Cool. So, hi everybody, this is my honor and privilege for this Padafus seminar. I forgot which one we are now on the fourth, the third of the year. To receive a sound to help from the CNRS, where he works at the Institut d'histoire et de philosophie des sciences, the technique of in Paris. And without further ado, I will give him the floor. After his presentation, we'll take a five-minute break. After that, I will have a brief comment to start the discussion. Thank you to Avacom. Thank you. And the Belgian train. Thanks to Alexandre for his kind invitation. I'm really happy to return to the ESP, where I spent very good years as postdoc. So, when Alexandre let me know the theme of the seminar, I wasn't sure what I could present. For a few years, I worked mostly on the question of the prediction of some dynamics and fluid mechanics on one side and statistical mechanics on the other. And I didn't immediately see how this research could fit into the theme. So, I thought that going back to my work during my PhD devoted to the discrete representation of time in physics was more relevant for this seminar. And I have to say that a part of this talk is a work in progress, so any comments will be welcome. So, just a few words about the context of the talk. My PhD was on the philosophical analysis of discrete mechanics, which is a formulation of classical mechanics with discrete time symbol. And in particular, I discussed if it was possible to dispense with continuous representation of time in physics to do without continuous time symbol and what consequences we could draw. I also investigated more practical issues. I discussed whether this discrete mechanics was preferable for computational physics to perform computer simulations. And these two kinds of questions are related to the two-fold origin of discrete mechanics on the one hand. The discrete mechanics was initially developed by the physicist Tung Daoli as a new physical theory. Tung Daoli is a Nobel Prize winner for works in particle physics for the charge parity symmetry. And on the other hand, discrete mechanics has also been developed as a computational mechanics to perform computer simulations and classical mechanics. And this is partly due to the mathematician Gerald Martin. In this talk, I will leave aside the computational aspects and I will focus on Lee's approach which aims to develop a new discrete physical theory. Okay, now just a few words of introduction. So here is this seminar paper published in Physics Letters in 1983. Can time be a discrete dynamical variable? And the abstract claims, the possibilities at the time can be regarded as a discrete dynamical variable is examined through all phases of mechanics, from classical mechanics to non-relative quantum mechanics and to relative quantum physics theories. So the project is quite ambitious, I mean Lee aims to offer a discrete reformulation of physical theories from classical to quantum physics. In another paper, we can read a different perspective on this project. For more than three centuries, we have been influenced by the precepts that fundamental laws of physics should be expressed in terms of differential equations. Different equations are always regarded as approximations. Here, I try to explore the opposite. Different equations are more fundamental and differential equations are regarded as approximations. So Newton claimed in his point of view that time is continuous and classical mechanics and actually all physics then rounded on this assumption. And afterwards, he used differential equations to express the fundamental laws of physics. Lee suggests starting from a discrete time variable and expressing all the fundamental equations of physics with different equations. And a few years later, Lee proposed to go even further and reformulate in a discrete way, not only classical quantum mechanics, quantum physics theory, but also the theory of quantum gravitation. So far, my research has been limited to the first step of Lee's project and then the classical mechanics. But I also try to investigate the discrete representation of time in this more fundamental theory, particularly quantum mechanics and quantum gravity. So time is usually represented as continuous in physics, classical mechanics, electromagnetism, hydrodynamics, relativity, quantum mechanics and so on. And by a continuous representation, I mean using the mathematical symbol t defined on real numbers in Java. Lee, on the other hand, tries to do three times as discrete, uses the symbol tk which runs through a finite set of elements. Each instant tk is indexed by a natural number k. So I have two main claims in this talk. First, I will argue that the continuous representation of time is not mandatory in physics, it's not necessary. And consequently, I will argue that it's a theoretical framework that physicists can choose to describe phenomena. But the discrete representation of time is another perfectly good theoretical framework. Physicists can choose one framework or the other. Secondly, I will argue that Lee's discrete physics theories fit well with the somehow relationist conception of time, namely the conception that time does not exist by itself. At least, I will argue that the time symbol does not play the same role in Lee's discrete theories compared to the usual classical and quantum mechanics. The time symbol is no longer the parameter responsible for the evolution of physical systems in Lee's discrete theories. So first, I will start by presenting some general stuff on time of physics before turning section 2 to my claim that what can do without a continuous representation of time. And the next three sections, section 3, 4 and 5, are then devoted to three discrete physical theories. And the purpose of these sections is first to support the claim 1. We can reformulate physical theories with a discrete time symbol. Moreover, leaving aside the question of discreteness, I will also analyse the word of the time symbol in these theories. And as I said, I will show that it's not an evolutionary parameter for physical systems. So to start with, why should we pay attention to the representation of time in physics? A task in the philosophy of science is to analyse and clarify the notion of time in physical theories. Why? Because we usually consider that physics has something to say about time. And following Will Frim Selaar's distinction between the manifest image of the word and the scientific image, correct calendar distinguishes between manifest time and physical time. Manifest time is the concept of time that comes from a commonsense picture of the word, informed by our sensible experience. It's an notion of time that we probably inherit from childhood. And some of its features are, for example, that the present is special, time flows, the past is fundamentally different from the future, time is harder, one-dimensional, continuous, absolute. So continuity is a feature of the manifest time. Of course, that's the one in which I am interested in. In contrast, physical time is a notion informed by our physical theories, Newtonian mechanics, special and general relativity, and so on. And even if no single conception of time arises from a studio of physics, correct calendar claims that there is a clash between manifest time and physical time. He says physical time departs dramatically from manifest time. No physical theory has ever required the property's characteristic of manifest time. For example, according to our physical theories, the present is not special for calendar. Classical physics, Einsteinian physics do not posit a special now. Similarly, there is no temporal flow in physical theories according to Craig's calendar. There is no past future asymmetry at the microscopic level. Most of the equations of physics are time universal invariants. So that's why Craig's calendar argues that if we assume that physical time is our best theory of time, then it turns out that manifest time paints a very misleading image of time. This claim might be controversial, but at least, even if in our ordinary life, we tend to consider the bigger time as continuous, I claim that we should be careful about the question whether physical time is continuous. I would like to stress another distinction that is important to the talk. We should distinguish between time, more precisely physical time, and