Minimum size of radiation with spin

ANNALS OF PHYSICS

112, 477-484 (1978)

Minimum Size of Radiation With Spin R. W. M. WOODSIDE Physics Department, McMaster University, Hamilton, Ontario, Canada L8S-4M1

Received February 10, 1977

It is shown that any massless body with conserved momentum and classical spin and positive energy density must have an infinite spatial extent. This is accomplished by extending the special relativistic centroid theorem to the two types of massless radiation. This theorem states that for radiation with everywhere positive energy density in Minkowski space and conserved momenta: (1) when the spin is parallel to the direction of motion, the radiation must always fill an entire spatial plane normal to this direction; (2) when there is a component of spin perpendicular to the direction of motion the radiation must always contain points having an infinite spatial separation in the direction of motion. The observation of finite light beams possessing angular momentum shows Classical Electromagnetism to be incomplete and a large class of Neoclassical theories untenable.

1. I N T R O D U C T I O N

The centroid theorem [1-4] states that if a body with positive energy density is to have a certain mass and rest frame spin, then it must also have a minimum size, ensuring that no part of the body travels faster than light during the rotational motion. Here and throughout the word "spin" refers to classical rotational angular momentum. The present work extends this purely kinematical result to massless bodies with well-defined momentum, centroid and spins both parallel and perpendicular to the direction of motion. Intuitively one sees that a massless body is frozen in time, so any rotational angular m o m e n t u m must be gained at the price o f an infinite m o m e n t arm. This is the necessary reason for the "terms at infinity" [5, 6] which give rise to the spin of electromagnetic radiation. It will be shown that radiation with only parallel spin must in any frame and at every instant occupy at least an entire spatial plane perpendicular to its direction of motion [17]. Though not proven here, this plane is the expected limit o f the centroid disk for massive rotating bodies. In a special frame where the radiation has only perpendicular spin, it will be shown that at every instant this sort of radiation must at least fill the interior o f a spatial parabola whose latus rectum is parallel to the direction of motion. The shape and orientation of this region changes from frame to frame, but in any frame and at every instant this radiation must contain points infinitely distant in the direction of motion. Such radiation cannot be causally absorbed or emitted in a finite time, which is the 477 0003-4916/78/1122-0477505.00/0 Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.

478

R. W. M. WOODSIDE

classical analog of Abbott's quantum field theoretic argument [7] against the existence of such particles. The author has not examined the case of rotating tachyons, but their "zero mass limit" should give rise to this radiation [8]. Any postulated radiation with conserved momentum, conserved nonzero spin, having a positive energy density and which is also finite in size would contradict the centroid theorem, so that all such radiation must be infinitely large. At least one important conclusion can be drawn from this work. Classical Electromagnetism is an incomplete theory as it cannot explain Beth's observation [9] that light beams with finite cross section do possess angular momentum. Before it can be claimed that the properties of light may essentially be understood within the bounds of classical electromagnetism, this theory must be enriched by some new physical concept which can give finite light pulses an equivalent to spin. The extension of the centroid theorem is divided into two parts. In the first section the centroid or center of energy is used to define a spin that is appropriate to a general body. This spin contains the same information as the more usual intrinsic spin tensor [11], but the differences should be noted. The definitions employed here make sense for finite bodies, though they will apply to certain infinite systems also. In the last section the results of the first are used to construct minimum sizes for the two types of radiation which are found to be infinite, contrary to expectation. The conventions of Misner et al. [10] will be used throughout.

2. SPIN AS THE TOTAL ANGULAR MOMENTUM ABOUT A CENTROID

A general body is to be described by a symmetric and divergence-free energy tensor, T % defined over a fiat space time. One then defines the body's total angular momentum by j.v ~

f xt.T~lO Jv

(2.1)

and its momentum by t"

P " ~ Jv T"°

(2.2)

where the integration is done on a constant time surface, v, of some inertial frame. These integrals are assumed to be finite. The properties of the T "~ ensure the conservation of these quantities, provided certain integrals involving T "~ vanish on the boundary of v. With a large enough volume these integrals will always be zero for a finite body, but this is not necessary for infinite systems [6]. The minimal desiderata of an isolated body are completed on introducing the center of energy or centroid

c" =-- f~ x"T°~° J" TOO

(2.3)

SIZE OF SPIN RADIATION

479

which is assumed to be well defined. At the heart of the theorem is the observation that if the energy density is everywhere positive, then the body must contain all possible centroids. Applying this definition to the conserved j0, one obtains Newton's first law [13] p.x o

c"--

p0

+a"

(2.4)

where the a" is the x ° = 0 centroid. These centroids always have the noncovariant form a" = (0, a 9.

