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**Contents:**

Hollands and R. Kratzert , Singularity structure of the two point function of the free Dirac field on globally hyperbolic spacetime Ann , Annalen der Physik , vol. Antoni and S. Dappiaggi, T. Hack, and N. Pinamonti , The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor Rev , Math. Phys , vol. Sanders , The locally covariant Dirac field Rev , Math. Rejzner , Fermionic fields in the functional approach to classical field theory Rev , Math. Zahn , The renormalized locally covariant Dirac field Rev , Math.

Fredenhagen and K. Brunetti, K. Fredenhagen, and K. Rejzner , Quantum gravity from the point of view of locally covariant quantum field theory arXiv , pp. Fredenhagen, and M. Hollands and W. Kay, M. Radzikowski, and R. Fewster , A general worldline quantum inequality , Classical and Quantum Gravity , vol. Ivrii , Wave fronts of crystallo-optics system solutions Soviet Math , Dokl , vol. Esser , Second analytic wave front set in crystal optics Applicable Anal , pp. In what follows, I will describe in more detail my part of the talk.

Traditionally, most of the basic intutitions on which foundations are based come directly from our experience with the natural world: forming collections of things, counting, observing natural phenomena, etc. This has ultimately led to set theory as a successful foundation of mathematics. However, the major ideas that have allowed such success are difficult to translate into the physical sciences, if possible at all. Now, for the first time in history, we start to have an important amount of basic intuitions coming from the interplay between mathematics and theoretical computer science, which gives us an opportunity to find new foundations.

More importantly, some of the basic features that would make such new foundations successful could be easier to translate into the physical sciences. As a real-life example of work in this direction, I will introduce the main ideas behind homotopy type theory and the univalent foundations program. More concretely, how our foundational understanding can benefit from the interplay between category theory, type theory, and logic. Categorical Quantum Mechanics CQM is a field that in particular aims to establish a new paradigm for the mathematical foundations of quantum theory by seeing it as an example of abstract theory of systems and processes, but, in order to avoid overlap, these ideas will mostly be covered by Pierre.

Discussion: How could the advantages of thinking categorically in mathematics be translated into physics? It seems possible to get a good notion of general theory of systems and processes, and to view certain parts of physics e. However, at the moment, in order to select quantum theory out of all possible theories, a number of ad hoc choices have to be made which are just motivated by the mathematical structure of standard quantum theory.

How could we find a set of physically or informatically motivated axioms that select quantum theory out of all possible theories? Could the fact that the univalent foundations program is an approach to the foundations of constructive mathematics represent a serious limitation? What are the philosophical implications of CQM? Since CQM is a new field, this question has not yet received a lot of attention.

How could dagger Frobenius algebras be interpreted physically? These algebras are a key ingredient of CQM, in relation to how classical and quantum information are modelled in this context. Should the state of physical systems depend only on boundary conditions in the past, and not in the future? At the macroscopic scale, it seems obvious that this answer to this question is 'yes': everyday events are always preceded in time by their causes. However, given the fundamental time symmetry of most of our physical laws, it is interesting to ask whether this is necessarily still true at the quantum level.

The question is compelling even at this abstract level, but there are also more concrete physical reasons to be interested. Many no-go theorems in quantum mechanics, such as Bell's theorem, rule out local hidden variable theories given certain assumptions. One of these is that the state of the quantum system is uncorrelated with the future state of the detector.

This is an explicitly time-asymmetric assumption — we do of course expect the past state to be relevant — and it is interesting to consider what the implications of relaxing it are. This idea is not new but has remained rather unexplored, perhaps partly because of its unintuitive nature. However it has been popularised to some extent by Huw Price, among others, and has started to attract more attention in quantum foundations — one recent example would be Leifer and Pusey, Is a time symmetric interpretation of quantum theory possible without retrocausality?

The possibility of a local, spacetime-based quantum theory that is still compatible with quantum no-go theorems is attractive enough that it seems worth investigating further. Contemporary physics seems to be inwrought with reductionist ideologies, which hold that everything, i. Whereas many physicists would concede that such a reduction has not yet been established, the idea that one could proof the opposite, namely that reductionism is wrong, seems false, if not ludicrous to most. Yet, in the first chapter of his book, David Chalmers, a famous philosopher from NY University, sets out to do exactly this: To proof that reductionism is simply false.

