M(r′ − r)t/¯h] ,
|
|||
|
|
(14)
|
|||
|
|
|
|||
|
|
up to a phase factor that is irrelevant for decoherence.
|
|||
|
|
Here �p2 is the Fourier transform of p2(r) ≡ ρ2(r, r), the
|
|||
|
|
probability distribution for the location of the environ-
|
|||
|
|
ment particle. M is the 3 × 3 Hessian matrix of second
|
|||
|
|
derivatives of the interaction potential of the two parti-
|
|||
|
|
cles at their mean separation. A slightly less general for-
|
|||
|
|
mula was derived in the seminal paper [45]. For roughly
|
|||
|
|
thermal states, ρ2 (and thus p) is likely to be well ap-
|
|||
|
|
proximated by a Gaussian [58,59]. This gives
|
|||
|
|
|
|||
|
|
f(r, r′, t) = e− 1
|
|||
|
|
|
|||
|
|
2 (r′−r)tMtΣM(r′−r)t2/¯h2,
|
|||
|
|
(15)
|
|||
|
|
|
|||
|
|
where Σ = ⟨r2rt
|
|||
|
|
2⟩ − ⟨r2⟩⟨rt
|
|||
|
|
2⟩ is the covariance matrix of
|
|||
|
|
the location of the environment particle.
|
|||
|
|
Decoherence
|
|||
|
|
is destroyed when the exponent becomes of order unity,
|
|||
|
|
i.e., on a timescale
|
|||
|
|
|
|||
|
|
τ ≡
|
|||
|
|
�
|
|||
|
|
(r′ − r)tMtΣM(r′ − r)
|
|||
|
|
�−1/2 ¯h.
|
|||
|
|
(16)
|
|||
|
|
|
|||
|
|
Assuming a Coulomb potential V = gq2/|r2 − r1| gives
|
|||
|
|
M = (3�a�at − I)gq2/a3 where a ≡ r2 − r1 = a�a, |�a| =
|
|||
|
|
1. For thermal states, we have the isotropic case Σ =
|
|||
|
|
(∆x)2I, so equation (16) reduces to
|
|||
|
|
|
|||
|
|
τ =
|
|||
|
|
¯ha3
|
|||
|
|
|
|||
|
|
gq2|r′ − r|∆x
|
|||
|
|
�
|
|||
|
|
1 + 3 cos2 θ
|
|||
|
|
�−1/2 ,
|
|||
|
|
(17)
|
|||
|
|
|
|||
|
|
where cos θ ≡ �a · (r′ − r)/|r′ − r|. To be conservative,
|
|||
|
|
we take ∆x to be as small as the uncertainty principle
|
|||
|
|
allows. With the thermal constraint (∆p)2/m <∼ kT on
|
|||
|
|
the momentum uncertainty, this gives
|
|||
|
|
|
|||
|
|
6
|
|||
|
|
|
|||
|
|
|
|||
|
|
∆x =
|
|||
|
|
¯h
|
|||
|
|
|
|||
|
|
2∆p ∼
|
|||
|
|
¯h
|
|||
|
|
√
|
|||
|
|
|
|||
|
|
mkT
|
|||
|
|
.
|
|||
|
|
(18)
|
|||
|
|
|
|||
|
|
Substituting this into equation (17) and dividing by the
|
|||
|
|
number of ions N, we obtain the decoherence timescale
|
|||
|
|
|
|||
|
|
τ ∼
|
|||
|
|
a3√
|
|||
|
|
|
|||
|
|
mkT
|
|||
|
|
|
|||
|
|
Ngq2|r′ − r|.
|
|||
|
|
(19)
|
|||
|
|
|
|||
|
|
caused by a single environment ion a distance a away.
|
|||
|
|
Each such ion will produce its own suppression factor f,
|
|||
|
|
so we need to sum the exponent in equation (15) over all
|
|||
|
|
ions. Since the tidal force M ∝ a−3 causes the exponent
|
|||
|
|
to drop as a−6, this sum will generally be dominated by
|
|||
|
|
the very closest ion, which will typically be a distance
|
|||
|
|
a ∼ n−1/3 away. We are interested in decoherence for
|
|||
|
|
separations |r′ − r| = h, the membrane thickness, which
|
|||
|
|
gives
|
|||
|
|
|
|||
|
|
τ ∼
|
|||
|
|
|
|||
|
|
√
|
|||
|
|
|
|||
|
|
mkT
|
|||
|
|
|
|||
|
|
Ngq2enh ∼ 10−19 s.
|
|||
|
|
(20)
|
|||
|
|
|
|||
|
|
The relation between these different estimates is dis-
|
|||
|
|
cussed in more detail in the Appendix.
|
|||
|
|
|
|||
|
|
C. Microtubules
|
|||
|
|
|
|||
|
|
Microtubules are a major component of the cytoskele-
|
|||
|
|
ton, the “scaffolding” that helps cells maintain their
|
|||
|
|
shapes.
|
|||
|
|
They are hollow cylinders of diameter D =
|
|||
|
|
24 nm made up of 13 filaments that are strung together
|
|||
|
|
out of proteins known as tubulin dimers. These dimers
|
|||
|
|
can make transitions between two states known as α
|
|||
|
|
and β, corresponding to different electric dipole moments
|
|||
|
|
along the axis of the tube. It has been argued that micro-
|
|||
|
|
tubules may have additional functions as well, serving as
|
|||
|
|
a means of energy and information transfer [20]. A model
|
|||
|
|
has been presented whereby the dipole-dipole interac-
|
|||
|
|
tions between nearby dimers can lead to long-range po-
|
|||
|
|
larization and kink-like excitations that may travel down
|
|||
|
|
the microtubules at speeds exceeding 1 m/s [60].
|
|||
|
|
Penrose has gone further and suggested that the dy-
|
|||
|
|
namics of such excitations can make a microtubule act
|
|||
|
|
like a quantum computer, and that microtubules are the
|
|||
|
|
site of of human consciousness [2]. This idea has been fur-
|
|||
|
|
ther elaborated [21–24] employing methods from string
|
|||
|
|
theory, with the conclusion that quantum superpositions
|
|||
|
|
of coherent excitations can persist for as long as a second
|
|||
|
|
before being destroyed by decoherence. See also [61,62].
|
|||
|
|
This was hailed as a success for the model, the interpre-
|
|||
|
|
tation being that the quantum gravity effect on micro-
|
|||
|
|
tubules was identified with the human though process on
|
|||
|
|
this same timescale.
|
|||
|
|
This decoherence rate τ ∼ 1 s was computed assuming
|
|||
|
|
that quantum gravity is the main decoherence source.
|
|||
|
|
Since this quantum gravity model is somewhat contro-
|
|||
|
|
versial [32] and its effect has been found to be more than
|
|||
|
|
|
|||
|
|
20 orders of magnitude weaker than other decoherence
|
|||
|
|
sources in some cases [46], it seems prudent to evalu-
|
|||
|
|
ate other decoherence sources for the microtubule case
|
|||
|
|
as well, to see whether they are in fact dominant. We
|
|||
|
|
will now do so.
