Using $\star$-calculus on the dual of the Borchers-Uhlmann algebra endowed with a combinatorial co-product, we develop a method to calculate a unitary transformation relating the GNS representations of a non-quasifree and a quasifree state of the free hermitian scalar field. The motivation for such an analysis and a further result is the fact that a unitary transformation of this kind arises naturally in scattering theory on non-stationary backgrounds. Indeed, employing the perturbation theory of the Yang-Feldman equations with a free CCR field in a quasifree state as an initial condition and making use of extended Feynman graphs, we are able to calculate the Wightman functions of the interacting and outgoing fields in a $ϕ^p$-theory on arbitrary curved spacetimes. A further examination then reveals two major features of the aforementioned theory: firstly, the interacting Wightman functions fulfil the basic axioms of hermiticity, invariance, spectrality (on stationary spacetimes), perturbative positivity, and locality. Secondly, the outgoing field is free and fulfils the CCR, but is in general not in a quasifree state in the case of a non-stationary spacetime. In order to obtain a sensible particle picture for the outgoing field and, hence, a description of the scattering process in terms of particles (in asymptotically flat spacetimes), it is thus necessary to compute a unitary transformation of the abovementioned type.
The theory of quasifree quantum stochastic calculus for infinite-dimensional noise is developed within the framework of Hudson-Parthasarathy quantum stochastic calculus. The question of uniqueness for the covariance amplitude with respect to which a given unitary quantum stochastic cocycle is quasifree is addressed, and related to the minimality of the corresponding stochastic dilation. The theory is applied to the identification of a wide class of quantum random walks whose limit processes are driven by quasifree noises.
We study the relative entropy, in the sense of Araki, for the representation of a self-dual CAR algebra $\mathfrak{A}_{SDC}(\mathcal{H},Γ)$. We notice, for a specific choice of $f \in \mathcal{H}$, that the associated element in $\mathfrak{A}_{SDC}(\mathcal{H},Γ)$ is unitary. As a consequence, we explicitly compute the relative entropy between a quasifree state over $\mathfrak{A}_{SDC}(\mathcal{H},Γ)$ and an excitation of it with respect to the abovely mentioned unitary element. The generality of the approach, allows us to consider $\mathcal{H}$ as the Hilbert space of solutions of the classic…
Unitary Representations corresponding to local shifts in reference frames are a key topic for progress in Quantum Gravity. The fibre bundle construct defined in our previous work in which quantum fields become liftings of; or sections through; a fibre bundle with base space curved space-time, continues to be the context for this paper. We investigate in more depth the subgroup T of the Poincare group consisting of translations of space-time as a gauge group of automorphisms. We define to be a representation of T as such automorphisms of the local fibre algebra A(x) which we assume to be isomor…
We propose a new axiom system for unitary quantum field theories on curved space-time backgrounds, by postulating that the partition function and the correlators extend analytically to a certain domain of complex-valued metrics. Ordinary Riemannian metrics are contained in the allowable domain, while Lorentzian metrics lie on its boundary.
The Riemann Hypothesis states that the Riemann zeta function $ζ(z)$ admits a set of ``non-trivial'' zeros that are complex numbers supposed to have real part $1/2$. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. Hilbert and Pólya suggested that the Riemann Hypothesis could be solved through the mathematical tools of physics, finding a suitable Hermitian or unitary operator that describe classical or quantum systems, whose eigenvalues distribute like the zeros of $ζ(z)$. A different approach is that of finding a c…
We revisit the representation theory of the quantum double of the universal cover of the Lorentz group in 2+1 dimensions, motivated by its role as a deformed Poincaré symmetry and symmetry algebra in (2+1)-dimensional quantum gravity. We express the unitary irreducible representations in terms of covariant, infinite-component fields on curved momentum space satisfying algebraic spin and mass constraints. Adapting and applying the method of group Fourier transforms, we obtain covariant fields on (2+1)-dimensional Minkowski space which necessarily depend on an additional internal and circular di…
We elaborate on conformal higher-spin gauge theory in three-dimensional (3D) curved space. For any integer $n>2$ we introduce a conformal spin-$\frac{n}{2}$ gauge field $h_{(n)} =h_{α_1\dots α_n}$ (with $n$ spinor indices) of dimension $(2-n/2)$ and argue that it possesses a Weyl primary descendant $C_{(n)}$ of dimension $(1+n/2)$. The latter proves to be divergenceless and gauge invariant in any conformally flat space. Primary fields $C_{(3)}$ and $C_{(4)}$ coincide with the linearised Cottino and Cotton tensors, respectively. Associated with $C_{(n)}$ is a Chern-Simons-type action that is bo…
This manuscript gathers and subsumes a long series of works on using QW to simulate transport phenomena. Quantum Walks (QWs) consist of single and isolated quantum systems, evolving in discrete or continuous time steps according to a causal, shift-invariant unitary evolution in discrete space. We start reminding some necessary fundamentals of linear algebra, including the definitions of Hilbert space, tensor state, the definition of linear operator and then we briefly present the principles of quantum mechanics on which this thesis is grounded. After having reviewed the literature of QWs and t…
