Quantum Mechanics of This Specific Universe / Extracting Predictions from the Wave Function (contd.)

0:00 Okay, this is the second session, which will be divided into two parts by the coffee break, amongst other things. The first three talks... These are going to be half an hour each, and I gather that it was the desire of the speakers as well as of the organizers to have each of those talks go about 20 minutes with a 10-minute discussion afterwards, or at least a 30-minute total of this to be internal discussion. After that, we'll have a coffee break. Then there will be a discussion after Bryce's talk of his talk before dinner. I think there's a little bit of leeway, but not all that much if you want a coffee break. That's where we charge our batteries. So let's try to stick more or less within that. I will signal to the various speakers. You'd better call me an asshole. Pardon? You'd better call me an asshole. Asshole is coffee. We will start with Jonathan Halliwell, who will speak about extracting predictions from the wave function. Well, I actually offered this title of extracting predictions from the wave function of the universe a few weeks ago, and I didn't have a very clear idea. What I was going to say, and in actual fact I'm going to be speaking about something which is seemingly unrelated, but I will make some contact with the original title towards the end, in particular I'll say a little bit about the question of what objects, what distribution do you use constricted from the wave function to make one's interpretations, to make predictions? So what I'm going to be discussing is the relationship between the path integral and the Witten equation for many superspace models in quantum cosmology. This is, of course, to a large extent a technical issue, although there are a number of conceptual points of interest, and it is these that I'm going to concentrate on. As is well known, the relationship between the path integral and the canonical quantization procedure has really been well understood for systems like non-relativistic quantum accounts, that is, there exists a derivation of the Schrodinger equation in the path integral.

2:30 However, most of the systems' field theories of current interests are constrained systems, the constraints reflecting the gauge of variance or the parameterization of variance. And for these systems, the canonical quantization procedure is the direct method. For all those systems, the relationship between the direct quantizational procedure and the part-lytical method is not as well explored. Now, one of the most interesting constraint theories is general relativity. There are four constraints reflecting the invariant theory and the four dimensions of theomorphism. The first three constraints, the momentum constraints, are linear in the momentum. These constraints generate the diffeomorphisms within the three surfaces, and the symmetry of the theory is very similar to the symmetry of the normal engagement. The fourth constraint, the Hamiltonian constraint, is quadratic in the momentum. It is this constraint that distinguishes general relativity from the normal engagement. This constraint has two roles. It generates the symmetry of the theory called pre-paracharization variance, pre-paracharizations in time. It also generates dynamics, so that dynamics and symmetry are inextricably entangled. Now because reparameterization variance is a distinguishing feature of general relativity, it is of interest to consider simple models whose only symmetry is reparameterization symmetry. So what I'm going to be considering is a quantization of systems described by The main examples I have in mind, of course, are many superspace models. These are simplified models for quantum gravity in which one freezes all the finite number of The degrees of freedom of the metric. So, for example, one could write the four metric in this form to restrict the lab's function to be homogeneous, restrict the three metric in such a way that it is described by a finite number of functions, q alpha and t. So, for example, one might write the three metric as a scale factor, this kind of thing, times some fixed three-speed metric.

5:00 This sort of restriction, the momentum constraints, are vicariously satisfied, and the Hamiltonian constraint just reduces to, after integration over three surfaces, just reduces to a single constraint of this form. This p-squared is a potential where we have some q-dependent metric, f-alpha-beta, the metric of mini-superspace, which generally has a hyperbolic signature. Now, in the direct quantization of this system, one introduces a wave matrix psi, a function on the super-space. This is annihilated by the operator version of this one's James. This is, of course, the Witte equation. There is the notorious operator ordering problem in essentially deciding how to organize the p's and the q's in this expression. And I'll be discussing that problem. Alternatively, one can use the power of integral method in which one represents the wave function, And it's also necessary to include gauge-fixing and Gauss terms because of the reprimetization symmetry. Let me briefly illustrate this reprimetization symmetry. So we look at actions of this form in this field acquaintance. This action is invariant in the reprimetization. That is, if one is following transformations on Q and P, supplemented by this transformation and relapse function, where here epsilon is an arbitrary function of time. The action changes by the following amount, at this quantity evaluated at the end points. If we were dealing with a gauge theory, in which the generator of the symmetry is linear from the momentum, then this quantity in green would just vanish identically, and so the action would be invariant. However, it is, as I emphasized, quadratic in momentum, and therefore this thing only vanishes if we choose this parameter epsilon to vanish at both end points. So the action is invariant under these re-parameterizations. Any function epsilon satisfying these vanishing conditions. So, that's the action. The Perth integral may be constricted using the very general method of Vitalin, Kretkin, and Bukowski, which is based on PRS invariants. This transparency shows all I'm going to say about the Perth integral constriction. I'm not going to get too much into these.

