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Quantum theory's irreducible empirical core is a probability calculus. While it presupposes the events to which (and on the basis of which) it serves to assign probabilities, and therefore cannot account for their occurrence, it has to be…
In a mathematical context in which one can multiply distributions the "`formal"' nonperturbative canonical Hamiltonian formalism in Quantum Field Theory makes sense mathematically, which can be understood a priori from the fact the so…
Computations in renormalizable perturbative quantum field theories reveal mathematical structures which go way beyond the formal structure which is usually taken as underlying quantum field theory. We review these new structures and the…
A pedagogical introduction to some of the main ideas and results of field theories on quantized spacetimes is presented, with emphasis on what such field theories may teach us about the problem of quantizing gravity. We examine to what…
As applied to quantum theories, the program of renormalization is successful for `renormalizable models' but fails for `nonrenormalizable models'. After some conceptual discussion and analysis, an enhanced program of renormalization is…
We consider the quantum Hall effect in terms of an effective field theory formulation of the edge states, providing a natural common framework for the fractional and integral effects.
The algebraic formulation of the quantum group gauge models in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider gauge groups taking values in the quantum groups and noncommutative gauge fields…
In this introductory article a brief description of Quantum Field Theories (QFT) is presented with emphasis on the distinction between strongly and weakly coupled theories. A case is made for using numerical simulations to solve QCD, the…
A generalization of the recently formulated nonlinear quantization of a parameterized theory is presented in the context of quantum gravity. The parametric quantization of a Friedmann universe with a massless scalar field is then considered…
Gravity is difficult to quantize. This is a well-known fact but its reason is given simply by non-renormalizability of the Newton constant and little is discussed why among many quantum gauge theories, gravity is special. In this essay we…
These lectures present some basic facts in field theory necessary to understand the quantum theory of the Standard Model of weak and electromagnetic interactions.
We propose a new formalism for quantum field theory which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e. it computes correlation functions through convergent rather…
"Physical theories of fundamental significance tend to be gauge theories. These are theories in which the physical system being dealt with is described by more variables than there are physically independent degree of freedom. The…
In this paper I offer an introduction to group field theory (GFT) and to some of the issues affecting the foundations of this approach to quantum gravity. I first introduce covariant GFT as the theory that one obtains by interpreting the…
We survey some results relating noncommutative geometry to the class field theory of number fields. These results appear within the context of quantum statistical mechanics where some arithmetic properties of a given number field can be…
In grand unified theories with large numbers of fields, renormalization effects significantly modify the scale at which quantum gravity becomes strong. This in turn can modify the boundary conditions for coupling constant unification, if…
Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -- the theory of…
The usual mathematical formalism of quantum field theory is non-rigorous because it contains divergences that can only be renormalized by non-rigorous mathematical methods. The purpose of this paper is to present a method of subtraction of…
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary --- or, in general, projective unitary --- representations implement the action of an abstract symmetry group on physical states…
The best mathematical arguments against a realistic interpretation of quantum mechanics - that gives definite but partially unknown values to all observables - are analysed and shown to be based on reasoning that is not compelling. This…