[Music] is [Music] [Music] so we start with the most basic concepts of all in physics mass length and time and i would like to start by asking you to imagine that we have no tools no mathematics no physics no equations but just our senses nothing more than that and we ask what kind of masses lengths and times can we perceive with just our senses and nothing more than that so no instruments of any kind and no analysis of any kind just our bare senses and let us start with mass for example and ask in terms of masses m and will use standard international units throughout what is the smallest mass that you think you can estimate by holding it and weighing it in this fashion for instance what do you think is the smallest mass a gram certainly a gram you can tell you can tell the difference between a gram and a kilogram with our intuition how about a nanogram and a picogram could you tell the difference you could do this certainly a fraction of a gram would be a safe estimate so let us put this down and mass in kilograms let us put this down and say that you can start estimating something which is of the order of a gram or maybe a fraction of a gram so that is 10 to the minus 4 kilograms and what is the heaviest mass you think you could estimate 10 kilograms you could certainly tell the difference between 10 kilograms and 100 kilograms certainly how about a thousand kilograms well you wouldn't be able to lift it that's a good point and in fact if i gave you a lump of metal and didn't tell you its density then you have no way of knowing whether its ten thousand kilograms or a hundred thousand kilograms or a million kilograms at all so whatever you can push just about is the upper limit may be 100 kilograms lets base safe and have another order of magnitude a thousand kilograms so thats 10 to the 3 kilograms the upper limit of what you can do with just your senses what about length and let us measure this in meters what do you think is the smallest length you could perceive with your eye with the naked eye a fraction of a millimeter maybe half a millimeter or something like that so we'll play it safe once again and i'm going to say that millimeter is 10 to the minus 3 and i say very sharp resolving power so 10 to the minus 4 meters for example up to and what is the longest distance that you could estimate without instruments well you cannot tell the difference between the distance to a planet and the distance to a star with the naked eye unless you have other pieces of information and on a clear day if you stand on a very clear place and look up from a mountain peak you could perhaps see 10 kilometers but you see the point is if you did not have other objects for reference you have no way of estimating how far things are if you had a blank wall with no texture on it and no sign no signs no way of distinguishing what relative scales are and so on then you have no way of knowing how far things really are but perhaps 10 kilometers is a good estimate you certainly can't tell the difference between a thousand kilometers and 100 kilometers pardon me you aren't allowed to learn you don't know i am saying with your bare senses that's it of course if you start running and you say what you know endurance is and so on that's a different story but let us just say you are standing and you are trying to do an experiment to see how far you can see 10 kilometers safe thing thats about 10 to the 4 these are just orders of magnitude 10 to the 4 meters and what about time what is the smallest time that you could perceive in seconds or the blink of an eyelid certainly your pulse is of the order of a second but you can measure time scale smaller than that the blink of an eye that's about how much a fraction of a second 10 to the minus 1 seconds 10 to the minus 1 seconds incidentally while we're talking about it why do we blink you would like to have a tear layer certainly you would like to clean the eye every now and then you would like to have a layer but how is it that when i blink i'm sorry i'm going to go off into digressions of this kind how is it that when you blink things don't go off yes but if i did this experiment we blink in the tenth of a second now and we blink maybe every five seconds or so every five seconds if i switched off the lights in this room it will be tremendously distracting to you this can be done this is a very easy experiment to do you find it tremendously distracting if every five or ten seconds lights went off for a tenth of a tenth of a second at a time that's what you do when you blink but it doesn't happen when we see when we look around why do you think that is the case it's a close answer it's a close answer the processing section of the brain closes down simultaneously otherwise you won't have this feature at all very cleverly designed it will be tremendously distracting otherwise so even the processing center closes down that's the reason why you don't feel that you're blinking unless you do it consciously of course close your eyes then of course you know that it's dark and so on but otherwise a reflex blink you do not see as a shutting off of the lights at all because even the processing is not done during this period but anyway to come back here it's a tenth of a second here and what's the longest time that you can actually perceive without any instruments without any external signal without the seasons without the stars and so on a little more than an hour people have done this experiment they've actually put people in cells with all the comforts of life provided with uniform lighting so they can't tell the difference between night and day with the same bland food so you can't tell whether it's you know breakfast or lunch or dinner and so on provided there in a cavity you open up this recess and you get this food very familiar from the hostel life right they're trying to get you used to this and then you are supposed to go and press a button every two hours or so and pretty soon you realize that the experimental subject starts pressing the button