einstein had a dream .he believed quantum mechanics was an incomplete description of reality and that its completion might explain the troublesome fundamental probabilities of quantum mechanics as emerging from some hidden degrees of freedom : probabilities would arise because of our ignorance of these `` hidden variables '' .his dream was that probabilities in quantum mechanics might turn out to have the same meaning as probabilities in classical thermodynamics , where they refer to our ignorance of the microscopic degrees of freedom ( e.g. the position and velocity of each gas molecule ) : he wrote , `` the statistical quantum theory would , within the framework of future physics , take an approximately analogous position to the statistical mechanics within the framework of classical mechanics ''  .  a decade after einstein s death , john bell shattered this dream : any completion of quantum mechanics with hidden variables would be incompatible with relativistic causality !the essence of bell s theorem is that quantum mechanical probabilities can not arise from the ignorance of _ local _ pre - existing variables . in other words ,if we want to assign pre - existing ( but hidden ) properties to explain probabilities in quantum measurements , these properties must be non - local : an agent with access to the non - local variables could transmit information instantly to a distant location , thus violating relativistic causality and awakening the nastiest temporal paradoxes .[ it is important to emphasize that we use `` local '' here in einstein s connotation : locality implies superluminal communication is impossible . in contrast , often quantum mechanics is deemed `` non - local '' in the sense that correlations among properties can propagate instantly , thanks to entanglement .this ` quantum non - locality ' can not be used to transfer information instantly as correlations can not be used to that aim . in the remainder of the paperwe will only use the former meaning of locality ( einstein non - locality ) and we warn the reader not to confuse it with the latter ( quantum non - locality ) . ]modern formulations of quantum mechanics must incorporate bell s result at their core : either they refuse the idea that measurements uncover pre - existing properties , or they must make use of non - local properties . in the latter case , they must also introduce some censorship mechanism to prevent the use of hidden variables to transmit information .an example of the first formulation is the conventional copenhagen interpretation of quantum mechanics , which ( thanks to complementarity ) states that the properties arise from the interaction between the quantum system and the measurement apparatus , they are not pre - existing : `` unperformed experiments have no results '' .an example of the second formulation is the `` de broglie - bohm interpretation '' of quantum mechanics that assumes that particle trajectories are hidden variables ( they exist independently of position measurements ) .bell s result is at the core of modern quantum mechanics , as it elucidates the theory s precarious co - existence with relativistic causality .it has spawned an impressive amount of research .however , it is often ignored in basic quantum mechanics courses since traditional proofs of bell s theorem are rather cumbersome and often overburdened by philosophical considerations . herewe give an extremely simple graphical proof of mermin s version of bell s theorem .the simplicity of the proof is key to clarifying all the theorem s assumptions , the identification of which generated a large debate in the literature ( e.g. see ) .here we focus on simplifying of the proof .we refer the reader that wants to gain an intuition of the quantum part to refs . , and to for a proof without probabilities .let us define `` local '' a theory where the outcomes of an experiment on a system are independent of the actions performed on a different system which has no causal connection with the first .[ as stated previously , this refers to locality in einstein s connotation of the word : the outcomes of the experiment can not be used to receive information from whoever acts on the second system , if it has no causal connection to the first . ] for example , the temperature of my room is independent on whether you choose to wear a purple tie today .einstein s relativity provides a stringent condition for causal connections : if two events are outside their respective light cones , there can not be any causal connection among them .let us define `` counterfactual - definite '' a theory whose experiments uncover properties that are pre - existing .in other words , in a counterfactual - definite theory it is meaningful to assign a property to a system ( e.g. the position of an electron ) independently of whether the measurement of such property is carried out .[ sometime this counterfactual definiteness property is also called `` realism '' , but it is best to avoid such philosophically laden term to avoid misconceptions . ]  bell s theorem can be phrased as `` quantum mechanics can not be both local and counterfactual - definite '' . a logically equivalent way of stating it is `` quantum mechanics is either non - local or non counterfactual - definite '' .  to prove this theorem, bell provided an inequality ( referring to correlations of measurement results ) that is satisfied by all theories that are both local _ and _ counterfactual - definite .he then showed that quantum mechanics violates this inequality , and hence can not be local and counterfactual - definite .it is important to note that the bell inequality can be derived also using weaker hypotheses than `` einstein locality '' and `` counterfactual definiteness '' : such a proof is presented in appendix a ( where einstein locality is relaxed to `` bell locality '' and counterfactual definiteness is relaxed to `` hidden variable models '' ) .however , from a physical point of view , the big impact of bell s theorem is to prove the incompatibility of quantum mechanics with local counterfactual - definite properties , and we will stick to these hypotheses in the main text ( see also appendix b for a schematic formalization of all these results ) .  a couple of additional hypothesis at the basis of bell s theorem are often left implicit : ( 1 ) our choice of which experiment to perform must be independent of the properties of the object to be measured ( technically , `` freedom of choice '' or `` no super - determinism '' ) : e.g. , if we decided to measure the color of red objects only , we would falsely conclude that all objects are red ; ( 2 ) future outcomes of the experiment must not influence which apparatus settings were previously chosen ( whereas clearly the apparatus settings will influence the outcomes ) : a trivial causality requirement ( technically , `` measurement independence '' ) . these two hypothesisare usually left implicit because science would be impossible without them .all experiments performed to date ( e.g.  ) have shown that bell inequalities are violated , suggesting that our world can not be both local and counterfactual - definite .however , it should be noted that no experiment up to now has been able to test bell inequalities rigorously , because additional assumptions are required to take care of experimental imperfections .these assumptions are all quite reasonable , so that only conspiratorial alternatives to quantum mechanics have yet to be ruled out ( where experimental imperfections are fine - tuned to the properties of the objects , namely they violate the `` freedom of choice '' ) . in the next couple of yearsthe definitive bell inequality experiment will be performed : many research groups worldwide are actively pursuing it .we use the bell inequality proposed by preskill , following mermin s suggestion .suppose we have two identical objects , namely they have the same properties .suppose also that these properties are predetermined ( counterfactual definiteness ) and not generated by their measurement , and that the determination of the properties of one object will not influence any property of the other object ( locality ) .we will only need three properties , , and that can each take two values : `` 0 '' and `` 1 '' .for example , if the objects are coins , then might mean that the coin is gold and that the coin is copper ( property , material ) , means the coin is shiny and it is dull ( property , texture ) , and means the coin is large and it is small ( property , size ) .suppose i do not know the properties because the two coins are a gift in two wrapped boxes : i only know the gift is two identical coins , but i do not know whether they are two gold , shiny , small coins or two copper , shiny , large coins or two gold , dull , large coins , etc .i do know that the properties `` exist '' ( namely , they are counterfactual - definite and predetermined even if i can not see them directly ) and they are local ( namely , acting on one box will not change any property of the coin in the other box : the properties refer separately to each coin ) .these are quite reasonable assumptions for two coins !my ignorance of the properties is expressed through probabilities that represent either my expectation of finding a property ( bayesian view ) , or the result of performing many repeated experiments with boxes and coins and averaging over some possibly hidden variable , typically indicated with the letter  , that determines the property ( frequentist view ) .for example , i might say the gift bearer will give me two gold coins with a probability ( he is stingy , but not always ) .bell s inequality refers to the correlation among measurement outcomes of the properties : call the probability that the properties of the first object and of the second are the same : and are both 0 ( the first coin is gold and the second is shiny ) or they are both 1 ( the first is copper and the second is dull ) . for example , tells me that with chance ( namely they are both 0 or both 1 ) .since the two coins have equal counterfactual - definite properties , this also implies that with chance i get two gold shiny coins or two copper dull coins . note that the fact that the two coins have the same properties means that : if one is made of gold , also the other one will be , or if one is made of copper , also the other one will be , etc ._ bell s inequality : _ under the conditions that three arbitrary two - valued properties , , satisfy counterfactual definiteness and locality , and that for ( i.e. the two objects have same properties ) , the following inequality among correlations holds , namely , a bell inequality .the proof of such inequality is given graphically in fig .[ f : p ] .the inequality basically says that the sum of the probabilities that the two properties are the same if i consider respectively and , and , and and must be larger than one .this is intuitively clear : since the two coins have the same properties , the sum of the probabilities that the coins are gold and shiny , copper and dull , gold and large , copper and small , shiny and small , dull and large is greater than one : all the combinations have been counted , possibly more than once .[ in fig .[ f : a ] the events to which the probabilities represented by the venn diagrams of fig .[ f : p ] refer are made explicit . ]= .7  this is true , of course , only if the two objects have same counterfactual - definite properties and the measurement of one does not affect the outcome of the other . if we lack counterfactual - definite properties , we can not infer that the first coin is shiny only because we measured the second to be shiny , even if we know that the two coins have the same properties : without counterfactual definiteness , we can not even speak of the first coin s texture unless we measure it . moreover , if a measurement of the second coin s texture can change the one of the first coin ( non - locality ) again we can not infer the first coin s texture from a measurement of the second : even if we know that the initial texture of the coins was the same , the measurement on the second may change such property of the first . both the `` counterfactual definiteness '' and the `` einstein locality '' hypotheses we used here can be relaxed somewhat , as shown in appendix a ( suggested only to more advanced readers ) .  