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What Is Spin?: Independent Scientific Research Institute Box 30, CH-1211 Geneva-12, Switzerland

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What is spin?

Andr Gsponer Independent Scientic Research Institute Box 30, CH-1211 Geneva-12, Switzerland e-mail: isri@vtx.ch ISRI-03-10.13 February 2, 2008
Abstract This is a late answer to question #79 by R.I. Khrapko, Does plane wave not carry a spin?, Am. J. Phys. 69, 405 (2001), and a complement (on gauge invariance, massive spin 1 and 1 , and massless spin 2 elds) to the 2 paper by H.C. Ohanian, What is spin?, Am. J. Phys. 54, 500505 (1985). In particular, it is conrmed that spin is a classical quantity which can be calculated for any eld using its denition, namely that it is just the non-local part of the conserved angular momentum. This leads to explicit expressions which are bilinear in the elds and which agree with their standard quantum counterparts.

arXiv:physics/0308027v3 [physics.class-ph] 10 Sep 2003

The problem of dening and calculating the intrinsic spin carried by plane waves is a recurring question (see, e.g., [1, 2]) even though it has been denitely claried sixty years ago by J. Humblet who recognized that such a denition is only possible for a wave-packet, that is a superposition of plane waves conned to a bounded four-dimensional region of space-time [3, p.596]. This concept is in agreement with standard text books which make clear that plane waves of innite extent are just mathematical idealizations and that physical waves correspond to elds produced a nite time in the past (and so localized to a nite region of space) [4, p.333], [5, p.99]. More generally, this concept agrees with Julian Schwingers fundamental idea of source theory, namely that a physical description refers only to the nite space-time region which is under the experimenters control [6, p.78]. Finally, in practical calculations, this concept leads to a simple recipe, applicable in many cases, which consists of approximating a wave-packet by a plane wave provided it is assumed that all contributions coming from innitely remote space-time points (i.e., from so-called surface terms) are zero. It is therefore not surprising that the paper of H.C. Ohanian [1], as well as both answers to R.I. Khrapkos question [2], i.e., [7, 8], are essentially based on this concept. These papers focus on the spin 1 electromagnetic eld: What about 1 spin 2 electrons, massive spin 1 particles, and possibly higher spin elds such 1

as spin 2 gravitation? Is there a simple, classical, i.e., non-quantical approach showing that these elds have indeed spin 1 , 1, and 2 respectively? 2 In the case of massless spin 1 particles, i.e., the electromagnetic eld, Ohanian showed that the angular momentum M of a wave-packet can be decomposed into two part, M (r) = L(r) + S, where by use of Gausss theorem one of the parts is found to be non-local, i.e., independent of the spacetime point r. This part, S, therefore corresponds to an intrinsic angular moment carried by the wave-packet as a whole. Written in terms of the electric eld E and the vector potential A, the dS volume density of this spin angular momentum is dV = E A, while the energy dU density is dV = 1 (E E + B B) where B is the magnetic eld. The normalized 2 spin content of a wave-packet is therefore dS EA = 1 dU (E E + B B) 2 (1)

which is essentially Humblets result [3], conrmed by Jackson [4, p.333], Rohrlich [5, p.101], Ohanian [1], and many others. The occurrence of the non-observable and gauge-dependent vector potential A in a measurable quantity deserves an explanation: The crucial step in the derivation of (1) is to express B as B = A in the denition dM = r (E B) of the dV total angular momentum density. This enables the use of the Gauss theorem to discard a surface term and to isolate the non-local term E A where, therefore, A can be considered as an abbreviation for the operational expression ()1 B, [9]. As for the non-invariance of the spin density E A in a gauge transformation A A + , the answer is that by using Maxwells equations and discarding surface terms the additional contribution to the total spin content of a wave-packet can be written either d3 x E or d3 x A . This means that t the denition (1) is consistent with Maxwells equations and the notion of a wavepacket if E = A , that is, only if the scalar potential is such that = 0. This t condition, which is equivalent to the statement = (t), denes a gauge which by Maxwells equations for a source-free electromagnetic eld yields the constraint A = 0. Therefore, while the separation of the total angular-momentum density into spin and orbital parts is not gauge invariant, the denition of spin for a wavepacket implies that the gauge = 0 must be used. This makes this denition unique because for a source-free electromagnetic eld the gauge function is then restricted by the requirement 2 = 0, whose only solution which is regular everywhere is a constant, so that = 0, and (1) is then gauge invariant. In the case of a massive spin 1 eld one has to use Procas equations instead of Maxwells. It turns out that the calculations made in the case of Maxwells 2

eld can be repeated with the difference that the vector elds E and B, as well as the vector and scalar potentials A and , are now complex. The normalized spin content of a Proca wave-packet is then
1 (E A + E A) dS 2 = 1 1 dU (E E + B B ) + 2 m2 ( + A A ) 2

(2)

which reduces to (1) for a real eld of mass m = 0.


