Haag's theorem

Rudolf Haag formulated a theorem that is now commonly known as the Haagsches theorem . It says that the interaction picture of a relativistic quantum field theory (QFT) is inconsistent, i.e. i.e., does not exist. Haag's proof of 1955 was then generalized several times. a. von Hall and Arthur Wightman , who came to the conclusion that a unique, universal Hilbert space representation that describes both the free and the interacting field does not exist. Reed and Simon showed in 1975 that an analog theorem also exists for neutral, interaction-free scalar fields of different masses, from which it follows that the interaction picture is not consistent even in the borderline case of a negligible interaction.

Mathematical formulation of Haag's theorem

In a modern variant, Haag's theorem can be formulated as follows:

Two representations of the canonical commutation relation (KVR) are given, as well as (whereby stand for the respectively valid Hilbert space and for the respective complete sentences of the operators in the KVR). Both representations are called unitary equivalent if and only if there is a unitary mapping between Hilbert space and in which there is an operator for each operator . The property of unitary equivalence is a necessary condition for the expectation values ​​of the observables , i.e. H. the predictions of physical measurements turn out to be identical in both representations. Haag's theorem states that - unlike in the case of conventional, non-relativistic quantum mechanics - such a unitary equivalence does not exist within the framework of QFT. The user of the QFT is therefore the so-called selection problem (engl .: choice problem confronted), d. H. with the problem of finding the correct (ie: physically meaningful) representation from a non-countable set of non-equivalent representations. To date, the problem of selection is one of the unsolved problems in QFT. ${\displaystyle (H_{1},\{O_{1}^{i}\})}$ ${\displaystyle (H_{2},\{O_{2}^{i}\})}$ ${\displaystyle H_{n}}$ ${\displaystyle \{O_{n}^{i}\}}$ ${\displaystyle U}$ ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle O_{1}^{j}\in \{O_{1}^{i}\}}$ ${\displaystyle O_{2}^{j}=UO_{1}^{j}U^{-1}\in \{O_{2}^{i}\}}$

Physical (descriptive) approach

As already mentioned by Haag in his original work, the phenomenon of vacuum polarization is the core problem on which Haag's theorem is based. Each interacting quantum field (including the non-interacting fields of different masses) polarizes the vacuum in such a way that it lies in a renormalized Hilbert space , which differs from the free Hilbert space . Of course, it is always possible to define an isomorphic mapping that mediates between the states in both Hilbert spaces. However, Haag's theorem states that in the context of such a mapping the KVR does not have the property of unitary equivalence, and physical measurement results are consequently not unambiguous. ${\displaystyle H_{\mathrm {R} }}$ ${\displaystyle H_{\mathrm {F} }}$

Cases not affected by Haag's theorem

One of the basic assumptions that lead to Haag's theorem is the translational invariance of the system. Such systems that can be formulated on a grid with periodic boundary conditions ('Box-QFT'), as well as systems that can be localized due to external potentials, are not affected by Haag's theorem. Haag and David Ruelle have presented a formalism of scattering theory , which is based on asymptotically free states, is known as the Haag-Ruelle theory and serves as the basis for the widely used LSZ reduction formula. However, the latter methods cannot be used for massless particles and do not yet provide satisfactory solutions even in the case of bound states.

Lack of acceptance among QFT users

Although Haag's theorem challenges the mathematical consistency of interacting QFT, it is largely ignored by physicists who practice QFT. This fact, which is surprising at first glance, is related to the impressive successes of QFT in predicting and verifying experimental measured values, which make a fundamental reformulation of the interaction picture superfluous. However, due to the uncertain axiomatic basis, it is unclear why or under what conditions the QFT with interaction leads to an accurate physical description of reality.

For further reading

• Doreen Fraser: Haag's Theorem and the Interpretation of Quantum Field Theories with Interactions , Ph.D. thesis, U. of Pittsburgh, 2006.
• A. Arageorgis: Fields, Particles, and Curvature: Foundations and Philosophical Aspects of Quantum Field Theory in Curved Spacetime , Ph.D. thesis, Univ. of Pittsburgh, 1995.
• J. Bain: Against Particle / field duality: Asymptotic particle states and interpolating fields in interacting QFT (or: Who's afraid of Haag's theorem?) . In: Knowledge . 53, 2000, pp. 375-406.

Individual evidence

1. ↑ R. Haag: On quantum field theories (PDF; 2.9 MB) In: Matematisk-fysiske Meddelelser. 29, 12, 1955.
2. ^ D. Hall, AS Wightman: A theorem on invariant analytic functions with applications to relativistic quantum field theory . In: Matematisk-fysiske Meddelelser . tape 31 , no. 1 , 1957.
3. ^ M. Reed, B. Simon: Methods of modern mathematical physics. Vol. II: Fourier analysis, self-adjointness . Academic Press, New York 1975.
4. ↑ John Earman, Doreen Fraser: Haag's Theorem and Its Implications for the Foundations of Quantum Field Theory . In: Knowledge . 64, 305, 2006 ( online at philsci-archive ).
5. ^ Reed, Simon, Scattering theory , Academic Press 1979
6. ^ R. Haag: Quantum field theories with composite particles and asymptotic conditions . In: Phys. Rev. . 112, No. 2, 1958, pp. 669-673. bibcode : 1958PhRv..112..669H . doi : 10.1103 / PhysRev.112.669 .
7. ↑ D. Ruelle: On the asymptotic condition in quantum field theory . In: Helvetica Physica Acta . 35, 1962, pp. 147-163.
8. ^ Klaus Fredenhagen, Quantum field theory , Lecture Notes, University of Hamburg 2009
9. ↑ Paul Teller: An interpretive introduction to quantum field theory . Princeton University Press, 1997.