7:30 The basic idea is the following. It takes the original action and adds on to it a gate-fixing condition. Here, the grand multiplier pi enforces the gate-fixing condition, m dot equals chi, where chi is an arbitrary multiple, sorry, an arbitrary function of the p's and q's. In that, would it be more convenient to use the n itself? It turns out that one is obliged to use gate-fixing conditions which involve n dots. We can't use anything wrong, for example, in a part of the book. This point was discussed in great detail by a title of the one, and the argument is that, generally, the conditions must be of this form. But you can use less, which is further at hand, than the gauge-fixing function, which contains the labor time as a specific function. Yes, an alternative form is to pick out a particular q, let's say q-normals, and require that that be equal to some function of time. That gives you a slightly different object. So, given the gauge-fixed action, VOS, sorry, the term in Franklin and Blachowiczki, then tell us the radon ghost terms, such that the total action is invariant in the global symmetry of VOS, and for this choice of gauge-fixing conditions, the ghost action has the following form. So this action is invariant in the BOS symmetry, and it also yields the original field equations, that is, that the ghosts are gone from cloud five, vanish identically. These two statements both subject to certain boundary conditions. It's essentially, of course, to give the correct measure. If I just, well, first of all, I need to break the symmetry, the reparameterization symmetry, by adding a gauge to it. And in the path integral, that would occur, of course, just as a delta function, because I need to ensure that I'm going to get a result that is independent of the stage-fixing function, chi. And the Gauss field, when you integrate them out, gives a p-factor, which ensures precisely that. It's a measure, which ensures that. What I can tell you in the book of this, what you can tell us to do now is to write down a path integral of this form, which was PBIS for this action.

10:00 Integrated over all fields, written fields, er, er, er, er, er, er, er, er, er, er, er, er, er, er, er, I'm going to show you a few examples of how you can use these terms in a very simple way. The chi is not the gauge variation. The gauge variation is n dot minus chi. So you need the gauge variation of that whole thing. And so the remaining terms of all the results, you can expect to convert to this. DR. Yes, the VRS is different from all of that. DR. Okay, so we go into both of them this way. This is essentially the measure of what mechanization theory is. Similarly, I can do the pi and n integrations. The pi integral just pulls down a delta function on n dots, and then the functional integral over n just collapses down to a single ordinary integration over, let's say, the n-point value of the lapse function. The remaining bit takes a very familiar form. It's just an ordinary functional integral representing a standard quantum mechanical propagator with time coordinates and t. So, you see, the final result boils down to something very familiar. It's just the interval over all proper times, in fact, of an ordinary quantum mechanical propagator. So, this is the main expression of what we're looking at. Since we... Is that an interval? It's a phase-phase, by the way.

12:30 If I enter... Say again? You let the P's at the ends go freely. Right. You let the P's at the ends go freely. So, the Frank-Wilkowski procedure fixes, shall we say, the local measure, but it doesn't fix the measure in the large, that is, it does not fix the range of integration of the lapse function. Let's investigate the significance of this. All but one of the n's. I did a, I have a, after integrating over the pi's you get a delta n dot, and that essentially says that n is equal on every time slice. What the range is. Yeah, so I've only got one endpoint value of n there. The choices are you take n to be half infinite or fully infinite. So supposing first of all I take, well I just defined, actually I've done that the wrong way, there should actually be t equals n times that, but it's slightly the wrong way then. So g is integral over all time separations of this thing. Well I suppose if we take t to have a half infinite range. If I operate on G with a Hamiltonian operator at one endpoint, then I end up operating on this with H, which, because it's a basic Schrodinger equation, just gives ID by DT, and then doing that integration, we get this thing evaluated at the endpoints, and assuming certain convergence, as T goes to infinity, I'm just going to get left with this at T equals 0, which will give a delta function. So I get a green function at the wheel of the width operator if I take T to have a half-inch of A. Similarly, if I take T to have a fully infinite range, then, just going through exactly the same again, I'm just going to get this propagator evaluated at minus infinity and infinity, and again, assuming certain convergence, as T goes to plus or minus infinity, I'd expect to get zero on this side, so I get a solution for a really good equation.

