Quantum Statistics

0:00 Anyway, so then the question becomes, well, why isn't classical statistics the same as Bose-Einstein's statistics? Well, we can understand why it's not the same as Fermi-Dirac. With Fermi-Dirac, after all, he's putting a very strange constraint in place, no more than the same parameters. but what about the Doseinstein case why is there a difference in practical concepts so that becomes a problem for me but that's a problem that I think most people know is easily solved and the reason it's easily solved is because if you just go through the Boltzmann theory look at the entropy, work out the statistics this switch the wrong place. This is exactly where one wants to lean. Not just me who wants to lean now, I'm sure everybody does. You can block this around. You can say, if you have people that fall on you, they'll make their elbows at the end. Anyway. So, in other words, if you just trundle the handle of possible statistical method, you actually get out. When you symmetrize permutational symmetry, you still get out possible statistics. For example, that's in David Albert's book, Time and Chance, and he got it from Nick Huggett, who published a paper from J. Phil about four years ago on this. But actually there are, of course, other people in physics who have said this, so it doesn't you need to hug it. I think Hugginian philosophers appreciate this better. Who's the physicist who said this? Hossman? Heston? Is this what it is? Anyway. There are various physicists who have said this. So, this ought to be well known. But the thing is, if that's true, then why still you've got the question

2:30 why quantum statistics differ from classical statistics? But it's true that when you don't count as distinct classically, the permutation of atoms, you know more of quantum mechanics. Then why is that? Classically, you get out of Boltzmann, you do get the Gauss. Despite the little argument I went through, you still get out of the Gauss. That's the wrong way to put it. What I mean is, you still get out the Boston entropy function. And, for example, in application to photons, you get the beam distribution, not the time distribution. So, in that sense, you get the classical statistics out when you have the permutation distribution. But then again, the question is why is quantum statistics different? And now, I think, there's a puzzle. because whilst all SS physicists know about this, then what do physicists say? And the answer is they historically have said that the difference between quantum and classical is because classical you do not have permutations in the truth. Schrodinger is a great example of that. Schrodinger is not just my hero here, Michel's hero too, but our hero does go on about exactly this distinction between you or don't you allowed permutations. He gives examples of giving two coins with portraits of Newton and Shakespeare on them to children competing for a prize. And you could give Newton, the Newton coin to Tom, Charles Tom, or you could give Shakespeare and so forth, you run through them. And he says, this is classical statistics. But really, the quantum mechanical business is you don't give these coins with portraits on them. You just give money, which is fungible. It's not like I did a pound to this to Tom, and a pound to Dick, but then I could have given a pound to Dick, and it's the same thing. It's fungible. So Schoen gives exactly this example. And Schoen was very knowledgeable. Schoen was worth quantum statistics. One of the things that led into the wave mechanics is very mysterious. And there are many other physicists who have said this. But what you do sometimes get into is physicists saying that,

5:00 ah, the difference between quantum and classical is localized and non-localized. and there you might think you're in with a chance because okay that is a difference the trouble is it doesn't really seem to make any difference where it counts it's just not really relevant it is relevant to tracking over time whether you've got the same article as read it in earlier time but that isn't actually relevant it's not relevant to statistical analysis if you're doing statistics space or no space because either way once you've got the state at a given time you've got it for all time okay where it would maybe make some sense is if you're just working configuration space classically and then you've got some initial classical configuration and then in a later time maybe they sort of swap around but because you contract them through the time you can which swapped around configurations in A-R-R from earlier configurations. So it might make sense in that context. But we don't work on configuration space, we work on phase space. Where, once you've got the point in phase space, that's it. You've got the entire history. So, you know, all you're going to do is have bunches of world lines in the swapping around the world lines. That's not what's going on if you think, oh, can we distinguish that from that? if you've got the entire world line then swapping around the world lines isn't giving you this it's just giving you this but with that world line called by a different name from this world line so that doesn't actually work and actually I don't think I've really come across a decent proposal I don't think I've come across a decent proposal So, in other words, what I'm saying is, and I'd love to be corrected, if you look at the physics literature, universally what we find in quantum literature is that the reason quantum of Haskell's statistics differ is either because quantum mechanically you have indistinguishability, or it's because quantum mechanically you've got non-localism.