1 Turning to the case of a spin 2 eld one meets the difculty that the Dirac eld is not a classical eld in the sense that it is usually not expressed (as the Maxwell and Proca elds) in terms of scalars and three-dimensional vectors. This is the reason why in the paper of Ohanian there is no explicit calculation showing that the 1 spin of Diracs eld for an electron is indeed 2 . Fortunately, there is a formulation of Diracs theory (available since 1929!) which enables such a calculations to be made in a straight forward manner, and which allows a direct comparison with the Maxwell and Proca elds. In this formulation [10] the four complex components of the Dirac eld correspond to a scalar s and a three-dimensional vector v, which are combined into a complex quaternion D = s + v obeying the Dirac-Lanczos equation D = mD i (3)

where is the 4-dimensional gradient operator and a unit vector. This equation, which is fully equivalent to Diracs equation, has many remarkable properties which stem from the fact that it is written in the smallest possible algebra in which the Dirac electron theory can be expressed [11, 12]. In the present application, following the procedure which lead to equations (1) and (2), and without using any quantum mechanical operator or rule such as projecting onto a subspace or calculating a Hilbert space scalar-product, one directly nds that the normalized spin content of an electron wave-packet is given by dS 1 = dU 2 V[DiD ] 1 S[( it D)iD Di( it D )] 2

(4)

1 which has an overall factor 1 explicitly showing that the electron eld has spin 2 . In 2 (3, 4) the operator ( ) means changing the sign of the vector part, i.e., s + v = sv, while S[ ] and V[ ] mean taking the scalar, respectively vector, part of the quaternion expression within the square brackets.

We see that there is a great similarity between expressions (1, 2) and (4). The denominator is always the energy density which is obtained from the scalar part of the Poynting 4-vector, while the numerator is the vector part of the spin angular momentum pseudo 4-vector, i.e., DiD in Lanczoss formulation of Diracs 3

1 theory [13]. Moreover, if 2 i is interpreted as the spin operator, expression (4) is exactly the same as the one that would be written in quantum theory for the ratio between the spin and energy densities.

In principle, the method used here can be generalized to any spin. This requires to know the explicit form of the corresponding energy-momentum tensor from which the analogue of the Poynting vector can be derived and used to calculate the non-local part of the angular momentum. Unfortunately, there is no unique denition for this tensor, and there is much debate even for the spin 2 case which corresponds to gravitational radiation. Nevertheless, using Einsteins energy-momentum pseudotensor, standard text books show how to derive the average rates of loss of angular momentum and energy of a point mass emitting gravitational waves in terms of its reduced mass quadrupole moments Qjk , e.g., [14, p.357] and [15, p.994]. This leads to the expressions ... ... ... dU 2 1 dSj = m2 jkl < Qkn Qnl > , = m2 < Qjk Qjk > (5) dt 5 dt 5 which after division by one another show that gravitational radiation has spin 2, as is conrmed by comparing with the corresponding expressions for the electromag netic angular momentum and energy loss rates of a point charge of 4-velocity Z and electric charge e, see, e.g., [14, p.175, p.206] dSj 2 = e2 jkl < Zk Zl > dt 3 , dU 2 = e2 < Zj Zj > . dt 3 (6)

In conclusion, the spin content of any eld can be dened and calculated without any reference to quantum theory. In particular, if a polarized plane wave is used to calculate expressions (1, 2) or (4) one obtains a result that is 1 or 1 times a 2 normalized unit of angular momentum. If the corresponding wave is attributed to a single quantum such as a photon, a weak interaction boson, or an electron, this unit can be taken as the measured value of . However, in order to consistently deal with elds containing a single or a small number of quanta, the classical theory is not sufcient: It must be supplemented by a quantum interpretation in which the elds themselves become dynamical variables [4, p.751]. Finally, it is clear that spin has nothing to do with a vortex or a whirl which would be carried by a wave or a wave-packet: It is simply the non-local part of the angular momentum that derives from the dynamics implied by the wave-equations dening the eld.

