Photon Position Margaret Hawton, Lakehead University Thunder Bay, Canada

It has long been claimed that there is no photon position operator with commuting components and, as a consequence, no basis of localized states and no position space wave function, just fields and energy density. In this talk I will argue that all of these limitations can be overcome! This conclusion is supported by our position operator publications starting in 1999.

Localizability 1)For any quantum particle ψ~e -i  t with +ve  = and localizability is limited by FT theorems. 2)If all k's are equally weighted to localize the number probability density, then energy density (and fields in the case of photons) are not localized. 3)For 3D localization of the photon, transverse fields don’t allow separation of spin and orbital AM and this is reflected in the complexity of the r-operator. The literature starts before 1930 and is sometimes confusing, in part because there are really 3 problems:

Classical versus quantum For a classical field one can take the real part which is equivalent to including +ve and –ve  's. Thus (1) does not limit localizability of a classical pulse, but the math of (2) and (3) are relevant to localizability of a classical field.

1) Localizability of quantum particles For positive energy particles the wave function ψ~e -i  t where  must be positive. Fourier transform theory then implies that a particle can be exactly localized at only one instant. This has been interpreted as a violation of causality. Also, the Paley-Wiener theorem limits localizability if only +ve (or –ve) k's are included.

Paley-Wiener theorem The Fourier transform g(r) of a square-integrable function h(k) that vanishes for all negative values of k (i.e. +ve k or +ve  only) must obey: This does not allow exact localization of a pulse travelling in a well defined direction but does allow exponential and algebraic localization, for example (Iwo Bialynicki-Birula, PRL 80, 5247 (1998))

For a particle localized at a where Hegerfeldt theorem Consider a photon, helicity, localized at r=0 at time t=0. The probability amplitude to find it at a at time t is: FT theory implies that an initially localized particle immediately develops tails that are nonzero everywhere.  depends on scalar product. In field theory.

Hegerfeldt causality paradox red particle localized at r=0 (or in any finite region) at t=0 can be found anywhere in space at all other times. wave fronts propagation direction

These problems are not unique to 3D. I’ll first consider the 1D analog of the Hegerfeldt causality problem. As an example consider an ultrafast photon pulse whose description requires only one spatial variable, z, if length<<area.

In 1D there is no problem to define a photon position operator, it the same as for an electron. The probability density that particle is at a is |  (a,t)| 2. Representations of the (1D) position operator are:

The exactly localized states are Dirac  -functions in position space and equally weighted in k-space: Exactly localized states cannot be realized numerically or experimentally so I’ll include a factor e -  k :

Consider a traveling pulse with peak at  z=0, center wave vector k 0 and width ~1/  If k 0 =0 we get the simple forms (PV is the principal value): A pair of pulses, one initially at –a travelling to the right (k's>0), and the other at a travelling to the left (k's<0) is : localized

1D ultrafast pulse imaginary part (tails go to 0 as 1/  z) real part (localizable) pulse propagation (k>0 only) peak at z=-a+ct 0 /2≈1

Causality paradox in 1D: photon at a=0 time t=0 can immediately be found anywhere in space (dark blue imaginary part). Resolution of the “causality paradox” in the recent literature is localizable states are not physically realizable, but is this the case? localizable (  -function) nonlocalizable PV~1/  z nonlocalizable PVs cancel (interfere destructively) when coincident.

At any t ≠ 0 the probability to find the photon anywhere is space in nonzero. Due to interference there exists a single instant when QM says that the photon can be detected at only one place. But this is just familiar spooky quantum mechanics, and I think the effect is physically real.

Let’s have a closer look as the pulses collide.

nonlocalizable PV tails

→0 as pulses collide, a QM interference effect

Have total destructive interference of nonlocalizable part when counter propagating pulses peaks are coincident.

Back to 3D (or 2D beam) causality paradox: red particle localized at r=0 at t=0 can be found any where in space at all other times. wave fronts propagation direction

. There is an outgoing plus an incoming wave.

In 3D have sum of incoming and outgoing spherical pulses:

A single quantum mechanical pulse is not localizable. For a pair of counter propagating pulses the probability to detect the photon can be exactly localized at the instant when their peaks collide. This gives a physical interpretation to photon localizability, it implies that we don’t know whether the photon is arriving or departing. Conclusion 1

2) Fields versus probability amplitudes and a pair of pulses initially at –a travelling to the right (k's>0) and at a travelling to the left (k's<0) is If n=0 (an integer in general) we get localizability. Recall that pulses were described by

For a monochromatic wave but this is ambiguous for a localized pulse that incorporates all frequencies, for which number and energy density have a different functional form.

I.Based on photodetection theory,the photon wave function is sometimes defined as the expectation value of the +ve energy field operator as below where |  > is a 1-photon state and |0> the vacuum: II.If we consider instead the probability amplitude to find a photon at z the interpretation is:

If E(z±ct) is LP along x,  t B y =-  z E x the magnetic field has the opposite sign for pulses travelling in the positive and negative directions. Thus if the nonlocalizable (PV) part of the E contributions cancel, the nonlocalizable contributions to B add. In a QM description, the photon energy density is not localizable. Energy density

We have a localizable position probability amplitude if k's equally weighted, electric field if weighted as k -1/2.

I don’t know, really, but consider the E-field due to a planar current source localized in z and approximately localized in t. What “wave function” should we consider? The important thing is what can be produced and detected. And does a photodetector see just the electric field?

current source This gives the same simple solution in the far field that I have been plotting and has a localized E-field. The source is localized in space but can’t be exactly localized in time since  >0. QED is required to do a proper job.

Plots with near field + far field source

Plots with near field + far field

In far field, get propagating pulses i/  z as previously plotted.

source detector emission/absorption should 2 nd quantize propagating free photon

detector

Photon position probability amplitude and fields are not simultaneously exactly localizable. Exponential localization of both is possible, but what matters is the field/probability amplitude that can be produced and detected. A localized current source in 1D produces a localizable E in the far field. Photon energy density is not localizable. Conclusion 2

3) Transverse fields in 3D It has long been claimed that there is no hermitian photon position operator with commuting components, and hence there is not a basis of localized eigenvectors. However, we have recently published papers where it is demonstrated that a family of position operators exists. Since a sum over all k’s is required, we need to define 2 transverse directions for each k. One choice is the spherical polar unit vectors in k-space.

kxkx k z or z kyky  

kxkx kzkz kyky More generally can use any Euler angle basis   

A unique direction in space and j z is specified by the operator so it is rather complicated. It does not transform like a vector and nonexistence proofs in the literature do not apply. Position operator with commuting components

  

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.

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.

Optical angular momentum (AM)

Interpretation for helicity , single valued, dislocation -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 = .

Position space

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:

The basis vectors contribute orbital AM.

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

. Conclusion 3 A transverse basis is required for the general description of pulses and beams, for example spherical polars. This necessarily singles out some direction in space, call it z. The transverse vectors form a screw dislocation with an associated definite total angular momentum, j z, which can’t in general be separated into spin and orbital AM.

Unidirectional pulses are not localizable, but counter propagating pulses can be constructed such that when they collide the particle can be detected in only one place. Relevance of field or energy density or probability amplitude depends on the experiment. Localized photons are not just fuzzy balls, they contain a screw phase dislocation. This applies quite generally, e.g. to optical beam AM. Summary