He establishes that there are phenomena which, according to his argumentation, cannot in principle be reduced to physical notions. I think his argumentation, and the underlying question, concern an important border-stone of physics and may have implications for the fundamental methodology of this science. Therefore, I would like to propose a critical discussion of Chalmers' argumentation which aims at elaborating the extend to which his conclusion applies to physics. As a preparation for this discussion, I would gladly offer to give a talk where I present Chalmers' argument in a detailed but brief way.

Furthermore, I would add my thoughts about his reasoning and could, if time permits, sketch a slightly different argumentation which is more adapted to foundations of physics. In the subsequent discussion session, depending on the preferences of the participants, we could focus on aspects of Chalmers' argument and its criticism, or also evaluate my different proposal.

I think this topic very much aligns with the goals of the workshop because it concerns and invites to rethink the most general, and hence most deep, paradigms of modern physics. The paradigm of black hole thermodynamics is likely to provide a stepping stone towards a theory of quantum gravity. But does black hole thermodynamics form an instance of genuine thermodynamics—as it is typically believed after the discovery of Hawking radiation—or merely an analogy to thermodynamics proper?

The question will be of interest to any one working on gravity, QFT in curved spacetime, foundations of thermodynamics and, more philosophically, the distinction between analogy and identity. It also nicely touches a general problem of physics at the frontier of current knowledge: what kind of theoretical principles to trust in when the right empirical data is missing. Symmetries are regarded as central components of all modern physical theories. However, it seems that not much can be said about them in general and the most important work in their analysis should be devoted to finding crucial differences between various types of symmetries.

All symmetries can be understood as functions determined on a set of kinematically possible states of a given theory. The first important distinction Caulton is between analytic and synthetic symmetries. Analytic symmetries are these which do not change any physically real properties of any state; they are merely redescriptions of a physical system and operate only on 'superfluous structure' of a theory which does not have any physical content. The term 'gauge symmetry' is often used in a similar way.

Synthetic symmetries change some physically real properties of at least some states, but there are another physically real properties which are left unchanged by them. The second important distinction Kosso is between symmetries that have direct empirical significance that is, can be tested by performing appropriate transformation and observing appropriate invariance and these that do not have direct empirical significance it does not mean that they do not have empirical significance at all — but in this case it is only indirect, e.

The aim of my presentation is to explain in more details these two abstract distinctions, consider possible relations between them my hypothesis is that under some plausible understanding of being physically real and of empirical significance these two distinctions should coincide and try to connect them with more 'concrete' divisions of symmetries, for which we can easily provide well known examples, namely discrete vs. Discussion: What is a relation between analytic vs. Which of known symmetries belong to which of these four types? How can we decide this question? Can we understand the notion of empirical significance in a non-anthropocentric way?

Does it make sense to consider a status with respect to the above distinctions of a symmetry in a theory which is only approximately true? The theory of categories, developed in particular by Grothendieck, offers a very general formalism that enables to draw links between physics, topology, logic and computation. At the opposite of the exponential rate of ramification of the tree of science, could it be a path for gathering over-specialized scientists by showing their problems are nothing but the same object watched from different point of views?

Is information a fundamental concept of Physics? Recent works have proposed that Gravity is an entropic force, one even suggesting that Dark energy can be explained from Quantum information principles E. Verlinde, , further cementing the relevance of the concept of information. In October we organized a compact course on quantum field theory on curved spacetimes at the University of Potsdam.

More than 40 participants with varying backgrounds came together to learn about the subject including its mathematical prerequisites. Thus prepared the participants then attended the lecture series on the main topic itself, quantum field theory on curved backgrounds.

This book contains the extended lecture notes of this compact course. Nicolas Ginoux 3. We give complete proofs. In Sects. Remark 1. Note that Axioms 1—5 are not independent. For instance, Axiom 4 can easily be deduced from Axioms 1,3, and 5. Example 1. Axioms 1—4 are easily checked.

This shows Axiom 5. Example 2. Let X be a locally compact Hausdorff space. Example 3. Let X be a differentiable manifold. If we complete this normed vector space, then we are back to the previous example of continuous functions. Definition 2. Definition 3. Remark 2. Example 4. Lemma 1. This easily follows from the triangle inequality and from homogeneity of the norm. By the used in the proof. Example 5.

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In particular, the spectrum of a is nonempty. Remark 4. In particular, self-adjoint elements are normal. In a commutative algebra all elements are normal. Proposition 2. Then the following holds: 10 C. Becker 1. We start by showing Assertion 1. To see Assertion 2 let a be invertible. To show Assertion 3 let a be normal. To show Assertion 6 let a be self-adjoint. Becker Since the factors in this product commute the product is invertible if and only if all factors are invertible.

