Does the Higgs Mechanism Exist?

0:00 Thank you. to Sunday the 4th of September 2005, speaking in the bedroom of the apartment, the 6000 Florence a Night apartment in Budapest, after meeting my little feline friend Zoltan this morning. Poor little Pushka. I'm hoping he'll be alright. Let's listen to this and see how well it's recorded. Let's get started with the pleasure to introduce Professor Hange, who is going to talk about Good morning, Mr. Gotti, it's better for me to be here as a place and like the thanks the organizers of the university conference. Now, I want to talk about the problem of locality in quantum physics, and we'll first make some general remarks about the meaning of the principle of locality. We'll then discuss some well-known local features as random physics, and then we'll describe in more detail the algebraic formulation of random physics, which was already mentioned in some other talks, but it emphasized in particular the notion of subsystems, the notion of independent systems, which is both already

2:30 discussions to the talk by Professor Peltz, and then I come to application of this general considerations to Penfield theory, where I want to describe a version of Penfield theory which is fully in agreement with this principle of locality, and which leads to a formulation the conditions of general covariance that can be a basis for treating general relativity in the sense of twin business. And I'll give you a short outlook. The talk is based on work that was published in recent years. So the main publication is a joint publication of Romeo Tronetti and I had him myself in publications a few years ago. There are papers by Hollands and Mort, which are more or less, which are also related to this. Actually the whole attempt has to be considered to be a joint attempt of these two groups. And I would also like to mention papers by Richard and myself, which are not on quantum field theory, on generic space science, but on Mikrovsky's space, but where also the principle of locality has been discussed and has been shown that quantum field theory admits such a global problem. Okay so let me start with the principle of locality. Now, the principle of locality appears in its most evident form in classical the field theory. For instance, if you look at the Klein-Gordon field with a phi to the 4 interaction, then it's determined by the field equation, the Klein-Gordon operator applied to the field

5:00 phi, is proportional to the field at the same point to the power, and this is a prototype for a local field equation and tells us that the physical law relates the value of the field at some point with the values of the same field at points which are immediate neighborhoods, so in this case, infinitesimal neighborhoods of the point x, which is expressed in terms of performing derivatives. When you ask what is the general meaning of this principle of locality, you can get the following abstract formulation, namely that the principle of locality says that it's possible to consist subsystems of a given system as systems in their own right, in such a way that they are isolated so far that the coupling to the environment can be controlled. And putting it in this very general form, it seems to me that the principle of locality is crucial for possibility to improve any science. So in this sense, it's not something we can give up easily, but we have to try to find the appropriate form of the principle of locality in different locations. Now, they're saying that the principle of locality is something that is very general. One should also see that the principle of locality is often somewhat hidden. It's not always obvious for the principle that cavity is satisfied. For instance, if you look at Newtonian mechanics, you have these interactions at a distance, and the whole framework of Newtonian mechanics relies on the fact that it's possible to isolate, to isolate, for instance, a two-particle system in an n-particle system, otherwise I think you would never have discovered alternative mechanics. And that it's possible to isolate

7:30 a two-particle system in an n-particle system is due to the fact that the relevant forces in physics decay at a distance. And there's one example in physics that this is not true, this is the force between quarks and this is connected with the problem of the confinement and certainly we cannot apply Newtonian physics to understanding the substructure of elementary particles. Another example are gauge theories. Although gauge theories, if you insist on formulating gauge theories in terms of observable quantities, you will meet a lot of problems for local formulation. So local formulation seems to be possible in gauge theories only, but introducing some redundant structure, which means in this case that you have to go from the observable quantities to the gauge potentials that are not directly observable. So you have to pay a price for the global formulation of the theory. It cannot be done on every level. Okay, now I come to quantum physics. And actually, quantum physics shows a lot of non-local features. So I missed a few of them, you are familiar with them. So for instance, you start with the Schrödinger equation. The Schrödinger equation is a local differential equation. But if you try to compute stationary states or energy line values, can count boundary conditions, so even on this level, quantum mechanics has some known local features. More severe is the problem of the reduction of a multi-particle system to, say, a system of two particles, because of the fact that the particles are identical

10:00 to the multiparticle wave function is subject to certain symmetry requirements, so it's not true that the two-partic system is an easily indescribable subsystem of the n-particle system. Also to mention the Feynman path integral, if you write the evolution operator in terms of the final path in the ground, you have to place an integral for all possible paths, and so this offers some quantity which involves local information, and has the Arano non-form effect, which tells us that an electron can be influenced by a magnetic field, which is non-zero on a place where the electron will never be. So this is a non-node effect. And of course, in those striking non-locality, you can actually have a better correlation, which tells us that composed systems have states which cannot be described in terms of states to subsystems. So, taking together all these non-local features is what we said that quantum physics is intrinsically non-local. but I think this is only due to the emphasis on looking at the state or the system. If you instead should pay attention to the algebra observed, with no localities disappear. So the general attitude is to identify physical system with its algebra of observance. And just a few remarks on the structure of the scientific form. It's this point of view that the system can be identified with the scientific observables.