15:00 So if half an infinite lapse, I get a green function, or fully infinite lapse, I get a solution for a really good equation. Now here I'm dealing with Lorentzian power integrals. Ultimately it would be of interest to look at Euclidean ones. The Euclidean action is only positive even after doing some kind of conformal notation, if necessary, for positive maps. So it seems that Euclidean functions automatically lead to Green functions. So that in outlines, the derivation, has still got to show that this quantity... And finally, the solution to the Schrodinger equation, which is surely an easy problem. So, H equals naught, or H equals the delta function, dividing that, first of all, I can write down some sort of skeletonization for this Bayes-based path integral. Schrodinger equation, and determine the operator, the Hamiltonian operator, where the classical Hamiltonian. So this is the question of deriving the time-dependent Schrodinger equation from Hilbert space back then. And this is a question that's been very well studied over many years by many authors, starting with DeWitt and ending most recently by Kirtash. In a canonical compensation procedure, there's obviously this ambiguity in going from the base and the P's by operators. This is normally alleviated, at least partially, by demanding covariance in Q's. And in which case, the Hamiltonian operator has the following form. It's del squared, and it's a really simple space metric, plus an arbitrary multiple of a curvature term. Here, covariance in the Q implies invariance of the P-multi-definitions of the P-metric before the Q is to represent the P-metric, which seems like a reasonable sort of condition to impose. It all goes to configuration space, not integral. Then, this ambiguity in the curvature term corresponds to an ambiguity in the choice of measure. When one writes down a configuration space, part of the rule, and the time slice, the definition, one has to specify two things, a covariant skeletonization and a covariant measure.

17:30 In configuration space, there's a unique skeletonization, namely one uses the Hamilton-Banker equation. There's no unique measure. One can then have a look at how this works in phase space, which is what Krikash did. People have often claimed that if one goes to phase space, these ambiguities cannot be fixed, because there is a unique measure known as the Hubel measure. However, the ambiguity is then transferred to the skeletonization. There's no unique skeletonization in the action in phase space. There's no analog in the Hamilton-Jackson function. So this ambiguity permeates both configuration space, the phase space properties, as well as the quantumization procedure. There is one thing you can do, and that is do what people do in math. In order in field theory, just do a loop expansion and that leads to a unique... That's all you need. That's all you need to do in order to get to the short one. I didn't want to actually save the next transparency for cosmology. There isn't actually a way of fixing this. So, this wasn't meant to be anything more than just do what field theorists normally do in their expansion. All values of time. Yeah, but the Schrodinger equation is not fixed. Except it can't be that arbitrary in field theory. Extra terms like that would give you divergences. It looks like it cuts on to the end of this particular bit. Well, what I've shown is that if we start with this action, dq dot minus nh, and demand covariance in the queues, that is, a preliminary definition of variance with respect to the pre-metric, then the Wheel of Ritiquin, corresponding to this, is of this form, Bielsberg's sign-on. However, the fourth field that I was interested in is not the pre-metric, but the formative, which also involves the lapse function, n. So it seems interesting to inquire what happens when one looks for invariance and preliminary definitions.

20:00 There are other terms which involve the lapse function also. So supposing I take this action and then just pull some arbitrary function, omega squared, out of here and absorb it into the lapse function. So I introduce a new lapse function, m tilde, into n omega squared. Just redefine f and v tilde, which will absorb the powers of omega to the minus 2. The associated real-equality equation is somewhat different. It's this alternator in the conforming relative metric, f tilde. This will not in general be the same as this, that they will not imply each other, but for any relationship between Psi and Psi tilde, unless one chooses Psi to be the conformal tripling. So if one chooses this operator to be the conformally covariant wave operator, if one chooses Psi tilde to be related to Psi by this pair of omega, then The two relativistic equations are the same and one gets full invariance under both the definitions of the magic and the math function also. This is a catch, because when one goes to the systems that are not homogeneous, then you go to the forum. When you have infinitely many conformal vectors which you can put in, you have then infinitely many constraints with various conformal vectors and you cannot make by the use of the conformal Laplace theorems for each one of them to make them consistent in each other because when you start taking commutators between the two things then they do not close. I would have meant that there are going to be problems in the full theory, but here I'm just trying to find a way of making it work in mini-superspace. But the problem initially arose when I observed that in various papers on mini-superspace, people often use different definitions of this lapse function, and they absorb various factors of the metric. And in order that all these different definitions did the same with the drift function, they were obliged to take this thing. But the extension to the full superspace definitely put them out there. So, that's one way of fixing the operator order. There's an associated problem, and this is where we make some kind of contact with the question of extracting predictions, and that is, what is the associated probability measure with the Willard-Witt equation?