Corollary 2. Let A be a C -algebra with unit. This is known as continuous functional calculus. Proposition 3. Moreover, the following holds: 1. Becker Proof. Assertion 2 clearly holds if f is a polynomial. Proposition 4. Lemma 2. Then the following three statements are equivalent: 1.

Remark 6. Definition 9. Remark 7. Definition Example 6. Lemma 3. The set of all states on A is denoted by S A. Example 7. Example 8. Proposition 5. Then we have the following: 1.

Remark 8. States of this form are called normal.

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Lemma 4. Then the following holds: 1. The other direction is obvious. Therefore, the pairing 20 C. Example Remark 9. The universal representation is faithful. Remark Theorem 1. Then the following two statements are equivalent: 1. By the same argument, b2 and hence b is a multiple of the identity. Becker Theorem 2. Therefore, such vector states are pure. Proof of Theorem 2. By Corollary 4, convex combinations of pointwise limits of states are states. Hence S A is a bounded closed convex set in the topology of pointwise convergence.

By the Banach—Alaoglu theorem from functional analysis, S A is thus a compact subset of the closed unit ball in the dual space of A in the topology of pointwise convergence. The Krein—Milman theorem then implies that S A is the closed convex hull of its extreme points, which by Remark 11 contain all pure states. It remains to show that all extreme points in S A are pure. The restriction of a pure state to a subalgebra need not be pure.

However, the norm making the algebraic tensor product of Hilbert spaces into a preHilbert space is unique. So it seems natural to study norms on the algebras by means 24 C. Becker of norms on representation spaces. The simplest way to do so is by using the universal representations. Lemma 7. In general this is not the case, so we set the following. Nor can it be written as a pointwise limit of such convex combinations. Hence a pure state is decomposable if and only if it is a product state.

One aims at a characterization of this convex set by inequalities. While a complete characterization is unknown, a simple such inequality has been deduced from the work of Bell in the late s on the Einstein—Podolsky—Rosen paradox. See also [9]. Lemma 8. Hence, 1. Taking pointwise limits of convex combinations, the inequality holds by continuity.

Let e1 , e2 be the standard basis of C2. The existence of entangled states may thus be considered as a characterizing phenomenon of quantum systems. In fact, if one of the observable algebras is abelian — e. Proposition 6. Our approach follows ideas in [7]. A different proof of this result may be found in [8, Sect. Note that since V is not given a topology there is no requirement on W to be continuous. In fact, we will see that even in the case when V is finite dimensional and so V carries a canonical topology W will in general not be continuous.

Proposition 7. Then 1.

By Assertion 5 of Proposition 2 the spectrum is contained in S 1 and by Proposition 1 it is nonempty. This shows part 2. This follows directly from the properties of a Weyl system. Lemma 9.

Lecture 1 Quantum field theory in curved spacetime

Estimate 1. Lemma Uniqueness also holds in the appropriate sense. Theorem 3. Becker Corollary 5. CCR-algebras of symplectic vector spaces are simple, i. Direct consequence of Corollary 2 and Lemma Corollary 6. Theorem 3 yields the result. Observe that symplectic linear maps are automatically injective. References 1. Bratteli, O. Springer, Berlin Heidelberg 1 3. Davidson, K. Dixmier, J. Murphy, G.

Dappiaggi, T. Effective action for time-dependent harmonic oscillator in quantum mechanics. What is the physical meaning of diffeomorphism invariance? It is argued that the theory is logically and mathematically consistent whereas semiclassical prescriptions for incorporating the back-reaction of the quantum field on the geometry lead to inconsistencies. Chicago: University of Press; Brunetti and K. Guillemin and S.

Takesaki, M. Manuceau, J.

Quantum Field Theory on Curved Spacetimes. Concepts and Mathematical Foundations. Editors: Bär, Christian, Fredenhagen, Klaus (Eds.) Free Preview. Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations (Lecture Notes in Physics) () [unknown] on ykoketomel.ml .

Springer, Berlin Heidelberg 29 9. Baez, F. In particular causality relations will be explained, Cauchy hypersurfaces and the concept of global hyperbolic manifolds will be introduced. Finally the structure of globally hyperbolic manifolds will be discussed. More comprehensive introductions can be found in [1] and [2]. This means one can find a basis e1 ,. We choose a time-orientation on V by picking one of these two connected components. See Fig. Hence, for both Lorentzian scalar products one gets the same set I 0. In cartesian coordinates x1 ,. This turns Minkowski space into a Lorentzian manifold.