12:30 It's much more, let's say, plausible in classical physics. where the observables are the functions, real-world functions of face space, and they are a multi-continent one corresponding to the point of face space, a set of points of space space. Now, what is the mathematical structure of the space of classical observables? on the real vector space, and on this real vector space we have two products, namely the point-wise product and the fossil product. Now, it turns out that in quantum physics the structure is very similar, namely what is the structure in quantum physics, There, the observables are certain elements of an algebra operator in some little space. And again, the set of certain operators is a... It's a set of servo-gen operators, it's a real vector space, and again we have two products, namely the Yorban product, just the symmetrized product, which is commutative, but non-assistative. And we have a second product which is a d-bracket, and it's given by the comitator divided by i times h-bar, and it's anti-symmetric and also non-associative. Now these two products satisfy two compatibility conditions. So one condition is related to the possibility of writing down Hamilton's or of the equation of motion, then we take one element of the space of observance, called

15:00 H, but it's an arbitrary element, and then we form the d bracket for every a, and the The V-record of H is A, and the condition is that this is a derivation for both parts, the long part of the V-record, which tells us that time evolution preserves the structure of the central zone. It's a very natural requirement. Of course, for the Lieb record, it just tells us that the Lieb record is Satisfacción de Programme. There is a second condition, namely the second condition is that the associators of these two products are related to that proportion of each other. So if I take the product of three factors, different orders, so if I take the yellow product of A, the yellow product of B and C, and here subtract the opposite, the other order of brackets, where I first compute the product of A and B, and then take the yellow product of C. Then I get up to a factor the same, so I will do the same thing as a d-brand. And so this is, you can verify this condition, and then one sees the structure, the structural similarity between the test and the quantum theory. and if you set h bar zero, you get the structure of the Poisson algebra, Poisson algebra, classical physics, and in case h bar is non-zero, you can combine all these relations in the condition that the following operation, a times b is the yellow product plus i, h by about two times the b bracket, is an associative product. But of course, then you have to leave the space of sub-atroning elements and you

17:30 have to go to the complexification of the algebra. So the fact that the algebra of observance and Frenn physics is a complex algebra with a star operation is just a very convenient way of combining these two problems. But there is one question, and I guess somebody of you has some idea about this, namely this condition two is satisfied in the exact mechanics that we know it, and it's of course nice from from a mathematical point of view, but what is the physical meaning of this? I have never found any discussion on this. I have no idea what it sounds. That would perhaps be a nice argument for the structure of quantum mechanics. Okay, so the general program is now to identify physical systems with complex star algebra, this unit, and the states, the states are the linear functionals on the algebra, which is associated with the every element of its expectation value, and these functions that satisfy the two conditions, that they are positive in the sense that omega of A star A is large of 0, and they should be normalized so that the unit element of this corresponds to So, that's always the expectation of 1. Now, the relation to the most standard Hilbert space formulation of 10 mechanics is very easy, namely, you can perform the G and S construction, which is falling in any state omega in the sense of a linear functional. Then you can construct a little space, h, a representation pi of the algebra of observance

20:00 on the silver space, and a unit vector omega on the same little space, such that the evaluation of the state omega on a is the same as the expectation value done in the usual 20 mechanical And actually this relation fixes the structure of these three objects So in this sense, the algebraic formulation contains the usual So diverse-based formulations are more general in the sense that not all states divide up to the same diverse-based. So you can get different diverse-based representations of your algebra corresponding to different And, okay, symmetries are, of course, representing the quantum offices of the actual problem. Now, I come to the notion of a subsystem. Now, what is a subsystem? A subsystem here is, of course, again, in terms of some algebra. algebra, the simplest possibility, and which seems not a joke, the possibility which is realized in the non-examples, is that the subsystem can be identified as a sub-algebra which shares the same unit element. This is a very convenient notion of subsystem, and in particular on your, on the large system, then the state on the subsystem is immediately, it can be immediately given, it's maybe nothing else as the restriction of the linear function of the subset. Okay, algebra A1, which is sub-algebra A, with the reduced state, omega 1, is just the state omega ever related on elements of society. Now, if you, if you, in case both algebraics

22:30 are isomorphic to full matrix algebraics, then this operation corresponds to what one often does in terms of so-called reduced density matrices. This then corresponds to the factorization of the element space, which takes a partial trace. But you see that this, having this algebraic formulation, these subsystems can be much more general. They are need not to implement themselves. And actually, in climate theory, we need this generalization. I think this is more important than when this is a kind of information of somewhat classical observables in addition to the canon observables, and then it's no longer meaningful to or assume that the plant is a full nature. The next interesting or important structure is the possibility of composed resistance. discussed in the one by some heads. Maybe if you're picking two algebras, A1, A2, you can always perform the tensor program and power the composed system. There's some subtle points if you want to extend this to C star algebras, which I won't want to discuss. But the algebra, the algebra, the tensor program And given this possibility to perform tensor products, you can now also say what it means to have independent subsystems. So we have A1, A2, about independent subsystems of A, both sub-alphas of A, And then you look at the algebra generated by the subpartum A1 and A2, and you require that the subpartum is isomorphic .

