 So yesterday I gave a very speedy, I apologise a too speedy introduction to the problem of compound systems, but we have a very tight budget of lectures and I wanted not only to talk explicitly about these composite systems and how you handle them, but also I think it's very valuable to discuss this classical Einstein-Badolski-Rosen experiment because it goes to the core of whether quantum mechanics is correct and what it really has to say about the universe. And there wasn't going to be time, so I could have spent two lectures doing what I did yesterday, but it's being compressed in order that we can talk about this experiment which is an important crucial application of this apparatus of how we apply quantum mechanics to compound systems. So Einstein is famous for saying that God is sophisticated, but he does not play dice. He disliked the probabilistic aspect of quantum mechanics, not that he disliked or disapproved of the use of probability in physics. His thesis work had been on kinetic theory and statistical physics, so he was quite comfortable with the idea that in classical physics you use statistical methods, probabilistic methods to do things like kinetic theory, but he understood that the reason you were doing, in that case, you were doing probability theory because you add incomplete information. So when you lack information, it's obvious that you have to assign probabilities. The thing that's worrying for him was worrying for him about quantum mechanics was that it asserted that even when you had complete information, which we know is embodied in a complete set of amplitudes, still the outcome of experiments is probabilistic and uncertain. And he felt that this was wrong because, and the relation to God there, I guess, is that an omniscient God would not have the future uncertain, would know what the future was, so God must know something that we don't know. We have an uncertain future because we are short of information, but the information must be there. We just don't know the information. So there must be some variables, some variables which encode the information about what's going to happen, which if you knew them and at some future time in physics perhaps you would know what these variables were and then you would be able to predict exactly what was going to happen. And in 1935 Einstein with Bedolski and Rosen proposed this, described this thought experiment, which they argue demonstrated that there must in fact be these sorts of hidden variables. In 1964, I guess it was, John Bell analysed a similar experiment and showed that the predictions of quantum mechanics are actually incompatible with the existence of these hidden variables. And then in 1972, 20 years after Einstein's death, an experiment of this type was actually conducted and the measure, and many, many since are being conducted, and these measurements vindicate the predictions of quantum mechanics and therefore prove these hidden variables cannot exist. So that's what the agenda is today to describe this. So what's the experiment that Bell describes is this, which is there are various versions of this, but the key idea is the same. Suppose you have some nucleus which is going to be unstable and it's going to emit a positron in this direction, well it does emit an electron in this direction. And Alice sits over here and measures the spin. She measures the spin of the component of spin of this electron as it comes by in a direction of her choice, we will call it A for Alice. And Bob sits over here and he measures the spin in the direction of his choice, the component of spin in the direction of his choice, which of course we will call B. So let's imagine Alice acts first, sorry, and the idea is that we know from nuclear physics that both before and after the decay, the spin on this nucleus is zero. We know of course that electrons and positrons are gyros, they carry their spinning particles, they carry angular momentum. And because the angular momentum change of the nucleus is zero, it must be that by conservation of angular momentum the spin of this is oppositely directed to the spin of that so that the angular momentum of the electron plus the positron is zero. So supposing Alice measures the spin first and Alice gets that A dot S turns out to be plus or half. So she finds that the component of spin along her chosen vector A is S. What she then says to herself is this, if Bob measures along A, if Bob chooses to put B equal to A, then he's guaranteed to find the answer minus a half, right? Because if he measures with B along B, the components of the spin along B, a vector B, which is close to the vector A, he's not very likely to get plus or half. He's most likely to get minus a half, but I can't guarantee that he'll get minus a half. Right, so the point of that is that Alice's thought process makes it absolutely clear that Bob's measurements are going to be correlated without Alice's measurements. And what we want to do now is put that on a quantitative basis and ask what does quantum mechanics have to say about this. OK, so what we want to do is talk about the correlations between the measurements that Bob makes and the measurements that Alice makes, given that Bob is going to choose vectors B at his discretion. So let's talk about the quantum mechanical predictions. So what we do is we choose, which we're free to do, the z axis to be along Alice's vector. We can do that without loss of generality. I'm about to write down a result that will emerge in the next couple of weeks from our work in angular momentum. Maybe it will emerge next week from our work in angular momentum, but for now I have to ask you to take on trust that because the electron plus the positron together have no angular momentum, it must be that their wave function can be written like this as E plus P minus E minus