every three hours or so or every four hours or so so she assumes that a certain amount of time two hours has elapsed when really three or four or five hours have elapsed and eventually even the rhythms of the body get knocked out and gradually all the cycles change and i think this experiment has been done and people have gone to the stage where they find they think a day is like 50 hours long so your perception of time goes off unless you have other indications you have other longer cycles and so on but even the physiological cycles change everything changes along with this so i would say maybe you can tell the difference between a month and a year but you certainly can't tell the difference beyond that i will play it safe once again and say this whole thing on the right hand side the upper limit is of the order of 100 days for instance and a day is about 10 to the 5 seconds so 100 days about 10 to the 7 seconds that's the world of middle dimensions the world in which we have some intuition but now we start asking what's the universe doing on what scale is that operating then of course you're in for a tremendous surprise this is the world of middle dimensions would like to put this in quotation marks the macroscopic world in which we live in which we have got used to there are other parameters that other physical variables like velocities and so on which also have comparable numbers but we have started with mass length and time and let us stick to that now what do you think is the smallest mass that we know of with our instruments whatever we have detected what is the smallest mass we have detected what do you think is the smallest mass that you know of uh we've gone smaller than that an electron certainly is less massive than a proton so it's how much is that about 10 to the minus 30 kilograms so let's say 10 to the minus 30 we don't know of particles with smaller masses than that at the moment but let us say 10 to the minus 30 this is the mass of an electron and there's a vast gap between these two guys here on the upper limit what's the largest mass you can think of the mass of the known universe certainly all the matter in the universe we don't know how to estimate this we do not know the size of the universe too well but what do you think we should write here how big do you think it is we could do the following we could assume that we have a whole lot of stars and try to estimate the total mass of the stars the total mass of all the stars would mean multiplying the number of stars by an average star like the sun for instance now what's the mass of the sun about 10 to 30 kilograms we could estimate this and remember the way we would do this is always to assume that you are on a desert island with no information no wikipedia no internet and you have only sand to write on and you need to make all these estimates for your survival this is the way to learn any subject you're compelled to do this then of course i could start by saying the sun is a hot ball of gas i estimate the average density of a gas multiplied by its size and then and so on and so forth but 10 to the 30 is about right 10 to the 30 kilograms and how many stars are there about 10 to the 22. there are 10 to 11 galaxies and about 10 to 11 stars in each galaxy so that gives us 10 to the 52. and this thing here is an estimate of course you could do this in another way you could start by asking what was the density what is the density of matter known by ordinary matter in the universe and then you could multiply this by the radius of the known universe you can multiply this by the size what is the size of this universe that's a very tricky question very tricky question it depends on what you mean by size you could say a very naive way of doing this would be to say well i know the age of the universe and what's the age of the universe 13.7 billion years old 0.7 that's important it's known it's established the first decimal point is known you multiply that by a light year because it's that old and this would mean that this is like the radius of the edge of the universe stars are receding from you at the speed of light essentially so this is a good estimate and when you make that estimate it turns out the answer is about 10 to the 52 once again it comes right there there are other ways of doing this of course this assumes that you are at the center of the universe and the rest of it is simply expanding away from you but that is not necessarily true at all but there are similar estimates which will tell you the co-moving radius of the universe and it's of the order of 10 to the so 10 to the it's of the order of 40 billion light years or so it's in the same ballpark and therefore the whole thing is we'll assume is 10 to the 52 on this side of course there's dark matter there's dark energy and so on and forget about that but whatever it is for the purposes of this argument it is of the order of ten to the fifty two nothing what about length what is the smallest length that you know of well certainly experiments on electrons have told you that they have no structure whatsoever we have done nuclear physics we know the size of a nucleus is 10 to the minus 15 meters a femtometer and go beyond that inside the nucleus etcetera but we can construct we can conceive of an extremely small length from the three fundamental constants of nature planck's constant speed of light in vacuum and newton's gravitational constant these are the three fundamental constants of nature and they are very very important so you have planck's constant the speed of light and newton's gravitational constant those are the natural constants mass length and time are some things which we have created in some sense but the constants we have available in nature have different dimensionalities not mass length and time necessarily what are the dimensions physical dimensions of planck's constant till second is units now what are the dimensions yeah its energy multiplied by time so it is m l squared t to the minus one and c is of course l t inverse