25:00 Now, actually, this is a little bit too weak, so I should make it more precise, so you have also to say which map should be the isomorphism, so what is the precise expression of these of these isomorphies that require that the two algegors commute with each other. And then you look at the map where you have linear combinations of products, tensor products of elements, and you map this into a linear combinations of products, and the requirement is that this is an injective map. Okay, then you see it's important, but well-known, the fact that A1 and A2 are independent in this sense, and nevertheless there exist states which show the bell correlations, and this is always, the bell correlations always exist if the light was contained in the light was contained. of two sub-algements, as an object of two-by-two matrices, so you see that the bell correlations, occurrence of bell correlations is a purely kinematical effect, this is a mathematical consequence of the positivity position of the linear function of the omega, and it has nothing to do with any I think, probably, you will talk about, will you ever mention the correlation of the planet field theory? Okay, so that, for instance, the planet field theory was shown by Steven Samuelson, by Natnala, are that you, if you have space-accelerated regions, you will have all those states that the observables in these two space-accelerated regions total of the .

27:30 OK, now I come to my field zero. On quantum field theory, you want to treat generic gravitational backgrounds, so you want to study quantum field theory on, say, walker spacetime, visitor spacetime, etc. forever. Now, if you try to do this and you talk to the solution in general, then you meet the problem that the generic spacetime has no non-frivial symmetry. And if you look at the standard combination of concrete theory, you need a lot of symmetries, and concrete theory and Minkowski's space relies heavily on the concept of concrete elements. You can do similar analysis for highly symmetric space distances. It was done for the zitter space, space, and something on anti-lisitor space, and also something on Robinson-Walker space times, which are large spatial symmetry, but in the generic case all these methods fail. And you see it most drastically, the phenomenon, that there is no state on the generic space called the vacuum state. There was a lot of discussion in the physics literature. What could be called the vacuum on a generic space time? I think the, it's my opinion, I think all of you will agree, is that there is no such concept of vacuum, which which applies to generic spacetime. The other concept, which also one would like to consider to be fundamental, turns out to be non-fundamental, is the concept of particles.

30:00 A particle on a generic curve of spacetime is not a low-defined object. So all attempts to base, say, physics of the concept of particles, become meaningless once you admit arbitrary curve backgrounds. And you see it very drastically in phenomena like the Hawking radiation or the Unruh effect. There's another problem which also seems not to be well known. Namely, if you try to formulate kind of field theory using the path that you wrote, of course you can write down on a sheet of paper these funny symbols, but they have no meaning. Why do they have no meaning? Of There are not measures, but there's some way to avoid them in a perturbative way, so why is it not possible to do the same thing on a curved space surface related to the absence of a natural vacuum state, namely what enters in the perturbative definition of the final part of the world is the vacuum expectation value of the time. product of two fields. And since there is no distinguished vacuum, the final propagator is not well defined. Of course, you can find states and look at the expectation of the time-model products, but the choice of these states is highly ambiguous, and so you cannot I would say that's the path of the goal of the current space-time. It involves some non-genetic, non-intrinsic conventions. Okay, now how can these difficult problems be overcome? And they can be overcome by something which looks like a trick. namely, instead of looking at one single spacetime, which has no reasonable symmetries, we look at all spacetimes, of course all spacetimes of a super-class, but this glass must be sufficiently general, and then one does a construction of planet-in-zero and all these spacetimes simultaneously in a coherent way.

32:30 The coherence between this planet-in-zero and this different spacetime replaces the notion of symmetry. That's the basic idea. So, how does it work? The general concept is the following. to look at subregions of space-time and consider these subregions as space-times in their own right. But if we do this, we have then to take into account that the same region might be considered to be a subregion of a completely different space-time. Now, for instance, a look at this diamond region, I can consider this to be a subregion of Minkowski spacetime, but I can also consider it as a subregion of a cylinder spacetime. There is no intrinsic information whether this is a subregion of this space-time or this space-time. So once I take serious that subregions are space-time in their own right, I lose the concept of a global space-time which is uniquely given. A lot of global space-time that could be related to this global space-time. Actually, I think that's very much, very near to what we can really do, I mean, if you look at solutions of the Einstein equations, you, you, which have, say, singularities, and you can often, often extend the solutions, you add certain parts of the space center which you cannot cause in excess, when it's in the Schwarzschild case, you can go to the Pruska extension of the Schwarzschild metric. space like separated region in which we know nothing. And if physics would depend on the question whether this extended region exists or not exists, we would have problems. So it's

35:00 important to have a structure which really includes only the intrinsic structure of the the friction which we are interested in. Okay, then, the relation to the discussion before this, the basic idea of Park's approach to the mental field theory, then we can see the idea that the sub-frictions label the subsystems of the district. So, and that seems to me to be the fundamental role of space-time, to label the subsystems of the theory of the country. Now we want to formalize this general structure. And so we have a space line M. And we associate to each space line M the algebra observables. Now we consider our space-time M to be a sub-region of another space-time M. And chi is the map which maps KM into M. Then we require that there is a corresponding map which maps the algebra A of M into A of N. And of course this should be a structure-preserving map, so it should be a homomorphism of the algebra, injective homomorphism of the algebra. So an embedding in the algebraic sense. Now we can of course do this step and we can iterate these embeddings. So we have to space them M1, embedded M2, and then embedded into M3. Because we could also do it in one step, M1 directly into M3. We want you to be sure that the structure doesn't depend on this choice. But this on the algebraic level just means that the corresponding homomorphism on algebra must satisfy this covariance condition. Actually, if you apply this to sub-regions of Rykovsky space, you just recover the condition

