karakostas_the conventionality of simultaneity in the light of the spinor representation of the...

8/14/2019 Karakostas_The Conventionality of Simultaneity in the Light of the Spinor Representation of the Lorentz Group

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Pergamon Siud. Hist. Phil. Mod. PhJs., Vol. 28. No. 2. pp. 249-276. 1997@ 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain13552198/97 $17.00 + .OO

The Conventionality of Simultaneity inthe Light of the Spinor Representation

of the Lorentz GroupVassilios Karakos tas *

The assessment of the conventionality of simultaneity has commonly takenplace so far within the traditional formulation of the special theory ofrelativity. The E-synchrony transformation is presented within this contextin a sufficiently general manner that explores the connection of spatio-temporal measures to the choice of an E-simultaneity relation. Subse-quently to the recent work of Zangari, the feasibility of the latter is theninvestigated in terms of the two-component spinor formulation of specialrelativity. This is motivated by the fact that the spinor formulation providesthe most fundamental expression of a spacetime theory that is consistentwith the principle of special relativity. It is shown within this contextthat the transformation elements of the spinor group (unlike its Lorentzcounterparts) prevent the groups representations being extended to arepresentation of the z-class of non-standard synchrony transformationsin four-dimensional space. The underlying reasons are traced down anddiscussed at length, whereas the compatibility of this finding with a generalversion of the principle of general relativity that is applicable to both tensorand spinor quantities is also demonstrated. It is finally established that thestandard simultaneity relations far from constituting just a sensible choicein a range of conventional possibilities, is uniquely and objectively singledout by the properties of a spinor structure in Minkowski spacetime. Thedesirability of such a structure is anticipated by its fundamental status.@ 1997 Elsevier Science Ltd.

1. IntroductionOne of the most debated questions in the foundations of special relativity (SR)concerns the precise status of the simultaneity relation of spatially distant eventswithin a single inertial frame. A stream of conflicting analyses of the issuesinvolved has been recently introduced into both the physics and the philosophy

(Received 24 Jume 1996; revised 5 November 1996)* Department of History and Philosophy of Science, University of Cambridge, Free School Lane,Cambridge CB2 3RH, U.K.

PII:S1355-2198(97)00006-3

249


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250 Stu dies in Hist ory and Philosophy of Modern Physi csliterature mainly by Winnie (1986), Havas (1987) Ungar (1991) Coleman andKort (1991), Anderson and Stedman (1992), Redhead (1993) Zangari (1994)and Gunn and Vetharaniam (1995).

In the tradition initiated by Reichenbach (1957; 1969) and carried on byGrunbaum (1973) Salmon (1977) and many others, the simultaneity relation istaken to contain an ineradicable element of convention which reveals itself inour freedom to select (within certain limits) the value to be assigned to the one-way speed of light in a given direction. The source of this thesis, which shall becalled hereafter the thesis of the conventionality of simultaneity (the CS thesis),is the special nature of light as a first signal and the subsequent absence of anymeans of instantaneously synchronizing distant clocks in SR.

In order to specify a value for the one-way speed of light in a givendirection, a criterion of simultaneity for distant events (and therefore for distantclock synchrony) is required. As instantaneous synchrony is not possible, anyassumption of one-way speed values is, a fortiori, equivalent to the assumptionof the criterion for synchrony (Reichenbach, 1957, p. 127). The essence of theCS thesis is that this circularity is inescapable and that no fact of nature allows aunique standard of either the synchronization of distant clocks within an inertialframe or the determination of the speed of light in a given direction. The onlyspeed of light that is free from this kind of underdetermination (the argumentgoes) is the round-trip speed, in which only one clock measures the starting andfinishing times of the trip. The circularity of distant clock synchrony does notenter.Aware of the circularity problem concerning the one-way speed of light,Einstein (1952, p. 40) proposed that while the round-trip speed is an indisputableempirical fact, the one-way speed of light may have to be determined priorto the adoption of any criterion for distant simultaneity by stipulating thatlight has the same speed in all directions. Under this stipulation, of course, thesymmetries of spatial homogeneity and isotropy are recognised as an integralpart of Minkowski spacetime.

On the basis of Einsteins definition, proponents of the CS thesis haveconstantly stressed that while the criterion of isotropic light propagation mayresult in a descriptively simpler formulation of SR, other criteria leading todifferent simultaneity judgements and consequently different unidirectionalspeeds of light may have been employed without undermining the empiricalsuccess of the theory. According to the CS thesis, any simultaneity definitionwill be physically acceptable, in so far as it is consistent with the round-triplight principle, holding that the average speed of a light ray over any closedpath is a universal constant independent of position and direction (e. g. Winnie(1970); compare with Brown (1990)). To provide this with an appropriatetheoretical framework, conventionalists have sought to replace the standardLorentz transformation which embodies Einsteins isotropy convention by anE-extended Lorentz group of transformations that adopts the round-trip sensefor temporal coordinates (Section 2).Opponents of the CS thesis (e. g. Mittelstaedt, 1977; Friedman, 1983; Torretti,


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Conventiona1it.vof Simultaneity 2511983) , on the other hand, have maintained that Einsteins isotropy stipulationcannot be consistently removed, since it is intimately connected with thegeometry of the Minkowski spacetime as described by its standard metricaland conformal structure-an argument that is normally attributed to Malament(1977). What Malament showed, is that Einsteins standard simultaneity relationis the only non-trivial simultaneity relation definable in terms of the (symmetric)light cone structure of SR. Essentially, this was achieved by showing that forany non-standard simultaneity relation that was transitive and adapted to areference frame F, there exists a (symmetric) causality-preserving automorphismof Minkowski spacetime that, while preserving F, would fail to preserve theassociated simultaneity relation.It is important to realise, however, as has recently been emphasised byColeman and Kort (1991), that the effectiveness of Malaments result is con-tingent upon an ontological interpretation of the isotropic cone structure ofMinkowski spacetime and consequently of its respective spacetime invariants. For if one holds, as defenders of the CS thesis do, that the cone structure ofa Minkowski spacetime may be generated by a variety of conventions, none ofwhich are infected with a particular simultaneity assumption, then this sufficesto weaken the significance of Malaments result. In fact, it has been argued (e. g.Ungar, 1991) that this result serves to expose an indisputable incompatibilitybetween the various geometric structures defined on non-standard (anisotropic)simultaneity relations and those of the standard (isotropic) relation when bothare estimated on the basis of the customary Minkowski norm for the relativisticline element.

It should be remarked that most of the assessment of the CS thesis hastaken place so far within the traditional 4-vector (or, more generally, tensor)formulation of SR. The scope and nature of the latter is however met with certainlimitations. The transformation elements of the proper Lorentz group L, whenacting on a real four-dimensional manifold, fail to exhaust all of its continuousrepresentations, whereas the topological property of double-connectedness ofL: forbids the existence of physical fields with half-integral spin in Minkowskispacetime (Section 3). The most general (fundamental) expression of a spacetimetheory that affords the description of spinorial fields and is consistent with theirreducible linear representations of the Lorentz group must be given in termsof a two-component spinor formulation (Sections 3 and 4). This motivatesour attempt to project the CS thesis into the spinor formulation of SR andexamine the status of the non-standard Lorentz transformations within thiscontext (Section 4). In order to make contact with the traditional viewpoint,however, we first present in a sufficiently general manner the Reichenbach-1 An appraisal of the significance of Malaments result especially in relation to its sensitivity toconstraints such as the need for the simultaneity relation to be an equivalence relation may befound in Norton (1992) and Redhead (1993). See also Janis (1983) for a general line of argumentin defence of conventionalism that exploits the possibility of using Malaments definition of astandard simultaneity relation, originally corresponding to an observer at rest, in establishing, byvirtue of that very definition, non-standard synchrony relations relative to an inertial frame inwhich the observer ceases to represent rest.


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252 Studies in Hist ory and Philosophy of Modern Phy sicsGriinbaum conventionalist interpretation of simultaneity and explore its directrepercussions for the formal (mathematical and logical) aspects of the specialtheory of relativity (Section 2).

Before starting off, we note that quite recently two papers have appeared inthe philosophy literature by Zangari (1994) and Gunn and Vetharaniam (1995)that seem to be relevant to our forthcoming considerations in relation to thespinor representation of SR. While Zangaris paper should be noted for havingexplicitly connected the problem of the conventionality of simultaneity with thespinor concept, it should nonetheless be underlined that Zangari fails to providea fully satisfactory analysis as to how Einsteins simultaneity relation is to beuniquely established in the presence of spinor fields in Minkowski spacetime.On the other hand, Gunn and Vetharaniams paper sets out, by way of refutingZangari, a derivation of a Dirac equation for the electron spin that allegedlyholds for reference frames described by arbitrary synchrony relations. We findGunn and Vetharaniams derivation of an arbitrary synchrony Dirac equationto be fundamentally flawed. Since a solid development of these issues would takeus far afield, our criticism of Zangari and refutation of Gunn and Vetharaniamis given in the Appendix.