the representation of time. This distinction is not always made in the literature, and it is absent perhaps because it is implicitly admitted, but it's a way of loosely speaking when we say time, we actually say the representation of time or time symbol. Or in contrast, it is absent because there is an assumption, an endorsed assumption about time. The assumption according to which is the representation of time, new words, time, and its features. I will go back to this point below in the talk. In any case, there are different ways to represent time in physics and in physical theories. Usually time is represented with the type with the symbol t, a continuous parameter. But in Galileo's or Newton's text, for example, we have geometrical quantities. Time is represented by a segment. In these figures, AC and CI and IPO are segments. And as I said, we can also represent time with a discrete symbol. Here is an example where the distinction is not made. It's a quotation of the calendar. We can read about the question of the dimensionality of time, that in classical physics, assuming that the instance of time forms a continuum, the set of instance is topology, one-dimensional. So how should we understand the sentence, the instance of time forms a continuum? In this case, I guess it's a way of speaking, the calendar actually speaks about the time symbol, but not about time per se. And the reason is that the calendar leaves elsewhere, opens the question of the structure of time. I quote him, Is time discrete or dense or continuous is not a question like are quarks ontologically more basic than protons? And yet have a physical theory, empirically well-confirmed, which tells us what is the structure of time, while we have physics theories on the structure of protons. So how should we interpret time symbols in physics? Here are some possibilities that I see. First, the continuous representation of time might be an idealization, for example, the continuous representation of mass density in fluid mechanics, but no, I don't think so, we have to exclude this possibility because an idealization is an hypothesis that we know to be false, but we use it for practical reasons. We know that fluid are molecules, but we don't know if time is continuous or discrete. It's not literally a false assumption. The second possibility is the representation of time could refer to physical content. It would mean that the structure of time would be an empirical question. Maybe, however, in the current state of science, this question is not confirmed and at best it's a physical hypothesis. The third possibility, the continuous representation of time, has metaphysical content. In that case, the structure of time is not an empirical question. And the last possibility is that the continuous representation of time is a neutral theoretical framework. It's a viewpoint that could be attributed to karma by these successors. According to these accounts, the continuous representation of time is somewhat conventional. I will go back to this account later. Now let's suppose we can replace our continuous representation of time with a discrete one. What would be the consequences of dispensing with a continuous representation of time? What would it change? To tackle these questions, I will use what is, to my knowledge, the only reference in the philosophy of science that directly, precisely addresses this question. It seems that the structure of time by William Newton Smith, a Canadian philosopher who has been in Oxford, in notably defends the thesis that if we could dispense with a continuous representation of time in physics by replacing it with a discrete one, then we would have at least as much reason to regard time as discrete as we now have for regarding it at the continuous. And I would like to discuss this claim. First, what does it mean by time is continuous and time is discrete? I quote him, if time is continuous, the set of all instances will be isomorphic to the real, the real numbers. So Newton Smith endorses a set theoretic definition of time. Of course, such a conception is criticized, notably by Michael Domet who suggests that continuous time should not be conceived as a set of elements. This is related to the view of Hermann Weider, from whom I quote, there is no point in the journey of the continuum. And we can also mention some recent related discussion by Nicolas Jeza, but this is not my point and I put aside these discussions. Similarly, Newton Smith defends discrete time as a set of instances isomorphic now to a finite subset of natural numbers. I will not also enter into a discussion about this definition. And for discussion on discrete time, I see for example the chapter, the Unbook of the Philosophy of Time by Jean-Paul von Bendegen for discrete time. Here, I would like to stress an important point in Newton Smith's claim. For me, he implicitly adopts a holistic confirmation for statements about time. To the question, why do we tend to regard time as continuous, he says, it's quite simply that the best physical theories we have in fact construct for the physical world require in the mathematical formulation a time parameter that ranges over the elements of the real number system. So in other words, using a continuous time symbol, a continuous representation of time in our best physical theories is a good reason for believing that time is continuous. As I will discuss below, we might disagree with this methodological criteria. But for now, let's continue with Newton Smith's discussion. So with this holistic criterion, let's assume that physical theories, the discrete and continuous parameter would be empirically equivalent. And I will argue for this claim later in the talk. So in that case, we could not decide between these two theories. We would face a non-decedibility result from which Newton Smith then argues for two options. On the one hand, time is continuous and time is discrete can be interpreted as an inaccessible empirical statement. So time is generally continuous or discrete. It's an empirical fact. But we cannot know if it is continuous or discrete. On the other hand, time is continuous and time is discrete can also be interpreted as non-amplical statements. And in that case, continuity or discreteness of time are not empirical facts. There are theoretical frameworks used to describe empirical facts. So Newton Smith considers two possibilities among the three I mentioned previously. The representation of time might refer to physical content, which is an empirical question, or it should be viewed as a neutral theoretical framework. And actually, he's inclined to this option three. And as Newton Smith acknowledges, he follows Karnab's view on time. For Karnab, the questions about the existence and the features of time, space and spacetime are indeed questions about the choice of using such or such linguistic framework to describe a physical phenomenon. Therefore, since Newton Smith lies on Karnab's analysis, it's not very surprising that he does not consider option two that statements about time have metaphysical content. So I mostly agree with Newton Smith's analysis. Nevertheless, I argue that we should amend this view. And the main reason is that I disagree with his holistic confirmation criteria. In contrast, I follow Penelope Maddi's analysis. She claims that also physicists widely use continuous representation of time. They still consider the question of its continuity as an open question, I quote her. Mathematical existence assumptions in science and accompanying assumptions about the structure of physical reality, such as time and space, are not treated on an epistemic ball with ordinary physical assumptions. The standards of their introduction are weaker and they work in successful theory lacks confirmatory force. So in other words, Maddi denies holistic confirmation for assumptions about the structure of time. One needs empirical investigations on the structure of time to believe in statements on time. I mean, we need scientific... What does she mean by the structure of time? Its metric? It's all statements on time. So... Is it topology? Actually, in this quotation, the context of this quotation is a discussion about continuity on time or discreteness. It's not very... But in this... The context of this quotation is just continuity. But I think we could extend to... That's