to prove bell s theorem , we now provide a quantum system that violates the above inequality .consider two two - level systems ( qubits ) in the joint entangled state , and consider the two - valued properties , , and obtained by projecting the qubit on the states where it is easy to check that is orthogonal to and is orthogonal to .it is also easy to check that so that the two qubits have the same properties , namely : the measurement of the same property on both qubits always yields the same outcome , both 0 or both 1 .we are now ready to calculate the quantity on the left of bell s inequality .just write the state in terms of the eigenstates of the properties , and .e.g. , it is easy to find the value of if we write in fact , the probability of obtaining zero for both properties is the square modulus of the coefficient of , namely , while the probability of obtaining one for both is the square modulus of the coefficient of , again .hence , .analogously , we find that and that by expressing the state as summarizing , we have found which violates bell s inequality .this proves bell s theorem : all theories that are both local and counterfactual - definite must satisfy inequality which is violated by quantum mechanics .then , quantum mechanics can not be a local counterfactual - definite theory : it must either be non - counterfactual - definite ( as in the copenhagen interpretation ) or non - local ( as in the de broglie - bohm interpretation ) .i acknowledge g.m .dariano , g. introzzi , p. perinotti , and w.h .zurek for interesting discussions and/or useful feedback .in particular , p. perinotti suggested the appendix .this appendix is addressed only to more advanced readers . in the spirit of the original proof of bell s theorem , one can relax both the `` counterfactual definiteness '' and the `` einstein locality '' hypotheses somewhat .in fact , instead of supposing that there are some pre - existing properties of the objects ( counterfactual definiteness ) , we can suppose that the properties are not completely pre - determined , but that a hidden variable exists and the properties have a probability distribution that is a function of .the `` hidden variable model '' hypothesis is weaker than counterfactual definiteness : if the properties are pre - existing , then their probability distribution in is trivial : there is a value of that determines uniquely the property , e.g. a value such that the probability and hence , namely it is certain that property for object has value for .we can also relax the `` einstein locality '' hypothesis , by simply requiring that the probability distributions of measurement outcomes factorize ( `` bell locality '' ) . call the probability distribution ( due to the hidden variable model ) that the measurement of the property on the first object gives result and the measurement of on the second gives , where denote the three two - valued properties , , and . by definition , `` bell locality '' is the property that the probability distributions of the properties of the two objects factorize , namely the factorization of the probability means that the probability of seeing some value of the property for object 1 is independent of which property one chooses to measure and what result one obtains on object 2 ( and viceversa ) .the `` bell locality '' condition is implied by ( and , hence , it is weaker than ) einstein locality .in fact , einstein locality implies that the measurement outcomes at one system can not be influenced by the choice of which property is measured on a second , distant , system .so , the probability of the outcomes of the first system must be independent of the choice of the measured property of the second system , namely .the same reasoning applies to the second system , which leads to condition .following , we now show that a bell - local , hidden variable model together with the request that the two systems can have identical properties , implies counterfactual definiteness .this means that we can replace `` counterfactual definiteness '' with `` hidden variable model '' in the above proof of bell theorem , which , with these relaxed hypothesis states that `` no local hidden variable model can represent quantum mechanics '' .  if two objects have the same property , then , namely the probability that a measurement of the same property on the two objects gives opposite results ( say , and ) is null . in formulas , where the emphasizes that we are averaging over the hidden variables ( since they are hidden ) : is the probability distribution of the hidden variable in the initial ( joint ) state of the two systems .note that in eq .we are measuring the same property on both objects but we are looking for the probability of obtaining opposite results . using the bell locality condition the probability factorizes , namely eq .becomes since , , and are probabilities , they must be positive .consider the values of for which : the above sum can be null _ only if _either or is null .namely if ( which implies that has the predetermined value ) or if ( which means that has predetermined value ) : we remind that counterfactual definiteness means that is either 0 or 1 : it is equal to 0 if the property of the object does not have the value , and it is equal to 1 if it does have the value .we have , hence , shown that eq .implies counterfactual definiteness for property : its value is predetermined for one of the two objects .  summarizing ,if we assume that a bell - local hidden variable model admits two objects that have the same values of their properties , then we can prove counterfactual definiteness .this means that we can relax the `` counterfactual definiteness '' and `` einstein locality '' hypotheses in the proof of the bell theorem , replacing it with the `` existence of a hidden variable model '' and with `` bell locality '' respectively , so that the bell theorem takes the meaning that `` no bell - local hidden variable model can describe quantum mechanics '' [ the hypothesis that two objects can have the same values for the properties is implicit in the fact that such objects exist in quantum mechanics , see eq . ] .namely , if we want to use a hidden variable model to describe quantum mechanics ( as in the de broglie - bohm interpretation ) , such model must violate bell locality .otherwise , if we want to maintain bell locality , we can not use a hidden variable model ( as in the copenhagen interpretation ) .we have given two different proofs of the bell inequality based on different hypotheses . in this appendixwe summarize the logic behind the bell inequality proofs .* `` counterfactual definiteness '' .* `` einstein locality '' . * `` no super - determinism '' * `` measurement independence '' * `` hidden variable model '' , implied by ( a ) and by the fact that systems with same properties exist ( see appendix a ) . * `` bell locality '' , implied by ( b ) ( see appendix a ) .  in the main text we have proven ( fig .[ f : p ] ) the following theorem : + ` ( a ) and ( b ) and ( c ) and ( d ) \rightarrow bell inequality \rightarrow not qm ` , + where with `` ` not qm ` '' we mean that quantum mechanics ( qm ) violates the bell inequality and is , hence , incompatible with it . usingthe fact that `` ` x and y \rightarrow not z ` '' is equivalent to `` ` z \rightarrow not x or not y ` '' ( _ modus tollens _ ) , we can state the above theorem equivalently as ` qm \rightarrow not ( a ) or not ( b ) or not ( c ) or not ( d ) ` . since one typically assumes that both ( c ) and ( d ) are true , they can be dropped and the theorem can be written more compactly as + ` qm \rightarrow not ( a ) or not ( b ) . `+ namely , ( assuming `` no super - determinism '' and `` measurement independence '' ) ` quantum mechanics implies that either counterfactual definiteness or einstein locality must be dropped ` .this is the most important legacy of bell .we have also seen that the hypotheses ( a ) and ( b ) can be weakened somewhat , so that the bell inequality can also be derived using only ( a ) and ( b ) .namely , we can prove ( see appendix a ) : + ` ( a ) and ( b ) and ( c ) and ( d ) \rightarrow bell inequality \rightarrow not qm ` .+ namely , ( assuming `` no super - determinism '' and `` measurement independence '' ) quantum mechanics is incompatible with bell - local hidden variable models .a. einstein , b. podolsky , n. rosen , _ can quantum - mechanical description of physical reality be considered complete ? _ , phys .rev . * 47 * , 777 ( 1935 ) .a. einstein , in _a. einstein , philosopher - scientist _, ed . by p.a .schilpp , library of living philosophers , evanston ( 1949 ) , pg . 671 .j. s. bell , _ on the einstein podolsky rosen paradox _, physics * 1 * , 195 ( 1964 ) ; bell j. s. , _ on the problem of hidden variables in quantum mechanics _ , rev .phys . * 38 * , 447 ( 1966 ) .j. s. bell , _ speakable and unspeakable in quantum mechanics _ ( cambridge univ . press , cambridge , 1987 ) .mermin , _ hidden variables and the two theorems of john bell _ , rev .phys . * 65 * , 803 ( 1993 ) .`` bell s telephone '' in r. werner , _ quantum information theory an invitation _ , springer tracts in modern physics * 173 * , 14 , ( 2001 ) , available from http://arxiv.org/pdf/quant-ph/0101061 .a. peres , _ unperformed experiments have no results _ , am . j. phys . *46 * , 745 ( 1978 ) .mermin , _ bringing home the atomic world : quantum , mysteries for anybody _ , am .* 49 * , 940 ( 1981 ) .j. preskill , lecture notes at http://www.theory.caltech.edu/people/preskill/ph229 .g. auletta , _ foundations and interpretation of quantum mechanics _( world scientific , singapore , 2000 ) . p.g .kwiat , l. hardy , _ the mystery of the quantum cakes _* 68 * , 33 ( 2000 ) .kurt jacobs and howard m. wiseman , _ an entangled web of crime : bell s theorem as a short story _, am . j. phys .* 73 * , 932 ( 2005 ) .a.l . macdonald , _ comment on " resolution of the einstein - podolsky - rosen and bell paradoxes _ , phys .49 * , 1215 ( 1982 ) .a. peres , _ existence of `` free will '' as a problem of physics , _ found .* 16 * , 573 ( 1986 ) .stapp , _ s - matrix interpretation of quantum theory , _ phys .d * 3 * , 1303 ( 1971 ) ; _ bell s theorem and world process _, nuovo cimento * 29b * , 270 ( 1975 ) ; _ are superluminal connections necessary ? _ , nuovo cimento * 40b * , 191 ( 1977 ) ; _ locality and reality , _ found .* 10 * , 767 ( 1980 ) ; w. de baere , _ on some consequences of the breakdown of counterfactual definiteness in the quantum world , _ fortschr . phys . * 46 * , 843 ( 1998 ) . y. aharonov , a. botero , m. scully , _ locality or non - locality in quantum mechanics : hidden variables without `` spooky action - at - a - distance '' _ , z. natureforsh .a * 56 * , 5 ( 2001 ) .a. aspect , p. grangier , g. roger , _ experimental realization of einstein - podolsky - rosen - bohm gedankenexperiment : a new violation of bell s inequalities _ , phys .* 49 * , 91 ( 1982 ) .kwiat , e. waks , a.g .white , i. appelbaum , p.h .eberhard , _ ultrabright source of polarization - entangled photons _a * 60 * , r773 ( 1999 ) .m. giustina , a. mech , s. ramelow , b. wittmann , j. kofler , j. beyer , a. lita , b. calkins , t. gerrits , s. woo nam , r. ursin , a. zeilinger , _ bell violation with entangled photons , free of the fair - sampling assumption _ , nature * 497 * , 227 - 230 ( 2013 ) .p.m. pearle , _ hidden - variable example based upon data rejection _ , phys .d * 2 * , 1418 ( 1970 ) .j.s . bell , _ atomic - cascade photons and quantum - mechanical nonlocality _ , comments atom .* 9 * , 121 ( 1980 ) ; reprinted as chap . 13 of . g. ghirardi , _ on a recent proof of nonlocality without inequalities _ , found .phys . * 41 * , 1309 ( 2011 ) , also at arxiv:1101.5252 .t. norsen , _ bell locality and the nonlocal character of nature _ , found .* 19 * , 633 ( 2006 ) .