P plus. So that is to say that the states we know that a spin-a-half particle, again this all needs to be justified properly when we do angular momentum, but we anticipated these results once before early in the course. The spin-a-half particle has a complete set of states in which you're guaranteed to get plus along some direction for the spin and minus along the direction for some spin. There are a complete set of states. So here are the complete set of states for the electron. Here are the complete set of states of the positron. This says that there is a probability of a half. This one over root two is the amplitude to find that the electron is plus in the z direction and the positron is minus in that direction. This is a similar amplitude with a minus sign for the opposite possibilities. So the origin of this expression will emerge shortly. I must ask you to take that on trust. The state of the system, the composite system of the electron and the positron taken together. So we talked about the collapse of the wave function in these circumstances yesterday. Alice makes her measurement. She finds plus, which means that she collapses the wave function into this. So this is before Alice makes her measurement after Alice has made her measurement and found plus. We have that psi is simply E plus P minus, which is to say that there is unit amplitude that Bob will find minus if he measures along the z axis. In other words, if he chooses B to be the z axis which we've established is which we've chosen to be the same direction that Alice chose. So that's consistent with what Alice said. What happens if he takes to be some other direction? Well, what we need to do is express some other directions. We need to write the ket. So we would like to calculate the amplitude or the probability that when B uses some other direction he finds it to be positive. He finds that s of his positron is along that direction, has plus a half along that direction. So in order to do that, I have to ask you to take something on trust that we will derive, but we've seen before, which is that this thing is equal to sine theta upon 2 e to the i phi upon 2 of positron down plus cos theta upon 2 e to the minus i phi on 2 for the positron plus. So what does that say? That says that the state of having of being certain to give you a half along the vector B for the positron is given by this amplitude, that's just some complex number, times the amplitude times the state where you will definitely get minus a half on the z axis plus this amplitude times the state in which you're guaranteed to get plus a half on the z axis for the positron and theta and phi are the polar angles of B. B is a unit vector so it is defined by a couple of angles and they're theta and phi, they give you the orientation of B with respect to the z axis and here we're using the complete set of states along the z axis. So that's a result we will derive but I'm asking you to take it on trust for now. So what's the probability that Bob measures plus on B? The answer is that we, it's according to the dogma of the theory, it's this because the state of the system, the state of the positron after Alice has made his measurement, her measurement is definitely this. So this is, that's how it works, the apparatus. So basically we flip this around, we take the Hermitian adjoint of this thing, bang it into minus and guess what, we get the complex conjugate of this coming out. Whoops, that's sine theta on 2 e to the minus i phi and I've written on 2 and I've written the probability which means I need to do a mod square, do a mod square, this factor goes away and we're looking at sine square theta on 2. It follows straight away, you could also calculate it that the probability that Bob finds minus on B is 1 minus the probability that he finds plus on B is equal to cos squared theta upon 2. So that puts precisely on a quantitative basis what Alice said, Alice said that if Bob chooses a vector B which is very similar to my A, which is the case when theta equals naught, then he's guaranteed, well if it's identical, he's guaranteed to find minus because this becomes 1 and that becomes naught. And if he chooses a vector B which is similar to my vector A, it's not guaranteed that he'll get minus but he has only a small probability getting plus and that's because theta will be small and his probability is looking like sine squared theta upon 2. So what Einstein, Podolski and Rosen said, well the question is why is the result that Bob gets somehow dependent on the measurements that Alice gets? And in particular it looks like the result of Bob's measurement depends on which direction Alice chose because this angle theta is the angle between Bob's vector and Alice's vector. And we can imagine that Alice, let's imagine that Alice goes first and chooses a direction, apparently Bob's, the probability of Bob's outcomes depends on theta therefore depends on Alice's choices. But supposing this positron and electron are sent out at relativistic speed, perfectly plausible that they are, then Alice and Bob, Alice makes a measurement and Bob can make a measurement in the rest frame of the nucleus at essentially the same time. And if Bob acts that quickly, then there is no time for a light signal from Alice to reach Bob setting out after Alice has made her measurement. So Bob definitely makes his measurement in complete ignorance, has to make his measurement in complete ignorance of what choices Alice may or may not have made. And indeed if in this relativistic case it's easy to see that who acts first, different observers, observers moving at different speeds with respect to the nucleus and Alice and Bob will disagree about who acts first. The whole question of who acts first is neither here nor, it clearly can't affect the physics because it's an observer dependent