7:30 And I'm saying neither of these works. Now, it's worth saying, on the side of indistinguishability, that there is an entire industry now of philosophy and physics focusing on this question. I'll just say Steve French's name because he's one of the most prolific people publishing on this sort of thing. But there's lots of others. And they look at things like quasi-set theory, they've got all kinds of ways of trying to say something about this and when I say to them well look you've got you've got permutation symmetry classically all I've got back is the statement that, well, we only understand that because we're forced to understand it in the quantum mechanical case. In other words, that to impose permutation symmetry classically is to put in place these things in the classical context. So it doesn't diminish the revolutionary of indistinguishability by saying we can put it into the classical context as well. So that's the sort of statement back. But I don't agree with that. I don't think it is revolutionary classically. And I think that the case, the fact that it is classical, shows that it's not. Well, it's not very deep in quantum mechanics either, actually. But of course, I have to give an account. Well, then what is the correct accounting for the difference between quantum mechanics? And let me just say the reason why I don't think the permutation symmetry is metaphysically

10:00 profound, I mean, it's interesting, but it's only interesting insofar as weak discernibility I think discernibility is interesting, but it's just, it's pretty ubiquitous. I mean, here's an example. Plus I and minus I, as mathematical objects, are only weakly discernible. I should perhaps say what absolute discernibility is. Absolute discernibility is, can you find a predicate with one free variable, that applies to one part of a collection of objects only, then it's absolutely similar. I mean, if for every object in the collection there's a predicate and one figure that applies to it in one, then all of those objects are absolutely similar. So there's no predicate that applies to class size, it doesn't apply to class size. Of course, I can say there's chalk marks on the blackboard. It's a bus chalk mark as opposed to the minus chalk mark. But you can't characterise that mathematically as distinct. You can characterise plus one as distinct from minus one mathematically. You need to have the natural numbers here not just as an additive group, but as a multi-pegamy field. and then the positive numbers are absolutely distinguished from the negative numbers by virtue of their square, being within, doesn't take you outside of the class of positive numbers whereas the square of any pair of negative numbers takes you outside of that class so mathematically you can characterize as distinct the positive from the negative and indeed the first automorphism of a number field you cannot, with genuine automorphism, genuine symmetry There is this automorphism of the complex numbers, which takes a complex number to its complex conjuality. And as such, pairs of complex numbers which are complex. It can only be a relatively weak, sorry, weakly discernment in mathematics. For an example of relative dissonability, take the points in the directive line, and then x greater than y, a relationship greater than relative dissonance points. So I think it's interesting

12:30 that there are objects, these are in mathematical examples of objects, there are objects that are not absolutely dissonable. But I don't think this isn't metaphysically very Or if it is, then it was obvious with Piano's extensitization that it was much longer. It hadn't been 20 years ago, and it was glaringly obvious already then. I mean, in all the examples that have been given of this sort of thing, it's not that many, but quite a little bit of a few, they're all in mathematics. But it has been missed. and I'll give you an example now there's been three or four papers in the last two years saying about Shapiro's structuralism in mathematics that there's a problem with it because you don't have a notion of object because you can't distinguish plus size from what it's there which is the answer is what they're relatively similar and a big deal so you can see there's a bit of an issue I mean the further reason why I say look it's not metaphysically profound Now it's values of variables, values of bound variables, it's classical, and it's the phrase, you know, form-on-age of algorithms. I don't think you can say that that is metaphysically revolutionary today. So I think this entire motivation is about nonsense. I really do. It's absolute nonsense. Well, I say that, I do have a way of rescuing it, but now it's got to be posed more focused in a more focused way. If you're going to really retain the case, then I think there's certainly more focus than it has to be said. All right, now, let me, before I give you my solution, because my solution probably won't be terribly interesting to you, I'll just give it to you. I'll just get you a bit more puzzled before you were interested in my solution. So you need to hear a bit more about the puzzles. Okay, so first of all, let's just see how Boltzmann calculates his entry. I won't go too much into that, because I know that... And is anyone here a specialist in thermodynamics? It's just two counts, maybe it was a... No, no, it's a funny thing. People don't really pay much attention to it.