2. The Conventionalist Interpretation of SimultaneityIt is well known that Minkowski spacetime does not admit an absolutesimultaneity relation. Due to the absence of a unique global time in SR,simultaneity becomes relativized to an inertial frame. This is of course the

uncontroversial notion of the relativity of simultaneity in Minkowski spacetime.What is controversial, as discussed in the introduction, concerns the status ofsimultaneity within a single inertial frame.In Einsteins definition of simultaneity, the partition of events into simul-taneity classes amounts to a foliation of Minkowski space into disjoint space-

like hypersurfaces-hypersurfaces of simultaneity-orthogonal to worldlines ofinertial observers that make up any given inertial frame. Coordinate systemsadapted to such a frame will be represented by (to, ~0) where the subscripts referto Einsteins synchronization. With respect to the coordinate system (to, x0) theinfinitesimal spacetime interval assumes the canonical Minkowski form

ds; = c2( dto)* - ( dxo). (1)Consider now inertial observers at rest with respect to each other in a frameparametrized by an E-synchrony coordinate system (t,, xE) defined, after Winnie(1970) by the following set of coordinate differential transformations:

dx, = dxodt, = dto - E/C. dxs. (2)

It is clear from relations (2) that such a system represents a change in thesynchronization of the clocks at each point in space, whereas the spatial


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Conventionality of Simultaneity 253

coordinates themselves remain unchanged. Relations (2) give rise to hypersur-faces of simultaneity that are oblique to worldlines of inertial observers, themagnitude of obliqueness being determined by a particular realization of theanisotropy parameter E. Hence, E-synchronizations yield alternative t,-obliquecoordinations of Minkowski space, which, in turn, introduce an asymmetrictemporal orientation into spacetime. In terms of Reichenbachs synchronizationparameter, ER, our E is equivalent to (l-2, ER) with 1E 1 < 1. Thus, Reichenbachsnon-standard definition of simultaneity reduces to the standard one of Einsteinwhen ER = 112, i.e. E = 0.

The spatial metrical properties of events on a t,-hypersurface are determinedby

ds; = ( dt,)2 + g dx, dt, + 9 ( dx,12, IE/ < 1.cThe form of equation (3) may be viewed as the direct effect of considering anE-type inner product of 4-vectors in oblique coordinates (e. g. Leighton, 1959,pp. 440448). The presence of the spacetime cross-terms dx,, dt, reflects, inaccordance with the CS thesis, the non-isotropic propagation of a light signalover a one-way spatial path.

With the distance definition of (3) the latter is given by c,(n) = c n1-E*nwhere c stands for the average speed of light of outgoing and return journeys

along an arbitrary direction of propagation n (InI = 1 , remaining thusindependent of any synchronization procedures (e. g. Anderson and Stedman,1982).The metric components associated with the E spacetime interval are given by .I

goo(E, E) = 1, &O(E,E) = Ei> gj.,(E, E) = Ej ' Ej - 6ij. (5)Note that both dsz and g, reduce to the standard Einstein-Minkowskian formwhen E = 0.Clearly, spatial distances betwen simultaneous events depend on the hyper-surface of simultaneity on which the events lie and thus on the choice ofan E-synchronization. The spacetime interval itself, however, is independentof an E-value, like any quantity in SR that represents an observable, i.e. anabsolute quantity. Thus, the relativistic spacetime interval (3), defined in termsof synchrony-free coordinates, is transformable into the standard Minkowskianline element

ds; = c2( dto)2 - (dxo)2 = ds; (6)via the linear coordinate transformations (2). Several proposals have appeared in the literature to experimentally establish a uniquedetermination of the one-way velocity of light and thereby the true simultaneity relation at adistance. Some of these proposals were aptly criticized by Salmon (1977); compare with Torretti(1983, p. 339).3 The synchrony-free metric form (5) is quite standard in studies of general relativity andderivations may be found in Basri (1965) and Moller (1972, Sec.).


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254 S tud ies in History and Philosophy of Modern Physics

The invariance of relation (6) is interpreted by conventionalists as a mani-festation of the action of an r-extended Lorentz transformation group, whoseindividual elements A, do not assign a preferred status to Einsteins isotropystipulation and upon which E-versions of SR may be based. The properties ofA, in the case of a two-dimensional spacetime have been recently studied byUngar (1986; 1991); compare with Winnie (1970) and Giannoni (1978).The individual E-Lorentz transformations A, are typically construed as coor-dinate transformations connecting two arbitrary spacetime coordinate systems(tE, x,) and (t:, , XI 1, in relatively moving frames F and F' , respectively, wherethe arbitrariness refers to the freedom of choosing a particular value of E ineach frame. Clearly, a single application of A, transforms (t,, x,) to (tE,, XL,directly. To realise, however, what is going on, we may define an E-Lorentztransformation by performing first a coordinate transformation of (t,, x,) tostandard coordinates (to, x0) in F by means of relations (2) [re-coordinationprocedure in F], then applying an isotropic (standard) Lorentz transformationto obtain the coordinates of (to, x0) in frame F and finally transforming (tb, x;),as in the first step above, to obtain (t:,, XL, [re-coordination procedure in F].To see this explicitly notice that relations (2) which essentially describe alinear transformation from standard to E-synchrony coordinates, can be mostnaturally expressed in matrix language with the aid of a 4x4 real matrix P(E) :

X(E) = P(E) X(O)

t

1 -E,, /c -E,,/C -E,, /c to fto-EmXoIC\0 1

0 0 Xl= 0 0 1 0 x200 0 1 Ii3 1 Xl=L I.2 (7)x3

If A designates the standard Lorentz transformation, it follows from the analysisconcerning its .r-Lorentz counterpart that the latter has the form of a similaritytransformation :

A, = P(g)AP- (E). (8)Relation (8) shows just how the standard Lorentz transformation changes witha change of coordinates ; hence, A, can be viewed as the result of a passivesimilarity transformation of A. Far from being a new transformation, it merelyrepresents the standard Lorentz transformation, realised between orthogonalspacetime coordinate systems, in oblique coordinates.From the viewpoint of the proponents of the CS thesis, this result, wheninterpreted within an operationalist positivist doctrine, becomes of crucialimportance. It shows that the observable consequences of the E-Lorentz trans-formations, for arbitrary choices of E, I E < 1, are one-to-one translatable intoconsequences of the standard Lorentz transformations, when the differences inthe temporal coordinates of the systems involved are taken into account. Thishas been regarded by conventionalists to indicating further that formulations ofSR containing different choices of E-simultaneity relations result in kinematicallyequivalent versions of SR, vindicating thereby the Reichenbach-Grtinbaum


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Conventionality of Simultaneity 255thesis of the conventionality of simultaneity. Moreover, as has been arguedby certain authors (e. g. Havas, 1987), such a view of the conventionality ofsimultaneity is compatible with what is involved in the case of a generallycovariant formulation of SR, namely, when the latter is viewed as a specialcase of the general theory on a flat spacetime for which the curvature tensor isassumed to vanish everywhere.To this extent even anti-conventionalists by and large have conceded thatnon-standard simultaneity relations, contrived as they are, may be employed inSR as a trivial consequence of an E-dependent general covariance, subject tothe proviso that from a physical point of view they are not equally significantto the standard choice. For all this rather remitting admission, we show inSection 4 that the standard simultaneity relation does not just represent anoverwhelmingly sensible choice within a range of conventional possibilities, but,rather, it is directly imposed by independent physical phenomena, theoreticalconsiderations, and their empirical consequences as they are naturally revealedin the spinor formulation of SR.

3. The Spinor Representation of the Lorentz GroupIn the traditional approach to SR, which permeated all of the conventionalityof simultaneity debate so far, the primitive terms are spacetime, represented bya four-dimensional real affine differentiable manifold, and events, designated

minimally by 4-component vectors at each point in the manifold. The Lorentzgroup of coordinate transformation matrices may accordingly be identified. inthe most general sense of that specification, with the group of all 4x4 real linearmatrices that leave the quadratic form of a Lorentz signature metric invariant.These matrices, however, do not exhaust all continuous representations ofthe (proper) Lorentz group L:; furthermore, they do not form the smallestmatrices endowed with the algebraic properties of _!I see e. g. Cartan, 1966). Inaddition, there are relativistically invariant phenomena in nature, most notably,physical fields of half-integral spin, whose spacetime description cannot be It should be underlined that the principle of general covariance hv itself bears no physicalinput, since any non-general-covariant theory of spacetime can be given a generally covariantform (e. g. Weinberg, 1972, p. 92). All one needs to do is to replace ordinary derivatives withcovariant derivatives, add an affine connection into the theory, replace the Minkowski metric by aRiemannian metric tensor that transforms under arbitrary general coordinate transformations, andsay that there exist coordinate systems in which the components of the affine connection happento vanish so that the original formalism is valid. If, then, one interpretes the time coordinateof any of these reformulations as providing a criterion of simultaneity, the most preposterousof simultaneity relations could be warranted, including non-standard simultaneity relations evenin Newtonian spacetimes that possess an absolute, global time! The possibility of expressing aspacetime theory in a coordinate independent (and thus generally covariant) manner can hardlytherefore lend support to the conventionality of simultaneity. What is required is some independentmeans of arguing that the time coordinate of a given formulation of spacetime theory doesrepresent a putative definition of a possible simultaneity relation, like the Reichenbach-Grlnbaumcausal theory of time attempts to establish for the z-synchrony relation.