granted in the context of this discussion. So what would be open to the physicists as to the structure of time? Its metric, its topology. Is it open time, closed time? I don't know. Yes, I think all these possibilities are... are okay. There is no specific... She does not specify what kind of structure of time she has in mind. But what she says is that one needs specific inquiries that are specifically dedicated to this question. So the question of topology of time, of discreteness of time, of granularity of times. And if we don't have empirical investigation of specific statement, we should not argue for this or one or the other structure of time. So consequently, I argue for amending Newton-Smith accounts by restricting the consequences to the representation of time and not by jumping to statements all the time. And in that case, I define the continuous representation of time and the discrete have to be interpreted as 30-card frameworks which can be equally used in physics. So the representation of time and not time itself. Using one or the other is a question of choice in the matter of decision. Nevertheless, this claim crucially depends on an assumption that I didn't support yet. I have assumed that we can reformulate our best physical series with a discrete time parameter and I have assumed that this series are empirically equivalent and the rest of the talk will address this assumption. But before that, just a last word, even if you disagree with Maddie's objection and if you rather agree with Newton-Smith, the rest of the talk is also interesting for you since in that case, we support the claim that the continuity of time and the discreteness of time are two theoretical frameworks. So let's begin with classical mechanics. Discrete classical mechanics. It's the first step in this account for discrete physics. So it's generally admitted that it's impossible to formulate classical mechanics without a continuous time symbol. In particular, the format allows our differential equations which require continuous time symbol and the formulation of classical mechanics with different equations seems generally impossible. According to Adolf Greenbaum, a physical COE whose format allows to take the form of different equations is at best a gleam in the eyes of up-food speculative theoreticians. So it is not very optimistic. However, we will see that this statement is no longer true. Today, classical mechanics can be formulated with a discrete time symbol. So there are dozens of discrete classical mechanics actually. It's a research topic in itself to ask which one is the best. I will consider the formulations that I believe to be preferable, which is some of the least discrete classical mechanics. Why? Because it's the most satisfactory in the sense that it maximizes the number of conservation laws that we have in the usual classical mechanics, the one with a continuous time symbol. There is indeed a theorem due to G and Marsden in 1988 which guarantees that to maximize this conservation laws, the number of conservation laws, we must use a discrete representation of time applied to a variational approach of classical mechanics. And I believe this is a very important theorem overlooked in the philosophical literature. So for example, Donald Winspan also produced a discrete classical mechanics in the 70s, which finds the conservation of energy, conservation of momentum. Nevertheless, the conservation of the area in the phase space was lost. And indeed, it was not actually a discrete mechanics based on the variational approach. So what is variational mechanics? It's a reformulation of Newtonian mechanics as a theory of variational principles and notably based on the least action principle. This reformulation is partly due to Lagrange and it is a very important achievement in physics and mathematics and Alexandre is an expert on Lagrangian mechanics. Roughly speaking, the idea is to describe the evolution of the physical system as the one that minimizes quantity S called the action. Among all its possible evolution, the actual evolution of the system is the one that minimizes this action. This action is defined as an integral of a function, the Lagrangian L, a function of the projection of q and the velocity q prime, the time derivative of the position. Tung Dao's least discrete mechanics is also based on the discrete least action principle. Similarly, a physical system will follow the evolution that minimizes a discrete action Sd among all its possible evolutions. This discrete minimization principle leads to discrete Euler-Lagrange equations. So they are another boost to the usual Euler-Lagrange equation, which are the equations of motion of Lagrange mechanics. There is also another equation, which is the conservation of the discrete energy. I will not discuss in more detail this term, I have done this in a couple of papers. Here, I would like to contact that some systems can be solved exactly within discrete mechanics, analytically. This is for example the case for the free particle or the harmonic oscillator. We can take discrete equations of motion and solve them. We can do physics with this equation and with these solutions. Okay. An important point for the talk is that discrete mechanics is empirically equivalent to the usual continuous classical mechanics. It's of course important to support Newton-Smitz claims undecidability claims about time symbols. Excuse me, but I will interrupt you. No. How does he make the calculations? Can you speak louder? How does he make the calculations? How does he find the solutions using discrete time? I mean, he uses the usual tools of calculus. Yes. Those usual tools of calculus, when you do this context with discrete time, they shouldn't work, because I think the tools of the calculus principles that you are going to explain to me, so does he have an instrument? Do you use these computing tools to find the solutions of those equations? No. Actually, I have discussed this pointer in some papers. I think we should distinguish between the representation of time and the use of continuum, mathematical continuum. So it's more instrumental than the use of the user? For example, the variable q, qk, is a real number. That's a solution. Yes, but the same with the equations. All variables here, even if they are discrete, we all value the quantities. So you can use all the the tools of continuum mathematics, even if you represent discrete variables. So about empirical equivalence, so we can argue for this empirical equivalence since we can argue for this empirical equivalence for if n, the number of steps is sufficiently large, since the discrete action tends to the continuous action in the limit when n goes to infinity. So this is the main reason for empirical equivalence. If you have a large n, sufficiently large n, you can always you can have equations and there are solutions which are the difference between the continuous equation and the discrete equations become empirically undetectable since there are always measurement errors with finite accuracy. I have also discussed this empirical equivalence in other papers, so I will not spend more time on it today, but if you have questions, you can discuss it after. Here I would like to discuss another and actually another point for this talk. This is actually the original part of the talk. Lee tricks time as I quote a discrete dynamical variable, so I think it's pretty clear what he means by discrete, but what does he mean by dynamical variable? And I will show that dynamical variable means coordinate in these discrete mechanics and I will argue that this treatment comes from the discretization of the extended theory of Lagrangian mechanics and I will argue that it's very important and importantly, I will show that this representation of time fits very well with the relationless viewpoint of time. So what is the extended theory of Lagrangian mechanics? It's a generalization of the usual theory of Lagrangian mechanics so what we find in textbooks for which time is treated on the same footing as positions, namely as a coordinate. And I think Jacobi proposed it in the 19th century. And here is how Jeremy Butterfield talks about this generalization. If one wishes, one can treat time in a manner more similar to the position of the particle. In both the Lagrangian Hamiltonian formatism, T is then