What happens to spin during the SO(3)->SE(2) contraction? (On spin and extended structures in quantum mechanics)

6 Pages·0.15 MB·English

by  M. Czachor

What happens to spin during the SO(3) SE(2) contraction?
→
(On spin and extended structures in quantum mechanics)
Marek Czachor∗
Research Laboratory of Electronics
Massachusetts Institute of Technology
Cambridge, MA 02139
Andrzej Posiewnik
Instytut Fizyki Teoretycznej i Astrofizyki
Uniwersytet Gdan´ski
ul. Wita Stwosza 57, 80-952 Gdan´sk, Poland
5
9
their momentum. Second, if one considers the action of
9 As is known, a Lie algebra of a little group of a timelike
a four-dimensional nonunitary representation of SE(2)
1 four vector is not equal to so(3) unless spacelike components
on the electromagnetic four-potential, it can be shown
n of thevectorvanish. Inspiteofthisfact thealgebra can still
thatthe“translation”operatorsgenerategaugetransfor-
a be interpreted as the angular momentum algebra, as can be
mations [3,4]. Such results are rather confusing as there
J shown with the explicit example of the Dirac equation. The
is no clear explanation for the continuous deformation
8 angularmomentumcorrespondstotheevenpartoftheDirac
of the angular momentum into position or a generator
1 spin operator. Its eigenvalues in directions perpendicular to
momentumdecaytozeroin theinfinitemomentum/massless of gauge transformations. Of course, it is possible that
1 formasslessfieldsvariousphysicallydifferentobservables
limit. This explains physically why only extremal helicities
v maysatisfythee(2)algebrasincemasslessunitaryrepre-
survive the massless limit. The effect can be treated as a
7 sentations of the Poincar´egroupare generatedby a one-
result of a Lorentz contraction of an extended particle. A
1
natural measure of this extension is introduced for massless dimensional representations of E(2) so that not much
0
1 particlesofanyspin. Itisshownthatsuchparticlescanbein- room is left for different possible algebraic structures.
0 terpretedascircular strings whoseclassical limit isdescribed We would like to show in this paper that it is possi-
5 by Robinson congruence. Finally, as an application of the ble to interpret the contraction so(3) e(2) also as the
→
9 even spin, we formulate the Einstein-Podolsky-Rosen-Bohm continuous deformation of the angular momentum of a
/ Gedankenexperiment for Dirac electrons. relativistic particle. The Lie algebra corresponding to
h
p the intermediate momentum p~, with 0 < p~ < , is
| | ∞
- neitherso(3)nore(2)but, asweshallsee,thereisa nat-
nt I. INTRODUCTION ural way of treating it as the relativistically generalized
a angular momentum algebra.
u Itisknownthatthetransitionfromtherotationgroup To make our analysis as explicit as possible we shall
q
SO(3) to the groupSE(2)of two-dimensionalEuclidean discus the mentioned limits in the context of the Dirac
:
v motions can be obtained as a continuous Ino¨nu-Wigner equation. It will be shown that relativistic spin can be
i contraction of respective Lie algebras [1,2,3]. This con- naturally represented by an “even” spin operator which
X
traction can be explained in physicalterms simply as an reduces to ordinary nonrelativistic spin for a particle at
r
a abstract counterpart of the relativistic Lorentz contrac- rest and whose components in directions perpendicular
tion of a moving body and it can be also shownthat the tomomentumdecaytozeroasthe momentumincreases.