Abstract: bell s theorem is a fundamental result in quantum mechanics : it discriminates between quantum mechanics and all theories where probabilities in measurement results arise from the ignorance of pre - existing local properties . we give an extremely simple proof of bell s inequality : a single figure suffices . this simplicity may be useful in the unending debate of what exactly the bell inequality means , since the hypothesis at the basis of the proof become extremely transparent . it is also a useful didactic tool , as the bell inequality can be explained in a single intuitive lecture .


the fermionic quantum many - body problem is believed to be in the class of problems whose full solution is exponentially hard .approximate methods have been developed , but many of these are also extremely computationally demanding .there is , therefore , an ongoing search for efficient approximate methods , useful , for example , in computational surveys of wide classes of materials , or to provide a first look at a complicated situation .+  + the predominant approach has been to use simplifying approximations , for example , truncated perturbation - theory series expansions , variants of mean - field theory , quasiclassical approximations , or analytical interpolation functions .the development of machine learning ( ml ) techniques in computer science motivates us to explore a complementary approach .ml provides an estimate of the result of a calculation based on interpolation from a statistical analysis of datasets of solved problems .ml is widely used in many big - data applications , and has recently been proposed as a method for obtaining approximate solutions of the equations of density functional theory ( dft) , of the molecular electronic schrdinger equation  , and of transmission coefficients for electron quantum transport  .ml is also used to construct force - fields from molecular dynamics .+  + in this paper we investigate ml techniques to infer solutions to the quantum many - body problem arising in applications of the dynamical mean - field theory ( dmft ) method. dmft has become widely used in condensed - matter physics and materials science for obtaining nonperturbative information about materials with strong local correlations .while dmft is an approximation to the full many - body problem , it does require the solution of a fully interacting quantum impurity model ( a quantum field theory defined in zero space but one time dimension ) , and accurate solutions require substantial numerical effort which is time consuming even with modern algorithms and hardware .a sufficiently accurate ml model of dmft could provide an inexpensive _ solver _ , useful for rapid preliminary screening of wide ranges of materials and as a method for identifying promising starting points for further refinement using more expensive and sophisticated methods or experiments .+  + in its conventional formulation , dmft maps one function of frequency into another .the input is the _ bare hybridization function _ , which encapsulates relevant information about the crystal structure and quantum chemistry of a material via a representation of what the local density of electronic excitations would be if many - body correlations were neglected .a small number of additional parameters , such as the on - site interaction strength , must also be specified .the output is the electron green s function ( or equivalently self - energy ) , which provides an approximation to the exact density of states ( dos ) obtained by including local effects of many - body correlations .+  + implementing a ml approach to dmft thus entails constructing a training set of physically reasonable hybridization functions , determining the spectral functions corresponding to the training examples , and constructing a model that provides the needed interpolation formula .such a ml procedure goes beyond previous applications of ml to electronic structure because we are mapping a function to a function whereas the ml approaches to dft or the schr equation provide only a small number of scalar outputs , such as the total energy of atomization , ionization potential , or excitation energy .a key issue is thus to devise an efficient representation of the functions of interest in terms of a reasonably sized set of parameters .the first application we foresee for real systems is a material science computation tool to optimize a desired property .+  + in this paper we address this key issue for the anderson impurity model ( aim ) , the archetypical quantum impurity model . for this modelthe input hybridization function is known _ a priori _ , and can be specified by few parameters .the focus therefore lies on the prediction of the output , namely , the electron green s function . in futurework we will discuss ml applications to the full dmft problem of determining the self - consistent relation between the green s function and hybridization function , starting from an arbitrary hybridization function . +  + this paper is organized as follows : in sec .[ ml_supervised ] , we summarize the supervised learning approach . in sec .[ ml_supervised_repres ] we discuss how to represent a function for ml and in sec .[ ml_supervised_kernel ] we discuss the kernel ridge regression that we use in this paper , while in sec .[ quality_ml ] we show how the ml procedure is tested . in sec .[ qi_aim ] the one site aim and its solution by exact diagonalization are presented . in sec .[ qi_ml_aim ] we present the machine learning solution .section [ qi_metho ] discusses the methodology of the calculation .section [ qi_repre_g ] presents in detail the four types of representation for the green s functions we study : in sec .[ qi_repre_g_cont_frac ] the continued fraction , in sec .[ qi_sec_gwn ] the matsubara frequency , in sec .[ qi_sec_gtau ] imaginary time , and in sec .[ qi_sec_gl ] legendre polynomials .the results for these different representations are shown in secs .[ qi_sec_cont_frac_results ] , [ qi_sec_gwn_results ] , [ qi_sec_gtau_results ] and [ qi_sec_gl_results ] . in secs .[ qi_sec_prediction_size ] and [ qi_sec_mass_enh ] we show how the size of the learning set affects the predictions of the dos and the mass enhancement . finally , in sec . [ qi_predic_min_set ]we look at how we can define an absolute minimal learning set for our problem .section [ sum_concl ] is the summary and conclusion .appendix [ ml_appen ] gives details on the kernel ridge regression , appendix [ ed_appen ] gives details of the exact diagonalization method , appendix [ legendre_appen ] gives details of the representation of the green s function using legendre polynomials , and appendix [ appen : eff_alpha ] presents the derivation of the effective matrix for the legendre polynomials approach .we wish to learn a function of one complex variable , , with representing the green s function or self - energy .the model whose solution gives is specified by a descriptor , , the set of input parameters needed to describe the model . where appropriate we denote the dependence of on the input parameters as .the ml approach is to infer from a given set of results .+  + the functions of physical interest have a spectral representation with a real number .the spectral function is non - negative , integrable , and typically nonzero only over a finite range of .+  + while is fully specified by , for technical reasons one often has data only for , or is only interested in , on the _ matsubara frequencies _ where is the temperature , and is an integer . is sometimes also studied on the _ imaginary time _( ) axis in the interval .the -dependent function is related to the matsubara frequency values by it is worth noting that the definition of the matsubara frequencies along with eq.  implies three important results that will be used later : where is the first moment of the spectral function which depends upon the particular hamiltonian .+  + equation  implies that the values of at different values are correlated .this presents a challenge since the machine learning algorithms we will use treat the different components of ( or the coefficients of its expansion ) as independent . therefore , by independently adding numerical errors on each point the machine may not fully respect the needed correlations and constraints . put differently , the predicted by our machine learning algorithm could arise from a spectral function with negative regions .this issue is not unique to ml and is related to the well - known _ analytical continuation _ problem of inverting eq .  to determine from the full set of values of . while general theorems from complex variable theory imply that the inversion is possible in principle , the kernel of eq.  has many very small eigenvalues so the inversion problem is ill conditioned .however , in all the cases we have considered , these issues seem not to pose any difficulties in practice , so we proceed straightforwardly , with the assumptions justified _ a posteriori_. +  + in any numerical approach to problems involving continuous functions , a choice of discretization must be made .any discretization expresses the continuous function in terms of a set of numbers which we assemble into an -dimensional vector : one must note that the s need not be direct points of the function but can also be coefficients that define the function .+  + we have considered four discretizations of : ( 1 ) a continued fraction representation ; ( 2 ) values on the matsubara axis up to some cutoff frequency ; ( 3 ) discrete values on the imaginary time axis , ; and ( 4 ) an expansion of in terms of orthogonal ( legendre ) polynomials .both the continued fraction and legendre polynomials representation lead to a correctly normalized and non - negative spectral function , but as will be seen the continued fraction representation is for other reasons not optimal .the other representations do not necessarily lead to a which has a spectral representation with a non - negative spectral function for small and random sets of results , but , as will be seen , they seem to work well in practice .in this paper we apply the kernel approach , which involves two ingredients : a _ distance kernel _ , a symmetric and positive definite function , fixed _ a priori _ , and a coefficient matrix which is to be determined . in terms of these, one generates an approximation to the desired given by the so - called kernel ridge regression ( krr ) , an expansion in an abstract kernel space . in terms of the expansion defined in eq . )we approximate the components of in terms of approximate components given by here we use the subscript to label the entries in ( i.e. , the different components in the discrete representation of the function : direct points or appropriate coefficients ) and to label the training examples . +  + the coefficients calculated by minimizing a cost function defined in terms of the difference between the approximations and the exactly known , with the addition of a set of lagrange multipliers introduced to regularize the problem . in our approachwe assume that each component of is learned separately and independently , so we define a cost function separately for each as minimization then gives at fixed ( see appendix [ ml_appen ] for details ) in eq . , is a column vector of length containing the different for fixed and is also a column vector of length but containing the different values of ( fixed ) on the training set ( ) .it is important to note that this vector is not the same as the vector in eq .  which is a vector of length containing the values of for one specific .finally , and are the kernel and identity matrices of size .+  + in our actual calculations we assume that the kernel and the lagrange multiplier are independent of , i.e. , the same for all components of the function to be learned. we may then assemble all of the examples into a matrix in which each column is for a different and combine eq .  