- Email: [email protected]
The conformal group and causality
ANNALS
OF PHYSICS:
47,468-480
(1968)
The Conformal
Group
and Causality*
JOE ROSEN+ Physics
Department,
Brown
University,
Providence,
Rhode Island
02912
Space-time transformations in Minkowski space are interpreted as neither “active” nor “passive” but as transformations of events and trajectories of processes with respect to the same observer and in the same flat space-time. The distinction is significant for the nonlinear special conformal transformations, which violate causality in the sense that the causal relations among events of a process are not always the same as among corresponding events of a transformed process. Aspecial conformal transformation canalways be found to transform the separation of any pair of events of nonlightlike separation from timelike to spacelike and vice versa. The transformation of trajectories under special conformal transformations is studied. Examples of reversal of temporal ordering by special conformal transformations are presented. Finally it is argued that none of this should prevent the conformal group from being physically useful.
I. INTRODUCTION
The space-time conformal group, the group of conformal space-time transformations listed in Section II, has certain features which make it especially interesting (Z)-(C~).~-~ It is the lowest dimensional semisimple group containing the PoincarC (=inhomogeneous Lorentz) group. This enters the conformal group in the list of candidates for a generalization of the PoincarC group as the space-time symmetry group of physics (3), (8)-(Z3).2 Another point is that the physics of massless particles is governed by the vanishing of the line element of their world lines, and the conformal group is the largest group preserving null line elements (8).2*3 So there arises the possibility that the conformal group is an exact space-time symmetry group for massless particles. This should have bearing not only on photons and neutrinos but also on massive particles of such high energies that their rest masses are negligible in comparison. The conformal group might then be a badly broken approximate symmetry for massive particles which improves as the energies involved increase (3).2 A similar situation exists for SU, as an internal *This research was supported in part by the U. S. Atomic Energy Commission No. NYO-2262TA-161). + On leave of absence from Tel-Aviv University. 1 Schouten (I) includes a historical review. * See references given by Kastrup (3) to his and other work. a See references given by Fulton, Rohrlich, and Witten (4), (5). 46%
(Report
THE CONFORMAL
GROUP AND CAUSALITY
469
symmetry for hadrons. An additional consideration, related to the last, is that relativistic wave equations become conformal-invariant when the mass term is removed (2) (3), (5), (9), (IO), as was long known for the Maxwell equations (II)-(13). An objection to the possible physical usefulnessof the conformal group has been raised based on the claim that the conformal group violates causality (14). This is the central theme of the present work. We show in the following that the space-time conformal group does violate causality in a certain sense,and we look into several aspectsof this violation. But, basedon our interpretation of space-time transformations, we conclude that the violation of causality by the conformal group should not prevent it from possibly being physically useful. In Section II the connected conformal space-time transformations are briefly reviewed. Section III discussesthe interpretation of space-time transformations. The effect on causality of the special conformal transformations is considered in general in Section IV. It is proved in Section V that for an arbitrary pair of events of timelike or spacelike separation a special conformal transformation can always be found which will change their separation to spacelike or timelike, respectively, and it is shown how to construct this transformation. Section VI treats the transformation of trajectories and especially their velocities under specialconformal transformations (IS). Two examples where reversal of the temporal ordering of events is brought about by a special conformal transformation are presented in Section VII. Section VIII resolves an apparent contradiction, pointed out by Giirsey,* between a result on the transformation of trajectories and a result on the transformation of events. Our conclusions are brought forth in Section VIII.
II. CONFORMAL
SPACE-TIME
TRANSFORMATIONS
The connected group of conformal space-time transformations consists of the following transformations:5 Lorentz transformations
4 I am grateful to Professor F. Giirsey for raising this point. 5 The inversion xp + f@ = -xyxp can be used together with the translations [Eq. (2)] to form the special conformal transformations [Eq. (4)1, but it itself is not an element of the connected group (i.e., that part of the group which is connected continuously with the identity transformation).
470
ROSEN
and special conformal
transformations
xu -+ 3 = (XN - C~X2)/O(X), (4)
u(x) = 1 - 2c * x + c2x2,
where indices run from 1 to 4, x“ = (t, x), and g,, = (+ 1, - 1, - 1, - 1). (The speed of light is taken as unity.) In the following we shall be concerned mainly with the 4-parameter Abelian subgroup of special conformal transformations [Eq. (4)], the only nonlinear transformations among Eqs. (l)-(4). Under these transformations for an infinitesimal interval dx2 -+ dE2 = dx2/a(x)2,
(5)
and for a finite interval (x1 - x2)2 + (Xl - X,)2 = (x1 - X&J(Xl)
III.
INTERPRETATION
OF
SPACE-TIME
a(x,).