treated as one of the configuration as well as of configuration coordinate Q, all of which are functions of temporal parameter Tau. So I am interested in Lagrangian mechanics but we can do the same things with Hamiltonian mechanics. The idea is to introduce a new parameter here is a grid Tau which describes the evolution of the system which is now represented by the new coordinates Q plus time. And I think it's even better to remove the adjective temporal about Tau. Tau is not another time symbol, just a mathematical parameter. I will discuss it. So with this extending theory of Lagrangian mechanics, a physical system is described by the coordinate Q, the position M, T and t. And the evolution of the system is parametricized by Tau. So here is an initial state with a Q initial tie. At the end we have the final state and you can describe all the states by evaluating the Tau parameter. So in the traditional or usual Lagrangian theory, the configuration space contains the position Q and its time derivative Q prime, the velocities. We can see the configuration space in the Lagrangian, Q and Q prime. Now the configuration space is larger, is extended. It contains the position Q, its derivative with respect to the new parameter Tau now, so dot Q. And also the time t and its derivative with respect to Tau dot t. And if we have N particles in the system, we have a 6N dimensional configuration space in the traditional usual Lagrangian theory. In contrast, we have 6N plus 2 dimensional configuration space in the extended Lagrangian theory. An important feature is that the extended Lagrangian mechanics is invariant under parameterization or reparameterization. It means that if we take another parameter, or more precisely if we modify the Tau parameter by applying a continuous transformation which changes Tau into Tau prime, the action is and the equations of our motions are unchanged, as they are not modified. In that sense, the Tau parameter is a free gauge. And that's why we can consider that Tau has no physical meaning and it's not a temporal parameter. This property is at the origin of debates in the philosophy of physics since it leads to the famous problem of time or timeless physics and this problem arises in the terms of unified quantum mechanics and general relativity and I'm not going to discuss this problem which is actually more concerned about quantum mechanics and I will focus on the extended Lagrangian mechanics. Before moving to the discrete case because here we are in the continuous case before moving to the discrete case let me discuss the consequences of this new treatment of time in continuous mechanics. I'm sympathetic to Karim's table analysis about this topic. According to him, the extended Lagrangian theory is a usual Lagrangian theory that more naturally allows for a Leibnizian viewpoint. So what does it mean? Roughly speaking, we can usually oppose two conceptions of time. Newtonian view or substantialism Roughly speaking, time is considered as an entity in itself responsible for the evolution of bodies. By contrast in Leibnizian viewpoint or relationism, time is not considered as an entity it's just a succession of events. And Ernst-Narth, notably different in this point this point of view, I said time is an abstraction at which we arrive by means of the changes of things. So what does Karim's table claim that the extended theory of Lagrangian mechanics leads to this view? The traditional or usual Lagrangian mechanics as well as the Newtonian mechanics actually is not spontaneously easy to interpret according to the Leibnizian viewpoint and it fits rather well with the Newtonian viewpoint because the time symbol plays a privilege role. I quote him the problem with the traditional Lagrangian mechanics for Leibnizian relation is the one of excess temporal structure. Indeed, time symbol plays a central role. The parameter that describes the evolution of physical systems the position is a function of the symbol t, the action and the integral of the time symbol t it's also involved in the definition of velocities which are derivative with respect to time. On the other hand the extended Lagrangian theory of time in the extended Lagrangian theory the time symbol t is no longer an evolution parameter it's just a coordinate in that sense it's a dynamic variable such as the position. In addition the tau parameter being a free gauge cannot be viewed as a temporal evolution so that's why the extended theory of Lagrangian mechanics fits well with the relationist point of view. Time does not play a privilege to be manager. So let's now return to the discrete case to the discrete Lagrangian mechanics I mean to the least discrete mechanics my point is that this discrete Lagrangian mechanics is actually the discretization of the extended theory of Lagrangian mechanics and this is why he claims that it treats time as a discrete dynamical variable the time symbol is a discrete coordinate and the transition from the continuous case to the discrete case is done by discretization of the tau parameter the continuous parameter tau is replaced by a discrete parameter n the position q and the time symbol t are then functions of the discrete parameter n which describes all the states of the system in a discrete manner accordingly Tsongdauli's discrete mechanics goes well with the relationist view point of time however this Leibnizian view point is even more constrained in the discrete case compared with the continuous case in the continuous case we can do from the usual Lagrangian mechanics to the extended law without any modifications in the content of the equations of motion in a sense it's just a change of perspective which is very important for some purpose but we have the same physics content in contrast in the discrete case it's in a sense mandatory to adopt the extended Lagrangian mechanics to have good equations again this follows from the G. Marsden C.O.M which states that we lose some conservation laws if we don't adopt these discrete mechanics so here is a non-trivial observation it seems that the use of a discrete representation of time is not completely independent of our view point on time on the relationist or substantialistic debate on time the Lagrangian view point seems to follow from the discrete approach and from the good discrete approach with the good equations good motion equations and conservation of energy and a related point in these discrete mechanics there is no discrete time step h or epsilon whatever you want and this may be surprising because when we think of the discretization of differential equations we think of the discretization with a time step this is not the case here and it's related to the fact that we discretized the extended theory of Lagrangian mechanics and not the usual Lagrangian mechanics here there is just a sequence of coordinate Qn and Tm but no time step which governs the evolution of the system so this is again compatible with the Lagrangian view point I think if a time step h a discrete increment governs the evolution of the discrete dynamics this time step could have been conceived in a Newtonian view point even discrete but and finally from a theoretical point of view we see that this is a very valuable advantage I mean to not have a time step for more formative physical theories particularly covalent loop quantum theory there is indeed only one parameter n to turn to infinity for the continuous limit and no time step parameter to turn to 0 I will skip this slide about invariance of the parameterization in this case if I have 10 minutes 5 minutes just a few words about the quantum mechanics case which is the second step in this project of discrete physics so maybe first let's have a look to the trajectory of our path in discrete classical mechanics this path on the left are piecewise continuous functions and we recover the continuous smooth path with the continuous limit when n turns to infinity and the transition to discrete to discrete quantum mechanics will be based on this notion of discrete path in classical mechanics how by using the path integral formalism of quantum mechanics developed by Feynman indeed this formalism already allows us to go from classical to quantum mechanics the quantum probability for particles starting to sum initial 0.0 to be at the finite point f is defined by considering all the possible