|
|||
|
|
Using coordinates where the x-axis is along the tube
|
|||
|
|
axis, the above-mentioned models all focus on the time-
|
|||
|
|
evolution of p(x), the average x-component of the electric
|
|||
|
|
dipole moment of the tubulin dimers at each x. In terms
|
|||
|
|
of this polarization function p(x), the net charge per unit
|
|||
|
|
length of tube is −p′(x). The propagating kink-like exci-
|
|||
|
|
tations [60] are of the form
|
|||
|
|
|
|||
|
|
p(x) =
|
|||
|
|
� +p0
|
|||
|
|
for x ≪ x0,
|
|||
|
|
|
|||
|
|
−p0
|
|||
|
|
for x ≫ x0,
|
|||
|
|
(21)
|
|||
|
|
|
|||
|
|
where the kink location x0 propagates with constant
|
|||
|
|
speed and has a width of order a few tubulin dimers.
|
|||
|
|
The polarization strength p0 is such that the total charge
|
|||
|
|
around the kink is Q = − � p′(x)dx = 2p0 ∼ 940qe, due
|
|||
|
|
to the presence of 18 Ca2+ ions on each of the 13 fila-
|
|||
|
|
ments contributing to p0 [60].
|
|||
|
|
Suppose that such a kink is in two different places
|
|||
|
|
in superposition, separated by some distance |r′ − r|.
|
|||
|
|
How rapidly will the superposition be destroyed by de-
|
|||
|
|
coherence?
|
|||
|
|
To be conservative, we will ignore colli-
|
|||
|
|
sions between polarized tubulin dimers and nearby water
|
|||
|
|
molecules, since it has been argued that these may be in
|
|||
|
|
some sense ordered and part of the quantum system [24]
|
|||
|
|
– although this argument is difficult to maintain for the
|
|||
|
|
water outside the microtubule, which permeates the en-
|
|||
|
|
tire cell volume. Let us instead apply equation (19), with
|
|||
|
|
N = Q/qe ∼ 103. The distance to the nearest ion will
|
|||
|
|
generally be less than a = R + n−1/3 ∼ 26 nm, where the
|
|||
|
|
tubulin diameter D = 24 nm dominates over the inter-
|
|||
|
|
ion separation n−1/3 ∼ 2 nm in the fluid surrounding
|
|||
|
|
the microtubule. Superpositions spanning many tubuline
|
|||
|
|
dimers (|r′ − r| ≫ D) therefore decohere on a timescale
|
|||
|
|
|
|||
|
|
τ ∼ D2√
|
|||
|
|
|
|||
|
|
mkT
|
|||
|
|
|
|||
|
|
Ngq2e
|
|||
|
|
∼ 10−13 s.
|
|||
|
|
(22)
|
|||
|
|
|
|||
|
|
due to the nearest ion alone. This is quite a conserva-
|
|||
|
|
tive estimate, since the other nD3 ∼ 103 ions that are
|
|||
|
|
merely a small fraction further away will also contribute
|
|||
|
|
to the decoherence rate, but it is nonetheless 6-7 orders
|
|||
|
|
of magnitude shorter than the estimates of Mavromatos
|
|||
|
|
& Nanopoulos [25–27]. We will comment on screening
|
|||
|
|
effects below.
|
|||
|
|
|
|||
|
|
1. Decoherence summary
|
|||
|
|
|
|||
|
|
Our decoherence rates are summarized in Table 1. How
|
|||
|
|
accurate are they likely to be?
|
|||
|
|
In the calculations above, we generally tried to be con-
|
|||
|
|
servative, erring on the side of underestimating the deco-
|
|||
|
|
herence rate. For instance, we neglected that N potas-
|
|||
|
|
sium ions also end up in superposition once the neuron
|
|||
|
|
|
|||
|
|
7
|
|||
|
|
|
|||
|
|
|
|||
|
|
firing is quenched, we neglected the contribution of other
|
|||
|
|
abundant ions such as Cl− to η, and and we ignored col-
|
|||
|
|
lisions with water molecules in the microtubule case.
|
|||
|
|
Since we were only interested in order-of-magnitude
|
|||
|
|
estimates, we made a number of crude approximations,
|
|||
|
|
e.g., for the cross sections. We neglected screening ef-
|
|||
|
|
fects because the decoherence rates were dominated by
|
|||
|
|
the particles closest to the system, i.e., the very same par-
|
|||
|
|
ticles that are responsible for screening the charge from
|
|||
|
|
more distant ones.
|
|||
|
|
|
|||
|
|
Table 1. Decoherence timescales.
|
|||
|
|
|
|||
|
|
Object
|
|||
|
|
Environment
|
|||
|
|
τdec
|
|||
|
|
|
|||
|
|
Neuron
|
|||
|
|
Colliding ion
|
|||
|
|
10−20s
|
|||
|
|
Neuron
|
|||
|
|
Colliding H2O
|
|||
|
|
10−20s
|
|||
|
|
Neuron
|
|||
|
|
Nearby ion
|
|||
|
|
10−19s
|
|||
|
|
Microtubule
|
|||
|
|
Distant ion
|
|||
|
|
10−13s
|
|||
|
|
|
|||
|
|
IV. DISCUSSION
|
|||
|
|
|
|||
|
|
A. The classical nature of brain processes
|
|||
|
|
|
|||
|
|
The calculations above enable us to address the ques-
|
|||
|
|
tion of whether cognitive processes in the brain consti-
|
|||
|
|
tute a classical or quantum system in the sense of Fig-
|
|||
|
|
ure 1. If we take the characteristic dynamical timescale
|
|||
|
|
for such processes to be τdyn ∼ 10−2 s − 100 s (the ap-
|
|||
|
|
parent timescale of e.g., speech, thought and motor re-
|
|||
|
|
sponse), then a comparison of τdyn with τdec from Table 1
|
|||
|
|
shows that processes associated with either conventional
|
|||
|
|
neuron firing or with polarization excitations in micro-
|
|||
|
|
tubules fall squarely in the classical category, by a mar-
|
|||
|
|
gin exceeding ten orders of magnitude. Neuron firing it-
|
|||
|
|
self is also highly classical, since it occurs on a timescale
|
|||
|
|
τdyn ∼ 10−3 − 10−4 s [53]. Even a kink-like microtubule
|
|||
|
|
excitation is classical by many orders of magnitude, since
|
|||
|
|
it traverses a short tubule on a timescale τdyn ∼ 5×10−7 s
|
|||
|
|
[60].
|
|||
|
|
What about other mechanisms?
|
|||
|
|
It is worth noting
|
|||
|
|
that if (as is commonly believed) different neuron fir-
|
|||
|
|
ing patterns correspond in some way to different con-
|
|||
|
|
scious perceptions, then consciousness itself cannot be
|
|||
|
|
of a quantum nature even if there is a yet undiscovered
|
|||
|
|
physical process in the brain with a very long decoherence
|
|||
|
|
time. As mentioned above, suggestions for such candi-
|
|||
|
|
dates have involved, e.g., superconductivity [12], super-
|
|||
|
|
fluidity [13], electromagnetic fields [14], Bose condensa-
|
|||
|
|
tion [15,16], superflourescence [17] and other mechanisms
|
|||
|
|
[18,19]. The reason is that as soon as such a quantum
|
|||
|
|
subsystem communicates with the constantly decohering
|
|||
|
|
neurons to create conscious experience, everything deco-
|
|||
|
|
heres.