(6)
TRANSFORMATIONS
It is usual to choose one of two interpretations for space-time transformations such as Eqs. (l)-(4): they can be considered either “passive” (coordinate) transformations or “active” (point) transformations (5). In the “passive” interpretation, space-time together with the world lines and events embedded in it are considered unafFected but expressed with respect to a new coordinate system, i.e., viewed through the eyes of a different observer. In the “active” interpretation, space-time and its contents are considered transformed by a mapping with respect to the same coordinate system or, one might say, under the nose of the same observer. We would like to suggest a third way of interpreting transformations of this kind. This new interpretation is most likely completely superfluous, if one is concerned only with linear transformations [Eqs. (l)-(3)] in flat space-time. But when considering the nonlinear special conformal transformations [Eq. (4)], the two usual interpretations exhibit rather unpleasant features. A “passive” transformation then introduces a curvilinear coordinate system, while an “active” transformation destroys the flatness of space-time. The unpleasantness lies in the fact that, since everyday physics experiments are interpreted “flatly”, the special conformal transformations should be consistent with flat space-time and rectilinear coordinates, if they are to be useful, say, in the realm of particles. To remedy this, we suggest interpreting space-time transformations as leaving both space-time and the coordinate system unaffected, the transformations serving to map only the world lines and events of a physical process under consideration. The same
THE CONFORMAL GROUP AND CAUSALITY
471
observer then seesa different processtaking place but in the sameAat space-time background. We call a transformation interpreted this way a “physical” transformation. So in summary, as long as we confine our attention to linear transformations in flat space-time, the distinction between “physical” and “active” transformations is quite academic. But nonlinear transformations make the distinction important, because then a “passive” transformation introduces curvilinear coordinates, an “active” transformation introduces space-time curvature, while a “physical” transformation leaves the same rectilinear coordinates and flat space-time with only the experiment being changed. This then is the way we shall interpret space-time transformations in the following. The sameobserver will observe a transformed processin the samespace-time. If every physically valid process thus happens to be transformed into another physically valid process by a given transformation or group of transformations, we have a symmetry of physics.6
IV. CAUSALITY
The causal relation of two events is determined by the two assumptions that no influence travels faster than light and that influences do not reach backwards in time. If x1 and x1 are the coordinates of two events, then their separation is called timelike, lightlike, or spacelike, if (x, - -x~)~ is positive, null, or negative, respectively. In the latter case the two events are not causally related; otherwise they are. When a causal relation exists, its full determination requires knowledge of the temporal ordering of the two events, Sgn(t, - fJ, specifying which event can inAuence the other. The group of Poincart transformations [Eqs. (1) and (2)] preserves (x1 - x2)” always and Sgn(t, - tJ when (x1 - x# > 0, so it preserves the causal relation of pairs of events. The group of dilatation transformations [Eq. (3)] always preserves both Sgn[(x, - x,)~] and Sgn(l, - t?), so that it too preservesthe causal relation of pairs of events. Combining these, the group of similitude transformations [Eqs. (l)-(3)] therefore preserves the causal relation of pairs of events. In fact, Zeeman (16) has shown that this is the largest connected group of space-time transformations with this property. Thus the special conformal transformations do not preserve the causal relation of pairs of events (2), (14). In fact, we show in the following section that for any given pair of events a special conformal transformation can be found to transform their separation from timelike to spacelike or vice versa. [A lightlike separation is preserved, according to Eq. (6), except when (i This is the way internal symmetries are of necessity always treated.
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the transformation is singular for x1 or xz , i.e., except when a(~,) = 0 or u(xz) = 0.1 In Section VII we show how reversal of temporal ordering under a special conformal transformation can occur. Does the conformal group then violate causality ? If by this question we mean, “Can the special conformal transformations change the causal relations of pairs of events ?” then from our point of view the answer is affirmative. According to the “physical” picture the causal relation is determined by Sgn[(xl - xJ2] and Sgn(r, - t2) before the transformation and by Sgn[(x, - ?2)2] and Sgn(f, - tz) after it and is not always preserved.’
V. CHANGE
OF SIGN OF (xl -- x$
Under a special conformal transformation events having coordinates x1 and x2 we have
[Eq. (4)] for an arbitrary
pair of
(Xl - x2)2 + (X1 - X,)2 = (x1 - x2)“/cr(xJ u(x2).
(6)
If (x, - x2)2 f 0, its sign will be changed under the transformation only if I and 0(x2) can be made to have opposite signs. We shall show that this can always be done; i.e., for (xi - x2)2 $ 0 it is always possible to find c such that U(Xl) 4x2) < 0. First express u(x) in various ways as follows: u(x) = 1 - 2c . x + c2x2
(7)
= c”(c/c” - x)”
for
c2 f 0
(8)
= x2(c - x/x”)”
for
x2 f 0.
(9)
Case I:
Assume c2 f 0. Then from Eq. (8) u(xl) u(x2) = c”(c/c” - x1)2 (c/c” - x2)2. Subcase 1: If both t, f
(10)
t, and x1 f x2, define
a = (t2 , Xl>, b = 01 , x2).
Subcase 2: If tl = t, = t but x, + x2 , put a = (t + T, x1), b = (t + T, x2),
where 0 < T2 -=c(x1 - x2)2.
Subcase 3: If x1 = x2 = x but t, f t, , define a = (t2 , x + X), b G (tl , x + X), where ’ Compare with Kastrup (3), who holds the “active” point of view.
0 < X2 < (tl - Q2.
THE
CONFORMAL
GROUP
AND
473
CAUSALITY
Now for all three subcases (a - .x1)2 > 0, (a - x# < 0, (b - x$ < 0, and (b - xJa > 0. So take c/c” = a or b, i.e., c = a/a2 or b/b2. If a2 = 0, take c = b/b2; if b2 = 0, take c = a/a2. If u2 = b2 = 0 and subcase 2 or 3 holds, change T or X within the allowed freedom to make a2 and b2 nonzero. But if a2 = b2 = 0 and subcase 1 holds, things are handled differently. Then, since a2 + b2 = xl2 + xz2, we have xl2 = --x22. Case ZZ Assume xl2 = -xZ2 f
0. Then from Eq. (9),
U(Xl) 0(x2) = X12X2”(C - X~/X~“)“(C - x2/x22)2 = -x14(c
- Xl/Xl”)“(C
- x2/x22)2.
(11)
This will be negative for any c with sufficiently large 1ca j. Case ZZL Assume a2 = b2 = xl2 = x 22 = 0. Then t,2 =
t,2 =
x 12 =
x22,
(12)
and from Eq. (7) 0(x1) U(X2) = (1 - 2c . x&l
- 2c * x2).
(13)
Take c = h(x, - xc), where h is a number. Then U&xl) 0(X2) = (1 + 2lq
* x&l
- 2h.q * x2).
(14)
This will be negative for any h with sufficiently large j h I, unless x1 . xZ = 0. But x1 . x2 = 0 and Eq. (12) imply (x, - Y~)~ = 0, which contradicts our assumption. This covers all cases and proves our assertion.
VI.
