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,
 then to orbital angular momentum of the
bases, connection with optical beams
 conclude
Notation: momentum space
zˆ
pz or z
q
f
px
ˆ
p
ˆ
θ
py
ˆ
φ
ˆ ~ zˆ  pˆ ; θˆ  φ
ˆ  pˆ
φ
Use CP basis vectors:
e(0)
 
1
2
 θˆ  iφˆ 
e0  pˆ
 is the p-space gradient.
(Will use  and  when in r-space.)
Is the position of the photon an observable?
In quantum mechanics, any observable
requires a Hermitian operator
1948, Pryce obtained a photon position operator,
pˆ  S
a
a
rP  i p p 
p
• 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
1) Adlard, Pike, Sarkar, PRL 79, 1585 (1997)
a
ˆ
A ~q r
a is an arbitrarily large integer; power law
2) Bialynicki-Birula, PRL 80, 5247 (1998)

ˆ exp (r / r0 )
Z~m

  1 to satisfy Paley-Wiener theorem,
r0 arbitrarily small; exponentially localized
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
De
pz
q
p
f
ˆ
θ
ˆ
φ

py
 iS p 
e
 iS zf
e
 iS yq
O  DOD 1
F  DF
i   r(  )  D i  D1
px
i  is the position operator for m>0;  ri , p j   i  ij
etc. are preserved by above unitary transformation.
New position operator becomes:
r
( )
a
a
(0)
( )
( )
 rP  a S p where S p  pˆ .S
cos
q
ˆ for θˆ / φ
ˆ basis

φ
p sin q
 a
( 0)
 
• its components commute
• eigenvectors are exactly localized states
• it depends on “geometric gauge”, , that is
on choice of transverse basis
Like a gauge transformation in E&M
A  A  
a  a  

a p   A r 
2
  a   p pˆ + string so this looks
exactly like the B-field of a magnetic
monopole, complete with the Dirac
string singularity to return the flux.
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.
Geometric gauge transformation
  0, a
(0)
cos q
ˆ

φ
p sin q

 eg  =-f
  f , a
( f )
cos q  1
ˆ

φ
p sin q
no +z singularity

ˆ 
ˆ

θ
φ

ˆ
since   p


p
p q
p sin q f
( )

e
e
e
is rotation by  about pˆ :
 θˆ  iφˆ   cos f  i sin f 
 cos fθˆ  sin fφˆ 
ˆ
 i  sin f θˆ  cos f φ
e(f ) 

i (0)

1
2
1
2
qp
1
2
Rotated about zˆ by
 f  pˆ .zˆ  f cos q
 f at q =0, =f at q =p
φˆ
f
q0
θˆ
Is the physics -dependent?
Localized basis states depend on choice of , e.g.
e(0) or e(-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 of
angular momentum (AM). Note: orbital AM rxp
involves photon position.
“Wave function”, e.g.
F=E+icB
Any field can be expanded in plane wave
using the transverse basis determined by :
F  r, t   
3
d p
 2p 
3
f  p  e e
(  ) i  p.r  pct  /
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.
For an exactly localized state f p   Np e
a ipr '
Optical angular momentum (AM)
Helicity : e( ) 
Spin sz : e( ) ~
1
2

xˆ  isz yˆ 

2
1
Usual orbital AM: Lz  i
If coefficient f  p  ~ e



ˆ e  i
θˆ  iφ
p   z

 i
f
ilzf
Lz eilzf  lz eilzf and lz is OAM
Interpretation for helicity 1, single
valued, dislocation on -ve z-axis
( f )
1
e
cos q  1  xˆ  iyˆ 


2
sz=1, lz= 0
2
cos q  1  xˆ  iyˆ 


exp
2
2


1
  2 sin q  exp  if 


 2if 
sz= -1, lz= 2
sz=0, lz= 1
Basis has uncertain spin and orbital AM, definite jz=1.
Position space
e

ipr /
2p
0
imf
 ;l
 pr 
 4p  i Yl  ,   Yl q , f  jl  
 
l  0;n  l
l
n
n*
Yl n* q , f  eimf df ~  n ,m eim
im
e dependence in p-space  e in r-space
There is a similar transfer of q dependence,
and the factor jl (pr / ) is picked up.
Beams
Any Fourier expansion of the fields must make use
of some transverse basis to write
F  r, t   
3
d p
 2p 
3
f  p  e e
(  ) i  p.r  pct  /
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:
Bessel beam, fixed q 0 , azimuthal and radial (jz =0):
Volke-Sepulveda et al, J. Opt. B 4 S82 (2002).
A has   zˆ and     zˆ terms.
e1(0)  e(0)
1
φˆ 
i 2
xˆ  iyˆ if 1 xˆ  iyˆ if
1
i 2
e i 2
e
2
2
ˆθ  1 cos q xˆ  iyˆ eif  1 cos q xˆ  iyˆ e if  sin q zˆ
2
2
2
2
The basis vectors contribute orbital AM.
e1( f ) and e(f1) have same l z  1
Nonparaxial optical beams
Barnett&Allen, Opt. Comm. 110, 670 (1994) get
xˆ  iyˆ
1 zˆ sin q eif
co
s
q

2
2
 1 2if ( f )
( f )
cos
q

1
cos
q

e1 +
e e 1
2
2
Elimination of e2if term requires linear combination of
RH and LH helicity basis states.
Partition of J between basis and coefficient
( ) ( )
r e  0 since eigenvector at r '  0.
L(  )  r (  )  p, L(  )e( ) =0, L(  ) acts only on coefficient.
S
( )
( )
 J L

 a
( )

 p  pˆ S p gives AM of basis.
J  S (  )  L(  ) is invariant under geometric gauge
( )
(  ) imf
transformations, e.g. e  e e
and f  fe
for a fixed F describing a physical state.
 to rotate axis is also possible, but inconvenient.
 imf
Commutation relations
 L(i  ) , L(j )   i ijk L(k ) ;
r, L(  )   0
( ) 

Si
( )
( )
 J i , rj   i ijk rk  i 
 p j 


J z r
( )
j
( )

S
1
z

2 p j
 0 since S
( )
z
= m
L() is a true angular momentum.
Confirms that localized photon has a definite
z-component of total angular momentum.
Summary
• 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.