|
|||
|
|
How extreme variations in the decoherence rates can
|
|||
|
|
we obtain by changing our model assumptions? Although
|
|||
|
|
the rates can be altered by a few of orders of magnitudes
|
|||
|
|
by pushing parameters such as the neuron dimensions,
|
|||
|
|
the myelination fraction or the microtubule kink charge
|
|||
|
|
|
|||
|
|
to the limits of plausibility, it is clearly impossible to
|
|||
|
|
change the basic conclusion that τdec ≪ 10−3 s, i.e., that
|
|||
|
|
we are dealing with a classical system in the sense of Fig-
|
|||
|
|
ure 1. Even the tiniest neuron imaginable, with only a
|
|||
|
|
single ion (N = 1) traversing the cell wall during firing,
|
|||
|
|
would have τdec ∼ 10−14 s.
|
|||
|
|
Likewise, reducing the ef-
|
|||
|
|
fective microtubule kink charge to a small fraction of qe
|
|||
|
|
would not help.
|
|||
|
|
How are we to understand the above-mentioned claims
|
|||
|
|
that brain subsystems can be sufficiently isolated to
|
|||
|
|
exhibit macroquantum behavior?
|
|||
|
|
It appears that the
|
|||
|
|
subtle distinction between dissipation and decoherence
|
|||
|
|
timescales has not always been appreciated.
|
|||
|
|
|
|||
|
|
B. Implications for the subject-object-environment
|
|||
|
|
decomposition
|
|||
|
|
|
|||
|
|
Let us now discuss the subsystem decomposition of
|
|||
|
|
Figure 2 in more detail in light of our results. As the
|
|||
|
|
figure indicates, the virtue of this decomposition into
|
|||
|
|
subject, object and environment is that the subsystem
|
|||
|
|
Hamiltonians Hs, Ho, He and the interaction Hamiltoni-
|
|||
|
|
ans Hso, Hoe, Hse can cause qualitatively very different
|
|||
|
|
effects. Let us now briefly discuss each of them in turn.
|
|||
|
|
Most of these processes are schematically illustrated
|
|||
|
|
in Figure 4 and Figure 5, where for purposes of illus-
|
|||
|
|
tration, we have shown the extremely simple case where
|
|||
|
|
both the subject and object have only a single degree of
|
|||
|
|
freedom that can take on only a few distinct values (3
|
|||
|
|
for the subject, 2 for the object). For definiteness, we
|
|||
|
|
denote the three subject states |¨- ⟩, | ¨⌣⟩ and | ¨⌢⟩, and in-
|
|||
|
|
terpret them as the observer feeling neutral, happy and
|
|||
|
|
sad, respectively. We denote the two object states |↑⟩
|
|||
|
|
and |↓⟩, and interpret them as the spin component (“up”
|
|||
|
|
or “down”) in the z-direction of a spin-1/2 system, say a
|
|||
|
|
silver atom. The joint system consisting of subject and
|
|||
|
|
object therefore has only 2 × 3 = 6 basis states: |¨- ↑⟩,
|
|||
|
|
|¨- ↓⟩, | ¨⌣↑⟩, | ¨⌣↓⟩, | ¨⌢↑⟩, | ¨⌢↓⟩. In Figures 4 and 5, we
|
|||
|
|
have therefore plotted ρ as a 6 × 6 matrix consisting of
|
|||
|
|
nine two-by-two blocks.
|
|||
|
|
|
|||
|
|
=
|
|||
|
|
+
|
|||
|
|
|
|||
|
|
Object�
|
|||
|
|
evolution
|
|||
|
|
Object�
|
|||
|
|
decohe-�
|
|||
|
|
rence
|
|||
|
|
Ho
|
|||
|
|
(Entropy�
|
|||
|
|
constant)
|
|||
|
|
(Entropy�
|
|||
|
|
increases)
|
|||
|
|
|
|||
|
|
Hoe
|
|||
|
|
|
|||
|
|
Observation/Measurement
|
|||
|
|
|
|||
|
|
(Entropy decreases)
|
|||
|
|
Hso
|
|||
|
|
�
|
|||
|
|
|
|||
|
|
2
|
|||
|
|
1_
|
|||
|
|
2
|
|||
|
|
1_
|
|||
|
|
|
|||
|
|
8
|
|||
|
|
|
|||
|
|
|
|||
|
|
FIG. 4. Time evolution of the 6×6 density matrix for the basis
|
|||
|
|
states |¨- ↑⟩, |¨- ↓⟩, | ¨⌣↑⟩, | ¨⌣↓⟩, | ¨
|
|||
|
|
⌢↑⟩, | ¨⌢↓⟩ as the object evolves in
|
|||
|
|
isolation, then decoheres, then gets observed by the subject. The
|
|||
|
|
final result is a statistical mixture of the states | ¨⌣↑⟩ and | ¨⌢↓⟩,
|
|||
|
|
simple zero-entropy states like the one we started with.
|
|||
|
|
|
|||
|
|
1. Effect of Ho: constant entropy
|
|||
|
|
|
|||
|
|
If the object were to evolve during a time interval t
|
|||
|
|
without interacting with the subject or the environment
|
|||
|
|
(Hso = Hoe = 0), then according to equation (1) its
|
|||
|
|
reduced density matrix ρo would evolve into UρoU † with
|
|||
|
|
the same entropy, since the time-evolution operator U ≡
|
|||
|
|
e−iHot is unitary.
|
|||
|
|
Suppose the subject stays in the state |¨- ⟩ and the
|
|||
|
|
object starts out in the pure state |↑⟩. Let the object
|
|||
|
|
Hamiltonian Ho correspond to a magnetic field in the y-
|
|||
|
|
direction causing the spin to precess to the x-direction,
|
|||
|
|
i.e., to the state (|↑⟩+|↓⟩)/
|
|||
|
|
√
|
|||
|
|
|
|||
|
|
2. The object density matrix
|
|||
|
|
ρo then evolves into
|
|||
|
|
|
|||
|
|
ρo = U|↑⟩⟨↑|U † = 1
|
|||
|
|
|
|||
|
|
2(|↑⟩ + |↓⟩)(⟨↑| + ⟨↓|)
|
|||
|
|
|
|||
|
|
= 1
|
|||
|
|
|
|||
|
|
2(|↑⟩⟨↑| + |↑⟩⟨↓| + |↓⟩⟨↑| + |↓⟩⟨↓|),
|
|||
|
|
(23)
|
|||
|
|
|
|||
|
|
corresponding to the four entries of 1/2 in the second
|
|||
|
|
matrix of Figure 4.
|
|||
|
|
This is quite typical of pure quantum time evolution: a
|
|||
|
|
basis state eventually evolves into a superposition of ba-
|
|||
|
|
sis states, and the quantum nature of this superposition
|
|||
|
|
is manifested by off-diagonal elements in ρo. Another fa-
|
|||
|
|
miliar example of this is the familiar spreading out of the
|
|||
|
|
wave packet of a free particle.
