Photon location in spacetime

1. Photon location in spacetime Margaret Hawton Lakehead University Thunder Bay, Canada

2. 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.

3. 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

4. TWO WAYS TO Locate a photon 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=awithtimelike normaln=(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=bwithspacelike normal n=(0,0,0,1). It will be counted in some pixel of the 2D array at time t.

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

6. 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' andx3'=g-1b+bt'. In the spacelike experiment the observer sees its normal rotated to nm (,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.

7. Photon counting as seen by observer with velocity -

8. Derivation of POVMs for photon counting experiments • notation:

9. Klein Gordon particles 1 • The k-space basis can be defined for k0= andk0=-which are orthogonal and it can be normalized covariantly to give: • for state vector |y & field • Defining • the 4-flux (particles/m2/s or particles/m3) is 1 J. Halliwell and M. Ortiz, Phys. Rev. D 48, 748 (1993)

10. J satisfies the continuity equation • so that a conserved inner product can be defined in spacetime as the integral of |J| over S that is

11. PHOTONS • The k-space invariant integral can be evaluated on any S by first integrating over its normal: • The k-basis states on S are orthonormal: • Defining , the wave function proportional to the 4-potential Ame(x) is

12. The 4-flux, J, is the expectation value of the contraction of the EM field tensor and 4-potential operators: 2 • 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. • 2 M. Hawton and T. Melde, Phys. Rev. A 51, 4186 (1995)

13. Localized particle basis on s • The generalized NW orthonormal particle basis is • where kSkmnm is invariant. From previous slide • This gives the spacetime orthonormality condition • and the projection of |y onto the localized particle basis as

14. The inner product can be expanded as • The integral over projectors onto the localized states on S is a partition of the identity operator: • In this basis the POVM is PVM,

15. 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.

16. Localized field basis • Nonlocality can be avoided by defining a biorthonormal basis on S consisting of a q-function potential and a field with alocalized component: • satisfy • so that • In 4D the basis state Ax',(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.

17. SPACELIKE AND TIMELILKE EXPERIMENTS • Spacelike particle basis • is the probability density. • Spacelike field basis • is probability density. • The probability to count a photon is its integral over a 3D spacelike hyperpixel of the detector array. • The biorthonormal (relativistic) field basis eliminates the nonlocality problem.

18. 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 cosq=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

19. 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.

20. 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|2w but penetration depth  w-1 so the w dependence drops out.4 • The particle and field bases count the same total number of photons. • 5 Bondurant (1985); M. Hawton, Phys. Rev. A 82, 012117 (2010)

21. 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.

22. CONCLUSION • 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.

23. Thank you!

24. Some comments on the nw basis 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)

25. 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.