into a unique matrix equation makes the process of learning a function very efficient since even if is a very large matrix , the solution needs to be obtained only once . for , we use the weighted exponential kernel where is the manhattan distance between the two parameter sets and gives the radius of effect that a particular point of the data set will have in the prediction process . +  + for numerical calculation , direct matrix inversion should be avoided and eq .is solved in the form . since is a square , real , symmetric , and positive definite matrix ,a standard cholesky solver can be used .the process is very fast ; for example , on a desktop computer with a intel core i7 - 4770 quad core at 3.40 ghz , using a dataset of 5000 points , it takes about 4 s to learn 2400 parameters defining in its continued fraction representation plus the ground - state energy .+  + the way we _ train _ our machine is to choose results from a database of solutions .this subset of solutions is called the training set and we keep the remaining solutions in the database apart and consider a subset of them as the test set .the training set is then used to construct and eq .  can be solved .we then use the test set to check if the predictions from the ml give accurate results and thus can be used to predict solutions not contained in the database .more details on the training process we use are given in sec .[ qi_metho ] .+  + as shown in this section , we can see that ml is an interpolation approach , never an extrapolation .the obvious question of how small the shift in parameters of a desired prediction from the must be ( i ) depends on the training set density ( the examples that will be used to obtain the ) , ( ii ) depends on it s distribution , and ( iii ) still has to be investigated in more rigorous ways than is done either this paper or in the literature in general .   in most ml studies ,the quality of the machine is assessed in terms of the mean absolute error ( mae ) of the predicted quantities  . because we are interested in predicting a function we introduce a different metric , the average relative difference ( ard ) , defined as where the factor is introduced so that the ard is given in percentage and denotes the average over a suitably defined set of function arguments . in practicewe take the set of arguments to be the first few matsubara frequencies corresponding to energies up to the typical scales of the problem ; the values at higher are typically controlled by sum rules and are small , with very small errors .the matsubara frequencies are an appropriate measure because ( i ) they contain the relevant physical information , ( ii ) there is a natural discretization , and ( iii ) the use of a common estimator permits straightforward comparison of different approaches .note that three of the four representations ( the continued fraction , the matsubara frequency , and legendre polynomials ) give in a direct way .we only compare .+  + however , the ard or any measure that give an error representation for the entire function as a single number can be misleading , because it can mask narrow but important regions of the function which might be badly predicted .therefore , as a further test of the quality of the machine , we examine in detail , for two specific cases the predictions of the ml for ( a ) ( b ) the density of states obtained by using pad analytical continuation , and ( c ) the low - frequency renormalization factor estimated as the extrapolation this renormalization factor is a unique number characterizing an important low - frequency property of the model .in this paper we apply machine learning techniques to the single impurity anderson impurity model .the hamiltonian is as shown in fig [ fig : aim](a ) , this represents one localized level ( with two - fold spin degeneracy ) , embedded in a bath of noninteracting electrons .the localized electronic states are represented by the creation ( annihilation ) operators ( the spin ) while the continuous bath states are represented by the operators .the on - site energy of the impurity is and the interaction term which penalizes double occupancy if positive , is .the bath dispersion is and the hybridization between the bath and the impurity is , where is a wave number ., the aim can be uniquely described by three parameters  ] . in the case of the green s function in positive imaginary time ,the time interval is so that we may define the variable .thus ( see appendix [ legendre_appen ] ) where the coefficients are formally given by the fourier transform to is given by [ ] as where and are the spherical bessel functions . may be directly measured in a ctqmc calculation and the maximum order is defined as the largest where is greater than the statistical noise . in the present casewe must calculate the and devise an alternative prescription for .we use a recently introduced algorithm based on a fast chebyshev - legendre transform that exploits the idea that smooth functions can be represented by polynomial interpolation in chebyshev points , i.e. , by expansions in chebyshev polynomials using fast fourier transform .this algorithm is implemented in a free matlab toolbox called chebfun .we still have to define .the is chosen by looking at the odd legendre coefficients .for every example we have investigated , we find that for odd decreases rapidly as increases . at some , changes sign and starts oscillating around zero .we set by finding the at which changes sign .for each example ( a particular set of parameters , , and ) , the value of for which the sign change happens is different .however , for the 5000 examples in the database , we found that the first sign change happens for at most around 111 .for security , we use as our definition of the expansion the first 121 ( ) terms . for each example , for for the odd use the values we have while we have to decide what values to use for ( for even we do not change anything ) .two easily implemented options are to replace the odd coefficients by zero for or to replace them by the last value before the first sign change of so that either for odd or .we verified that both solutions work very well to reconstruct from eq .however , for machine learning there is a difference . indeed , in ml, having pure zeros in the learning set makes the learning process much more difficult .we thus use the second solution where for odd is a very small constant .+  + the great advantage of the legendre polynomial representation is that either by obtaining from ctqmc or directly from , the number of coefficients is very limited , making the learning process much smaller , and , if after learning all coefficients at once , we see that some fine tuning is needed , the maximum number of new machines that will be necessary is limited and manageable . also , as we do not directly learn the function but reconstruct it from eq .( or eq . ) it helps smooth things out .+  + therefore what is directly learned is the vector of coefficients . this section we present the results both of computation of the full and , for two particular examples , of the renormalization factor .for the two examples we will use parameters similar to those used in [ ] and [ ] . in one case [ ] , , and . in the second case [ ] , an infinite aim is considered with a flat conduction - band dos with and . in the case of infinite [ ] ,this gives an occupation .we will use the two examples in our database that are the closest to these parameters .for the first case , we have , , and . for the second case , as infinite , we will use , , and .we present the results both in and .for real frequencies , we need to choose a small imaginary part for the frequency .since our ed fitting procedure relies on choosing a fictitious temperature to fit on the matsubara axis defined by this temperature , we take the minimum - energy unit to be twice the difference between two ( ) . for small dataset length , we generated many random combinations for the learning set , predicted the results for every combination and averaged out .we first use ml to learn the coefficients of the continued fraction representation of the green s function ( eq . ) .we learned the first hundred of each type since , as we mentioned in sec .[ qi_aim ] , the remaining coefficients do not contribute to within our accuracy . using test sets of length 100 and generating learning and test sets multiple times , from the reconstructed calculate the ard , and we show the results as a function of training set size in fig . [fig : ard_coeffs ] as dots .we see that the ard decreases from about 6.5% for a random learning set of 500 examples to about 0.016% for a learning set of 4900 .+  +  , squares ( ) denote the predictions of , and diamonds ( ) denote the legendre polynomial expansion . ]  for the comparison with the two specific examples , the results are presented in fig .[ fig : predic_wn_w_coeffs ] .only the first 50 matsubara frequencies are shown for .+  +  for the half - filled case , fig .[ fig : predic_wn_w_coeffs]-(a ) and ( b ) , we see that the prediction for randomly chosen learning sets of 500 in is not very good . in terms of real frequency , we see that , however , even with the size 500 random learning set , the high - frequency regions are well predicted . however , around the fermi level , the prediction is wrong .for example , the prediction is not particle - hole symmetric even though the model is .however , at the largest learning set size ( 4999 ) the prediction is correct .the doped case is similar . from fig .[ fig : predic_wn_w_coeffs](c ) we see that for this particular example , the value at the lowest matsubara frequency is well predicted for a small random learning set but the values at the next three frequencies are not good .turning now to real frequencies we see that the high - frequency regions and the region around the fermi level are qualitatively fairly predicted . for a large learning set ,the doped case is also well represented .+  + thus , if the green s function is learned from its continued fraction coefficients , a small and random learning set is able to capture the high - frequency physics for the half - filled case , but it is not reliable for the low which carry most of the interesting physical information at .the ards are denoted by x s in fig .[ fig : ard_coeffs ] .the values are systematically smaller than the equivalent from the continued fraction coefficients , but still the same order of magnitude .the real frequency result at half filling of fig .[ fig : predic_wn_w_gwn](b ) shows that this time , around the fermi level , the ml predicted dos has the correct qualitative behavior of particle - hole symmetry . if we compare the matsubara frequency results of fig .[ fig : predic_wn_w_gwn](a ) and ( c ) with those of fig .[ fig : predic_wn_w_coeffs](a ) and ( c ) we see that for these two examples , learning directly is better . however , the analytically continued results for the learning set of length 500 are of poor quality , essentially because analytical continuation is sensitive to small errors in .this is thus hard to assess if the prediction really contains noncausality ( fig .[ fig : predic_wn_w_gwn](b ) ) ( ) , gives such shape at high frequency ( fig .[ fig : predic_wn_w_gwn](b ) and ( d ) ) and give spurious states where the dos should be zero ( fig .[ fig : predic_wn_w_gwn](d ) at about ) .once again , for a large learning set , the predicted results are very good .it is important to note that since we need numerical analytical continuation to obtain the real frequency results , 100% perfect matching is impossible .+   the ards for imaginary time representation are denoted by squares in fig .