TRANSFORMATION
OF
Under a special conformal transformation into a trajectory ji = Z(i). The transformation by putting c” = (6. c):
TRAJECTORIES
(15)
a trajectory x = x(t) is transformed formulas are obtained from Eq. (4)
t -+ f = [t - b(t’ - X2)]/@, x), s + x = [x - c(t2 - x2)]/u(t, x),
(1%
474
ROSEN
the inverse being f - t = [i + b(i2 - jP)]/a(--7, f - x = [E + c(i2 - nq/+r,
-ji), -jz),
(16)
where a(t, x) = 1 - 2(bt - c * x) + (b2 - c2)(t” - x2) = (1 - bt + c * x)” - (bx - ~t)~ + (c x x)“.
(17)
The following relations hold: u(-i,
-f-i) = l/cr(t, x),
1 + bi -
c
. z = (1 - bt +
c
bjz - ct = (bx - ct)/o(t,
.
X)/U@,
x),
x),
c x 2 = c x x/o@, x).
The velocities of the original and transformed trajectories are v = dx/dt and t = dji/di, respectively. The relation between them is given by the transformation v+t
= N/D
= N/ii,
(19)
where N = [a@, x) - 2(c x x)“] v + 2[(1 - bt + c * x) + v * (bx - ct)](bx - ct) + 2[v, c, x] c x x + 2[(bx - ct) + (1 - bt + c * x) v] x (c x x), (20) D = a(t, x) + 2(bx - ~t)~ + 2[(1 - bt + c . x)(bx - ct) - (bx - ct) x (c x x)] * v,
and, using the relations of Eq. (18), m = [u(-i, +
v
-ji) l
- 2(c x E)“]v + 2[(1 + bi - co 5)
(bjz - ci)](bz -
ct)
+ 2[v,
+ 2[(bP - ct) + (1 + bi - c i3 = u(-i,
-9
+ 2(bz -
ci)”
l
Z)V]
C,
ji]
c
x z
x (c x E),
+ 2[(1 + bi -
- (bE - cf) x (c x ji)] . v.
c
l
Z)(bji - ci) (21)
From Eqs. (20) and (21) it follows that D2 - N2 = (1 - v2) a(t, x)~, iP - R2 = (1 - v2) 0(-i, -jz)z,
(22)
THE
CONFORMAL
GROUP
AND
CAUSALITY
475
from which we obtain 1 -
y2 -+
1 -
82 = (1 - v”) a(t, x)2/D2
= (1 - v”) a(-&
-Z)Z/B”.
(23)
Equations (19) and (23) then give, for v2, G2 # 1, v/(1 - v2)li2 + 3/(1 - 82)1/2 = &N/c@, x)(1 - v2)1/2 = -&tm/u(-i, --jz)(l - v2)1/2.
(24)
The sign in Eq. (24) depends on which branch of the transformed trajectory is being considered. Either as a byproduct of Eq. (23) or more simply from Eq. (5) we obtain the result that, except when the special conformal transformation under consideration becomes singular, (25)
That is, the property of a motion that it is slower than, equal to, or faster than the speed of light is preserved under special conformal transformations.* Considering now geodesic trajectories (straight lines) x = vt + x0, where v and x0 are constant vectors, the two interesting cases are v2 = 1, null or lightlike geodesics corresponding to trajectories of light rays and free massiess particles, and v2 < 1, timelike geodesics corresponding to trajectories of free massive particles. In the first case it is not difficult to see that under a special conformal transformation a null geodesic trajectory is transformed into a null geodesic trajectory. This suggests the possible relevance of the conformal group to the physics of light rays, massless particles, and extremely relativistic particles whose masses are negligible in comparison with their energies (3),2 as mentioned in the introduction. In the case of timelike geodesics it has been shown that the transformed trajectories under special conformal transformations correspond to those of charged particles in constant electromagnetic fields of electric type (6). In the nonrelativistic limit this is motion of constant acceleration. In genera1 the function G(i) is rather complicated, but under certain conditions V/(1 - V2)1/2becomes a linear function of i. The following are such cases: (a) For the trajectory x = 0, a particle at rest at the origin, F/(1 - 92)1/2 = f2ci. 8 I would like to thank Dr. L. Caste11 for comments concerning this point.
(26)
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ROSEN
If c = 0, b arbitrary, the transformed trajectory again represents a particle at rest at the origin, j2 = 0, and i = t/(1 - bt). (b) For the trajectory x = x0 f 0, a particle at rest at x0 , when b = 0, C/(1 - 92)1/2 = &2[(1 + 2c x&c - C2X& l
(27)
The transformed trajectory will again represent a particle at rest, C = 0, only when (1 + 2c . x&c = c2x, . Besides the trivial solution c = 0 (the identity transformation) the only other solution to this equation is c = -x,,/x,,~, The transformed trajectory then becomes jz = -x0 , and t = -xo2/t. (c) For the trajectory x = C-t, v # 0, a free massive particle passing through the origin, when c I/ v, say c = av, t/(1 - t2)li2 = -J-[1 + 2(b - a)t] v/( 1 - v2)lj2.
(28)
Note that 5 = v only when b = a. The transformed trajectory will then again represent free massive particle motion through the origin, P = vt.
VII.
EXAMPLES
OF
REVERSAL
OF
TEMPORAL
ORDERING
Starting with the trajectory describing a light ray or free massless particle x = t,
y=z=o
we perform a special conformal transformation 0, 0) and obtain
7
(29)
characterized by cU = (0, c, > 0,
t = t/( 1 + 2&X) = t/(1 + 2&J), x = x/(1 + 2C&) = t/(1 + 2c,t) = f, y=z=o.
(30)
The resulting trajectory is the same as the original one. But the images of events along the original trajectory do not retain the same temporal ordering. This is illustrated in Figs. l(a) and l(b), where Fig. l(a) shows the original trajectory and Fig. l(b) shows the transformed one. Events are marked off along the original trajectory and their images are indicated along the transformed trajectory. Note the reversal of temporal ordering of events B and D for example. Another interesting example is afforded by the trajectory of a massive particle at rest at the origin x = 0.