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256 Studies in History und Philosophy of Modern Physicscaptured by the irreducible representations of L (see e. g. Kim and Noz, 1986).The minimal simple representation of the characteristic invariance of Lorentztransformations, satisfying all aforementioned conditions, manifests itself in theconstruction of the SL(2, c) group, i.e. the spinor representation of the Lorentzgroup. The group SL(2, c)-the special linear group in two dimensions


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Conventionality of Simultaneity 257Given a 2x2 matrix C one can get a 4-vector xP by considering, with the aidof (lo), the inverse of formula (9),

xU = 1/2Tr(Ca,). (12)In this way one defines a one-to-one linear correspondence between all 4-vectors and all 2x2 hermitian matrices. Thus every point in space xv leads toits corresponding hermitian matrix C( = c,,, ), which may be understood as ahermitian spinor of the second rank (spinor indices are, in general, suppressedin this paper).Now, corresponding to every element A of the group SL(2, c) consider thefollowing transformation in the two-dimensional space of hermitian matrices C:

C = ACA+, (13)where C = x;uP and t denotes the conjugate transpose or hermitian conjugate.By virtue of relation (12), this expresses a linear transformation of the 4-vectorxP. It is also real since the hermitian nature of the matrix C is preserved. Hence,real xP transforms into real x;.

The corresponding operation in the Minkowski space of 4-vectors is atransformationx; = A&4)x,, (14)

where the transformation matrix A of the Lorentz group is expressed in terms ofthe matrix A of X(2, c). The general Lorentz transformation (14) is normallycharacterized by the fact that it preserves the scalar product, i.e. it leaves thespacetime interval invariant, xi2 -x I2 = xi -x2. Note that the determinant of thematrix C has the value det(C) = xi - x2, i.e. it is just the length of the 4-vectorx,. Consequently, the requirement for the invariance of the length under thetransformation (13) can be put in the form det (C ) = det (C) , which is indeed thecase since by the unimodularity condition of X.(2, c), det( A) = det(A+ 1 = 1.Thus, the transformation induced by the matrix A of SL(2, c) describes acoordinate transformation of the Lorentz group on a 4-vector xv.Let us now briefly inquire into the relation between these two transformations.To determine the Lorentz transformation corresponding to a given matrix ofSL(2, c), we write by virtue of equations (14) (12) (13) and (9):

XI = A,,(A)x, = 1/2Tr(CoP)= 1 2Tr(ACAtcr,)= 1 2Tr(Ax,avAt u,,) (15)= 1/2Tr(AoVAtaP)x-,.

From this equation, since x,, is arbitrary, we getA,,(A) = 1/2Tr(Aa,A+cr,). (16)

The map A - A,, of the group SL(2, c) onto the group L, constructed inthis way, is called the spinor map or spin transformation and the representationof the 4x4 matrix A,, E _!Iby means of the second-order matrix A E SL(2, c)


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258 Studies in History and Philosophy of Modern Physicsis called its spinor representation. Notice that the spinor map is not one-to-one,since it is clear from (16) that A,, (A) is quadratic in A and therefore both Aand -A have the same image in L. In fact, it can be shown that it is preciselytwo-to-one, i.e. if A(A) = A(B), then A = +B. Thus, to each spin matrix Athere corresponds a unique proper Lorentz transformation A and to each suchtransformation are related two matrices differing only in sign, +A and -A.

This ambiguity in the sign forms an integral part of the situation and cannotbe consistently removed by making either choice. Such an attempt would violatethe property of the spin transformation matrices of forming a group (see e.g.Gelfand et al., 1963). In other words, making a consistent choice of signamounts to falling upon a non-trivial two-fold representation of the Lorentzgroup L which does not descend to L.

The essential two-to-one homomorphism of SL(2, c) onto LY wes its existenceto the fact that the spinor and Lorentz groups, though locally isomorphic-their infinitesimal generators obey the same commutation relations (see e. g.Kim and Noz, 1986) --differ in their global topological properties. While theproper Lorentz group is doubly connected, SL(2, c) is simply connected withthe identity and thus provides the universal covering group of L (see e. g.Penrose and Rindler, 1984). Clearly, starting with the light-cone parametricrepresentation of (9) one is able to derive everything else from this includingall faithful representations of SL(2, c), that is, both single- and double-valuedrepresentations of L.The single-connectedness of SL(2, c) [as well as of its rotational subgroupSU(2)] becomes especially important in relativistic physics when the latter isreconciled with the principles of quantum mechanics. To obtain, for instance,physical fields in Minkowski space that carry spin degrees of freedom, onehas to consider the continuous unitary representations of the Lorentz (orPoincare) group up to phase, yielding group-isometric representations up tosign (Wigner, 1939). These representations, as shown by Bargmann (1954) arein one-to-one correspondence with the irreducible representations of SL(2, c).This means that both integral and half-integral values of spin representa-tions correspond to faithful, single-valued representations of the spinor groupSL(2, c). Consequently, only spinors and spinorial fields of even rank may befully accounted for in terms of the ordinary Lorentz group. Spinors of oddrank are quantities which cannot be unambiguously obtained in the doublyconnected space of L; there remains an overall sign underdetermination. Theseconsiderations imply that the relativistic invariance of physical systems withhalf-integral spin is a direct manifestation of the two-valued, two-dimensionalcomplex representation of L. Hence, the condition of relativistic invariance inmicrophysics and its phenomenological consequences suggest that the kinemat-ics of SR, in its most general (unifying) and fundamental expression, must beamenable to the universal covering representations of the Lorentz group, thespinor representations of SL( 2, c) .


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Conventionality of Simultaneity

4. The Demise of Conventionality259

We saw in the preceding section that the group SL(2, c) associates every4-vector xv with a spinor matrix C of the second order and in the space ofthese matrices a standard Lorentz transformation X = Ax is given as a lineartransformation C = ACAt induced by an element A of the group SL(2, c). Wehave also seen in Section 2 that non-standard, namely, E-Lorentz transforma-tions, may be admitted on 4-vectors through similarity transformations of non-canonical coordinates that include a direction dependent parameter E. They wereinterpreted as passive t,-coordinate transformations on a real four-dimensionalvector space. Furthermore, we demonstrated in Section 3 that the kinematicsof SR, in its most general expression, requires invariant transformations tobe determined in terms of the spinor rather than just the 4-vector (or, moregenerally, tensor) transformations of the Lorentz group.

It is only natural then to inquire whether E-transformations are admissibleon the spinor representative C of a Minkowski vector. To this end, let A(E)be an element of a hypothetical E-extended representation of the spinor group,whose entries may depend appropriately upon the E parameter so that theunimodularity condition, detA(E)= 1, is satisfied. The latter is supposed toguarantee, together with the hermiticity properties of the associated 2 x 2(complex) matrix, that the components of the corresponding vector are real.

Were such an A(E) to exist, one would then be able to define an E extendedspinor transformation by means of the relationc, = A(E)CA+(E). (17)

Such a transformation would essentially describe (in spinor terms) an E-coordinate transformation of the extended Lorentz group on t,-functions of4-vectors, as their length would be preserved by

det(C,) = det[A(E)CAt(E)l. (18)To determine the form of A(E) recall that the transformation (17) oughtto preserve the one-to-one and linear correspondence between all E-class 4-

vectors and their 2 x 2 hermitian representative matrices. Thus, the matrix C,ought to be associated with the vector xE, the latter being defined in termsof oblique coordinates as in equation (7). Since a choice of the value of Ein the specification of a Minkowski vector sinply amounts to appropriatelychanging the synchronization of clocks along the spatial coordinate axes of areference frame, while the spatial coordinates themselves remain unchanged,we shall consider as an example of an E-synchrony transformation the case ofa passive transformation that is associated with an E-change in the scale ofthe measurement of time, precisely as required by the CS thesis6 (Section 2).We shall further assume, for simplicitys sake, that the said E-transformation(i.e. what amounts to a time transformation of the spatial axes of a coordinate6 The typical relativistic case of a boost that takes one reference frame to another is ignoredhere, since the content of the CS thesis restricts considerations to a single frame.


