Free Quantum Field Theory and Lorentz transformations from Quantum Theory of numerable systems and a little bit more (homogeneity, isotropy, and locality).

by Prof. Giacomo Mauro D'Ariano (Università di Pavia)

Thursday 9 Feb 2017, 14:30 → 15:30 Europe/Rome

0M03 (Dip. Fisica, Monte S. Angelo)

I will show how a recent axiomatic derivation of quantum theory and of quantum field theory leads to a quantum cellular automata theory of fields. Free Quantum Field Theory (QFT) can be derived without quantization rules as a "quantum ab initio" theory of numerable systems, with general assumptions as homogeneity, isotropy, locality and linearity of the interactions, without assuming special relativity and any mechanical notion. What directly follows from the principles is a theory of quantum walks on the Cayley graph of a group G. Virtually abelian G corresponds to QFT in Euclidean space, whereas relaxing linearity leads to interacting QFTs. The new axiomatization is purely mathematical, with no physical primitives, nevertheless it has a thorough physical interpretation. The usual free quantum field theory (Weyl, Maxwell, and Dirac) is recovered in the so-called "relativistic limit” of small wave-vectors. The purely mathematical adimensional theory contains the standards for mass, space, and time through the nonlinearities intrinsic to the theory (maximum wave-vector, frequency, and mass, the latter coming from from unitarity). The relativistic limit connects these standards to the speed of light and to the Planck constant, whereas at the maximum value for the particle mass the dispersion relation becomes flat, with interpretation as a mini-black hole, thus setting the scale at Planck’s. The Galilean relativity principle can be semantically translated for a general dynamical system, and for the case of a quantum walk it corresponds to the invariance of the walk with the representation. The Lorentz transformations make a nonlinear group (the theory is a model for doubly special relativity), and the usual linear transformations are recovered in the small wavevector regime (corresponding to the whole physical domain experimented so far!) The notion of particle is still that of Poincaré invariant. A new emerging feature is that for Planckian boosts/masses also the rest-mass get involved in the transformations, leading to a De Sitter covariance. As a consequence of unitarity, the mass is bounded to values belonging to the unit circle, and consistently the Fourier-conjugated variable—the proper time—belongs to Z. This facts, along with other features having general-relativistic flavor, are promising for an approach to quantum gravity. The construction poses to theoretical physicists problems of geometric group theory, a hot field in contemporary pure mathematics.