(31)
THE CONFORMAL GROUP AND CAUSALITY
Performing the same special conformal transformation
477
as above, we obtain
i = t/(1 - C,V), x = -c,t”/(l - C,2P), yzz=().
(32) b
i
FIG. 1. (a) a massless particle trajectory with events marked off along it. A special conformal transformation characterized by cfi = (0, c, > 0, 0, 0) transforms it into the same massless particle trajectory, shown in (b), where the images of the original events are indicated. Note the reversal of temporal ordering of, e.g., events B and D.
a
b t
FIG. 2. (a) a trajectory of a massive particle at rest with events marked off along it. A special conformal transformation characterized by CC = (0, cz > 0, 0,O) transforms it into the hyperbolic trajectory describing two massive particles shown in (b), where the images of the original events are indicated. Note the reversal of temporal ordering of, e.g., events B and D. Also note the possibility of events such as B and E becoming simultaneous.
478
ROSEN
Eliminating I from Eq. (32), the transformed trajectory proves to be a hyperbola in the x-t plane (X -
1/2c,)~ - t2 = l/4&2.
(33)
The velocity C = djz/df along this trajectory obeys [from Eq. (26)] C,/(l - Ce2)l/2 = *2c,t.
(34)
The situation is pictured in Figs. 2(a) and 2(b), where Fig. 2(a) shows the original trajectory and Fig. 2(b) shows the transformed one. In this case the transformed trajectory describes a process involving two particles, where the motion is that of two noninteracting particles of equal and opposite charge-to-mass ratios in the same constant electric field (6). Events are again marked off along the original trajectory and their images are shown along the transformed trajectory. Note the reversal of temporal ordering of events B and D, for example. Also note that events such as B and E can become simultaneous.
VIII. RESOLUTION
OF APPARENT
CONTRAJXCTION
Giirsey4 raised the following question: Any two events on a timelike trajectory (and therefore having necessarily a timelike separation) can be transformed by a special conformal transformation so that their images have a spacelike separation (proved in Section V). The image of the trajectory remains timelike, though (proved in Section VI), apparently forcing the images of the two events to have a timelike separation. How can these two facts be reconciled ? Examine the effect on (F1 - J2)z in Eq. (6) of a continuous sequence of special conformal transformations from the identity to the one mentioned above. Any change of sign of (Zl - X2)2 is brought about by its passing through infinity, i.e., by either a(~,) or u(x2) passing through zero. Therefore, there must exist on the original trajectory between the two events under consideration at least one event for which the special conformal transformation becomes singular. The transformed trajectory does not then run directly from the image of the earlier event to the image of the later, but from the image of the earlier event it runs off to infinity and reaches the image of the later event along another branch. It is this unconnectedness that allows the transformed trajectory to remain timelike (except, it might be added, at images of singular events, where it tends to lightlikenessg) without the necessity of a timelike separation between all pairs of events on it. The situation might be made clearer with the help of Figs. 2(a) and 2(b). The original trajectory in Fig. 2(a) is timelikepar excellence. The transformed trajectory O See
Section
VI.
THE CONFORMAL GROUP AND CAUSALITY
479
in Fig. 2(b) is also timelike, but tends to lightlikeness at the images of events C and F, the two singular events for the special conformal transformation in effect. Note that pairs of transformed events having spacelike separation are always located with each member on a different branch of the transformed trajectory, and the corresponding original pairs of events therefore have a singular event between each two of a pair. Consider for example events B and E.
IX. CONCLUSIONS
In the “physical” approach to space-time transformations, as described in Section III, the special conformal transformations violate causality in the sense of Section IV. But there seemsto be no reason why this fact in itself should invalidate the conformal group as a possible symmetry of physics. In fact, from the “physical” point of view violation of causality is quite harmless.It meansthat the causal relations among events of a processmight differ from the causal relations among corresponding events of a conformal-transformed process, and there is nothing so terrible about that. The useful transformation of time reversal, for example, doesjust such a thing. Under conformal transformations there can even occur caseswhere a processinvolving particle creation is transformed into a process involving particle annihilation (which is familiar from the time reversal transformation), or where a process involving a certain number of particles is transformed into a process involving a different number of particles (see Section VII). But from the “physical” viewpoint there is no objection to this happening. The validity of the conformal group as a symmetry of physics should, we feel, be tested by examining whether under it every physically valid processis transformed only into physically valid processes,however different (from the causal or any other aspect) the transformed processesare from the original or from each other.
ACKNOWLEDGMENTS
I would like to express my thanks to Professors Nathan Rosen and John Stachel for discussions. RECEIVED: September 20, 1967
REFERENCES I. J. A. SCHOUTEN, Rev. Mod. Phys. 21, 421 (1949). J. WESS, Muovo Cimento 18, 1086 (1960). 3. H. A. KASTRIJP, Phys. Rev. 150, 1183 (1966).
2.
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Nuovo Cirnento 26, 652 (1962). Rev. Mod. Phys. 34, 442 (1962). de Investigacibn y de Estudios Avanzados del Instituto Polit&nico National, Mexico City, 1966). E. L. HILL, Phys. Rev. 84, 1165 (1951). L. CASTELL, Nuovo Cimento 46A, 1 (1966). J. A. MCLENNAN, JR., Nuovo Cimento 3, 1360 (1956). L. GROSS, J. Math. Phys. 5, 687 (1964). E. CUNNINGHAM, Proc. London Math. Sot. 8, 77 (1910). H. BATEMAN, Proc. London Math. Sot. 8,223 (1910). P. A. M. DIRAC, Ann. Math. 37, 429 (1936). A. GAMBA AND G. LUZZATTO, Nuovo Cimento 33, 1733 (1964). J. ROSEN, Bull. Am. Phys. Sot. 12, 489 (1967). E. C. ZEEMAN, J. Math. Phys. 5, 490 (1964).
4. T. FULTON, F. ROHRLICH, AND L. WITTEN, 5. T. FULTON, F. ROHRLICH, AND L. WITTEN, 6. J. PLEBANSKI, unpublished work (Centro 7. 8. 9.