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260 Studies in History and Philosophy of Modern Physics

system) takes place along the x1 -axis, so that, E= (E,, , 0 , O) . The loss of generalityentailed by this assumption is trivial, and by virtue of relation (17) we have thefollowing equivalence:c = (Xo-cc,~1)+X3

(x1 - ix2E x1 + ix2 (x0-&,X1) -x3 )

= (z;; w) ( x0 + x3 x1 - ix2 ) (;yj21) (19)XI + ix2 x0 - x3 a22/

where the a,, designate the entries of the transformation matrix A(E) and theoverbar denotes complex conjugation.To obtain then the explicit form of A(E), is a matter of solving the linearequations resulting from (19). A tedious calculation shows that a solution toequation (19) exists only if the matrix elements of A (E) are all real and havethe values at\ = ~22 = 1 and ~12 = a21 = 0 ; that is, if A(E) is the identitymatrix. It follows then that E = 0, or, equivalently, in terms of the Reichenbachsynchronization parameter, that ER = l/2.It should be noted in this connection that an expression analogous torelation (19) appears also explicitly in Zangari (1994). However, it has beenproduced here on the basis of Gelfand et al. (1963, p. 252) sharply defineddistinction between the spinor and tensor coordinate transformations of theLorentz group, independently of Zangari. In fact, the vanishing of the epsilonsynchronization parameter with respect to Lorentz-spinor transformations canbe demonstrated in a computational easier way by making use of equation (18)and the unimodularity condition of A(E). Recall that both of these prerequisitesare essential for carrying out a Lorentz transformation in the space of thesecond-order hermitian matrices of the spinor group. If therefore the resultingrelation det(C,) = det(C) is written out long-hand, one realises that theequality holds-i.e. the time-space cross terms (of the kind E,,XOX~)on theleft-hand side cancel out-if and only if E= 0. Hence, the possibility of re-coordinatizing the temporal component of a local coordinate system accordingto t,-synchrony relations no longer exists, if the well-defined transformationproperties of the spinor representation of the Lorentz group are to be preserved.Thus the standard synchronization relation, contrary to the CS thesis, seems tobe singled out by the very nature of things when the spinor (universal covering)group of Lorentz transformations is taken into consideration.What we have established so far is that the defining properties of the carriermatrices of the spinor group prevent its representations to be extended toa representation of the E-class of non-standard synchrony transformations infour-dimensional space. Since in any four-dimensional formulation of relativitytheory the temporal and spatial components of a Minkowski vector are treatedon an equal footing under the action of the Lorentz group, our result may beviewed as a special consequence of the fact that the spinor representations of As will be shown in the Appendix, Zangaris reasoning in justifying the objectivity of the standardsimultaneity relation solely on the basis of relation (19) is inadequate and also substantiallymistaken.


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Convenlionality of Simultaneity 261the Lorentz group of coordinate transformations cannot be consistently definedunder linear substitutions to arbitrary non-standard (oblique) coordinate sys-tems, regardless of the type of dependence (temporal or spatial) involved in theensuing transformations.Note in this respect that by comparison to the tensor representations ofthe Lorentz group of general linear 4 x 4 matrices, the spinor representationsare restricted by their uniquely prescribed matrix properties (e.g. hermiticity),their relationship to the non-commutative algebra of Paulis fundamental spinmatrices, and their topological condition of possessing a simply connected space.There just exists no element of the spinor group (i.e. a complex hermitian uni-modular matrix of the second order) that would relate in a consistent fashion thecomponents of a spinor under a change to an oblique reference system. If theselatter were the case, then, as Cartan (1966, p. 15 1) emphasised, the representingspinor matrix would undergo under the resulting rotation of the Cartesian axesa continuous linear substitution, on following the continuous variation of anassociated point in the groups topological space that forms a suitable closedcontour, by starting from the identity substitution and terminating at its negativeimage. We would thus have a multivalued representation of a spinor-like groupthat may be extendable to oblique coordinate transformations, but we haveseen in Section 3 that this is not possible since all linear representations of thecomplex unimodular group, be it SL(2, c) or SU(2), are one-valued. Hencespinors which we defined in the preliminary sections in terms of an orthogonalbasis system cannot be consistently defined in an oblique coordinate system, ifthe spinor group is to maintain its definite topological properties. 8It may be worthwhile to consider briefly at this point any possible repercus-sions Lorentz-transformed spinor quantities may have upon the content of theprinciple of general relativity. This is of immediate interest since one may tendto think that the inability of consistently defining a Lorentz spinor in otherthan rectangular coordinate systems leads to a discrepancy with the norm offormulating physical quantities in a generally relativistic manner. Let me first ofall say (and the following analysis intends to make it clear) that in the presenceof a spinor structure in a spacetime manifold the expression of the principle ofgeneral relativity acquires the following form:(SGR) The fundamental equations of physics involving spinor fields are formu-lable in spacetime in a Lorentz-covariant and coordinate-invariant manner.To understand this, one has to realise that while spinor- and Lorentz-covariantderivatives are equivalent when applied to spinors of even rank that (by default)are in one-to-one correspondence with tensor fields, in the case of odd rankspinor fields (i.e. of truly spinorial objects) it is not possible to transform8 In fact, Geroch (1968) [see also Magnon (1987)] has shown, in the form of an existencetheorem, that a necessary and sufficient condition for a non-compact spacetime-i.e. a spacetimewith normally no closed timelike curves that excludes therefore causality violation-to admit aspinor structure is that the spacetime manifold carries a global field of orthonormal tetrads. Itis worth noticing that this theorem depends critically on the condition alluded to above that thesign ambiguity in the two-to-one homomorphism of X(2, c) onto the Lorentz group vanishesover the entire manifold.


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262 Studies in Hist ory and Philosophy of Modern Phy sicsLorentz-covariant equations into coordinate-covariant equations without spinorindices (see e. g. Treder er al., 1980; Gockeler and Schticker, 1987). In otherwords, for odd rank spinors no coordinate-covariant representation exists, as istypically the case for tensor quantities.This is a consequence of the fact that spinors, due to their peculiar Lorentztransformation properties, are incorporated into general relativity by introduc-ing a system of orthonormal tetrads at the tangent space of each spacetimepoint in the manifold. The set of the 16 components of the tetrad define what isvariably called a tetrad, frame or v ierbein$eId [see e. g. Dewitt (1961, Sec. 14)and Weinberg (1972, Sec. 12.5)]. One has to think of it as forming a set offour linearly independent vector fields, not as a single tensor. To clarify theimplications of this in relation to the statement of (SGR) above, let ht be such afield at each point x of a Riemannian spacetime manifold -n/14 aving arbitrarymetric structure. The Latin index a numbers the tetrad base vectors and theGreek index p designates tensor components; both have the range 0 , 1,2,3. Theformer refers to an orthonormal basis of the tangent space at each spacetimepoint, while the latter is associated with a general coordinate system.

Provided that the ht vary continuously and differentiably with x, they definefour mutually orthogonal differentiable vector fields whose normalisation is fixedby

gab = h:h;g~v. (20)Here g,, is the spacetime metric tensor, g& the Minkowski metric of thetangent space, whereas the h! are considered to be functions of the spacetimecoordinates.In any given coordinate system on 3M4,however, the components of a vierbeinfield are determined by the above description only up to a transformation

h; = A;(A)h;, (21)where the matrix coefficients Ai of the Lorentz group are expressed in termsof an element A of its universal covering group, the spinor group SL(2, c).Then, the hf can be used to transform, through contraction with the spacetimemetric tensor, coordinate indices of an arbitrary coordinate system in JA&into tetrad indices of a local Minkowskian system, and vice versa. Thus,arbitrary coordinate representations of the Lorentz transformation group canbe converted into representations of the spinor group at each point of spacetime.The ensuing conversion takes place evidently in a Lorentz-covariant andcoordinate-invariant manner, disclosing thereby the content of our initial (SGR)statement. The latter, to sum it up, constitutes a codification of the fact that Note, however, that since in a general spacetime LM4, it is not possible to set up a continuouscorrespondence between the tangent space of each point of 3f4 and that of a single fixed pointx E %I, the above picture can be described accurately only in terms of a second principal fibrebundle structure, a spinor bundle, whose base manifold is 3M4 and whose fibre is isomorphicto the universal covering manifold of the bundle of oriented, time oriented orthonormal tetradframes with structure group X(2, c). The interested reader may consult Geroch (1968) for details;further explanatory comments may be found in 54 of the Appendix.


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Conventionality of Simultaneity 263spinor fields (due to their algebraically restricted transformation properties)can consistently be introduced into general relativity by associating a spinorwith an orthonormal tetrad basis of the tangent space at each spacetime point,{h,(x)]; i.e. by introducing arbitrary local inertial frames having as metricstructure the standard form of the Minkowskian metric tensor, g(h,, hb) =diag( 1, - 1, - 1, - 1). For general tensor fields, there still exists of course the dualversion of (SGR) that pure spacetime tensor quantities should allow Lorentz-invariant and coordinate-covariant formulations, according to the classicalRiemannian techniques.