15:00 It's really interesting. Anyway, so the key point to make is that entropy, and basically how do you measure entropy? A little quick. How do you measure entropy? Did anyone read just UFIC's paper on Bluff Your Way and the Second World? It's a great paper, a classic paper. It starts off with this big complaint about the two cultures, the scientific culture and the humanities. And people expect it to have heard about Macbeth or Moliere and so forth. have you heard of the Second Law? Well, no, I've heard of it. Can you say it? Can you tell me anything about it? And the entire humanity's culture shuts down and can't say a damn thing. But my worries, the philosophers of physics, to a great neglect, are getting to the same position. The Second Law is hugely neglected. And entropy is hugely neglected. What is entropy? Entropy, classically, is defined for a reversible motion in thermodynamic phase space, between some point A and some point B. This space is parameterized by things like pressure, volume, temperature, and mass. And the point about reversibility, the idea here is that if you've got a thermal system, you can very slowly modify it, compress it, let it expand, and warm it up very, very slowly, cool it down again. And you can do so so that you can not just play with it as you wish, you can put it into any point you wish in the face face, and then bring it back again. So, now, if you measure the heat flow into and out of the system as you do this divided by its absolute temperature along the trajectory then you get the entropy difference between the value here and the value here as a function of state Okay, and what you find is it doesn't matter what path you take

17:30 And that's the critical dimension. And you could have just done that just with dq. But then you find it doesn't matter what part you take. You divide it down into the temperature as you do it, it doesn't matter what part it takes. It's very, very fundamental. That's what tells you, because part independent, there must be just a function of these things, such that there may be a difference in values of the function. That's what you say when you say that ds is a perfect differential. and dq is not dq is not a perfect differential there is no function q of these things such that some change in q is equal to a difference in the times this is a perfect differential ok now that's very important so differences in entry are a difference in the times now there's something further that comes out of that which is extensibility if you look at this thing it's kind of obvious if you've got this as a function of it's a function of these things if you double the mass and the volume you've got one system you stick it right next to another identical system exactly the same Same temperature, same temperature, same mass, everything's the same. And then you take it through these sequences of changes, and you're evaluating this thing along the way. Now, unless these two systems interact in some interesting way, like if they were two black holes, if they don't interact, then, and this is thermally conducted, then it's got to be that the heat flows into and out of them. the net heat flows into now this one is the same as the net heat flows into now this one. And this one doesn't affect this one. So in other words, the heat flow into the whole thing will be twice the heat flow into a half unit, one part of it. Now you've got to be very careful here. A great deal of metaphysical importance rests on this question. you're sure. I'll go into some of the analysis. But anyway, so, there seems to be a good case to see that the heat flow into an atom of this guy will be the same as the heat flow

20:00 into an atom of this guy. And furthermore, it will be the same as if this guy didn't exist. Which tells you... Actually, I don't think you need the second statement. Just the mere fact that the heat flow into an out of this would be the same as the heat flow into an out of this. And any heat going from this guy into this one will be matched by the heat going from this guy into this one. So you can neglect that. All you have to consider is the heat flow into an out of the environment. What that tells you is that the heat flow into an out of the composite system is exactly twice the heat flow into an out of just one composite system. That's the classic. And that tells you that this thing, because the temperature is the same at every point as it's moved along, the temperature is exactly the same. The entropy change for the composite is just double the entropy change for one part of it. That's extensivity of the entropy. So, this thing for 2m, 2v, is equal to twice the entropy of m, okay? And you can see how that's going to tell you that it'll be the same if you have any, well, rational number, and that's universally thought to be the case prior to a puzzle what Boltzmann's work what Boltzmann's work is he wanted to compute the entropy on the basis of a statistical analysis of molecules, and the way he did it, he said, look, what I'm going to look at is an input in the energy. He looked at the energy interval, he looked at... What he considered was a cool screening of phase space, A bunch of numbers, natural numbers, which gives you the number of particles, molecules, in energy range E1 to E1 plus delta E1.

22:30 N2 to E2, E2 is delta E2, a circle, and NS in this energy range. And then he said that the equilibrium distribution is going to be the most likely such cross-quaint distribution. and what did he get for most likely is the distribution is more likely one rather than another if it has more ways of sorting the NS particles over and now he supposed that there were CS cells in phase space corresponding to this energy and I think the clearest way of thinking about this is imagine the one particle phase space Imagine the Es to Es plus delta S As that region of the one particle phase space Then the CS cells So let's Break this down into CS little cells And now let's imagine distributing NS particles Over that How many ways can I do it? How many distinct ways can I do it? And the answer is, just for ns particles, there's ns, the cs to the power ns. Well, if I've got ns particles, and I just want to distribute them over the cs cells, that's the answer. I've got CS choices for the first particle, CS choices for the second, CS choices for the third, and I keep going to NS and CS to the end. Because I can put them all into one cell, so you can get them straight to it. It's got more than one cell if I'm wrong. So that's the answer. But then, if I've got NS particles into this, that region of one particle in the beta space, Then I've got another ns plus 1 particles into this region of n states, and so forth. And I'm going to get a product of all of these things. But now, of course, one of the particles I put into this region, I could have put it into this region.