and g you can find from newton's force equation so with these three constants it's clear you can create three numbers three combinations which have dimensions of mass length and time and they would be called the plant length the plant mass and the planck time and the plant length turns out to be of the order of 10 to the minus 35 meters construct this as an exercise a combination of h c and g which has dimensions of length and that is of the order of 10 to the minus 35 seconds so this is l planck on this side much much smaller than this and what is the longest largest length we can think of for the radius of the universe we have got various estimates for the size of the universe but it is an upper bound we do not even know if it is finite or not but let us write this down as 10 to the power maybe 25 or 26 meters very roughly but is enormous as you can see what about this what about time again planck time is the smallest time we could construct we could think of as time itself and that turns out to be of the order of 10 to the minus 42 seconds and this is t plus this here i should write mass of the universe and this is radius these are in quotation marks and what is the longest time that you could think of the age of the universe which is 13.7 billion years therefore this thing here is about 10 to the 10 years which is 10 to the 10 multiplied by 10 to the 7 so 10 to the 17 seconds of the order of 10 to the 17 seconds [Music] age of the universe but remember these are orders of magnitude and now just think this range here is invariably about seven to eight orders of magnitude give or take one or two orders of magnitude very tiny window but nature is operating on a much much bigger scale it is operating on a length scale mass scale which is like 80 orders of magnitude to the extent we know about it a length scale which is at least 60 orders of magnitude possibly much much more maybe even infinite on a time scale which as of now is already about 60 orders of magnitude it could become much much larger and these are orders of magnitude these are not just factors you are not just doubling or tripling the range you are actually multiplying by 10 each time and this is absolutely incredible and it is just mind blowing to see the range on which nature is operating therefore i put it to you any intuition which you develop to conduct your daily lives in this world of middle dimensions there is no reason why that should continue to hold in this range there is no reason at all and the so-called understanding of classical physics the intuition you develop and so on the physical intuition is a myth it's simply something which has been hardwired into your brains for reasons quite different from the understanding of nature therefore there is no reason to expect that whatever laws whatever regularity you find in this range of orders of magnitude should continue to hold good in this range on either side and indeed it doesn't so with the induction of instruments microscopes on the one hand and telescopes on the other you really able to broaden your range in which you can probe nature all the way from this side to that side and simultaneously it turns out you need other tools you need mathematical tools it turns out in order to understand how nature operates in this scale so the real miracle is that you have a language and you have the means to understand this range in some sense even though it's not necessary for survival in some now you could ask why is it that i can't perceive of times smaller than this why is it that i can't look at i can't tell the difference between a picosecond and a nanosecond why do you think that's the case you don't need it you don't need it for survival you don't need it evolutionarily for survival the time skills you needed to put it very crudely this is a delicate question but you can give a very rough very crude argument after all it was all about survival and what you needed was to have a reflex time that was sufficiently fast to ensure your survival to put it in very very graphic terms is certainly not to be taken literally our primate ancestors had to make sure they didn't fall down into the jaws of predators depending on the gravity the rate of fall is controlled entirely by the gravity depending on that you needed time when you let go of your mother ape and you started falling you needed time for the muscles of the hand to send a message to your brain and the brain to send a message back to the muscle to say hold on or you're going to fall down a fraction of a second was sufficient for that you didn't need a picosecond you didn't need a femtosecond so you didn't have to waste brain power and neurons processing that information you got hardwired into this world and that was enough on this side all these things are guided by that mass for example you do not need to know the difference between a microgram and a picogram but you do need to know the difference between a gram and a kilogram and once again in this metaphorical language it's essential to know it was essential for our ancestors to know that it would be much more effective to throw a rock at a predator rather than a leaf so they had to know the difference between a gram and a kilogram they didn't need to know the difference between a microgram and a picogram and that's why the brain didn't waste any time trying to process information from this range or on that range and this is why you think you understand newtonian physics because you see masses you see things you see times you can actually push these rocks around and so on but i put it to you that there's nothing intuitive about it it took a long long time to discover that when you push an object you change its velocity and not its position you change its position as a consequence of changing its velocity after all newton's law says that the force is proportional to the rate of change of the momentum or the velocity not the position only bacteria which are swimming in a nutrient fluid at terminal velocity only for them is the force