10. II. 12. 13. 14.
15. 16.
OF PHYSICS:
47,468-480
(1968)
The Conformal
Group
and Causality*
JOE ROSEN+ Physics
Department,
Brown
University,
Providence,
Rhode Island
02912
Space-time transformations in Minkowski space are interpreted as neither “active” nor “passive” but as transformations of events and trajectories of processes with respect to the same observer and in the same flat space-time. The distinction is significant for the nonlinear special conformal transformations, which violate causality in the sense that the causal relations among events of a process are not always the same as among corresponding events of a transformed process. Aspecial conformal transformation canalways be found to transform the separation of any pair of events of nonlightlike separation from timelike to spacelike and vice versa. The transformation of trajectories under special conformal transformations is studied. Examples of reversal of temporal ordering by special conformal transformations are presented. Finally it is argued that none of this should prevent the conformal group from being physically useful.
I. INTRODUCTION
The space-time conformal group, the group of conformal space-time transformations listed in Section II, has certain features which make it especially interesting (Z)-(C~).~-~ It is the lowest dimensional semisimple group containing the PoincarC (=inhomogeneous Lorentz) group. This enters the conformal group in the list of candidates for a generalization of the PoincarC group as the space-time symmetry group of physics (3), (8)-(Z3).2 Another point is that the physics of massless particles is governed by the vanishing of the line element of their world lines, and the conformal group is the largest group preserving null line elements (8).2*3 So there arises the possibility that the conformal group is an exact space-time symmetry group for massless particles. This should have bearing not only on photons and neutrinos but also on massive particles of such high energies that their rest masses are negligible in comparison. The conformal group might then be a badly broken approximate symmetry for massive particles which improves as the energies involved increase (3).2 A similar situation exists for SU, as an internal *This research was supported in part by the U. S. Atomic Energy Commission No. NYO-2262TA-161). + On leave of absence from Tel-Aviv University. 1 Schouten (I) includes a historical review. * See references given by Kastrup (3) to his and other work. a See references given by Fulton, Rohrlich, and Witten (4), (5). 46%
(Report
THE CONFORMAL
GROUP AND CAUSALITY
469
symmetry for hadrons. An additional consideration, related to the last, is that relativistic wave equations become conformal-invariant when the mass term is removed (2) (3), (5), (9), (IO), as was long known for the Maxwell equations (II)-(13). An objection to the possible physical usefulnessof the conformal group has been raised based on the claim that the conformal group violates causality (14). This is the central theme of the present work. We show in the following that the space-time conformal group does violate causality in a certain sense,and we look into several aspectsof this violation. But, basedon our interpretation of space-time transformations, we conclude that the violation of causality by the conformal group should not prevent it from possibly being physically useful. In Section II the connected conformal space-time transformations are briefly reviewed. Section III discussesthe interpretation of space-time transformations. The effect on causality of the special conformal transformations is considered in general in Section IV. It is proved in Section V that for an arbitrary pair of events of timelike or spacelike separation a special conformal transformation can always be found which will change their separation to spacelike or timelike, respectively, and it is shown how to construct this transformation. Section VI treats the transformation of trajectories and especially their velocities under specialconformal transformations (IS). Two examples where reversal of the temporal ordering of events is brought about by a special conformal transformation are presented in Section VII. Section VIII resolves an apparent contradiction, pointed out by Giirsey,* between a result on the transformation of trajectories and a result on the transformation of events. Our conclusions are brought forth in Section VIII.
II. CONFORMAL
SPACE-TIME
TRANSFORMATIONS
The connected group of conformal space-time transformations consists of the following transformations:5 Lorentz transformations
4 I am grateful to Professor F. Giirsey for raising this point. 5 The inversion xp + f@ = -xyxp can be used together with the translations [Eq. (2)] to form the special conformal transformations [Eq. (4)1, but it itself is not an element of the connected group (i.e., that part of the group which is connected continuously with the identity transformation).
470
ROSEN
and special conformal
transformations
xu -+ 3 = (XN - C~X2)/O(X), (4)
u(x) = 1 - 2c * x + c2x2,
where indices run from 1 to 4, x“ = (t, x), and g,, = (+ 1, - 1, - 1, - 1). (The speed of light is taken as unity.) In the following we shall be concerned mainly with the 4-parameter Abelian subgroup of special conformal transformations [Eq. (4)], the only nonlinear transformations among Eqs. (l)-(4). Under these transformations for an infinitesimal interval dx2 -+ dE2 = dx2/a(x)2,
(5)
and for a finite interval (x1 - x2)2 + (Xl - X,)2 = (x1 - X&J(Xl)
III.
INTERPRETATION
OF
SPACE-TIME
a(x,).
