After a brief summary of the mathematical problems of the main stream quantum field theory, a way out of this difficulties is presented. This proposal implies a revolution of mathematics, i.e. the mathematics based on the quantum set theory of G.
I will argue that proposal of establishing operational foundations of Quantum The- ory should have top-priority, and that Lucien Hardy's program on Quantum Gravity should be paralleled by an analogous program on Quantum Field Theory (QFT), which needs to be reformu- lated, notwithstanding its experimental success. In this paper, after reviewing recently suggested operational principles of quantumness, I address problem on whether Quantum Theory and Special Relativity are unrelated theories, or instead, if one implies other. I show how Special Relativity can be indeed derived from causality of Quantum Theory, within computa- tional paradigm the universe is a huge quantum computer, reformulating QFT as a Quantum- Computational Field Theory (QCFT). In QCFT Special Relativity emerges from fabric of computational network, which also naturally embeds gauge invariance. In this scheme even quantization rule and Planck constant can in principle be derived as emergent from under- lying causal tapestry of space-time. In this way Quantum Theory remains only theory operating huge computer of universe. Is computational paradigm only a speculative tautology (theory as simulation of reality), or does it have a scientific value? The answer will come from Occam's razor, depending on mathematical simplicity of QCFT. Here I will just start scratching surface of QCFT, analyzing simple field theories, including Dirac's. The number of problems and unmotivated recipes that plague QFT strongly motivates us to undertake QCFT project, since QCFT makes all such problems manifest, and forces a re-foundation of QFT.
An important way in which quantum field theory (QFT) differs from non-relativistic quantum mechanics is that in QFT systems generically possess an infinite number of degrees of freedom. This chapter examines the proposals for non-fundamental roles that the particle concept plays in particle physics. The fields which feature in the QFT formalism are mathematical expressions; work needs to be done to determine the ontological properties of the entities that are represented by the mathematical field formalism. One difficulty for the quanta interpretation of QFT is that the analysis in terms of quanta offered in the preceding paragraphs is restricted to free systems. The no-go results for interacting systems inspire a different approach to regarding particles as non-fundamental entities. The fields which feature in the QFT formalism are mathematical expressions; work needs to be done to determine the ontological properties of the entities that are represented by the mathematical field formalism.
The topic of measurement in relativistic quantum field theory is addressed in this article. Some of the long standing problems of this subject are highlighted, including the incompatibility of an instantaneous ``collapse of the wavefunction'' with relativity of simultaneity, and the difficulty of maintaining causality in the rules for measurement highlighted by ``impossible measurement'' scenarios. Thereafter, the issue is considered from the perspective of mathematical physics. To this end, quantum field theory is described in a model-independent, operator algebraic setting, on generic Lorentzian spacetime manifolds. The process of measurement is modelled by a localized dynamical coupling between a quantum field called the ``system'', and another quantum field, called the ``probe''. The result of the dynamical coupling is a scattering map, whereby measurements carried out on the probe can be interpreted as measurements of induced observables on the system. The localization of the dynamical coupling allows it to derive causal relations for the induced observables. It will be discussed how this approach leads to the concept of selective or non-selective system state updates conditioned on the result of probe measurements, which in turn allows it to obtain conditional probabilities for consecutive probe measurements consistent with relativistic causality and general covariance, without the need for a physical collapse of the wavefunction. In particular, the problem of impossible measurements is resolved. Finally, there is a brief discussion of accelerated detectors and other related work.