(2.5)

To find the relation between this kind of centroid for two different frames it must be expressed generally in terms of quantities with known transformation rules. This is accomplished by substituting (2.4) via (2.3) into (2.1) for the j0v and with (2.2) one obtains the perspicuous relation (2.6)

j o y ~_ atOp,j.

Remembering (2.5) one can then solve for the a", giving a" =

(2.7)

__jo,/po.

This equation determines the x ° = 0 centroid for any frame, so writing ~ for the x~ = 0 centroid of boosted frame gives ?t~ =

(2.8)

_jo~/po.

Regarding the ~ as a point, its coordinates can be found in the unboosted frame by applying the Lorentz transformation A f which connects the two frames d" =

--A~J",/AjP

~.

(2.9)

The a" can be spatially or temporally different from a". To see the differences clearly (2.6) must be used to motivate the definition of spin as s .~ ~

(2,10)

J," -- at"P'L

In any frame this spin is the total angular momentum about a centroid for that frame. In the rest frame of a body it is the intrinsic spin tensor, but it does not transform like a tensor, because due to (2.6) it appears in every frame with the general form 0 s.

.

.

0

--s z S ~'

595]II212-I7

.

Sz

0 --Sz

--S u

(2.11)

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R.W.M.

WOODSIDE

Unlike the intrinsic spin tensor it does not require a preferred frame for its definition. Within any frame it transforms as an axial 3-vector and is the remains of the total angular momentum after all the orbital angular momentum has been removed. The polarization vector (2.12)

W . ~ ½E,~vooJv°P~

also removes all the orbital angular momentum. This allows the components o f s "v to be found in any frame as (2.13)

si = Wt/P °

which is obtained on evaluating the components of W" by using (2.10) and (2.11), and solving for the s i. With this decomposition of the total angular momentum and the explicit form of the AT [12] and by lapsing into obvious vector notation, (2.9), which relates the two centroids, can now be written as d° =

(flyn.

a/A~¢P ~) po,

: a + (flyn. a / A j P ~) P + ( f l v / A ~ P 0 s

(2.14) × n

where fl, V are the usual boost parameters and n is the boost direction. The difference between the two centroids is transparent. The ~" has been shifted from the a" by a certain amount parallel to the momentum and by a purely spatial amount perpen dicular to the spin vector. Since the world line c" is parallel to the momentum and passes through a ", a vanishing spin requires all possible centroids to lie on this line. A nonzero spin destroys the uniqueness of the c" and multiplies it into a bundle o f world lines all parallel to the momentum which pierce the x ° = 0 plane at b =a

+ (fly/A~6P ") s × n.

(2.15)

As remarked earlier a positive energy density forces the body to contain all these points. Thus from a knowledge of the body's momentum and polarization vector, (2.15) enables one to find the minimum size of a general body in an arbitrary frame. This equation makes use of all six independent quantities in the J"~ and the four remaining quantities P " determine the direction of motion, so that no further independent information can be had from the Poincar6 symmetry.

3. THE CENTROID THEOREM FOR RADIATION WITH SPIN The methods of the previous section are completely general, but since the massive case is well known it will not be treated. The massless case naturally subdivides into

481

SIZE OF SPIN RADIATION

two types of radiation depending on whether the polarization vector is space-like or parallel to t h e momentum. 3.1. P . P . = O, W . W . > 0 For this radiation a frame can always be found where the momentum and polarization vectors have the values P - = E(1, 1, 0, 0),

(3.1.1)

W" = Es(O, O, O, 1).