37:30 of Pogoricoinus. So in case you have symmetries, you just recover the symmetries that you originally postulate it. But it's much more general. Here in general you relate spacetimes which don't have any incentives. Okay, now the general structure which we have is that of a functor Well, there are two categories. One is the category of spacetimes with embeddings as morphisms and the category of algebras of observables where the injective homomorphisms are the morphisms and the Kernel Field Theory is an association of spacetimes to algebras of embeddings of spacetimes to algebras satisfying the condition of a So, Bernard K was joking when I told him this idea that this is something that's a categorical imperative form. Now, what about the locality possible? It's a more traditional kind of field-theoretical sense. That means that the algebra, the fields that space-like separated bones combine with each other. Now actually, this is already contained in the structure a few at mid-tensor products in these both categories. So you have the category of space-time, and there's a natural product, Maybe it's a disjoint union. It can have two different spacetimes. It just takes a disjoint union, which is again a spacetime. On the first side, it looks like a spacetime which is not very useful. But nevertheless, it's a structure which can be given. Now, what is the corresponding algebra to a disjoint union? So the natural tensor structure on the level of calculus is a tensor product.

40:00 And then you require that the Sphunker respects tensor products, and this is equivalent to requirement that the observables in space like separated fictions are independent. So in particular, thank you. Okay, now just a quick look. I can now run on our use this structure and ask other questions. So we can again ask the question, does that exist the vector state? Of course, we need a definition of what we would like to call a vector state. And in different kind of field theory, we would require the necessary condition for a vacuum state that it's invariant under concrete transformation. Now here we could require the following thing. We say a vacuum state is just a family of states. omega m is one of these spacetimes, and it satisfies the condition of local covariance which says that if chi is an embedding of m into n, then the state omega m is obtained by combining the state omega m with the homomorphism of a chi. So, it just tells us that if I have an association of space, of states to space times, then if I restrict my attention to a sub-region, I get the space-time as in the corresponding state of the sub-region. And when you try to find such an object in examples, for instance, you can look at the free scale up here, and we require an addition that all states satisfy certain regularity conditions, the sort of Kalama condition, then we can prove that no such family exists. So this is a precise statement on the non-existence of a vacuum in the generic space intercepts this definition. Okay, so this justifies the procedure not to base the theory on the vacuum.

42:30 On the other hand, it poses problems. Mainly, if not only the mathematical formalism of quantification, we also need an interpretation of the formalism. So, it's a differentiation underneath, you really cross the space, relies heavily on the concept of a vacuum and the concept of particles. So, what can we do in a generic space-time? We need there something by which we can compare observance in different points of space-time. Or, which is this general framework, I think it is the same, on different space lines. And one object by which such a comparison can be done is the concept of a locally covarian field. So what's with the local equilibrium field? The local equilibrium field is a family of fields in the traditional sense, so these are operator-building distributions. Smear them with test function f, get the element of the algebra, because this moment is one of the only greatest, I have a boundary where it is, affiliation, but that's not crucial at the moment. And the crucial condition is that these fields have to satisfy a condition which relates the fields of different space-time. It's the same condition that we require for sustains. then we require that the field of the manifold M at the point X is a field of the manifold N, if I consider the point X as a point in the manifold N. This is the consistency condition for the fields. And fortunately, these locally covariant fields exist. They exist for the free field, can position the functions of the three fields, like the momentum tensor, or other big polynons of the three fields. So there are a lot of fields that satisfy this tradition, and it remains even true. If we introduce interactions and we renormalize the basis of the space

45:00 and then we find a large family of such fields. And so it seems that the interpretation of The global topology can be different, but globally, of course, these embeddings should be satisfying all nice conditions, particularly, I would say, are isometries. So, so, so, so, so, so M and its image are geomorphic to each other. But, but, of course, the manifold M might have some global features which are not present under the plan. So, so, here you see that you can formulate the theory without looking at global properties of the spacecraft. Yeah, so global properties are not visible on the level of the spacecraft. Okay. Now what about dynamics? dynamics. Now what, how can you formulate dynamics in the case where you have no time symmetry? Normally we use, say, space terms which have, which time translation in the sense that the slums are a killing vector field, time by killing vector field, and we have vector, one parameter we call it. Isometry is generated by this time vector field, and then you can discuss time evolution, but what is the meaning of time evolution on an imaginary case? Now actually this can be done using an additional axiom, which I will mention up to now, namely the so-called time-slash axiom. The time-slash axiom has a long history theory, but it was only very seldom used, I like to call it a stepchild of maximatic theory. It now becomes a very important condition, namely if the condition is the following, that if I have a manifold M and I'm embedded into another manifold N such that the image contains the portion surface of the larger space line,

