I L NUOVO CIMENTO

VOL. XII, N. 6

lO Dicembre 1954

Conformal Geometry and Elementary Particles. ]~. L. INGI~AHAI~I

Institute /or Advanced Study, Princeton, New Jersey (ricewlto il 2 S e t t e m b r e ]954)

Summary. --- The kinematical consequences of basing (classical or quantum) field theory on the eonformal gcometry are examined in this paper. The space in question is t h a t of all spheres in R4 (fiat 4-space of signature ( - [- ~ - - ) ) ; the fundamental invariant, the angle under which two spheres intersect. In the mathematical prelimilLaries (w 1 ) a convenient inhomogelmous formalism i.~ developed, p e r m i t t i n g the sphere to be treated as a point in a 5-dimensinnal Rienmnnian space of constant unit curvature whose length element is the infinitesimal angle between two neighboring spheres. The c[)nformal group (proper Lorentz transformations, translations, uniform dilations, and inversions in spheres) is just the ]5-parameter group of motions (metric-preserving transfnrmations) of this space. In w2 ]:he spheres are interpreted as finite (non localized) test particles. Physical fields are thus defined on test particles rather than (e.g. in position space) on the events occul)ied by these test particles alone. The st)heres can be labelled either with the 5-position (test particle's spacetime position and size) in ,u q-frame, or with the 5-momentum (test particle's 4-momentum and rest mass) in ,~ p-frame. There exists a motion transforming q- into p-sp,~ee and conversely (in other words, q- and p-observers are physically equivaleut), in v i r t u e of which any conformal theory is shown to exhibit an automatic Born-reciprocity between q- and p-space. The 5-position and 5-momentum satisfy an uncertainty-relation type equ'~timi; i.e., the non-localiz'~tions in q- and p-space are in an inverse relation with Plam:k's h measuring the intrinsic correlation. All the o t h e r motions can be built up from subgroups taking q- and p-space into themselves. These are systematically interl)reted as changes of frame. Those involving relative motion represent uni/orm relative ac('elerations of Lorentz observcrs (Lorentz group*---* zero accelerations). Test particle mass and size are invariant under the Lorentz group but nonuniformly renormalized by the accelerative motions. In w3 the geodesics of sphere space (motion equations of (elementary) test particles) are

826

R.L.

INGRAHAM

shown to describe uniform motion in the present (force-free) case. Test particles tend to dislocalize in q space with increasing time andin p-space with increasing energy. The time-constancy of the 5-momenta, their inter-dependence as given by special relativity dynamics follow from these motion equations. Free elementary particles are shown to maintain a state of uniform velocity under all motions, in particular the accelerative ones. This contradicts ordinary relativity and suggests an experiment capable in principle of choosing between the conformal and Lorentz geometries for physics.

1.

-

Introduction.