path all the classical path between the two points so that's the state of the continuous case when we go from classical to quantum mechanics in the usual continuum theory the particle take on any smooth path and he explains how we move on to discrete quantum mechanics in the corresponding discrete theory we again restrict the particle to move only along the discrete path so here is a figure that illustrates Feynman's idea so we have to consider all the paths between initial state and finite step to define the transition amplitude and an important point for my purpose is is that to be able to compute the quantum probability time has to be discretized in Feynman path in integral formalism time is sliced with a discrete time epsilon so in Feynman path integral formalism time has to be discretized and at the end of the calculation we take the continuous limit these discrete quantum mechanics yes so in the last slide you discretized the time as like the variable but you made a distinction before between time and the parametrization tau and the left point of view was that you discretized the parameter tau but not the time is that not kind of cheating and saying you discretized time but then you discretized tau which is explicitly separated from time it's a very good question actually Feynman path integral formalism yes there is the time that you discretized but not necessarily the parametrization tau in between I perfectly agree Feynman path integral formalism is not based on the extended Lagrangian theory it's based on the usual Lagrangian theory we have and T plays a role in evolution parameter so here when Feynman defines the path integral I mean at the beginning of his paper he used all the classical tools of Lagrangian theory so that's why we have time and we discretized time so we are here in the usual framework of Lagrangian mechanics but you are perfect you anticipate what I want to say these quantum mechanics will be based on the discrete version of the extended Lagrangian mechanics applied to path integral but if we just focus before on the Feynman framework we have exactly a discrete time with a time step epsilon so I will skip some slides to be so I think you understood what I wanted to say about these discrete quantum mechanics I want to discuss different things but one thing I would like to discuss is the fact that if we focus on discrete Feynman path integral and not yet in least discrete quantum mechanics I would like to show that the discretization is crucial and even indispensable even in the case of the Feynman path integral and after that I would like to I would like to move on to these quantum mechanics the continuous limit used by Feynman is not mandatory except we cover exactly the usual quantum mechanics predictions but for n the number of step n sufficiently large we expect a discrete quantum mechanics to be equivalent to the usual quantum mechanics and to define this claim it is useful to present an analogy that Feynman makes between the women integral and this path integral so the women integral is the usual integral it corresponds to the area A under the curve and similarly the path integral is and this Feynman integral is defined by considering small rectangles with h and we make the sum of these rectangles and we take the limit at the end of the calculation and similarly the path integral is defined by considering a time step epsilon and at the end of the calculation epsilon goes to 0 so in this sense both the definition of the women integral and the path integral are based on discretization and limiting process however there is an important difference in the women integral the limit when h tends to infinity is well defined the limit does exist so we can in that sense discretization disappears but it is not the case with path integrals the limit when epsilon turns to 0 is not well defined in general we have to keep the discretization step epsilon and Feynman was fully aware of this problem already in the seminal paper he said there are very interesting mathematical problems involved in the attempt to avoid the subdivision and limiting processes these curious mathematical problems are sidestepped by the subdivision process however one feels as cavalry must affect calculating the volume of pyramid before the inversion of calculus so even if it leads to complicated calculations involved by discretization we have to keep this discretization time step it's true to define a path integral here I took a quotation in a paper by Kevin Davey on the mathematical rigor in physics and he says it is unclear however how to construct the underlying measure dx so relative the measure in the path integral relative to which the path integral needs to be defined in fact there are even theorems showing that in certain cases one cannot define a path integral measure satisfying the requirement of the physics so in the current state of science it seems that we need a subdivision process and a discretized representation to define the path integral so to define and also to compute path integral some textbooks decide just to mainly use the lattice definition of the path integral to make a computation just maybe two minutes and I'm done I would like to answer your question with these two slides does the path integral lie on these discrete measurements? so we have seen that these discrete mechanics does not involve a discrete time step epsilon however the Feynman path integral is based on the discrete time step epsilon so for that reason the Feynman integral does not generally lie on these discrete mechanics as I said it is based on the discretization of the traditional usual Lagrangian theory not the extended one so in that time is an evolution parameter and not a dynamical variable however we are not doomed to use the traditional or usual Lagrangian theory we can also formulate the path integral with a discrete extended Lagrangian mechanics and as a matter of the history of physics Feynman actually suggested this approach at the end of his paper in 1948 he claims that the subdivision of time into equal intervals is not necessary the total action s must be now represented as a sum and this sum this discretization is precisely that of these discrete mechanics it is a discretization of the extended Lagrangian theory in this condition it seems that we can recover the relation u0 for time symbol with path integral formalisms we have a sequence of steps where the position and time are coordinates in the action used to define the path integral integrals two slides and I stop and again one advantage of this approach is that time is treated on the same footing as position there is no longer a discrete time step but only a number n of discrete steps and as Rovelli and Vito stress look into covariant, look into gravity books at first this looks magic this looks magic the discretization of the Lagrangian does not depend on the lattice spacing the number n appears only in the upper limit of the sum indeed we recover the continuous case to recover the continuous case it is not required to that epsilon goes to 0 but only that the number of steps remains the same so in Li's discrete quantum mechanics we keep n large large enough but finite and we expect to have an empirical equivalent theory and to finish just a diagram provided by Rovelli explaining the links between classical mechanics quantum mechanics on the one hand and continuous and discrete mechanics on the other hand so two limits are involved so the classical limit which is the limit when the constant the Planck constant turns to 0 and the continuous limit which is the limit where n the number of steps turns to infinity so you should recognize here Li's discrete mechanics discretized classical theory as n this is discrete classical mechanics when n goes to infinity you recover the continuous extended Lagrangian mechanics with a Hamilton function which is a function of q and t if you go to the quantum case you have the so you have discrete quantum mechanics Li's discrete quantum mechanics which is the path integral with the discrete extended Lagrangian mechanics and you recover the when you take the continuous limit you recover the usual Feynman path integral so I will not discuss the case of quantum gravity and just maybe to sum up the two claims that I defended but I think it's quite obvious now so I'm done okay and come