[ fig : ard_coeffs ] .the values are systematically smaller than the corresponding ards from the continued fraction coefficients but still of the same order of magnitude and similar to those from the representation in matsubara frequency .most comments made regarding fig .[ fig : predic_wn_w_gwn ] can also be made for fig .[ fig : predic_wn_w_gtau ] .+  + based on the ard and the results in we can conclude that directly learning is a good procedure and of general applicability .the ards of the legendre orthogonal polynomials representation are denoted by diamonds in fig .[ fig : ard_coeffs ] .the values are very similar to those from and .the results for the two examples in matsubara and real frequency presented in fig .[ fig : predic_wn_w_gl ] are comparable to what is obtained from learning directly , but this time the pad continuation is less problematic because the coefficients are learned and then the green s function is reconstructed , giving a with less independent error on each . moreover , in sec .[ qi_sec_gtau ] , we had to learn 2048 slices of plus while here we only needed to learn 121 coefficients .+  + we may therefore conclude that the representation by an expansion in terms of legendre polynomials is the most efficient way to learn many - body green s functions using machine learning .it is important to show how our prediction of the dos evolves with increasing learning set length .[ fig : ard_coeffs ] displays the learning set length dependence of the .now we look at the dos around the fermi level for the half - filled case .the conclusions we show for the dos around the fermi level are generally applicable to the high - frequency results also .one exception is the continued fraction representation where much smaller learning sets can predict the high - frequency behavior ( see fig .[ fig : predic_wn_w_coeffs ] ) . in fig .[ fig : predic_dos_nd1_4_representations ] , we show the dos around for different random learning set lengths and for the four representations : in fig .[ fig : predic_dos_nd1_4_representations](a ) the continued fraction , in fig .[ fig : predic_dos_nd1_4_representations](b ) the matsubara frequency , in fig .[ fig : predic_dos_nd1_4_representations](c ) imaginary time , and in fig .[ fig : predic_dos_nd1_4_representations](d ) legendre polynomials . the dots ( . )denote the exact result , the red dashed lines ( - - ) denote the result for a learning set of length 500 , the blue dot - dashed lines ( - . )denote the result for a learning set of length 1000 , the cyan circles ( o ) denote the result for a learning set of length 2000 , and the magenta solid lines ( - ) denote the result for a learning set of length 3000 .we can see that convergence to the correct prediction is attained before the learning set is maximum ( 4999 ) .we now consider how well machine learning predicts the renormalization factor , equivalent in this model to the mass enhancement .in addition to being an important physical property for fermi liquids , also acts here as a unique number that can be used to estimate the quality of the ml prediction of low - frequency properties .the results are shown in fig . [fig : z ] .overall , the predicted from ml learned continued fraction coefficients never completely converges ( the black line is the exact result ) and is quite inaccurate for learning set of 500 and 1000 when . even if it never perfectly converges , the relative difference at the larger learning set is small , consistent with the reasonable visual appearance of the dos ( fig [ fig : predic_wn_w_coeffs ] ) . for is a peculiar concordance where the prediction is better for 500 than 1000 and 2000 .this is clearly accidental and the overall shape of the dos around is better for 1000 and 2000 .it is also important to note that for fig .[ fig : z]-(b ) the exact is small and thus we predict very small numbers . for the other three representations , they all systematically converged to the correct answer , around a learning set of 2000 for fig . [fig : z](a ) , which also can be seen from fig . , and around a learning set of 3000 for fig .[ fig : z]-(b ) .again , these minimal converging lengths are for totally random sets of examples as learning sets .   in this sectionwe study how to introduce a selection bias into the learning set such that a new system is predicted as accurately as possible with the smallest learning set possible . because our database is very homogenous ( dense coverage of , , and ) we can ask how good the learning would be if we find the members of the dataset that are closest to the , , and and form the learning set as combinations thereof .since the descriptor has three components and , at most , the new parameters can be between two values of the database , this would give us a maximally localized learning set of minimal size 8 or less .let us once again choose two examples for which we will do the predictions .the first example is , , and .the second example is , , and .+  + for the first case , none of the results in the 5000 database share any of these parameters so that none of the components of the difference between the descriptor of the example and any descriptors of the database is ever zero .this means that the learning set will be of length 8 . in the secondexample , since we look at half - filling and the database contains half - filled results , the learning set size is 4 .this means that we are really looking at how our ml scheme can predict new results .we show the results in fig . [fig : predic_wn_w_gl_min ] using the legendre polynomial representation and we see that with a very small learning set we can predict quite precisely new results .+  +  this possibility of prediction with very small learning enables us to look closely at how the machine itself behaves .indeed , what we call the machine is given by the parameters of eq . .only in the cases of the representations in terms of matsubara frequencies and imaginary time does the really represent the green s function we are trying to predict itself .however , within the legendre polynomial representation , using our approximation that the kernel functions have fixed parameters , we can define an effective -like parameter for the reconstructed green s function in matsubara frequency from the predicted legendre polynomials coefficients .we show the derivation in appendix [ appen : eff_alpha ] , and we obtain where the effective , called , is given by the results are shown in fig .[ fig : alpha_gl ] .the case of , , and is presented in figs .[ fig : alpha_gl](a ) and ( b ) while the case , , and is presented in fig .[ fig : alpha_gl](c ) . for the half - filled example, only the imaginary part of is shown , as the real part of is zero for the particle - hole symmetric case . despite the discrete nature of functions of matsubara frequency, we present the curves as continuous here .it is also interesting to note ( not shown ) that the curves for obtained from the matsubara frequency representation correspond exactly to the curves of in fig .[ fig : alpha_gl ] as expected .one very important result is that is , from quite a low frequency , a smooth function of .this opens up the possibility of using interpolation from curves as a another way to perform very efficient ml .  for completeness , the other representation that directly learns is the imaginary time one and thus is really a representation of in the ml space .we thus show the curve for both examples in fig .[ fig : alpha_tau ] .once again the curves are smooth and in this case fundamentally continuous .we have proposed a machine learning scheme to learn the electron green s function .our method should apply to any other correlation function of interest .we have reduced the problem of learning a function of a single variable into one of learning a relatively small set of independents numbers , either direct slices of the function or coefficients used to define it . for the green s function of the single - site anderson impurity model ,we have tested four different representations of : the continued fraction , the matsubara frequency , imaginary time and legendre polynomial expansion . directly learningthe function in imaginary time is a well - defined operation as long as the learning set is not too small and random .however , replacing the direct function in imaginary time by its legendre polynomial expansion is clearly superior because the need to learn only a small number of coefficients improves the accuracy .this way is even more promising for the context of learning correlation functions for real materials as new powerful ctqmc algorithms directly measure these coefficients .we also observe that the matsubara frequency representation may be problematic for more general dmft calculations . in the aim studied here ,the system is always a metal and thus the dataset of solved problems in is very homogeneous . in the general case of the dmft , where the aim serves as an intermediate problem, there is an interaction driven metal to insulator transition . in matsubara frequency, the green s function at lower frequencies changes qualitatively from the metallic to the insulating phase .this creates a dataset much less homogeneous at low .this is why , even if this representation yields accurate predictions in the present study , we do not think that it can be efficiently used in general for ml+dmft .contrary to , changes much less drastically from a metal to an insulator .therefore , this representation should perform much better for ml . finally ,as we already discussed , the continued fraction representation can only be really used when the aim is solved via ed and thus is a very limited representation .+  + before handling real materials , learning dmft for model hamiltonians is the next logical step .concomitant to this , the next logical step for ml of a function itself is to look for a scheme where learned numbers are not considered to be totally independent .this is highly nontrivial and perhaps could be achieved by adding appropriating new constraints to the minimization problem inherent in kernel ridge regression of eq . .for example , constraints concerning moments could perhaps be integrated in the ml scheme for learning functions that have a spectral representation .this research was supported by the office of science of the u.s .department of energy under subcontract no .l - f.a . thanks ara go for numerous discussions on implementing an exact diagonalization code for the anderson impurity model .we thank peter b. littlewood for discussions and critical reading of the manuscript .this research used resources of the argonne leadership computing facility at argonne national laboratory , which is supported by the office of science of the u.s .doe under contract no.de-ac02-06ch11357 .o.a.v.l acknowledges funding from the swiss national science foundation grant no .ppoop2 _ 138932 .in kernel ridge regression , the coefficients of the expansion in the kernel space of eq .  are found by minimization with respect to of the cost function ( eq . ) the are determined as the solution of the set of equations . by using the definition of given by eq .  and also the fact that the kernel is a symmetric function , we obtain \nonumber \\ & & \times k_m(\mathbf{d}_l,\mathbf{d}_q)= 0 \ ] ] a sufficient condition for eq .  