We show that the Connes-Rovelli thermal time associated with the quantum harmonic oscillator can be described as an (unsharp) observable, that is, as a positive operator valued measure. We furthermore present extensions of this result to the free massless relativistic particle in one dimension and to a hypothetical physical system whose equilibrium state is given by the noncommutative integral.

In this article, we present a detailed analysis of two famous delayed choice experiments: Wheeler’s classic gedanken-experiment and the delayed quantum eraser. Our analysis shows that the outcomes of both experiments can be fully explained on the basis of the information collected during the experiments using textbook quantum mechanics only. At no point in the argument, information from the future is needed to explain what happens next. In fact, more is true: for both experiments, we show, in a strictly mathematical way, that a modified version in which the time-ordering of the steps is changed to avoid the delayed choice leads to exactly the same final state. In this operational sense, the scenarios are completely equivalent in terms of conclusions that can be drawn from their outcomes.

We consider operators acting on a UMD Banach lattice X that have the same algebraic structure as the position and momentum operators associated with the harmonic oscillator (Formula Presented) acting on L²(R^d). More precisely, we consider abstract harmonic oscillators of the form (Formula Presented) for tuples of operators (Formula Presented), where i A_j and iB_k are assumed to generate C₀ groups and to satisfy the canonical commutator relations. We prove functional calculus results for these abstract harmonic oscillators that match classical Hörmander spectral multiplier estimates for the harmonic oscillator (Formula Presented). This covers situations where the underlying metric measure space is not doubling and the use of function spaces that are not particularly well suited to extrapolation arguments. For instance, as an application we treat the harmonic oscillator on mixed norm Bargmann–Fock spaces. Our approach is based on a transference principle for the Schrödinger representation of the Heisenberg group that allows us to reduce the problem to the study of the twisted Laplacian on the Bochner spaces L²(R^2d; X). This can be seen as a generalisation of the Stone–von Neumann theorem to UMD lattices X that are not Hilbert spaces.

In this chapter we address a couple of topics in the theory of H^∞-calculus centering around the question what can be said about an operator of the form A+B when A and B have certain “good” properties such as being (R-)sectorial or admitting a bounded H^∞-calculus.

In this chapter we address two strongly interwoven topics: How to verify the boundedness of the H^∞-calculus of an operator and how to represent and estimate its fractional powers. For concrete operators such as the Laplace operator or elliptic partial differential operators, the fractional domain spaces can often be identifed with certain function spaces considered in Chapter 14 and the imaginary powers of the operator are related to singular integral and pseudo-differential operators treated in Chapters 11 and 13.

The mapping properties of T will of course heavily depend on the assumptions made on the kernel K that we will discuss in more detail in this chapter.

In this chapter, we complement the discussion of three major themes of Fourier analysis that we have studied in the previous Volumes.

Before addressing this question for the Calderón{Zygmund type operators of the kind studied in Chapter 11, we investigate a number of related objects in a simpler dyadic model. Besides serving as an introduction to some of the key techniques, it turns out that these dyadic operators can be, and will be, also used as building blocks of the proper singular integral operators towards the end of the chapter.

This chapter presents an in-depth study of several classes of vector-valued function spaces defined by smoothness conditions.

As we have seen in the preceding sections, in the context of inhomogeneous linear evolution equations, maximal regularity enables one to set up an isomorphism between the space of data (initial value and inhomogeneity) and the solution space.

We prove a new Burkholder–Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if (S(t, s)) _⩽s≤t⩽T is a C-evolution family of contractions on a 2-smooth Banach space X and (Wt)t∈[0,T] is a cylindrical Brownian motion on a probability space (Ω , P) adapted to some given filtration, then for every 0 < p< ∞ there exists a constant C_p,X such that for all progressively measurable processes g: [0 , T] × Ω → X the process (∫0tS(t,s)gsdWs)t∈[0,T] has a continuous modification and Esupt∈[0,T]‖∫0tS(t,s)gsdWs‖p⩽Cp,XpE(∫0T‖gt‖γ(H,X)2dt)p/2.Moreover, for 2 ⩽ p< ∞ one may take Cp,X=10Dp, where D is the constant in the definition of 2-smoothness for X. The order O(p) coincides with that of Burkholder’s inequality and is therefore optimal as p→ ∞. Our result improves and unifies several existing maximal estimates and is even new in case X is a Hilbert space. Similar results are obtained if the driving martingale gtdWt is replaced by more general X-valued martingales dMt. Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes (including splitting, implicit Euler, Crank-Nicholson, and other rational schemes) we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs dut=A(t)utdt+gtdWt,u0=0,where the family (A(t)) _t∈[,T] is assumed to generate a C-evolution family (S(t, s)) _⩽s⩽t⩽T of contractions on a 2-smooth Banach spaces X. Under spatial smoothness assumptions on the inhomogeneity g, contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes.

We present an analysis of the Frauchiger–Renner Gedankenexperiment from the point of view of the relational interpretation of quantum mechanics. Our analysis shows that the paradox obtained by Frauchiger and Renner disappears if one rejects promoting one agent’s certainty to another agent’s certainty when it cannot be validated by records from the past. A by-product of our analysis is an interaction-free detection scheme for the existence of such records.

We generalise the classical Weyl pseudo-differential calculus on R^d to the setting of two d-tuples of operators A = (A₁,..., A_d) and B = (B₁,..., B_d) acting on a Banach space generating bounded C0-groups satisfying the Weyl canonical commutation relations. We show that the resulting Weyl calculus extends to symbols in the standard symbol class S⁰ provided appropriate bounds can be established. Using transference techniques we obtain boundedness of the H^∞-functional calculus (and even the Hormander calculus), for the abstract harmonic oscillator.

Extending results of Pardoux–Peng and Hu–Peng, we prove well-posedness results for backward stochastic evolution equations in UMD Banach spaces.