back for the demonstration because if you refuse holistic confirmation why should you accept this local confirmation well if holistic confirmation does not give me information about ontology why this pseudo-local one should give me more about ontology I think it's your skeptical your skeptical completely and if you agree about if you're a realist you should be a realist completely should trust empirical techniques so I don't understand I see that the argument is quite strong if you say you don't even know how to test structure of time even if you but that's not a question between holistic and holistic it's just that we don't even it's difficult to imagine an expert I'm not sure that that if we reject holistic confirmation you should also reject local confirmation so your question is on Mady's objection if I understand the way I I'm pretty agree but maybe we need more arguing but I'm pretty agree with the the idea that some excuse me some mathematical tools in physical theories does not play the role of physical assumptions and so I but actually the distinction between holistic confirmation and local confirmation is of course not so easy to do because in the sense all empirical confirmation are holistic confirmation because you but when you for example when you want to test if protons are made of quarks or if you want to test if water is made of water cubes you don't just see oh since freed mechanics works so it means that freed is continuous you will to to probe directly yeah exactly no actually yes yes even if it yes so it's a strange metaphor between the structure of matter and the structure of time because structuring of matter can do something I can throw atoms through molecules and that's what happens I cannot test structure of time directly I agree with you so what's the point for this weird discussion of matter because I'm surprised that it's not okay if it's someone else but many is a good philosopher she would not say that for no good reason so I try to understand so you would say that we just don't have any possible confirmation about the structure of time I think direct confirmation of the structure of time it will always be indirect to look at dynamics of stuff probe directly the structure of time I have no idea how you can do that yes but does it mean that you so it's quite holistic all the time but you it's not because I agree with you there is no local confirmation about the structure of time but does it prove that we should endorse holistic confirmation of the structure of time to trust our best representation knowing what you said about equivalence and its tricky and maybe we should be sometimes on the side a little bit it's a very strange kind of empirical inquiry to prove the structure of time I would be very if someone asked me how to do it maybe there is it's not there are some discussions about some local some local investigations about LHC there were some ideas that show some collision maybe you could have some indirect effect about this critical time I just know that there are some speculative works but I have one express life that I could have thought because I knew that you really had this question you saw it a lot of express lives a lot of express lives I see that you work in the same laboratory as Philippine Man so you brought 100 slides just in case so this year so Marius Christou Doulou and colleagues published a paper which is quite interesting they but it's speculative they they tried to to imagine an experiment to test in the context of quantum gravity and they showed that it's not so far from our technology our technology set ups so it's they use an entanglement between two mass particles so they show that the difference of phase between the two the two pass can be proportional to the Planck constant and they show that this kind of experiment could work maybe 30 maybe 50 years it's not so far but it might exist a local experiment about the structure of time but I don't know exactly how to justify it but I think this kind of confirmations are more convincing that just a holistic confirmation by saying we have a theory which uses a continuous parameter this theory works so time should be continuous again because I agree with Penelope and Madi about space and time which are not all actually which are not exactly physical hypothesis in this series they are mathematical tools to use in physical theory but of course it depends on the interpretation of the mathematical variables but I think that she has in mind and they make the distinction between between physical hypothesis mathematical assumptions I have another question coming that I can express so I really like your talk and especially the Marsden there but I didn't know I have to check because this is huge this is major this is something I don't know why in quantum gravity for example this would be a strong reason to go to prioritize quantum loop contrary to super-strength but in a sense Lee and all these guys they just changed subject so it's not the problem in a certain amount of time anymore that was important for our people people that were interested in discretization of time in the 40s for example they had in mind discrete succession of states so there is a continuous evolution of states and physics quantum mechanics will be a little bit more complicated there will be some discretization of succession of states so that's Hamiltonian formalism when you go to Lagrangian formalism Lagrangian formalism the essence of it is history beginning and then stuff happening there is no notion of states in Lagrangian mechanics in fact you can find when you do the to do a objection with the Hamiltonian to get the notion of states and succession of states but in Lagrangian it's why it's so open to the extended version it's very natural because you have the basic entities history it's already something extended in states and times so to discrete that through the extended version it's extremely useful for all kind of reason to go to quantum gravity but this is not exactly the project of the people that wanted discretization of time before it's it's discretization of tau and tau is a free gauge so it's something else so I see I like the new project because the new project will give us maybe to quantum gravity but I don't see how it's related to the ancient project about time as a real thing is discrete now it's the free gauge parameter that is discrete which will get us to interesting this greatness in the quantum field theory and in the quantum gravity theory after that that we need because it's a lattice not this time that it was before it looks like they changed the subject they just didn't say it they just said follow us but it's not the same subject at all maybe it's a better subject you won't say to me but it's not what was motivating people but what is the ancient project I thought that the ancient project was people saying ok we have a lot of problem with continuous time because it's differential equation now we have to go to another project which will be discrete time time itself like space will be discrete like a lattice now it's a lattice but in an abstract abstract space of gauge parameters I'm sorry but so is it about the least project no I think the least project is not the continuation of the ancient project it's a new project yeah but what do you mean exactly by the ancient project the projects where space and time were real discrete entities but who for example who people prime but the fine man himself was joking with that at the beginning I think it's why people were fascinated by the discretization of time and space because they thought it was real time and space when it's the gauge parameter as you said Q and T take their values in the real they are they are they are not continuous they are to the parameters but they take their value in a continuous real yes perfectly so it's another project again I'm not sure it's two projects but again this is the same answer to Michel even if variables are real valued parameters they can be continuous power discrete in this project Q and T are discrete even if they are real even if they are real numbers be careful they are not discrete the dynamics is discrete because now they are dynamical variables that have steps in this gauge free parameter which is not time so it contrasts with discrete Hamiltonian discrete Hamiltonian you have a