Is a momentum space of quantum fields Euclidean?

The Wick rotation provides the standard technique of computing Feynman diagrams by means of Euclidean propagators. It is the Laplace transform from a Minkowski coordinate space to the Euclidean momentum one.

Then one can suppose that a momentum space of quantum fields in an interaction zone really is Euclidean.

Reference:

G.Sardanashvily, hep-th/ 0511111

Hierarchy of Noether identities (from my Scientific Biography)

Every Euler - Lagrange operator obeys Noether identities which, however, can be trivial. If they are not trivial, a Lagrange system is called degenerate. Nontrivial Noether identities always satisfy first-order Noether identity, which also may be trivial, and etc. If first-order Noether identities are not-trivial, a degenerate Lagrangian system is called reducible. A problem was to separate trivial and non-trivial Noether identities of zero and higher orders.

This separation was effected in terms of cohomology. As a result, we described a generic reducible degenerate Lagrangian system, whose Euler - Lagrange operator satisfies non-trivial Noether identities which are not independent, but are subject to non-trivial first-order Noether identities, satisfying, in turn, non-trivila second-order Noether identities, etc. Under a certain cohomology condition, the hierarchy of these Noether identities is described by the exact cochain complex of odd and even antifields, called the Kozul - Tate complex [119,127].

This description was extended to Noether identities for an arbitrary differential operator [120].

Having received the hierarchy of Noether identities for reducible degenerate Lagrangian systems, I believed natural to generalize to it the second Noether theorem, linking the Noether identities with the gauge transformations of zero and higher order. This has been done. Generalized second Noether theorem corresponds to the Koszul – Tate complex some cochain sequence. Its ascent operator, called the gauge operator, consists of a gauge symmetry of a Lagrangian and gauge symmetries of first and higher orders, which are parameterized by odd and even ghost fields [129,133]. This cochain sequence and the Kozul - Tate complex of Noether identitie fully characterize the degeneration of a Lagrangian  system, which is necessary for its quantization.

References:

My Scientific Biography

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory  (WS, 2009)

Can contemporary mathematics describe quantum physics?

Created by humans, the existent mathematics is anthropomorphic, but not universal. Even in the basics of mathematical logic and set theory, it emanates from the everyday experience of people dealing with classical macroscopic objects. This mathematics meets fundamental challenges when trying to describe, for example, quantum systems.

Indeed, it is the mathematics of sets. Based on this mathematics, theoretical physics treats any physical system as a set. This treatment, a posteriori, seems to be adequate in the case of macroscopic classical systems.

However, what is about quantum systems? Is any quantum system a set? Does it consist of elements? For instance, is a photon a set? The concepts of an element, a subset, the complement of a subset, an empty set, the union and intersection of sets etc are not so evident in a quantum world. In particular, if a quantum system (e.g., a hydrogen atom) is made up by two quantum systems A and B, it does not contains neither A nor B as a subsystem. This is the well known entanglement problem in quantum theory.

Reference:

G.Sardanashvily's site: Frontier problems

Why only electromagnetic and gravitational interactions are in classical physics?

In fact, any system of classical physics (classical mechanics, classical field theory) is a system of fermions interacting with classical electromagnetic field and gravitational field. Why nothing more?

It is surprising that the strict mathematical formulation of relativistic mechanics (in the terms of one-dimensional submanifolds of a world manifold) describes relativistic particles interacting only with electromagnetic and gravitational fields, and nothing more, too.  An equation of motion of such a particle is a geodesic equation in a pseudo-Riemannian space with a torsion, which is a strength of an electromagnetic field.

Thus, one can conclude that space-time transformations are compatible only with gravitational and electromagnetic interactions, and no others, e.g., a scalar field.

References:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)

G.Sardanashvily, Relativistic mechanics in a general setting, arXiv: 1005.1212