(6)
TRANSFORMATIONS
It is usual to choose one of two interpretations for space-time transformations such as Eqs. (l)-(4): they can be considered either “passive” (coordinate) transformations or “active” (point) transformations (5). In the “passive” interpretation, space-time together with the world lines and events embedded in it are considered unafFected but expressed with respect to a new coordinate system, i.e., viewed through the eyes of a different observer. In the “active” interpretation, space-time and its contents are considered transformed by a mapping with respect to the same coordinate system or, one might say, under the nose of the same observer. We would like to suggest a third way of interpreting transformations of this kind. This new interpretation is most likely completely superfluous, if one is concerned only with linear transformations [Eqs. (l)-(3)] in flat space-time. But when considering the nonlinear special conformal transformations [Eq. (4)], the two usual interpretations exhibit rather unpleasant features. A “passive” transformation then introduces a curvilinear coordinate system, while an “active” transformation destroys the flatness of space-time. The unpleasantness lies in the fact that, since everyday physics experiments are interpreted “flatly”, the special conformal transformations should be consistent with flat space-time and rectilinear coordinates, if they are to be useful, say, in the realm of particles. To remedy this, we suggest interpreting space-time transformations as leaving both space-time and the coordinate system unaffected, the transformations serving to map only the world lines and events of a physical process under consideration. The same
THE CONFORMAL GROUP AND CAUSALITY
471
observer then seesa different processtaking place but in the sameAat space-time background. We call a transformation interpreted this way a “physical” transformation. So in summary, as long as we confine our attention to linear transformations in flat space-time, the distinction between “physical” and “active” transformations is quite academic. But nonlinear transformations make the distinction important, because then a “passive” transformation introduces curvilinear coordinates, an “active” transformation introduces space-time curvature, while a “physical” transformation leaves the same rectilinear coordinates and flat space-time with only the experiment being changed. This then is the way we shall interpret space-time transformations in the following. The sameobserver will observe a transformed processin the samespace-time. If every physically valid process thus happens to be transformed into another physically valid process by a given transformation or group of transformations, we have a symmetry of physics.6
IV. CAUSALITY
The causal relation of two events is determined by the two assumptions that no influence travels faster than light and that influences do not reach backwards in time. If x1 and x1 are the coordinates of two events, then their separation is called timelike, lightlike, or spacelike, if (x, - -x~)~ is positive, null, or negative, respectively. In the latter case the two events are not causally related; otherwise they are. When a causal relation exists, its full determination requires knowledge of the temporal ordering of the two events, Sgn(t, - fJ, specifying which event can inAuence the other. The group of Poincart transformations [Eqs. (1) and (2)] preserves (x1 - x2)” always and Sgn(t, - tJ when (x1 - x# > 0, so it preserves the causal relation of pairs of events. The group of dilatation transformations [Eq. (3)] always preserves both Sgn[(x, - x,)~] and Sgn(l, - t?), so that it too preservesthe causal relation of pairs of events. Combining these, the group of similitude transformations [Eqs. (l)-(3)] therefore preserves the causal relation of pairs of events. In fact, Zeeman (16) has shown that this is the largest connected group of space-time transformations with this property. Thus the special conformal transformations do not preserve the causal relation of pairs of events (2), (14). In fact, we show in the following section that for any given pair of events a special conformal transformation can be found to transform their separation from timelike to spacelike or vice versa. [A lightlike separation is preserved, according to Eq. (6), except when (i This is the way internal symmetries are of necessity always treated.
472
ROSEN
the transformation is singular for x1 or xz , i.e., except when a(~,) = 0 or u(xz) = 0.1 In Section VII we show how reversal of temporal ordering under a special conformal transformation can occur. Does the conformal group then violate causality ? If by this question we mean, “Can the special conformal transformations change the causal relations of pairs of events ?” then from our point of view the answer is affirmative. According to the “physical” picture the causal relation is determined by Sgn[(xl - xJ2] and Sgn(r, - t2) before the transformation and by Sgn[(x, - ?2)2] and Sgn(f, - tz) after it and is not always preserved.’
V. CHANGE
OF SIGN OF (xl -- x$
Under a special conformal transformation events having coordinates x1 and x2 we have
[Eq. (4)] for an arbitrary
pair of
(Xl - x2)2 + (X1 - X,)2 = (x1 - x2)“/cr(xJ u(x2).
(6)
If (x, - x2)2 f 0, its sign will be changed under the transformation only if I and 0(x2) can be made to have opposite signs. We shall show that this can always be done; i.e., for (xi - x2)2 $ 0 it is always possible to find c such that U(Xl) 4x2) < 0. First express u(x) in various ways as follows: u(x) = 1 - 2c . x + c2x2
(7)
= c”(c/c” - x)”
for
c2 f 0
(8)
= x2(c - x/x”)”
for
x2 f 0.
(9)
Case I:
Assume c2 f 0. Then from Eq. (8) u(xl) u(x2) = c”(c/c” - x1)2 (c/c” - x2)2. Subcase 1: If both t, f
(10)
t, and x1 f x2, define
a = (t2 , Xl>, b = 01 , x2).
Subcase 2: If tl = t, = t but x, + x2 , put a = (t + T, x1), b = (t + T, x2),
where 0 < T2 -=c(x1 - x2)2.
Subcase 3: If x1 = x2 = x but t, f t, , define a = (t2 , x + X), b G (tl , x + X), where ’ Compare with Kastrup (3), who holds the “active” point of view.
0 < X2 < (tl - Q2.
THE
CONFORMAL
GROUP
AND
473
CAUSALITY
Now for all three subcases (a - .x1)2 > 0, (a - x# < 0, (b - x$ < 0, and (b - xJa > 0. So take c/c” = a or b, i.e., c = a/a2 or b/b2. If a2 = 0, take c = b/b2; if b2 = 0, take c = a/a2. If u2 = b2 = 0 and subcase 2 or 3 holds, change T or X within the allowed freedom to make a2 and b2 nonzero. But if a2 = b2 = 0 and subcase 1 holds, things are handled differently. Then, since a2 + b2 = xl2 + xz2, we have xl2 = --x22. Case ZZ Assume xl2 = -xZ2 f
0. Then from Eq. (9),
U(Xl) 0(x2) = X12X2”(C - X~/X~“)“(C - x2/x22)2 = -x14(c
- Xl/Xl”)“(C
- x2/x22)2.
(11)
This will be negative for any c with sufficiently large 1ca j. Case ZZL Assume a2 = b2 = xl2 = x 22 = 0. Then t,2 =
t,2 =
x 12 =
x22,
(12)
and from Eq. (7) 0(x1) U(X2) = (1 - 2c . x&l
- 2c * x2).
(13)
Take c = h(x, - xc), where h is a number. Then U&xl) 0(X2) = (1 + 2lq
* x&l
- 2h.q * x2).
(14)
This will be negative for any h with sufficiently large j h I, unless x1 . xZ = 0. But x1 . x2 = 0 and Eq. (12) imply (x, - Y~)~ = 0, which contradicts our assumption. This covers all cases and proves our assertion.
VI.