Such a Lorentzian metric g is called a warped product metric Fig. This example covers Robertson—Walker spacetimes where one requires additionally that N , h is complete and has constant curvature. In particular Friedmann cosmological models are of this type. In general relativity they are used to discuss big bang, expansion of the universe, and cosmological redshift; compare [2, Chaps. Hence, one can illustrate the set of timelike vector in the tangent spaces T r,t PI , resp. One uses PI , g and PI I , g to discuss the exterior and the interior of a static rotationally symmetric black hole with mass m, compare [1, Chap.

This definition is not exactly the one given in [1, Chap. A time-orientation Fig. Definition 1. In what follows time-oriented connected Lorentzian manifolds will be referred to as spacetimes. It should be noted that in contrast to the notion of orientability which only depends on the topology of the underlying manifold the concept of time-orientability depends on the Lorentzian metric. A continuous piecewise C 1 -curve in M is called timelike, lightlike, causal, spacelike, future-directed, or past-directed if all its tangent vectors are timelike, lightlike, causal, spacelike, future-directed, or past-directed, respectively.

We consider Minkowski space R2 , g Mink. By Example 1 every open subset of Minkowski space forms a Lorentzian manifold. Let M be two-dimensional Minkowski space with one point removed. Definition 4. One important feature of the exponential map is that it is an isometry in radial direction which is the statement of the following lemma.

Lemma 1 Gauss Lemma. A proof of the Gauss lemma can be found, e. Suppose in addition that c is smooth. This 2 Lorentzian Manifolds 49 finishes the proof if one supposes that c is smooth. For the proof in the general case see [1, Chap. Remark 3. Every point of a Lorentzian manifold which need not necessarily be a spacetime possesses a convex neighborhood, see [1, Chap. Furthermore, for each open covering of a Lorentzian manifold one can find a refinement consisting of convex open subsets, see [1, Chap. Sometimes sets that are geodesically starshaped with respect to a point p are useful to get relations between objects defined in the tangent T p M and objects defined on M.

For the moment this will be illustrated by the following lemma. We can find a future-directed timelike curve c from p to q. Definition 6. Causal domains appear in the theory of wave equations: The local construction of fundamental solutions is always possible on causal domains provided their volume is small enough, see Proposition 3 on page The last notion introduced in this section is needed if it comes to the discussion of uniqueness of solutions for wave equations: Definition 7.

Similarly, one defines future-compact subsets Fig. If now the spacetime is compact something strange happens. If the spacetime M is compact, there exists a closed timelike curve in M. Therefore one excludes compact spacetimes, for physically reasonable spacetimes one requires the causality condition or the strong causality condition Fig. Definition 8. A spacetime is said to satisfy the causality condition if it does not contain any closed causal curve.

A spacetime M is said to satisfy the strong causality condition if there are no almost closed causal curves. Obviously, the strong causality condition implies the causality condition. Example 9. In Minkowski space Rn , g Mink the strong causality condition holds. Remark 5. U V forbidden! Consider the spacetime M which is obtained from the Lorentzian cylinder by removing two spacelike half-lines G 1 and G 2 whose endpoints can be joined by a short lightlike curve, as indicated in Fig.

Then the causality condition holds for M, but the strong causality condition is violated: For any p on the short lightlike curve and any arbitrarily small neighborhood of p there is a causal curve which starts and ends in this neighborhood but is not entirely contained. The notion of global hyperbolicity has been introduced by J. Leray in [5]. Globally hyperbolic manifolds are interesting because they form a large class of spacetimes on which wave equations possess a very satisfying global solution theory; see Chap. In Example 9 we have already seen that for Rn , g Mink the strong causality condition holds.

Hence, Minkowski space is globally hyperbolic. G1 p Fig. In general one does not know much about causal futures and pasts in spacetime. For globally hyperbolic manifold one has the following lemma see [1, Chap. In Fig. A subset S of a connected time-oriented Lorentzian manifold is called achronal or acausal if and only if each timelike or causal, respectively curve meets S at most once.

A subset S of a connected time-oriented Lorentzian manifold is a Cauchy hypersurface if each inextendible timelike curve in M meets S at exactly one point. Obviously every acausal subset is achronal, but the reverse is wrong. However, every achronal spacelike hypersurface is acausal see [1, Chap.