same effect is obtained both in the massless (m 0) In the infinite-momentum/massless limit the operator
and infinite momentum (p~ ) limits. This res→ult is “points” in the momentum direction. The three com-
| | → ∞
intuitively quite natural since a light cone is the large- ponents ofthe new spin commute with the free Hamilto-
momentum asymptoticsofany m>0 masshyperboloid. nian so that one can consider projections of spin in any
Still, even though the SO(3) SE(2) transition is now direction even for a particle moving with some well de-
well understood from a math→ematical point of view, it fined and nonvanishing momentum. This property will
seemsthatnodefinite physicalinterpretationofthis fact beusedforderivingarelativisticversionofthe Einstein-
is generally accepted. Podolsky-Rosen-Bohm paradox for two spin-1/2 parti-
We know that the Lie algebraso(3) correspondsphys- cles. We shall also discuss the relationship between the
ically to an angular momentum. On the other hand evenspinoperatorandthe Pauli-Lubanskivector,show-
there exist at least two physical interpretations of the ing that the infinite-momentum and massless limits are
translation subalgebra of e(2). First, the generators of equivalent for the first one but not for the latter.
translations can be naturally associated with the two- We begin our analysis with a brief summary of prop-
component position operator for massless fields [2] lo- erties of little algebras corresponding to any timelike or
calizing massless particles in a plane perpendicular to null four-momentum.
1
II. LITTLE ALGEBRAS FOR ANY P WITH follow from the m 0 and p~ contractions of (7).
PαPα ≥0 The two contractio→ns are eq|ui|v→ale∞nt and the contracted
algebra is e(2). A reader interested in a geometrical in-
A Lorentz transformation terpretation of these contractions is refered to the book
by Kim and Noz (p. 200 in [3]).
Λ(~µ,~ν)=e−iµ~J~−i~νK~ d=efe−iL(µ~,~ν), (1) The form (6) of the generators leads to a difficulty
in a direct physical interpretation, as L and L are
1 2
where J~ and K~ are, respectively, the generators of rota- not Hermitian for finite dimensional representations of
tions and boosts, leaves a four-vector p unchanged if SL(2,C) and nonvanishing ~p. For photons represented
byafour-potentialtheseoperatorsgenerategaugetrans-
L(~µ,~ν)p=i(~ν p~, ~νp0+~µ ~p)=0. (2) formations. For the rest frame four-momentum of mas-
· ×
sive particles such representations correspond to spin.
Ifp =0theneitherpismasslessandp=0orpisspace-
0 However, what is a physical meaning of this algebra for
like. Both cases can be regarded as physically meaning-
particles that are not at rest? In order to answer this
less. So let us assume that p = 0. (2) is then satisfied
0 6 question we must first understand in what way the spin
if
operator enters relativistic quantum mechanics.
~νp = ~µ ~p. (3)
0
− ×
III. SPIN OF THE ELECTRON AND THE DIRAC
Substituting (3) into (1) we find that the little group is
EQUATION
three-parameter and generated by
L~ =J~ p~ K~. (4) The Uhlenbeck andGoudsmith idea of an internalan-
− p × gularmomentumoftheelectronwasformallyintroduced
0
to quantum mechanics by Pauli in 1927 [5]. Pauli added
(4) satisfies the algebra
anewinteractiontermtotheSchr¨odingerequationinor-
dertoexplainabehaviorofelectronsinamagneticfield.
p~
L~ L~ =iJ~ i (p~ K~). (5) Thisapproachwassuccessfulbut,infact,nootherjustifi-
× − p2 ·
0 cationoftheconceptofspinexistedatthattime. Ayear
later Dirac formulated his relativistic wave equation [6].
Let now ~n = p~/p~, m~ be orthogonal to ~n, m~ = 1, and
| | | | Heassumedthattheequation(1)hasaSchr¨odingerform
~l=~n m~ and let A =A~ m~, A =A~ ~l and A =A~ ~n
for an×y three-compo1nent v·ector A2~. We·get 3 · i∂tΨ = HΨ where H does not contain time derivatives,
(2)factorizesthe Klein-Gordonequation,(3)isrelativis-
tically covariant, and found that no single-component
p~
L1 =J1+K2| | wave function can satisfy such requirements. The ad-
p
0 ditional degrees of freedom present in the multicompo-
p~
L =J K | | (6) nent wave function could be interpreted physically by a
2 2 1
− p0 non-relativistic approximation where positive energy so-
L =J lutions of the Dirac equation were shown to satisfy the
3 3
equation postulated by Pauli. The Pauli spin operator
and (5) implies was found to be an “internal” part of the generator of
rotations restricted to “large” components of a bispinor.
pαp
[L ,L ]=i αL In this way the spin operator was identified, from a
1 2 p2 3
0 mathematicalviewpoint,withthespinorpartofthegen-
[L3,L1]=iL2 (7) erator of rotations.