TRANSFORMATION
OF
Under a special conformal transformation into a trajectory ji = Z(i). The transformation by putting c” = (6. c):
TRAJECTORIES
(15)
a trajectory x = x(t) is transformed formulas are obtained from Eq. (4)
t -+ f = [t - b(t’ - X2)]/@, x), s + x = [x - c(t2 - x2)]/u(t, x),
(1%
474
ROSEN
the inverse being f - t = [i + b(i2 - jP)]/a(--7, f - x = [E + c(i2 - nq/+r,
-ji), -jz),
(16)
where a(t, x) = 1 - 2(bt - c * x) + (b2 - c2)(t” - x2) = (1 - bt + c * x)” - (bx - ~t)~ + (c x x)“.
(17)
The following relations hold: u(-i,
-f-i) = l/cr(t, x),
1 + bi -
c
. z = (1 - bt +
c
bjz - ct = (bx - ct)/o(t,
.
X)/U@,
x),
x),
c x 2 = c x x/o@, x).
The velocities of the original and transformed trajectories are v = dx/dt and t = dji/di, respectively. The relation between them is given by the transformation v+t
= N/D
= N/ii,
(19)
where N = [a@, x) - 2(c x x)“] v + 2[(1 - bt + c * x) + v * (bx - ct)](bx - ct) + 2[v, c, x] c x x + 2[(bx - ct) + (1 - bt + c * x) v] x (c x x), (20) D = a(t, x) + 2(bx - ~t)~ + 2[(1 - bt + c . x)(bx - ct) - (bx - ct) x (c x x)] * v,
and, using the relations of Eq. (18), m = [u(-i, +
v
-ji) l
- 2(c x E)“]v + 2[(1 + bi - co 5)
(bjz - ci)](bz -
ct)
+ 2[v,
+ 2[(bP - ct) + (1 + bi - c i3 = u(-i,
-9
+ 2(bz -
ci)”
l
Z)V]
C,
ji]
c
x z
x (c x E),
+ 2[(1 + bi -
- (bE - cf) x (c x ji)] . v.
c
l
Z)(bji - ci) (21)
From Eqs. (20) and (21) it follows that D2 - N2 = (1 - v2) a(t, x)~, iP - R2 = (1 - v2) 0(-i, -jz)z,
(22)
THE
CONFORMAL
GROUP
AND
CAUSALITY
475
from which we obtain 1 -
y2 -+
1 -
82 = (1 - v”) a(t, x)2/D2
= (1 - v”) a(-&
-Z)Z/B”.
(23)
Equations (19) and (23) then give, for v2, G2 # 1, v/(1 - v2)li2 + 3/(1 - 82)1/2 = &N/c@, x)(1 - v2)1/2 = -&tm/u(-i, --jz)(l - v2)1/2.
(24)
The sign in Eq. (24) depends on which branch of the transformed trajectory is being considered. Either as a byproduct of Eq. (23) or more simply from Eq. (5) we obtain the result that, except when the special conformal transformation under consideration becomes singular, (25)
That is, the property of a motion that it is slower than, equal to, or faster than the speed of light is preserved under special conformal transformations.* Considering now geodesic trajectories (straight lines) x = vt + x0, where v and x0 are constant vectors, the two interesting cases are v2 = 1, null or lightlike geodesics corresponding to trajectories of light rays and free massiess particles, and v2 < 1, timelike geodesics corresponding to trajectories of free massive particles. In the first case it is not difficult to see that under a special conformal transformation a null geodesic trajectory is transformed into a null geodesic trajectory. This suggests the possible relevance of the conformal group to the physics of light rays, massless particles, and extremely relativistic particles whose masses are negligible in comparison with their energies (3),2 as mentioned in the introduction. In the case of timelike geodesics it has been shown that the transformed trajectories under special conformal transformations correspond to those of charged particles in constant electromagnetic fields of electric type (6). In the nonrelativistic limit this is motion of constant acceleration. In genera1 the function G(i) is rather complicated, but under certain conditions V/(1 - V2)1/2becomes a linear function of i. The following are such cases: (a) For the trajectory x = 0, a particle at rest at the origin, F/(1 - 92)1/2 = f2ci. 8 I would like to thank Dr. L. Caste11 for comments concerning this point.
(26)
476
ROSEN
If c = 0, b arbitrary, the transformed trajectory again represents a particle at rest at the origin, j2 = 0, and i = t/(1 - bt). (b) For the trajectory x = x0 f 0, a particle at rest at x0 , when b = 0, C/(1 - 92)1/2 = &2[(1 + 2c x&c - C2X& l
(27)
The transformed trajectory will again represent a particle at rest, C = 0, only when (1 + 2c . x&c = c2x, . Besides the trivial solution c = 0 (the identity transformation) the only other solution to this equation is c = -x,,/x,,~, The transformed trajectory then becomes jz = -x0 , and t = -xo2/t. (c) For the trajectory x = C-t, v # 0, a free massive particle passing through the origin, when c I/ v, say c = av, t/(1 - t2)li2 = -J-[1 + 2(b - a)t] v/( 1 - v2)lj2.
(28)
Note that 5 = v only when b = a. The transformed trajectory will then again represent free massive particle motion through the origin, P = vt.
VII.
EXAMPLES
OF
REVERSAL
OF
TEMPORAL
ORDERING
Starting with the trajectory describing a light ray or free massless particle x = t,
y=z=o
we perform a special conformal transformation 0, 0) and obtain
7
(29)
characterized by cU = (0, c, > 0,
t = t/( 1 + 2&X) = t/(1 + 2&J), x = x/(1 + 2C&) = t/(1 + 2c,t) = f, y=z=o.
(30)
The resulting trajectory is the same as the original one. But the images of events along the original trajectory do not retain the same temporal ordering. This is illustrated in Figs. l(a) and l(b), where Fig. l(a) shows the original trajectory and Fig. l(b) shows the transformed one. Events are marked off along the original trajectory and their images are indicated along the transformed trajectory. Note the reversal of temporal ordering of events B and D for example. Another interesting example is afforded by the trajectory of a massive particle at rest at the origin x = 0.