This p a p e r is a n a t t e m p t to indicate t h e c o m m o n features of all physical theories b a s e d on the conformal group in Minkowski space as f u n d a m e n t a l group. B y f u n d a m e n t a l group is u n d e r s t o o d t h a t group of coordinate transformations u n d e r which the field laws are f o r m - i n v a r i a n t , or, e q u i v a l e n t l y stated, t h a t group connecting the preferred coordinate s y s t e m s r e p r e s e n t i n g the physically equivalent observers of the theory. The f u n d a m e n t a l group of t h e s t a n d a r d classical and quantized field theories of t o d a y is (at m o s t ) t h e tenp a r a m e t e r one of Lorentz t r a n s f o r m a t i o n s a n d space-time translations; to a t t a i n the f i f t e e n - p a r a m e t e r conformal group one adjoins the u n i f o r m dilations and inversions in space-time h y p e r s p h e r e s (1). The l a s t - m e n t i o n e d transformations are non-linear and, roughly speaking, it is to their presence in t h e f u n d a m e n t a l group t h a t m o s t of t h e novel f e a t u r e s are due. Our subject is thus pure kinematics ~ independent of the p a r t i c u l a r f o r m of the field laws considered. Ah'eady a t this stage m u c h with physical c o n t e n t can b e said. The physical i n t e r p r e t a t i o n of the g e o m e t r y (2"3) shows t h a t t h e finite size of t e s t (elementary) particles t a k e s its place alongside space-time as a bonafide dimension of physical space, for example. This new degree of f r e e d o m n a t u r a l l y brings with it t h e a p p e a r a n c e of a n o t h e r f u n d a m e n t a l dimensional constant, Planck~s h, alongside c in t h e angle metric. Again, a n y such t h e o r y m u s t exhibit a p e r f e c t reciprocity b e t w e e n position and m o m e n t u m space (2"3). The five i n d e p e n d e n t dimensions in the l a t t e r picture are t e s t particle mom e n t u m , energy, and r e s t mass. Or again, free particle behavior is d e t e r m i n a b l e a t this stage (3"2). The persistence of u n i f o r m m o t i o n u n d e r certain accelerative changes of frame, contradicting r e l a t i v i t y kinematics, suggests a simple e x p e r i m e n t capable of choosing b e t w e e n the conformal a n d L o r e n t z geometries for physics. A sequel t o this p a p e r t r e a t s t h e consequences of a p a r t i c u l a r set of (hnearized) field laws in this g e o m e t r y . G r a v i t a t i o n , electron, a n d light in p a r t i c u l a r are described. (1) Our terminology will in general ignore signature. Thus a space-time (chypersphere )) is a hyperboloid as a real locus.

CONFORMAL GEOMETRY AND ELEMENTARY PARTICLES

$2"]

The conformal group is a c t u a l l y a group of projective t r a n s f o r m a t i o n s on the X5 (2) of spheres in Minkowski space, in which the events t h e m s e l v e s a,re r e p r e s e n t e d as null spheres. Physical fields get 6P c o m p o n e n t s wi~h respect to t h e six homogeneous ((~ hexaspherical ~) coordinates of these spheres and convenient tensor f o r m a l i s m in which t h e f u n d a m e n t a l group is r e p r e s e n t e d b y linear t r a n s f o r m a t i o n s with c o n s t a n t coefficients is usable (2"1). An equivalent inhomogeneous f o r m a l i s m of tensors with 5" c o m p o n e n t s is developed (2"2) and used in preference to the homogeneous f o r m a l i s m because of its m o r e i m m e d i a t e physical meaning. Because of their relative u n f a m i l i a r i t y in physics, these necessary m a t h e m a t i c a l preliminaries are worked out systematically at some length. I n the conventional t r e a t m e n t (a) one t h u s has various differential laws for homogeneous 6-tensors whose domain of definition is restricted to the X4 of null spheres. One essential difference of the present t h e o r y f r o m t h e conventional t r e a t m e n t is t h a t not only the null, b u t also t h e non-null, spheres f o r m the domain of definition of t h e physical fields; i.e., t h e physical c o n t i n u u m f r o m which we s t a r t is five-dimensional. (Indeed, this t h e o r y shows t h a t the X, of null spheres, far f r o m sufficing as the physical domain, plays a r a t h e r exceptional role.) We need the X5 of all t h e spheres for b o t h for m a t h e m a t i c a l and physical reasons. M a t h e m a t i c a l l y , the force-free a n d weak field theories t r e a t e d in this p a p e r a n d the sequel resp. m u s t . b e considered as a degenerate case of the rigorous theory, for which the proper curvilinear generalization of C4 (~) is the geometrical framework. The unique correct generalization of C, (the p r o t o t y p e being the generalization of euclidean b y R i e m a n n i a n g e o m e t r y ) m u s t s t a r t f r o m an underlying X~; t h e a r g u m e n t s , which are not directly r e l e v a n t to this paper, cannot be given here (~). Previous a t t e m p t s (5) to generalize the ordinary conformal g e o m e t r y C, invari a b l y s t a r t e d f r o m an u n d e r l y i n g X~ and were on these grounds alone, (3) Alphabet conventions: late roman letters k, l, m , . . . - - 1 , ... 4; early greek letters ~, fl, y . . . . . l, ..., 5; late greek letters p, r, ~, ... = 0, 1,..., 5. Notation:

X , = n-dimensional manifold

('~ ~ R, = P~ = 1~,~ =

,) ,) ,) .)

conformal geometry flat (Riemannian) space ordinary projective space Riemannian space.

