 All the calculations with all the states allowed to be inserted. So let me do more precisely the following thing. Let me choose a physical state on the sine state, psi, frist, and then I do any capital nature I want. And choose the initial state here to be an obituary downstate. By an obituary downstate, I mean a state which has any value of a, not necessarily a value such that k squared plus x squared is equal to 0. It's a obituary downstate. Because this state was exact for every value for which k squared plus x squared is not equal to 0, every such calculation would be 0 for those k's that were not k squared plus x squared equal to 0. And therefore, if we did an accurate of this form, we'd get an answer that was proportional to a delta function, to a basically proportional to an x squared. You see, in the situation of the point particle, unlike in string theory, you can actually use the schoemanism to do off-shell calculations. You can actually use the schoemanism to do calculations of correlation functions of operators that are not just S matrix elements. This is calculating field theory, that's actually what you do. And it's a property of field theory, that all correlation functions of operators have certain reasonable analytic properties. In case space you have poles and cuts and various other structures, we never have delta functions like this. This kind of analytic behavior in field theory coordinators is just never present. In fact, the general theorem is about the analytic structure of operators in a loving quantum field theory, which disallows a signal. If you were to treat this state seriously, it doesn't seem like this thing. Correct. If you treat these states seriously, they don't suffer from the problem that there is an ambiguity to change the choice of gauges. That's not the problem. The problem is that they have amplitudes. You would get answers for amplitudes that are very singular. That were not the answers that are allowed by general rules of quantum field theory. We know that we've got twice the space. We know that we've got twice the physical space in the question. We know that we've got 400 methods of quantumizing the theory, all of which are much more physical than this. And we know that we've got two spaces. We've done this VRS in quantumization. We've got twice this physical space. One of the two must be this cloud. One of these has a strange property that we use it to do calculations. And we get answers that are performant known that are just not consistent with any sort of field theory behavior. Especially right now, if you throw away this guy, what we say is the physical state of our system consists of operative scaling. So I know this is not terribly satisfactory. I know this is not terribly satisfactory because it's not some systematic procedure we've used, which works out round to the end to give you the answer. But that's basically the way physics is. What was the logic behind this VRS in quantumization? This VRS in prescription that this was physically in the space? It was whether it seems reasonable. It seems like a property we want. This is a property. But here in this situation, we've got a subset of states that had all the reasonable requirements. That's also a reasonable thing to do. And in reason, the right answer. The right answer, meaning the answer that we get by, other means of what? So there may be a more formal way to understand why these states are not good states. But we're just going to be pragmatic. Because if we were to do a VRS in quantumization with this point particle, we would do it by throwing out these states. We would just by having to say that physical states are states that are annihilated by, this was the one, had it been. So b is one side of this, physical state. So the end result of this whole procedure is a prescription that for a quantum particle, physical states are the homology of radiation beyond side of the state. And we answer the right answer, well, at least this gives us the right spectrum. We know the right answer to spectrum is with that many different ways. And it's also these states that will give us reasonable answers to correlation functions. To stay out of the match, not just what we know to be correct. In some cases, we know what's correct, but we've written the right interactions with this point particle. We haven't discussed why we have to do interactive point particles. Just an interesting problem, which we won't get into in this. In this format, we won't get into this concept. It reproduces a defining diagram of quantum field theory. OK, so the states you have thrown out are the states that don't get the case rate of the state. Right. The states that would have been exact, that weren't exact everywhere else, but not exactly at case rate of the state. Case rate of the state, you know? No states. Can this be some kind of I epsilon? You see, maybe what you're saying is re-burning where I say. You know, in an I epsilon prescription, as you've said, you've got reasonable logic criteria. These are things that are exactly zero, except at case rate plus x-ray rate. So even if this is the language of quantum field theory, you won't be able to do it. Now, these states just don't exist a little bit of a shape. They don't exist in the sense