1. LASER PHYSICS 2014 SOFIA, BULGARIA 16 July 2014 PONDEROMOTIVE POTENTIAL and the INTENSE-FIELD MASS SHIFT H. R. Reiss Max Born Institute, Berlin, Germany American University, Washington, DC, USA 1

2. 2 Preview History of the intense-field mass shift Definition of the basic quantity: the ponderomotive potential U p Properties of U p Ponderomotive energy in transverse and longitudinal fields Gordon-Volkov mass shell Provenance of the Gordon-Volkov alteration of the mass shell The legendary Lamb-Jaynes-Franken (LJF) wager Resolution of the LJF problem New unresolved problems

3. Historical Background The usual electron mass shell: was found to be altered to in the strong-field theory of Compton scattering: N. D. Sengupta, Bull. Math. Soc. (Calcutta) 44, 175 (1952) and (independently) in the strong-field theory of pair production: HRR, PhD Dissertation, U of Maryland (1958) HRR, J. Math. Phys. 3, 59 (1962). The extra term came to be known as the strong-field mass shift. 3

4. The quantity U p is the ponderomotive potential. Suggestions for means to measure the strong-field mass shift have been made, starting with: HRR, PRL 17, 1162 (1966), and continuing to the present: C. Harvey et al., PRL 109, 100402 (2012). No clear experimental confirmation has been found. There is a fundamental reason for this failure. It will be shown that there is no intense-field mass shift; the extra term in the mass-shell expression arises from a mechanism other than a mass change. 4

5. Ponderomotive Potential U p U p is defined as where the angle brackets denote a cycle average and the absolute value is needed because A μ is a spacelike 4-vector. The product A μ A μ is obviously Lorentz-invariant, and it is also gauge-invariant if it describes a transverse field. (Shown on the next slide.) The dimensionless intensity parameter z f = 2U p /mc 2 is basically just a dimensionless way to express the ponderomotive potential. 5

6. Gauge Invariance of U p Propagating fields satisfy the necessary condition that any occurrence of the basic spacetime 4- vector x µ can only be in the form of the scalar product: Schwinger, PR 82, 664 (1951); Sarachik and Schappert, PRD 1, 2738 (1970). Gauge transformation: 6

7. The last result uses the light-cone property of the propagation vector: and transversality of a PW field: Hence A μ A μ is gauge invariant and thus U p and z f are gauge invariant – for transverse fields only. If the dipole approximation is applied a priori, then none of these statements are true. The dipole approximation amounts to replacing a transverse field by a longitudinal field. This is unacceptable under relativistic conditions. 7

8. Strong-Field Coupling Constant The coupling constant of ordinary QED is α, the fine structure constant. Perturbation theory in α has a zero radius of convergence. Dyson, PR 85, 631 (1952). What about strong-field QED (SFQED) based on the relativistic Volkov solution? Investigation of convergence of strong-field expansions led to several results: HRR, PhD Dissertation, 1958; J. Math. Phys. 3, 59 (1962); J. Math. Phys. 3, 387 (1952). SFQED has, in general, a non-zero radius of convergence. Intensity dependent singularities occur in the complex coupling-constant plane that put an upper limit on convergence of perturbation theory in SFQED: z f < 1 (or U p < mc 2 /2). The perturbative expansion parameter is not α, but an intensity-dependent multiple of α. The coupling constant of SFQED is: 8

9. That is, it is the ordinary QED coupling constant multiplied by the number of photons contained in a cylindrical volume with radius of the electron Compton wavelength and a length equal to the field wavelength. 9

10. PROPERTIES OF U p, z f z f is the dimensionless statement of U p. Both quantities are: Lorentz invariant Gauge invariant Measures of the coupling of an electron to a strong PW field This (and other factors) support the hypothesis that U p is a true potential energy. That is, an electron in a strong field must possess U p just because it is in the field. That energy must come from the field, so a momentum U p /c must also be acquired. That is, there is a potential 4-momentum U μ associated with the electron in the strong field: 10

11. The mass shell of an electron in the field is thus The last product is Lorentz invariant, so it can be evaluated in the frame where 11

12. 12 The final result is: This is exactly the mass shell found in SFQED from relativistic Volkov methods. It does not arise from a mass shift as such, but from the acquisition by a charged particle in a transverse field of a component from the photon mass shell. That is, the charged particle really has a mixed mass shell; part is ordinary, but there is an additional ponderomotive 4-momentum that is on the light cone. This has immediate implications.

13. 13 mc 2 vs. U p Dominance It was noted earlier that the radius of convergence of a perturbation expansion of SFQED in terms of the expansion parameter z f is This is consistent with the mass shell Were it true that then the ponderomotive potential would be the dominant influence, not the rest energy.

14. 14 Basic Intensity Measure The basic measure of intensity of a longitudinal field is the magnitude of the electric field E; directly, or non-dimensionalized with respect to the Sauter- Schwinger critical field. The basic measure of the intensity of a transverse field is the ponderomotive potential U p ; directly, or non-dimensionalized with respect to the electron rest energy. Some investigators use a measure equivalent to z f 1/2, apparently in the attempt to put the electric field E in evidence. This is not useful. What appears is not E but E/ω, which is just the vector potential A. Transverse fields are a good practical example of the primacy of potentials over fields.

15. 15 Longitudinal and Transverse Fields (very briefly) Longitudinal fields are static or quasistatic electric fields. Transverse fields are plane-wave propagating fields. Their measures of intensity are qualitatively very different.