25:00 In other words, how did I, my original n particles, how did I break them down into n1, n2, ns, and so forth? Okay, I could have done it in any large number of different ways. How many different ways could I have sorted n particles into n1 particles into that? Once I've done that, then how many ways to put the n1 particles into this, answer this. How many ways to put the n2, answer this, et cetera. So I've got to multiply this thing by the number of ways of breaking down n particles into these collections. answer to that is, how many are there? N factorial over Ns factorial. N factorial because I can just, I can just permute any product over the collection, but I'm not going to count permutations within a class. That's not going to give me a different way of breaking down the n-object into this collection. I think that's really really fair. Okay, so now what do you do with that? Well, as I said, he took the equilibrium distribution as the set of numbers ns, such that this thing is a maximum. So that's the most probable distribution. And then you can find the entropy as k times the logarithm of that maximum number, w is the maximum number. So I'll write this down as n factorial. Cs to the ns over ns to the 1. That's the total number of ways of communicating all of these particles. The total number of different ways of distributing the name of the particles. So as to get these as the equation numbers. To maximize that, stick it in here, and go, you've got the entropy. So how do you maximize this? Well, this is a well-defined procedure for doing it. We're looking actually at the logarithm of this.

27:30 There's lots of good questions about why the logarithm, but look at what happens to the logarithm. Suppose I've got W max for one system, and I've got W max for another. What is the total number of ways of distributing particles so as to get this number here and this number here. And I'm asking now, what if I consider the composites? If for every distribution of a particle in all of these numbers of ways, I could choose any one consisting of this. It's just going to be a product of the two. And that's going to be the total. And then what's the logarithm of the total? it will just be the logarithm of the product, which is the sum of the logarithms, and what have you got? You've got the additivity method, okay? So you've seen something quite fundamental about additivity method, in this picture, something to do with independence. And indeed, Einstein was led to this, at least in his 1905 paper, 100 years ago, almost to the month. It was March. Yeah, it was March, So, he was looking at this, and seeing that we've got this working at the veen limit of the black-looking distribution. And there's a bit more that he was doing, but let me not go too much into it. So it's pretty fundamental stuff, as there's quite a lot. Anyway, so going back to this now, so you take the logarithm and you maximize it, and the logarithm gives you n log n minus n. This is Stirling's approximation. Logarithm of x factorials of x log x minus x. We can use that repeatedly for large numbers. So the n factorial here will give you n log n. Then the product will give you a sum. The logarithm of this thing will just give you ns log cs. And now we're subtracting off again the sum of these guys, which is ns log ns minus ns.

30:00 So that's what the logarithm of value is. And now you've maximized this thing, subject to constraints. Once the epsilon ns is a total energy, there you go. and the sum of the numbers of DNS is the total number of points. So these things are fixed. So you can vary DNS arbitrarily here as long as you respect these two constraints. Now there's a very standard method which tells you how to do this. And what that gives you is that first of all, the s of the ns's, just cancels out that term. Now let's vary. We have to vary this, so that doesn't enter into it. We want the maximum value, so we want delta of w to equal 0. What would it equal? Vary the ns's, and here we're going to get the sum over s, delta of ns, nor c s. We're not going to vary the Cs, because those are fixed. Vary the Ns's, you'll pick up another term, delta Ns log Ns. You'll pick up a term from varying this, which will give you 1 over Ns times the variation of delta Ns, the 1 over the counts of this. So you'll do that with delta Ns sum of the Ns. And then... That's it, isn't it? Yeah, that's it. No, sorry. Yeah, when we vary this bit, we're going to get, so that's minus delta n into 1 plus log x. And now you add on this under-determined multipliers business, put one under-determined multiplier for variation, which holds the total number constant, and you've got another one times the psi n s, also times delta n s, so we're holding the total energy constant. And now, the delta ns is totally arbitrary, so you can take them right outside and you've got an expression that multiplies by the delta ns. The delta ns is totally arbitrary, therefore the expression stands for every s. That gives you those