In returning to our original (flat space) considerations, we recall that spinors,conceived of as geometrical objects, are constructed purely in terms of conformalgeometry (stereographic projection, preservation of angles). An immediateconsequence is that spinors can essentially be defined on a spacetime manifoldwhen only a conformal structure is assumed, i.e. when account is taken only ofthe equivalence class of metrics which can be obtained from a given metric bya conformal resealing. Under the latter operation, of course, no transformationof points is involved, and as one may expect, two conformally equivalentmetrics necessarily share their light cone directions. The information containedin the construction of spinors, therefore, is that of the light cone structure.Moreover, this structure can be shown to remain invariant under a group ofactive transformations that is isomorphic to the conformal group SU (2,2), i.e.the group acting on what is known as a twistor space [2-component spinorsbeing replaced by 4-component bispinors; see, for instance, Penrose (1967)].So when an element of the spinor group acts on the hermitian matrix C,that is essentially the spinor equivalent of a Minkowski vector, the conformal-orthogonality relation, inherent in the construction of C, is preserved under theresulting transformation by the isometries of the group. Given, as we explicitlyshowed in the context of the CS thesis, that no E-transformation matrix A (E)exists that would extend the spinor group to an E-representation of it, it becomesevident that standard simultaneity is picked out by the orthogonal-conformalstructure of spinors as determined by their very geometric construction, thetopological properties of the group SL(2, c) and its spinor map that associatesa second-rank hermitian spinor with an orthogonal basis of four unit 4-vectorsat each point in a Minkowski spacetime.One is no longer allowed to modify appropriately the spacetime metric of theisotropic formulation of SR and still produce experimentally indistinguishableversions of it by making judicious compensatory adjustments in the rest of thetheory. The existence of a spinor structure in a Minkowski spacetime (that isuniversally tangential to itself) fixes the metrical relation as an orthogonalityrelation with respect to time-like geodesics. This is also tied to the fact that anysynchrony-free reformulation of SR that incorporates an E-coordinatization ofMinkowski spacetime as a basic metrical relation would mutilate the symmetriesnecessary for the description of physical processes under a unified scheme ofspecial relativity and quantum mechanics. It would truncate for instance theisotopic spin symmetry governed by the SU (2) group that is characteristic of the


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264 Studies in History and Philosophy of Modern Physics

experimentally established lo strong (proton-proton) interactions in elementary-particle dynamics. Clearly, any choice of the value of E other than that suggestedby Einstein would be deficient in unifying power. Standard simultaneity thereforediffers from non-standard choices of E not just in descriptive simplicity, asdefenders of the CS thesis have constantly maintained, but also in actual physicalsignificance that is readily revealed in the spinor formulation of SR.As pointed out in Section 3, the latter may be regarded as applying at a deeperlevel of theoretical structure than that described by the standard formulation of4-component vectors. In a relativistic theory of matter the spacetime coordinatesof events are determined in terms of a set of parameters with a built-in logic(based on the algebraic properties of the irreducible representations of the groupsof automorphisms involved) that is used to express the fundamental laws ofnature. The latter, in turn, are given in the form of field equations whose solutionsare the dependent variables that relate to the prediction of physical quantities.The most general type of field variable to underlie a theory that is consistentwith the principle of special (or general) relativity -irrespective of whether thetheory is quantised or not-turns out to be the spinor variable (minimally, aspinor of rank 1 or its univalent complex conjugate) (see e. g. Sachs, 1967). Thisis also the most fundamental type of variable in the sense that while spinors maybe combined (with their complex conjugates and duals) to form real spacetimevectors, which subsequently may be combined to form tensors (of any rank), noother type of covariant variable may form or be part of a first-rank spinor (e. g.Penrose and Rindler, 1984; Stewart, 1991). The consequence in physical theoryconstruction is that a spinor formalism can yield all of the physical predictionsof vector or tensor formalisms, but the spinor formalism is able to generateextra predictions that have no natural counterpart in the other formalisms ofcovariant theories (a collection of examples can be found in Misner et al. (1973,Sec. 41.11)). In this respect, one may say, that the most primitive (irreducible)expression of a covariant theory must be in terms of a 2-component spinorformulation.

In summary, the existence of spinor fields in contemporary physics seems tosuggest strongly that a spacetime manifold, to be of general physical interest,must possess the requirements of a spinor structure. The mathematical meaningof the latter is made entirely clear on a four-dimensional manifold of aLorentzian signature that admits a conformal structure and carries a globalorientable orthonormal basis field. The isomorphism between a Minkowski 4-vector and a second-rank hermitian spinor, formed by the tensor product ofa 2-spinor and its complex conjugate, provides the fundamental link betweenspin- and Minkowski-space. In view of this proviso, the real 4-dimensionalrepresentations of the Lorentz group decompose into the direct sum of two2-dimensional (complex, hermitian) irreducible representations, whose basiselements obey the defining condition of an anti-commutative (Clifford) algebra.It is essentially the physical-geometrical realisation of the properties of thelatter in conjunction with the irreducibility of the simply connected topology ofSee, for instance, Weinberg (1995, Ch. 3.3) and references therein.


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Conventionality of Simultaneity 265the spinor group that irrevocably fix the aforementioned orthogonality relationbetween space-like and time-like directions of relativistic events in particlephysics. Within this context the space-like hyperplanes intersect orthogonallythe worldlines of a particles inertial trajectories and are uniquely fixed. This isestablished by the fact that conformal-orthogonality, the relation that determinesthe simultaneity hypersurfaces at each point of an inertial trajectorys world line,is invariant with respect to the actions of the spinor group. Thus one cannotdispense with the standard simultaneity relation without dispensing with theorthogonal-conformal structure of spinors and thereby with the entire spinorrepresentation of the Lorentz group, whose implications for the relativisticallyinvariant description of physical processes are paramount. This is indicativeof the fact that an adequate relativistic description of the Dirac equation forthe free electron in terms of tE re-coordinatized reference systems, that onemay attempt to associate with E Dirac matrices, is impossible (see Appendix59). These considerations establish therefore that the standard simultaneityrelation, far from constituting just a sensible choice in a range of conventionalpossibilities, firmly embedded in the structure of the spinor (universal covering)representation of the Lorentz group, a fact that undercuts decisively the CSthesis in the context of the special theory of relativity.

5. AppendixThis appendix aims at accommodating certain critical comments on two

recently published papers by Zangari (1994) and Gunn and Vetharaniam ( 1995)that appear to be relevant to our preceding considerations.Zangaris paper is, to the best of our knowledge, the first published referenceto have connected the problem of the conventionality of simultaneity with thespinor concept. It is apparent from this work that we entirely align ourselveswith Zangaris main thesis that .s-transformations are not possible on spinors.It should be underlined however that this result, in its common form, is not aliento the basic literature of the theory of spinors. It is expressed, for instance, byCartan (1966, p. 15 1) in his account of the relativistic covariance of spinor fieldequations, and explicitly stated by Gelfand et al. (1963, p. 252) in drawing anacute distinction between the spinor and tensor coordinate transformations ofthe Lorentz group. As pointed out in Section 4, the latter reference provided themotivation for us to reproduce this result within the context of the CS thesis,independently of Zangari.We note in this connection that, in spite of our common point of departure,Zangari fails in his further analysis of the mathematical underpinning of thisresult. He explains:

If one [ ] transforms SL(2, c) to 0(1,3), then four-vector representations arepossible because O( 1,3) is decomposable into S the three-dimensional generators ofO(3) [ matrix formulae of S ] and K the pseudo-one-dimensional generatorsof Lorentz boosts [ matrix formulae of K 1. his is why time can be separated


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266 Studies in History and Philosophy of Modern Physicsfrom the space coordinates in four-vectors, even though this is not possible in theLorentz transformation itself. However, in SL(2, c), the generators are related viaS = iK nd are not decomposable. Thus, time is inextricably linked to the spacecoordinates and one cannot single it out for special mathematical attention. Hence,both Galilean and E-transformations cannot be defined on the spinor X, althoughboth are perfectly permissible on the four-vector x. (p. 274)

We shall show that the premisses and auxiliary results involved in this argumentare wrong. That the final semi-conclusion E-transformations cannot be definedon the spinor is right, just indicates the overall confusion surrounding Zangariscritical argument.

1. Contrary to what Zangari initially asserts, it is a well known mathematicalfact that the infinitesimal rotation and boost generators of the groups SL(2, C)and 0(1,3) obey the same set of commutation relations. I1 i.e.

[si, sj] = ieijk sk , [S ;, Kj] = iE ijkKk , [Ki, Kj] = -iEijk S k . (22)This is a consequence of the fact that the two groups have the same structure,

i.e. they have the same Lie algebra, and are therefore locally isomorphic. It isfor this reason that a 4 x 4 Lorentz transformation matrix can be explicitlyconstructed from the 2 x 2 matrices of SL(2, c). It is therefore incorrect withinthe context of Zangaris argument to assert that the

(A) four-vector representations are possible because O(1, 3) is decomposable intoS [ ] and K.

Due to the aforementioned mathematical equivalence, so is SL(2, c) itself.However one has to note in addition, following up our considerations ofSection 3, that the two groups, though locally isomorphic, differ dramatically

in their global topological properties due to the two-to-one homomorphism ofSL(2, c) onto O(1, 3). It is also due to the latter that the boost operators in theSL(2, c) regime can take on two different signs, while the associated Lie algebraSL(2, c) of the infinitesimal transformation generators will remain invariantunder the sign change. This is most clearly encapsulated in the Dirac equationwhich represents a direct sum of the representations with the two different signsof the boost operators.

2. If (A)s just incorrect, it is fundamentally wrong to conclude from (A)hat(B) this is why time can be separated from the space coordinates in four-vectors, eventhough this is not possible in the Lorentz transformation itself

Zangaris assertion (B)uns thoroughly against the essence of the theory ofrelativity. For the structure of a relativistic spacetime (and this is over and abovethe Lorentz transformation) is not isomorphic to the structure E3 x E1 of thel1The first commutation relation indicates that transformations generated by S; form a rotationsubgroup of the Lorentz group; the second one implies that boosts alone do not form a group,whereas, the third commutator suggests that a multiplication of two pure boosts results in amultiplication of a boost and a rotation.