[L ,L ]=iL . The first difficulty met in relativistic interpretation of
2 3 1
this operator was the fact that, contrary to the nonrela-
Eqs. (6) and (7) mean that the Lie algebra of the little tivistic case, the components of spin were not constants
group is parametrized by p. In the massive case we, in ofmotionevenforafreeparticle(unlessinarestframe).
general, do not obtain so(3); in fact this is the case only In the Heisenberg picture the spin operator of the free
if p~ = 0, that is when we consider the little group of electron satisfies the following precession equation
| |
a rest frame four-momentum. Such a rest frame always
exists for m>0. For m=0 we have S~˙ =~ω S~, (9)
×
p0 =±|p~|. (8) where ~ω = 2γ5~p and only the projection of S~ on the
−
“precession axis” ~p, the helicity, is conserved. The to-
and (6) is indeed the Lie algebra e(2). Note also that
|p~| 1 in the infinite momentum limit and, as ex- tal angular momentum J~ also commutes with the Dirac
p0 → ±
pected, the algebra is again e(2). The same conclusions
2
Hamiltonian and the purely spin part of J~ can be ex- where ǫklm is the three dimensional Levi-Civita symbol.
tracted by ~n J~ = ~n S~, where ~n = ~p/p~. The exis- TheexplicitformofthegeneratorsofthePoincar´egroup
· · | |
tence of the conserved helicity is sufficient for represen- is
tation classification purposes and physical applications P0 =H =α~ ~p+mγ0
in, for instance, the Clebsh-Gordancoefficients problem. ·
P~ =p~
The components of spin in directions other than ~n are
rarely needed. In the last section of this paper we shall J~=~x p~+S~
×
consider one such case, namely the Einstein-Podolsky- i
Rosen-Bohm Gedankenexperiment and the Bell inequal- K~ =tp~ ~xH+ α~
− 2
ity for relativistic electrons.
The Pauli-Lubanski vector is
The general theory of representations of the Poincar´e
group shows that the group possesses two Casimir op- W0 =J~ p~
·
erators: the mass PαP and the square WαW of the 1
α α W~ = (S~H +HS~). (10)
Pauli-Lubanski vector 2
W = 1ǫ MβγPδ In the rest frame we find W0 = 0 and W~ = mS~γ0. W~
α αβγδ
2 is therefore spin times the rest mass operator. In the
where Mβγ are generators of SL(2,C). W commutes subspace of the “large”, positive energy two component
α
with Pβ for all β hence, in particular, with P0. The spinors it is indeed proportional to Pauli’s spin.
Pauli-Lubanski vector appears naturally in the theory In order to understand the physical meaning of the
because it, in fact, generalizes the generators of the lit- Pauli-Lubanskivectorletusmultiply (10)fromthe right
tle group described in Sec. II. The Casimir WαW has by H−1 (this operator is well defined for massive fields;
α
eigenvalues m2j(j+1) where m and j are, respectively, for massless fields it exists in the subspace of nonzero
the rest mass and the modulus of helicity of the irre- momenta). We get
ducible representation in question. 1
In standard approaches to relativistic field theories it S~p =W~ H−1 = (S~+λS~λ) (11)
2
is oftenstatedthatW is the covariantgeneralizationof
α =Π S~Π +Π S~Π
spin. Itssquareisdefinedasthefollowingelementofthe + + − −
enveloping field of the Poincar´eLie algebra whereλisthesignofenergyoperatorandΠ = 1(1 λ)
± 2 ±
WαW projectonpositive (+) or negative( ) energy solutions.
