## Abstract

Scattering in a model of a massive quantum-mechanical particle, an ‘‘electron’’, interacting with massless, relativistic bosons, ‘‘photons’’, is studied. The interaction term in the Hamiltonian of our model describes emission and absorption of ‘‘photons’’ by the ‘‘electron’’; but ‘‘electron-positron’’ pair production is suppressed. An ultraviolet cutoff and an (arbitrarily small, but fixed) infrared cutoff are imposed on the interaction term. In a range of energies where the propagation speed of the dressed ‘‘electron’’ is strictly smaller than the speed of light, unitarity of the scattering matrix is proven, provided the coupling constant is small enough; (asymptotic completeness of Compton scattering). The proof combines a construction of dressed one–electron states with propagation estimates for the ‘‘electron’’ and the ‘‘photons’’.

This is a preview of subscription content, access via your institution.

## References

Ammari, Z.: Asymptotic completeness for a renormalized nonrelativistic Hamiltonian in quantum field theory: the Nelson model. Math. Phys. Anal. Geom.

**3**(3), 217–285, (2000)Arai, A.: A note on scattering theory in nonrelativistic quantum electrodynamics. J. Phys. A

**16**(1), 49–69 (1983)Bach, V.: Fröhlich, J., Sigal, I.M.: Quantum electrodynamics of confined nonrelativistic particles. Adv. Math.

**137**(2), 299–395 (1998)Bach, V., Klopp, F., Zenk, H.:

*Mathematical analysis of the photoelectric effect*. Adv. Theor. Math. Phys.**5**(6), 969–999 (2001)Bloch, F., Nordsieck, A.: Note on the radiation field of the electron. Phys. Rev.

**52**, 54–59 (1937)Chen, T.:

*Operator-theoretic infrared renormalization and construction of dressed 1–particle states*. Preprint, http://www.ma.utexas.edu/mp-arc/01-310, 2001Davies, E.B.: The functional calculus. J. London Math. Soc. (2),

**52**(1), 166–176 (1995)Dereziński, J., Gérard, C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys.

**11**(4), 383–450 (1999)Dereziński, J., Gérard, C.: Spectral and scattering theory of spatially cut-off

*P*(φ)_{2}Hamiltonians. Commun. Math. Phys.**213**(1), 39–125 (2000)Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field. Adv. Math.

**164**(2), 349–398 (2001)Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Rayleigh scattering. Ann. Henri Poincaré,

**3**, 107–170 (2002)Fröhlich, J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. H. Poincaré, Sect. A

**XIX**(1), 1–103 (1973)Fröhlich, J.: Existence of dressed one-electron states in a class of persistent models. Fortschr. Phys.

**22**, 159–198 (1974)Gérard, C.: On the scattering theory of massless Nelson models. Rev. Math. Phys.

**14**, 1165–1280 (2002)Hunziker, W., Sigal, I.M.: The quantum

*N*–body problem. J. Math. Phys.**41**(6), 3448–3510 (2000)Jost, R.: The general theory of quantized fields. In: M. Kac, (ed.),

*(Proceedings of the Summer Seminar, Boulder, Colorado, 1960)*, Volume**IV**, Lectures in Applied Mathematics, 1965Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys.

**5**, 1190–1197 (1964)Pauli, W., Fierz, M.: Zur Theorie der Emission langwelliger Lichtquanten. Nuovo Cimento

**15**, 167–188 (1938)Pizzo, A.: One particle (improper) states and scattering states in Nelson’s massless model. http://arxiv:org/abs/math-ph/0010043, 2000

Spohn, H.: Asymptotic completeness for Rayleigh scattering. J. Math. Phys.

**38**(5), 2281–2296 (1997)Spohn, H.: The polaron model at large momentum. J. Phys. A: Math. Gen.

**21**, 1199–1211 (1988)Sigal, I.M., Soffer, A.:

*Local decay and propagation estimates for time–dependent and time–independent Hamiltonians*. Princeton University preprint, 1988Reed, M., Simon, B.:

*Methods of modern mathematical physics: Scattering Theory*. Vol. 3. New York: Academic Press, 1979Dereziński, J., Gérard, C.:

*Scattering theory of classical and quantum N-particle systems*. Berlin-Heidelberg-New York: Springer, 1997Kato, T.:

*Perturbation theory for linear operators*. New York: Springer-Verlag, 1966Yennie, D., Frautschi, S., Suura, H.: The infrared divergence phenomena and high-energy processes. Ann. Phys.

**13**, 379–452 (1961)Spencer, T., Zirilli, F.: Scattering states and bound states in Commun. Math. Phys.

**49**(1), 1–16 (1976)Spencer, T.: The decay of the Bethe-Salpeter kernel in

*P*(φ)_{2}quantum field models. Commun. Math. Phys.**44**(2), 143–164 (1975)

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

Communicated by H. Spohn

*Dedicated to Freeman Dyson on the occasion of his 80th birthday*

Work partially supported by U.S. National Science Foundation grant DMS 01-00160.

*Acknowledgement.* We thank V. Bach for his hospitality at the University of Mainz, where part of this work was done, and we are indebted to Gian Michele Graf for pointing out a serious gap in an earlier version of this paper. We also thank one of the referees for pointing out many typos and some small errors.

## Rights and permissions

## About this article

### Cite this article

Fröhlich, J., Griesemer, M. & Schlein, B. Asymptotic Completeness for Compton Scattering.
*Commun. Math. Phys.* **252**, 415–476 (2004). https://doi.org/10.1007/s00220-004-1180-x

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00220-004-1180-x

### Keywords

- Neural Network
- Statistical Physic
- Complex System
- Nonlinear Dynamics
- Electron State