32:30 Minus Ns Minus 1 Plus Ns Well that counts back to Plus alpha Minus Bx But this thing, I should have a log ns left somewhere. Oh yeah, I've got one here, and one of them counts, but one doesn't. So I've got an ns log s here. Sorry, I've just got a log ns here. Okay, and that vanishes for every s. So that tells you, that gives you expression for log ns in terms of these other things. It tells you that log ns is equal to, I don't know if you have that one actually, but let me not try to take my errors, I know I need that one. So this thing is going to be log Cs plus alpha minus b sub s. Got that right, so ns exponentiate is going to give you, yeah, so then ns is equal to So, if connecting on both sides, I'll get e to this thing, which is e to the log Cs, which is just Cs, times e to this thing, which is e to the alpha minus Bs, okay? which is started through a canonical distribution of phase space and so forth. But e to the minus alpha. So what I should end up with is Cs e to the minus alpha, that's sort of normalization, and e to the minus p is epsilon s. And then, Peter, what is that? I'm just going to multiply. You have to go through further kind of analysis here to get it out. But it's got to be proportional to the temperature and the constant proportionality to k and so forth. So, you know, we're sort of working the handles here, I've just done a little bit of it, I don't want to carry on it this way, because this is absolutely standard textbook stuff. But what you find at the end of the day is when you stick all of this back into the entropy function, you find that the entropy is something like this, n log v plus v, and then there's something in the temperature, I think it's actually about 2 thirds, plus some other stuff.

35:00 And this thing, when v is doubled and n is doubled, this thing does not double. The problem comes from the way to cure it is to subtract n log n. Subtract n log n from this, and divide that by n. And now, you double n, you double n. It's okay. But what is subtracting n log n? You're subtracting a logarithm of something. What is it? It's n fact n log n. So, in other words, if only we define it, the original expression by in factorial, we don't know. The original factorial, remember, is n, n, s, factorial, n, s, n, n, s, so just divide by in factorial. I mean, for one thing, you'll put the CES for equal to one, that is not a way, that is a small number. So that's not a number of any ways of distributing anything. So there's something wrong with the whole picture, this is a way, just, you think, well, stick the CS back in there, well, right, stick the CS back, just choose some examples, and you can see, it's not a natural number, it's not a number, it's not a count of anything. so this is the sort of problem now I think the right answer to this is indeed Boltzmann got it wrong you don't start with a combinatoric count and of course the combinatoric count anyway was dependent on some choice of these cells remember my original picture we divided this up into cells and what's the extent of each of these cells

37:30 where both of these cells are true choice, talk. Who says that's right? I mean, we're back to an old problem now. What is equitable? What is the right thing to say? I think that the answer to all of this is that it's not a count of equitable things that you start off with. What you do instead is you're looking at the volume, So the phase-space volume is going to measure phase-space. And it's the phase-space measure associated with a given distribution of n-one-partic or two points, and so forth. And your most probable distribution is the one with the largest volume. But now, this is the fundamental point. The volume we're talking about is in the reduced phase-space. Taking the quotient and the fermentation move. And what that does is exactly give you that expression. Volume in space space, unreduced, goes to D over N factorial in the reduced space space. That's the total. Now what we're looking at here is volumes associated with a certain specification, namely that you're in some range like that. There's no further division through from permuting the particles between this region and some other regions. So the right analysis of Boltzmann's method is this one. And it gives you the right answer with an extensive entropy as you need to classify it. Now, what happens quantum mechanically? Well, quantum mechanically, he didn't write down this, nor with the n factorial expression. What he did was he said, I've got NS, and this is one of these little tricks that just completely destroys you from a historian. It can stop you for years. How on earth do you just get floored by it? He called the number of particles P. He called the C's N.

40:00 The P's he called the number of energy quantum. and then he asked, how many ways of distributing these energy elements over these ns particles? He didn't just use the symbol, he called them particles, it even resonated. So the entire picture got shifted about, and these should be in the CSs, these should be Ns, these should be particles, and then he had been doing one balsam, but he didn't. He had this different notation, and he had this picture of the cells that had been resonating between real particles and analysis of electrical analogies in the Maxwell-S theory, the classical theory, to reduce the relationship, actually, and a number for these, a number of resonators. Which for us, CS is the number of cells. This is the thing that Boltzmann just fixed by choosing an arbitrary tool. He analyzed it, effectively, by the number of normal modes in a blackboard cavity, right? and then he multiplied that, and it's really torturous when you look at it, just how, you know, one arrived at it from the point of view of a clear understanding of quantum statistics. anyway but in asking the question this way and having this as a number of energy elements divided over these particles he didn't think about interchanging the energy elements as being into a different situation that was the point so how many ways are out there of distributing PS energy elements over RS resonators answer well Think of the resonators as being boxes Let's have RS-1 boxes RS-1 lines, because his box, his box is what's in this And then we'll have a box at the end I don't need to stick another line to sort of end it Now I've got the PS energy out of this