PβPα. TheoperatorS~pisthereforetheso-ca−lledevenpartofthe
β
generatorofrotationsS~. Thedecompositionofoperators
We have, therefore, two possibilities of introducing the
into even and odd parts is well known in first quantized
spin operator in Poincar´e invariant theories. The Pauli-
approachestotheDiracequation[7,8]. Theevenpartsof
Lubanski vector has the advantage of having four con-
operatorsareeffectivelythepartsthatcontributetoaver-
served components and is closely related to the genera-
agevaluesofobservablescalculatedinstatesofadefinite
tors of the little group of a momentum four-vector. The
signofenergy. Theevenspinoperatoroccursnaturallyin
dimension of W is, however, energy times angular mo-
α the contextof Zitterbewegung and the magnetic-moment
mentum so its relationship to Pauli’s 1927 spin is not
operator of the Dirac electron [9,10].
evident.
All these facts suggest that the even spin operator,
which is somehow in between the two notions discussed
above, might be the correct candidate for the electron’s
IV. THE PAULI-LUBANSKI VECTOR AND
spin. All the three components of S~ commute with H
ELECTRON’S SPIN p
so a projection of S~ in any direction is a constant of
p
motion.
Inthefollowingdiscussionwewillworkinthemomen-
The explicit form of S~ for free electrons moving with
tum representation and units are chosen in such a way p
momentum ~p is the following
that c = 1 = h¯. The bispinor parts of generators are
Sαβ = i[γα, γβ] and the generators of SL(2,C) are m2 p~2 im
4 S~ = S~ + | | (~n S~)~n+ ~p ~γ (12)
Mαβ =xαpβ xβpα+Sαβ. p p20 p20 · 2p20 ×
−
where m2 =pαp . Its components
Let S~, α~, J~ and K~ be defined by α
m2 imp~
Skl =ǫklmSm m~ S~ = m~ S~ | |~l ~γ =S
· p p2 · − 2p2 × p1
i 0 0
S0k = 2αk ~l S~ = m2~l S~+ im|p~|m~ ~γ =S (13)
Jm =ǫmklMkl · p p2 · 2p2 × p2
0 0
Km =M0m ~n S~ =~n S~ =S
p p3
· ·
3
satisfy the Lie algebra (7) for any ~p and w = 0 for m = 0. It follows that the
~a
Pauli-Lubanskivector,as opposedto S~ , cannotbe used
p
m2 foraunifiedtreatmentofspininbothmassiveandmass-
[S , S ]=i S
p1 p2 p2 p3 less cases. The same concernsthe seemingly naturaland
0
[Sp3, Sp1]=iSp2 (14) covariantchoiceof(W0/m,W~ /m)astherelativisticspin
[S , S ]=iS . four-vector. This property of W~ explains the following
p2 p3 p1
apparent paradox. The “polarization density matrix”
Itfollowsthattheevenspinoperator(13)isaHermitian (normalizedbyTrρ=2m)fortheDiracultra-relativistic
representation of the algebra (7) although a direct sub- electron can be written as [16]
stitution of generators of (1,0) (0,1) to (6) would not
2 ⊕ 2 1
leadto Hermitianmatrices. The eigenvaluesof~a·S~p,for ρ= 2pµγµ 1−γ5(ζk+ζ~⊥·~γ⊥)
any unit~a, are
(cid:16) (cid:17)
whereζ~ = 2 W~ (W~ ~n)~n , and denotesanaver-
1 (p~ ~a)2+m2 ⊥ mh − · i h·i
s = · (15) age. For helicity eigenstates ζ~ = 0 and ζ equals twice
~a ±2 p ⊥ k
p | 0| the helicity so that
and the corresponding eigenvector in a standard repre- 1
sentation is ρ= pµγµ(1 γ5). (16)
2 ±
Ψ~a =N |p0|+m (|s~a|+ 12~a·~n)w±± m2|~ap·0~n|w∓ (16) is identical to the expression for the density matrix
± p m (s + 1~a ~n)w m~a·~nw oftheDiracneutrino. However,forsuperpositionsofdif-
p| 0|− (cid:0)± | ~a| 2 · ±− 2|p0| ∓(cid:1) ! ferenthelicities ζ~ =0 whichseems to suggestthat even
⊥
p (cid:0) (cid:1) 6
where w satisfies ~n ~σw = w . in the infinite momentum limit some “remains”of spin’s
± ± ±
In the rest frame t·he eigenv±alues s are 1 for any~a. components in directions perpendicular to the momen-
~a ±2
s tend to 0 for both m 0 and p~ , if~a p~=0. tum may be found. Still, there is no contradiction with
~a
The transition from p~→= 0 to|p~|→=∞ defo·rms con- our analysis if we treat S~ and not W~ as the relativistic
| | | | ∞ p
tinuously su(2) into e(2) and the spin operator S~p be- spinoperator. TheremainsarethoseofW~ andnotofS~p.