statement according to relativity. So how is it that the result of Bob's measurements depend on Alice's choices when it's not logically possible for a signal to go from here to there in order to affect it? Well, Einstein, Podolski and Rosen said what it must be is that actually the result of Alice's measurement is preordained. We don't know what the result is going to be but that's because we're pig ignorant. But God knows it's foreordained because the result is encoded somehow in the state of the electron, not written on the board here because we're using this clapped out quantum mechanical rubbish. And similarly inside the positron there's also this magic information, this DNA, this whatever, which foreordains the result of Bob's measurements and then everything is okay. That was their interpretation of this problem. So now let's talk about Bell's inequality. So that was the state of affairs for I guess 30 years, right? 1935 until 1964. So John Bell said, okay, so let's calculate something. Let's sigma A be the result of Alice's measurement. And it's obviously going to be plus or minus a half, right? Whatever number she comes up with is going to be either plus or half or minus a half. And similarly sigma B being plus or minus a half is the result of Bob's measurement. And let's calculate the expectation value of sigma A sigma B. So there are four cases to consider because they can both measure plus or half, they can both measure minus a half, one can measure plus and a half, one measure minus a half and that in two different ways. So this thing is going to be, there are four possible values that, sorry, sigma A times sigma B could be either plus or minus a quarter. And the possibilities to consider are the probability that Alice gets plus and times the probability that Bob gets plus, given that Alice gets plus, plus the probability that Alice gets plus and Bob gets, sorry, Alice gets minus. And Bob gets minus the probability that Bob gets minus given that Alice has got minus, right? So in both these cases, in both these cases that product is going to be plus, right? Because in this case this is going to be plus or half and that's going to be plus or half. In this case Alice, sigma A is going to be minus a half and sigma B is going to be minus a half, the product is going to be plus or half. And then we have some minus cases which is the probability that Alice gets plus say and Bob gets minus given that Alice got plus. And then we have minus the probability that Alice gets minus and Bob gets plus given that Alice got minus. OK, we have to make Bob's probabilities conditional on Alice's because we've seen that they're correlated. We can argue that the probability that for Alice to get plus is the probability for Alice to get minus, namely it's a half, right? When Alice makes that measurement we don't know a blind thing. So both possibilities are equally likely. So that must be what these probabilities are for Alice. And we've just worked out what the probability for Bob to get plus was. We've worked out these probabilities. So we know that the probability for Bob to get plus given that Alice got plus, we found that that was sine squared theta on 2, right? So that's sine squared theta on 2. By symmetry, you could work it out, but by symmetry this will be sine squared theta upon 2, right? Because if Alice has got minus, we know that Bob is jolly unlikely to get minus if he chooses an angle, a vector which is close to A. And we figured that these, that's this one here, we've already shown that the probability for Bob to get minus given that Alice got plus, we've already shown is cos squared theta on 2. So both of these are going to be cos squared theta upon 2, and both of those are going to be sine squared theta on 2, which means that sigma A, sigma B expectation value is a quarter of sine squared theta on 2 twice over because we get two times like that minus twice cos squared theta upon 2. Oh, times a half. Sorry, sorry, sorry. So here's a half, and here's a half from the probabilities of A, so those twos are not really there. And what is this? Cos squared minus sine squared is cos twice the angle, so this is minus a quarter of cos theta. And what is cos theta? Cos theta is actually A dot B. It's the angle between A and B, so this is minus a quarter of A dot B. So that's what quantum mechanics predicts is the expectation value of the product of these two measurements. So now what Bell did was calculate what this would be in a hidden variable theory. So draw a line, and now we're into another conceptual framework. What we're going to say is that, so there is some function, sigma E, which will depend on, so what's this? This is, this thing here is the value that you will find for the spin, the component of spin of the electron along the vector A. We think this is a random variable because we don't know the values taken by the hidden, this is a set of hidden variables. This is an N vector with components which are the hidden variables that we don't know, but Einstein, Podolski and Rosen claim must exist to make the outcome of these experiments causal. So, so this is not a probabilistic quantity, this is something, this is either a half or it's minus a half, right? Depending on the values that these variables hidden from us, about which we do not know, and of course on the direction in which you measure the component of spin. So this is equal to plus or minus a half in a causal way. And similarly, there must be sigma P, this is the positrons, this is the positrons spin, that's also going to be plus or minus a half, depending causally on these things. We don't know what this function is, we don't know what these variables are, we don't know how many of these variables there are or anything. But what we generally do know by conservation of angular momentum is minus