47:30 then the corresponding altruist should be equal. It's not just a sub-altruist one, but it's equal to the larger altruist. This is an abstract way of saying that the dynamics in the larger space-time is completely fixed once I know everything in the neighborhood of the Scotian surface. And how does it give concrete information, this principle, or we'll spread this with this picture, so I have two spacetimes, M1 and M2, I have two other spacetimes, small ones, N minus and N plus, at embeddings of N minus into M1 and into M2, such that the image contains pushy surfaces, the same as with the manifold N plus. Now, I use the information that I have to construct an automathism of the algebra of A of M1, which tells us in how far these two spacetons are left from. How far is the time evolution from this kosher surface to this... kosher surface is different in this space and from this space. So I have the formula. Let me... I start from M1. then, or maybe I should have, oh yeah, I think I have not an error, I should first do this method and this method, so I should write it, because it's obviously all right, I don't know. Okay, so first go from here to here. I use the fact that these, in this case, because because of this time size, I assume, the corresponding homomorphism is involved, so I can take the inverse. I go then from here to here, this is the wood direction, then I use the, again, the opposite direction,

50:00 using again the time-starts exit, and finally I come back to M0. So this shows that there is some intrinsic notion of time-reputation, at least I can compare time-reputation so this was the best time. Okay. So, let's move to the last sheet of outlook. So we have seen that the principle of locality is crucial for a general liquid-variant formulation of field theory. theory. We have seen that this principle allows the comparison of fields in different space times and this is important for the interpretation of the theory. So one interpretation for instance is that one can look at something called the global temperature. This is a recent concept developed by Ujima and Rose. And another thing is that one can also look for a local particle adaptation. So we know what a particle state of the Ecopsy space is, When we look at expectation values of fields in such a particle state, and then if we find the same behavior of a state in the third space term, we can say locally this state looks like a particle state. So there seems to be some possibility to give a local particle interpretation. Now, the most important question is, of course, can we go beyond a fixed gravitational background? Can we quantize the background? Now, if you approach this formally, you can just consider the fluctuations of the background as a quantum field in its own right. So you get the same formalism that you have just one additional field in the fluctuation field of the metric tensor of the space-time.

52:30 But you need an additional requirement, namely this decomposition of the gravitational field in the background field, the fluctuation field, is physically meaningless, but only in the formalism. So you need a condition that this does not affect the theory. But in this abstract framework, this is a very natural formulation, namely And then it tells you that these relative time evolution detail which I constructed must be the identity. Because the different backgrounds can be compensated by the fluctuating fields. If you look at what this means in terms of equations of the fields, it turns out that these are just the Einstein equations for the interrupting kind of fields. So the gravitational field, the contrast gravitational field has to satisfy the Einstein equation. Okay, so let's make, give some hope that consistent background-independent polarization of gravity is possible. Of course, one has to add the warning. I think there's this big problem that gravity is retroverted to be non-renormalizable and this difficulty is still present, so the best we can expect is that we get a consistent construction of gravity as an effective field. Thank you. Okay, I'm sure there are some questions. I actually have two. One is purely linguistic, but the second has a different answer. So the linguistic question is, since the phrase principal locality already has a completely fixed meaning in algebraically, but also from the point of view of my taste of language, What is the benefit of all your principal locality and principal localizability? Uh, yeah. Okay. Isn't that what it's really about? Um, yeah. Just a minute.

55:00 Maybe, but I would not agree that I think the principle of locality in the original intention of So I think it's in the same spirit, but I agree it's a stronger principle and maybe much a good thing. So the second question is quite good. The time size requires, of course, that you are able to talk about short-term cultures, doesn't it? Not really. Are you thinking of... Not exactly. Are you thinking of the same limits or...? Yeah, okay. This depends of course on your theory. So if you have, say, the three fields, you have to know the fields and the derivatives of the fields. So it's a little bit more than the portion service itself. So in the abstract framework, I just admit the neighborhoods. So you have a... I have a whole neighborhood. Yeah, but you can certainly replace the neighborhood by a genre of neighborhoods, and perhaps you can do some jet structure or so and so. It depends on your model. Right, but you're not looking at this. No, I think it's here for free fields. Yeah, so there's a short time option. Yes, you have a short time option. But this is more general, so you don't need short time option. I still don't really understand this locality concept. It seems to me that this definition actually ignores the actual causal structure of the of the space-times, this is just, the localization is just what we are defining, kind of, so we can restrict these objects from smaller and smaller, I don't know, from regions of the space-times, but absolutely lowering the actual life-home structure. No, no, no, no. I did not present all details, so I should add that, of course, the spacetimes are differential manifolds with the metric tensor of the Lorentz form. And these embeddings always respect the metric tensor, so they preserve the metric tensor.

57:30 And in addition, this is important, I did not mention this, they have to respect the causal structure. Which is an additional requirement. So if two points in the image can be joined by causal curves, the whole causal curve must be contained in the image. So you don't get new causal relations by the image. This is very important otherwise the formalism becomes inconsistent. I see. Next, I'm still curious about this pathological question, even vocally. You can remove points from a normal metaphor or cut of edges etc. So you can make very tricky pathologies even in the vocation. So how can you manage these local beddings, in this case, first? Secondly, even the global topology of the space-time can appear in the field theory in the form of different Casimir attacks, etc. So how can we ignore these global structures? Okay, so the first question. Again, I did not present all details. So the space times which are admitted here are globally hyperbolic space times. So these are space times which are mid to quotient surface. In such a space time, you cannot just remove a point. No, not asymptotic. The name globally hyperbolic does not mean that the space time is large. It can be arbitrarily small, but it emits a portion service. So you cannot just remove a point. This would destroy a globular velocity. And the other question was the Casimir effect. Yeah, but the Casimir effect is actually a nice example. Where mainly in the Casimir effect you can look at the local algebras and the local algebras are the same. But you have different states. And actually, an old paper by Donald K. on the Casimir effect, where essentially these