that they can use zero correlation, zero amplitude for any physical process. Exactly. Even a little bit of a shape. So all the language of field theory, I can I epsilon, should we apply to these states? Add to the imaginary part of the world. And then case rate, if it's always the same, it will never be, case rate of the state will never be zero. So if all of these states are exact, maybe you would say. OK. Please. If we do that, it's one of the first ones. I mean, we add an I epsilon to the mass. We'll be flying around the ground states, of course. But then what will happen to the up states? The up states will have the case rate that's actually different. Slightly modified, much like relation states. Never? I mean, I don't know. I think the way of treatment is just pragmatic. In a situation where you can solve the theory in two different ways, one way is certainly correct. The second way was some ansatz. You know what the right answer is. You must do what you do. You see, what we're doing. We've got this enhanced Hilbert space that was not physical. And we want to find a way of cutting it down from this VRSD procedure, which was that whatever states we have must be in VRSD. But we're no guarantee that that space itself would not be bigger than the physical Hilbert space of interest. There's no guarantee any of them. We saw that, in this case, is bigger. Now, we want to do the extra cutting down that will get us out of the states. You see, we have your basically two options. You have to divide on this, and it's because we're really in a combination of two. And we see that these states are very funny properties. Just throw that away. You can't do that because they will be a little bit complicated. I completely agree. So if we can't do that, it's OK. But at the level of a free theory, apart from OK, OK, like that, right? At the level of the free theory, all we want is the Hilbert space. The question will be, when you get to an interactive theory, when you use this to generate interactions, then is it a consistent thing to do? For instance, this is the first thought that you keep when they try to mix with states at this point, when they try to produce these states. That's a very good question. That kind of thing, we will have to address carefully in any situation where we use this. Now, the reason I found this monoparticle so much in detail is that we're going to do something very similar. So in string theory, we will do a BRSD quantization of our action. And we will put two conditions on physical states. The conditions will be first, that they could have to be BRSD comma 1, G. And second, that they may be annihilated by the 0 model of the B field. Then the second condition will be B0 and field 0. But more or less the same reasons as this. Now, when we use this in string theory, we will perform careful analysis of, for instance, issues like our states that are not annihilated by being produced in the new system. Then we will be very careful. Then we use it to study interactions. And the level of the free theory, of course, we would justify this more or less by the same reasons. Yeah. But also, again, by a comparison, with another way of computing the spectrum, the way we've already done it in class, it's like, we have to get the same answer. Once again, we find double the answer. So my answer to your question is that it's an ansatz. And while that, we will be very careful to check for consistency. But I haven't seen a careful, principled analysis of this ansatz beyond analytic probabilities. OK, I think we should stop this class here about some questions about this. So what have we done in this class? And we just summarized. What we did was to understand the BRSD procedure in generality. We understood how we can be general constructed BRSD chart. And then we applied this to the quantum particle. We applied this to the quantum particle with one extra prescription that was put in by hand, namely that the physical states, in addition to the BRSD code 1 and B, the BRSD code 1 and J are added by B. With one extra prescription, we recovered the Hilbert space that we know would be correct to the quantum particle. Now, the next class we're going to do is to extend this analysis to string theory. We're going to write down the BRSD charge on the worksheet of the string theory. And then in the next class, we will begin a careful analysis of BRSD co-homology of string theory, and the BRSD co-homology together with this condition of E0 and J. It states a string theory. What we will do is prove what's called the no-host theorem of string theory. That is, we will prove that this co-homology coincided with the Hilbert space that we obtained through light code quantum quantization in the previous class. This plus one or two more lectures were in the initial form of our initial formal development of string theory, which is essentially the boring part of the string theory. So all this for man is over here again. Sooner or later, we will start to move into this. We will start calculating scattering amplitude. That will be in a voluntary vector. So just to begin with our schedule, we will have one more class is created. And then in the next class, we'll be on the second day, the 30th of December, the next day, the 30th of January. OK.