sigma electron at V and B. Because we know that the positron spin is oppositely directed to the electron spin by conservation of angular momentum, so if you get plus or half here, you are certain to get minus or half here. So this is, this equality is conservation of angular momentum. So what we do now is evaluate the expectation value which quantum mechanics told us. So we do sigma E depending on A times sigma P depending on B, expectation value, we write this out as in classical probability theory. Now what's that going to be? Well, this expectation value means averaged over all possible values of the hidden variables, the things that we don't know. So the reason that this thing seems uncertain to us is because this thing is unknown to us and we therefore think of this as a random variable. So what's this expectation going to be? It's going to be an integral over the components of V, we have to sum over all possible values of what we don't know, times some probability density that we don't know times sigma E of V, A times sigma P of V, B. So basically we just take an average of this product which is completely determined by V and then what we take an average with this appropriate weight over all the possible values of V to get the experimental expectation value, standard probability theory. The next thing that we do is we replace this by the corresponding sigma E using that switch of sign business. So we argue that this is minus the integral D to the end V rho of sigma E V, A sigma E V, B. So anything that's changed here is we've acquired a minus sign and that P has become an E. Now we say, okay, now let's imagine that we make this measurement with some other vector, right? Supposing we now calculate the same expectation value between A and the vector C, just some other vector. And then we have that the expectation value of sigma E, A, sigma P, B minus, that's a complete expectation value, minus the expectation value of sigma E, A, sigma P, C, some other vector C, and that's going to be, according to this apparatus, it's going to be minus the integral D to the end V rho, depending on V, open a bracket. No, sigma E of V, A will be a common factor and then we will have sigma E of V, B minus sigma E of V, C. Right? Because the right-hand sides are both going to have this factor because we've taken the expectation value using sigma E of A, sigma E of A in both cases and what will differ in the two cases is that term in the back. So one time it will be B and one time it will be C. So that's what we get. Now Bill does something slightly nifty. He makes the observation that, well, but he knows that sigma squared V, B is a quarter because he knows that this number is either plus or minus a half depending on the values taken by V and B. The square of this number is guaranteed to be a quarter. So we can, we can say, we can insert into here, we can insert a four sigma squared E of V, B without any harm, right? Because we're just inserting a one. So he says that this expectation value, this commodity, I'm going to write it out again, this expectation value on the left is minus the integral d to the nv rho sigma E of V, A. I better write it out. Four sigma squared E of V, B brackets sigma E of V, B minus sigma E of V, C. Very helpful, I'm sure. What we now do is we take, we break this sigma squared into a sigma and sigma and we take one of the sigmas inside here. When one of these sigmas comes in here, we get a sigma squared again, which is a quarter, times four is one. So we get a one appearing here and then we, and then of course this sigma, this sigma that I brought in appears there as well. So the next line is this is equal to minus d to the nv rho sigma E of V, A sigma E of V, B brackets one minus four times sigma E of what? We've carried this one in V, B and we've already got one there, which is a sigma E of V, C. So this is what that expectation value at the top is, it's this. So why's Bell done this? What we now argue is that this bracket, so this product of things here is going to be either plus or minus a quarter, right? Because all of these things, they're causal functions and they're either equal to plus or half or they're equal to minus half. So this product is equal to either plus or half a quarter or minus a quarter, we don't know. But whatever happens, so this bracket is either equal to zero or something positive. The bracket is equal to two or nothing. So what we really need is that this bracket is greater than or equal to zero. It's not negative. And this thing in the front here is a fluctuating quantity. It's equal to plus or minus a quarter. So what we can argue now is let's take the modulus of both sides. The modulus of the left side is whatever it is. The modulus of the right side just means we drop this. And we can argue that this integral, this integral is going to be smaller than, the actual integral here is going to be smaller than, what we would get if we replaced this with plus a quarter because sometimes that is minus a quarter and we'll be taking away from the integral given that this thing here is never, this thing here is never negative. There's no way that we can never get a positive result, a positive contribution to the integral when this is negative. So if we assume that this is always positive, we're going to overestimate this integral. So let me write that down. We will overestimate, overestimate integral if we replace sigma e v, a sigma e v, b by plus a quarter. Because sometimes it's minus a quarter and that minus sign is never cancelled by a minus sign over here. So then I can argue that the modulus of the left side, which unfortunately I now have to write out again, that's p, sorry, sigma e of a sigma p of c, modulus, that's an expectation value now I need a modulus sign, is less than or equal to, because I'm going to write down something