42:30 Just made a arbitrary assignment But how many different Sequences of chalk marks Are they like that Or how many chalk marks have I made Plus NS Minus 1 That's how many chalk marks I made So I can move all the chalk marks Arbitrarily sequences of chalk marks, answers this number vectorially. But now, I'm not going to count as distinct sequences where I just commute the dots among themselves, and I'm not going to count as distinct sequences where I just commute the vertical slashes among themselves. So I'm going to divide how many dots are there, answer ps factorial, I'm going to divide like this. How many slashes are there, answer ns minus 1. So that's what we've got. Instead of this, instead of this, we've got this thing. And therein lay the entire mystery as it may inform statistics, because we couldn't see what was going on between these two things. And the conclusion, of course, what concludes eventually, for 20 years, today, is a big event. But eventually the conclusion was, well, these things were recognized as being later later, eventually. And the idea would like to do not have distinct, you know, they need to distinguish in ways that Maskell practices are not. And that's supposed to explain it. And indeed, when you then stick this in and you run through all of the stuff that I've got here, you get two things. You get an extensive N should be, which can be in different sentences. So now you see I've got a problem, because this is what one ought to have, right? And I hope it's clear, isn't it? Well, sorry. This is what you have classically, but this is what Planck has in sort of a common sense. How come the difference?

45:00 I can't do this in phase space. I mean, even for the simplest space, I need four dimensions of two particles. So, imagine that this line is two-dimensional and this line is two-dimensional. So, now, the classical phase space is this, and I divide it into four cells. And I can stick two particles in that cell, so I can... A point in that cell means both particles have got this energy range. So I want to have two points, this is really, this is the one particle space, so I can have one here, so it's in mu space, so I can have two particles in there, I can have two particles in there, or one here, one there, and I weight each of the cells equally, that's what Watson But if I go to the reduced phase space, I factor out the permutations, I'm looking at the diagonal instead. And the other thing is, if I look at the volume, what is the volume of this square in the reduced phase space? What is the volume of that, and what is the volume of that? They're not the same. if I just take the volume and divide by impact which I say is the right thing to do in fancy I end up with only a half weighting for that that square what was before a whole square this one gets a full weighting and you can see what happens if I had more this one gets a full weighting this one gets a full weighting Now this one is the ones on the diagonal that don't get the full weight in. That's if you do it by the volume measure. It's inevitable. If volume measure is the right thing, this is the right thing.

47:30 That's what you want to have. If this volume measure is the right thing. Hilbert space is a discrete structure. That's important comparison. They're not squares at all. Now I go to the reduced phase, I just get these phase, but I wait for them all the same. I don't care that I was half way, because it's on the back. I give it its goal. And then you ask how many different points are there. And the answer is this thing. Okay, so I should have worked this out with a detailed result with you. How many different ways are there of distributing particles on human space where I'm symmetrizing. So this is two particles three cells. So CSC you can just do this with can I give these their proper names? So that's N and that's C and that's that's N and that's C So now we've got N equal to 2 I think I've got C plus 3 and then that was 2. Doing it this way. And in how many ways are they distributing, I guess equals nine, nine cells? I think what I've got wrong is that this octagon is not the one particle phase space.

50:00 So I've got nine cells and two particle phase space, and a single point represents the state of the system. And the point about doing the diagonal is I'm not distinguishing that point on the state system from that point on the state system. So this is a two-particle system with nine cells in phase space. That would give me, if I was asking where to put, how many different ways are there in saying the two points? Ah, it's three cells in the one particle. That's right, it's three cells in the one particle state space. So that would give me 9 in the state space where I don't factor out the permutations. And 9 possibilities for the representative point of the system. Okay, I can have both particles in one, both particles in the other, both particles in the third one, so it's 6, and then I can place the point of each. Moving on blank, there's 2 possibilities. That's right. And now, if I divide this down the middle and I don't count as distinct these two points, how many different possibilities are there? One, two, three, four, five, six. And that should be what this thing gives me. Two plus five minus one is four factorial over two factorial, that's twelve, times three minus one, two factorial, that's two, which is six. whereas this is what I would use if I was just weighting the volume because this thing will not give me a natural number this isn't a number of ways of counting anything it's just going to be 3 squared which is 9 over 2 factorial which is 2, that's 4.5 that's not a count of anything but of course it is, 4.5 is exactly the volume exactly So that's the difference between classical systems and quantum systems. There's nothing to do with permutations, nothing to do with indistinguishability. It's entirely to do with, essentially, a discrete structure to help the space in comparison to the continuous measure that we have on a classical phase.