1:00:00 ideas, at least for the flood case, already contained. Yeah. Then he looks at sub-regions, at the diamond trees in a Casimir situation where you have, let's say, about boundary conditions, and the Newcastle space. And the requirement is that you use the same definition of the energy convention in both cases. if the normal pathology is different, then you cannot avoid the differences in the best way the public patients. Because you speak only on the streets, and not only on the issues of children, that's the point. And at this level, you are just talking about the... The locality, as you said, manifests itself on a level of the general books. But in the states, there are 100 divisions, and they already trust, but there can be many other things that make it local. So you get this locality, realization of the locality or localizability principle only if you pay attention, draw attention to the algebras, not on the level of states. In the level of states, the locality is final. I still have a very short question, but I am usually asking me about this in the barrel of correlations, which you can follow up into a very short answer to this question, if it is a correlation among the law, what are the events here, what are the events here, what are the events here? Or is it better correlation? We have this correlation between observables and this independent sub algebraics. Maybe these are just these operators which we presented, of course there are other ones, but these are maybe those observables which show the effect most drastically. Oh, no, okay, you call it correlation, but there's a name, you look at, you have a state, and you look at the expectation value, and you can show that the state is not a convex

1:02:30 combination of product states this is i think there's a sense of relative equality can you say more about what states there are in the framework because when you had the theorem that there isn't this family of animal states i don't know there are other states but they are not there's not this covariant them sorry yeah so what is known about what states there are Yeah, so what you can use instead of a single state is a family of sets, so-called folium of states. So there's a consistent association of these folia of states to each space-time. So you can associate a family of states to every space-time, and this set of states satisfies this condition of local covariance, but not the single one. Yes, so there is no representative, in a sense some equivalence class of states, but you cannot make a consistent choice. I think you can formulate it as a problem in homology. So some of these problems can be solved and some are not. The case for the state is that the problem cannot be solved. It's just a positivity condition which is violated. If you give up the positivity condition on the state, there is a so-called which satisfies the condition of logical variance. At least locally, if you restrict the two to, say, the journalism function that satisfies this function. I have a question for the quantiquality extension of the program. You said that you still face the non-anomalizability, therefore it would still be just an effective theory. I wondered why are you bounded, since the non-anomalizability just arises when one tries to quantize the relative version of VR. Why are you bounded to do that? Why couldn't you? Okay, you say one could try to proceed a la Ashtekar. Well, not necessarily. Yeah, yeah, yeah.

1:05:00 It's very difficult to construct . That's the only reason. No, no, I don't think that this is an absolute difficult. But it's a difficulty for our present knowledge. Our technical abilities admit us to construct alternatives to be the best thing. There's another one more question. Yeah. You said that you are only restricted to the very hyperbolic space-time and remain In this principle, we're looking at only at the distributors locally, which is a goal of this time. Could you extend this to more general space, or? Yeah. Because it's really restrictive in the end-size for the . Yeah. You are right. First, why do we restrict ourselves to globally hyperbolic space? Just because if you do the construction for, say it's a pre-scalar field, you need to have the existence of a unique solution of the quotient problem. Once you have this unique solution, you get a unique algebraic structure. So, therefore, it's really important to have global hyperbolism. But what can be done in case of a non-globally hyperbolic space? The idea is that, in this case, look at the globally hyperbolic subregions. So, each globally hyperbolic subregion is used by this method. And if they are contained in each other, they have to satisfy all this condition. But then, you don't get the full algebra. So, the general idea is now to look at the free algebra generated by all of these globally hyperbolic sub-algebras, multiple all relations which are used for this privilege. This gives, of course, a very huge algebra, which potentially includes all possible boundary conditions. boundary condition then implemented by the choice of an ideal in this algebra. So typically

1:07:30 a maximal ideal. So the maximal ideals of this large algebra, which you can associate to a non-globally hyperbolic spacetime, characterize the possible boundary conditions. So in a And no problem you have a body space, so you need to specify boundary conditions. And the abstract way of fixing the boundary conditions is to fix an ideal in this hard time. I'm not sure, I understand. Can you really be sure that there will only be a maximal ideal in this hard time? Oh, I don't think the existence of maximal ideals, they probably inform some kind of song's lemma. Yes, yes, yes. So that's probably not a, maybe a more problem in practice to find such ideas. But the existence of these ideas, I think, is not a problem. Well, to find them in examples is difficult, and we have done it in only very simple cases. So we look at theory on a half space, and characterizes them in terms of such an ideal. So it seems, in the examples, the idea works, and in general it seems to be consistent. because this has been another one that we need to take a moment. Okay, let's thank. And we have a coffee break for 12 and a half minutes. Thank you. So he is a finalist.