proportional to the velocity but not for us so even newton's law is counter intuitive very counter intuitive and its consequences can be equally dramatic so don't confuse facility in a certain range of mass length and time with understanding of this range of length mass and time the very different things and it turns out that the language you need for understanding making predictions is mathematical inherently mathematical we do not know why we do not know the deep reason why we have no reason we do not understand as yet why it is that our brains which were hardwired evolutionarily for survival in a certain range of parameters has been able to come out with deductions using a language and abstraction called mathematics which enables it to probe the rest of the region and why it is that with the aid of these instruments which we have developed we are actually able to investigate the rest of this range a good bit of this rain and understand it in some codified sense so it's not a that's a surprise that's a surprise not the fact that electrons behave like waves or like particles or whatever you might have heard not the fact that newton's inverse square law of gravitation although universal is actually an approximation these are not surprises really it would be a surprise if it weren't so if everything was decided by a few simple equations that would be a surprise and no reason why that should be so and it isn't so just isn't so so classical physics is one portion of this range which we've been able to uncover using rather simple rules but not necessarily trivial rules fairly complicated rules and as the course goes along i will show you that classical dynamics is actually quite intricate has a very very precise structure a very interesting structure and that quantum mechanics is a very non-trivial extension of this and the relationship between the two is not yet fully understood we do not fully understand what is going on in quantum mechanics in a certain sense but i will also point out that in a particular sense quantum mechanics is easier than classical dynamics which is much more intricate mathematically much more intricate and then of course there are other problems associated with the rest of the curriculum such as statistical physics what happens when you have large collections of objects how you understand them why you need probabilistic concepts why you at all need statistical concepts this is something which is worth understanding and it will turn out that we will acquire at the end of this course hopefully some perspective on why all this is happening and where we stand today and what kind of progress we could expect and this would of course come after you finish the next course but for the moment we will stick to classical physics in which we will switch off planck's constant and this curriculum also does not have much about gravitation although i will mention this once in a while so it will be non relativistic it will be non quantum mechanical and of course we will ignore gravitation for some part of the course at least so it turns out that you might as well set you eliminate g set c equal to infinity and h equal to 0 and that would be the classical physics we are going to look at but you must be aware then these are boundary these are limiting cases that the rest of it is really part of a much bigger hole so this is what i would like to convey in the rest of this course so is there any question on what i have done said so far wherever necessary i'll use orders of magnitude estimates and wherever we think we need to do something more rigorously we'll look at those things we'll work out things much more explicitly i'll try and do everything on the board so that all the equations are understood understandable but there may be cases where i might just code a result or quote especially mathematical results i might once in a while quote them and say these are well-known theorems and i won't bother about proving them but we will understand them we will assume that the rigorous proofs are available and we will try to understand them on some more physical terms now when i say physical i would like to state we explain that i do not mean necessarily mechanistic everything need not be mechanistic at all for example electric and magnetic fields exist we know that they have classical limits classical electric and magnetic fields exist but you cannot give a mechanical model for it not in terms of wheels gears pulleys and so on this is not possible their fields every point in space and time would have a field and they may or may not be detectable or perceptible to you with one instrument or another but it doesn't mean they don't have a reality they don't exist they're not hard objects like this so the universe does not just consist of rigid bodies it just does not consist of classical waves there is much more to it than that for example electromagnetic waves which are already in that sense non mechanical they already go beyond your normal mechanical intuition and yet they are completely classical in a certain sense so you have to allow for larger possibilities as you go along so this is the sense in which i would say something is physical i assume also i might as well say this right in the beginning i assume also you are familiar with complex numbers we will freely use as much mathematical tools as we can and i have been asked this question in the past if all measurements are real why do you need complex numbers that's a good question but it should really be asked in class 12 in that stage it is just that you need these numbers you need matrices not real numbers necessarily you need combinations of numbers you need n tuples multiples of numbers and so on and complex numbers is one such thing so the word imaginary number does not mean anything as far as i am concerned so if you ask me what is the physical meaning of two plus three i i would ask you what is the physical meaning of minus three for that matter what's the physical meaning of three halves or what's the physical