Any Cauchy hypersurface is achronal. Moreover, it is a closed topological hypersurface and it is hit by each inextendible causal curve in at least one point. Any two Cauchy hypersurfaces in M are homeomorphic. Furthermore, the causal future and past of a Cauchy hypersurface is past- and future-compact, respectively. This is a consequence of, e. In particular, in any Robertson—Walker spacetime one can find a Cauchy hypersurface. Of course, if S is achronal, then every inextendible causal curve in M meets S at most once.

For a proof of the following proposition, see [1, Chap. From this we conclude that a spacetime is globally hyperbolic if it possesses a Cauchy hypersurface. The following theorem is very powerful and describes the structure of globally hyperbolic manifolds explicitly: they are foliated by smooth spacelike Cauchy hypersurfaces. Let M be a connected time-oriented Lorentzian manifold. Then the following are equivalent: 1 M is globally hyperbolic.

The crucial point in this theorem is that 1 implies 3. This has been shown by A. Bernal and M. Geroch [7, Theorem 11]. See also [8, Preposition 6. Note that a Cauchy time function is strictly monotonically increasing along any future-directed causal curve. We conclude with an enhanced form of Theorem 1, due to A. Let M be a globally hyperbolic manifold and S be a spacelike smooth Cauchy hypersurface in M. Any given smooth spacelike Cauchy hypersurface in a necessarily globally hyperbolic spacetime is therefore the leaf of a foliation by smooth spacelike Cauchy hypersurfaces. Academic Press, San Diego 39, 41, 42, 43, 47, 49, 50, 54, 55 2.

Wald, R. University of Chicago Press, Chicago 39, 41, 57 3. Friedlander, F. Cambridge University Press, Cambridge 51 5. Leray, J. Mimeographed Lecture Notes, Princeton 53 6. Bernal, A. Geroch, R. Ellis, G. We first recall the physical origin of that equation which describes the propagation of a wave in space. However, if one prescribes the height and the speed of the wave at some fixed time then it is well known see also Sect. In particular, we want to discuss the local and global existence of solutions as well as give a short motivation on how those provide the fundamental background for some quantization theory.

They are called advanced or retarded fundamental solutions according to their support being contained in the causal future or past of the origin. Ginoux spacetimes, i. Nevertheless using normal coordinates it is always possible to transport Riesz distributions from the tangent space at a point to a neighbourhood of this point. Those fundamental solutions are in some sense near to the formal series from which they are constructed Corollary 3.

The global aspect of the theory is based on a completely different approach. First it would be illusory to construct global fundamental solutions on any spacetime; therefore, we restrict the issue to globally hyperbolic spacetimes, which can be thought of as the analogue of complete Riemannian manifolds in the Lorentzian setting. After discussing uniqueness of fundamental solutions Sect. Here it should be pointed out that the local existence of fundamental solutions Sect.

This chapter is intended as an introduction to the subject for students from the first or second university level. Only the main results and some ideas are presented; nevertheless, most proofs are left aside. We shall also exclusively deal with scalar operators, although all results of Sect. For a thorough and complete introduction to the topic as well as a list of references we refer to [1], on which this survey is widely based.

Examples 1. For the sake of simplicity we do not deal with vector bundles, hence we restrict the whole discussion to scalar operators, i. From now on any differential operator will be implicitly assumed to be scalar. We want to prove existence and uniqueness results — locally as well as globally — for waves, i. In this context we recall the central role played by fundamental solutions. For the definition of the topology of D M, K we refer to Sect.

Here and in the following we denote by d x the canonical measure associated with the metric g on M. We denote this distribution again by f , i. We state this in a bit more precise but purely formal manner, see, e. Proposition 6 for a situation where the following computation can be carried out under some further assumptions. In other words, u is some kind of convolution product of f with F. Therefore we momentarily forget about the wave equation itself and concentrate on the search for fundamental solutions.

As we shall already see in the next section, if there exists one fundamental solution then there exist many of them in general, hence one has to fix an extra condition to single one particular fundamental solution out. In physics this condition has to do with the finiteness of the propagation speed of a wave.

The most naive condition would be to require the support to be compact. Definition 5. The identity 3. We now prove 3. This proves 3. Equation 3. This shows the first step of the induction and achieves the proof of Lemma 1. The advanced resp. The second important properties for our purpose are the following. The Riesz distributions satisfy 1. As a consequence of Lemmas 1 and 2 we obtain the following.

Corollary 1. This shows evidence that there exist significantly more than one fundamental solution as soon as there is one.