Content-Type: multipart/mixed; boundary="-------------1611240625855" This is a multi-part message in MIME format. ---------------1611240625855 Content-Type: text/plain; name="16-92.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="16-92.keywords" Blockchain, Bitcoin, Proof-of-Work, quantum mechanics, Heisenberg Uncertainty Principle, neuroscience ---------------1611240625855 Content-Type: application/x-tex; name="Heisenberg.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Heisenberg.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% This is the LaTeX2e file for %% Blockchain time and Heisenberg Uncertainty Principle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \date{14th November, 2016} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{amsart} \usepackage{latexsym,amsmath,amsfonts,amscd,amssymb} \usepackage{graphics} \textwidth 6in \oddsidemargin.2in \evensidemargin.2in \parskip.2cm \textheight20cm \baselineskip.6cm \newtheorem{theorem}{Theorem} %[subsection] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} %\theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{counterexample}[theorem]{Counter-example} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} %\numberwithin{equation}{section} %\renewcommand{\thesubsection}{\textbf{\Alph{subsection}}}Dirichlet series \renewcommand{\thesubsection}{\textbf{\arabic{subsection}}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\too}{\longrightarrow} \newcommand{\surj}{\twoheadrightarrow} \newcommand{\inc}{\hookrightarrow} \newcommand{\bd}{\partial} \newcommand{\x}{\times} \newcommand{\ox}{\otimes} \newcommand{\iso}{\cong} \newcommand{\isom}{\stackrel{\simeq}{\too}} \newcommand{\quism}{\stackrel{\sim}{\too}} \newcommand{\Arg}{\text{Arg}} \newcommand{\Res}{\text{Res}} \newcommand{\unit}{\textbf{1}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\CP}{{\mathbb C \mathbb P}} \newcommand{\Map}{\operatorname{Map}} \newcommand{\End}{\operatorname{End}} \newcommand{\Diff}{\operatorname{Diff}} \newcommand{\Div}{\operatorname{Div}} \newcommand{\Hilb}{\operatorname{Hilb}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\GL}{\operatorname{GL}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Int}{\operatorname{Int}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\ind}{\operatorname{ind}} \newcommand{\PD}{\operatorname{PD}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\torsion}{\operatorname{torsion}} \newcommand{\im}{\operatorname{im}} \newcommand{\id}{\operatorname{id}} \newcommand{\Hol}{\operatorname{Hol}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\length}{\operatorname{length}} \newcommand{\VarC}{\mathrm{Var}_\CC} \newcommand{\mhs}{\mathfrak{m}\mathfrak{h}\mathfrak{s}} \newcommand{\Gr}{\mathrm{Gr}} \newcommand{\hs}{\mathfrak{h}\mathfrak{s}} \newcommand{\bn}{\mathbf{n}} \newcommand{\ba}{\mathbf{a}} \newcommand{\Jac}{\operatorname{Jac}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\Ker}{\operatorname{Ker}} \newcommand{\cA}{{\mathcal A}} \newcommand{\cB}{{\mathcal B}} \newcommand{\cC}{{\mathcal C}} \newcommand{\cD}{{\mathcal D}} \newcommand{\cE}{{\mathcal E}} \newcommand{\cF}{{\mathcal F}} \newcommand{\cG}{{\mathcal G}} \newcommand{\cH}{{\mathcal H}} \newcommand{\cM}{{\mathcal M}} \newcommand{\cI}{{\mathcal I}} \newcommand{\cJ}{{\mathcal J}} \newcommand{\cK}{{\mathcal K}} \newcommand{\cN}{{\mathcal N}} \newcommand{\cL}{{\mathcal L}} \newcommand{\cO}{{\mathcal O}} \newcommand{\cP}{{\mathcal P}} \newcommand{\cR}{{\mathcal R}} \newcommand{\cS}{{\mathcal S}} \newcommand{\cT}{{\mathcal T}} \newcommand{\cU}{{\mathcal U}} \newcommand{\cV}{{\mathcal V}} \newcommand{\cX}{{\mathcal X}} \newcommand{\cY}{{\mathcal Y}} \newcommand{\cW}{{\mathcal W}} \newcommand{\cZ}{{\mathcal Z}} %\renewcommand{\AA}{{\mathbb A}} \newcommand{\CC}{{\mathbb C}} \newcommand{\DD}{{\mathbb D}} \newcommand{\HH}{{\mathbb H}} \newcommand{\NN}{{\mathbb N}} \newcommand{\PP}{{\mathbb P}} \newcommand{\LL}{{\mathbb L}} \newcommand{\QQ}{{\mathbb Q}} \newcommand{\RR}{{\mathbb R}} \newcommand{\TT}{{\mathbb T}} \renewcommand{\SS}{{\mathbb S}} \newcommand{\ZZ}{{\mathbb Z}} \renewcommand{\a}{\alpha} \renewcommand{\b}{\beta} \renewcommand{\d}{\delta} \newcommand{\g}{\gamma} \newcommand{\e}{\varepsilon} \newcommand{\eps}{\epsilon} \newcommand{\f}{\epsilon} \newcommand{\h}{\theta} \renewcommand{\l}{\lambda} \renewcommand{\k}{\kappa} \newcommand{\s}{\sigma} \newcommand{\m}{\mu} \newcommand{\n}{\nu} \renewcommand{\o}{\omega} \newcommand{\p}{\phi} \newcommand{\q}{\psi} \renewcommand{\t}{\tau} \newcommand{\z}{\zeta} \newcommand{\G}{\Gamma} \renewcommand{\O}{\Omega} \renewcommand{\S}{\Sigma} \newcommand{\D}{\Delta} \renewcommand{\L}{\Lambda} \renewcommand{\P}{\Phi} \newcommand{\Q}{\Psi} \newcommand{\fro}{{\mathfrak{o}}} \newcommand{\frs}{{\mathfrak{s}}} \newcommand{\frl}{{\mathfrak{c}}} \newcommand{\fru}{{\mathfrak{u}}} \newcommand{\frg}{{\mathfrak{g}}} \newcommand{\frM}{{\mathfrak{M}}} \newcommand{\frm}{{\mathfrak{m}}} \newcommand{\frh}{{\mathfrak{h}}} %\newcommand{\frs}{{\mathfrak{s}}} \newcommand{\bk}{{\mathbf{k}}} \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}})