1:10:00 Well, I don't know. But that's the winner. The human health case. Yeah. OK, so I'm only talking about aspects of cultural development on the series, because, well, for one thing, I'm simply because of lack of time, all that plays a role, but also because I'm rather unapologetically a mathematical physicist. And when a mathematical physicist examines a notion, ordinarily it disintegrates into many different sub-notions. The notion, for example, like independence, which disintegrated into many sub-notions of which Klaus posited the strongest possible notion of independence of subsystems in his setup. There were many other versions of independence which were much weaker and which are actually also obtained, not only the mathematical results, but disobedience of systems. But that's not the topic. The topic is causality. So I'm going to be speaking today of three aspects of the physical slash self-promotion of causality which have yielded themselves to a rigorous mathematical population and treatment. Now, I shall only be talking about four-dimensional Kosky space, but everything that I say is valid for arbitrary, static, globally, metabolic space-times, probably more general than that. So let me begin. There are three different aspects of causality. The first one is an expression in the setting of causal field theory of a very strong causality which is present in classical field theory. So just as follows, they begin with classical field theory for different reasons, however. So, let me give you a concrete example of the classical field theory, a solution of the

1:12:30 time-moment equation. So, 5x here is a function. This is a strong differential, partial differential equation. All right, now, in this four-dimensional Minkowski space, let me talk about some region which is going to be some ball in free space across a period of time. So we're looking at some ball at a short time in the constant space. Okay, and then associated with that ball, there are these additional regions. this one, consists of all the points such that the backward-like cone of that point intersects with that region NT0, the entire backward-like cone. Which means, in particular, every past-directed like-like or time-like line emanating from that point that region NT0. And similarly, this, every forward region, every forward target, every future derivative line emanating from each point in here, must intersect NT0. Okay? This is called the domain independence often. So the, and this is the unit of those two. Now, since I started with a ball, that region that emerges in an empty knot is what we call a double comb with base empty knot, but we can also talk more gently about it. I'm going to stick with this one. Now, if you have a subset of Mikoski space, if you consider the set of all points which a space like separated from all points in O, that's called the causal complement of O, and then you can consider the causal complement of the causal complement of O. That is usually called the causal closure or the causal hold or the causal completion of O. Please note that the causal completion of NT0 is precisely this double cone. Okay, now, the causality property that I want to remind you of is that if you have a solution of the Klein-Borden equation,

1:15:00 then its values in this double cone are completely determined by the restrictions of the field and its time derivative to the base of the double cone. So if you're given the value of the field of the solution, in this case the solution, phi restricted to this base, along with the values of the derivative of the field, the solution restricted to this base, then the information on this base alone completely determines the values of the solution everywhere throughout this region. All right? It's a very powerful form of validating. Okay, now, because I'm going to have to deal with situations where we cannot make restrictions in short times, I also want to present this formulation of the same class of a situation. For a large class of open subregions of R4, so of course N is not open. For a large class of open regions of R4, the values of phi in its causal completion are completely determined by the values of phi in that open subset of the completion. Now, if we keep this in mind, let's turn now to quantum field theory. By quantum field theory, 5x is no longer a function of any kind. It is an operator value distribution. So, in general, you have got to smear this object with a suitable test function Finally, this test function has got to be a test function of all four variables. You have to smear it over an entire open set in a Cosby space in order to get a density fine part rid of some Hilbert space. If you're dealing with a free field, a free Bose field of mass gamma, then you're dealing

1:17:30 with such a distribution which satisfies the Klein-Borne equation now in this form. It's going to be for some dense subset of vectors in H psi, when you apply the operator that you get by smearing the operator with this test function, then you get zero. This is true for every test function and every vector in that Hilbert space, or in a subset of the Hilbert space. So, in other words, phi is, in the distributional sense, a solution of the final order equation. All right, now, in the special case of free fields, you can, in fact, smear it with just a function in the special variables, and you do end up, as you can find, not quicker. So there are short-time free fields. Just as when you look at the time-ordinate, a solution by time-ordinate as a function, we concentrated on the short-time values of the solution and the short-time values of the derivative of the solution, yes? Okay, so we have these corresponding short-time values of the free-point field. And we can also introduce this object, this is the commutator of this operator with the freehamton operator in this signboard view. I'll write it down. It's simple, however. Okay, and, but last night, thinking about this, it's probably an idea to remind you that the free field at time t0 smeared with h is obtained from the free field at time 0 precisely through this edge-on-action. And therefore, when you look at this particle pi at t0h, which is defined this way, you're recognizing as precisely the derivative of phi divided t to t0. Alright, so in other words, phi at t0h is the, for the free field, the exact power part of the derivative institution of the Klein-Gordon field, restricted to a sharp tire.

