Haag-Ruelle Scattering Theory

These are lectures delivered at the Higgs Centre School of Theoretical Physics 2025 from 9th to 13th June 2025. To read more about Haag-Ruelle scattering theory, I strongly recommend the book by Duncan [1], besides the classic, but more complex, book by Haag [2].

Preface

When talking about scattering in particle physics one has in mind the following situation:

• at a certain time in the far past (in the idealized case, at \(t = -\infty\)), two particles are prepared very far apart from each other and with velocities such that they move towards each other;

• as the particles approach each other, they interact possibly in a very complicated way; a number of particles is produced (they may be different from the particles in the initial state);

• the particles in the final state move away from each other and are detected at a certain time in the far future (in the idealized case, at \(t = +\infty\)).

The states at \(t = -\infty\) (resp. \(t = +\infty\)) are called the incoming (resp. outgoing) asymptotic states or, briefly, in (resp. out) states. Cross sections, which are the main observables in collider experiments, are defined in terms of the probabilities of transitions from the in states to the out states, which are called scattering amplitudes.

In standard QFT courses, scattering amplitudes related to time-ordered n-point functions of field by means of the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula (see for example this Wikipedia article. The LSZ formula is the main tool to compute scattering amplitudes in perturbation theory, since time-ordered n-point functions are the fundamental objects calculated in perturbation theory. In particular, the representation of time-ordered n-point functions in terms of Feynman diagrams induces a representation of scattering amplitudes in terms of Feynman diagrams as well.

Haag-Ruelle (HR) scattering theory [3][4] provides a different approach to scattering amplitudes and is the subject of these lectures. It is useful to highlight the main points of interest of the HR theory and some of the differences with the LSZ theory:

1. Logically, HR is more fundamental than LSZ, in the sense that it provides a definition of the asymptotic states. On the other hand, LSZ provides only matrix elements between the asymptotic states, essentially assuming their existence. In fact, the LSZ formula has been derived from the HR theory [5], but not vice versa.

2. Both HR and LSZ theories are supposed to be valid in non-perturbatively defined QFTs. HR theory is inherently non-perturbative and is constructed on top of Wightman axiomatic framework [6]. Wightman axioms are essentially a set of minimal properties that all non-perturbatively defined QFTs should satisfy. In contrast, common proofs of the LSZ formula (which do not assume HR) are based on perturbation theory in one way or another.

3. The HR theory yields scattering amplitudes in terms of so-called Wightman functions, i.e. vacuum expectation values of products of field operators without time ordering. This is in contrast to the LSZ formula, which uses time-ordered n-point functions.

The last point is the reason why I have personally got interested in the HR theory. In fact, in the past few years, some proposals on how to approximate Fourier transforms of Wightman functions (aka spectral densities) in terms of Euclidean correlation functions have been put forward. This opens the way to use lattice QCD to compute hadronic scattering amplitudes by means of HR-based methods [7].

[1]

A. Duncan, The Conceptual Framework of Quantum Field Theory, Oxford University Press, 2012. https://doi.org/10.1093/acprof:oso/9780199573264.001.0001.

[2]

R. Haag, Local Quantum Physics, Springer, Berlin, 1996. https://doi.org/10.1007/978-3-642-61458-3.

[3]

R. Haag, Quantum field theories with composite particles and asymptotic conditions, Phys. Rev. 112 (1958) 669–673. https://doi.org/10.1103/PhysRev.112.669.

[4]

D. Ruelle, On the asymptotic condition in quantum field theory, Helvetica Physica Acta 35 (1962) 147–163. https://doi.org/10.5169/seals-113272.

[5]

K. Hepp, On the connection between wightman and lsz quantum field theory, in: 8th Brandeis University Summer Institute in Theoretical Physics: Particle symmetries and axiomatic field theory, 1966: pp. 135–240. https://link.springer.com/article/10.1007/BF01646494.

[6]

A.S. Wightman, Quantum Field Theory in Terms of Vacuum Expectation Values, Phys. Rev. 101 (1956) 860–866. https://doi.org/10.1103/PhysRev.101.860.

[7]

A. Patella, N. Tantalo, Scattering amplitudes from Euclidean correlators: Haag-Ruelle theory and approximation formulae, JHEP 01 (2025) 091. https://doi.org/10.1007/JHEP01(2025)091.