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Photon angular momentum and geometric gauge. Margaret Hawton, Lakehead University Thunder Bay, Ontario, Canada William Baylis, U. of Windsor, Canada. Outline. photon r operators and their localized eigenvectors leads to transverse bases and geometric gauge transformations,

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Photon angular momentum and geometric gauge

Photon angular momentum and geometric gauge

Margaret Hawton, Lakehead University

Thunder Bay, Ontario, Canada

William Baylis, U. of Windsor, Canada


Outline
Outline

  • photon roperators 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


Notation momentum space

pz or z

q

py

f

px

Notation: momentum space


Is the position of the photon an observable

Is the position of the photon an observable?

In quantum mechanics, any observable requires a Hermitian operator

a =1/2 for F=E+icB ~ p1/2 as in QED to normalize

last term maintains transversality of rP(F)

but the components of rP don’t commute!

thus “the photon is not localizable”?


A photon is asymptotically localizable

A photon is asymptotically localizable


Photon angular momentum and geometric gauge

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!


Euler angles of basis

p components and z

q

c

py

f

px

Euler angles of basis


New position operator becomes

New position operator becomes: components and

its components commute

eigenvectors are exactly localized states

it depends on “geometric gauge”, c,that is on choice of transverse basis



Photon angular momentum and geometric gauge

Topology: You can’t comb the hair on a fuzz ball without creating a screw dislocation.

Phase discontinuity at origin gives d-function string when differentiated.


Photon angular momentum and geometric gauge

Geometric gauge transformation creating a

no +z singularity


Photon angular momentum and geometric gauge

q=p creating a

f

q=0


Photon angular momentum and geometric gauge

Is the physics creating a c-dependent?

Localized basis states depend on choice of c, e.g. el(0) or el(-f) 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 ofangular momentum (AM). Note: orbital AM rxp involves photon position.


Wave function e g f e ic b

For an exactly localized state creating a

“Wave function”, e.g. F=E+icB

Any field can be expanded in plane wave using the transverse basis determined by c:

f(p) will be called the (expansion) coefficient. For F describing a specific physical state, change of el(c) must be compensated by change in f.



Photon angular momentum and geometric gauge

Interpretation for helicity creating a l=1, single valued, dislocation on -ve z-axis

sz=1, lz= 0

sz= -1, lz= 2

sz=0, lz= 1

Basis has uncertain spin and orbital AM, definitejz=1.


Photon angular momentum and geometric gauge

Position space creating a


Photon angular momentum and geometric gauge

Beams creating a

Any Fourier expansion of the fields must make use of sometransverse basis to write

and the theory of geometric gaugetransformations presented so far in the context of exactly localized states applies - in particular it applies to optical beams.

Some examples involving beams follow:



Photon angular momentum and geometric gauge

Elimination of creating a e2if term requires linear combination of RH and LH helicity basis states.


Photon angular momentum and geometric gauge

Partition of creating a J between basis and coefficient

Dc to rotate axis is also possible, but inconvenient.


Photon angular momentum and geometric gauge

Commutation relations creating a

L(c) is a true angular momentum.

Confirms that localized photon has a definite z-component of total angular momentum.


Summary
Summary creating a

  • Localized photon states have orbital AM and integral total AM, jz, 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 jz invariant.

  • These considerations apply quite generally, e.g. to optical beam AM. Position and orbital AM related through L=rxp.