(3.1.2)

On taking the coordinate origin of the frame at the x ° = 0 centroid for this frame, (2.15) simplifies to b =

(-

/3n~

/3n,

O) s

(3.1.3)

so that in the x ° = 0 surface all the possible centroid world lines occupy a twodimensional region which is perpendicular to the spin vector and contains the direction of motion. Since the choice of the constant time surface was arbitrary, this region is the same at any instant. The actual region is determined by the values of/3 and n which describe permissible Lorentz boosts. There are many boosts which will shift the centroid by the same amount; but if it is claimed that a certain point is in the region, then at least one boost which does move the centroid to that point must be exhibited. Since there are only two boost directions present in (3.1.3), one need only consider boosts in their plane and a single parameter 0 suffices to specify the boost direction. Expressing the region in polar coordinates (R, ~), now allows Eq. (3.1.3) to be written as (R cos ~b, R sin ~, 0) =

flsin0 /3 c o s 0 ) s 1 -- /3 cos 0 ' 1 -- fl cos 0 ' 0 -~

(3.1.4)

where sin 0 is nu and cos 0 is n~. By taking ratios of components from both sides of this equation one has cot ~ = --tan 0

(3.1.5)

3~0 = ¢ + -~--.

(3.1.6)

which is always true provided

Substituting this value for 0 back into (3.1.4) and solving for/3 yields R

/3 = R sin ff + (s/E)"

(3.1.7)

482

R, w .

M. WOODSIDE

Although (3.1.6) shows the centroid can be moved in all directions perpendicular to the spin vector, the restriction that/3 be less than one, allows it to be shifted only certain distances. On rewriting (3.1.7) as /32 ~_ R2(c°s2 ~ q- sin2 4) (R sin q~ ÷ (s/E)) 2

(3.1.8)

and reverting to rectangular coordinates after applying the inequality for/3, gives 1

1

Y > ~ (-~-) x~ -- ~ (~--E)-

(3.1.9)

Thus any point in this parabolic disk is a possible centroid and the assumption of positive energy density requires the radiation to be at least as large as the disk at any instant in this frame. The radiation is traveling in the x direction and as the parabola opens out there will be points in the disk that have an indefinitely large separation in the direction of motion. Consideration of the Lorentz invariant projection of this separation on the momentum vector shows the property of an infinite size in the direction of motion to persist in all physical frames. 3.2. p u p , = 0, W "I[ P" The spin of this radiation is a Lorentz invariant and in any frame the momentum and polarization vectors can have the values P" = E(I, 1, 0, 0),

(3.2.1)

W ~ = Es(1, 1, 0, 0).

(3.2.2)

Again setting the coordinate origin at the x ° = 0 centroid for this frame, (2,15) works out to fln~ ~n~_ ~ s (3.2.3) b (\0, 1 -- /3nx ' 1 -- fin x~ -E" Unfortunately all three boost directions appear here, so to find which boosts shift the centroid whither, a spherical parameterization of the boost direction is introduced n~ ~

c o s p,

n~ ~ sin p cos O,

(3.2.4)

n~ ~ sin p sin O, where p lies between 0 and ~', and 0 between 0 and 27r. Using polar coordinates for the y - z plane then gives

(o,

R

COS

R sin

¢)

[0,--

fisinpsinO flsinpcosO]~ s 1--flCOSp ' l--flcosp E-"

(3.2.5)

SIZE OF SPIN RADIATION

483,

As before, this implies 3~r 0 ~ ~ -}- 2

(3.2.6)

so that the centroid may be moved from the origin in any direction in the y - z plane. Putting this value of 0 into (3.2.5) yields

R--

fi sin p 1 -- f l c o s p

(3.2.7)

allowing the centroid to be sent any distance in the y - z plane. To see this consider (fl, p) as polar coordinates of some unphysical boost space with rectangular coordinates (~/, ~). The physically possible boosts then correspond to the interior points of the unit semidisk in the upper half of this boost plane. On the other hand (3.2.7) describes lines in the boost plane with negative slope R and a constant ~/intercept of one. Such lines even with arbitrarily large values of R always intersect the semidisk of physical boosts, so in fact there are boosts which can move the centroid to any point in the plane perpendicular to the direction of motion at x ° ----0. Since the choice of frames merely alters the value of E, this is true for the x ° -----0 surface of all frames. Finally, since the choice of constant time surface is arbitrary, the assumption of positive energy density demands that radiation with parallel spin must in all frames and at all times occupy at least an entire plane that is normal to the direction of motion.