(31)
THE CONFORMAL GROUP AND CAUSALITY
Performing the same special conformal transformation
477
as above, we obtain
i = t/(1 - C,V), x = -c,t”/(l - C,2P), yzz=().
(32) b
i
FIG. 1. (a) a massless particle trajectory with events marked off along it. A special conformal transformation characterized by cfi = (0, c, > 0, 0, 0) transforms it into the same massless particle trajectory, shown in (b), where the images of the original events are indicated. Note the reversal of temporal ordering of, e.g., events B and D.
a
b t
FIG. 2. (a) a trajectory of a massive particle at rest with events marked off along it. A special conformal transformation characterized by CC = (0, cz > 0, 0,O) transforms it into the hyperbolic trajectory describing two massive particles shown in (b), where the images of the original events are indicated. Note the reversal of temporal ordering of, e.g., events B and D. Also note the possibility of events such as B and E becoming simultaneous.
478
ROSEN
Eliminating I from Eq. (32), the transformed trajectory proves to be a hyperbola in the x-t plane (X -
1/2c,)~ - t2 = l/4&2.
(33)
The velocity C = djz/df along this trajectory obeys [from Eq. (26)] C,/(l - Ce2)l/2 = *2c,t.
(34)
The situation is pictured in Figs. 2(a) and 2(b), where Fig. 2(a) shows the original trajectory and Fig. 2(b) shows the transformed one. In this case the transformed trajectory describes a process involving two particles, where the motion is that of two noninteracting particles of equal and opposite charge-to-mass ratios in the same constant electric field (6). Events are again marked off along the original trajectory and their images are shown along the transformed trajectory. Note the reversal of temporal ordering of events B and D, for example. Also note that events such as B and E can become simultaneous.
VIII. RESOLUTION
OF APPARENT
CONTRAJXCTION
Giirsey4 raised the following question: Any two events on a timelike trajectory (and therefore having necessarily a timelike separation) can be transformed by a special conformal transformation so that their images have a spacelike separation (proved in Section V). The image of the trajectory remains timelike, though (proved in Section VI), apparently forcing the images of the two events to have a timelike separation. How can these two facts be reconciled ? Examine the effect on (F1 - J2)z in Eq. (6) of a continuous sequence of special conformal transformations from the identity to the one mentioned above. Any change of sign of (Zl - X2)2 is brought about by its passing through infinity, i.e., by either a(~,) or u(x2) passing through zero. Therefore, there must exist on the original trajectory between the two events under consideration at least one event for which the special conformal transformation becomes singular. The transformed trajectory does not then run directly from the image of the earlier event to the image of the later, but from the image of the earlier event it runs off to infinity and reaches the image of the later event along another branch. It is this unconnectedness that allows the transformed trajectory to remain timelike (except, it might be added, at images of singular events, where it tends to lightlikenessg) without the necessity of a timelike separation between all pairs of events on it. The situation might be made clearer with the help of Figs. 2(a) and 2(b). The original trajectory in Fig. 2(a) is timelikepar excellence. The transformed trajectory O See
Section
VI.
THE CONFORMAL GROUP AND CAUSALITY
479
in Fig. 2(b) is also timelike, but tends to lightlikeness at the images of events C and F, the two singular events for the special conformal transformation in effect. Note that pairs of transformed events having spacelike separation are always located with each member on a different branch of the transformed trajectory, and the corresponding original pairs of events therefore have a singular event between each two of a pair. Consider for example events B and E.
IX. CONCLUSIONS
In the “physical” approach to space-time transformations, as described in Section III, the special conformal transformations violate causality in the sense of Section IV. But there seemsto be no reason why this fact in itself should invalidate the conformal group as a possible symmetry of physics. In fact, from the “physical” point of view violation of causality is quite harmless.It meansthat the causal relations among events of a processmight differ from the causal relations among corresponding events of a conformal-transformed process, and there is nothing so terrible about that. The useful transformation of time reversal, for example, doesjust such a thing. Under conformal transformations there can even occur caseswhere a processinvolving particle creation is transformed into a process involving particle annihilation (which is familiar from the time reversal transformation), or where a process involving a certain number of particles is transformed into a process involving a different number of particles (see Section VII). But from the “physical” viewpoint there is no objection to this happening. The validity of the conformal group as a symmetry of physics should, we feel, be tested by examining whether under it every physically valid processis transformed only into physically valid processes,however different (from the causal or any other aspect) the transformed processesare from the original or from each other.
ACKNOWLEDGMENTS
I would like to express my thanks to Professors Nathan Rosen and John Stachel for discussions. RECEIVED: September 20, 1967
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2.
480
ROSEN
Nuovo Cirnento 26, 652 (1962). Rev. Mod. Phys. 34, 442 (1962). de Investigacibn y de Estudios Avanzados del Instituto Polit&nico National, Mexico City, 1966). E. L. HILL, Phys. Rev. 84, 1165 (1951). L. CASTELL, Nuovo Cimento 46A, 1 (1966). J. A. MCLENNAN, JR., Nuovo Cimento 3, 1360 (1956). L. GROSS, J. Math. Phys. 5, 687 (1964). E. CUNNINGHAM, Proc. London Math. Sot. 8, 77 (1910). H. BATEMAN, Proc. London Math. Sot. 8,223 (1910). P. A. M. DIRAC, Ann. Math. 37, 429 (1936). A. GAMBA AND G. LUZZATTO, Nuovo Cimento 33, 1733 (1964). J. ROSEN, Bull. Am. Phys. Sot. 12, 489 (1967). E. C. ZEEMAN, J. Math. Phys. 5, 490 (1964).
4. T. FULTON, F. ROHRLICH, AND L. WITTEN, 5. T. FULTON, F. ROHRLICH, AND L. WITTEN, 6. J. PLEBANSKI, unpublished work (Centro 7. 8. 9.
10. II. 12. 13. 14.
15. 16.