22:30 It was suggested by Hawking and Page that the probability measure to use is mod pi squared times the square root of n. And this was supposed to provide some kind of measure on the full new super space. But one can think of other reasons why this is not very satisfactory. But one main reason here is that this thing is not invariant into these restalings that I've described. This is not the same as mod pi tilde squared times square root of f squared. However, the Klein-Gordon one is, yes, which you'll perhaps guess anyway because... This is more appropriate for the Schrodinger equation, and surely the Klein-Gordon current is the one to use. The above one isn't conserved either, is it? The Klein-Gordon one is. So, this quantity, J alpha times the square root of f, is conserved. It does have the problem, however, of giving negative probabilities. If one takes, for example, a wave function, i, c, d, i, s, Then J has the following form. Y functions are typically of this form in regions where the y functions, where the universe is a part of the classical. And if you take some component like J naught, which people often do, then this will be positive or negative depending on whether or not S is increasing or decreasing in its function. And generally S will actually, in all directions, it will be increasing or decreasing. So it would not be possible to get some global probability density from this. Well, typically, Q naught is the scale factor, for example. Yes, right. However, a recent suggestion by the Lenkins is the following. One takes some surface sigma, which is not parallel to the direction of j. It could be, for example, the surface is a constant s. And then one takes j dot d sigma. And he claims that this thing ought to be positive. This is a conjecture, it's not going to be shown to be true or false. So that might be one way of making sense of it.

25:00 So you do have to use some idea like that because the potential, the V, that's in that equation, it doesn't have a constant sign. It can change sign, whereas the mass squared term in the kind-ordinate equation doesn't change sign here, hence you can go in space-like directions and you can also turn around in the reverse direction of time for these flows. So you need to slice them through with some surface that's appropriate for that flow. Yes, there still needs to be worked out more carefully to see if it works or not, but it's a possibility. Where was I? Two minutes? Okay, finally, armed with a particular, just on a slightly different point there, armed with a particular definition of a particular explicit expression for the propagator, it's possible to investigate whether or not things like a naive composition rule or naive normalization of the wave functions Actually, one could surely think of reasons why they wouldn't work from the very beginning, but let me try and show explicitly what goes wrong. So this is the quantity that we've got from the particle integral, the integral over all times in ordinary quantum mechanical propagating. Now, how the period of those times in the literature claims that if you take two of these things and then integrate over all the intermediate queues, then you're going to get another propagator like this. But if you just plug in this definition, it's getting these two quantum mechanical ones. And then perform the integral over Q. These two bits, using the dependent properties of the ordinary quantum mechanics propagator, these two bits come together and we get two integrals over time of one of these things. And one of the integrals over time gives the desired results, namely this thing. Then this is multiplied by a second integral over time, times a constant, and therefore infinity is the result. So you get the desired result times infinity. One can argue that the waveforms aren't normalisable. Supposing I take g, which I know to be a solution to the Turing equation, if I take an infinite range of the lapse, multiply it by some arbitrary function, phi of q prime, and integrate it over all q prime, this thing, psi, will be a solution to the Hilbert equation. If I try to take log psi squared and then integrate it over all q, for precisely the same reason that this guy's edges, it gets infinity here also.