1:20:00 Alright? So, these data here, phi t0h, pi t0h, in the case of the free field, of the shock time data in the Kochi phone in the classical case. All right, now with that in mind. For any open subset of Nikoski space, let me define two values. Well, actually, why not? You might know them. So let's, this will denote the formula that was generated by all of the fully smeared field operators whose support is contained in an open subset. Now, what that means exactly is mechanically rather complicated, but as you can think of it, this algebra as being the set of bounded functions of the field values in the real problem. Okay? Similarly, for that region, nt0, that Schottheim wall, I'd like to define the algebra a nt0 as the phenomenon algebra generated by the sharp time field and the sharp time time derivative. With Smear-Bowell test functions with support in that spatial wall, the base. Alright? Okay. Then, this is a theorem for the free-moron field. This algebra, which is the bounded functions of the field and its time derivative at the sharp time t-naught, alright, so this is the algebra you get by considering only the sharp time values, gives you the entire algebra a, d, and t-naught, alright, this is a double cone. This is the cause of completion of empty nut. And here we have, here we call, these are the bounded functions of the field throughout this double cone. Alright? This is the exact quantum mechanical color part of that first part of that theorem that I had in the report

1:22:30 from the classical case. That the values of the classical solution in the double cone generated by the base is exactly determined by the field and its derivatives restricted to the base. All right? This is the exact counterpart. All right, but because one doesn't always have these sharp-timed things, I want to also formulate it here for open sets. So, if I have any open subset of Mikasi space which contains that sharp-time ball, that base of the double cone, and which is also still contained in that double cone, all right? So this is a little neighborhood of that ball in Mikasi space-time. then it follows, this is a theorem, that the algebra generated by this possibly much smaller open subset of this double cone coincides with the whole algebra of the double cone. Or, let me point out that for any open set that can be satisfied this condition here, you can show that the causal completion of that open set coincides with this double column. So for the purposes of my next definition, what we know about the free field is that for any open set like this, the algebra associated with this smaller open set coincides with the algebra are associated with this possibly much larger open set, the causal completion of the goal. All right, so the field values in this larger region are completely determined by the field values that is possibly much solid. All right, this is the exact kind of part of the second part of the first theory of lecture. All right, now for general quantum fields, This, as was briefly alluded to in a question I posed to Truss, in general one of the fields, those sharp-time objects do not exist as operators. And therefore, that algebra A, NT0 has no content.

1:25:00 But, when you steer the general part deals with test functions over a whole open subset, those are perfectly good operators and you can certainly define the algebra as A or O for open subsets in general, up to technical conditions which are well understood when I'm not talking about. So, in light of that last line of the theorem I had on the previous slide, A formulation of a type of causality for general point of view theory is this. A net of algebras, of localized observables, satisfies the condition of no imperative causality if the algebra associated to an open set already coincides with the algebra associated with the causal flaw, the causal completion of that open set. So, rephrasing the theorem on the previous slide, the three quantum fields satisfies local primitive causality. And you understood, I hope, with my long preparation through the classical field into the pre-mobile, that this condition of multiple quirk of continuity is an expression of an abstraction of the idea we have some hyperbolic wave equation, and things are propagating according to some hyperbolic wave equation like characteristics, all right? So this is, as Chris is going to get, in point of view theory, you're not going to be able to, without specifying a particular propagation, a particular wave of the value. This is the abstract content of the very strong cause

1:27:30 in the classical field theory. Now, so, this is one aspect of pathology, and it captures, as I said, this idea of hyperbolic propagation within led-like characteristics. So, what is the status of this? Well, it holds for many models. as it holds for free fields but it holds also for many other fields. So it's known to hold in many balls of interest. But it has some, at first glance, at first consideration, anti-intuitive consequences. So, let me point out to you, consider a small double cone O2 and consider a region O1, an open one like this, okay, and consider the causal complement, the causal completion, rather, of O1, that gives you this much larger double comb, O1, W5, yes? So as you can see, the double comb O2 is contained in double comb O1, W5. Everybody can follow? Alright, so if I take any observable localized in O2, then, now remember, this algebra was generated by... this algebra was generated by any field operator with support in this logic. So a particular field operator with support here are among those. So this has got to be a subalgebra in this. So this is also an element of this. But if you have local causality, this algebra is precisely this algebra. That means that the same observable is localized simultaneously in O2 and in O1. Alright, let me damage your minds further by considering this. same setup. Now I'm adding this further region O3. You can see that the causal completion

1:30:00 of O3 includes O2. And now I've got the same operator A localized here, here, and here. They can be clearly continued at it in the night. Alright, so, I know how much this is probably close at one time. What sleight of hand is going on here? Alright, so the point is that you must recall that observables do not represent single measuring apparatuses in a given laboratory being taken in an experiment period over some fixed time interval. Rather, an observable represents an equivalence class of measuring devices. Observables, respectively states, in algebraic monocube theory are equivalence classes of measurement devices, respectively preparation devices in the laboratory. Because as you know perfectly well, many different apparatus, with the same experimental result, with the same preparation, etc. Okay? So, if you're not familiar with this already, then I would recommend to you learn to look at this discussion of what observables really are, and what the real relation is to and what states really on what their relation is to preparation of us. So, in other words, there's no conceptual problem whatsoever. It's simply that there are devices in these disjoint space-time regions which may be precisely the same experimental results in all possible preparations. That is the content of local primitive causality. And that local primitive causality is proven to hold in models is therefore a non-trivial operational assertion. Okay, second manifestation of causality, quantum field theorists call this local commutability, algebraic quantum field theorists call this locality, up until this morning.

1:32:30 So what we usually mean when we say the principal locality is this, if the space-time region O1 is space-like separated from the space-time region O2, then every element in the algebra of observables localized in O1 commutes with every element of the algebra of observables localized in Rome 2. That is a principle. What is the justification for this principle? Well, actually.