1. Photon angular momentum and geometric gauge Margaret Hawton, Lakehead University Thunder Bay, Ontario, Canada William Baylis, U. of Windsor, Canada

2.  photon r operators and their localized eigenvectors  leads to transverse bases and geometric gauge transformations,  then to orbital angular momentum of the bases, connection with optical beams  conclude Outline

3. Notation: momentum space pxpx p z or z pypy  

4. Is the position of the photon an observable?  =1/2 for F=E+icB ~ p 1/2 as in QED to normalize last term maintains transversality of r P (F) but the components of r P don’t commute! thus “the photon is not localizable”? In quantum mechanics, any observable requires a Hermitian operator

5. A photon is asymptotically localizable

6. Is there a photon position operator with commuting components and exactly localized eigenvectors? It has been claimed that since the early day of quantum mechanics that there is not. Surprisingly, we found a family of r operators, Hawton, Phys. Rev. A 59, 954 (1999). Hawton and Baylis, Phys. Rev. A 64, 012101 (2001). and, not surprisingly, some are sceptical!

7. pxpx pzpz pypy Euler angles of basis   

8. New position operator becomes: its components commute eigenvectors are exactly localized states it depends on “geometric gauge”,  that is on choice of transverse basis

9. Like a gauge transformation in E&M

10. Topology: You can’t comb the hair on a fuzz ball without creating a screw dislocation. Phase discontinuity at origin gives  -function string when differentiated.

11. Geometric gauge transformation no +z singularity

12.   

13. Is the physics  -dependent? Localized basis states depend on choice of , e.g. e (0) or e (-  ) localized eigenvectors look physically different in terms of their vortices. This has been given as a reason that our position operator may be invalid. The resolution lies in understanding the role of angular momentum (AM). Note: orbital AM r x p involves photon position.

14. “Wave function”, e.g. F=E+icB For an exactly localized state Any field can be expanded in plane wave using the transverse basis determined by  f(p) will be called the (expansion) coefficient. For F describing a specific physical state, change of e (  ) must be compensated by change in f.

15. Optical angular momentum (AM)

16. Interpretation for helicity , single valued, dislocation on -ve z-axis s z = , l z =  s z = -1, l z =  s z =0, l z =  Basis has uncertain spin and orbital AM, definite j z = .

17. Position space

18. Beams Any Fourier expansion of the fields must make use of some transverse basis to write and the theory of geometric gauge transformations presented so far in the context of exactly localized states applies - in particular it applies to optical beams. Some examples involving beams follow:

19. The basis vectors contribute orbital AM.

20. Elimination of e 2i  term requires linear combination of RH and LH helicity basis states.

21. Partition of J between basis and coefficient  to rotate axis is also possible, but inconvenient.

22. Commutation relations L (  ) is a true angular momentum. Confirms that localized photon has a definite z-component of total angular momentum.

23. Summary Localized photon states have orbital AM and integral total AM, j z, in any chosen direction. These photons are not just fuzzy balls, they contain a screw phase dislocation. A geometric gauge transformation redistributes orbital AM between basis and coefficient, but leave j z invariant. These considerations apply quite generally, e.g. to optical beam AM. Position and orbital AM related through L=r x p.