state and you have another state and it's discrete and you have another state in this gauge gauge formulation which is the good one we know for doing quantum gravity you don't have coordination between space and time coordination between space and time but since between them it's not even time it's a gauge free parameter it's not maybe there's an order these are not states these are order elements of reality whatever that means and when you switch to quantum mechanics you have these transition elements and you go to the discrete transition element which okay and when you go to the transition element in the quantum one on the left on the square that you present you don't have you don't have evolution at all you have only correlations between final and first state first state beginning and end of history that's it you don't have evolution anymore it's why there's some passages that are very possible there is change in this version in the corner left of your slides because change is supposed to be true in time or at least in time steps traditionally maybe more wrong I don't have to think about it I don't know it's up to you I don't have a good question because I agree with you it's very nice so first of all I had the same question because it seems to me that when we discretize the parameter tau what you're saying is the space Q is the time Q is continuous but we are sampling at a discrete rate but fundamentally the mechanics behind is still continuous and when you discretize the gauge you're just discretizing the sampling time and since you could change the parameterization and discretizing is this new parameter discretizing the trajectory because you could in theory always change your parameter enough to get all the continuous time input so it seems to me that the thing that you made a good account of there is a description of time that you remove time from being a fundamental thing that acts on the system and instead being simply a descriptor of the order of relation of things but at least to me the linear notion is still compatible with continuous time and then I'm not convinced much by having a discretization of the sampling or of the parameterization tau instead of simply taking the continuous time because if you're still relying on the assumption that there's a fundamental continuous physics behind you you don't gain much or conceptually you don't gain much thank you I think there are a lot of things what you said first I perfectly agree that linear view point is compatible with continuous time what I said is if you discretize if you use a discrete representation of time it seems that you and you want to have a good physical series with good equations you should adapt the linear view point but if you perfectly agree with the continuous representation of time you can either adapt Newtonian or linear view point but about the experiment about just sampling or with discrete series I think it's a very important question and it is related to the question whether discrete series are autonomous or independent from continuous series in other words does discrete mechanics need a continuous theory to exist to be formulated I think it's related to the quotation when he says that let's start with discrete equations and consider continuous equations are approximation of discrete equations but is it possible or not so the question is discrete mechanics is autonomous independent with respect to continuous mechanics and I think it is and again in my PhD I defended the claim that discrete mechanics is an autonomous theory it's independent of continuous classical mechanics and to support this claim you just a small argument you just since you discretize the first principle the list action principle you don't need all the stuff of the continuous classical mechanics if you if we tend to regard discrete mechanics as just a sampling continuous mechanics maybe just because we learn before continuous mechanics we of course we think with a concept of continuous mechanics but try to remove all this background and just start with discrete list action principle you can derive discrete other Lagrange equations and we don't need the continuous mechanics to to derive discrete Lagrange equations excuse me you draw the ladder after climbing it I don't think so you draw the ladder after climbing it it's a catch time code so you use a tool then okay yes maybe just to complement in the Lagrange in the Lagrange but in the Lagrange this is the more natural way to see Lagrange it's not defining history to time defining history to anything because it's an order because time is inside the picture is not from outside contrary to the Lagrange mechanics so this is the natural way to understand Lagrange even the continuous one really continuous one that's the right way to understand Lagrange it's to see time as a dynamic variable so the discreteness as you said could be from the start but it's maybe not the discreteness that someone is looking for when they want discreteness of all the quantities or something like that but if you're a Lagrange or a Lagrange or a Lagrange this is the way to go it's abdomen that you define you define change by change of states here there's no states so why not doing that it makes sense the more a good mathematical way to see Lagrange mechanics but the problem is that we say to every student in physics that these two formulas are exactly the same because they are isomorphic but they are completely different even if most of the time they give us the same answer they are structurally different mathematically different and now you can convince us with the master period that we should all be Lagrange mechanics if we want discreteness thank you for your talk very interesting as part of some of the questions because I agree with what you said I really like what you emphasized the different decision presentation of time physical time most of the issues you have which is because of this because of this lack of distinction but my question is what is the point that we didn't have time to talk about contourality more and I wanted to know if this program goes end up in the same direction as Roveisman because they start on the same same idea and they have more or less the same end that you have discreet the artist of the front so I don't know if it's exactly the same problem but you haven't yet or if there are differences that are interesting too good good question and I have to say that I'm not yet familiar with Liz with the contourality in this project so it's for future but just maybe I can say that they are these are two different directions in a sense why I mean for several reasons reasons but maybe I don't want to say a first difference that I see is that in Liz contouring gravity we have a fundamental length which is introduced to remove the ultraviolet divergences and I think it's not the case in Luke contouring gravity after the recalculation we have all these spines he introduced some cut-off length to remove the divergences in what? in the recalculation after all the recalculation he has to introduce some cut-off to avoid the space of the divergences we have a natural way to get rid of that I have to give you a good answer but just another answer is it seems to me that but it's a question of interpretation and not a question maybe of theoretical results Liz is time to interpret statistically the discretization of spacetime in the discrete contouring gravity conversely I think it's not the case in Hoverly's view and there are a couple of quotations of Hoverly when he says that contouring gravity is often confusing when boundary amplitudes is often confused with the quantum discreteness of space the use of discreteness of space is not given by the fact that we use discretization in the theory so if we compare Liz and Hoverly's quantum gravity it seems that unlike Hoverly does not see any possible realistic interpretation of discretization but maybe it's just a matter of interpretation and not theoretical differences and maybe maybe a complete one but I think that Liz's quantum gravity corresponds to is close to lattice quantum gravity while Hoverly's quantum gravity does not fit with lattice theory because there is no lattice step in Hoverly's theory so we have to learn more about that but it's a little point discretization comes later in Hoverly but at the beginning of the