comes parallel to the momentum direction. The latter The“transversepolarization”vectorζ~ hastobetreated
⊥
phenomenon can be deduced from either (12) and (13)
as a measure of superposition of the two helicites.
or the discussed limits of (15).
The above limits must be understood in terms of Lie
algebra contractions. Physically the infinite momentum V. LORENTZ CONTRACTION... OF WHAT?
limit is more reasonable than m 0. It means that
→
the greatervelocityofaparticle,the less “fuzzy”arethe
The decomposition of operators into even and odd
components of spin in directions perpendicular to mo-
parts can be used for rewriting the Dirac Hamiltonian
mentum. Intuitively, the particle becomes flattened by
in a form which is rather unusual but especially suitable
the Lorentz contraction so that contributions to the in-
forinvestigationofitsultra-relativisticandmasslesslim-
trinsic angular momentum from rotations around direc-
its. Let us consider the angular velocity operator ~ω de-
tions perpendicular to p~ become smaller the greater is
fined by (9). For m = 0 and p~= 0 ~ω does not commute
the flattenning. 6 6
with H. Its even part, commuting with H, is given (in
For m equals exactly zero, two of the three compo-
ordinary units with c=1) by
nents of spin vanish which agrees with the fact that the 6
onlyself-adjointfinitedimensionalrepresentationsofe(2) c2+mc3~γ ~n/p~
are one dimensional. Physically this effect can be again Ω~ = · | |~ω.
c2+m2c4/p~2
explained by the Lorentz contraction: A massless par-
ticle is completely flattenned and its “intrinsic” angular We cansee thatΩ~ reducesto ~ω in bothlimits. AHamil-
momentum can result only from rotations in the plane tonianofaparticlemovingwithvelocity~v =cβ~ cannow
perpendicular to ~p.
be expressed as
Equations (11) and (15) imply that the eigenvalues of
~a W~ are m2c4
· H = 1+ Ω~ S~ =β~−2Ω~ S~ =β~−2Ω~ S~ (17)
c2~p2 · p · · p
1 p
w =p s = 0 (p~ ~a)2+m2 (cid:16) (cid:17)
~a 0 ~a
±2 p0 · where each of the operators appearing in H is even and
| |
p commuting with H. The limiting form H = ~ω S~ is
and the eigenvectors are identical to those of~a S~ . For ·
p characteristicofallmasslessfields,whereforhigherspins
·
W~ the massless and infinite momentum limits are not the equation(9) is still valid, but angularvelocities for a
equivalent. Indeed, let ~a·p~ = 0. Then w~a = ±21m 6= 0 given momentum are smaller the greater the helicity.
4
The new form of the Hamiltonian leads to the follow- TheRobinsoncongruencepictureistypicalofclassical
ing interesting observation[11]. Notice that for massless twistors. Itsuggeststhatclassicalmasslessfieldsmaybe
fields the Hamiltonian can be written in either of the related naturally to classical strings whose radii would
following two forms have to be different for different inertial observers. The
quantized twistor formalism does not have such a picto-
H =ω~ S~ (18) rial representation since even for a spin-0 particle whose
·
momentumis given noworldlineexists,butadditionally
or
because of the difficulties with the relativistic position
operator.
H =~c ~p=~v p~ (19)
· · Thestring-likepictureofmasslessfieldsresultingfrom
where ~v is the velocity operator for a general massless themomentofinertiaformulasandfromtheiragreement
field(cα~ incaseoftheDiracequation)and~c=(~v p~)p~/p~2 with the classical Robinson congruence is a physical in-
is its even part. We recognize here the classicalm·echan- dicationthatafundamentalroleshouldbeplayedinthis
ical rule for a transition from a point-like description to context by the conformal group. Indeed, a transition
theextended-object-likeone: linearmomentumgoesinto from one inertial reference frame to another not only
angularmomentum,linearvelocityintoangularvelocity, transforms the particle’s four-momentum, but simulta-
andviceversa. Thethirdpartofthisrule(mass–moment neously rescales the radius rs of the congruence.
of inertia) can be naturally postulated as follows Finally, the fact that the massive Dirac Hamiltonian
writtenintermsofevenoperatorshasthe“energyofpre-
H =mk~c2 =Ik~ω2. (20) cession”form(17)suggeststhatsomekindofanextended
structure can be associated also with massive spinning
where (20) defines the kinetic mass (m ) and the kinetic
k particles. Structures of this type were constructed ex-
momentofinertia (I )ofthe masslessfield. The explicit
k plicitly by Barut and collaborators [13,14,15].
formofI formasslessfieldsofhelicitys=m n[corre-
k
−
spondingtothe(m,n)spinorrepresentationofSL(2,C)]
is, in ordinary units, VI. AN APPLICATION OF S~P: THE BELL
THEOREM FOR DIRAC’S ELECTRONS
s¯h~p S~
I = · . (21)
k cp~2
Let us consider two electrons with opposite momenta
The equation ~p1 andp~2 = p~1(wechoose,inthisway,acenterofmass
−
reference frame).
Ik =mkrs2 (22) The squared total even spin operator
characteristic, by the way, of circular strings (here with (S~ 1+1 S~ )2 (25)
mass m ) defines some radius which is equal to p1 ⊗ ⊗ p2
k
in the helicity basis is given by the matrix
¯hs
r = (23)
s
|p~| (1+ mp22)1 0 0 0
0
which can be expressed also as an (operator!) form of  0 m21 m21 0 
the “uncertainty principle”  0 mpp202021 mpp202021 0 .
|p~|rs =h¯s. (24)  0 0 0 (1+ mp22)1 
 0 
It is remarkable that this radius occurs also naturally  
where1isthe4 4identitymatrixandp istheenergyof
in the twistor formalism [12]. A twistor is a kind of a oneofthepartic×les. Itseigenvaluesare: 01+m2,2m2 and
“square root” of generators of the Poincar´e group on p2 p2
0 0
a light cone and belongs to a carrier space of a repre- 0. The firsttwocorrespondto the non-relativistictriplet
sentation of the conformal group. It is known that al- stateandthethirdonetothesinglet(degeneraciesofthe
though spin-0 twistors can be represented geometrically eigenvalues are, respectively, 8, 4 and 4). An important
bynullstraightlines,thisdoesnotholdforspin-s,s=0, property of the definition (25) is the usage of squared
twistors[12]. Insteadofthestraightlinewegetacon6gru- even operators (this is not the same as the even part of
enceoftwisting,null,shear-freeworldlines,theso-called the ordinary squared two-particle spin operator).