meaning of three okay these are all abstractions as you can understand and we'll try to put them into correspondence with physical objects and that's all that's being done here so it's simply a code and this is the sense in which we would like to understand things now the first part of this course has to do with dynamics but before i do dynamics i would like to mention here something along the lines of what i have already said and what comes beyond dynamics and this is a very famous picture very famous diagram which i try to reproduce here in some sense you have ah will put the axis in a little later you have at very very slow velocities compared to the speed of light and on scales length scales much bigger than atomic length scales or microscopic length scales you would have non relativistic physics in some sense of macroscopic objects so let us put a little box here and say this is the region this is roughly the region of non relativistic classical mechanics or newtonian mechanics you could continue to remain non relativistic but you go go into domains which are extremely small atomic dimensions for instance and then you would have quantum mechanics so you would have here adjoining it very blunt very crudely you would have non relativistic quantum mechanics this side on the other hand you could look at fairly large objects but go into high velocities in this direction and then you would have to take into account relativistic corrections this would happen typically in astrophysics and certainly have to take into account relativistic corrections you would also might have to do this along with this thing here along with this for if you look at motions of electrons for example but we will come to that in a minute so on this side you have relativistic mechanics or relativistic physics you keep going could include gravitational fields very intense gravitational fields and so on you would have special relativity then you might have to make general relativistic corrections and so on on this side very schematically but then you could also go to very high speeds and very small objects and then you would need relativistic quantum mechanics on this side and that is where a little bit of surprise comes in because it turns out that when you do relativistic quantum mechanics you could start with this kind of problems we do in physics academic problems very idealized problems just as in thermodynamics you start by studying the ideal gas there is no ideal gas in nature of course everything interacts with everything else but just as you use that as a useful model similarly in quantum mechanics you might study for example the hydrogen atom a single hydrogen atom in the universe is a very very simple idealized model and you could study its quantum mechanics as well so you could look at the problem of particle orbiting around an attraction as for an attraction center here and then you could do this quantum mechanically you could do this relativistically but the moment you go to relativistic quantum mechanics it turns out that a new phenomenon occurs and this of course is the famous equivalence between matter and energy it turns out that mass and energy are just two ways of saying more or less the same thing since matter can be intercon matter and energy can be interconverted into each other things like the number of particles is no longer a sacred concept when it comes relativistic quantum mechanics because matter and energy could go on getting interchanged now once that happens you can easily see you could expect that there is no such thing as relativistic quantum mechanics for a single particle it is not likely to be consistent simply because you could have many many particles created if the single particle is very very energetic and then annihilated once again and reconverted to energy this happens under certain constraints and conditions but the principle is clear that this could happen and therefore it turns out relativistic quantum mechanics is intrinsically not really consistent and what you need is the possibility that you can have any number of particles included in one single theory and that leads you to relativistic quantum field theory so really this thing is superseded by quantum field theory and that has turned out to be the most successful language of all its necessarily very intricate very involved but in principle it explains all that we know so far and its like the culmination is like the ultimate theory at this level of understanding that we have now of all the universe around us but there are very very important gaps one of them of course is familiar to you from reading popular literature we found no way in which you can consistently combine relativistic quantum field theory with gravitation we do not yet know how to quantize gravity that is one problem the other problem is the theory that we have now for elementary particles at the fundamental level its called the standard model of particle physics is itself an ad hoc theory it is obviously and it is very obvious from the fact that it has many undetermined parameters many parameters which you put in by hand many constants it is clear it cannot be a final theory of particle physics there must be an underlying structure to it we do not know what that is so its still incomplete but the day is young this whole thing is about 100 years old as you can see we started here about 400 years ago we reached this fairly fast and the day is young there is a lot more to come but it is a good idea to have this perspective that this is where it is at the moment but you do not necessarily need all the complications of relativistic quantum field theory if you want to look at sub portions of this for instance if you want to design and this is brings me to an important point the whole thing is layered in such a way that depending on the regime of physical parameters depending on the physical problems you are looking at you might find it sufficient to use an effective theory you do not have to use first principles theories all