which is too large. I've deliberately made it too big of the integral rho. That factor has been replaced by a quarter, this quarter could be taken outside, and then we're staring at 1 minus 4 sigma e, this is v, b, sigma e of v, c. Now we make the observation of the integral, so we break this integral into two parts. It's this stuff times 1, but that integrates up to 1 because this is a probability density, and a probability density has to be structured so that if you integrate a probability density time over all parameter space you get 1. So this and this make a quarter, so this thing I'm going to write down what it's equal to, which is it's equal to 1 from the quarter brackets of 1 from here, then now let's consider this onto this stuff here. This onto this stuff here is roughly speaking where we came into this, that this times this was the expectation value of sigma on sigma. And this minus sign we can soak up by changing that back into a p. That's retracing logic that we did up there. So this becomes 1 plus 4 times the expectation value of sigma e, b, sigma p of c, expectation value, whether v has disappeared from here because we've done an expectation value operation. We've averaged away all the v dependence in the proper way. So this is Bell's inequality that we have here now. It's a statement about expectation values associated with the two particles and three possible vectors, a, b and c. So the next thing to do is to ask are the predictions, we've calculated the predictions of quantum mechanics for these expectation values. We've already done that. So the question to ask now is are the predictions of quantum mechanics consistent with this inequality? I guess we need to be able to see everything simultaneously and I've not handled that right. So let's write down here. Let's find the predictions of quantum mechanics. OK. This is the crucial thing. The prediction of quantum mechanics is that this product, which in the other calculation for reasons which if you stare hard at it, you'll realise that there's a notational issue. There's a reason for this. This in the hidden variable calculation is called sigma e, sigma p because remember Bob is measuring the positron, Alice is measuring the electron. So this is actually the same physical quantity that we've calculated down there and it's equal to minus a quarter of a dot b. So we can go straight back. So now we put in sigma e, a, sigma e, p, b expectation value is minus a quarter a dot b, which is from quantum mechanics. What does that do? Well, let's check out the left-hand side. What does the left-hand side look like? It's going to be the modulus of a quarter a dot c, well, minus a dot b. So the overall minus sign gets lost but it's going to be the modulus of this thing here. What's the right-hand side going to be? It's going to be a quarter of one minus a dot b, sorry b dot c. So now we need to ask ourselves is it true that this right-hand side is bigger than this left-hand side? And in this matter we can choose a, b and c exactly as we will, right? Because Bell has shown that for any vectors a, b and c, his inequality has to hold if there are hidden variables. There's so far no restriction on a, b and c. There are any three vectors. And if the quantum mechanical results violate Bell's inequality, for any vectors a, b and c, then quantum mechanics will be inconsistent with these hidden variables. So at this point we do a choice. We choose a dot b equals naught and we choose c is equal to a, say cos of psi plus b sin of psi. So what are we doing? We simply have psi at some angle. We're just choosing a and b to be orthogonal vectors and we're choosing c to be a vector that lies between a and b and we've got ourselves a parameter of psi which allows us to move c from pointing along a to pointing along b in a continuous way. So just concretely the picture is here is a. We're choosing a to be this way, we're choosing b to be that way and we're choosing c to be like that somewhere in the plane. Stuff it in and what do we get? We find that the left hand side is the modulus of a quarter. a dot c is cos of psi, a dot b is naught and the right hand side is a quarter of a one minus sin of psi. Plot these up and what do you find? Sorry can we change these back to sin of psi because my diagram will look better if I do cos of psi, sin of psi, cos of psi. Right, okay. So obviously there's nothing in that. It's just a change in the figure two unfortunately. Then what do we get? We find that the right hand side looks like when psi is small the right hand side is looking like psi squared on eight or something. Anyway it's writing quadratically and it goes to one. This is pi by two. Meanwhile the left side is basically a sine curve so we know what that looks like. It looks like this. So this is the left hand side. This is the right hand side and Bell has shown that the left hand side is smaller than the right hand side. So for only two values smaller than or equal to the quantum mechanical results are consistent with Bell's inequality for only a psi is naught and a psi is pi by two. The quantum mechanical results violate this inequality for all values of psi basically. So we conclude QM is inconsistent with these hidden variables. Once you've got a nice clean statement of this sort that quantum mechanics is inconsistent with something which EPR reasoned should be the case. The indications were that it was the case. Clearly the right thing to do is to go out and make a measurement and allow nature to decide for you whether quantum mechanics is right or hidden variables are right. So in 1972 this was first done using not an electron and positron pair but using pairs of photons. That's usually how this is done. The analysis is slightly more complicated if you use photons than if you use spinarhoff particles. So we followed Bell in using spinarhoff particles. But basically many of these experiments have now be conducted and the experiments vindicate the quantum mechanical predictions with a level of precision that you know it's clear that the experimental results are inconsistent with hidden variables. So the experimental results, and that's from 1972 onwards, there be many always refined experiments, are consistent with QM and inconsistent with hidden variables. So that means that quantum mechanics is not going to be replaced by a hidden variable theory at some time in the future because you cannot construct a hidden variable theory along these lines is not going to be consistent with experiments that are already conducted as there's no point speculating about it. So to come back now to Ein Spine, Podolski, and Rosen, what are wrong with the arguments which indicate that somehow B's measurements knew about A's measurement? I think a lot of the... Well, sorry. So the things that you should take away from this are that when you measure something, you do two things. You disturb the system and you gain information about the system. So when Alice measured that electron and found it plus a half for the spin in her direction A, she disturbed the electron, but she didn't disturb the positron because the positron was somewhere else and the positron couldn't possibly be disturbed by anything done to the electron until there had been time for a light signal to go from her operations to wherever the positron was. So she definitely doesn't disturb the positron, but she does disturb the electron, therefore she physically changes the state of the electron-positron system and that's why she's collapsed the wave function from that linear combination to this here. But she has gained information about B because of the correlation that existed in the original setup between her electron and the positron by having discovered what was the state of affairs with the electron, she was able to make some quite strong predictions about what B might find, what Bob might find on measuring the positron. This experiment emphasises a theme that's quite common, it's quite recurrent one in quantum mechanical calculations, and it's very important to think holistically. To do this problem you have to think about the electron-positron system. It's no good thinking, oh I can deal with the electron or I can deal with the positron. Both together have to be considered because of these correlations in the system. A lot of the confusion that I think Einstein, Rolski a Rosenhad, and that is in many treatments of this experiment, arises from slipping into the era of thinking that because Alice has found plus a half for the components of spin on her vector A, that the spin is pointing along A. As we shall see, a spin a half particle has always plus a half of spin in the directions of all three coordinate axes at the moment of the z component, you can know that the answer, you can know that it has a positive value for sigma z, but you don't know what the values of sigma x and sigma y are, but you know that they have, you don't know the values, but you know that they do have values, which are comparable to that of sigma z. What we physically think of is that Alice has determined that the spin of her electron points in the northern hemisphere, well in the hemisphere that has her vector A for its pole. She does not know it's pointing, that it's aligned with A, she only knows it's in the northern hemisphere of that. So when Bob makes his measurement, and then she can say, aha, she then knows for certain that the positron has its spin in the southern hemisphere of her vector A, right? But she does not know where it points there, because she does not know where her electron points in her hemisphere, she doesn't know where the positron points in its hemisphere. She only knows now, all she's learned is which hemisphere the positron is pointing in. So she can exclude, as Quantum Mechanics says, only one result of B's measurement, namely if Bob chooses to measure along the vector A, then he will not find plus a half, because the top hemisphere has no point in common with the bottom hemisphere, and Alice knows that the positron is in the bottom hemisphere. So I think the bottom line is that there isn't a logical problem if we just keep focused on the idea that what is preordained is which hemisphere the electron or the positron is pointing in, not the direction. It's an error to think of these spins as pointing in a particular direction. It's difficult to escape from the idea that a vector points in some direction, but then it's difficult when we do relativity to get used to the idea that time is relative and that two events that are simultaneous or one event that happens before another event in our frame of reference, in somebody else's frame of reference, reverses the order of the events. So the absoluteness of time is something that's very difficult to escape from, but we all grow up, we get used to it. Time isn't absolute, and Quantum Mechanics is telling us that no vectors don't point in particular directions. In the case of spin-a-half particles, the best you can say is that they have particular hemispheres in which to point, and we'll, as we go on, so the next item on the agenda is angular momentum and that will enable us to look at this a little bit more closely about under what circumstances it is the case that a gyroscope or whatever seems to point pretty much in a definite direction and we'll find, in just the same way that things move only because they have ill-defined energy, things point in a definite direction only because they have ill-defined angular momentum and electrons do not have ill-defined angular momentum and they have well-defined angular momentum and that stops them pointing in any particular direction. Okay, all done.