Haag's theorem

Rudolf Haag formulated a theorem which is commonly known as haag cal theorem today. It states that the interaction picture of a relativistic quantum field theory ( QFT ) is inconsistent, that is, does not exist. Hague Evidence from 1955 was then repeatedly generalized, including by Hall and Wightman, which led to the conclusion that a unique, universal Hilbert space representation, both the free and the interacting field describes does not exist. Reed and Simon showed in 1975 that an analogous theorem exists for neutral, non-interacting scalar fields of different masses, from which it follows that the interaction picture not even in the limit of a negligible interaction is consistent.

Mathematical formulation of the Hague between theorem

In a modern variant allows the haagsche theorem formulated as follows:

If its two representations of the canonical commutation relation ( KVR), and (where for the applicable Hilbert space and for each complete sets of the operators in the KVR stand ). Both representations are said to be unitarily equivalent if there is a unitary map between Hilbert space and, when at any operator exists an operator. The property of the unitary equivalence is a necessary condition that the expectation values of the observables, ie the predictions of physical measurements, fail identical in both representations. The haagsche theorem states that - unlike in the case of the conventional non - relativistic quantum mechanics - such a unitary equivalence in the context of QFT is not present. The user of the QFT is therefore the so-called selection problem ( engl.: choice problem) faced, that is, with the problem of a non - countable set of non- equivalent representations of the correct (ie: physically meaningful ) to find representation. To date, the selection problem is one of the unsolved problems in QFT.

Physical ( descriptive ) approach

As already mentioned by Hague in his original paper, the phenomenon of vacuum polarization is the core problem, which underpins the haagsche theorem. Each interacting quantum field (this includes the non-interacting fields of different masses) polarizes the vacuum such that it is in a renormalized Hilbert space which is different from the free Hilbert space. Of course it is always possible to define an isomorphic mapping, which provides both Hilbert spaces between the states. However, says the haagsche theorem that under such a figure, the KVR not possess the property of unitary equivalence, physical test results are not therefore turn out clearly.

Cases that are not affected by the Hague between Theorem

One of the basic assumptions that lead to the hague between theorem includes the translational invariance of the system. Such systems can be formulated on a lattice with periodic boundary conditions ( 'Box - QFT '), as well as systems that can be localized due to external potentials are not affected by the Hague between theorem. Hague and David Ruelle have presented a formalism of scattering theory, which is based on asymptotically free states, as Haag- Ruelle theory is well known and serves as the basis for the widespread LSZ reduction formula. However, the latter methods are not applicable to massless particles and deliver even in the case of bound states still no satisfactory solutions.

Lack of acceptance among users of QFT

Although the haagsche theorem, the mathematical consistency of the interacting QFT is questioning, it is largely ignored by physicists who practice the QFT. This surprising at first glance fact is linked to the impressive successes of QFT together in the prediction and verification of experimental measurements that have a fundamental reformulation of the interaction picture seem superfluous. However, it is unclear due to the uncertain axiomatic basis, why or under what conditions the QFT with interaction leads to an accurate physical description of reality.

Haag's theorem

Mathematical formulation of the Hague between theorem

Physical ( descriptive ) approach

Cases that are not affected by the Hague between Theorem

Lack of acceptance among users of QFT

Further reading