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Conventionality of Simultaneity 261Cartesian product of Euclidean space and time. I. e. the location of an eventin Minkowski spacetime, designated by a four-component vector, cannot beanalysed as an ordered pair of independent individuals, a spatial location anda temporal moment. It makes no sense to ask in SR-in terms of coincidencesof space and time points-for either the temporal or spatial separation of twoevents, or even to ask whether the two events occurred simultaneously or atthe same place. All one can meaningfully discuss is the spatiotemporal intervalbetween them.3. The author continues by saying:

(C) However, in SL(2, c), the generators are related via S = iK and are notdecomposable.

This is again wrong, as pointed out in !j 1. That the rotation generators S,can be expressed as i times the boost generators K;, means only that we canstudy the SL(2, c) group with just three forms of generators, instead of six, bycomplexifying the group parameters. This is a typical case of the mathematicalflexibility offered by working on the complex plane and bears no relationwhatsoever on issues of decomposability.

4. Nevertheless, Zangari, on the basis of(C), reaches the conclusion:(D) Thus, [in SL(2, c)] time is inextricably linked to the space coordinates and onecannot single it out for special mathematical attention.

By way of reasoning, Zangari apparently implies here that time is not inextricablylinked in the four-dimensional real representation of the Lorentz group, which,as we commented in (2), is fundamentally wrong. Moreover, in relation to theSL(2, c) group, Zangari fails to realise that its elements reside in an internaltwo-dimensional complex vector space, spin space, whose tensor product withits complex conjugate, nonetheless, comprises a four-dimensional vector spacethat is isomorphic to the structure of the tangent space at each point inMinkowski spacetime with real coordinates; hence, the identification of second-rank hermitian spinors with Minkowski vectors of Section 3.The SL(2, c) group elements themselves act, in general, on the space of smoothsections of a fibre bundle structure whose fibres are spin space and whose basemanifold is spacetime. In the case for instance of DiracS equation for the electronspin (to be discussed below), the spinor ID= [ 91 (x), . . . , (C/~(X)] may be viewedas a section of a fibre bundle whose base space is Minkowski space and whosetypical fibre is a complex four-dimensional space with principal structure groupSL(2, c) 8 SL(2, c) and symmetry gauge group U( 1). The latter is the case,because a point in the fibre [The value of the Dirac spinor ID] s physicallyrelevant only up to a phase factor eiJ(-r)which is normally considered as anelement of the unit circle in the complex plane.The essential point is that while any conceivable coordinate transformations(not just E-synchrony transformations)-which would be permissible by agenera& covariant formulation of an arbitrary vector field in spacetime-may


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268 Studies in History and Philosophy of Modern Physicstake place on the base space, only isotropic light-cone (orthogonal) coordinatesystems are admitted on the tangent space spanned by second-rank Hermitianspinors. This is necessitated by the fact that in order to construct a spacetimefunction whose squared differential increment will remain invariant with respectto the actions of the Lorentz transformation group in both ordinary space andspin space, one should introduce at each spacetime point a 16-component fieldof orthonormal tetrads (a vierbein field) that transforms locally according tospinor representations of the Lorentz group, or, alternatively, consider a 4-quaternion field whose components transform globally as a contravariant four-vector in spacetime and as a second-rank spinor (of the type q,fi = QI Q @iwith o( = 1,2, B = l, 2 denoting the spinor variables of P and its complexconjugate) in spin space. The appropriate mechanism of the former was brieflydiscussed in Section 4; the latter is analysed in Karakostas (1996). Furtherexplanations are also given in 59 below.5. It is rather peculiar that on the basis of the conjunction (A)A(B)A(C)A(D),whose individual components are either false or lack truth value, Zangari finallyconcludes:

(E) Hence, both Galilean and z-transformations cannot be defined on the spinor.Although the inclusion of Galilean transformations spoils (E), the rest of it iscorrect. Spinor considerations are trivially unaffected by Galilean transforma-tions.The errors and inconsistencies in Zangaris paper--especially the failure toaccount for the relationship between ordinary space and spin space relative tothe spinor transformations of the Lorentz grouphave given rise to the paper ofGunn and Vetharaniam, who, by way of refuting Zangari, claim to have deriveda Dirac equation that is expressible in terms of arbitrary synchrony relations.We shall show that the authors derivation of an arbitrary synchrony Diracequation is substantially mistaken. This is explicitly demonstrated in 9 below.First, a brief evaluation of the overall outlook of the authors paper is in order.6. Central to Gunn and Vetharaniams paper is the following claim that repeatsitself throughout the text; in its stronger, conclusive form, it reads as follows:

Specifically,we have found to be false [ ] that a complex representation of spacetimepoints is required to describe spin-l/2 particles. (pp. 607-608)The aforementioned claim is regarded to be a direct consequence of the authorsderivation of a Dirac equation that takes place

(F) In the 4-dimensional real representation of spacetime points associated withSO(1,3), rather than in the complex representation associated with SL(2, c). (p. 605)This is, however, unfortunate. The authors alleged derivation of a Diracequation on the basis of(F) contradicts deeply established facts of fundamental


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Cowvenfionulify of Simultaneily 269

physics. Assertion (F ) is impossible, within the context of Gunn and Vethara-niams considerations, on both physical and mathematical grounds.The physical impossibility of(F) arises out of the fact that the group SO( l(3)

is incapable of accounting for the transformation properties of half-integralspin particles under Lorentz transformations, since it contains no double-valuedrepresentations. The mathematical impossibility of(F) is due to the fact that thetransformation properties of a Dirac spinor can only be accounted for in termsof the unimodular representations of the group SL(2, c) 8 SL(2, c) defined overa four-dimensional complexified Minkowski space.

Contrary to what Gunn and Vetharaniam assert, the essentiality of the latterbecomes apparent at an even basic mathematical level by noting that the operatory~a/ax~ occurring in the Dirac equation (ya/axj - ill)+ = 0, (p = m&z)simply amounts to the operation of first-order differentiation in the complexplane. This stems from the fact that the Dirac operator is algebraically a bi-quaternion, which, by definition, is scaffolded by means of the quantity oPxj [cf.equation (9)], and which, in turn, constitutes a four-dimensional generalisationof the complex number, x + iy. I27. Gunn and Vetharaniam make further the claim that

the complex representation is [ ] less powerful than the [real] 4-dimensional one(P. 606)

This we dispute. Even leaving aside topological considerations of the relevantaction groups, we have conclusively argued in Section 4 (see also referencestherein) that spinors-being complex 2-vectors-are the most general andfundamental mathematical objects to underlie a theory that is consistent withthe principle of special or general relativity. The basic point is simple: all scalars,real space-time vectors or tensors (of any rank) can be constructed in termsof these complex 2-vectors, their complex conjugates and their duals, whereas,spinors themselves need not (and for odd-rank spinors cannot) be describablein terms of the aforementioned conventional quantities.8. It is interesting to see why Gunn and Vetharaniam thought that the complex

t2To avoid any misapprehension, we note that if one imposes the condition yp = -yp. where theoverbar denotes complex conjugation, all four y-matrices are purely imaginary and thus the Diracoperator is real. In this way, one arrives at the so-called Majorana representation of the Diracequation which is utilised to bring out the particleeantiparticle symmetry of the field operators.That is, if I&.({ s a solution of the Dirac equation in a Majorana representation, so is its complexconjugate rS/,v. It follows then that there exists the possibility of defining real four-componentspinors (Majordna spinors) in a Lorentz covariant fashion, However, this should not be regardedas dismissive of the complexification of Minkowski space, but rather as a contingency arising outof it and which is characteristic of the indefinite signature of its metric. The rich structure ofa complexified Minkowski space (MC = A4 x M), whose scalar product is in general defined byg(st + iyt. x2 + ~JQ) = g(xt, x2) - g(yt,y2) + i[g(.q.y2) + g(yl, x2)]. allows for the considerationof purely real, purely imaginary, complex and complex conjugated expressions of vector fields. Inrelation to Majorana spinors, however, we stress that a Majorana-Dirac equation yielding realeigenvalues cannot, except in the trivial case of a zero eigenvalue, be obtained from a variationalprinciple.