to holdis that the quantity in the square brackets vanishes , i.e. in eq ., is a dummy index . for fixed , we can represent and as vectors and of length given by the number of examples and the kernel as a matrix so that with the identity matrix .in this appendix we describe what is meant by exact diagonalization ( ed ) in the case of the anderson impurity hamiltonian . here , eq .is the target hamiltonian . to solve the problem by ed, we map to a new hamiltonian where the effect of the continuous bath is approximated by a few poles and weights .this gives a hamiltonian with a finite number of site given by where the and are chosen to reproduce as much as possible the effect of eq . .we have now a finite size hamiltonian with a hilbert space of size that can be solved using matrices diagonalization techniques .we use the usual lanczos approach .of course , by using a finite bath , it is impossible to reproduce the continuous case to perfect match . only in the casewhere the number of bath sites was equal to infinity could one recover the continuous case exactly .therefore , the idea proposed by caffarel and krauth is to define a distance function from the matsubara axis representation .note that the matsubara axis representation of a problem requires the use of a fictitious temperature , which we choose as . here, we use the inverse of the noninteracting green s function to define the distance function  is the maximum number of frequencies used to define .it is important that .we typically use . since the bath is fixed with a half bandwidth of 1 , . at an energy corresponds to approximately and thus is large enough .finally , is the inverse of the noninteracting green s function of the hamiltonian of eq .  andis written as we specify the set of parameters as those that minimize .this is a problem of unconstrained optimization in several variables .   for our ( eq . ) with hybridization fixed , the ground state is nondegenerate and always in the sector where and .once the energy and many - body wave function are obtained , the green s function can be calculated from the electron and hole part . it is most convenient to represent as a continued fraction +  + a second lanczos procedure can be used to obtained the coefficients . the algorithm to calculate these coefficients from the ground state is as follows : +  +  * we construct a starting vector * we construct the so - called lanczos space with the recursion relation of orthogonal vectors * it is easy to obtain the coefficients as and with .the equivalent can be done for the and .the are orthogonal by construction if evaluated exactly . with double precision floating point arithmetic and a recursion relation ( eq . )that only forces three vectors to be orthogonal , at some point ( typically after 50 ) the orthogonality will be lost . for the green s function, it does not matter much . if necessary, there are slightly more complicated algorithms that do partial orthogonalization to keep a good orthogonality of our set of vectors .note that for proper numerical calculation eqs ., and must be modified so that the states in eq .are normalized .the legendre polynomials are defined on an interval  $ ] and thus an arbritrary function in can be written as the coefficients are found by multiplying both sides of eq .  by and then integrating over . the orthogonality relation for the legendre polynomialsis used , . hence , we define to be consistent with ref .[ ] . finally , we can also change the integration variable and therefore get the result of eq .  and the expansion of eq . .in the legendre polynomial expansion , the green s function on the matsubara axis is given by eq . in this approach , the ml procedure is for the coefficients and not itself .therefore , using eq  we may write therefore , putting eq .  in eq .we obtain in the most general case , this is as far as we can go .however , in our approach , we have chosen to consider that the parameters of are such that is independent of , i.e. , .this enables us to factor out in eq . and thus obtain we therefore obtain in this approximation a renormalized parameter in the ml expansion .this can be compared with the obtained from the ml directly on .this is not possible for the other two representations since the reconstruction of is done using highly nonlinear relations ( fourier transform and the continued fraction ) .

Abstract: machine learning methods are applied to finding the green s function of the anderson impurity model , a basic model system of quantum many - body condensed - matter physics . different methods of parametrizing the green s function are investigated ; a representation in terms of legendre polynomials is found to be superior due to its limited number of coefficients and its applicability to state of the art methods of solution . the dependence of the errors on the size of the training set is determined . the results indicate that a machine learning approach to dynamical mean - field theory may be feasible .


presentism is a metaphysical thesis about what there is .it can be expressed as ( e.g. crisp 2003 ) :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ presentism_. it is always the case that , for every , is present . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  the quantification is unrestricted , it ranges over all existents . in order to make this definition meaningful , the presentist must provide a specification of the term ` present ' .crisp , in the cited paper , offers the following definition :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ present_. the mereological sum of all objects with null temporal distance . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  the notion of temporal distance is defined loosely , but in such a way that it accords with common sense and the physical time interval between two events . from these definitions it follows that the present is a thing , not a concept .the present is the ontological aggregation of all present things .hence , to say that ` is present ' , actually means is part of the present " .the opposite thesis of presentism is eternalism , also called four - dimensionalism .eternalists believe in the existence of past and future objects .the temporal distance between these objects is non - zero .the name four - dimensionalism comes form the fact that in the eternalist view , objects are extended through time , and then they have a 4-dimensional volume , with 3 spatial dimensions and 1 time dimension .there are different versions of eternalism .the reader is referred to rea ( 2003 ) and references therein for a discussion of eternalism .several philosophers have defended presentism from various critiques , notably chisholm ( 1990 ) , bigelow ( 1996 ) , zimmerman ( 1996 , 2011 ) , merricks ( 1999 ) , hinchliff ( 2000 ) , crisp ( 2007 ) , and markosian ( 2004 ) . some of these authors have considered criticisms arisen from the alleged incompatibility of presentism with relativity theory . inwhat follows we review the main objections posed to presentism in the framework of both relativities , special and general , and then we move on to consider a new type of argument based on the existence of compact objects in the universe .we show that the presentist can not accept the standard physical understanding of these objects , without introducing changes in his ontological views or rejecting current astrophysics .special relativity is the theory of moving bodies formulated by albert einstein in 1905 ( einstein 1905 ) .it postulates the lorentz - invariance of all physical law statements that hold in a special type of reference systems , called _inertial frames_. hence the ` restricted ' or ` special ' character of the theory .the equations of maxwell electrodynamics are lorentz - invariant , but those of classical mechanics are not .when classical mechanics is revised to accommodate invariance under lorentz transformations between inertial reference frames , several modifications appear .the most notorious is the impossibility of defining an absolute simultaneity relation among events .the simultaneity relation results to be frame - dependent .then , some events can be future events in some reference system , and present or past in others . since what there is can not depend on the reference frame adopted for the description of nature, it is concluded that past , present , and future events exist .then , presentism is false .this argument has been formulated with different levels of sophistication by philosophers such as smart ( 1963 ) , rietdijk ( 1966 ) , and putnam ( 1967 ) . in particular ,both rietdijk and putnam argued independently that special relativity implies determinism .this position was revisited by maxwell ( 1985 ) in the eighties .he maintained that probabilism and special relativity were incompatible .strong objections to rietdijk , putnam , and maxwell can be found in stein ( 1968 , 1991 ) . inhis 1991 paper stein showed that `` the openness of the future relative to the past for any region of space - time is , given special relativity , logically incompatible with the objective global temporal structure '' ( harrington 2009 ) . in later yearsharrington ( 2008 , 2009 ) defended a theory of local temporality , also called point - present theory , which assuming special relativity allows for the possibility of an open future , i.e not fully determined ( see also ellis ( 2006 ) and ellis and rothman 2010 ) .the argument for eternalism from special relativity seems particularly strong in the reformulation of this theory proposed by hermann minkowski .minkowski ( 1907 , 1909 ) introduced the concept of space - time : the mereological sum of all events .space - time can be represented by a 4-dimensional manifold endowed with a metric that allows to compute distances between events . these are spatio - temporal distances or _intervals_. the differential ( arbitrarily small ) interval for minkowski space - time , which is invariant under lorentz transformations , is : where the minkowskian metric tensor is pseudo - euclidean , with rank 2 and trace .the coordinates with the same sign are called _ spatial _ ( the usual convention in notation is , , and ) and the coordinate is called _time coordinate_. the constant is introduced to make the physical units uniform , and can be shown to be equal to the speed of light in empty space .minkowski s interpretation allows to separate space - time , at each point ( every point represents an event ) , in three regions according to ( space - like region ) , ( light - like or null region ) , and ( time - like region ) .particles that go through the origin can only reach time - like regions .the null surface can be inhabited only by particles moving at the speed of light , like photons .points in the space - like region can not be reached by material objects from the origin of the _ light cone _ that can be formed at any space - time point ( see fig .[ fig1 ] ) .the introduction of the metric allows to define the future and the past of a given event .once this is done , all time - like events can be classified by the relation ` earlier than ' or ` later than ' .the selection of a ` present ' event - or ` now ' - is entirely conventional . to be present is not an intrinsic property of any event .rather , it is a secondary , relational property that requires interaction with a conscious being .the extinction of the dinosaurs will always be earlier than the beginning of the second world war .but the latter was present only to some human beings in some physical state .the present is a property like a scent or a color .it emerges from the interaction of self - conscious individuals with changing things and has not existence independently of them ( for more about this point of view , see grnbaum 1973 , chapter x ) .this sounds convincing , but there is a strong ontological assumption : the system of all events is assumed .this provides a simplification that pleases the physicist , but can terrify the philosopher .presentists , surely , will plainly reject this assumption as outrageous . butdo they have an alternative interpretation of special relativity with the same explanatory and predictive power as that of minkowski s ?yes , they have .the alternative was already suggested by poincar ( 1902 ) in the form of the conventionalist thesis .poincar remarked that it is always possible to save an interpretation based on an euclidean physical geometry if additional fields are introduced in the theory to adjust all physical quantities ( lengths , masses , speeds ) to the adopted geometry . in the case of special relativity , this yields lorentz s theory of moving bodies .physicist will always prefer the simplest theory , but the presentist can argue that from an ontological point of view , minkowski s interpretation is far more costly than the presentist one : a 3-dimensional euclidean and lorentz - invariant theory .an euclidean space - time is consistent with presentism since the interval can be written as :    hence , since in a proper system , and using the invariance of the interval , we get :    where from eq .