27:30 So, not surprisingly, the wave functions are not memorisable on the full super space, but that is really not a surprise, because what one would expect would be that if these things are to work at all, is that contained in these queues is some kind of time coordinate, so that what one really ought to be doing is taking some hypersurface within the super space, a dimension one less than the full super space, and then perhaps integrating that. And possibly getting normalization, normalizability about hyperspace, but certainly not in the full surface space. So that's all I have to say. In the middle of the practice, as a speaker, let's take a few minutes to answer the questions that I gave. Yeah, but to do a very simple case of a single, say, relativistic particle, how do you calculate the normalization in that context? In a relativistic particle, you don't integrate over the... For the relativistic particle, it would be like doing an integral over time as well as space. Well, what I know for sure is that if I take a parameterized relativistic particle, fixed against conditions, and then give it integration over the full extended phase space, then everything drops out and we can adjust the integral to be two-scale. And the particular DQT that I get depends on my choice of gain, according to Gordon's conclusion, and I don't get it. What do you mean by the extended phase base? I mean that's including... You go back there and you have a hand and you have a time and you have a seat and a seat bar and a row and a row bar. So at that stage you haven't integrated over the lap function? No, that's the lap function that's integrated over the lap function. I'd expect that things would get online and it would make sense. It's when you go to the physical space, you have to go to the physical space.

30:00 All of these terms are related to the idea of i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. What would be the justification for integrating the lapse over negative values? It's not, well, what does that, can you actually say what that really means in general? Is it absolutely obvious to you? Well, it's the distinction between sort of putting one sentence in half of another. Yeah. Right, that's the whole issue is whether you want G to be, you know, to know what the first argument is in half of the second. Mm-hmm. It's a question of giving degrees over a positive number of times or positive number of times. Yeah, but there's also actually a possibility that if you take 20 ranges for the variables, and that's like you've got the Vizcay theorem that shows gage independence, that's an independence of choice of gage conditions, but it's often difficult to skeletonize far-fetched rules when you have far-fetched rules. It's probably okay with the particular gage sources used in general. As long as you don't get out of many superspaces, you don't have to ask the hard questions. And as a simple counterexample to this result, I can give you a solution which is normalizable.

32:30 Let the Q0 be the scale factor of the universe and let the rest be some finite system with a bound Hamiltonian sigmoid from below, which it usually is, and then you can, what you have is a separable system, you have a sum of two Hamiltonians, or it's a difference of two Hamiltonians, but they have the same spectrum, and you can choose... Icon function of one multiplied by the icon function of the other. The total h acting on them gives zero. Each one individually is normalized together. And we call it the minus sign. Well, you have the minus sign. It's the difference of two actions. The difference of two actions? Well, yeah, that's the minus sign. But that's already there. That would be a solution. That's probably just a generally true statement, not an always true statement. Well, this is, I guess, true. I just felt the solution was generated from the conference room. I don't know if this is completely genuine. Well, there are lots of other questions one might ask about the full theory, but time stops us. Thank you very much, Julien. Our second speaker is David Bower, who will tell us about the status of hierarchy and gravity. Do you need help? Well, I need something to write on. I've already had a pen. Oh, here it is. I received a note after I was well committed to the fact that I wasn't allowed to use the blackboard. I was asked to talk about the status of higher derivative theories, and I said, well, But I have some work in progress. I can give you a progress report. And I say, well, we've heard that there's a couple of papers out there. Can't you tell us about those? And I said, well, yeah, I can do that. So this is partly a progress report of what I'm doing and partly a report on the work of some other people. So let me remind you what the issues are.

35:00 We're starting with quantum theory superficially is disastrous. We want to do it because if we quantize the gravitational theory alone and take that as an indicator that that theory by itself is incomplete and inconsistent and cannot be taken seriously. Bryce will disagree with me. He'll do that on his own time. He'll do that on his own time. Taking that as an assumption, if I go to first order, Kelly Spella, first from 15 years ago approximately, discussed the expanded theory in which you have a Riemann square term and a curvature scalar term squared. You don't need to have a Ricci term in addition. I can do it either way. I don't care which way he did it. I can do it either way. I rewrite his thing in that form. I personally prefer to write it as C squared plus R squared in order to separate the transverse and the scalar parts cleanly. That term, the renormalizations imposed by the existence of this term require that I have these terms present in order to be renormalized. Once I have those terms, I have to reanalyze the theory, and if I'm doing perturbation theory around flat space, the canonical structure of the theory, or the canonical structure of the theory in general, if I do so, I obtain a theory that's exactly the same form as I had before, that is, there are some pi's and some q dots minus mu. H mu, which are exactly the same kind of constraint structures. The only difference is that instead of having one anti-symmetrical, symmetrical three-tensor