the time if you would like to design the bet a better carburetor for your car there is no reason why you should know the underlying organic chemistry of the fuel there is even less reason to know that each molecule is made up of atoms and each atom in fact is made up of a nucleus and electrons and inside the nucleus there are nucleons the neutrons and protons and inside the neutrons there are quarks and they held together by things called gluons its not necessary for you to know that in order to design a better carburetor although thats true so this business of reductionism should also be carefully examined while you would like to understand how reductionism go to first causes in order to understand things from very basic principles you must realize that at every level of organization there is a set of effective laws and this is really all you need the boundaries between two regimes is very interesting always these boundaries are interesting where classical mechanics stops and quantum mechanics starts is it a fuzzy boundary is it sharp these are very interesting questions where non relativistic physics stops and relativistic physics starts off this is again a very interesting question on the other hand inside a domain you could have an effective theory this is what happens if you study elementary chemistry if you would like to understand reactions you have a law called the law of mass action you can derive the law of mass action from more fundamental considerations but you do not effectively need to do this if you want to understand chemical reactions in the large so this is another thing one must understand and appreciate that although there are these fundamental theories they may not be necessary in all cases in most cases they are not what you need is an effective theory an effective model and above all you must recognize that when you take a whole lot of objects and put them together the sum the product the end product may be greater than the sum of the parts there might be properties which come out for a collection which don't exist in each individual component individual atoms do not have colors but when i put a sufficient number of atoms together of an object it acquires a colour for example and this is a property that emerges due to the fact that you have a collection and then a very interesting question is at what stage does the color emerge and that is true for every one of these properties physical properties so we must understand that this is also possible individual water molecules do not do anything very interesting but when you put a lot number of them together it turns out depending on the external conditions they could exist in different phases same interaction the same interaction between two water molecules under different conditions could give you either ice or steam or water so clearly this is a property of a collection and not a property of an individual molecule and that is another crucial thing we must bear in mind that there are these properties called emergent properties which emerge when you put a lot of things together and one of our preoccupations is going to be with these properties so this is another thing which i would like to write down here emergent properties and effective effective models i use these phrases once in a while to mean precisely what i have just explained for example newtonian mechanics is an effective model in a certain regime or of parameters of length mass time velocities angular momentum and so on so is quantum mechanics as far as we know it is possible that one day it subsumed into a larger theory or a larger framework but the difference between physical laws and mathematical laws is precisely this whereas mathematical laws once you lay the axioms down would appear to be applicable in some absolute sense physical laws are always applicable in some range of physical parameters you go beyond the range the law may or may not extrapolate in a smooth way and this is also something which one should bear in mind because this is very often lost sight of that's what leads to questions of the kind is the electron a wave or a particle as we go along i will convince you that the question itself is meaningless an electron is what it is and what properties it displays in fact that's going to be your attitude if i ask you what this piece of chalk is every description every definition that you think you give for what this piece of chalk is will be a statement of one of its properties so i am not going to worry about what this chalk really is and use the word a piece of chalk for a pre agreed upon collection of properties of this object so that is going to be a shorthand for a collection of properties which you have agreed upon then of course you see that there is no difficulty with defining any object an electron is a shorthand for the collection of its properties it might turn out that these properties would depend on how you probe these properties and indeed it does in this bigger range that i talked about and therefore i have no conflict at all and then the question of whether things like electrons or waves or particles becomes a question of semantics it becomes a question of failure of the words wave and particle to apply in that regime in an unambiguous manner i need another language but i have one its called quantum mechanics so i do not regard this wave particle duality as a mystery i just think its a failure of ordinary language and of course it would be a very unhappy situation if i didn't have a language instead of ordinary language but fortunately for us we have discovered that there is such a language and we will use that language when the time comes so i hope we have got our philosophy right because the rest of it is not going to be so descriptive is going to get considerably more quantitative and let me stop once again and pause and ask you is there any comment or question that you would like to ask now so why is it that the combination of h c and g gives the lower limits of the blank length and it's a wonderful