( [ eq : eucl2 ] ) follows that iff .all events have zero temporal distance in the 3-dimensional euclidean plane .a full discussion of such an alternative is presented by craig ( 2001 ) .an euclidean geometry can be preserved ( to a high price , i.e. introducing distorting fields ) in special relativity since both euclidean and minkowskian geometries have zero intrinsic curvature .the curvature of a given manifold is given by the riemann tensor , formed with second derivatives of the metric .this tensor is identically zero for any space - time with a constant metric .since the intrinsic curvature is the same for both euclidean and minkowskian geometries , it is always possible to find a _ global _ transformation between them . in the case of more general space - times , represented by manifolds with intrinsic curvature , such transformations can only be local .this leads us to general relativity .a basic characteristic of minkowski space - time is that it is ` flat ' : all light cones point in the same direction , i.e. the local direction of the future does not depend on the coefficients of the metric since these are constants .more general space - times , however , are possible . if we want to describe gravity in the framework of space - time , we introduce a pseudo - riemannian manifold , whose metric can be flexible , i.e. a function of the material properties ( mass - energy and momentum ) of the physical systems that produce the events represented by space - time points .the key to relate space - time to gravitation is the _ equivalence principle _ introduced by einstein ( 1907 ) : at every point of the manifold that represents space - time , there is a flat tangent surface .  in order to introduce gravitation in a general space - timewe define a metric tensor , such that its components can be related to those of a locally minkowskian space - time defined by through a general transformation :    in the absence of gravity we can always find a global coordinate system ( ) for which the metric takes the form given by eq .( [ eq : eta ] ) everywhere . with gravity ,conversely , such a coordinate system can represent space - time only in an infinitesimal neighborhood of a given point .the curvature of space - time means that it is not possible to find coordinates in which at _ all points on the manifold_. it is always possible , however , to represent the event ( point ) in a system such that and .the key issue to determine the geometric structure of space - time , and hence to specify the effects of gravity , is to find the law that fixes the metric once the source of the gravitational field is given .the source of the gravitational field is represented by the energy - momentum tensor that describes the physical properties of material things .this was einstein s fundamental intuition : the curvature of space - time at any event is related to the energy - momentum content at that event .the einstein field equations can be written in the simple form ( einstein 1915 ) :    where is the so - called einstein s tensor .it contains all the geometric information on space - time .the constants and are the gravitational constant and the speed of light in vacuum , respectively .einstein s field equations are a set of ten non - linear partial differential equations for the metric coefficients . in newtonian gravity , otherwise , there is only one gravitational field equation .general relativity involves numerous non - linear differential equations . in this factlays its complexity , and its richness .a crucial aspect of general relativity is that 4-dimensional space - time with non - zero curvature is not dispensable anymore .is this implying that presentism should be abandoned ?not necessarily .thomas crisp ( 2007 ) has proposed a `` presentist - friendly '' model of general relativity .he suggests that the world is represented by a 3-dimensional space - like hypersurface that evolves in time .this interpretation requires to introduce a preferred foliation of space - time , and to consider the usual decomposition for the dynamics of space - time in such a way that ` the present ' is identified with the evolving hypersurface ( see fig .[ fig2 ] ) .    in order to formulate such a model of space - time ,some global constraints must be imposed : there should be a possible foliation into cauchy hypersurfaces in order to allow for global time - like continuous vector fields that can be used to introduce a `` global time coordinate '' .this is the case , for instance , of the friedmann - robertson - walker - lematre metric .general conditions for such a metric requires the absence of cauchy horizons and the fulfillment of the so - called _ energy conditions _ ( hawking and ellis 1973 ) .although a case can be made for general inhomogeneous metrics and massive violations of the energy conditions on the basis of recent cosmological data ( e.g. plebaski and krasiski 2006 ) , in what follows we focus on local aspects that should be accommodated in any cosmological model compatible with current astrophysics .we shall now provide a general definition of a black hole , independently of the coordinate system adopted in the description of space - time , and even of the exact form of the field equations . a space - time is represented by an order pair , formed by a manifold and a metric , i.e. .let us consider a space - time where all null geodesics that start in a region end at .then , such a space - time , , is said to contain a _ black hole _ if _ is not _ contained in the causal past . in other words , there is a region from where no null geodesic can reach the _ asymptotic flat _ future space - time , or , equivalently , there is a region of that is causally disconnected from the global future . the _ black hole region _ , , of such space - time is $ ] , and the boundary of in , , is the _event horizon_.  a black hole can be conceived as a space - time _ region _, i.e. what characterizes the black hole is the metric and , consequently , space - time curvature .what is peculiar of this space - time region is that it is causally disconnected from the rest of the space - time : no events inside this region can make any influence on events outside .hence the name of the boundary , event horizon : events inside the black hole are separated from events in the global external future of space - time . the events in the black hole , nonetheless , as all events , are causally determined by past events .a black hole does not represent a breakdown of classical causality .the simplest type of black hole is described by the schwarzschild metric ; this metric characterizes the geometry of space - time outside a spherically symmetric matter distribution .the schwarzschild metric for a static mass can be written in spherical coordinates as :   the radius is known as the _ schwarzschild radius_. it corresponds to the event horizon if all the matter that generates the curvature is located at .the light cones can be calculated from the metric ( [ schw ] ) imposing the null condition . then : where we made .notice that when , , as in minkowski space - time .when , , and light moves along the surface .the horizon is therefore a _null surface_. for , the sign of the derivative is inverted .the inward region of is time - like for any physical system that has crossed the boundary surface . in fig .[ fig3 ] we show the behavior of light cones in schwarzschild coordinates .similar figures are given in classical textbooks such as misner et al .( 1973 , p. 848 ) and carroll ( 2004 , p. 219 ) .as we approach to the horizon from the flat space - time region , the light cones become thinner and thinner indicating the restriction to the possible trajectories imposed by the increasing curvature . on the inner side of the horizon the local direction of timeis ` inverted ' in the sense that null or time - like trajectories have in their future the singularity at the center of the black hole .there is a very interesting consequence of all this : an observer on the horizon will have her present _ along _ the horizon .all events occurring on the horizon are simultaneous .the temporal distance from the observer at any point on the horizon to any event occurring on the horizon is zero ( the observer is on a null surface so the proper time interval is necessarily zero ) .if the black hole has existed during the whole history of the universe , all events on the horizon during such history ( for example the emission of photons on the horizon by infalling matter ) are _ present _ to any observer crossing the horizon .these events are certainly not all present to an observer outside the black hole .if the outer observer is a presentist , she surely will think that some of these events do not exist because they occurred or will occur either in the remote past or the remote future .but if we accept that what there is can not depend on the reference frame adopted for the description of the events , it seems we have an argument against presentism here . before going further into the ontological implications ,let us clarify a few physical points .  what happens to an object when it crosses the event horizon ?according to eq .( [ schw ] ) , there is a singularity at .however , the metric coefficients can be made regular by a change of coordinates .for instance we can consider eddington - finkelstein coordinates .let us define a new radial coordinate such that radial null rays satisfy .( [ schw ] ) it can be shown that : then , we introduce : the new coordinate can be used as a time coordinate replacing in eq .( [ schw ] ) .this yields : or where   occurs at the event horizon , represented by the surface of the cylinder .the global time direction is in the direction of the axis of the cylinder . ]notice that in eq .( [ ef ] ) the metric is non - singular at .the only real singularity is at , since the riemann tensor diverges there . in order to plot the space - time in a -plane, we can introduce a new time coordinate . from the metric ( [ ef ] ) or from fig .[ fig4 ] we see that the line , , and constant is a null ray , and hence , the surface at is a null surface .this null surface is an event horizon because inside all cones have in their future . in figure [ fig4 ]we see how the light cones tilt in eddington - finkelstein coordinates . _ all _light cones of events at have their null surfaces coincident with the horizon .the horizon itself is a null surface common to all light cones associated with physical systems that are entering the black hole .we remark , then , that the horizon 1 ) does not depend on the choice of the coordinate system adopted to describe the black hole , 2 ) it is an absolute null surface , in the sense that this property is intrinsic and not frame - dependent , and 3 ) is non - singular ( or ` well - behaved ' , i.e. space - time is regular on the horizon ) .more complex black holes can be considered .for instance , rotating ( kerr ) , charged ( reissner - nordstrm ) , or both rotating and charged ( kerr - newman ) black holes . in all of them the outer event horizons are null surfaces , so the above considerations remain valid . rotating black holes introduce additional problems to presentism , related to the existence of cauchy horizons and closed timelike curves , but we do not discuss these issues here .rather , we focus on the implications of the existence of null surfaces that can be crossed by massive particles or observers for presentism .in a world described by special relativity , the only way to cross a null surface is by moving faster than the speed of light .as we have seen , this is not the case in a universe with black holes .we can then argue against presentism along the following lines .