37:30 here and one conjugate symmetrical three-tensor, I have two sets of them corresponding to the fact that this is a higher derivative theorem. So, the sum of this quantity, when I look at the asymptotic behavior of infinity to get the Hamiltonian, is no longer has a positive definite energy structure. It has a manifestly indefinite energy structure. The excitations around flat space are the ordinary graviton excitations plus a set of massive modes. There's a massive scalar associated with this term and a massive . Spin 2 excitation associated with that term. Because of the indefiniteness of the signs of the terms that appear in here, those terms can be construed as either having negative energy or negative probabilities. Therefore, I at this point sort of stop and say, well that theory is not terribly satisfactory either, and at that point I have to enlist with Murray. In the position that there is really only one available theory that has any pretense to be an acceptable theory of gravity and a quantum mechanical theory of gravity. There has appeared a putative loophole in this argument. The putative loophole is given by Antoniades and Tomboulis in a paper in Physical Review, a date that does not appear on this montage. This all comes from two pages in the physical review, and it's about two years ago. Is that visible? I'm afraid not. Antoni Anis and Tom Boulis do a BRST quantization, or a BRS quantization, of this higher derivative theorem.

40:00 They introduce the ghost loops of the kind that were discussed in the previous talk, Not the propagators, but they calculate the way in which the mass of these massive excitations varies. And by that I mean they look at the inverse propagator, And calculate how that mass changes when they change the gauge. That is, they change the parameters that appear in the longitudinal parts of the propagator, the gauge parts of the propagator. Now they don't do so in this purely higher derivative theory. They do this in a theory in which it's coupled to matter. It's coupled to additional fermion fields, n of them. They then do a one over n approximation, taking the first loop corrections and calculating the inverse propagator. They do that so that these massive excitations will become unstable. And since they're doing a one over n approximation, they claim that this is a consistent approximation to that order. And that indeed takes those massive poles and splits them into a pair of complex conjugate poles on the physical sheet. And it's on the physical sheet because of the instability of the theory. It's not on the unphysical sheet in the zero-thorner theory, whether it's it or not. Once they do that, then they find that the dependence of the mass on the gauge parameter is given by this rather unpleasant expression. The parameter r is the residue of the pole, of that unphysical pole. Alpha squared is one of the parameters that appears in the Lagrangian. These are things that are calculated from the ghost loops. They then evaluate that quantity. These are the graphs that they calculated where those are ghost loops and these are various kinds of external particles coming in. When they calculate them, they obtain for the vertex that appears here, this horrible mess down here, which I want to move up like so.

42:30 The third term with the NF, NF is the number of fermions times alpha squared, is the quantity that comes from the virtual fermions. This term is there whether the virtual fermions are there or not. When they put this result and that result into this expression, they find a non-vanishing result, and they've reintroduced the number of fermions by multiplying NF alpha squared and dividing by NF. The reason they do that is that in this large N approximation, NF alpha squared is supposed to be held constant in the limit as NF goes to infinity. However, I can simply cancel that term out, and if I understand their paper correctly, it's a very long and detailed paper, and I have not reproduced their calculations to make sure, this term would be there whether the fermions were there or not. There are a number of different types of equations that can be used to calculate the mass of these fields. That excitation is therefore a gauge excitation, and is therefore not a physically observable quantity, and I could in particular choose to move that off arbitrarily far, and I needn't include it in the unitarity of the theory. Well, the claim is that if you hold nf alpha squared constant and let nf go to infinity, it is in fact zero. Oh, this is the contribution of the graph. I mean, this is the ghost loop.

45:00 And this contribution right here doesn't come from the fermiont loops. That's right. The whole calculation, the change in the mass, comes from the Gauss loop. You use a Ward identity, a Ward-Slavnov identity, to calculate the change in the inverse propagator. You evaluate it at the pole. At the pole, the only thing that you can have is the change of the mass. That's all that's left, because since there's a zero at that point, if you aren't changing the location of the zero, You're going to get zero. If you are changing the location, you get the change in the location of the zero, which is what's calculated here. Well, this... No. You're using the Wartes-Lavnov identity to calculate the change in the graviton propagator, which is just a ghost loop. So all you have to evaluate is the ghost loop. And instead of having to calculate... The real loops involving the exchange of the tensor excitations, all you have to do is calculate loops involving vectors, and it's much easier. I can assure you, it's much easier. That's right. That I'm a little unclear on. Terry says I have to do 1 over n. If I look at it, it seems to me his calculation goes through. Even if I drop the fermion loops, because the term it comes from is not a term involving the fermions. Look at that term. It didn't come from the fermions. I can calculate it. It doesn't depend upon ns.