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270 Studies in History and Philosophy of M odern Physicsrepresentation is less powerful than the [real] 4-dimensional one. Because, theyassert

the complex representation [ ] is not able to distinguish between parity conservingand parity violating physics. (p. 606)On this basis, they

conclude that the 4-dimensional real representation is to be preferred over the complexrepresentation. (p. 607)

It is well known that no discrete transformation symmetries, like the parity-inversion operation mentioned by the authors, are included in the irreducible(complex) representations of either the Einstein group of general relativityor the Poincare group of special relativity, since irreducibility of a groupsrepresentation within the context of this theory implies covariance only in termsof a continuous (analytic) group of transformations-a Lie group. This originatesin the fact that the irreducible representations of the latter are generated by thesuccessive application of infinitesimal transformations starting from the identity.Consequently, discrete symmetries like parity inversion, or time reversal (or,charge conjugation in relativistic QFT) are undefined within the context ofthe irreducible two-component spinor formalism of the theory of relativity. Ofcourse, all these discrete transformations, as well as any combinations of themtwo or three at a time, do exist within the four-component Dirac bi-spinorformalism of a relativistically covariant field theory, which, by construction, isbuilt up from the two-component spinor variables that characterise the structureof the SL(2, c) group. Thus, the authors reasoning can hardly be used toquestion the fundamental status of the spinor representation of the Lorentzgroup, which, after all, is an indisputable fact of theoretical physics.9. The crux of Gunn and Vetharaniams paper rests with their claim ofhaving derived a Dirac equation for the electron spin that allegedly holds inreference frames described by arbitrary synchrony. We anticipated in Section 4,on general grounds concerning the topological characteristics of the spinorgroup, that such an endeavour cannot be consistently carried out if the Diracspinor is to be regarded as a well-defined mathematical object that satisfiescertain specified transformation properties under rotations, most notably, therelationship of the two-to-one homomorphism of the spinor group onto theLorentz group. We stressed in particular (see also Cartan, 1966, p. 152)that the spinor representation of the Lorentz transformation group cannot beconsistently defined under linear substitutions to oblique coordinate systems,independently of temporal or spatial dependence, if the spinor group is toretain its fundamental topological property of single-valuedness. However, notopological considerations appear in the paper of Gunn and Vetharaniam.Confining ourselves to the deliberations of the latter, we shall demonstratethat the authors derivation of an arbitrary synchrony Dirac equation is basedon a fallacy concerning Diracs defining condition of the relativistic invariance


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Conventionality of Simultaneity 271of his equation that is intimately related to the anti-commuting properties ofthe Y-matrices, i.e. employing the covariant format,

with YiYj + YjY/i= 0 fori f j (i, j = 0, 1, 2. 3) (23)

(Y)* = -(Y/C)* = 1, Y0+ = Y0, Y/X+ -yk (24)or, written in a compact form as

IYip Yjl = 2gijl, (25)where 1 designates the fourth-order identity matrix, the y; are the basis elementsof the matrix representations of SU(2) @ SU(2) [direct products of a certainrepreSentatiOn of the Pauli matrices of equation (9)] and gii (go0 = 1 = -gkk,k=l, 2, 3; gij = 0 if i # j) are the components of the Minkowski metric inits canonical (flat spacetime) form. Note that the anti-commutation relations:(1) derive exclusively from the factorization condition of a Dirac Hamiltonian(or, a Klein-Gordon equation) and the requirement that the Hamiltonian behermitian. Together with the Dirac equation, these are the basic relativisticequations for a free spin-l/2 particle.As one may suspect, Gunn and Vetharaniam derive their arbitrary synchronyDirac equation by considering synchrony-free forms of the yo and gij elements-parametrized by a synchronization variable n [our E of equation (2)]-and whichare reproduced here for the convenience of the reader [equations (7) and (4) oftheir paper, respectively] r3

(26)gOO= 1 _ 6ijninj, gOi = _ni6ii, $j = -6ji, (27)

We shall first demonstrate that the authors expressions of the y-matrices,defined in terms of arbitrary synchrony coordinates, violate the fundamentalproperty of Diracs matrices of forming a group. Note that the latter consistsof 16 independent products of the yi comprised of not more than four factorsand in which no two of the y; are the same. We denote collectively the elementsof Gt6 by

{YR} = 113 YisYiYjr Ys~YiYsIJ (28)where y5 = yoyt yz Ys. The complete irreducible Dirac group, because of its two-valued representation, is formed by the monomials {YR} and ( - ye } . They forma group GQ of 32 elements with centre Z2 = { 1, - 11. One also checks that theI3Note that an accurate reading of relation (27) demands that no Kronecker delta should appearin the first term; it is mathematically meaningless. In addition, the presence of the delta factorin the second term is redundant and ill-used. Note also that relation (27) is the contravariantform of our E-synchrony Minkowski metric; as such, it codifies essentially the kinematics of aGalilean-transformed observer in relation to reference frames moving relatively to each other withdirection dependent velocity EC (see e. g. Anderson and Stedman. 1992).


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212 Studies in Hist ory and Philosophy of Modern Phy sicsquotient group Gi6 = G32/z2 is abelian, since the products YR,YR2and Y&YR,in the original group simply differ in sign. Consequently, the product of any twoelements of G32 yield a member of the group within a factor of -1 or +i. Thisproperty, together with certain well-known properties of representations of finitegroups, lead essentially to Paulis fundamental theorem, according to which, theDirac (Clifford) algebra of Minkowski space (generated by the yi matrices ofG32) has a unique (up to equivalence) irreducible complex representation (e.g.Messiah 1961, p. 818).

It is not difficult to see that the authors non-canonical expressions of the y-matrices fail to conform with an E-variation of Paulis theorem; try, for instance,the combination yooysof (26) to realise that its product lies outside the rangeof its corresponding G32. The origin of this failure is located in the fact thatthe authors y-matrices do not form a mutually anticommuting set of matricesypyv + y,,yfl = 0 (/A f v = 0, 1,2, 3), being unable thereby of closing an E-extended Dirac algebra. In fact, a firm inspection of the authors derivation ofan arbitrary synchrony Dirac equation-and which is based on a pedagogicalsketch of a standard derivation given by Griffiths (1987, pp. 214215)-revealsthat Gunn and Vetharaniam treat Diracs anti-commutation relations y;~~ +yiyi = 2gij 1 as being merely the sum of the product of two independent factors,disregarding the anti-commutativity of y/iyj + yjyi = 0 that is essential for thefactorisation of a Dirac Hamiltonian [check equations (3), (5) and (6) of theauthors paper]. l4One cannot refrain from observing, on the other hand, that Diracs anti-commutative relations, by their very nature, and by virtue of the bijective mapY> - y(x), require the y; to be isomorphically related to a four-dimensionalorthonormal basis, {y(ei) }, spanning Minkowski space ; only then its differentelements would form a mutually anti-commuting set having squares unity, sothat their distinct products would be able to satisfy the relations y(ei)y(ej) +y(ej)r(ei) = 2g(ei, ej) = 2gij with gij = 0 if i + j and Y2(ei) = g(ei, ei) = 1otherwise. Thus, the authors purely formal construction of a set of yi as inequation (26) which might superficially seem to satisfy Diracs fundamentalrelation riyj + yjyi = 2gijl for suitably chosen non-standard forms of ~0 andgij-without however satisfying the algebraic anti&commutative properties ofthe yi themselves that are essential for their role as group generators, mostnotably, YOY~ J+YO = 0, (k = 1,2,3)- guarantees neither faithfulness norirreducibility or uniqueness (in short, existence) of the associated Lie algebraSL(2, c). The severity of failing in this cannot be over-emphasised: we obtaincomplete knowledge of every free fermionic states and their behaviour once allthe irreducible representations of the (Poincare-extended) SL(2, c) group arefound.r4 In fact, Gunn and Vetharaniams derivation of their equation (5) is incorrect, since contrary tothe authors assertion, their equation (3) would have contained linear cross terms in the momentumoperators pp,pv, if the arbitrary-synchrony form of the Minkowski metric had been explicitlyused; those terms would have derived their origin from the non-zero off-diagonal elements of themetric.


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Conventionality of Simultaneity 273The reader may also be assured that the extension of Diracs equation toa Riemannian (general relativistic) space is naturally realised by means of the

vierbein formalism discussed in Section 4. Briefly, in considering a spacetimemanifold with arbitrary metric structure g,, (not merely with arbitrary syn-chrony dependence) both the Dirac algebra generators and the spacetime metriccomponents are assumed to be functions of spacetime

{YP(X)>Y,(X)1 = 2g,,(x)l. (29)However, by use of orthonormal tetrad frames {h,(x)}, functioning as anholo-nomic coordinates at the tangent space of each point in the manifold, thespacetime dependence of both y and g is shifted entirely into the vierbein fieldh :v

Y,,(X) = h,Wya/, g,,(x) = h~bdhh,(x)g,~. (30)Thus, the four point-dependent matrices of equation (30) satisfy the anti-commutation rule

Ih,(x)ya, &x&,1 = 2h;(x)h:(x)g,A (31)provided that the y0 are the usual, flat-space, generators of the Dirac algebra

{ya, yb} = &?,bl, (32)and the g,h = diag(1, -1, -1, -1) are the components of the Minkowski metric ofthe tangent space (see e.g. Hehl et al., 1976).Just as the vierbeins may be considered as constituting the transformationmatrix between the coordinate basis and an orthonormal basis of the tangentspace, the Dirac algebra generators may be regarded as the objects which connecta spin space basis to the orthonormal tangent space basis of spacetime. For thisreason they are sometimes referred to as soldering forms. Once a particularmatrix representation for the Dirac algebra generators has been chosen, itremains unaltered under the action of a local Lorentz transformation, fixingthereby the relationship between the tangent space basis and the spin space basis.The various matrix representations of the Dirac algebra are related by similaritytransformations y0 = SyiS-, with S being a non-singular 4 x 4 matrix elementof the algebra. The Dirac spinor field of four complex components is thenregarded to be lying in the carrier space of a y-representation of the algebra.retaining thus its well-defined properties.It may be worth adding in this connection that a full exposition of thepreceding analysis, which would allow us to consider the global behaviour ofthe Dirac field, involves introducing the Christoffel affine connection Ttfi todefine the covariant derivatives of vector field components in the coordinatebasis, D,, V,, = a,, V, - I$, I+ and the Fock-Ivanenko spin-affine connectionOg,, to define the covariant derivatives of spinorial vector field components inthe orthonormal basis of the tangent space, V,WQ = a,& - flZ,h,qb. Sinceboth the vierbein field and the fundamental structures of the Dirac spinorfield have vanishing covariant derivative V,h$ = 0 = Vpyv, the Riemanncondition safeguarding the covariant constancy of the spacetime metric is