Acknowledgments
The author thanks Prof. J.D. Jackson, Prof. F. Rohrlich, Dr. J.-P. Hurni, and Dr. A.B. van Oosten for valuable comments and suggestions. 4

References
[1] H.C. Ohanian, What is spin?, Am. J. Phys. 54 (1986) 500505. See also: W. Gough, The angular momentum of radiation, Eur. J. Phys. 7 (1986) 8187. [2] R.I. Khrapko, Does plane wave not carry a spin?, Am. J. Phys. 69 (2001) 405. [3] J. Humblet, Sur le moment dimpulsion dune onde lectromagntique, Physica 10 (1943) 585603. [4] J.D. Jackson, Classical Electrodynamics (Wiley, 1975) 848 pp. [5] F. Rohrlich, Classical Charged Particles (Addison-Wesley, 1965) 305 pp. [6] J. Schwinger, Particles, Sources, and Fields, Volume I (Addison-Wesley Pub. Co., 1969) 425 pp. [7] L. Allen and M.J. Padgett, Response to question #79, Am. J. Phys. 70 (2002) 567568. [8] V.B. Yurchenko, Answer to question #79, Am. J. Phys. 70 (2002) 568569. [9] I am indebted to Professor Rohrlich for an illuminating correspondence on this point. See equation (4-182) in his book [5, p.100]. [10] C. Lanczos, Die tensoranalytischen Beziehungen der Diracschen Gleichung, Z. f. Phys. 57 (1929) 447473, 474483, 484493. Reprinted and translated in W.R. Davis et al., eds., Cornelius Lanczos Collected Published Papers With Commentaries (North Carolina State University, Raleigh, 1998) Vol. III pages 2-1133 to 2-1225. http://www.physics.ncsu.edu/lanczos . [11] In Diracs formulation the 4-complex-component electron eld is taken as a 4 1 column vector , and the linear operators are 4 4 complex matrices. In Lanczoss formulation the same 4-complex-component eld is a biquaternion D B M2 (C) C1,2 C3,0 . The linear operators are = = = then linear biquaternions functions of biquaternions, which are isomorphic to the algebra of 4 4 complex matrices M4 (C) M2 (B) C4,1 . = = In both formulations the operator space has 4 4 2 = 32 dimensions over the reals. The difference is that in the Dirac formulation the eld is an abstract 4-component column vector, while in the Lanczos formulation the 5

eld is directly related to the algebraic structure of spacetime because any biquaternion D = s + v is the direct sum of a scalar s and a 3-component vector v. Lanczoss formulation is therefore more suitable than Diracs for studying and demonstrating the classical aspects of the electron eld, and for making comparisons with the Maxwell and Proca elds which are usually expressed in terms of scalars and vectors. Finally, in terms of Clifford algebras, the Dirac eld is a degenerate 8real-component element of the 32-dimensional Clifford algebra C4,1 (i.e., an element of an ideal of that algebra) while the Lanczos eld D is any 8real-component element of the 8-dimensional Clifford algebra C1,2 B, = which is therefore the smallest algebra in which Diracs electron theory can be fully expressed. For more details, see: A. Gsponer and J.-P. Hurni, Comment on formulating and generalizing Diracs, Procas, and Maxwells equations with biquaternions or Clifford numbers, Found. Phys. Lett. 14 (2001) 7785. Available at http://www.arXiv.org/abs/math-ph/0201049 . For an introduction and overview of the modern use of quaternions in classical and quantum physics, see: A. Gsponer and J.-P. Hurni, The physical heritage of Sir W.R. Hamilton, presented at the conference commemorating the sesquicentennial of the invention of quaternions, Trinity College, Dublin (1719 August 1993) 35 pp. Available at http://www.arXiv.org/abs/math-ph/0201058 . [12] A. Gsponer, On the equivalence of the Maxwell and Dirac equations, Int. J. Theor. Phys. 41 (2002) 689694. Available at http://www.arXiv.org/abs/math-ph/0201053 . [13] A. Gsponer and J.-P. Hurni, Lanczos-Einstein-Petiau: From Diracs equation to non-linear wave mechanics, in W.R. Davis et al., eds., Cornelius Lanczos Collected Published Papers With Commentaries (North Carolina State University, Raleigh, 1998) Vol. III pages 2-1248 to 2-1277. http://www.physics.ncsu.edu/lanczos . [14] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields (Pergamon, 1975) 402 pp. [15] C.W. Misner, K.S. Thorn, and J.A. Wheeler, Gravitation (Freeman and Co., San Francisco, 1973) 1279 pp.

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