52:30 And, of course, that goes over mechanically into how do you compute the analogy of volume of phase space. It's kind of mechanical what you do, that's the sort of volume of space, but what is the volume of phase space? It's a counter of the states. It's a partition function. All right, so I guess that's my solution, okay, and I do think it clears up all of the issues. But there's one little fly in the eye, which is that on this way of thinking, you really ought to always be classically just working with the reduced space space value rather than the hundred space. And that, almost all the time, it doesn't check out what the partition function is. And if you look at the partition function classically, if the sum over E to the I of the psi of an AC of a KT if these are energy levels then you have a degeneracy chroma in there if it's just if they're not energy levels but you allow repetitions then it's just all states meaning all cells in phase space that's the energy of phase space now if you do that then you get out an expression for the classical entropy in terms of position it's this S equals the n kt log partition function plus the total energy. And you might think, well, isn't the right thing to do when you divide that by a factorial? I think it probably is the right thing to do. Because this sum over all states, you're summing now over all cells, And if you were to factor out the computations, you'd have 1 over n factorial, smaller number of cells. So I think that ought to be right. But if you plug that into this, you do not get the Maxwell's resistance.

55:00 So that's a flying noiton for me. I'm not quite sure what to do about it. I think the right answer is that there's nothing magical about this form this form itself doesn't mean anything the only reason we write that here is because it works it's not that S ought to be equal NKT times the number of partition function it's not that it's now there's some conceptual basis for that it's just that it works so if you want to work with the reduced partition function if you want to get the correct entropy well work with the partition function and then subtract off the game n log n. And then that will work. What I can't do is sort of stick that into that by dividing by my own factorial. So I'm not sure if that's a problem or not. But let me just close with one other thing, right now, I'm going to ask somebody, and that is, to claim the literature by people like David Oldham is that The one over n factorial business is metaphysics. Okay? Because, classically, it doesn't actually change the entropy. It does give you extensibility that the claim is that this is going to make Huggit, but that's just a convention. It's a very interesting question. Is there a reason of a convention? And what it boils down to is that question. Is there a reason of a convention? So if I take a gas, and I slide in a barrier into there, so I go from this, where it varies in place, and then I take it out again, is this reversible or is it? If it's reversible, the next density of the entropy is an empirical fact. If it isn't reversible, then you can't measure it. Because remember, you only measure changes of entropy through reversible processes.

57:30 Right on the answer is an if-reversible, because the whole issue is that it's phonological. It's about experiments we could do. Something that's called reversible or not, if you could reverse it as goes phonology. You know, the thermodynamic quantities. And there's no thermodynamic quantities changing, so this is reversible. So I think this is measurable. So, here's a word, Hexotism, having straightforward empirical retritation. Well, I think there's further things to say about that, actually. But another thing to say is, well, one of the factorial Hexotism, it's not that it can't be purely metaphysical. Because when you do it in quantum, it changes the statistics dramatically. And the point may be, you can have Maxwell-Boltzman statistics in quantum, absolutely not. You just don't do the solution for the space. But that's entirely consistent with what I've been saying. The point about discretizing that goes on with human space, you replace continuous things with these two points. Well, a measure in terms of a count of points gives you approximately the same answer as a measure in terms of continuous measure. The number of points is a proportion to volume. As long as you don't commute out the question about permutation, now the kind of points is not approximately the same as well. Okay, so if you don't commute and have a space, if you don't commute out the permutation, then you've just got all the points to build the space, and that's a good problem. It's in the same measure as classical space spaces without commuting out. So, by the way, permutations and hensitings of itself is empirically settled. There's a little flaw in that piece. I'm not sure if there's a problem. But maybe I just wish to ask you a general question about the place of science as a type of derivation of quantum statistics and classical statistics.