16. 16 Longitudinal

17. 17 Transverse

18. 18 Ponderomotive Potential vs. Quiver Energy The U p that has been discussed here relates only to transverse fields, where it is a potential energy. It is possible to have an electron immersed in a very strong field with a large U p and yet have a very small kinetic energy. This interpretation is consistent with experiment. In a short-pulse ionization, a photoelectron emerges from the laser pulse too quickly to transform its potential energy into kinetic energy. If U p were a kinetic energy, then the measured spectrum should peak at about the maximum U p. This doesn’t happen. Numerical example: A well-known experiment on ionization of helium with U p about 230 eV, gave a spectrum peaking at about 9 eV. See Mohideen et al. PRL 71, 509 (1993). With longitudinal fields, a free electron is driven by the field to have a kinetic “quiver energy” that (for linear polarization) is about the same as the potential energy with a transverse field of the same amplitude. However, transverse fields cannot “drive” a free electron, whereas longitudinal fields must have virtual sources that can transfer energy and momentum to a free electron. HRR, JPB 46, 175601 (2013)

19. 19 New Questions That Arise It is well-known and obvious that a single photon (zero mass) cannot interact with a single free electron (nonzero mass) because it is impossible to satisfy energy and momentum conservation conditions. The same result is true for any number N of identical photons, even for N uncountably large. Then how is it possible for a laser beam to exert radiation pressure on a free electron as is predicted by SFQED and as measured in the laboratory? HRR, JOSA B 7, 574 (1990); Smeenk et al., PRL This will be answered in several steps: The poles of the propagator in ordinary QED are shown. These are contrasted with the families of poles that appear in SFQED. An attempt was made to exactly sum families of Feynman diagrams to relate the two results. A discrepancy was found that is shown to be due to a family of apparently divergent diagrams. These diagrams have a finite closed-form sum. That sum gives the extra mass-shell term.

20. 20 Poles of the Ordinary Electron Propagator Ordinary QED: Gives mass shell: From Sakurai, Advanced Quantum Mechanics

21. 21 Poles of the Volkov Electron Propagator The Volkov propagator has families of poles with accumulation points through which a contour can be passed: HRR & Eberly, PR 151, 1058 (1966).

22. 22 Volkov Mass Shell The amplitude for the contribution of each of these poles is intensity-dependent, with n = 0 dominant as the intensity → 0. The extra mass-shell term is an inherent property of the Volkov solution. This is still not a complete explanation for why this term does not appear in any order of perturbation theory, but the term can be very large even in a nonrelativistic theory. Much insight into this comes from 2 papers: Fried and Eberly, PR 136, B871 (1964) Eberly and HRR, PR 145, 1035 (1966).

23. 23 Exact Sums of Feynman Diagrams Fried and Eberly took a simple case: Compton scattering by a spinless electron in a circularly-polarized monochromatic field. They were able to sum exactly all the classes of Feynman diagrams that can occur. The result was identical to the Volkov-solution treatment of the same problem, with one exception: the modified mass-shell term did not occur. However: A class of diagrams with zero denominators occurs from “seagull” diagrams where absorption and emission of a photon occurs at a single vertex (that is possible with spin-zero “electrons”). These divergent diagrams were omitted by Fried and Eberly. Eberly and HRR found that it was possible to exactly sum all of the divergent diagrams, obtaining a finite result that gave exactly the modified mass shell of the Volkov-based calculation.

24. 24 Thus the very important result is obtained that the modified mass shell does not occur in any finite order of perturbation theory, but an exact sum of an infinite number of divergent diagrams leads to the modified mass shell. This explains why no countable number of photons can transfer energy and momentum to a free electron, but an uncountably large number (obtained from the Volkov solution) makes it possible for a beam of light to exert radiation pressure on a free electron. This is explicitly nonperturbative. It is possible because the electron acquires a ponderomotive potential that contains a component of the photon mass shell in addition to the normal mass shell. This is the component from the ponderomotive 4-momentum U µ.

25. 25 The Lamb-Jaynes-Franken Wager At a Rochester Conference on Quantum Optics in 1966(?), a dispute arose about whether the photon was real and necessary, or whether semiclassical electrodynamics was sufficient. Willis Lamb (of the Lamb shift) said the photon is necessary; Ed Jaynes (of the Jaynes-Cummings model) said it is not; Peter Franken (of nonlinear optics) arbitrated and held the money on a bet made by Lamb and Jaynes until such time as the answer became clear. (The wager was not settled; none of the 3 principals survives.) The answer is now available: both Lamb and Jaynes were partly wrong and partly right. QED with photons amounts to Feynman diagrams (covariant perturbation theory). This is incomplete: it misses the inherently strong-field incorporation of the ponderomotive potential. SFQED is incomplete because it fails to connect the electromagnetic field with the Standard Model and the Electroweak Model.

26. 26 Lamb, Jaynes, and Franken would have been surprised and probably pleased at the outcome of their basic question. However, an entirely new category of questions now appears. If SFQED has properties that lie outside the scope of Electroweak theory, then exactly where does the electromagnetic field fit into the Standard Model?

27. 27 SUMMARY The “intense-field mass shift” of a free electron in a laser field is not an actual mass change. Rather, it is an artifact of the ponderomotive potential U p of an electron in a strong transverse field. U p is Lorentz-invariant, gauge invariant, and measures the coupling of a charged particle to a transverse field. U p is a true potential energy for a transverse (i.e., laser) field. U p is a “quiver energy” (a kinetic energy) for a longitudinal field. The strong-field alteration of the electron mass shell does not arise in any finite order of perturbation theory. The strong-field alteration of the electron mass shell comes from a closed form sum of divergent Feynman diagrams that sum to a finite result; it does not exist in ordinary QED. How does this fit into electroweak?