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274 Studies in Hist ory and Philosophyof Modern Physicssatisfied, V,,gKn = 0. This ensures that the Riemann-Christoffel curvaturetensor is identical to the field strength tensor induced by the spin-gauge field,Rkuv = Kfpv, the latter being now interpretable as intrinsic spin space curvature.In this sense, general relativity may be viewed (in the tradition initiated byUtiyama, Kibble and Sciama ; Ivanenko and Sardanashvily (1983) provide anexcellent review) as a gauge theory of local Lorentz invariance in the tangentspace, with the spin connection functioning as the gauge field (see e.g. Chisholmand Farwell, 1989). The gauge is related to our freedom of choosing a differentrepresentation of the spinors and the spin basis at each spacetime point. Thisis equivalent to the physically reasonable requirement that the predictions ofa theory involving spinor fields should not depend on the particular localrepresentation of the Dirac algebra generators.Acknowledgements- It is a pleasure to acknowledge useful conversations with Michael Redhead,Michael Green and John Stewart.

ReferencesAnderson, R. and Stedman, G. E. (1977) Dual Observers in Operational Relativity,

Foundations of Physics 7, 29-33.Anderson, R. and Stedman, G. E. (1992) Distance and the Conventionality ofSimultaneity in Special Relativity, Foundations of Physics Letters 5, 199-220.Bargmann, V. (1954) On Unitary Ray Representations of Continuous Groups, Anna/sof Mathematics 59, 1116.Basri, S. A. (1965) Operational Foundation of Einsteins General Theory of Relativity,

Rev iew s of Modern Physics 37, 288-3 15.Brown, H. R. (1990) Discussion: Does the Principle of Relativity Imply Winnies (1970)Equal Passage Time Principle?, Philosophy of Science 57, 3 13-324.Cartan, E. (1966) The Theory of Spinors (Cambridge, MA: MIT Press).Chisholm, J. S. R. and Farwell, R. S. (1989) Unified Spin Gauge Theory of Electroweakand Gravitational Interactions, Journal of Physics A 22, 1059-1071.Coleman, R. A. and Kortt?, H. (1991) An Empirical, Purely Spatial Criterion for thePlanes of F-Simultaneity, Foundations of Physics 21,417-437.Dewitt, B. (1961) Dynamical Theory of Groups and Fields, in B. Dewitt and C.Dewitt (eds) Relativit y, Groups and Topology (London: Blackie & Son), pp. 587-820.Einstein, A. (1952) The Principle of Relativity, W. Perrett and G. B. Jeffery (trans.) (NewYork: Dover).Friedman, M. (1983) Foundations of Space-Time Theories Princeton, NJ: Princeton

University Press.GelFand, I. M. Milnos R. A. and Shapiro, Z. Y. (1963) Representations qf the Rotationand Lorent z Groups and t heir Applications (New York: Pergamon).Geroch, R. (1968) Spinor Structure of Space-Times in General Relativity I, Journalof Mat hemat ical Physics 9, 1739-l 744,Giannoni, C. (1978) Relativistic Mechanics and Electrodynamics Without One-WayVelocity Assumptions, Philosophy of Science 45, 1746.Giickeler, M. and Schiicker, T. (1987) DifSerent ial Geometry, Gauge Theories, and Grav it yCambridge: Cambridge University Press.Griffiths, D. (1987) Int roduction t o Element ary Particles New York: Wiley.Griinbaum, A. (1973) Philosophical Problems of Space and Time, 2nd enlarged edn(Dordrecht: Reidel).Gunn, D. and Vetharaniam, I. (1995) Relativistic Quantum Mechanics and theConventionality of Simultaneity, Philosophy of Science 62, 599-608.


27/28

Conventionality of Simultaneity 275Havas, I? (1987) Simultaneity, Conventionalism, General Covariance, and the SpecialTheory of Relativity, General Relativity and Gravitation 19, 435453.Hehl, F., Heyde, P., Kerlick, D. and Nester, J. (1976) General Relativity with Spin and

Torsion: Foundations and Prospects, Reviews of Modern Physics 48, 3933416.Ivanenko, D. and Sardanashvily, G. (1983) The Gauge Theory of Gravity, PhysicsReports 94, 145.Janiq A. I. (1983) Simultaneity and Conventionality, in R. S. Cohen and L. Laudan(eds), Physics, Philosophy and Psychoanalysis (Dordrecht: Reidel), pp. 101-l 10.Karakostas, V. (1996) Spinor Fields, Covariance Principles, and the Conventionality ofSimultaneity, Philosophy of Science.Kim, Y. and Noz, M. (1986) Theory and Applications ofthe Poincare Group (Dordrecht:Reidel).Leighton, R. B. (1959) Principles of Modern Physics (New York: McGraw-Hill).Magnon, A. (1987) Existence and Observability of Spinor Structure, Journal ofMathematical Physics 28, 13641369.Malament, D. (1977) Causal Theories of Time and the Conventionality of Simultaneity.Nazis 11, 293-300.Messiah, A. (1961) Quantum Mechanics ZZ (Amsterdam: North-Holland).Misner, C. W., Thorne, K. S. and Wheeler. J. A. (1973) Gravitation (San Francisco:Freeman).Mittelstaedt, P. (1977) Conventionalism in Special Relativity, Foundations of Physics7, 573-583.Moller, C. (1972) The Theory of Relativity, 2nd edn (Oxford: Clarendon Press).Naimark, M. A. (1964) Linear Representations of the Lorentz Group (New York:Pergamon).Norton, J. D. (1992) Philosophy of Space and Time, in W. C. Salmon et al. (eds)Introduction to the Philosophy of Science (Englewood: Prentice-Hall), pp. 179923 1.Penrose, R. (1967) Twistor Algebra, Journal of Mathematical Physics 8, 345-366.Penrose, R. and Rindler, W. (1984) Spinors and Space-Time Z Two-Spinor Cakulus andRelativistic Fields (Cambridge: Cambridge University Press).Redhead, M. L. G. (1993) The Conventionality of Simultaneity, in At the Cutting Edgeof the Philosophy of Science (Pittsburgh: University of Pittsburgh Press), pp. 103-l 28.Reichenbach, H. (1957) The Philosophy of Space and Time (New York: Dover). (OriginalGerman edition published in 1928.)Reichenbach, H. (1969) Axiomatization of the Theory of Relativity, (Berkeley: Universityof California Press). (Original German edition published in 1924.)Sachs, M. (1967) On Factorization of Einsteins Formalism Into a Pair of QuaternionField Equations, Nuovo Cimento 47A, 759-769.Salmon, W. C. (1977) The Philosophical Significance of the One-Way Speed of Light.No& 11, 253-292.Stewart, J. (1991) Advanced General Relativity (Cambridge: Cambridge University Press).Torretti, R. (1983) Relativity and Geometry (Oxford: Pergamon).Treder, H. J., von Borzeszkowski, H., van der Merwe, A. and Yourgrau, W. (1980)Fundamental Principles of General Relativity Theories: Local and Global Aspects ofGravitation and Cosmology (New York: Plenum Press).Ungar, A. (1986) The Lorentz Transformation Group of the Special Theory of RelativityWithout Einsteins Isotropy Convention, Philosophy of Science 53, 395402.Ungar, A. (1991) Formalism to Deal with Reichenbachs Special Theory of Relativity.Foundations of Physics 21,691-726.Wald, R. (1984) General Relativity (Chicago: University of Chicago Press).Weinberg, S. (1972) Gravitation and Cosmology: Principles and Applications of the GeneralTheory of Relativity (New York: Wiley).Weinberg, S. (1995) The Quantum Theory of Fields I: Foundations(New York: CambridgeUniversity Press).


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2 1 6 Studies in Hist ory and Philosophyof Modern Phys icsWigner, E. P (1939) On Unitary Representations of the Inhomogeneous LorentzGroup, Annals of Mathematics 40, 149-204.Winnie, J. (1970) Special Relativity Without One Way Velocity Assumptions, Philosophyof Science 37, 81-99; 223-238.Winnie, J. (1986) Invariants and Objectivity, in R. G. Colodny (ed.), From Quarks toQuasars: Philosophical Problems in Modern Physics, Pittsburgh Series in Philosophyof Science, Vol. 7, pp. 71-180.Zangari, M. (1994) A New Twist in the Conventionality of Simultaneity Debate,

Philosophy of S cience 61, 267-275.Editors Note-A reply to this paper by D. Gunn will be published in the nextissue of this journal together with a rejoinder by V. Karakostas.

karakostas_the conventionality of simultaneity in the light of the spinor representation of the...

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