1:00:00 I think that it's very convincing as an alternative to any other interpretations. At this stage, I cannot see why you should prefer it to the other possibilities. And in this case, it would not be just one more case of under-determination of interpretation, ontological interpretation, by the expression of algebra. What are the alternatives? That's the point. I don't see any. What is the alternative? You get some, but you said they are not acceptable, but I understand. But the non-locality... I understand. Given the indistinguishable... I don't see why... How does the distinction mean? There's a chemical difference in class, and lots of statistics. When you're using a distinction, then you get at least for statistics. Yeah. Well, I think that's a bit... So, my own way of doing things was to speak of impermitted rather than extinguished. In this case, it gives the same result essentially as what people say. So, in other words, if you do, you should still get an answer to lots of statistics. So, how can the presence or absence of limitation become a difference in a class of multiple statistics? We have to discuss that. I think it's possible. I can't. I'm sorry. That's categorical, really. It's just no possibility. in here, if, by the beginning, you still get out my sources of positive statistics, then how can people really make for the difference? See, where people can always be in Australia and think, well, if you can help this space, if you don't commute, you get out of positive statistics, and if you do commute, you get there later. So they're thinking so clearly permutation makes the difference. But that's explained on my story.

1:02:30 And what isn't explained about that story is why a muting doesn't make a difference in the maximum possible case. And what you've got on the standard line is a connotation of some of these things. Mysteriously, it doesn't make any difference classically. But yes, it doesn't make a difference when it happens. And that's totally unexplained. Yeah, yeah, okay, but maybe it's exactly what people mean when they speak of it, maybe that it's in post, it's in formative, that the idea is that... No, I don't accept that, no, I think you were right to do that, that was a good move, and I like the reference, I mean, I played that move too, I mean, I take that in distinguishments just to be, is it in distinguishments, and that's what I'm doing. As I say, I don't see an alternative explanation for what's going on, unless this business is non-mechanaly. You could say that it's just a name, people have strict on this difference. Non-mechanaly. Yeah. Okay. I don't think it's accurate, because when you really see what they're saying about locality, it's all to do with being able to generate it. It won't make it as a story, but I don't see it. I've got a drink, actually. I'm so sorry. I've got to be back in short do you have anyone I want to check this with you next week, that's my new trip see if I can get something wrong How does it work with bone?

1:05:00 Well, in bone... Can you get out of the box with a bone? Sure you can. You use non-symmetraised states of bone to get out of the box with a bone. Yeah, I suppose that's... Because you've got permutational symmetry at the level of the particles. Yeah, I mean, in Bohm, yes, you have, yeah, actually, yes, you have permutational symmetry for the particles. And what are the states? Um, no, let me see. No, if you have commutation symmetry at the level of it, if you impose that the trajectories have to remain the same, you know, the same that I have to put trajectories under the location of the particle. Well, you inevitably do, because the perfect particles just is to be introduced. Well, that enforces the symmetry or anti-symmetry of the states. That's why I'm going to talk. That's what I'm going to talk about. No, no, no. But that's as opposed to Paris' existence of the states. Well, it means that the wave function has to be symmetric or anti-symmetric, so each particle becomes non-symmetric. I mean, it doesn't just exclude, you know, you're right, you're right. that we have symmetry of the Hamiltonian, assuming the usual quantum symmetries, so symmetry, anti-symmetry, or parent statistics, if you impose symmetry, you predict with that But you shouldn't think this in terms of the switch of Hamiltonian. I mean, look, in Boltzmann's mistaken approach, you have the symmetry of Hamiltonian.

1:07:30 It's just the one exploiting in his statistical method. So the issue here, when you have the mathematical statistics in public candidates, it's not that you change the Hamiltonian. You know, it's the same simultaneous, it's just that you don't symmetrize the states once. And this, of course, is well known, the expectation values. Even, as long as you've got symmetrized operators, you can use unsymmetrized states and the expectation values to still permutation. Okay, so let me see anything. Okay, you have all symmetrized operators. Well, there's something very strange about that, because the point about commuting the trajectories, just think about it, anti-Hexa, you just don't have a difference. I mean, the trajectories, the collection of trajectories is exactly the same. just calling them by different names. Now, why that should force the music to synergize this wisdom from the synergize steps, I don't know how you've worked, I know you've worked Thank you very much. Can you just very quickly put your number in there for me before you go, because I haven't made a note of it with me. I'll give you a call tomorrow. That's OK. Brilliant. Thank you so much. I'd love to put it in the case. I'm sorry I'll go back to the ceiling. That was great. Thank you.