question why is it as a deep question why is it that the combination of these three gives the lower limits of what we know and not the upper limits and so on this is not a very ah it is not a serious problem in the sense that i do not know the upper limits i do not know for example if the universe is finite or not its unbounded but what i do know is that there is a fundamental velocity the speed of light there is a fundamental quantum of action planck's constant and there is g now in the case of planck's constant it turns out that this thing here actually describes certain fluctuations a certain indeterminacy which you are familiar with in the guise of the uncertainty principle and its numerical value is such that it happens to give you very applicable in the range of the small the very small this is where fluctuations would play a role very rough answer very rough answer but we will get more precise on this as we go along now of course it is not always true that it gives a lower bound if i look at mass for example i was careful not to use this i didn't use the planck mass the planck mass if i calculate turns out to be of the order of ten to the minus five grams it's enormous compared to elementary particles it's huge so we believe that planck's constant has something to do with fundamental at the lowest at the reductionist ultimate level it has to do with fluctuations at some microscopic level sub microscopic level and you would expect the mass also to do this but the actual masses of elementary particles that we know of are much much smaller so in a sense the plant mass is a very large mass it's a huge mass of course we put objects together you get much larger masses like people and galaxies and you know collections of galaxies and so on and it could be infinite on the other end so it is not always true that it does give the lower limit but in the case of length and time it does because of a deeper reason we believe that the planck length and the planck mass are in fact the length and time scales on which the plank length and the blank time are the length and time scales on which the concept of space time as a continuum itself breaks down so really we believe that below this below the plant length and below the plant time thinking of time and length as continuous objects itself is suspect we think the structure of space time itself would be very different from what we know of it on much longer time scales and length scales this is where quantum fluctuations in space and time themselves would start playing a role and therefore the meaning of space and the meaning of length the meaning of time is not very clear so that's the reason why it's it's not a reason that's the happenstance that is why it is on the lower lower end here simply because below that is fluctuation dominated and after that in some sense these fluctuations are smoothed out and on much larger scales you do not see these fluctuations at all its like saying that if i give you a piece of paper and tear it the edge is jagged but then of course you look at it from sufficiently far away it looks quite like a straight line but i go closer and closer i start seeing the fluctuations more and more so its on the lower level that you start seeing fluctuations rather than on a gross resolution that's a very crude answer but that's roughly what it is any other thoughts so as you can see these are not yeah this is precisely the point i gave this example of water going into ice or steam or liquid water it is quite clear that you need a sufficiently large number of molecules for this to happen for you to be able to distinguish these phases it is equally clear that the phase of water of the you know whether its eye is solid liquid or gas is not a property of a single molecule that's obvious that it's not it's the same molecules even the interaction between the molecules is exactly the same and yet when you put an aggregate together it exists in these three phases the interesting question is how many should i put together before i can tell whether it is liquid or gas and so on can i have an ice crystal which has only 10 water molecules or 50 or 100 and so on that's a much much harder question it turns out there is no clear boundary in this sense you could go on subdividing matter and then there comes a stage when you lose the concept of this phase it happens fairly smoothly in most cases so this is not a we will talk a lot more about this when i discuss short range order and long range order in liquids also a very rough answer would be if you take for example water at low temperatures but before it freezes you discover the system is trying to become crystalline so neighboring molecules are arranging themselves in a regular array but then it is perturbed by thermal fluctuations and it dissipates but as you lower the temperature thing becomes more sluggish and eventually it clicks into place as a crystal but you do need a collection for this and the study of emergent properties of course is very very vibrant its phase transitions is just one example of it and many many other such properties which are very interesting it's also called collective behavior it's got many many other terms coherent structures and so on and so forth another example for instance is if you took individual photons nothing much happens but if you put them all together in the right conditions they could get coherent and produce a laser beam and thats not a property of a single photon ok so many delicate things happen simply because you have these collections the collections could be extremely weakly interacting but still produce order a very crude example would be if there is a distraction at the other end of the room and just a few people in that corner start looking there only the neighbors are influenced they start looking there and pretty soon that propagates so even if you do only what your neighbors do you can have very long range coherence even though the interaction is short range that is an emergent property [Music] so [Music] you