|
|||
|
|
|
|||
|
|
2. Effect of Hoe: increasing entropy
|
|||
|
|
|
|||
|
|
This was the effect of Ho alone. In contrast, Hoe will
|
|||
|
|
generally cause decoherence and increase the entropy of
|
|||
|
|
the object. As discussed in detail in Section III and the
|
|||
|
|
Appendix, it entangles it with the environment, which
|
|||
|
|
suppresses the off-diagonal elements of the reduced den-
|
|||
|
|
sity matrix of the object as illustrated in Figure 4. If Hoe
|
|||
|
|
couples to the z-component of the spin, this destroys the
|
|||
|
|
terms |↑⟩⟨↓| and |↓⟩⟨↑|. Complete decoherence therefore
|
|||
|
|
converts the final state of equation (23) into
|
|||
|
|
|
|||
|
|
ρo = 1
|
|||
|
|
|
|||
|
|
2(|↑⟩⟨↑| + |↓⟩⟨↓|),
|
|||
|
|
(24)
|
|||
|
|
|
|||
|
|
corresponding to the two entries of 1/2 in the third ma-
|
|||
|
|
trix of Figure 4.
|
|||
|
|
|
|||
|
|
3. Effect of Hso: decreasing entropy
|
|||
|
|
|
|||
|
|
Whereas Hoe typically causes the apparent entropy of
|
|||
|
|
the object to increase, Hso typically causes it to decrease.
|
|||
|
|
|
|||
|
|
Figure 4 illustrates the case of an ideal measurement,
|
|||
|
|
where the subject starts out in the state |¨- ⟩ and Hso is of
|
|||
|
|
such a form that gets perfectly correlated with the object.
|
|||
|
|
In the language of Section II, an ideal measurement is a
|
|||
|
|
type of communication where the mutual information I12
|
|||
|
|
between the subject and object systems is increased to its
|
|||
|
|
maximum possible value. Suppose that the measurement
|
|||
|
|
is caused by Hso becoming large during a time interval so
|
|||
|
|
brief that we can neglect the effects of Hs and Ho. The
|
|||
|
|
joint subject+object density matrix ρso then evolves as
|
|||
|
|
ρso �→ UρsoU †, where U ≡ exp
|
|||
|
|
�
|
|||
|
|
−i
|
|||
|
|
�
|
|||
|
|
Hsodt
|
|||
|
|
�
|
|||
|
|
. If observing
|
|||
|
|
|↑⟩ makes the subject happy and |↓⟩ makes the subject
|
|||
|
|
sad, then we have U|¨-↑⟩ = | ¨⌣↑⟩ and U|¨-↓⟩ = | ¨⌢↓⟩. The
|
|||
|
|
state given by equation (24) would therefore evolve into
|
|||
|
|
|
|||
|
|
ρo = 1
|
|||
|
|
|
|||
|
|
2U(|¨- ⟩⟨¨- |) ⊗ (|↑⟩⟨↑| + |↓⟩⟨↓|)U †
|
|||
|
|
(25)
|
|||
|
|
|
|||
|
|
= 1
|
|||
|
|
|
|||
|
|
2(U|¨-↑⟩⟨¨-↑|U † + U|¨-↓⟩⟨¨-↓|U †
|
|||
|
|
(26)
|
|||
|
|
|
|||
|
|
= 1
|
|||
|
|
|
|||
|
|
2(| ¨⌣↑⟩⟨ ¨⌣↑| + | ¨⌢↓⟩⟨ ¨⌢↓ |),
|
|||
|
|
(27)
|
|||
|
|
|
|||
|
|
as illustrated in Figure 4.
|
|||
|
|
This final state contains a
|
|||
|
|
mixture of two subjects, corresponding to definite but
|
|||
|
|
opposite knowledge of the object state.
|
|||
|
|
According to
|
|||
|
|
both of them, the entropy of the object has decreased
|
|||
|
|
from one bit to zero bits.
|
|||
|
|
In general, we see that the object decreases its en-
|
|||
|
|
tropy when it exchanges information with the subject
|
|||
|
|
and increases when it exchanges information with the
|
|||
|
|
environment.4 Loosely speaking, the entropy of an ob-
|
|||
|
|
ject decreases while you look at it and increases while
|
|||
|
|
you don’t5.
|
|||
|
|
|
|||
|
|
4If n bits of information are exchanged with the environ-
|
|||
|
|
ment, then equation (3) shows that the object entropy will
|
|||
|
|
increase by this same amount if the environment is in ther-
|
|||
|
|
mal equilibrium (with maximal entropy) throughout. If we
|
|||
|
|
were to know the state of the environment initially (by our
|
|||
|
|
definition of environment, we do not), then both the object
|
|||
|
|
and environment entropy will typically increase by n/2 bits.
|
|||
|
|
5 Here and throughout, we are assuming that the total
|
|||
|
|
system, which is by definition isolated, evolves according to
|
|||
|
|
the Schr¨odinger equation (1). Although modifications of the
|
|||
|
|
Schr¨odinger equation have been suggested by some authors,
|
|||
|
|
either in a mathematically explicit form as in [55,56] or ver-
|
|||
|
|
bally as a so-called reduction postulate, there is so far no
|
|||
|
|
experimental evidence suggesting that modifications are nec-
|
|||
|
|
essary. The original motivations for such modifications were
|
|||
|
|
|
|||
|
|
1. to be able to interpret the diagonal elements of the
|
|||
|
|
density matrix as probabilities and
|
|||
|
|
|
|||
|
|
2. to suppress off-diagonal elements of the density matrix.
|
|||
|
|
|
|||
|
|
The subsequent discovery by Everett [64] that the probability
|
|||
|
|
interpretation automatically appears to hold for almost all
|
|||
|
|
observers in the final superposition solved problem 1, and is
|
|||
|
|
discussed in more detail in, e.g., [29,66–74]. The still more
|
|||
|
|
|
|||
|
|
9
|
|||
|
|
|
|||
|
|
|
|||
|
|
=
|
|||
|
|
+
|
|||
|
|
|
|||
|
|
Subject�
|
|||
|
|
evolution
|
|||
|
|
Subject�
|
|||
|
|
decohe-�
|
|||
|
|
rence
|
|||
|
|
Hs
|
|||
|
|
(Snap �
|
|||
|
|
decision)
|
|||
|
|
�
|
|||
|
|
Hse
|
|||
|
|
|
|||
|
|
�
|
|||
|
|
|
|||
|
|
2
|
|||
|
|
1_
|
|||
|
|
2
|
|||
|
|
1_
|
|||
|
|
|
|||
|
|
FIG. 5. Time evolution of the same 6 × 6 density matrix as in
|
|||
|
|
Figure 4 when the subject evolves in isolation, then decoheres. The
|
|||
|
|
object remains in the state |↑⟩ the whole time. The final result is
|
|||
|
|
a statistical mixture of the two states | ¨⌣↑⟩ and | ¨
|
|||
|
|
⌢↑⟩.
|
|||
|
|
|
|||
|
|
4. Effect if Hs: the thought process
|
|||
|
|
|
|||
|
|
So far, we have focused on the object and discussed
|
|||
|
|
effects of its internal dynamics (Ho) and its interactions
|
|||
|
|
with the environment (Hoe) and subject (Hso). Let us
|
|||
|
|
now turn to the subject and consider the role played by
|
|||
|
|
its internal dynamics (Hs) and interactions with the en-
|
|||
|
|
vironment (Hse).
|
|||
|
|
In his seminal 1993 book, Stapp [3]
|
|||
|
|
presents an argument about brain dynamics that can be
|
|||
|
|
summarized as follows.
