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Blockchain, Bitcoin, Proof-of-Work, quantum mechanics, Heisenberg Uncertainty Principle, neuroscience
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\date{14th November, 2016}
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\title[Blockchain time and Heisenberg Uncertainty Principle]{Blockchain time and Heisenberg Uncertainty Principle}
\subjclass[2010]{Primary: 91B55, 91B82, 91B80, 81S99, 92C20.}
\keywords{Blockchain, Bitcoin, Proof-of-Work, quantum mechanics, Heisenberg Uncertainty Principle, neuroscience.}
\author[R. P\'{e}rez Marco]{Ricardo P\'{e}rez Marco}
\address{CNRS, IMJ-PRG, Labex R\'efi\footnote {\tiny This work was completed through the Laboratory
of Excellence on Financial Regulation (Labex ReFi)
supported by PRES heSam by the reference ANR10LABX0095. It benefited from a French government
support managed by the National Research Agency (ANR) within the project Investissements d'Avenir
Paris Nouveaux Mondes (investments for the future Paris New Worlds) under the reference ANR11IDEX000602.}
, Labex MME-DDII, Paris, France}
\normalsize\email{ricardo.perez.marco@gmail.com}
%\thanks{}
\begin{document}
\begin{abstract}
We observe that the definition of time as the internal blockchain time of a network based on a Proof-of-Work
implies Heisenberg Uncertainty Principle between time and energy.
\end{abstract}
\maketitle
%\noindent \emph{We dedicate this article to }
\section{Introduction.}
The role of time in Physics remains mysterious. A proper and unified formalization of time (and of observer's
time) is lacking in modern physical theories.
In General Relativity time has a geometric meaning as the fourth coordinate in the $3+1$ Lorenzian spacetime.
The status of time in Quantum Theory is uncertain and subject to controversies. A fundamental observation by
W. Pauli \cite{P} is that there is no well behaved observable operator representing time, thus it is not an observable
in the classical sense. There are indeed various
interpretations of time. One can consult the classical references \cite{M}, \cite{VN}, and
\cite{M1} \cite{M2} for more information and an updated bibliography.
\medskip
Time in Quantum Mechanics is not just another spacetime coordinate as is particularly visible in Heisenberg
Uncertainty Principle \cite{H}. Usually stated for the standard deviation of corresponding
canonical Hamiltonian variables, as position and momentum,
$$
\Delta q \, .\, \Delta p \sim \hslash \ ,
$$
where $\hslash$ is the reduced Planck constant.%\footnote{$\hslash = 1.05\ldots 10^{-34} J.s$}.
We have sometimes a similar relation between energy and time,
$$
\Delta t \, .\, \Delta E \sim \hslash \ ,
$$
but, as is often observed (see \cite{MT}, \cite{H}), this is usually proved in Quantum
Mechanics in situations where $t$ is a proxy
for another canonical Hamiltonian variable. There is no general proof of this type of uncertainty relation
since time does not appear as an observable. The natural result in Quantum Mechanics is the Mandelstam-Tamm
inequality for an observable $R$ which states that
$$
\tau_R \, .\, \Delta E \geq \hslash/2 \ ,
$$
where $\tau_R$ is the characteristic time variation of $R$,
$$
\tau_R =\frac{\Delta R}{\left |\frac{d }{dt}\right |} \ .
$$
Other more general interpretations have been proposed of time and energy uncertainty relation
in general Quantum Systems, as stated by J. Von Neumann in \cite{VN} (p. 353): If we want to measure
the energy of a system with precision $\Delta E$ we need a minimal time $\Delta t$ and
$$
\Delta t \, .\, \Delta E \sim \hslash \ .
$$
Some criticisms and controversy surround this interpretation, as for instance the one in \cite{H}
assuming the that no minimum time would be necessary for measurements of observables in
Quantum Systems. An example of this is given by Aharonov-Bohm energy measurement model \cite{AB}. However,
more recently, Aharonov and Reznik \cite{AR} reviewed the result when the time measurement is made internally, with an
internal time. Then the uncertainty of the internal clock provides the time-energy Uncertainty Relation,
exactly as in the situation considered here with the ``blockchain time'' defined in this article.
A nice account of this research and more information about quantum clocks can be found in \cite{B}.
\medskip
For all these reasons, we believe that it is not without interest to have some non-standard models for time
that shed some light on these problems and the nature of time, energy, and their Heisenberg Uncertainty Relation.
\section{Bitcoin network.}
On January 9th 2009 the Bitcoin network started operating as the first decentralized
peer-to-peer (P2P) payment network, using bitcoin as the virtual currency.
The protocol was presented by an anonymous author (or group of authors) by the name of
Satoshi Nakamoto in the paper
\cite{N} \textit{``Bitcoin: A peer-to-peer electronic cash system''}. The
protocol relies on a major breakthrought:
The first \textit{Decentralized Consensus Protocol} (DCP): An open group of anonymous
and unrelated individuals can reach honest consensus
if a majority of the resources are provided by honest participants\footnote{We don't use nor give
here a precise definition for ``consensus'', as for example exists in the theory of Distributed
Systems. What we mean by ``consensus'' is the empirically observed agreement of the participants in the network,
that allows a ``trust system'' to function. Very much in the Quantum Theory spirit, the ``consensus'' reached
in the Bitcoin network is not deterministic but probabilistic, with certainty improving with time.}
\medskip
The DCP is made possible by a web of nodes interacting P2P via
communication channels through the Internet. Nodes in the network are
constantly synchronizing between themselves. The protocol requires computational power, thus
energy, to function properly. For a quick introduction to Bitcoin protocol we refer to \cite{PM1}.
\medskip
\section{Blockchain time.}
\medskip
A remarkable consequence of the protocol
is the creation of a proper internal chronology to the network. All bitcoin
transactions are recorded on a cryptographically secured database called \textit{the blockchain}. This database
is regularly updated by the DCP by the validation of new blocks of transactions. Each new validated
block provides a ``tick" of the internal clock. Since the blockchain is untamperable and unfalsifiable,
this clock is a universal untamperable and unfalsifiable clock with a precision of the order of magnitude of the
time it takes to validate one new block. The probability to alter the blockchain chronology decreases
exponentially with the number of validations \cite{N}.
\medskip
\medskip
Moreover, the precision of the internal
clock is directly related to the average validation time $\Delta t$ between blocks.
If the latency $\tau_0$ of synchronization
of the network is negligible compared to $\Delta t$, $\tau_0 << \Delta t$, then $\Delta t$ is directly related to
hashrate of the network and the difficulty set by the Proof-of-Work.
\section{Proof-of-Work.}
\medskip
The DCP used by the bitcoin protocol is based on a \textit{Proof-of-Work} (PoW) that needs an external
input of energy. The \textit{Thermodynamic Conjecture} states that this should be necessary in fairly
general conditions, as it follows from general physical thermodynamical principles (see \cite{PM2}).
\medskip
The proof of work consists in iterating hashes of the block header of the block in course of validation by
some particular nodes of the network (the miners).
More precisely he computes $hash({\hbox{\rm HEADER}})$ where $hash(x)=\hbox{\rm {SHA256}}(\hbox{\rm {SHA256}}(x))$ where
${\hbox{\rm HEADER}}$ in the block header with a varying nonce. The goal is to find a nonce for which
$hash({\hbox{\rm HEADER}})