+ argument :  * : there are black holes in the universe . * : black holes are correctly described by general relativity . * : black holes have closed null surfaces ( horizons ) . * therefore , there are closed null surfaces in the universe .  argument :  * : all events on a closed null surface are simultaneous with any event on the same surface . * : all events on the closed null surface are simultaneous with the birth of the black hole .* : some distant events are simultaneous with the birth of the black hole , but not with other events related to the black hole . *therefore , there are events that are simultaneous in one reference frame , and not in another .simultaneity is frame - dependent . since what there existcan not depend on the reference frame , we conclude that there are non - simultaneous events .therefore , presentism is false .let us see which assumptions are open to criticism by the presentist .an irreducible presentist might plainly reject .although there is significant astronomical evidence supporting the existence of black holes ( e.g. casares 2006 , camenzind 2007 , paredes 2009 , romero and vila 2013 ) , the very elusive nature of these objects still leaves room for some speculations like gravastars , and other exotic compact objects .the price of rejecting , however , is very high : black holes are now a basic component of most mechanisms that explain extreme events in astrophysics , from quasars to the so - called gamma - ray bursts , from the formation of galaxies to the production of jets in binary systems .the presentist rejecting black holes should reformulate the bulk of contemporary high - energy astrophysics in terms of new mechanisms . in any case , is susceptible of empirical validation through direct imagining of the super - massive black hole `` shadow '' in the center of our galaxy by sub - mm interferometric techniques in the next decade ( e.g. falcke et al . 2011 ) . in the meanwhile ,the cumulative case for the existence of black holes is overwhelming , and very few scientists would reject them on the basis of metaphysical considerations only .the presentist might , instead , reject .after all , we _ know _ that general relativity fails at the planck scale .why should it provide a correct description of black holes ?the reason is that the horizon of a black hole is quite far from the region where the theory fails ( the singularity ) .the distance , in the case of a schwarzschild black hole , is ( see eq .[ rs ] ) . for a black hole of 10 solar masses ,as the one suspected to form part of the binary system cygnus x-1 , this means km . andfor the black hole in the center of the galaxy , the schwarzschild radius is of about 12 million km .any theory of gravitation must yield the same results as general relativity at such distances .so , even if general relativity is not the right theory for the classical gravitational field , the correct theory should predict the formation of black holes under the same conditions .there is not much to do with , since it follows from the condition that defines the null surface : , where is the proper temporal separation . ]; similarly only specifies one of the events on the null surface .a presentist might refuse to identify ` the present ' with a null surface .after all , in minksowskian space - time or even in a globally time - orientable pseudo - riemannian space - time the present is usually taken as the hyperplane perpendicular to the local time ( see figs .[ fig1 ] and [ fig2 ] ) .but in space - times with black holes , the horizon is not only a null surface , it is also a surface locally normal to the time direction .this can be appreciated in figure [ fig5 ] , where the angle is the angle between the null surface and the hyperplane of the present . in a minkowskian space - timesuch an angle is 45 when the speed of light is measured in natural units ( ) .in such a space - time , certainly the plane of the present is not coincident with a null surface .however , close to the event horizon of a black hole , things change , as indicated by eq .( [ cones - schw ] ) .as we approach the horizon , goes to zero , i.e. the null surface matches the plane of the present . on the horizon ,both surfaces are exactly coincident : when .a presentist rejecting the identification of the present with a _ closed _ null surface on an event horizon should abandon what is perhaps her most cherished belief : the identification of ` the present ' with hypersurfaces that are normal to a local time - like direction .  the result mentioned above is not a consequence of any particular choice of coordinates but , as mentioned in sect .[ bh ] , an intrinsic property of a black hole horizon .this statement can be easily proved .the symmetries of schwarzschild space - time imply the existence of a preferred radial function , , which serves as an affine parameter along both null directions .the gradient of this function , satisfies ( ) :    thus , is space - like for , null for , and time - like for . the 3-surface given by is the horizon of the black hole in schwarzschild space - time .( [ ra ] ) it follows that over , and hence is a null surface[multiblock footnote omitted ] .premise , perhaps , looks more promising for a last line of presentist defense .it might be argued that events on the horizon are not simultaneous with any event in the external universe .they are , in a very precise sense , cut off from the universe , and hence can not be simultaneous with any distant event .let us work out a counterexample .the so - called long gamma - ray bursts are thought to be the result of the implosion of a very massive and rapidly rotating star .the core of the star becomes a black hole , which accretes material from the remaining stellar crust .this produces a growth of the black hole mass and the ejection of matter from the magnetized central region in the form of relativistic jets ( e.g. woosley 1993 ) .approximately , one of these events occur in the universe per day .they are detected by satellites such as _ swift _( e.g. piran and fan 2007 ) , with durations of a few tens of seconds .this is the time that takes for the black hole to swallow the collapsing star .let us consider a gamma - ray burst of , say , 10 seconds . before these 10 seconds , the black hole did not exist for a distant observer . afterwards, there is a black hole in the universe that will last more than the life span of any human observer .let us now consider an observer collapsing with the star . at some instant she will cross the null surface of the horizon .this will occur within the 10 seconds that the collapse lasts for .but for all photons that cross the horizon are simultaneous , including those that left long after the 10 seconds of the event and crossed the horizon after traveling a long way . for instance , photons leaving the planet of one million years after the gamma - ray burst , might cross the horizon , and then can interact with .so , the formation of the black hole is simultaneous with events in and , but these very same events of are simultaneous with events that are in the distant future of .the situation is illustrated in fig .[ fig6 ] .   is . ]the reader used to work with schwarzschild coordinates perhaps will object that never reaches the horizon , since the approaching process takes an infinite time in a distant reference frame , like that of .this is , however , an effect of the choice of the coordinate system and the test - particle approximation ( see , for instance , hoyng 2006 , p.116 ) . if the process is represented in eddington - finkelstein coordinates , it takes a finite time for the whole star to disappear , as shown by the fact that the gamma - ray burst are quite short events .accretion / ejection processes , well - documented in active galactic nuclei and microquasars ( e.g. mirabel et al .1998 ) also show that the time taken to reach the horizon is finite in the asymptotically flat region of space - time .our conclusion is that presentism , as usually stated , provides a defective picture of the ontological substratum of the world .what kind of ontological view is compatible with black hole astrophysics ?we suggest that one where what we call ` present ' has a local rather than a global character .the detachment between local and global time occurring in a black hole results , as we have seen , in a checkmate for the presentist position .there is no way to re - identify the present to give it back a global meaning in a universe with black holes .the intuitive ontology adopted by most practising astrophysicists is one where there are things , and these things change relative to each other .one can speculate that space - time is an emergent property of the system of all things ( e.g. bunge 1977 , perez - bergliaffa et al .the exact formulation of such an ontological theory to encompass a relativistic view of the world , taking into account the peculiarities of non - local effects in quantum mechanics , is an open problem . a problem that cries for attention from both philosophers and scientists alike .falcke , h. , markoff s. , bower g. c. , gammie , c. f. , moscibrodzka , m. & maitra , d. 2011 .the jet in the galactic center : an ideal laboratory for magnetohydrodynamics and general relativity . in jets at all scales , proceedings of the international astronomical union , iau symposium , eds .g. e. romero , r. a. sunyaev t. belloni , volume 275 , 68 - 76 .minkowski , h. 1907 .die grundgleichungen f die elektromagnetischen vorg in bewegten k , nachrichten von der gesellschaft der wissenschaften zu g , mathematisch - physikalische klasse , 53111 .mirabel , i.f . , dhawan , v. , chaty , s. , rodriguez , l. f. , marti , j.,robinson , c. r.,swank , j. , geballe , t. 1998 .accretion instabilities and jet formation in grs 1915 + 105 .astronomy astrophysics 330 : l9-l12 .rea , m. c. 2003 . four - dimensionalism . in the oxford handbook of methaphysics , eds .m. j. loux & d. w. zimmerman , 246 - 80 .oxford : oxford university press .rietdijk , c. w. 1966 . a rigorous proof of determinism derived from this special theory of relativity .philosophical papers 33 : 341 - 344 .saunders , s. 2002 .how relativity contradicts presentism . in time , reality experience , royal institute of philosophy , supplement , ed . c. callender , 277 - 292 .cambridge , new york : cambridge university press .

Abstract: presentism is , roughly , the metaphysical doctrine that maintains that whatever exists , exists in the present . the compatibility of presentism with the theories of special and general relativity was much debated in recent years . it has been argued that at least some versions of presentism are consistent with time - orientable models of general relativity . in this paper we confront the thesis of presentism with relativistic physics , in the strong gravitational limit where black holes are formed . we conclude that the presentist position is at odds with the existence of black holes and other compact objects in the universe . a revision of the thesis is necessary , if it is intended to be consistent with the current scientific view of the universe .   example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore