- Email: [email protected]
Spin and gravitation
ANNALS
OF PHYSICS:
31, 64-87 (1965)
Spin
and HEINZ
Institute
of
Theoretical
Physics,
Gravitation* PAGELS
Department of Physics, California
Stanford
University,
Stanford,
The presence of a gravitational or electromagnetic field can be regarded as altering the symmetries of flat space-time in such a way that the group of translations for states with charge and spin is no longer Abelian, the displacement operator obeying i[D. , DB] = eF,s - !/~Qx&~‘~~. Furthermore, just as invariance under gauge transformations implies current conservation, invariance under general coordinate and general similarity transformations imply that the stress-energy tensor must satisfy T”@lo = 0 and (Tayls - Tm61y),,. = 0, ten conditions which lead to the Einstein equations. It is also possible to construct the gravitational analogue of the electromagnetic current, J, = (~/2)ax6T~~1’, whichjdepends on the way in which the distribution of matter changes, and is divergence-free because of the Einstein equations. In this way we emphasize the dynamical content of symmetry principles, and show that Einstein’s equations essentially follow from covariance alone. I. INTRODUCTION
In this paper we shall investigate the properties of certain classical fields as revealed by breaking the symmetry of groups of transformations (displacemen&) which characterize the physical geometry. ‘Geometry’ is here understood in the senseof Klein’s Erlanger program (1) and Weyl’s extension thereof (Z-4) as being completely characterized by the group of transformations (Klein) or the abstract group of automorphisms ( Weyl). In other words, specific knowledge of the geometry is only accessible through an investigation of the transformation properties of physical states, and in particular t’he transformations of sbatesunder the displacement group. In this way one can emphasize the close connection between symmetry principles and the dynamical interpretabion of geometry as envisioned by Riemann and Einstein. We are primarily concerned here with gravit,ational and electromagnetic fields as classical phenomena and their role in the space-time description of the world. From the point of view of quantum field theory we know that gravitons and photons, becauseof their statistics, t’heir electrical neutrality, and the long range nature of their force, can build up a measurable field strength over macroscopic * This work was supported by the U.S. Air Force through the Air Force Office of Scientific Research Grant AF-AFOSR-62-452. 64
SPIN
.4ND
GRAVITATION
ti.5
regions of space-time. Consequently, as classical fields, they play a particular role in the displacement properties of physical states. The macroscopic, Euclidean geomet’ry is altered by the presence of an electromagnet’ic or gravit,ational field which manifests itself as a specific change in the transformation properties of st,ates under displacements. It, is precisely the nleasure of such a change in rh(l group properties t’hat interests us. If we denote by D,(z) t,he infinitesimal generator of translations at. t’he point s, then the presence of a field is reflect)ed in t’he non-Abelian prop&y of the group, and gravitation can be understood as iPa , Do] ~0. That, electromagnetism compensating fields for groups of gauge transformat6ons is well known (5, 6 ). It is, in fact,, the Lie bracket, i[D, , Da], which provides the measure of the syrrtmetry breaking of flat space and characterizes t,he geometry in t,he sense of Klcill and Weyl. h word is necessary clarifying the status of the particle in this paper. The particle is here seen as e&&sic to the geometry, a test particle, whose spin, charge, and mass are specified by the appropriate operators X, Q, M on thr state /$(.x)) prior to any statement about the geomet’ry. Only after we have specified the spin, charge, and mass of the particle can we, from the translational propcrt ies of t’he state I+(X)) representing the test particle, completely determine the Riemann tensor and the electromagnetic field strength. With the underst,anding that S, Q, and M are to be given in advance one does not, arrive at a diffcrrnt geometry for a different t’est particle. In other words, the specification of the s( atcl l#(zj) is prior to any knowledge of the geometric manifold and the specific geomet,ry emerges in a purely regulative fashion with respect to t,he translatfional properties of t,he state. Thus, as is shown in this paper, knowledge of the geomt+ry is completely given from the symmet,ry properties of t’he state under the group of translations. Adopting this view we will show in Section VI that if one demands a fully covariant, description of t’he state then t)he geometry nmst be determined by the Einstein equat8ions. Nowhere in this paper do we consider the more fundamental problem of finding an int~uhsic charact’erization of spin, charge, or mass in termsofpure geonletry. In this view (geometrodynamics) the geometry is understood as con.stit&ng t,he spin, charge, and mass of the part#icle. The problem of determining such a pllrely geometrical descript.ion of spin is not touched upon in Ihis paper. The rcadrr interest’ed in t’he exposition of t’his problem is referred to refs. 18 and 2/t. The next three sections we present by way of introduction and review. In the first, of these sections we consider the propertics of flat space as characterized by the PoincarB group and following this section we disruss the case of’ a Pparc endowed with electromagnetism as an example of a synnnetJry breaking or PO~Ipensating field. In Section IV we give a brief review of the formal apparatus of spinors in a Riemann space along with the properties of the spin connection or
66
PAGELS
Fock-Ivanenko coefficient. We emphasize, in Section V, the central role that spin has in the manifestation of gravitational phenomena through the translational properties of spin states. A principal result of Section VI is that if one demands the general covariance of physical theories under spin as well as coordinate transformabions then the energy-momentum tensor and the electromagnetic current must satisfy
These condit)ions on the energy-momentum tensor lead to Einstein’s connection between the metric geometry and the distribut,ion of matter. In this manner one can extract dynamical content from symmetry demands, a point emphasized by J. J. Sakurai (6) and DeBroglie (7). In Section VII we discuss the gravitational analogue to the Pauli interaction in electron theory. II.
FLAT
SPACE-TIME
The Poincarh group, the group of translations and rotations in Euclidean four-space, we shall assume describes the geometry of empty space for which the metric tensor is globally Lorentzian and the electromagnetic field vanishes. Our concern is not with the representations of this group and their classification (8) but rather how it characterizes the homogeneity and isot’ropy we associate with flat space-time. Consider a single particle state I+(Z)) an d compare it with the state I+(z))’ obtained from I+(Z)) by transporting it about any closed circuit C. The geometry of space-time is said to be homogeneous if I$> - I+)’ = 0 for all circuits C. The property of homogeneity is thus understood as a global property of the geometry. If P, is the infinitesimal Hermitian generator of translations (23) the homogeneity of space-time implies the Abelian property, i[Pa ) PO] = 0.
(2.1)
The operator, 1 + iP,8xa, physically translates the state 1$(z)) by 6x” relative to some fixed coordinate system (in the special case of flat space this operation is the same as translating the coordinate system by --6x”). From the definition of P, it is easily seen that for any geometric object &+... (2)
(x> . i Pm, ha... (x)1 = ah&. dam We introduce the operator X, ) not in the problematic operator, but as that operator satisfying
status
of a position
SPIN
AKD
GRAVITATIO’U’
(ii
where we have demanded of the uletric teusor g,,, that it be syrnruetric, t h(bCY)II ditiou that the geometry will be Riemaunian. I“oJ. our present c40nsideratioilsI II~ statement of the absence of a gravitational field becomes, iu ter~ns of the 1,ie brackets,
assunling that we perform only linear coordinate trausforn~atious. If we introduce the angular niorneuturn operator (here we do riot musklet spiu ) Lmp= SJ’p - LYpl’o
i “.(i )
whenit is easily seeu from (3..i), (2.1), arid (2.3 ) that IJobsatisfies iLy
, LPBI = g,,L? + gwL,, - !3YPL. - SpaL,
i[Pb , LpyJ = gvorl, - gp2, .
l2.i) (2.8 )
The equat,iou (3.7) is the formal statement of t,hc isotropy of space-tulle which Eve see follows from the conditions of homogeneity and the abseuc*eof gravitatiou. The secoud equation, (2.8), states that P, transform like a vectjor under Loreritz transformations. The task of classifying all the irreduc4ble (unitary) rcpreseutatious of the Poincark group-the group which we undersiaud as describing the symmetries of ernpt,y space-time-is formally ecluivalrni. to finding all the rcpreseutations of the commutation relations (2.1), ( 2.i), and (2.8) by self-adjoiut operators, a problem which has received int crisivc iuvestigat,ion. \nie shall JJOU concern ourselves with how we can break these syuuuetries of houiogeueity and isotropy. 111 the uext sectiou, we assumethe abscuer of gravitation so that the metric tensor is globally Lorentzian. III.
ELECTROXZAG~\;I’TISnT
Let us drop the demand that’ space-time be hornogeucous for certaiu states, rharged states, where --e is the charge. Then the transport of such a state &j about a closed circuit, C will not in general yield the samestate and the right hand side of ( 2.1) will not in geueral be zero. Now deriote 11, as the generator of I railslations for our state I+(x)) where D, is the “kiuetic~” nloment UIJI which sati&es I>” = ill’, where M is t’he mass of the particle. Kext d&e the mtnniutatol i[D, , DO] to be a c-number and denote it as follows:
68
PAGELS
4Da, 41 = eF,dx),
(3.1)
which reflects the non-Abelian character of translations. If we examine t’he properties of the object F&z) it follows from the definition (3.1) and the fact that D, is a translation operator (2.2 ) :
Fa,+ + F,s = 0
(3.2)
!FgP+~+$=o. In obtaining
(3.3)
(3.3) we have used the Jacobi identity
in the form
PA, Pa, Dal1+ Pa, D , Dxll + Pa, [DA, Dorll= 0. Hence F,,B(x) is identified with the electromagnetic field strength at x. The net effect of transporting a charged state about an unknotted circuit C is to change its phase by an amount Aq5 = e
s8
Fap dx” A clx@
(3.4)
where s is the surface spanning C. If C is unknotted, then the single surface s exists. The condition (3.3) then guarantees that A+ will depend only on C and not the particular surface spanning C. We note that to satisfy the commutation relations it is sufficient to make the “minimal” replacement Pa --f Pa - eA,(x)
= D,
(3.5)
so that
F 4 - aAa-aA, ad ap and so (3.2) and (3.3) are automatically fulfilled. The existence of the potential A,(x) now enables us to consistently define the phase change of a changed state for transport about any circuit C via the formula A$ = e
se A,(x)
dx”.
(3.7)
It is in this manner that one seeshow electromagnetism manifests itself as a simple suspensionof the demand of the homogeneity of space-time. Moreover if we understand this suspension of homogeneity as a global property of states under transport about a circuit, that is to say that the change in a state in transporting it about any circuit is for it to undergo a simple change in phase, then we are necessarily led to the existence of the potential A,(z). It is thus the way in which the homogeneity is broken that characterizes electromagnetism. That
SPIN
AND
60
GRAVITATION
invariance under the gauge transformation I#) + cxp (ie+(z ) )jJ/) which can take on different, values at different points implies the exist enw of a compensating field A,(z) is the fundamental recognition of Yang and Mills. To further emphasize the characterization of the electromagnetic field by the Lie bracket i[l), , L&l = eFap let, us consider a charged particle slowly nloving it 1 a region of constant, magnetic field strength, If ?“. Then in t,he statistical interprctation of quantum mechanics (3.1 j would imply that if we performed a series of cxpcriments to determine simultaneously 111e.I’ aiid !/ components of the m~nnwt~~~~ of t,he charged part,iclc WC would discover that. I hc uncert aint iw obeyed the Heiscnberg relation AD,Af), >= eH-’ raker t ban Al~,AD, - 0 which would be 1he case if there mere no field present. or thc part iclc wwc uncharged. We can understand this as follows. Any attempt’ to measure the :c componwt of the momentum of the particle in the field will dist,urh it,s position in 11~~.I direction. I
>= ~‘(l~m/, ?.
(2.S I
Thus there is an uncertainty relation assoeiatcd wi t,h the breaking of the holnogeneity of space. Such uncertainty relat,ions as (3.8 ) refer to 1.1~~kinetic I~LOmenturn, D2 = M2, and not the tot,al moment,um, 1’, , which includes the cnergy-moment~um of the external field and always coinmutes with itself. As a consequenceof (3.1) the condition of the isot,ropy of space-time (2.7) is no longer fulfilled. However we have supposedno gravit,ational field to hc prcscnt and hence there is a consistency condiGon on the operator -L’x and F&X ) &ir:h WCtind using t,he Jacobi idtntii y :
Nothing has been said so far about the dtterniination field. To this end one introdtices the current j,(a)
=
i
[II,
) p’,“]
zz
of the electromagnetic
2;;; :
which is conserved in virtue of (3.2 ) : ( 3.11 )
70
PAGELS
If jm(x) is given satisfying (3.11) along with appropriate boundary conditions then the field F&x) is uniquely specified. We have seen in the example of electromagnetism that knowledge of the properties of states under displacements characterizes the geometry in the sense of Klein and Weyl. A similar situation holds in the case of gravitation; however to expose this in the spirit of this investigation it is necessary to introduce intrinsic spin. To this end we shall give a brief review of the covariant spinor calculus in as much as we require it. For a more complete treat’ment of this subject the reader is referred to refs. 9, IO, and 11. IV.
SPINORS
IN
A RIEMANN
SPACE
We begin this review from an infinitesimal point of view and to start let us consider two points A and B infinitesimally close to one another. We label the coordinates of A and B in some suitable coordinate system xAa and xBa so that t’he difference of coordinates is dx” = xADl - xLIc(. Metric geometry begins by asking: what is the relation between the infinitesimal difference of the coordinates, dxa, and the distance between them? The answer to this question gives us the definition of distance. The simplest relation is arrived at by assuming the linear form ds = ya(x) where the yrr are Clifford the quadratic form
numbers.
In
which is the starting that the ya obey
dx”dg
(4.2)
geometry,
+ -YdxhdJ:)
it is necessary and sufficient.
= 2&4(x)
which defines a Clifford algebra. The existence of the y,(x), generalized to curved space, is germane to every Riemannian The r=(x) can be determined from the metric tensor gas(x) ordinate system up to a general similarity transformation (a mation which can take on different values at different points) d(x)
to
= sdx>,
point of Riemann’s %(xhB(~)
(4.1)
order that this linear form corresponds
(ds)2 = g&x) gdx>
dx”
= S(x>r&>S-‘w.
(4.3) the Dirac matrices space. in a particular cosimilarity transfor: (4.4)
There are thus two kinds of transformations that we can perform quite independently of one anot’her. (A) General coordinate transformations, and (B) General similarity (spin) transformations, each of which has a special
role to play in general relativity. In this review WV are c*ottcc>rtted with ~‘hc transfornlations of t,ype (B) and will consider S(x) to hc unitary, ‘St = 1. The linear fornl (4.1) as a starting point for geontrtry explicsitly ackttow1cdgc.s t~attsfortnatSiotts of both type (A) and (B), while rhc, cluadratic forn1 (1.2 ) hax trivial lrattsforntation propcarties under (H ). Undo a gt>nc>ral sitnilarily lraltsfortttatiott, S’(.r ), the line eletncnt (1.1 ) becotncs r/s’ = S’dsS~~‘, i.c., it (rattsfortns lilac>a tttat ris in the spin space. A further irnportattt property of the line rlrt\~cbr~i c~ntc~grs if w c+ottsider two ittfinitesitnal displac~c~ntent vcc*tors d.rla and tlx2” ;L( I~C poittt .r. The f’ortns we associate with 1hrsc, tlisplac,c,trr~,ttts, f/s, and t/s, , Itar-ci the propcrt y I hat 1hey do not comniutc i[f/S, , f/Q] = u,(j( s) r/s,”
A fh”
( 4..J I
\\-llc~rc~
uao= ; ha , Ysl
c4.t;1
is the spin-knsor and (11~~ h dx2’ = - ds$ A rlst” is the ittfinit esin~al surfarac clcn~nt. detertnined by the displacement vectors. Distance tneasurcnlent s carried out with rods and c*lorka tneasure the quantifies (C/S, )” attd (f/s, )” which of N~IIIN~ c01111nut
(‘, [ids,)‘,
(dSq)2]
=
0
11.7
1
inrplyittg w can nleasure both ~(d,+ and v’(d,s# with arbitrary precisiott. Thr spin tensor u,&.r), like the metric tensor gmp(.r), is an object of geontctric~al ittterest’, although its geonletrical significance is not’ as well understood. In this section we will only consider the formal properties of g,p and cati . In the following sc&ons we will see that gas(x) is detertnined front the tnatter tensor, ?‘,p( .r 1. and gap refers to the spin of the test particle. If one specifics hoth g-a(L) and u,~~(.L‘1 theta tht> ra(r) are completely detrrtnined up to a cahiral transforntatiott 7X’ = ew,a yk . That the metric tensor and the spin tensor are not indepcnden1 ot OIW another is seen from the relation
ilums, ml = g&w - gpPcTa + gaim - gpuax
i -1.8I
which expresses how a change in one tensor, say 6gaa( .u), induces a change in the other, &J,~( 2). The tnetric tensor has the irnportJant property that it remains unchanged by a gchneral siinilarity transfortnation g&p = sg,p s-’
= gap.
(4.9 )
The spin tensor cbhanges according to lJ&fi = S(z)
uafl(2) s-‘(xi
(4.10)
72
PAGELS
while a spinor transforms like d(x) = fJ(xM(x) #t’(x)
= lj+(x>s’(x>.
An essential complication is introduced if we now consider the covariant derivative of geometrical objects with spin transformation properties. In the case of general coordinate transformations one introduces a connection, for which we shall take Christoff el’s choice. (4.12) in order that the covariant derivative the second rank under (A) :
of a vector, Emls, transforms like a tensor of
To obtain the generalization of ordinary differentiation of a spinor in a space which is not flat it is necessary to introduce the Fock-Ivanenko coefficients (the spin connection), r,(x), which are matrices of the same rank and dimension as the TV (12, 13). The covariant derivative of a spinor V& is accordingly defined as
va* = g+
ir*Ij, (4.14)
va++ = g - ilC;Fr+, and the covariant derivative
of a spin matrix
(as a,~ or 7~) by
(4.15) l6 + Ws , &...I. transformation the rmdoes not transform like a spin
VaB,,q... = B+.
Under a general similarity matrix. Its law of transformation is easily obtained from (4.15) by observing that B-0 and VaB,8 transform like spinors:
r,‘(z) = X(z>r,(z>s(z>-l + i F
bY’(x>.
The Fock-Ivanenko coefficients are determined from the g&z) via the ro(x) by demanding that the covariant derivative of ru(z) shall vanish:
V~Y~= yals+ i[n , ~1 = 0. This condition
determines
the I’,(z)
from the y,(x)
up t,o an arbitrary
(4.17) multiple
SPIN AXD
7:;
GRAVITATION
of t,he identit,y, F,‘(Z) = I’,(Z) + a,(x)l. J’or explicit expressionsof ~‘,(:r I in terms of ya consult refs. 14 and 16. Now t.hat the formalism of the covariant differentiation of spinors has been displayed we can formally introduce the corresponding curvature tensor. This is done in Ibe usual fashion by covariantly differentiating the spin mat#rix B, twiw : GA-~& = N,~alx+ i[ra:x , &I + i[r, , Bali] + i[rx , &al - [I’A , [I’a , Ball
t.i.18 )
and then fornling Viva - OaCx. The result is
(,VxV’s - t’sI-~‘xB~ = R,Raba~ + ,I[&
, I&l
i 1.1!1!
where R”flahi,q the Riemann curvature tensor,
formed from the derivat,ives of the Christoffel symbols, and & is a tensor whose components are matrices in spin-space, the spin-rurvat we tensor, fornwd from the derivat ivcs of t,he Fock-Ivanenko coefkkni s
. arx Rax= -an dxh- -dx6+ i[rx 1I’,]. The spin-curvature tensor will be an object of fundanwntal importance for ow investligation. We end this brief review citing several of the formal propertics of of this tensor which follow from it)s definitioll. I’irst we note the antisymmetry ( A:‘.) -- )
REX+ Rxs = 0
which follows from (4.21). It is also czlcarfrom its deficit ion 1hal f& is invariant under a gauge transformation of I‘, :
and also satisfies V&o
-I- V&X
-I- v:sKxa = Q.
t -42-l j
The invariance of i&i under (4.23 j we will S(V is simply invariance wdcr tlrr electromagnetic gauge transformation. 1’. GlLBVIThTION
A. THE
CHARACTERIZATION
ANI,
SPIN
OF THE GEOMETRV
We now ret#urn to our project of characterizing the gconlctry by t,hc t.ransformat,ion properties of states representing the test particlc. The possihilityof
74
PAGELS
both spin and charge is now included in our understanding of physical statesand our interest will be restricted to Fermions. Here we denote such states by j#, q, sj, q = fe, 0, s = L!z>#, or more briefly I+) which under general coordinate transformations transforms like a scalar and under similarity transformations transforms like I$) = X(z)lrc/). Supp ose that, quite generally, the space is endowed with both electjromagnetism and gravitation. The effect of both these symmetry breaking fields is to replace the generator of translations P, of our test particle according to the rule (5.1)
p, ---f pa + I‘,(z)
where I’,(Z) is the Fock-Ivanenko coefficient. This replacement follows from the rule for differentiation (4.14 ). Hence we define for the general generator of translations for our state I$) Da = Pa + r,(x)
(5.2)
where P, obeys the flat space commutation rules [P, , Pp] = 0. When we compute the commutator of this translation operator with itself we have (5.3) or ipa,
Da] = -&Y(X).
The fundamental significance of the spin-curvature tensor thus becomes clear through (5.3) for it is a measure of the non-Abelian character of translations in a space endowed with gravitation and electromagnetism. The relation between the spin-curvature, space-curvature, and electromagnet,ism is obtained from the condition V,yh = 0. By forming the expression (Viva - V~V~)*/B we find using (4.19) : mva - van )YP = YoEL%x+ i[Ezsx) YSI so that
~
, rd = 0.
(5.4)
and to this end we write ^ Rax = - ji~rR~‘a -I- Max
where Mai is an arbitrary spin-matrix, Mah = - ML6 , and a,~ Substituting (5.5) into (5.4) we have ~0 R”px + ; bar , “/aWax + i[Ma~ , ~1 = 0.
(5.5) q
(iP>[m , WI(5.6)
SPIN
Now
AND
GRAVITATIOS
(i/3)[ uac , 701 = goart - gatrcl so that (5.6) ~aR’m + 92 (~&‘n
-
hwornes
R”aax~a ) + i[Jfsx , ral = 0
01’ (5.7)
wax , %I = 0.
Sinye A16x commutes with all t,he “/s it must he a nlultiple of t,he identity d/ax = ?nax(.r)l. Using the Rianchi identities on the spin-curvature (4.24) and t hc l%ianc*hi identities on Rapi : Rng~s,~ + Raasx~y + Ragw ( .5.*5) will then imply that m&~(x) must satisfy
= 0
the differential
equations (5.8)
Conlbining
the result. (Ti..5) and (5.3) we have
WC shall now make the identification of ,ma,3(.r) with elcct~romagnet~ic ticld strength, nl,p = -eeF,b . This must he t’he case if in the limit of vanishing gravitat,ional cff cct s, Rap76 -+ 0, we are to recover tQc rule for displacement’ operat#ors in the presenc~e of an electromagnetic field, i[U, , I!01 ----f ~F’,p(.r ). The Fock-Ivanenko cocfficient~s in virtue of this identification are no longer arbitrary up to a mult iplc of the ident(ity for we now demand that for r&(x ) = 0 that, r, = -~1,t.r )I. The illvariance of fiaS under the transformation I’, - I’, + &#J call now b(, understood as gauge invarianc*cb. Conscqucntly the fundamental connection MWWII spin-space, the liichnlanrl t ever, and thr clectromagnet~ic field is kail = J.&rXeRhCaJ- rl”uBi,
( 5.10 i
wherc~ (JX~and --e arc the spin and charge of the tclst parlic*lc. This intrgrahility condition was first obtained by Schrodinger in 18:3:! ( 1A ). I’ronl fi,,( x) and the Dirac matrices WI can CYIIISI nwt the quantities 12hColil atd Pa;1 This (aan he seen from (.5.-I) for IV{‘ niust hav(a, tlsifig 2gap = { ya , ral, i, R @6X= - :, ( Yt , h
, ral ;
( 5.1 1. 1
The elec,t,roInagrlet,ic part of &?a~is obtained by suhstif utiorl of (.5.11 ) int,o ( L.10 )
76
PAGELS
From (5.12) and (5.11) we see that the necessary and sufficient condition that the space be void of gravitation, Rolbxs = 0, and electromagnetism Fap = 0, is II& = 0. B. EQUATIONS OF MOTION Gravitation makes its appearance as an intimate connection between the spin of a state and its properties under translation. Simply speaking if we displace a spinning particle around a loop in a space for which Raaka + 0 its spin vector will not point in the same direction, a consequence of the nonintegrability of djrection in a Riemannian space. The formal consequence of this observation is that the translation operator, D, , and the Bargmann-Wigner spin vector, Wk = ( -g)“2[Xa/3y]u@‘D”/M, do not commut’e i[De , WA] = i( -g>““[XcyP~]cf’[D,
, Dal/M (5.13)
where M = mass of the particle and we have assumed Fap = 0, as is done for the remainder of this discussion, and used i[Dh , rB] = 0 as follows from (4.17). We now direct our interest to the problem of determining the equations of motion in a curved space for a massive, charged, spinning test particle. A particle here is thought of as moving along a trajectory with a tangent vector dx”/ds which is the velocity of bhe particle. In the case of a chargeless, spinless particle of mass m the equation of motion is the geodesic equation (5.14) In the case that there is an electromagnetic field present and the particle charge e then the geodesic equation becomes the Lorentz force law
has a
(5.15) Now recall that in the case where the particle is endowed with spin the commutator becomes i[P, , Pp] = eF,o ---f - & ; hence one might propose on the strength of induction that the appropriate generalization of the Lorentz force law (5.15) is --
r
c,dxa dx@ = -& udz” d----ds ds a ds = -- 1 C-JRx6a’ %a + eFauC?& 4
(5.16)
SPIN AND GRAVITATION
ii
which in the case of a spinlessparticle, oars= 0, reduces to the Lorentz force lag and in the case where the particle is uncharged and has spin compare to the equations Papapetrou has derived for a classically spinning body.’ In the casewhere the particle is masslessand possesses a spin like I he phot.on or ncut,rino we conclude t’hat its trajectory must he a null geodesic:.This we can see as follows. Relativistic invariance demands that the spin, TV, of a partick propagating at the speed of light must point either parallel or antiparallel lo its direction of propagation. If we next consider the path I’ of a photon in spaccltime then WCknow t,hat the spin of the photon Wa makes a constant angle equal to zero with the trajectory at all points so that the spin vector must bc parallelly propagated along I’, 6W” -zz.
o
i5.17)
6s
Homcver the spin of t’he photon and its direction are related by
where A= f 1
(helicit’y )
so (5.17) implies that P is a geodesic: --6 ax= zx 6s ds ds2
(:;.I!,)
Consequently, it automatically follows from relativistic invariance alone that a massless,spinning parMe must follow a null geodesic. E‘or example, as a photon propagates in a curved spacethe dirtc+ion of its spin will change in acocord with the nonintegrabilit’y of direction in such a space. The direction vector of the propagating photon, dx:“/ds, must follow the spin, pointing in the same dirt!?tion as the spin for all the points along the path. This is only possible if the direct#ion vector satisfies the geodesic equation (V5.19). c. CARTAN
ROTATION
AND GRAVITATIONAL
FLUX
Central to the manifestation of gravitaGon is how it effects the spin of a particle. In the case of electromagnetism the net result of transforming a charged state by transporting it about an infinitesimal unknot,ted circuit C was to change its phase by an amount given by the generator of the transformation, Aphase = 1 See Papapetrou (17). Papapetrou’s recognition that gup = -S,,q and that have the dS,p/ds term.
equation 5.7, p. 258, is the same as (5.16) upon the along a path V~u&dxX/ds) = 0 so that we do not
78
PAGELS
6F, = +eFap dx” A dx’ where dx” A dxB is the surface element spanning C. In the language of Cartan’s exterior calculus we call f = Fpp dx” A dx’ a twoform. Denoting the operation of exterior differentiation by d we have that f is exact f = da, since Fop = d,Afl - apA,, and therefore closed df = 0. The statement df = 0 is simply one-half of Maxwell’s equations (18). If there is also a gravitational field present the spin components of the state will change upon transport about a small circuit C. In the presence of gravitation we make the replacement -eFus ---) i&a so that the appropriate generator of the transformation is now SF, = --J+s&~~
+ eF,p dx” A dx”
where Q@ is the Cartan rotation matrix, the form 0,’ = $$iRa8h~dx’ A dx6 associated with the cycle C (29). We write the above generator as SF, = g + ef where g = -Q&a~RaBX6 dx” A dx6 is the gravitational flux treading the surface dx’ A dx6 spanning C, and ef = eF,B dx” A dx’ is the electromagnetic flux. The Bianchi identities in CarDan’s notation become dD”B = 0 so that one may write the combined Bianchi and one-half of Maxwell’s equations as d&F = dg + e df = 0, which is simply the statement that the total flux t#reading C depends only on C and not on the particular surface spanning C. We also note that at a point x0 we can always introduce a coordinate system so that the gravitational field vanishes at xo , r&(x0) = 0. We cannot, of course, transform away the electromagnetic field by either a coordinate transformation or a similar&y transformation. We still have the freedom to choose our representation so t,hat a,rb(xo) = 0 and from (4.17) we have that [I”,(xo), y~(x0)1 = 0. Hence l?,(xo) is a multiple of the identity so [I’,(zo), IYfi(x,,)] = 0. For this coordinate system and choice of representation the spin curvature tensor takes on the simple form &~(zo) = dJb(x0) - 8$,(x,). So we writ’e a = g + ef; fi = --&& dx* A dx” which defines the spin-curvature form. D. UNCERTAINTY RELATIONS GRAVITATIONAL FIELD
ASSOCIATED
WITH
THE
PRESENCE
OF
A
As in the case of electromagnetism we can also associate a Heisenberg uncertainty relation with the symmetry breaking of the homogeneity of space-time by the presence of a gravitational field. For example if we attempt’ to measure simultaneously the x component of the momentum and the energy of a Fermion at t,he point x in the presence of a gravitabional field then the measure of the difficulty in resolving these measurements will be given by APAE 2 ?,&,gRmozt(x),
SPIN
or,
GRAVITATION
79
ill general,
where
,k.
AND
THE
I), is the kinetic
EINSTEIN
momentum,
that is, D2 = Ill’.
&x~~~~oss
WC have seen in the preceding section that gravitation artd spin enjoy an ittCntatc ittt cwelationship and cuter our considerations together. It is now our iw ttwtiott, in mtsidering the dynamics of the tttrtric*al field y,p( L), to exploit 1his itttcrrclation further. We shall find that the cxistc~tw of spilt and spin trattsfotmtat ions is intimately bound up with the validity of th(s 32nstrin quatiorw RS :t dynattticAa1 basis for gravit~ational theory. (;cttcral relativit)y as originally conwived by Einstein stood upon tmw pritlc4plcs, one essentially a physical princ~iplr, i 1~~other ntat hctrtat~ical : I. The> Id an act ion drttsity C( s) and a corresponding total act,iort L
L = / (tlr)4d:(ErC).
( ci.1 )
80
PAGELS
We shall not, in what follows, make any specific assumptions about the form of J?(Z) but simply use the action principle to display the content of general covariance. Next consider the change in L due to any change in the metric g&x). In accord with the ideas of Einstein we recognize that the generator of these changes T,o( z ), is the symmetric stress-energy tensor:
6, L =
(czz)“; (-g)““T”qZ)sg@3.
s
(6.2)
The tensor Z’“‘(x) describes the distribution of matter of the dynamical system. We now specifically consider those changes in the metric due to a general coordinate transfromation, 2” = xx + 6x’(x). Such a change will induce a change in the metric at the point x: GQaJT(X)
= hQq9~X” +
Qh&JXh
+
(6.3)
Q~&~XX.
The demand of covariance under general coordinate transformations is thus formally equivalent to the demand that 6,L = 0 for changes in the metric induced by coordinate transformations. So we have
0 G 6, L = / (d3$6xx [; and since the 6xX are arbitrary
- a,((-g)“2gh~
G-Q>""TaBa,ga@
TP”>
1
we must have
Tmala = 0,
(6.4)
which constitute a set of four conditions on acceptable T”@(x) for a dynamical system. That general covariance under coordinate transformations imposes these conditions, the conservation of energy-momentum, is, of course well known. Next consider the similarity transformations. The change induced in the total action due to an arbitrary change in the Fock-Ivanenko coefficient 61’,(x) is
6rL
= Tr
s
(dx)4( -g>““17”(x>sr,(x>
(6.5)
where we have written for the generator of these changes s”(x) and have taken the trace to insure invariance of the integral. We now shall introduce certain assumptions in order that we might specify F(x) in terms of the distribution of matter Trrb( x) and the electromagnetic current j,( 2). Write P(x)
= ej”(x)I
+ J”(x)
thus separating ,7”(x) into a part which has trivial transformation spin space, j,(x)r, and a part which does not contain the identity,
(6.6) properties in In the
J”(x).
SPIN
AND
GRAVITATION
81
limit of Aat space E,(s) 3 -dA,(.c)I so t’hat,ja(z) is the generator of that part of *f,(x) which is a multiple of the identity, and hence, J’~(z) is identified wit’h the electromagnetic current. The term J”(X) is the generator of changes in the gravitational part of I’,(z) and so we assume t)hat it depends only on ~‘,o(x). Moreover let us assume that J”(z) transforms like a vector under reflections, depends linearly on T&z), and does not contain derivat,ivcs of Yap(.r) highci than the first. If we also assume, in line with thinking a,8(x j and g4a( x ) as fundamental geometrical objects, that J”(z) is chiral invariant, Jn( L) -+ .Iai.~.) it ye - e2y5”ya , then the most general form for 7 (.e) is J,(a)
= uu,pT’@ + bux*T::a.
( 6.7 i
Consistent, 1vit.h the above assumptions me have the specification terms of the matter distribution and the currents: Ja(z)
= eja(s)
+ aa,flT(z)”
of Jai.?) in
+ buksTaA!“.
i 6.8 )
Now examine the content of the demand for general covariance under similarit? transformations. The change in I’,(s) induced by a general infinitesimal similarity transformat8ion, X(X) = 1 + i&S(z), F’(X) = 1 - i&S(z) is given by
m,(z) = ips, r,] - dds.
(6.9)
The demand that physics be independent of t’he representation of ya we shall understand to imply that 6rL z 0 for changes in ro,induced by similarity transformations. Subst’ituting (6.9) int,o (6.5) we have 0 = 6rL
= Tr = Tr
s s
(cl&---g)lVyibs, (d~)~[d,(
r,l - d, 6s)
(-g)“‘?)&S
+ ,i( -g)“‘J”“[ss, r,]] = Tr = Tr
s s
(dx)*[(
-g)“‘Piu
(6.10) + ,i( -g)““[r,,
J”]]sX
(dx:)4(-g)1’2V~iS
where we have performed the usual partial integrations and used the cyclic property of the trace, Tr (ABC) = Tr (,BC’A). Since 6X is arbitrary (6.10) implies T-,9” = 0 or substituting
(6.8)
82
PAGELS
This can only be true if j”,, = 0 (Taxi* - Tns”),a
(6.11) = 0.
(6.12)
The first condition expresses the conservation of current for the dynamical system which follows from gauge invariance, r, + I’, - a,&.9where SS is a multiple of the identity. The infinitesimal electromagnetic gauge transformations are a subgroup of the group of infinitesimal spin transformations. The second set of equat’ions (6.12) is an additional set of six conditions which covariance imposes on acceptable matter distributions. In the case of flat space these conditions are trivially satisfied since differentiation is commutative and T”‘,s = 0. In 0th er words, in $at space the demand *for the independenceof physics on the representation of the ya is trivial, however in a non&t space the conditions (6.12) are decidedly nontrivial. We are now in a position to state our result. Suppose, in accord with the equivalence principle, that there exists a relationship between the distribution of matt,er and the geometry. Next apply the principle of general covariance which demands physics to be independent of the coordinate system and the representation of the y,(x). This we have seen leads to the conditions (6.4) and (6.12), TuBla= 0 ( Tx”i@- TxB’“),x = 0.
(6.13a) (6.13b)
Moreover assumethat Tab(x) depends only on the geometry in the neighborhood of X. If we now demand that the ten conditions, (6.13), be fulfilled for any matter distribution T,a(x) in any Riemannian space, then, as was shown in a previous paper (21), there is only one relationship between T,@(x) and the geometry satisfying the above conditions and which is the only relation true for every Riemannian space. That relation is given by the system of Einstein equations
~Tap = Rns - ?4g,pR -I- Xg,p
(6.14)
where K and X are phenomenological constants. It is in this manner that the dynamics of the gravitational field emerges from consideration of general covariance to include spin as well as coordinate transformations. The dynamics of the gravitational field essentially follows from symmetry principles. Gravitation can here be understood as a compensating field for both general coordinate transformations and spin transformat’ions, S( 2 ) = exp [~cT&Y*‘(z )I. B. THE
MAXWELL
EQUATIONS
We know that in the case of electromagnetism gauge invariance implies the existence of a conserved current j, = d,F”i which is interpreted as a source of
SPI?;
AND
83
GRAVIT.4TION
the field. Thus we are encouraged to examine the possibility of a conserved wrrent in the case of gravitation. To this end we take t,he divergence of fiiuo =
1&n~RXL,e -
eFap : Vak”~ = ,&q.rRXfag~a - eFapl, = -Yp
( (i.l.5 1
- ejo
where ~‘0 is t.he usual electromagnetic
current
and
( 6.16 1
-3, = -l&sxrRXLaa,a .
From t,he contracted Bianchi identities in the form RXfaPlu =
R” :
and hence by t’he Eiust’ein equat’ions
For every space endowed with matter and for every represeutjatiou of the ye t’here is a current giveu by (6.18). That this current is indeed divergence free is easily seeu from our condition (6.lXb) on acceptable matter distrihutious
It is difficult siIw
to associate a conserved
Tr
ss
quantity
f, tlX” = 0
with
this divergence
coudit iou
( (i.20 )
for any closed surface A S and any !!‘aa(r). This follows from Tr a,@ = 0. The gravitational curreut J, we can interpret as a source for the field 1‘,(z) aud heuc~c its fornial analogy with the electromagnetic current is clear. However the existence of t,his divergence-free current does riot enable us to develop au int cgral formulation of Einstein’s equations in analogy with the integral formulat,iou of ?tlaxwell’s equations. The syst,enl of Einst’ein-Maxwell equations uow has the form Rae -
41g,R
-I- Xgaa = ~Tap
( (i.‘)la 1
84
PAGELS
from which one can determine the I’,(x). The physical significance of gravitational current jfi = (~/2)flkTt”. 1sproblematic; however it might well be the starting point for a quant.um theory of gravitation in virtue of its formal analogy with the electromagnetic current for which there is a developed quantum theory. A quantum theory developed from the properties of the currents 3~ = (~/2)&‘;” and jb and the field I’,(x) could be seen as embracing both gravitation and electromagnetism on a similar footing. One is here faced with the problem of quantizing a system of nonlinear equations (6.21). To remove this nonlinearity by a perturbation approach is simply to remove the interesting feature of gravitation (22). VII.
GRAVITATIONAL
FIELD
INTERACTING
WITH
AN
ELECTRON
In order to examine in greater detail how the gravitational field interacts with the spin of a particle we consider those states which satisfy Dirac’s equation: (r”D,
- MO>1 ti > = 0
(7.1)
where D, is the translation operator previously discussed and Mo is the bare mass. If we operate on the left of (7.1) with r’Dp + MO we obtain (r”r@D,D,e - Mt)/
9 >= 0
(7.2)
or noting ~“7’ = go’ - ia”@ so that y”#Doi
DB = g”‘Da DB - ; oaBIDa, D,d = D2 + ~~=%aa
and hence (7.2) becomes 0” - MO2 - 5 u,~ F 4 2
asp ns R heaB ) +> = 0. >
(7.3)
The term (e/2)cT,pF”” is the familiar Pauli term with the appropriate g factor for the electron. The term ~.(T~~A~R~‘~’ is an additional interaction of the spin with the surrounding gravitational field. It is possible to show using the commutation relations for the ys and the algebraic symmetries of RX”@ that JL&T,,Y.TX~R~‘~~ = +$R where R is the scalar curvature, R = R”, . Using the Einstein equation R = - ( K/~)T $2X where T = T”, , the trace of the stress tensor, and substituting in (7.3) we obtain D2 _ MO2 2 A similar (1963)).
result
has been
e asp F”’ - ; T + ; 1I)) = 0. 2 >
obtained
by A. Peres
(Nuovo
Cimento,
Ser.
(7.4) X,
Vol. 28,
1091
SPIN
AND
85
GRAVITATION
First WC note that for our considerations t’he cosmological term is unessential in (7.6) for we can define a new mass tio2 = MU2 - x/2, assuming X/2 < M” , so tjhatJ (7.4) reads
Sext wc consider the contributions due to the T tc>rnl. Let us suppose that, OIP (aan write for the total stress-energy tensor !!‘(20 = YTF + T”,“Bt where 2’:: rcpresenk the self-stress energy of the elect’ron which we shall take t’o be T self a0
=
:$($rcrDatCI
+
( i.ti
ihsl~k)
and “‘Xtrm’l 1 &S is the stress tensor due to t,he presence of other fields including elet*tromagnetic field, Fab , T ext a&9 =
pl~tromagnetie
If only electromagnet8ism is present Tht”” that, t’ot,al stress-energy is additive T
=
Tself
+
rp;;xt.
)
1.h~ (7.7)
= 0, so we have in t,he approximation +
T”“‘.
Ci.8)
Now l’yi = FaxFaX - ,l~~ga~F,F”’ and hence T”‘” = 0. Ts”f, using the I)irac* equat,ion and choosing for the normalization of the wave function q$ = ?A/,, , is simply , self r
zx
2Jf””
so that the net effect of gravitation-spin irkeraction aualogous interaction is to change the effective mass of the electron t,o U:
( 7.9
)
t,o t.hc Pauli
and Eq. (7.5 ) becomes
We thus identify the gravitational g factor as g = -x, t,he coefficient of ~7”“~~. For laboratory situations t.hese spin-gravitat’ion effects are very smail. VIII.
CONCLUSIONS
In line with thinking of electromagnetism and gravit,ation as compensating fields, t’he presence of such a field can be understood as dest’roying t.he homogeneity and isotropy of flat space-time. The symmetry breaking of these classical fields is revealed in the alteration of the translational transformation properties
86
PAGELS
implies the conservation of currentj”,, = 0. For the case of gravitation under similarity transformations, S = exp [i~,~~~~(z)] r, -+ sr,s-’
leads to a divergenceless
invariance
+ ia,ss-’
current vd”=o 3, = ~KQhcTaA’f,
and the conditions
on T”‘,
CT
-/ala - TYB’a),r = 0.
These six conditions along with TUB1s = 0 and several ot’her assumptions led to the Einstein equations. If physical theories are to possess covariance under general coordinate and general spin transformations we are forced to consider the Einstein equations as the dynamical basis for gravitation. In this way one can point to the close connection between spin, spin transformations, and gravitation. ACKNOWLEDGMENTS The author wishes to thank Professor partment for his vital interest in this work John A. Wheeler of the Princeton Physics RECEIVED
: January
M. Schiffer of the Stanford Mathematics Deand his many critical suggestions, and Professor Department for several points of clarification,
31, 1964 REFERENCES
“Vergleichende Betrachtungen ifber Neuere Geometrische Forschungen.” KLEIN, Erlangen, 1872. 2. H. WEYL, “Classical Groups, ” 2nd ed. Princeton Univ. Press, Princeton, New Jersey, 1946. 9. H. WEYL, “Philosophy of Mathematics and Natural Science,” Translated by 0. Helmer. Atheneum, New York, 1963. 4. H. WEYL, “Gruppentheorie und Quantenmechanik,” 2nd ed. Hirzel, Iteipzig, 1939. 1. F.
SPIN 5. It.
\;TIY.4MA,
I'hys.
AND
GRAVITATION
87
Rev. 101, 1597 (1956).
6. J. J. SAKURAI, Ann. Phys. (S.Y.) 11, 1 (1966). i', LOVIS BR~CLIE, “Introduction to the Vigier Theory of IClcmentary Particles,” Translat,ed by A. J. Knodel. Elsevier, Amsterdam, 1963. 8. 13. P. WAGNER, Anna. Alath. 40, No. 1 (1939). 9. I’. CARTAN, “Lecons sur la Theorie des Spineurs Actualites Scicntifiqucs et Industriellos, No. 643 and 701. Hermann, Paris, 1938. Wiss., I’hysik-Math. JO. L. INFELI) ASI) R. L. VAN DEH WAEILUEN, Sit&~. Pvu~~rss. Llk(~tl. Kl.
380
(1933).
1,. I3.4~12 AXD H. JEHLE, Rev, Mot/. Phys. 26, 711 (l!k%). FOCK AND 1). IVANENKO, C'owpl. Rend. 188, 1470 (l!r29). Foes, 2. I’hysik 57, 2Gl (1929). (;. FLETCHER, Stcozw cimento 8, -Gl (1958). C;. LOOS, :l’,ctoao ci??!ento 30, 901 (1963). fG. 11:. S~CHRODINGER, Sit&r. I’reuss. Akad. Wiss. Kl. 105 (1932). 17. it. PAPAPETROU, PrOC. no{/. &C. 209, 218 (1%1). 18. J. .4. WHEELER AND C. MISNER, dnn. Phys. (,V.l*.) 2, 525 (1957). 19. F. ~.~AJU'AN, “Geometric des Espaces de I~icmann.” Gauthier-\illars, Paris, 1946. 20. H. WEYL, “Classical Groups,” p. 2i3. Princeton Iiniv. Press, Princeton, New Jersey, 1946. 9f. El. It. I'AOELH, J. Muth. Whys. 4, 731 (1963). $2. .J. A. WHEELER, Ann. Php. (A-.1-.j 2, 604 (1957). u?Q. I’. A. M. I)IRAC, “The Principles of Quantum Mechanics,” 3rd cd. Oxford Univ. Press, London, 1917. 94. J. A. WHEELER, “Geometrodynamics, ‘Neutrinos, Gravitation, and Geometry.’ ” Academic Press, New York, 1963. W. 12. \.. IS. \.. 14. J. 15. H. Ii.
OF PHYSICS:
31, 64-87 (1965)
Spin
and HEINZ
Institute
of
Theoretical
Physics,
Gravitation* PAGELS
Department of Physics, California
Stanford
University,
Stanford,
The presence of a gravitational or electromagnetic field can be regarded as altering the symmetries of flat space-time in such a way that the group of translations for states with charge and spin is no longer Abelian, the displacement operator obeying i[D. , DB] = eF,s - !/~Qx&~‘~~. Furthermore, just as invariance under gauge transformations implies current conservation, invariance under general coordinate and general similarity transformations imply that the stress-energy tensor must satisfy T”@lo = 0 and (Tayls - Tm61y),,. = 0, ten conditions which lead to the Einstein equations. It is also possible to construct the gravitational analogue of the electromagnetic current, J, = (~/2)ax6T~~1’, whichjdepends on the way in which the distribution of matter changes, and is divergence-free because of the Einstein equations. In this way we emphasize the dynamical content of symmetry principles, and show that Einstein’s equations essentially follow from covariance alone. I. INTRODUCTION
In this paper we shall investigate the properties of certain classical fields as revealed by breaking the symmetry of groups of transformations (displacemen&) which characterize the physical geometry. ‘Geometry’ is here understood in the senseof Klein’s Erlanger program (1) and Weyl’s extension thereof (Z-4) as being completely characterized by the group of transformations (Klein) or the abstract group of automorphisms ( Weyl). In other words, specific knowledge of the geometry is only accessible through an investigation of the transformation properties of physical states, and in particular t’he transformations of sbatesunder the displacement group. In this way one can emphasize the close connection between symmetry principles and the dynamical interpretabion of geometry as envisioned by Riemann and Einstein. We are primarily concerned here with gravit,ational and electromagnetic fields as classical phenomena and their role in the space-time description of the world. From the point of view of quantum field theory we know that gravitons and photons, becauseof their statistics, t’heir electrical neutrality, and the long range nature of their force, can build up a measurable field strength over macroscopic * This work was supported by the U.S. Air Force through the Air Force Office of Scientific Research Grant AF-AFOSR-62-452. 64
SPIN
.4ND
GRAVITATION
ti.5
regions of space-time. Consequently, as classical fields, they play a particular role in the displacement properties of physical states. The macroscopic, Euclidean geomet’ry is altered by the presence of an electromagnet’ic or gravit,ational field which manifests itself as a specific change in the transformation properties of st,ates under displacements. It, is precisely the nleasure of such a change in rh(l group properties t’hat interests us. If we denote by D,(z) t,he infinitesimal generator of translations at. t’he point s, then the presence of a field is reflect)ed in t’he non-Abelian prop&y of the group, and gravitation can be understood as iPa , Do] ~0. That, electromagnetism compensating fields for groups of gauge transformat6ons is well known (5, 6 ). It is, in fact,, the Lie bracket, i[D, , Da], which provides the measure of the syrrtmetry breaking of flat space and characterizes t,he geometry in t,he sense of Klcill and Weyl. h word is necessary clarifying the status of the particle in this paper. The particle is here seen as e&&sic to the geometry, a test particle, whose spin, charge, and mass are specified by the appropriate operators X, Q, M on thr state /$(.x)) prior to any statement about the geomet’ry. Only after we have specified the spin, charge, and mass of the particle can we, from the translational propcrt ies of t’he state I+(X)) representing the test particle, completely determine the Riemann tensor and the electromagnetic field strength. With the underst,anding that S, Q, and M are to be given in advance one does not, arrive at a diffcrrnt geometry for a different t’est particle. In other words, the specification of the s( atcl l#(zj) is prior to any knowledge of the geometric manifold and the specific geomet,ry emerges in a purely regulative fashion with respect to t,he translatfional properties of t,he state. Thus, as is shown in this paper, knowledge of the geomt+ry is completely given from the symmet,ry properties of t’he state under the group of translations. Adopting this view we will show in Section VI that if one demands a fully covariant, description of t’he state then t)he geometry nmst be determined by the Einstein equat8ions. Nowhere in this paper do we consider the more fundamental problem of finding an int~uhsic charact’erization of spin, charge, or mass in termsofpure geonletry. In this view (geometrodynamics) the geometry is understood as con.stit&ng t,he spin, charge, and mass of the part#icle. The problem of determining such a pllrely geometrical descript.ion of spin is not touched upon in Ihis paper. The rcadrr interest’ed in t’he exposition of t’his problem is referred to refs. 18 and 2/t. The next three sections we present by way of introduction and review. In the first, of these sections we consider the propertics of flat space as characterized by the PoincarB group and following this section we disruss the case of’ a Pparc endowed with electromagnetism as an example of a synnnetJry breaking or PO~Ipensating field. In Section IV we give a brief review of the formal apparatus of spinors in a Riemann space along with the properties of the spin connection or
66
PAGELS
Fock-Ivanenko coefficient. We emphasize, in Section V, the central role that spin has in the manifestation of gravitational phenomena through the translational properties of spin states. A principal result of Section VI is that if one demands the general covariance of physical theories under spin as well as coordinate transformabions then the energy-momentum tensor and the electromagnetic current must satisfy
These condit)ions on the energy-momentum tensor lead to Einstein’s connection between the metric geometry and the distribut,ion of matter. In this manner one can extract dynamical content from symmetry demands, a point emphasized by J. J. Sakurai (6) and DeBroglie (7). In Section VII we discuss the gravitational analogue to the Pauli interaction in electron theory. II.
FLAT
SPACE-TIME
The Poincarh group, the group of translations and rotations in Euclidean four-space, we shall assume describes the geometry of empty space for which the metric tensor is globally Lorentzian and the electromagnetic field vanishes. Our concern is not with the representations of this group and their classification (8) but rather how it characterizes the homogeneity and isot’ropy we associate with flat space-time. Consider a single particle state I+(Z)) an d compare it with the state I+(z))’ obtained from I+(Z)) by transporting it about any closed circuit C. The geometry of space-time is said to be homogeneous if I$> - I+)’ = 0 for all circuits C. The property of homogeneity is thus understood as a global property of the geometry. If P, is the infinitesimal Hermitian generator of translations (23) the homogeneity of space-time implies the Abelian property, i[Pa ) PO] = 0.
(2.1)
The operator, 1 + iP,8xa, physically translates the state 1$(z)) by 6x” relative to some fixed coordinate system (in the special case of flat space this operation is the same as translating the coordinate system by --6x”). From the definition of P, it is easily seen that for any geometric object &+... (2)
(x> . i Pm, ha... (x)1 = ah&. dam We introduce the operator X, ) not in the problematic operator, but as that operator satisfying
status
of a position
SPIN
AKD
GRAVITATIO’U’
(ii
where we have demanded of the uletric teusor g,,, that it be syrnruetric, t h(bCY)II ditiou that the geometry will be Riemaunian. I“oJ. our present c40nsideratioilsI II~ statement of the absence of a gravitational field becomes, iu ter~ns of the 1,ie brackets,
assunling that we perform only linear coordinate trausforn~atious. If we introduce the angular niorneuturn operator (here we do riot musklet spiu ) Lmp= SJ’p - LYpl’o
i “.(i )
whenit is easily seeu from (3..i), (2.1), arid (2.3 ) that IJobsatisfies iLy
, LPBI = g,,L? + gwL,, - !3YPL. - SpaL,
i[Pb , LpyJ = gvorl, - gp2, .
l2.i) (2.8 )
The equat,iou (3.7) is the formal statement of t,hc isotropy of space-tulle which Eve see follows from the conditions of homogeneity and the abseuc*eof gravitatiou. The secoud equation, (2.8), states that P, transform like a vectjor under Loreritz transformations. The task of classifying all the irreduc4ble (unitary) rcpreseutatious of the Poincark group-the group which we undersiaud as describing the symmetries of ernpt,y space-time-is formally ecluivalrni. to finding all the rcpreseutations of the commutation relations (2.1), ( 2.i), and (2.8) by self-adjoiut operators, a problem which has received int crisivc iuvestigat,ion. \nie shall JJOU concern ourselves with how we can break these syuuuetries of houiogeueity and isotropy. 111 the uext sectiou, we assumethe abscuer of gravitation so that the metric tensor is globally Lorentzian. III.
ELECTROXZAG~\;I’TISnT
Let us drop the demand that’ space-time be hornogeucous for certaiu states, rharged states, where --e is the charge. Then the transport of such a state &j about a closed circuit, C will not in general yield the samestate and the right hand side of ( 2.1) will not in geueral be zero. Now deriote 11, as the generator of I railslations for our state I+(x)) where D, is the “kiuetic~” nloment UIJI which sati&es I>” = ill’, where M is t’he mass of the particle. Kext d&e the mtnniutatol i[D, , DO] to be a c-number and denote it as follows:
68
PAGELS
4Da, 41 = eF,dx),
(3.1)
which reflects the non-Abelian character of translations. If we examine t’he properties of the object F&z) it follows from the definition (3.1) and the fact that D, is a translation operator (2.2 ) :
Fa,+ + F,s = 0
(3.2)
!FgP+~+$=o. In obtaining
(3.3)
(3.3) we have used the Jacobi identity
in the form
PA, Pa, Dal1+ Pa, D , Dxll + Pa, [DA, Dorll= 0. Hence F,,B(x) is identified with the electromagnetic field strength at x. The net effect of transporting a charged state about an unknotted circuit C is to change its phase by an amount Aq5 = e
s8
Fap dx” A clx@
(3.4)
where s is the surface spanning C. If C is unknotted, then the single surface s exists. The condition (3.3) then guarantees that A+ will depend only on C and not the particular surface spanning C. We note that to satisfy the commutation relations it is sufficient to make the “minimal” replacement Pa --f Pa - eA,(x)
= D,
(3.5)
so that
F 4 - aAa-aA, ad ap and so (3.2) and (3.3) are automatically fulfilled. The existence of the potential A,(x) now enables us to consistently define the phase change of a changed state for transport about any circuit C via the formula A$ = e
se A,(x)
dx”.
(3.7)
It is in this manner that one seeshow electromagnetism manifests itself as a simple suspensionof the demand of the homogeneity of space-time. Moreover if we understand this suspension of homogeneity as a global property of states under transport about a circuit, that is to say that the change in a state in transporting it about any circuit is for it to undergo a simple change in phase, then we are necessarily led to the existence of the potential A,(z). It is thus the way in which the homogeneity is broken that characterizes electromagnetism. That
SPIN
AND
60
GRAVITATION
invariance under the gauge transformation I#) + cxp (ie+(z ) )jJ/) which can take on different, values at different points implies the exist enw of a compensating field A,(z) is the fundamental recognition of Yang and Mills. To further emphasize the characterization of the electromagnetic field by the Lie bracket i[l), , L&l = eFap let, us consider a charged particle slowly nloving it 1 a region of constant, magnetic field strength, If ?“. Then in t,he statistical interprctation of quantum mechanics (3.1 j would imply that if we performed a series of cxpcriments to determine simultaneously 111e.I’ aiid !/ components of the m~nnwt~~~~ of t,he charged part,iclc WC would discover that. I hc uncert aint iw obeyed the Heiscnberg relation AD,Af), >= eH-’ raker t ban Al~,AD, - 0 which would be 1he case if there mere no field present. or thc part iclc wwc uncharged. We can understand this as follows. Any attempt’ to measure the :c componwt of the momentum of the particle in the field will dist,urh it,s position in 11~~.I direction. I
>= ~‘(l~m/, ?.
(2.S I
Thus there is an uncertainty relation assoeiatcd wi t,h the breaking of the holnogeneity of space. Such uncertainty relat,ions as (3.8 ) refer to 1.1~~kinetic I~LOmenturn, D2 = M2, and not the tot,al moment,um, 1’, , which includes the cnergy-moment~um of the external field and always coinmutes with itself. As a consequenceof (3.1) the condition of the isot,ropy of space-time (2.7) is no longer fulfilled. However we have supposedno gravit,ational field to hc prcscnt and hence there is a consistency condiGon on the operator -L’x and F&X ) &ir:h WCtind using t,he Jacobi idtntii y :
Nothing has been said so far about the dtterniination field. To this end one introdtices the current j,(a)
=
i
[II,
) p’,“]
zz
of the electromagnetic
2;;; :
which is conserved in virtue of (3.2 ) : ( 3.11 )
70
PAGELS
If jm(x) is given satisfying (3.11) along with appropriate boundary conditions then the field F&x) is uniquely specified. We have seen in the example of electromagnetism that knowledge of the properties of states under displacements characterizes the geometry in the sense of Klein and Weyl. A similar situation holds in the case of gravitation; however to expose this in the spirit of this investigation it is necessary to introduce intrinsic spin. To this end we shall give a brief review of the covariant spinor calculus in as much as we require it. For a more complete treat’ment of this subject the reader is referred to refs. 9, IO, and 11. IV.
SPINORS
IN
A RIEMANN
SPACE
We begin this review from an infinitesimal point of view and to start let us consider two points A and B infinitesimally close to one another. We label the coordinates of A and B in some suitable coordinate system xAa and xBa so that t’he difference of coordinates is dx” = xADl - xLIc(. Metric geometry begins by asking: what is the relation between the infinitesimal difference of the coordinates, dxa, and the distance between them? The answer to this question gives us the definition of distance. The simplest relation is arrived at by assuming the linear form ds = ya(x) where the yrr are Clifford the quadratic form
numbers.
In
which is the starting that the ya obey
dx”dg
(4.2)
geometry,
+ -YdxhdJ:)
it is necessary and sufficient.
= 2&4(x)
which defines a Clifford algebra. The existence of the y,(x), generalized to curved space, is germane to every Riemannian The r=(x) can be determined from the metric tensor gas(x) ordinate system up to a general similarity transformation (a mation which can take on different values at different points) d(x)
to
= sdx>,
point of Riemann’s %(xhB(~)
(4.1)
order that this linear form corresponds
(ds)2 = g&x) gdx>
dx”
= S(x>r&>S-‘w.
(4.3) the Dirac matrices space. in a particular cosimilarity transfor: (4.4)
There are thus two kinds of transformations that we can perform quite independently of one anot’her. (A) General coordinate transformations, and (B) General similarity (spin) transformations, each of which has a special
role to play in general relativity. In this review WV are c*ottcc>rtted with ~‘hc transfornlations of t,ype (B) and will consider S(x) to hc unitary, ‘St = 1. The linear fornl (4.1) as a starting point for geontrtry explicsitly ackttow1cdgc.s t~attsfortnatSiotts of both type (A) and (B), while rhc, cluadratic forn1 (1.2 ) hax trivial lrattsforntation propcarties under (H ). Undo a gt>nc>ral sitnilarily lraltsfortttatiott, S’(.r ), the line eletncnt (1.1 ) becotncs r/s’ = S’dsS~~‘, i.c., it (rattsfortns lilac>a tttat ris in the spin space. A further irnportattt property of the line rlrt\~cbr~i c~ntc~grs if w c+ottsider two ittfinitesitnal displac~c~ntent vcc*tors d.rla and tlx2” ;L( I~C poittt .r. The f’ortns we associate with 1hrsc, tlisplac,c,trr~,ttts, f/s, and t/s, , Itar-ci the propcrt y I hat 1hey do not comniutc i[f/S, , f/Q] = u,(j( s) r/s,”
A fh”
( 4..J I
\\-llc~rc~
uao= ; ha , Ysl
c4.t;1
is the spin-knsor and (11~~ h dx2’ = - ds$ A rlst” is the ittfinit esin~al surfarac clcn~nt. detertnined by the displacement vectors. Distance tneasurcnlent s carried out with rods and c*lorka tneasure the quantifies (C/S, )” attd (f/s, )” which of N~IIIN~ c01111nut
(‘, [ids,)‘,
(dSq)2]
=
0
11.7
1
inrplyittg w can nleasure both ~(d,+ and v’(d,s# with arbitrary precisiott. Thr spin tensor u,&.r), like the metric tensor gmp(.r), is an object of geontctric~al ittterest’, although its geonletrical significance is not’ as well understood. In this section we will only consider the formal properties of g,p and cati . In the following sc&ons we will see that gas(x) is detertnined front the tnatter tensor, ?‘,p( .r 1. and gap refers to the spin of the test particle. If one specifics hoth g-a(L) and u,~~(.L‘1 theta tht> ra(r) are completely detrrtnined up to a cahiral transforntatiott 7X’ = ew,a yk . That the metric tensor and the spin tensor are not indepcnden1 ot OIW another is seen from the relation
ilums, ml = g&w - gpPcTa + gaim - gpuax
i -1.8I
which expresses how a change in one tensor, say 6gaa( .u), induces a change in the other, &J,~( 2). The tnetric tensor has the irnportJant property that it remains unchanged by a gchneral siinilarity transfortnation g&p = sg,p s-’
= gap.
(4.9 )
The spin tensor cbhanges according to lJ&fi = S(z)
uafl(2) s-‘(xi
(4.10)
72
PAGELS
while a spinor transforms like d(x) = fJ(xM(x) #t’(x)
= lj+(x>s’(x>.
An essential complication is introduced if we now consider the covariant derivative of geometrical objects with spin transformation properties. In the case of general coordinate transformations one introduces a connection, for which we shall take Christoff el’s choice. (4.12) in order that the covariant derivative the second rank under (A) :
of a vector, Emls, transforms like a tensor of
To obtain the generalization of ordinary differentiation of a spinor in a space which is not flat it is necessary to introduce the Fock-Ivanenko coefficients (the spin connection), r,(x), which are matrices of the same rank and dimension as the TV (12, 13). The covariant derivative of a spinor V& is accordingly defined as
va* = g+
ir*Ij, (4.14)
va++ = g - ilC;Fr+, and the covariant derivative
of a spin matrix
(as a,~ or 7~) by
(4.15) l6 + Ws , &...I. transformation the rmdoes not transform like a spin
VaB,,q... = B+.
Under a general similarity matrix. Its law of transformation is easily obtained from (4.15) by observing that B-0 and VaB,8 transform like spinors:
r,‘(z) = X(z>r,(z>s(z>-l + i F
bY’(x>.
The Fock-Ivanenko coefficients are determined from the g&z) via the ro(x) by demanding that the covariant derivative of ru(z) shall vanish:
V~Y~= yals+ i[n , ~1 = 0. This condition
determines
the I’,(z)
from the y,(x)
up t,o an arbitrary
(4.17) multiple
SPIN AXD
7:;
GRAVITATION
of t,he identit,y, F,‘(Z) = I’,(Z) + a,(x)l. J’or explicit expressionsof ~‘,(:r I in terms of ya consult refs. 14 and 16. Now t.hat the formalism of the covariant differentiation of spinors has been displayed we can formally introduce the corresponding curvature tensor. This is done in Ibe usual fashion by covariantly differentiating the spin mat#rix B, twiw : GA-~& = N,~alx+ i[ra:x , &I + i[r, , Bali] + i[rx , &al - [I’A , [I’a , Ball
t.i.18 )
and then fornling Viva - OaCx. The result is
(,VxV’s - t’sI-~‘xB~ = R,Raba~ + ,I[&
, I&l
i 1.1!1!
where R”flahi,q the Riemann curvature tensor,
formed from the derivat,ives of the Christoffel symbols, and & is a tensor whose components are matrices in spin-space, the spin-rurvat we tensor, fornwd from the derivat ivcs of t,he Fock-Ivanenko coefkkni s
. arx Rax= -an dxh- -dx6+ i[rx 1I’,]. The spin-curvature tensor will be an object of fundanwntal importance for ow investligation. We end this brief review citing several of the formal propertics of of this tensor which follow from it)s definitioll. I’irst we note the antisymmetry ( A:‘.) -- )
REX+ Rxs = 0
which follows from (4.21). It is also czlcarfrom its deficit ion 1hal f& is invariant under a gauge transformation of I‘, :
and also satisfies V&o
-I- V&X
-I- v:sKxa = Q.
t -42-l j
The invariance of i&i under (4.23 j we will S(V is simply invariance wdcr tlrr electromagnetic gauge transformation. 1’. GlLBVIThTION
A. THE
CHARACTERIZATION
ANI,
SPIN
OF THE GEOMETRV
We now ret#urn to our project of characterizing the gconlctry by t,hc t.ransformat,ion properties of states representing the test particlc. The possihilityof
74
PAGELS
both spin and charge is now included in our understanding of physical statesand our interest will be restricted to Fermions. Here we denote such states by j#, q, sj, q = fe, 0, s = L!z>#, or more briefly I+) which under general coordinate transformations transforms like a scalar and under similarity transformations transforms like I$) = X(z)lrc/). Supp ose that, quite generally, the space is endowed with both electjromagnetism and gravitation. The effect of both these symmetry breaking fields is to replace the generator of translations P, of our test particle according to the rule (5.1)
p, ---f pa + I‘,(z)
where I’,(Z) is the Fock-Ivanenko coefficient. This replacement follows from the rule for differentiation (4.14 ). Hence we define for the general generator of translations for our state I$) Da = Pa + r,(x)
(5.2)
where P, obeys the flat space commutation rules [P, , Pp] = 0. When we compute the commutator of this translation operator with itself we have (5.3) or ipa,
Da] = -&Y(X).
The fundamental significance of the spin-curvature tensor thus becomes clear through (5.3) for it is a measure of the non-Abelian character of translations in a space endowed with gravitation and electromagnetism. The relation between the spin-curvature, space-curvature, and electromagnet,ism is obtained from the condition V,yh = 0. By forming the expression (Viva - V~V~)*/B we find using (4.19) : mva - van )YP = YoEL%x+ i[Ezsx) YSI so that
~
, rd = 0.
(5.4)
and to this end we write ^ Rax = - ji~rR~‘a -I- Max
where Mai is an arbitrary spin-matrix, Mah = - ML6 , and a,~ Substituting (5.5) into (5.4) we have ~0 R”px + ; bar , “/aWax + i[Ma~ , ~1 = 0.
(5.5) q
(iP>[m , WI(5.6)
SPIN
Now
AND
GRAVITATIOS
(i/3)[ uac , 701 = goart - gatrcl so that (5.6) ~aR’m + 92 (~&‘n
-
hwornes
R”aax~a ) + i[Jfsx , ral = 0
01’ (5.7)
wax , %I = 0.
Sinye A16x commutes with all t,he “/s it must he a nlultiple of t,he identity d/ax = ?nax(.r)l. Using the Rianchi identities on the spin-curvature (4.24) and t hc l%ianc*hi identities on Rapi : Rng~s,~ + Raasx~y + Ragw ( .5.*5) will then imply that m&~(x) must satisfy
= 0
the differential
equations (5.8)
Conlbining
the result. (Ti..5) and (5.3) we have
WC shall now make the identification of ,ma,3(.r) with elcct~romagnet~ic ticld strength, nl,p = -eeF,b . This must he t’he case if in the limit of vanishing gravitat,ional cff cct s, Rap76 -+ 0, we are to recover tQc rule for displacement’ operat#ors in the presenc~e of an electromagnetic field, i[U, , I!01 ----f ~F’,p(.r ). The Fock-Ivanenko cocfficient~s in virtue of this identification are no longer arbitrary up to a mult iplc of the ident(ity for we now demand that for r&(x ) = 0 that, r, = -~1,t.r )I. The illvariance of fiaS under the transformation I’, - I’, + &#J call now b(, understood as gauge invarianc*cb. Conscqucntly the fundamental connection MWWII spin-space, the liichnlanrl t ever, and thr clectromagnet~ic field is kail = J.&rXeRhCaJ- rl”uBi,
( 5.10 i
wherc~ (JX~and --e arc the spin and charge of the tclst parlic*lc. This intrgrahility condition was first obtained by Schrodinger in 18:3:! ( 1A ). I’ronl fi,,( x) and the Dirac matrices WI can CYIIISI nwt the quantities 12hColil atd Pa;1 This (aan he seen from (.5.-I) for IV{‘ niust hav(a, tlsifig 2gap = { ya , ral, i, R @6X= - :, ( Yt , h
, ral ;
( 5.1 1. 1
The elec,t,roInagrlet,ic part of &?a~is obtained by suhstif utiorl of (.5.11 ) int,o ( L.10 )
76
PAGELS
From (5.12) and (5.11) we see that the necessary and sufficient condition that the space be void of gravitation, Rolbxs = 0, and electromagnetism Fap = 0, is II& = 0. B. EQUATIONS OF MOTION Gravitation makes its appearance as an intimate connection between the spin of a state and its properties under translation. Simply speaking if we displace a spinning particle around a loop in a space for which Raaka + 0 its spin vector will not point in the same direction, a consequence of the nonintegrability of djrection in a Riemannian space. The formal consequence of this observation is that the translation operator, D, , and the Bargmann-Wigner spin vector, Wk = ( -g)“2[Xa/3y]u@‘D”/M, do not commut’e i[De , WA] = i( -g>““[XcyP~]cf’[D,
, Dal/M (5.13)
where M = mass of the particle and we have assumed Fap = 0, as is done for the remainder of this discussion, and used i[Dh , rB] = 0 as follows from (4.17). We now direct our interest to the problem of determining the equations of motion in a curved space for a massive, charged, spinning test particle. A particle here is thought of as moving along a trajectory with a tangent vector dx”/ds which is the velocity of bhe particle. In the case of a chargeless, spinless particle of mass m the equation of motion is the geodesic equation (5.14) In the case that there is an electromagnetic field present and the particle charge e then the geodesic equation becomes the Lorentz force law
has a
(5.15) Now recall that in the case where the particle is endowed with spin the commutator becomes i[P, , Pp] = eF,o ---f - & ; hence one might propose on the strength of induction that the appropriate generalization of the Lorentz force law (5.15) is --
r
c,dxa dx@ = -& udz” d----ds ds a ds = -- 1 C-JRx6a’ %a + eFauC?& 4
(5.16)
SPIN AND GRAVITATION
ii
which in the case of a spinlessparticle, oars= 0, reduces to the Lorentz force lag and in the case where the particle is uncharged and has spin compare to the equations Papapetrou has derived for a classically spinning body.’ In the casewhere the particle is masslessand possesses a spin like I he phot.on or ncut,rino we conclude t’hat its trajectory must he a null geodesic:.This we can see as follows. Relativistic invariance demands that the spin, TV, of a partick propagating at the speed of light must point either parallel or antiparallel lo its direction of propagation. If we next consider the path I’ of a photon in spaccltime then WCknow t,hat the spin of the photon Wa makes a constant angle equal to zero with the trajectory at all points so that the spin vector must bc parallelly propagated along I’, 6W” -zz.
o
i5.17)
6s
Homcver the spin of t’he photon and its direction are related by
where A= f 1
(helicit’y )
so (5.17) implies that P is a geodesic: --6 ax= zx 6s ds ds2
(:;.I!,)
Consequently, it automatically follows from relativistic invariance alone that a massless,spinning parMe must follow a null geodesic. E‘or example, as a photon propagates in a curved spacethe dirtc+ion of its spin will change in acocord with the nonintegrabilit’y of direction in such a space. The direction vector of the propagating photon, dx:“/ds, must follow the spin, pointing in the same dirt!?tion as the spin for all the points along the path. This is only possible if the direct#ion vector satisfies the geodesic equation (V5.19). c. CARTAN
ROTATION
AND GRAVITATIONAL
FLUX
Central to the manifestation of gravitaGon is how it effects the spin of a particle. In the case of electromagnetism the net result of transforming a charged state by transporting it about an infinitesimal unknot,ted circuit C was to change its phase by an amount given by the generator of the transformation, Aphase = 1 See Papapetrou (17). Papapetrou’s recognition that gup = -S,,q and that have the dS,p/ds term.
equation 5.7, p. 258, is the same as (5.16) upon the along a path V~u&dxX/ds) = 0 so that we do not
78
PAGELS
6F, = +eFap dx” A dx’ where dx” A dxB is the surface element spanning C. In the language of Cartan’s exterior calculus we call f = Fpp dx” A dx’ a twoform. Denoting the operation of exterior differentiation by d we have that f is exact f = da, since Fop = d,Afl - apA,, and therefore closed df = 0. The statement df = 0 is simply one-half of Maxwell’s equations (18). If there is also a gravitational field present the spin components of the state will change upon transport about a small circuit C. In the presence of gravitation we make the replacement -eFus ---) i&a so that the appropriate generator of the transformation is now SF, = --J+s&~~
+ eF,p dx” A dx”
where Q@ is the Cartan rotation matrix, the form 0,’ = $$iRa8h~dx’ A dx6 associated with the cycle C (29). We write the above generator as SF, = g + ef where g = -Q&a~RaBX6 dx” A dx6 is the gravitational flux treading the surface dx’ A dx6 spanning C, and ef = eF,B dx” A dx’ is the electromagnetic flux. The Bianchi identities in CarDan’s notation become dD”B = 0 so that one may write the combined Bianchi and one-half of Maxwell’s equations as d&F = dg + e df = 0, which is simply the statement that the total flux t#reading C depends only on C and not on the particular surface spanning C. We also note that at a point x0 we can always introduce a coordinate system so that the gravitational field vanishes at xo , r&(x0) = 0. We cannot, of course, transform away the electromagnetic field by either a coordinate transformation or a similar&y transformation. We still have the freedom to choose our representation so t,hat a,rb(xo) = 0 and from (4.17) we have that [I”,(xo), y~(x0)1 = 0. Hence l?,(xo) is a multiple of the identity so [I’,(zo), IYfi(x,,)] = 0. For this coordinate system and choice of representation the spin curvature tensor takes on the simple form &~(zo) = dJb(x0) - 8$,(x,). So we writ’e a = g + ef; fi = --&& dx* A dx” which defines the spin-curvature form. D. UNCERTAINTY RELATIONS GRAVITATIONAL FIELD
ASSOCIATED
WITH
THE
PRESENCE
OF
A
As in the case of electromagnetism we can also associate a Heisenberg uncertainty relation with the symmetry breaking of the homogeneity of space-time by the presence of a gravitational field. For example if we attempt’ to measure simultaneously the x component of the momentum and the energy of a Fermion at t,he point x in the presence of a gravitabional field then the measure of the difficulty in resolving these measurements will be given by APAE 2 ?,&,gRmozt(x),
SPIN
or,
GRAVITATION
79
ill general,
where
,k.
AND
THE
I), is the kinetic
EINSTEIN
momentum,
that is, D2 = Ill’.
&x~~~~oss
WC have seen in the preceding section that gravitation artd spin enjoy an ittCntatc ittt cwelationship and cuter our considerations together. It is now our iw ttwtiott, in mtsidering the dynamics of the tttrtric*al field y,p( L), to exploit 1his itttcrrclation further. We shall find that the cxistc~tw of spilt and spin trattsfotmtat ions is intimately bound up with the validity of th(s 32nstrin quatiorw RS :t dynattticAa1 basis for gravit~ational theory. (;cttcral relativit)y as originally conwived by Einstein stood upon tmw pritlc4plcs, one essentially a physical princ~iplr, i 1~~other ntat hctrtat~ical : I. The> I
L = / (tlr)4d:(ErC).
( ci.1 )
80
PAGELS
We shall not, in what follows, make any specific assumptions about the form of J?(Z) but simply use the action principle to display the content of general covariance. Next consider the change in L due to any change in the metric g&x). In accord with the ideas of Einstein we recognize that the generator of these changes T,o( z ), is the symmetric stress-energy tensor:
6, L =
(czz)“; (-g)““T”qZ)sg@3.
s
(6.2)
The tensor Z’“‘(x) describes the distribution of matter of the dynamical system. We now specifically consider those changes in the metric due to a general coordinate transfromation, 2” = xx + 6x’(x). Such a change will induce a change in the metric at the point x: GQaJT(X)
= hQq9~X” +
Qh&JXh
+
(6.3)
Q~&~XX.
The demand of covariance under general coordinate transformations is thus formally equivalent to the demand that 6,L = 0 for changes in the metric induced by coordinate transformations. So we have
0 G 6, L = / (d3$6xx [; and since the 6xX are arbitrary
- a,((-g)“2gh~
G-Q>""TaBa,ga@
TP”>
1
we must have
Tmala = 0,
(6.4)
which constitute a set of four conditions on acceptable T”@(x) for a dynamical system. That general covariance under coordinate transformations imposes these conditions, the conservation of energy-momentum, is, of course well known. Next consider the similarity transformations. The change induced in the total action due to an arbitrary change in the Fock-Ivanenko coefficient 61’,(x) is
6rL
= Tr
s
(dx)4( -g>““17”(x>sr,(x>
(6.5)
where we have written for the generator of these changes s”(x) and have taken the trace to insure invariance of the integral. We now shall introduce certain assumptions in order that we might specify F(x) in terms of the distribution of matter Trrb( x) and the electromagnetic current j,( 2). Write P(x)
= ej”(x)I
+ J”(x)
thus separating ,7”(x) into a part which has trivial transformation spin space, j,(x)r, and a part which does not contain the identity,
(6.6) properties in In the
J”(x).
SPIN
AND
GRAVITATION
81
limit of Aat space E,(s) 3 -dA,(.c)I so t’hat,ja(z) is the generator of that part of *f,(x) which is a multiple of the identity, and hence, J’~(z) is identified wit’h the electromagnetic current. The term J”(X) is the generator of changes in the gravitational part of I’,(z) and so we assume t)hat it depends only on ~‘,o(x). Moreover let us assume that J”(z) transforms like a vector under reflections, depends linearly on T&z), and does not contain derivat,ivcs of Yap(.r) highci than the first. If we also assume, in line with thinking a,8(x j and g4a( x ) as fundamental geometrical objects, that J”(z) is chiral invariant, Jn( L) -+ .Iai.~.) it ye - e2y5”ya , then the most general form for 7 (.e) is J,(a)
= uu,pT’@ + bux*T::a.
( 6.7 i
Consistent, 1vit.h the above assumptions me have the specification terms of the matter distribution and the currents: Ja(z)
= eja(s)
+ aa,flT(z)”
of Jai.?) in
+ buksTaA!“.
i 6.8 )
Now examine the content of the demand for general covariance under similarit? transformations. The change in I’,(s) induced by a general infinitesimal similarity transformat8ion, X(X) = 1 + i&S(z), F’(X) = 1 - i&S(z) is given by
m,(z) = ips, r,] - dds.
(6.9)
The demand that physics be independent of t’he representation of ya we shall understand to imply that 6rL z 0 for changes in ro,induced by similarity transformations. Subst’ituting (6.9) int,o (6.5) we have 0 = 6rL
= Tr = Tr
s s
(cl&---g)lVyibs, (d~)~[d,(
r,l - d, 6s)
(-g)“‘?)&S
+ ,i( -g)“‘J”“[ss, r,]] = Tr = Tr
s s
(dx)*[(
-g)“‘Piu
(6.10) + ,i( -g)““[r,,
J”]]sX
(dx:)4(-g)1’2V~iS
where we have performed the usual partial integrations and used the cyclic property of the trace, Tr (ABC) = Tr (,BC’A). Since 6X is arbitrary (6.10) implies T-,9” = 0 or substituting
(6.8)
82
PAGELS
This can only be true if j”,, = 0 (Taxi* - Tns”),a
(6.11) = 0.
(6.12)
The first condition expresses the conservation of current for the dynamical system which follows from gauge invariance, r, + I’, - a,&.9where SS is a multiple of the identity. The infinitesimal electromagnetic gauge transformations are a subgroup of the group of infinitesimal spin transformations. The second set of equat’ions (6.12) is an additional set of six conditions which covariance imposes on acceptable matter distributions. In the case of flat space these conditions are trivially satisfied since differentiation is commutative and T”‘,s = 0. In 0th er words, in $at space the demand *for the independenceof physics on the representation of the ya is trivial, however in a non&t space the conditions (6.12) are decidedly nontrivial. We are now in a position to state our result. Suppose, in accord with the equivalence principle, that there exists a relationship between the distribution of matt,er and the geometry. Next apply the principle of general covariance which demands physics to be independent of the coordinate system and the representation of the y,(x). This we have seen leads to the conditions (6.4) and (6.12), TuBla= 0 ( Tx”i@- TxB’“),x = 0.
(6.13a) (6.13b)
Moreover assumethat Tab(x) depends only on the geometry in the neighborhood of X. If we now demand that the ten conditions, (6.13), be fulfilled for any matter distribution T,a(x) in any Riemannian space, then, as was shown in a previous paper (21), there is only one relationship between T,@(x) and the geometry satisfying the above conditions and which is the only relation true for every Riemannian space. That relation is given by the system of Einstein equations
~Tap = Rns - ?4g,pR -I- Xg,p
(6.14)
where K and X are phenomenological constants. It is in this manner that the dynamics of the gravitational field emerges from consideration of general covariance to include spin as well as coordinate transformations. The dynamics of the gravitational field essentially follows from symmetry principles. Gravitation can here be understood as a compensating field for both general coordinate transformations and spin transformat’ions, S( 2 ) = exp [~cT&Y*‘(z )I. B. THE
MAXWELL
EQUATIONS
We know that in the case of electromagnetism gauge invariance implies the existence of a conserved current j, = d,F”i which is interpreted as a source of
SPI?;
AND
83
GRAVIT.4TION
the field. Thus we are encouraged to examine the possibility of a conserved wrrent in the case of gravitation. To this end we take t,he divergence of fiiuo =
1&n~RXL,e -
eFap : Vak”~ = ,&q.rRXfag~a - eFapl, = -Yp
( (i.l.5 1
- ejo
where ~‘0 is t.he usual electromagnetic
current
and
( 6.16 1
-3, = -l&sxrRXLaa,a .
From t,he contracted Bianchi identities in the form RXfaPlu =
R” :
and hence by t’he Eiust’ein equat’ions
For every space endowed with matter and for every represeutjatiou of the ye t’here is a current giveu by (6.18). That this current is indeed divergence free is easily seeu from our condition (6.lXb) on acceptable matter distrihutious
It is difficult siIw
to associate a conserved
Tr
ss
quantity
f, tlX” = 0
with
this divergence
coudit iou
( (i.20 )
for any closed surface A S and any !!‘aa(r). This follows from Tr a,@ = 0. The gravitational curreut J, we can interpret as a source for the field 1‘,(z) aud heuc~c its fornial analogy with the electromagnetic current is clear. However the existence of t,his divergence-free current does riot enable us to develop au int cgral formulation of Einstein’s equations in analogy with the integral formulat,iou of ?tlaxwell’s equations. The syst,enl of Einst’ein-Maxwell equations uow has the form Rae -
41g,R
-I- Xgaa = ~Tap
( (i.‘)la 1
84
PAGELS
from which one can determine the I’,(x). The physical significance of gravitational current jfi = (~/2)flkTt”. 1sproblematic; however it might well be the starting point for a quant.um theory of gravitation in virtue of its formal analogy with the electromagnetic current for which there is a developed quantum theory. A quantum theory developed from the properties of the currents 3~ = (~/2)&‘;” and jb and the field I’,(x) could be seen as embracing both gravitation and electromagnetism on a similar footing. One is here faced with the problem of quantizing a system of nonlinear equations (6.21). To remove this nonlinearity by a perturbation approach is simply to remove the interesting feature of gravitation (22). VII.
GRAVITATIONAL
FIELD
INTERACTING
WITH
AN
ELECTRON
In order to examine in greater detail how the gravitational field interacts with the spin of a particle we consider those states which satisfy Dirac’s equation: (r”D,
- MO>1 ti > = 0
(7.1)
where D, is the translation operator previously discussed and Mo is the bare mass. If we operate on the left of (7.1) with r’Dp + MO we obtain (r”r@D,D,e - Mt)/
9 >= 0
(7.2)
or noting ~“7’ = go’ - ia”@ so that y”#Doi
DB = g”‘Da DB - ; oaBIDa, D,d = D2 + ~~=%aa
and hence (7.2) becomes 0” - MO2 - 5 u,~ F 4 2
asp ns R heaB ) +> = 0. >
(7.3)
The term (e/2)cT,pF”” is the familiar Pauli term with the appropriate g factor for the electron. The term ~.(T~~A~R~‘~’ is an additional interaction of the spin with the surrounding gravitational field. It is possible to show using the commutation relations for the ys and the algebraic symmetries of RX”@ that JL&T,,Y.TX~R~‘~~ = +$R where R is the scalar curvature, R = R”, . Using the Einstein equation R = - ( K/~)T $2X where T = T”, , the trace of the stress tensor, and substituting in (7.3) we obtain D2 _ MO2 2 A similar (1963)).
result
has been
e asp F”’ - ; T + ; 1I)) = 0. 2 >
obtained
by A. Peres
(Nuovo
Cimento,
Ser.
(7.4) X,
Vol. 28,
1091
SPIN
AND
85
GRAVITATION
First WC note that for our considerations t’he cosmological term is unessential in (7.6) for we can define a new mass tio2 = MU2 - x/2, assuming X/2 < M” , so tjhatJ (7.4) reads
Sext wc consider the contributions due to the T tc>rnl. Let us suppose that, OIP (aan write for the total stress-energy tensor !!‘(20 = YTF + T”,“Bt where 2’:: rcpresenk the self-stress energy of the elect’ron which we shall take t’o be T self a0
=
:$($rcrDatCI
+
( i.ti
ihsl~k)
and “‘Xtrm’l 1 &S is the stress tensor due to t,he presence of other fields including elet*tromagnetic field, Fab , T ext a&9 =
pl~tromagnetie
If only electromagnet8ism is present Tht”” that, t’ot,al stress-energy is additive T
=
Tself
+
rp;;xt.
)
1.h~ (7.7)
= 0, so we have in t,he approximation +
T”“‘.
Ci.8)
Now l’yi = FaxFaX - ,l~~ga~F,F”’ and hence T”‘” = 0. Ts”f, using the I)irac* equat,ion and choosing for the normalization of the wave function q$ = ?A/,, , is simply , self r
zx
2Jf””
so that the net effect of gravitation-spin irkeraction aualogous interaction is to change the effective mass of the electron t,o U:
( 7.9
)
t,o t.hc Pauli
and Eq. (7.5 ) becomes
We thus identify the gravitational g factor as g = -x, t,he coefficient of ~7”“~~. For laboratory situations t.hese spin-gravitat’ion effects are very smail. VIII.
CONCLUSIONS
In line with thinking of electromagnetism and gravit,ation as compensating fields, t’he presence of such a field can be understood as dest’roying t.he homogeneity and isotropy of flat space-time. The symmetry breaking of these classical fields is revealed in the alteration of the translational transformation properties
86
PAGELS
implies the conservation of currentj”,, = 0. For the case of gravitation under similarity transformations, S = exp [i~,~~~~(z)] r, -+ sr,s-’
leads to a divergenceless
invariance
+ ia,ss-’
current vd”=o 3, = ~KQhcTaA’f,
and the conditions
on T”‘,
CT
-/ala - TYB’a),r = 0.
These six conditions along with TUB1s = 0 and several ot’her assumptions led to the Einstein equations. If physical theories are to possess covariance under general coordinate and general spin transformations we are forced to consider the Einstein equations as the dynamical basis for gravitation. In this way one can point to the close connection between spin, spin transformations, and gravitation. ACKNOWLEDGMENTS The author wishes to thank Professor partment for his vital interest in this work John A. Wheeler of the Princeton Physics RECEIVED
: January
M. Schiffer of the Stanford Mathematics Deand his many critical suggestions, and Professor Department for several points of clarification,
31, 1964 REFERENCES
“Vergleichende Betrachtungen ifber Neuere Geometrische Forschungen.” KLEIN, Erlangen, 1872. 2. H. WEYL, “Classical Groups, ” 2nd ed. Princeton Univ. Press, Princeton, New Jersey, 1946. 9. H. WEYL, “Philosophy of Mathematics and Natural Science,” Translated by 0. Helmer. Atheneum, New York, 1963. 4. H. WEYL, “Gruppentheorie und Quantenmechanik,” 2nd ed. Hirzel, Iteipzig, 1939. 1. F.
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AND
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87
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