Robinson congruence. A three-dimensionalprojectionof The singlet state takes in the helicity basis Ψ± the
this congruence consits of circles, whose radii are given usual form
exactly by our formula (23) (cf. the footnote at p. 62
1
in [12]). The circles propagate with velocity of light in Ψ= (Ψ Ψ Ψ Ψ ).
+ − − +
√2 ⊗ − ⊗
the momentum direction and rotate in the right- or left-
handed sense depending on the sign of helicity.
5
Inorderto provethe Belltheoremwemustcalculatethe [8] B. Thaller, The Dirac Equation (Springer,Berlin, 1992).
singlet state average of an analog of the nonrelativistic [9] A. O. Barut and A. J. Bracken, Phys. Rev. D 23, 2454
operator~a ~σ ~b ~σ. Here we find (1981).
· ⊗ · [10] A. O. Barut and A. J. Bracken, Phys. Rev. D 24, 3333
~a S~ ~b S~ 1 m2 (1981).
hΨ| ·s p1 ⊗ ·s p2|Ψi=−4s s (~ak·~bk+ p2~a⊥·~b⊥) [11] M.Czachor,inProblemsinQuantumPhysics;Gdan´sk’87,
| ~a| | ~b| | ~a ~b| 0 edited by L. Kostro et al. (World Scientific, Singapore,
(26) 1988).
[12] R.PenroseandW.Rindler,SpinorsandSpace-Time,vol.2
where the symbols and denote projections on, re-
(Cambridge University Press, Cambridge, 1986).
k ⊥
spectively, the momentum direction and the plane per- [13] A. O. Barut and N. Zhangi, Phys. Rev. Lett. 52, 2009
pendicular to it. For ~a and ~b perpendicular to ~p1 (26) (1984).
equals ~a ~b, the formulaknownfromthe nonrelativistic [14] A. O. Barut and W. D. Thacker, Phys. Rev. D 31, 1386
quantu−mm·echanics,andtheBelltheoremcanbeformu- (1985).
lated. For other directions the formula (26) differs from [15] A. O. Barut, in Conformal Groups and Related Symme-
the nonrelativistic one so might be used for an experi- tries,editedbyA.O.BarutandH.D.Doebner(Springer,
Berlin, 1986).
mental verification of the even spin concept [17].
[16] W. B. Berestetski, E. M. Lifshitz, and L. P. Pitayewski
Relativistic Quantum Theory (Nauka,Moscow 1968).
[17] Crucial role would be played by measurements made on
VII. CONCLUSIONS
massiveparticles;anexperimenttestingtheBellinequality
for pairs of spin-1/2 nuclei is prepared by E. Fry, but the
Wehaveshownthatthealgebraofalittlegroupofany
velocities of the particles will not be probably relativistic.
physical four momentum is isomorphic to the algebra of [18] M. H. L. Pryce, Proc. Roy.Soc. A 195, 62 (1948).
theevenspinoperator. Thisresultexplainsqualitatively [19] H. Bacry, Localizability and Space in Quantum Physics
the fact that massless fields can exist only in extremal (Springer, Berlin, 1988).
helicity states since eigenvalues of the even spin’s com-
ponentsperpendiculartomomentumtendtozeroinboth
infinite momentum and massless limits. A physical ori-
gin of this phenomenon can be explained by the Lorentz
flattenning of the Dirac particle provided the particle is
extended. For massless fields (or ultra-relativistic elec-
trons) the flattenned picture can be naturally associated
withtheclassicalRobinsoncongruenceofnullworldlines,
leading to a string-like classical limit of spinning parti-
cles. In this way we have returned to the old problem
of localization of spinning particles [1,8,18,19] and have
foundanotherargumentfortheirextendedstructureand
usage of noncommuting position operators.
∗
Permanentaddress: Wydzial FizykiTechnicznejiMatem-
atyki Stosowanej, Politechnika Gdan´ska, Gdan´sk, Poland;
Electronic address: [email protected]
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(1974).
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