4. CONCLUSIONS

The work presented in Section 2 permits the extension of the centroid theorem to general bodies in arbitrary frames and in Section 3 the theorem is given for massless bodies. The result that these bodies must be infinitely large is purely kinemetical and rests on three assumptions: Poincar6 symmetry of space-time; the body has a positive energy density, conserved momentum, and angular momentum; and all the angular momentum comes from the integrated antisymmetric moment of the momentum density. The free fields for radiation pulses in both Classical Electromagnetism and the linear approximation to General Relativity satisfy these assumptions. Consequently such radiation having parallel spin must also have an infinite size perpendicular to its direction of motion. This is not observed for light. In 1936 Beth [9] found a torque due to the change in polarization of a finite light beam. The idea of the experiment was to pass light through a quartz plate and measure the resultant torque as the light changed its polarization. Assuming the conservation of angular momentum at the crystal's boundary, Beth concluded that he was observing torques due to changes in the angular momentum of the electromagnetic field. The measured torque agreed well with the calculated torque per unit area exerted on an infinite plate by plane waves. The theorem clarifies the theoretical need for infinite

484

R.W.M.

WOODSIDE

plates and infinite waves, but they do not correspond to the experimental situation. It is natural to introduce plane waves for the resultant simplicity o f calculation, yet in the end one must be able to properly superpose them and recover the realistic situation. A n y possible superposition satisfying the above assumptions and still maintaining a spin will also maintain at least an infinite plane o f radiation. This means that Classical Electromagnetism is incomplete as it cannot possibly produce the observed localized light beams which have a polarization dependent angular momentum. The Neoclassical position conjectures the possibility o f recovering the basic facts o f electrodynamics without resorting to field quantization [14] and several provisional theories have been suggested [15]. Unless a Neoclassical theory alters the connection between mechanical quantities and the field quantities in Classical Electromagnetism (e.g., permitting singularities) it cannot hope to recover the basic fact o f localized light beams possessing angular m o m e n t u m . On the other hand, the difficulties involved with a relativistic position operator for massless particles [16] leave open the question o f whether field quantization will avoid or a c c o m o d a t e this purely classical theorem.

ACKNOWLEDGMENTS It is a pleasure to thank W. G. Unruh, D. W. Taylor, J. Reid, N. L. McKay, and E. C. Ihrig for useful criticisms in various stages of this work.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

M. H. L. PRYCE,Proc. Roy. Soc. Ser. A 195 (1948), 62-81. C. MOLLER,Comm. Dublin Inst. Adv. Studies Ser. ,4 5 (1949), 1-42. G. N. FLEMING,Phys. Rev. 137 (1965), B188-B197. H. P. ROBERTSONAND T. W. NOONAN, "Relativity and Cosmology," pp. 141-144, Saunders, Philadelphia, 1968. J. M. JAUCHANDF. ROHRLICH,"The Theory of Photons and Electrons," 2nd ed., p. 34, SpringerVerlag, New York, 1976. F. ROHRLICH,"Classical Charged Particles," pp. 99-101, Addison-Wesley, Reading, Mass., 1965. L. F. ABBOTT,Phys. Rev. D 13 (1976), 2291-2294. D. W. ROBINSON,Helv. Phys. Acta 35 (1962), 98-112. R. A. B~TH, Phys. Rev. 50 (1936), 115-125. C. W. MISNER,K. S. THORN,ANDJ. A. WHEELER,"Gravitation," W. H. Freeman, San Francisco, 1973. Reference [10, p. 158]. Reference [10, p. 69]. C. LANCZOS,"The Variational Principles of Mechanics," 4th ed., pp. 393-394, Univ. of Toronto Press, Toronto, 1970. E. T. JAVN~S,Survey of the present status of neoclassical radiation theory, in "Coherence and Quantum Optics" (L. Mandel and E. Wolf, Eds.), p. 36, Plenum, New York, 1973. P. W. MILONNI,Phys. Rep. C25 (1976), 1-81. A. J. K.~LNAY,Phys. Rev. D 3 (1971), 2357-2363. R. PENROSEAND M. A. H. MAcCALLUM,Phys. Rep. C 6 (1973), pp. 253-256.