(a) E.g., P. DIRAr Ann. o/ Math., 37, 429 (1936). (a) R. INGRAHAM: Proc. Nat. _tcad. Sci., 38, 921 (1952); Nuovo Cimento, 9, 886 (1952). (.~) H. ~VE'ZL: 8itz. d. Preuss. Akad. d. Wiss., 465 (1918), R a u m , Zeit, Materie 5. Aufl. Berlin, 1923), w167 40, 41; E. CARTAN:Ann. Soe. Pol. Math., 171 (1923); T. Y. THOMAS: Proc. Nat. Acad. Sei., 18, 103 (1932); O. V]~BLEN: Proc. Nat. Acad. Sci., 21, 168 (1935); J. A. SC}~OI~TEN and J. }tAANTJES: Math. Ann., 112, 594 (1936); 113, 568 (1936).

828

R.L.

INGRAHAI~

m a t h e m a t i c a l l y unsatisfactory. The lack of success of their application to physics is thus not surprising. The physical a r g u m e n t s , on t h e o t h e r hand, for the X5 are w h a t this p a p e r aims, in p a r t , to s u p p l y in some detail. A final general r e m a r k : the necessary a d j u n c t i o n of a fifth dimension to space-time, the expansion of the s t a n d a r d class of equivalent observers, the tensors with 6, components, etc., all t h e features t h a t point t o w a r d the possibility of a real unification of physical fields flow a u t o m a t i c a l l y f r o m the one a s s u m p t i o n t h a t the f u n d a m e n t a l physical g e o m e t r y is conformal. The unification is, so to speak, an incidental b y - p r o d u c t of this a s s u m p t i o n . I t is not a question of examining i m p a r t i a l l y a whole s p e c t r u m of logical possibilities. Physicists who distrust the aimless f o r m a l a d v e n t u r i n g of m u c h of so-called ((unified field ~) t h e o r y should t a k e c o m f o r t f r o m the fact t h a t here the m o t i v a t i o n is quite different. 2. -

The

Force

Free-World.

2"1. T h e sphere s p a c e M ~ . - L e t g,~n(gll~g22zg33-----g44--1, g , n n z O ( r e # n ) ) be the metric referred to o r t h o n o r m a l cartesian coordinates x m ( m z l , .... 4) of the R4 (2) of special relativity. Then the sphere (~) of center x m and radius (real or pure imaginary) R can be assigned six homogeneous coordinates (~) X ~ (It ~ O, 1,..., 5) via

(2.1)

] ~x ~ = 1,

t [(x) ~ ~

~x,,, = x~

~x~ =

89

R 2)

9 # 0 a p r o p o r t i o n a l i t y const.].

g~x"x",

Since t h e y are homogeneous, t h a t is, under the modular group (2.1)'

X ~ -~ ~X ~

[~)#0, const.]

the set of n u m b e r s X" goes into a n o t h e r set ~X" representing the s a m e sphere, the X ~ can be i n t e r p r e t e d as cartesian ((( preferred ,)) projective coordinates of a point in an ordinary five dimensional projective space P~ (2). A t r a n s f o r m a t i o n of cartesian coordinates in /)5 (2.2)

X ~' --~ A,,t,'X ,.

corresponds to a linear change of sphere basis. null spheres ( R = 0 ) is b y (2.1) the quadric Q: (2.3)

gm~X"~X ' ~ - 2 X ~

[Det A~ -/= 0, A ~ ' = c o n s t a n t s ] I n this P~ the locus of the

5 -~ 0 .

The quadric Q defines the projective tensor G,~ (Det G~,,#0) via (2.4)

aGo~ = - - 1 , other G~,,,~ 0 ; [a # 0 a p r o p o r t i o n a l i t y eonst.]

aGm~ = gin,,

(e) On polyspherical coordinates, see say F. KLEIN : Vorlesungen iiber HShere Geometrie, 3 Aufl. (Berlin, 1926), p. 49.