|
|||
|
|
|
|||
|
|
1. Since the brain contains ∼ 1011 synapses connected
|
|||
|
|
together by neurons in a highly nonlinear fashion,
|
|||
|
|
there must be a huge number of metastable rever-
|
|||
|
|
berating patters of pulses into which the brain can
|
|||
|
|
evolve.
|
|||
|
|
|
|||
|
|
2. Neural network simulations have indicated that the
|
|||
|
|
metastable state into which a brain does in fact
|
|||
|
|
evolves depends sensitively on the initial conditions
|
|||
|
|
in small numbers of synapses.
|
|||
|
|
|
|||
|
|
3. The latter depends on the locations of a small num-
|
|||
|
|
ber of calcium atoms, which might be expected to
|
|||
|
|
be in quantum superpositions.
|
|||
|
|
|
|||
|
|
4. Therefore, one would expect the brain to evolve
|
|||
|
|
into
|
|||
|
|
a
|
|||
|
|
quantum
|
|||
|
|
superposition
|
|||
|
|
of
|
|||
|
|
many
|
|||
|
|
such
|
|||
|
|
metastable configurations.
|
|||
|
|
|
|||
|
|
5. Moreover, the fatigue characteristics of the synap-
|
|||
|
|
tic junctions will cause any given metastable state
|
|||
|
|
|
|||
|
|
recent discovery of decoherence [11,36,37] solved problem 2,
|
|||
|
|
as well as explaining so-called superselection rules for the first
|
|||
|
|
time (why for instance the position basis has a special status)
|
|||
|
|
[44].
|
|||
|
|
|
|||
|
|
to become, after a short time, unstable:
|
|||
|
|
the
|
|||
|
|
subject will then be forced to search for a new
|
|||
|
|
metastable configuration, and will therefore con-
|
|||
|
|
tinue to evolve into a superposition of increasingly
|
|||
|
|
disparate states.
|
|||
|
|
|
|||
|
|
If different states (perceptions) of the subject correspond
|
|||
|
|
to different metastable states of neuron firing patterns, a
|
|||
|
|
definite perception would eventually evolve into a super-
|
|||
|
|
position of several subjectively distinguishable percep-
|
|||
|
|
tions.
|
|||
|
|
We will follow Stapp in making this assumption about
|
|||
|
|
Hs. For illustrative purposes, let us assume that this can
|
|||
|
|
happen even at the level of a single thought or snap de-
|
|||
|
|
cision where the outcome feels unpredictable to us. Con-
|
|||
|
|
sider the following experiment: the subject starts out
|
|||
|
|
with a blank face and counts silently to three, then makes
|
|||
|
|
a snap decision on whether to smile or frown. The time-
|
|||
|
|
evolution operator U ≡ exp
|
|||
|
|
�
|
|||
|
|
−i � Hsdt
|
|||
|
|
�
|
|||
|
|
will then have
|
|||
|
|
the property that U|¨- ⟩ = (| ¨⌣⟩ + | ¨⌢⟩)/
|
|||
|
|
√
|
|||
|
|
|
|||
|
|
2, so the sub-
|
|||
|
|
ject density matrix ρs will evolve into
|
|||
|
|
|
|||
|
|
ρs = U|¨- ⟩⟨¨- |U † = 1
|
|||
|
|
|
|||
|
|
2(| ¨⌣⟩ + | ¨⌢⟩)(⟨ ¨⌣| + ⟨ ¨⌢|)
|
|||
|
|
|
|||
|
|
= 1
|
|||
|
|
|
|||
|
|
2(| ¨⌣⟩⟨ ¨⌣| + | ¨⌣⟩⟨ ¨⌢| + | ¨⌢⟩⟨ ¨⌣| + | ¨⌢⟩⟨ ¨⌢|),
|
|||
|
|
(28)
|
|||
|
|
|
|||
|
|
corresponding to the four entries of 1/2 in the second
|
|||
|
|
matrix in Figure 5.
|
|||
|
|
|
|||
|
|
5. Effect of Hse: subject decoherence
|
|||
|
|
|
|||
|
|
Just as Hoe can decohere the object, Hse can decohere
|
|||
|
|
the subject. The difference is that whereas the object can
|
|||
|
|
be either a quantum system (with small Hoe) or a classi-
|
|||
|
|
cal system (with large Hoe), a human subject always has
|
|||
|
|
a large interaction with the environment. As we showed
|
|||
|
|
in Section III, τdec ≪ τdyn for the subject, i.e., the ef-
|
|||
|
|
fect of Hse is faster than that of Hs by many orders of
|
|||
|
|
magnitude. This means that we should strictly speaking
|
|||
|
|
not think of macrosuperpositions such as equation (28)
|
|||
|
|
as first forming and then decohering as in Figure 5 —
|
|||
|
|
rather, subject decoherence is so fast that such superpo-
|
|||
|
|
sitions decohere already during their process of forma-
|
|||
|
|
tion. Therefore we are never even close to being able to
|
|||
|
|
perceive superpositions of different perceptions. Reduc-
|
|||
|
|
ing object decoherence (from Hoe) during measurement
|
|||
|
|
would make no difference, since decoherence would take
|
|||
|
|
place in the brain long before the transmission of the ap-
|
|||
|
|
propriate sensory input through sensory nerves had been
|
|||
|
|
completed.
|
|||
|
|
|
|||
|
|
C. He and Hsoe
|
|||
|
|
|
|||
|
|
The environment is of course the most complicated sys-
|
|||
|
|
tem, since it contains the vast majority of the degrees of
|
|||
|
|
|
|||
|
|
10
|
|||
|
|
|
|||
|
|
|
|||
|
|
freedom in the total system. It is therefore very fortu-
|
|||
|
|
nate that we can so often ignore it, considering only those
|
|||
|
|
limited aspects of it that affect the subject and object.
|
|||
|
|
For the most general H, there can also be an ugly
|
|||
|
|
irreducible residual term Hsoe ≡ H − Hs − Ho − He −
|
|||
|
|
Hso − Hoe − Hse.
|
|||
|
|
|
|||
|
|
D. Implications for modeling cognitive processes
|
|||
|
|
|
|||
|
|
For the neural network community, the implication of
|
|||
|
|
our result is “business as usual”, i.e., there is no need
|
|||
|
|
to worry about the fact that current simulations do not
|
|||
|
|
incorporate effects of quantum coherence. The only rem-
|
|||
|
|
nant from quantum mechanics is the apparent random-
|
|||
|
|
ness that we subjectively perceive every time the subject
|
|||
|
|
system evolves into a superposition as in equation (28),
|
|||
|
|
but this can be simply modeled by including a random
|
|||
|
|
number generator in the simulation. In other words, the
|
|||
|
|
recipe used to prescribe when a given neuron should fire
|
|||
|
|
and how synaptic coupling strengths should be updated
|
|||
|
|
may have to involve some classical randomness to cor-
|
|||
|
|
rectly mimic the behavior of the brain.
|
|||
|
|
|
|||
|
|
1. Hyper-classicality
|
|||
|
|
|
|||
|
|
If a subject system is to be a good model of us, Hso
|
|||
|
|
and Hse need to meet certain criteria: decoherence and
|
|||
|
|
communication are necessary, but fluctuation and dissi-
|
|||
|
|
pation must be kept low enough that the subject does
|
|||
|
|
not lose its autonomy completely.
