Quantum field theory: Functional integrals are not true integrals

In contrast with classical field theory, there is no strict mathematical formulation of quantum field theory. In particular, perturbative quantum field theory is phrased in the terms of functional integrals. However, these are not true integrals. They, by hand, are provided with properties of integrals over finite-dimensional vector spaces, but some of these properties are not extended to measures on infinite-dimensional topological vector spaces.

For instance, functional integrals are assumed to be translationally-invariant, i. e.,

d(f(x)+c)=d(f(x)), c=const.

However, infinite-dimensional topological vector spaces do not admit the translationally-invariant Lebesgue measure, as a rule.

This fact is important, e.g., for describing a Higgs vacuum, usually represented by a constant field. Namely, quantum field systems in the presence of different Higgs vacuums seem to be non-equivalent.

References:

G.Sardanashvily, O.Zakharov, On functional integrals in quantum field theory, Rep. Math. Phys. 29 (1991) 101-108; arXiv:  hep-th/9410107

G.Sardanashvily, Higgs vacuum from the axiomatic viewpoint, Nuovo Cimento 104A (1991) 105-111