Photon location in spacetime

Margaret Hawton Lakehead UniversityThunder Bay, Canada

INTRODUCTION

Newton and Wigner defined a basis of exactly localized states |x,t for all x at a fixed t. I will show that the NW states can be generalized to a localized photon basis on any hyperplane in spacetime.

This covariant formulation was motivated by hypothetical and real experiments: a spacelike measurement of probability density at a fixed time and a timelike photon counting experiment.

OUTLINE

Photon counting experiments

Derivation of POVMs for these experiments

recent work on Klein Gordon particles

covariant generalization for photons

spacelike and timelike experiments

Conclusion

Spacelike: At any time t=a the photon must be somewhere in space. Imagine an array of transparent photon counting detectors throughout space turned on at time t=a with timelike normal n=(1,0,0,0). The photon will be detected at some position x.Timelike: The photon is detected when it arrives at a photon counting array detector in the plane x3=b with spacelike normal n=(0,0,0,1). It will be counted in some pixel of the 2D array at time t.

TWO WAYS TO LOCATE A PHOTON

Spacelike and timelike experiments viewed at rest in 1+1 D

As seen by an observer with velocity - (c=1) parallel to the x3 axis the spacelike detection events are not simultaneous and the timelike photon counting array is in motion. The detector coordinates are Lorentz transformed to x3'=g(x3+bt) and t'=g(t+bx3) giving t'=g-1a+bx3' and x3'=g-1b+bt'. In the spacelike experiment the observer sees its normal rotated to n

m (,0,0,). As 1 the light cone t'= x3' at rotation angle p/4 is approached so no observer sees a timelike experiment as spacelike and vice versa, but any n has a physical interpretation.

Photon counting as seen by observer with velocity -

DERIVATION OF POVMs FOR PHOTON COUNTING EXPERIMENTS

notation:

KLEIN GORDON PARTICLES 1

Defining

the 4-flux (particles/m2/s or particles/m3) is

The k-space basis can be defined for k0= and k0=- which are orthogonal and it can be normalized covariantly to give:

for state vector |y & field

1 J. Halliwell and M. Ortiz, Phys. Rev. D 48, 748 (1993)

so that a conserved inner product can be defined in spacetime as the integral of |J| over S that is

J satisfies the continuity equation

PHOTONSThe k-space invariant integral can be evaluated on any S by first integrating over its normal:

Defining , the wave function proportional to the 4-potential Am

e(x) is

The k-basis states on S are orthonormal:

Analogous to KG particles the spacetime inner product is the integral of |J| over S. Integration over ds gives its simpler k-space form that will be used here.

The 4-flux, J, is the expectation value of the contraction of the EM field tensor and 4-potential operators: 2

2 M. Hawton and T. Melde, Phys. Rev. A 51, 4186 (1995)

LOCALIZED PARTICLE BASIS ON S

The generalized NW orthonormal particle basis is

and the projection of |y onto the localized particle basis as

where kS kmnm is invariant. From previous slide

This gives the spacetime orthonormality condition

The integral over projectors onto the localized states on S is a partition of the identity operator:

In this basis the POVM is PVM,

The inner product can be expanded as

The relationship between the potential and the NW number amplitude is nonlocal as can be seen by substitution of a NW basis state in the 4-potential:

The factor 1/ |kS| results in a wave function that is not localized at the particle position x' as recognized by NW. apparently describes a photon created at x but this may be an artefact of basis selection.

satisfyso thatIn 4D the basis state Ax',e(x) is the response to a localized matter source or sink at x', i.e. it is a Green function. It is equal to the relativistic propagator1.

LOCALIZED FIELD BASISNonlocality can be avoided by defining a biorthonormal basis on S consisting of a q-function potential and a field with a localized component:

Spacelike particle basis

is the probability density.

SPACELIKE AND TIMELILKE EXPERIMENTS

Spacelike field basis

is probability density.

The biorthonormal (relativistic) field basis eliminates the nonlocality problem.

The probability to count a photon is its integral over a 3D spacelike hyperpixel of the detector array.

Timelike field basis: This describes an ideal photon counting array detector with good time resolution:

integrated over 2D pixel area and time (again a 3D hyperpixel) is the probability to count a photon. If the “wrong” spacelike probability density basis is used to calculate the flux across a timelike hypersurface an additional factor cos =q k3/ w arises and the formalism is not covariant3. The result derived here is more like Fleming’s covariant generalization of the NW basis4.3 A. Mostafazadeh and F. Zamani, Annals Phys. 321, 2183 (2006) 4 Gordon N. Fleming, Reeh-Schlieder Meets Newton-Wigner

These bases provide a convenient tool for locating an event in spacetime but the modes that contribute are only those in |y that can be the initial or final state. describes absorption and describes emission. Each localized state must act as both detector and source.

5 Bondurant (1985); M. Hawton, Phys. Rev. A 82, 012117 (2010)

Either probability density on x3=b describes photon counting: A semiconductor pixel counts photons because it is thick enough to absorb any photon incident on its surface. The atomic absorption probability m2|E|2w but penetration depth w-1 so the w dependence drops out.4

The particle and field bases count the same total number of photons.

Moving observer:According to an observer with velocity -b parallel to the x3 axis, the normal to S is rotated to (g,0,0,gb) for the spacelike probability density experiment and to (gb,0,0,g) for the timelike photon counting experiment. The (bi)orthonormal basis on any S has a physical interpretation as the POVM describing a photon counting array viewed by an observer that may be in motion relative to the array.

An orthonormal localized particle basis or biorthonormal field basis can be defined on any spacetime hyperplane. These bases provide a covariant description of an ideal spacelike or timelike photon counting array detector or source. The photon will be counted in one of its hyperpixels with probability unity.To avoid overcounting in 4D, particles should be counted on a 3D hypersurface.

CONCLUSION

Thank you!

Fleming5: “..being NW localized at x on 1 is incompatible with being localized at x on 2. This is a strange kind of localization! Does NW localization have any physical significance?!” He argues that it describes center of energy.Halvorson6:“..neither Segal nor Fleming has offered a conceptually coherent description of the physical meaning of Newton-Wigner localization.” …Conclusion reached here: A generalized (NW) basis provides a covariant description of a photon counting experiment. This (bi)orthonormal basis predicts absorption or emission somewhere on S with probability unity and so counts photons.

5 Hans Halvorson, Reeh-Schlieder Defeats Newton-Wigner (2000)

SOME COMMENTS ON THE NW BASIS

According to the Reeh-Schlieder theorem there are no projection operators in a finite region O but Halverson6 states that “It is essential for the proof of the RS theorem that the region O has some .. extension .. in 4 independent directions.”Conclusion reached here: In an ideal photon counting experiment the event that is detected is the first crossing of a hypersurface that has finite extent in only 3D so the RS theorem does not apply.