theory but if they get to the same theory that would be interesting interesting philosophical question of how which one should be interpreted in a much way yes because the interesting point is that they start both with time as a dynamical variable maybe maybe zooming out just a bit trying to see what the take-home message here would be because it seems that you are supporting this project that it's possible to reformulate all these physical theories whether classical quantum or quantum gravity by discretizing time would you then say that whether time is continuous or discrete is a convention would that be is that kind of a claim my claim is that the representation that was considered as discrete is a convention yeah I think I'm still confused between that distinction because I was always making the analogy with the discussion on whether in a special activity whether simultaneously this and simultaneously is a convention or not and there those who seem to defend the idea that the notion of this and simultaneously this is conventional can read it in two ways that it can either read it epistemically or on the claim it seems to be similar to one of the slides you showed by Newton-Smith I think where he said that either there's just no apparent way for us to to verify this claim and we'll never know whether two events are simultaneous or not but there is a fact of the matter as to whether they are simultaneous or not whereas the on the claim would say that there's just no such thing as distance simultaneously there's no matter of effect whether two events are simultaneous as we could apply the same here either we just have no way of verifying the structure of time or you may say think it ultimately like most people subscribing to the convention of 1782 that there's just no matter of effect about the discrete or continuous nature of time which in this case seems to me strange something when I when I push me to say that push you to say something that the representation of time is a continuous representation of time is a convention maybe it's a way of speaking I mean the the continuous representation of time is maybe see as a convention framework to describe physical framework so it's not a convention in itself the continuous representation of time but it's a framework and this framework we can use it or not we can use this framework or another framework in that sense it might be used as convention even if it's not a convention to say time is continuous representation of time is continuous yeah but it's just me that you want to resist going from there to any better physical conclusion about time itself yes yes and is it the same really? I'm not sure I think I think Lee is taking a much more realistic yes can you show the the theorems the theorems the theorems the theorems the master okay why is it that's right before I think so because I I don't know I'm sorry because I have no slide about the theorem you mentioned one yes this one in another world okay so according to Marsden Marsden's theorem the usual formulation which presupposes a continuous time the preferable formulation is continuous time or not no no I mean among different way of different discretizations the preferable discrete formulation is the variational okay alright yes more precisely but I have a paper and a plan B and we have a section maximum number of symmetries and conservation laws with respect to other discrete discrete formulations but then these symmetries and conservation laws are the same as the ones that we get from a continuous approach of time so there is equivalence at this level there is an equivalence between the discrete preferable formulation which uses a discrete time on the one hand on the usual formulation yes so there is no way to decide on the basis of symmetries and conservation laws for which we have some empirical evidence to decide between the discrete and continuous I have another question maybe just one difference but it's not an empirical an empirical difference is in the discrete case actually tau or the discrete tau is no longer a free gauge so you break the invariance of the reparameterization when you discretize it but you can recover it in three cases in the infinity means in the free particle system or in another case which is pretty tricky because it's perfect you can always find a way to express your discrete way to recover invariance of the reparameterization but it's not a natural way to have a gauge of the reparameter but the other question is continuous with what Alexandre said you said that the discrete approach is more congenial with the relation of view of time and you say a luminescent view of time but if you take the customary luminescent approach then the relation of time is based on states of events which are states and here the the discretizing is not based on states it's based on something else so you can ask what kind of reality are these correspond to something else because of course I've worked in a conventional framework there were states following each other but the lagrangian formulaism is a holistic a holistic approach with no states so what does it mean to have a relational theory of time within this holistic and regent approach good point yes that's what you said if you define a philosophy where time is related to change for example maybe they are not the same but they are closely related and you have something that it's not obvious what is change in the deformation of mechanics it's not obvious you can interpret some kind of change but it's not obvious because now the order of step constituting a history are just a name an order so someone that would say for me time and change must be together not obvious when you put the four theories there's a weird an interesting but weird passage of idea about discussing exactly the new conception of transition element discrete transition element where it says we recover a new notion of change no explanation there but he obviously felt felt in the book that he had to say something because it looks like change disappeared of the theory and he wants to say that physics is about change so he said we have a new notion of change which are represented by these transition, discrete transition amplitude what does it typically mean what kind of change is it I don't know at least it's very interesting but for us it's difficult to connect it with other notions of change maybe another interesting project would be to study what happens when you discretize extending Hamiltonian formalism maybe you're you convince us it's not the right way to go I agree with you last question before the beer really just I have a last question the tables are turned because I have to be fair and actually I think there is an objection to what I said I have an extra slide I would like to talk so in a recent paper Beethoven and Docher just mentioned the Feynman path and they say that it's not discrete at all it's just piecewise continuous function so it's not there is no discrete physics there is just non-differentiated physics but everywhere continuous because a function everywhere continuous even not differential at the corner so maybe I have to answer to this objection because I think it's almost the same discrete mechanics do we have discrete sequence or do we have piecewise straight line ok so just we can just answer that first I do not argue for discrete physics but just discrete representation of time but is it really a discrete representation of time or just maybe a discontinuous representation of time so yes I have to think about it in what sense it is really discrete and not just discontinuous but not differentiable that's my last question if you have an answer ok I was wondering what's the domain of the foundation that we are considering because as an example here technically an internal foundation is a discrete foundation of mechanics can take as a because you have states of discrete so as a matter of fact it was part of the less non-disformational discrete recognition how does it feel out of the domain of the field I'm not sure I have to guess maybe we are using non-disformational mechanics all the time so if the theory says that the discrete foundation mechanics is better than the non-disformational non-disformational mechanics it's kind of a strong business yes I have to look at it sorry sorry thank you thank you thank you thank you