|
|||
|
|
In our study of neural processes, we concluded that the
|
|||
|
|
subject is not a quantum system, since τdec ≪ τdyn. How-
|
|||
|
|
ever, since the dissipation time τdiss for neuron firing is
|
|||
|
|
of the same order as its dynamical timescale, we see that
|
|||
|
|
in the sense of Figure 1, the subject is not a simple clas-
|
|||
|
|
sical system either. It is therefore somewhat misleading
|
|||
|
|
to think of it as simply some classical degrees of freedom
|
|||
|
|
evolving fairly undisturbed (only interacting enough to
|
|||
|
|
stay decohered and occasionally communicate with the
|
|||
|
|
outside world). Rather, the semi-autonomous degrees of
|
|||
|
|
freedom that constitute the subject are to be found at a
|
|||
|
|
higher level of complexity, perhaps as metastable global
|
|||
|
|
patters of neuron firing.
|
|||
|
|
These degrees of freedom might be termed “hyper-
|
|||
|
|
classical”:
|
|||
|
|
although
|
|||
|
|
there
|
|||
|
|
is
|
|||
|
|
nothing
|
|||
|
|
quantum-
|
|||
|
|
mechanical about their equations of motion (except that
|
|||
|
|
they can be stochastic), they may bear little resemblance
|
|||
|
|
with the underlying classical equations from which they
|
|||
|
|
were derived.
|
|||
|
|
Energy conservation and other familiar
|
|||
|
|
concepts from Hamiltonian dynamics will be irrelevant
|
|||
|
|
for these more abstract equations, since neurons are en-
|
|||
|
|
ergy pumped and highly dissipative. Other examples of
|
|||
|
|
such hyper-classical systems include the time-evolution
|
|||
|
|
of the memory contents of a regular (highly dissipative)
|
|||
|
|
|
|||
|
|
digital computer as well as the motion on the screen of
|
|||
|
|
objects in a computer game.
|
|||
|
|
|
|||
|
|
2. Nature of the subject system
|
|||
|
|
|
|||
|
|
In this paper, we have tacitly assumed that conscious-
|
|||
|
|
ness is synonymous with certain brain processes. This is
|
|||
|
|
what Lockwood terms the “identity theory” [66]. It dates
|
|||
|
|
back to Hobbes (∼1660) and has been espoused by, e.g.,
|
|||
|
|
Russell, Feigl, Smart, Armstrong, Churchland and Lock-
|
|||
|
|
wood himself. Let us briefly explore the more specific
|
|||
|
|
assumption that the subject degrees of freedom are our
|
|||
|
|
perceptions. In this picture, some of the subject degrees
|
|||
|
|
of freedom would have to constitute a “world model”,
|
|||
|
|
with the interaction Hso such that the resulting commu-
|
|||
|
|
nication keeps these degrees of freedom highly correlated
|
|||
|
|
with selected properties of the outside world (object +
|
|||
|
|
environment). Some such properties, i.e.,
|
|||
|
|
|
|||
|
|
• the intensity of the electromagnetic on the retina,
|
|||
|
|
averaged through three narrow-band filters (color
|
|||
|
|
vision) and one broad-band filter (black-and-white
|
|||
|
|
vision),
|
|||
|
|
|
|||
|
|
• the spectrum of air pressure fluctuations in the ears
|
|||
|
|
(sound),
|
|||
|
|
|
|||
|
|
• the chemical composition of gas in the nose (smell)
|
|||
|
|
and solutions in the mouth (taste),
|
|||
|
|
|
|||
|
|
• heat and pressure at a variety of skin locations,
|
|||
|
|
|
|||
|
|
• locations of body parts,
|
|||
|
|
|
|||
|
|
are tracked rather continuously, with the corresponding
|
|||
|
|
mutual information I12 between subject and surround-
|
|||
|
|
ings remaining fairly constant.
|
|||
|
|
Persisting correlations
|
|||
|
|
with properties of the past state of the surroundings
|
|||
|
|
(memories) further contribute to the mutual information
|
|||
|
|
I12. Much of I12 is due to correlations with quite subtle
|
|||
|
|
aspects of the surroundings, e.g., the contents of books.
|
|||
|
|
The total mutual information I12 between a person and
|
|||
|
|
the external world is fairly low at birth, gradually grows
|
|||
|
|
through learning, and falls when we forget. In contrast,
|
|||
|
|
most innate objects have a very small mutual informa-
|
|||
|
|
tion with the rest of the world, books and diskettes being
|
|||
|
|
notable exceptions.
|
|||
|
|
The extremely limited selection of properties that the
|
|||
|
|
subject correlates with has presumably been determined
|
|||
|
|
by evolutionary utility, since it is known to differ between
|
|||
|
|
species: birds perceive four primary colors but cats only
|
|||
|
|
one, bees perceive light polarization, etc. In this picture,
|
|||
|
|
we should therefore not consider these particular (“classi-
|
|||
|
|
cal”) aspects of our surroundings to be more fundamental
|
|||
|
|
than the vast majority that the subject system is uncor-
|
|||
|
|
related with. Morover, our perception of e.g. space is as
|
|||
|
|
subjective as our perception of color, just as suggested
|
|||
|
|
by e.g. [50].
|
|||
|
|
|
|||
|
|
11
|
|||
|
|
|
|||
|
|
|
|||
|
|
3. The binding problem
|
|||
|
|
|
|||
|
|
One of the motivations for models with quantum co-
|
|||
|
|
herence in the brain was the so-called binding problem.
|
|||
|
|
In the words of James [75,76], “the only realities are the
|
|||
|
|
separate molecules, or at most cells. Their aggregation
|
|||
|
|
into a ‘brain’ is a fiction of popular speech”. James’ con-
|
|||
|
|
cern, shared by many after him, was that consciousness
|
|||
|
|
did not seem to be spatially localized to any one small
|
|||
|
|
part of the brain, yet subjectively feels like a coherent
|
|||
|
|
entity. Because of this, Stapp [3] and many others have
|
|||
|
|
appealed to quantum coherence, arguing that this could
|
|||
|
|
make consciousness a holistic effect involving the brain
|
|||
|
|
as a whole.
|
|||
|
|
However, non-local degrees of freedom can be impor-
|
|||
|
|
tant even in classical physics, For instance, oscillations
|
|||
|
|
in a guitar string are local in Fourier space, not in real
|
|||
|
|
space, so in this case the “binding problem” can be solved
|
|||
|
|
by a simple change of variables. As Eddington remarked
|
|||
|
|
[77], when observing the ocean we perceive the moving
|
|||
|
|
waves as objects in their own right because they display a
|
|||
|
|
certain permanence, even though the water itself is only
|
|||
|
|
bobbing up and down. Similarly, thoughts are presum-
|
|||
|
|
ably highly non-local excitation patterns in the neural
|
|||
|
|
network of our brain, except of a non-linear and much
|
|||
|
|
more complex nature.
|
|||
|
|
In short, this author feels that
|
|||
|
|
there is no binding problem.
|
|||
|
|
|
|||
|
|
4. Outlook
|
|||
|
|
|
|||
|
|
In summary, our decoherence calculations have in-
|
|||
|
|
dicated that there is nothing fundamentally quantum-
|
|||
|
|
mechanical about cognitive processes in the brain, sup-
|
|||
|
|
porting the Hepp’s conjecture [33]. Specifically, the com-
|
|||
|
|
putations in the brain appear to be of a classical rather
|
|||
|
|
than quantum nature, and the argument by Lisewski [78]
|
|||
|
|
that quantum corrections may be needed for accurate
|
|||
|
|
modeling of some details, e.g., non-Markovian noise in
|
|||
|
|
neurons, does of course not change this conclusion. This
|
|||
|
|
means that although the current state-of-the-art in neu-
|
|||
|
|
ral network hardware is clearly still very far from be-
|
|||
|
|
ing able to model and understand cognitive processes as
|
|||
|
|
complex as those in the brain, there are no quantum me-
|
|||
|
|
chanical reasons to doubt that this research is on the
|
|||
|
|
right track.
|
|||
|
|
|
|||
|
|
Acknowledgements:
|
|||
|
|
The author wishes to thank
|
|||
|
|
the organizers of the Spaatind-98 and Gausdal-99 win-
|
|||
|
|
ter schools, where much of this work was done, and
|
|||
|
|
Mark Alford, Philippe Blanchard, Carlton Caves, Angel-
|
|||
|
|
ica de Oliveira-Costa, Matthew Donald, Andrei Gruzi-
|
|||
|
|
nov, Piet Hut, Nick Mavromatos, Henry Stapp, Hans-
|
|||
|
|
Dieter Zeh and Woitek Zurek for stimulating discussions
|
|||
|
|
and helpful comments. Support for this work was pro-
|
|||
|
|
vided by the Sloan Foundation and by NASA though
|
|||
|
|
|
|||
|
|
grant NAG5-6034 and Hubble Fellowship HF-01084.01-
|
|||
|
|
96A from STScI, operated by AURA, Inc. under NASA
|
|||
|
|
contract NAS5-26555.
|
|||
|
|
|
|||
|
|
APPENDIX: DECOHERENCE FORMULAS
|
|||
|
|
|
|||
|
|
The quantitative effect of decoherence from both short
|
|||
|
|
range interactions (scattering) and long-range interac-
|
|||
|
|
tions was first derived in a seminal paper by Joos & Zeh
|
|||
|
|
[45]. Since our application involved scattering between
|
|||
|
|
particles of comparable mass, we used a generalized ver-
|
|||
|
|
sion of these results that included the effect of recoil [46].
|
|||
|
|
In this Appendix, we derive a slightly generalized formula
|
|||
|
|
for long-range interactions, and briefly comment on the
|
|||
|
|
relation between these short-range and long-range limit-
|
|||
|
|
ing cases.
|
|||
|
|
|
|||
|
|
1. Decoherence due to tidal forces
|
|||
|
|
|
|||
|
|
Even if the dissipation and fluctuation caused by Hint
|
|||
|
|
is dynamically unimportant, H1 and H2 can be neglected
|
|||
|
|
in equation (2) when calculating the decoherence effect in
|
|||
|
|
the many cases where the interaction Hamiltonian deco-
|
|||
|
|
heres the object on a timescale far below the dynamical
|
|||
|
|
time. In this approximation, we consider two particles
|
|||
|
|
with an interaction H = Hint = V (r2 − r1) for some
|
|||
|
|
potential V . According to equation (1), the two-particle
|
|||
|
|
density matrix ρ therefore evolves as
|
|||
|
|
|
|||
|
|
ρ(r1, r′
|
|||
|
|
1, r2, r′
|
|||
|
|
2, t0 + t)
|
|||
|
|
|
|||
|
|
= ρ(r1, r′
|
|||
|
|
1, r2, r′
|
|||
|
|
2, t)e−i[V (r2−r1)−V (r′
|
|||
|
|
2−r′
|
|||
|
|
1)]/¯h.
|
|||
|
|
(A1)
|
|||
|
|
|
|||
|
|
Following [45], we assume that the two particles are fairly
|
|||
|
|
localized near their initial average positions
|
|||
|
|
|
|||
|
|
r0
|
|||
|
|
i ≡ ⟨ri⟩0 = tr [riρi(t0)],
|
|||
|
|
(A2)
|
|||
|
|
|
|||
|
|
i = 1, 2, and approximate the potential by its second
|
|||
|
|
order Taylor expansion
|
|||
|
|
|
|||
|
|
V (r2 − r1) ≈ V (a) − F · (x2 − x1)
|
|||
|
|
|
|||
|
|
+ 1
|
|||
|
|
|
|||
|
|
2(x2 − x1)tM(x2 − x1).
|
|||
|
|
(A3)
|
|||
|
|
|
|||
|
|
Here F ≡= −∇V (a) is the average force, M is the Hes-
|
|||
|
|
sian matrix Mij ≡ ∂i∂jV (a) and a ≡ r0
|
|||
|
|
2−r0
|
|||
|
|
1. We have in-
|
|||
|
|
troduced relative coordinates xi ≡ ri−r0
|
|||
|
|
i . Assuming that
|
|||
|
|
the two particles are independent initially as in [45], i.e.,
|
|||
|
|
that ρ(t0) takes the separable form ρ(x1, x′
|
|||
|
|
1, x2, x′
|
|||
|
|
2, t0) =
|
|||
|
|
ρ1(x1, x′
|
|||
|
|
1, t0)ρ2(x2, x′
|
|||
|
|
2, t0), this gives
|
|||
|
|
|
|||
|
|
ρ1(x1, x′
|
|||
|
|
1, t0 + t) = tr 2ρ(t0 + t) =
|
|||
|
|
�
|
|||
|
|
ρ(x1, x′
|
|||
|
|
1, x, x, t0 + t)d3x = ρ1(x1, x′
|
|||
|
|
1, t0)f(x1, x′
|
|||
|
|
1, t), (A4)
|
|||
|
|
|
|||
|
|
where
|
|||
|
|
|
|||
|
|
12
|
|||
|
|
|
|||
|
|
|
|||
|
|
f(x1, x′
|
|||
|
|
1, t) ≈
|
|||
|
|
|
|||
|
|
eiφ(x1,x′
|
|||
|
|
1,t)
|
|||
|
|
�
|
|||
|
|
ρ2(x2, x′
|
|||
|
|
2, t0)e−it(x′
|
|||
|
|
1−x1)tMx2/¯hd3x2 =
|
|||
|
|
|
|||
|
|
eiφ(x1,x′
|
|||
|
|
1,t)�p2[M(x′
|
|||
|
|
1 − x1)t/¯h].
|
|||
|
|
(A5)
|
|||
|
|
|
|||
|
|
Here the phase factor
|
|||
|
|
|
|||
|
|
eiφ(x,x′,t) ≡ e
|
|||
|
|
i
|
|||
|
|
¯h[F·(x′−x)+ 1
|
|||
|
|
|
|||
|
|
2 x′tMx′− 1
|
|||
|
|
|
|||
|
|
2 xtMx]
|
|||
|
|
(A6)
|
|||
|
|
|
|||
|
|
is of no importance for decoherence, since it does not
|
|||
|
|
suppress the magnitude |ρ1(x1, x′
|
|||
|
|
1, t)| of the off-diagonal
|
|||
|
|
elements – it merely causes momentum transfer related
|
|||
|
|
to fluctuation and dissipation.
|
|||
|
|
It is the other term
|
|||
|
|
that causes decoherence. �p2 is the Fourier transform of
|
|||
|
|
p2(x) ≡ ρ2(x, x, t0), the probability distribution for the
|
|||
|
|
location of the environment particle.
|
|||
|
|
|
|||
|
|
2. Properties of the effect
|
|||
|
|
|
|||
|
|
Let us briefly discuss some qualitative features of equa-
|
|||
|
|
tion (A5).
|
|||
|
|
Since �p2(0) =
|
|||
|
|
�
|
|||
|
|
p2(x2)d3x2 = tr ρ2 = 1,
|
|||
|
|
ρ1(x, x′) remains unchanged on the diagonal x = x′.
|
|||
|
|
This is because Hint is not changing the position of our
|
|||
|
|
our object particle, merely its momentum.
|
|||
|
|
Since the
|
|||
|
|
mean position ⟨x2⟩ =
|
|||
|
|
�
|
|||
|
|
p2x2d3x2 = tr [x2ρ2] = 0 van-
|
|||
|
|
ishes (using equation (A2)), we have ∇�p2(0) = 0.
|
|||
|
|
In
|
|||
|
|
fact, |f| takes a maximum on the diagonal, and the
|
|||
|
|
Riemann-Lebesgue Lemma shows that |f| = |�p2| ≤ 1
|
|||
|
|
whenever x ̸= x′, with equality only for the unphys-
|
|||
|
|
ical case where p2 is a delta function, i.e., where the
|
|||
|
|
location of the environment particle is perfectly known.
|
|||
|
|
∂i∂j|f(0)| = −M⟨x2xt
|
|||
|
|
2⟩Mt2/2¯h2, so so the larger ⟨x2xt
|
|||
|
|
2⟩
|
|||
|
|
is (i.e., the more spread out the environment particle is),
|
|||
|
|
the closer to the diagonal decoherence will suppress our
|
|||
|
|
density matrix.
|
|||
|
|
Since M is the shear matrix of the force field −∇V , we
|
|||
|
|
see that it is tidal forces that are causing the decoherence
|
|||
|
|
— the average force F simply contributes to the phase
|
|||
|
|
factor eiφ. Specifically, the rate at which our object de-
|
|||
|
|
grees of freedom r1 decohere grows with the tidal force
|
|||
|
|
that it exerts on the environment: if the environment
|
|||
|
|
particle is spread out with ⟨x2xt
|
|||
|
|
2⟩ large, experiencing a
|
|||
|
|
wide range of forces from the object, object decoherence
|
|||
|
|
is rapid. In the opposite situation, where the object is
|
|||
|
|
spread out and the environment is not, the object will
|
|||
|
|
experience strong classical tidal forces but no decoher-
|
|||
|
|
ence.
|
|||
|
|
|
|||
|
|
3. Relation between long-range and short-range
|
|||
|
|
decoherence
|
|||
|
|
|
|||
|
|
Above we derived the effect of decoherence from long-
|
|||
|
|
range tidal forces. Another interesting case that has been
|
|||
|
|
solved analytically [45] is that of short-range interactions
|
|||
|
|
|
|||
|
|
that can be modeled as scattering events. If the scatter-
|
|||
|
|
ing takes place during short enough a time interval that
|
|||
|
|
we can neglect the internal dynamics of the object, then
|
|||
|
|
its reduced density matrix changes as [46]
|
|||
|
|
|
|||
|
|
ρ1(r, r′) �→ ρ1(r, r′)�p
|
|||
|
|
�r′ − r
|
|||
|
|
|
|||
|
|
¯h
|
|||
|
|
|
|||
|
|
�
|
|||
|
|
,
|
|||
|
|
(A7)
|
|||
|
|
|
|||
|
|
where p(q) is the probability distribution for the momen-
|
|||
|
|
tum transfer q in the collision. This equation generalizes
|
|||
|
|
the scattering result of [45] by including the effect of re-
|
|||
|
|
coil. The larger the uncertainty in momentum transfer,
|
|||
|
|
the stronger the decoherence effect becomes, since widen-
|
|||
|
|
ing p narrows its Fourier transform �p. Changing the mean
|
|||
|
|
momentum transfer ⟨q⟩ does not affect the decoherence,
|
|||
|
|
merely contributes a phase factor just as F did above.
|
|||
|
|
Typically, the last factor in equation (A7) destroys coher-
|
|||
|
|
ence down to scales of order the de Broigle wavelength
|
|||
|
|
of the scatterer, with directional modulations from the
|
|||
|
|
angular dependence of the scattering cross section. Gen-
|
|||
|
|
eralization to a steady flux of scattering particles [46]
|
|||
|
|
gives equation (6).
|
|||
|
|
Equation (A7) has striking similarities with the tidal
|
|||
|
|
force result of equation (A5): in both cases, the density
|
|||
|
|
matrix gets multiplied by the Fourier transform of a prob-
|
|||
|
|
ability distribution.
|
|||
|
|
If fact, up to uninteresting phase
|
|||
|
|
factors, we can rewrite our equation (A5) in exactly the
|
|||
|
|
form of equation (A7) by redefining p to be the probabil-
|
|||
|
|
ity distribution for momentum transfer q = M(x2 −x1)t
|
|||
|
|
due to tidal forces for a fixed x1, i.e.,
|
|||
|
|
|
|||
|
|
p(q) ≡ p2(x2)d3x2
|
|||
|
|
|
|||
|
|
d3q = p2(x1 + M−1q/t)
|
|||
|
|
|
|||
|
|
t3 det M
|
|||
|
|
.
|
|||
|
|
(A8)
|
|||
|
|
|
|||
|
|
Fourier transforming this expression and substituting the
|
|||
|
|
result into equation (A7), we recover equation (A5) up
|
|||
|
|
to a phase factor.
|
|||
|
|
Perhaps the simplest way to understand all these re-
|
|||
|
|
sults is in terms of Wigner functions [79]. If W(x1, p1) is
|
|||
|
|
the Wigner phase space distribution for the object parti-
|
|||
|
|
cle, then any of the momentum-transferring interactions
|
|||
|
|
that we have considered will take the form
|
|||
|
|
|
|||
|
|
W(x1, p1) �→
|
|||
|
|
�
|
|||
|
|
W(x1, p1 − q)p(q, x1)d3q
|
|||
|
|
(A9)
|
|||
|
|
|
|||
|
|
for some probability distribution p that may or may not
|
|||
|
|
depend on x1. Since the density matrix
|
|||
|
|
|
|||
|
|
ρ1(x1, x′
|
|||
|
|
1) =
|
|||
|
|
�
|
|||
|
|
W
|
|||
|
|
�x1 + x′
|
|||
|
|
1
|
|||
|
|
|
|||
|
|
2
|
|||
|
|
, p
|
|||
|
|
�
|
|||
|
|
e−i(x−x′)·pd3p
|
|||
|
|
(A10)
|
|||
|
|
|
|||
|
|
is just the Wigner function Fourier transformed in the
|
|||
|
|
momentum direction (and rotated by 45◦), the convolu-
|
|||
|
|
tion with p in equation (A9) reduces to a simple multi-
|
|||
|
|
plication with �p in equation (A7).
|
|||
|
|
|
|||
|
|
13
|
|||
|
|
|
|||
|
|
|
|||
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|
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|
|
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|
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|
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14
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15
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|||
|
|
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|||
|
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|