C O N F O R M A L G E O ~ I E T R Y AN]:) E L E M E N T A R Y

829

PARTICLES

t h a t is, u n d e r the ,nodular group (2.4)'

V~ --> ~G,,,

[~ ~ 0 , const.]

the c o m p o n e n t s G~ go into the c o m p o n e n t s 2G,, r e p r e s e n t i n g the s a m e quadric, and u n d e r t h e coordinate group (2.2) G , t r a n s f o r m s cogrediently to X ' :

(~u'v'-----d ~ ' J ; ' o~

(2.5)

[ A~'A ~.= A~; ~ ; ] .

The space P5 falls into two disjoint regions Q+ a n d Q-, the exterior a n d interior of Q, resp.. X" non-null s Q+ or Q- according as Sgn GqeXqXe : ~- 1 or - - 1, resp.. L e t X ~ and Y~ be two non-null spheres in the same region; t h e n the f u n d a m e n t a l invariant, the angle 0~ u n d e r which t h e y intersect, is defined b y

(2.6)

I

G.vX"

1"" cos 0~ ~-e }(X) l !(Y) I [(X) 2-= G ~ X ~ X ~ ~ 0 ,

etc.; Sgn (X) -~~- Sgn (y)o_~ e].

The invariance of (2.6) u n d e r the coordinate group (2.2) a n d b o t h m o d u l a r groups (2.1)' and (2.4)' shows t h a t this angle is a well defined sphere geometrical concept. The classical sphere space of R4 is now i n v a r i a n t l y defined as t h e space P~ with the tensor G,~ r e p r e s e n t i n g the quadric of null spheres; its f u n d a m e n t a l angle i n v a r i a n t is given b y the f o r m u l a (2.6). The fiat MSbius space M~ used b y the author is the s a m e sphere space trivially generalized to allow the p r o p o r t i o n a l i t y factors ~, Q, a, and ~ in (2.1), (2.1)', (2.4) and (2.4)' resp. to be (non-vanishing, differentiable) functions of the sphere (7). We shall be working exclusively in this g e o m e t r y for the rest of the paper. We recall the connection of M5 with conformal t r a n s f o r m a t i o n s of R4: e v e r y eonformal (~) t r a n s f o r m a t i o n Y " ' : / " ~ ( x ' ) in R4 is r e p r e s e n t e d b y a constant linear sphere t r a n s f o r m a t i o n X ~ -- P~',.X ~, where the P'~ satisfy

(2.;) [for some ~: p-~l, p~.

b~',; Det P~'~, ~ 0J,

in tile sense t h a t the null sphere X t ' - ~ (x'", R = 0 ) via (2.1) goes into the null sphere ~ t , _ ~(~" = j"'(x"), R = 0) via (2.1). This correspondence b e t w e e n tile (7) Tile reason for this is that the curvilinear M~ (cf. (~)) is defined as an X~ provided with the variable tensor GI~,(Xo) homogeneous of degree zero whose tangent spaces are classical sphere spaces as defined above. When coordinates Xe can be found for which G~,v(Xo) = a( Ye)g~v, gt,, constants, a homogeneous of degree zero, the space i~ said to be flat; in this case, all the tangent spaces can be identified. The resulting flat sphere space, M 5, is then identical with the classical sphere space except for the difference mentioned above. (s) Angle preservation in both sense and magnitude is implied.

830

R.L.

INGRAIIAM

conformal group and M, the group of sphere transformations P~' satisfying (2.7), is in fact an isomorphism. M will be called the group of eon/ormal sphere t r a n s f o r m a t i o n s - - b y (2.7) and (2.6) the transformations of non-null spheres are conformal also in the sense t h a t t h e y preserve the angle (8) between spheres. M is the group of motions of M 5 for b y (2.7) it takes the tensor Gt.. into itself. M is thus also the f u n d a m e n t a l group (see Introduction) of any physi(.al t h e o r y built on the space M~. The angle metric, which expresses the infinitesimal angle between two neighboring non-null spheres in terms of their coordinate differences, is arrived at in the following way. Suppose t h a t X ~' and X " + d X ~' are coordinates of the neighboring spheres in one point gauge, i.e., for some definite function ~(X Q) in (2.1). (Of course, it will have to be proved a posteriori t h a t our result is independent of the point gauge, t h a t is, invariant u n d e r the modular group (2.1)'.) Then the application of (2.6) yields (App. I) (2.8)

dOS = Y/,,dX~' d X " ,

where (2.9)

a)

yu,,-~ S . , , - - XI,X,. ,

b)

G.. , Sin,-= (X)~

c)

X t, -- ,S'~,~-~:~

and (X) 2 has been defined in (2.6). We note at once the invariance of (2.7) against the coordinate and quadric-modular groups (2.2) and (2.4)' resp.. The less trivial invariance against the point-modular group (2.1)': d X ~ ~ - ~ d X ~ + + d ~ X ~ follows from (2.1(})

yu~X~= 0 .

We shall call the tensor y,, the angle metric. I t is singular of rank 5 (cf. (2.10)), invariant under the group (2.4)', and under the group (2.1)', it gets the factor ~-2. S,, is non-singular, invariant u n d e r the group (2.4)', gets the f a c t o r e-~ under the group (2.1)', and satisfies the i d e n t i t y S , , . X ' X ' - ~ I . For the physical application only S,~ and Yu,. are r e l e v a n t ; no f u r t h e r mention n e e d be made of G,~ and its modular group (2.4)'. Note the close analogy between the angle metric (2.8) and Riemannian length metric. In the n e x t section we show t h a t dO S is actually the element of length in a simple Riemannian space. Moreover this inhomogeneous forrealism seems to be the most convenient for the physical interpretation. 2"2. The R i e m a n n i a n ]ormalism. - To express dO S in terms of a convenient set of inhomogeneous sphere coordinates, say x "~ and x S ~ I R l , let us go to

C ONFOR- ~[ AL G E O : ~ I E T R Y A N D

a point g a u g e in which T-----1 in (2.1). d X ~ ---

0,

ELEMENTARY

831

PARTICLES

I n this point g a u g e

d X ~ = - d x "~ ,

d X 5 ~-- , c , , d x "~ - -

sxSdx 5

[x,~ ~ -- g m n x ~]

and ,~ ~ = s(x~)-~g., , ~

= ~(~)-2g~,

w h e r e ~ a n d g,,, are t h e r i g h t m e m b e r s in (2.1) a n d (2.4) resp., e - - S g n R 2 is of course t h e s a m e s t h a t occurs in (2.6). S u b s t i t u t i n g these values in (2.8), we get (2) (2.11)

d 0 2 ~ 7:z d ~ d~ ~ = s(x~)-2((dx),~-- e(dx~)~).

[(dx)~ -

g,n.dx'~dx

~]

dO'- is t h u s expressed as t h e e l e m e n t of l e n g t h in a c e r t a i n five-dimensional R i e m a n n i a n space (2) Vs. A c t u a l l y t h e r e are t w o Vs's: V + , whose p o i n t s are all t h e spheres l y i n g in Q + ( s = - + - l ) a n d V [ , whose points are all t h e spheres l y i n g in Q - ( s = - - l ) . Their m e t r i c s are 7 ~+---- ~,/ x~ j5~-~-+ ~ a n d ~,~ ---- - - (xS)-~g~ resp. in t h e s e coordinates, w h e r e (2.11')

g~ ,,~ ~

g~.

,

g~5~ ~

0 ,

g~55 ~

s .

These m e t r i c s are infinite for a+----0, so t h e null spheres are b o u n d a r y points, limit spheres of n o n - n u l l spheres w h i c h b e l o n g to n e i t h e r Vs. I t is e a s y to show (9) t h a t t h e 175 are of c o n s t a n t c u r v a t u r e + 1 . H e n c e we are using t h e device of s t u d y i n g a n g l e - g e o m e t r y b y s t u d y i n g t h e m e t r i c g e o m e t r y of t h e u n i t sphere a b o u t t h e origin (9). W e shall call t h e x ~ n a t u r a l c o o r d i n a t e s for t h e lr5~. T h e y h a v e direct geoluetrical m e a n i n g for t h e sphere (xm----center, xS~-radius), t h e null spheres are i m m e d i a t e l y recognizable in t h e m (xS----0), a n d t h e angle m e t r i c has a n especially simple f o r m in t h e m . W h a t n o w does a c o n f o r m a l sphere t r a n s f o r m a t i o n look like in n a t u r a l c o o r d i n a t e s ? T h e general f o r m is easily w o r k e d o u t f r o m (2.7) a n d (2.1). W e give here t h e 1 5 - p a r a m e t e r g r o u p M b r o k e n down, for t h e sake of t h e p h y s i c a l a p p l i c a t i o n , i n t o subgroups. (9) Precisely: if R~ are the flat spaces with metrics sg~v in the cartesian coordiltates Zz~, then the V~ are the unit spheres X~: eg~Z~Z~=

1

in the R~ resp. via the imbedding Zo_

U+~,

Zm = xWx~

Z~ = 1/2(xs)-~((x)~-- ~(x~)~).

( ~f. E~SENtIART: R i e m a t t u i a ~ G e o m e t r y (Princeton, 1926), p. 204. The constant curvature § 1 of the V~ follows immediately from this.

832

R.L. INGRAttAM a)

x. . . .=. .L

b)

5~~ = x'" + a " ,

e)

5~~ - - t ~ x ' ,

d)

--1

x

~x~,

._7'5 - - .c'~

[prol)er L o r e n t z

~ = f f [ s p a c e - t i m e t r a n s l a t i o n s , ~t p a r a m e t e r ]

/~ > 0 , const.

---- / ~ _ I ( x 1 _L 0 ~ [ 2 1 2 / 2 ) ,

[ u n i f o r m dilations, 1 p a r a m e t e r ] ,~t =

[[x]2= g,~,~x"~x.... ~(~)~,

(2.12)

group, 6 p a r a m e t e r ]

A - % ' (i = 2, 3,4), .~5 = A-I~~ A:-1+~l+a2[x]2/4] [1 p a r a m e t e r ]

e-f)

f o r m u l a d) with the p e r m u t a t i o n s of t h e space indices (123) a n d (132) resp. [2 p a r a m e t e r ]

g)

~'] :

D - - l f ) (] :

l , .O 3 ) ,

,r

=

I n :- 1 - - fix 4 - fi2[x]2/4]

h)

~'=-

_5 J' :

D_I(,F a ~_/~[x]2/2),

D_l~tz

[1 l)arameter]

[fundamental inversion]. Ix] ~

I n d), e), f) a n d g) a, ..., etc., a n d fi are p a r a m e t e r s . T h e inverses are given b y a - + - - ~ , etc., fl--~--fi. These are l - p a r a m e t e r g r o u p s i n v o l v i n g inversions, h e n c e their non-linearity. A n y t r a n s f o r m a t i o n of M c o n n e c t a b l e t o t h e g r o u p i d e n t i t y can be d e c o m p o s e d into a p r o d u c t of t r a n s f o r n m t i o n s a), ..., g); adjoining t h e f u n d a m e n t a l inversion, we g e t t h e whole group. F o r instance, a n y inversion (inversion in a n y sphere) is t h e p r o d u c t of a t r a n s l a t i o n , a dilation, a n d t h e f u n d a m e n t a l inversion. The basic f a c t a b o u t M for t h e R i e m a n n i a n f o r m a l i s m is t h a t it is t h e g r o u p of m o t i o n s of t h e V5 ~, t h e m e t r i c is f o r m - i n v a r i ' m t u n d e r M : (2.1.3)

d 0 2 ~ - s(x5 ) 2((dx) ~_. e(dxS)2) : -- 2

m

e.(,v_5) -,

((dr)_ .~--e(d:~5) 2)

--n

[(d.r) :g,,~, d~ d.r ] . The proof is obvious for t h e s u b g r o u p s a), b) a.nd e): for d), ..., h) it will be o m i t t e d here (1o). This completes t h e m a t h e m a t i c a l preliminaries. 2"3. P h y s i c a l interpretation of the sphere space. - The m e t r i c (2.]1) implies t h a t we are c o n f r o n t e d in n a t u r e w i t h a five-dimensional p h y s i c a l c o n t i n u u m , i.e., t h a t five i n d e p e n d e n t n u m b e r s are n e e d e d f o r t h e c o m p l e t e specification of t h e p h y s i c a l (( p o i n t ~>. This s t a t e m e n t m a y be a r g u e d pro a n d con on a priori g r o u n d s ; t h e o n l y m e a n i n g f u l criterion is t h e success or failure of t h e p h y s i c a l t h e o r y built u p o n this g e o m e t r y . A c c o r d i n g l y , we m a k e c e r t a i n i d e n t ifications w i t h o u t f u r t h e r justification, and r e m a r k their various p h y s i c a l c o n s e q u e n c e s as t h e y are developed. (10) These inotious are the proper rotations about tile ori'gin in the spaces ll~ carrying the unit spheres X ~ into themselves, cf. footnote (9).

CONFORMAL GEOMETRY AND ELEMENTARY

PARTICLES

S3~{

Say the x -~ have the physical dimension L - - w i t h this interpretation we will call our space position space (q-space). (By (2A1) dO 2 is dimensionless, as it should be.) We connect t h e m with the physical variables - - the directly measured q u a n t i t i e s - of dimensions L , T, and M -1 as follows

(2.14)

.r a-~ x, etc.,

x 4=ct,

x5=

]tO)

,

C

where c and h ( = 2 ~ h ) are light velocity and Planck's constant resp.; x, y, z, t are position and time relative to a Lorentz frame; the physical significance of co, the mass-gauge, will be treated shortly. The f u n d a m e n t a l angle form in terms of these variables is

dO 2 = s

dr 2 - c2dt ~ -

--h'~d ~ 2

~C2

[dr ~_- dx ~ § dy 2 + dz 2].

The f u n d a m e n t a l inversion (2.12) h takes x ~ of a n y physical dimension into ~ of the inverse dimension. Therefore if we have assumed Dim x ~ = L , we m u s t also admit the interpretation Dim x ~ = L - L W e will call the x ~, when interpreted with the dimension L-~, k~: (contravariant) 5-wave number and the space with this interpretation, wave number space (k-space). Or, equally well, defining, p~: (eontravariant) 5 - m o m e n t u m of dimension M L T -~ p~ = h k ' ,

we (.an call it mome~tum space (p~space). I n this conjugate interpretation we introduce the physical variables of dimensions L -a, T -a, and M via (2.16)

k ~-- k,,

etc.,

k~=~,/c,

C

k~=~m.

R e a d : k~, etc., and v are wave n u m b e r and frequency (radians time -~) relative to a Lorentz frame, and m is rest mass. Or, for physical variables of dimensions M L T 1, 3 I L ~ T - 2 M , vi~ (2.16)'

pl=p,.,

etc.,

p~ = E/c ,

pS = em .

R e a d : p,, etc., and E are m o m e n t u m and energy relative to a Lorentz f r a m e ; m as before. I t follows from these definitions t h a t p,, = h k , , etc., E = ~v: if one set of physical variables is taken as primary, the other is defined b y these relations. I n terms of the physical variables of the wave n u m b e r or

834

R. L, INGRAHAM

m o m e n t u m space i n t e r p r e t a t i o n dO 2 takes the form

/

(2.17)

dO -~-- e \ ~ ]

(2.17)'

dO2=e(em)

( d k ~~ c ~dr 2--

2(dp2--c

~c2/l~2d,m~),

"-dE"--- ~c2dm-~) .

[dk2_~ dk2,+ dk~+ dk~] [dp ~= d p ~ + d p ~ § dp~].

The a b o v e implies t h a t in a n y t h e o r y built on sphere space, the physical fields are defined not just over position or w a v e - n u m b e r space, b u t over b o t h spaces. I f we wish to consider position and m o m e n t u m as the conjugate pair (as we shall generally do), substitute (2.1s)

p~ = _

~x~/[x] ~ ,

x~ = _

hp~/[p] ~

for (2.12)h). Because the t r a n s f o r m a t i o n s (2.18) are motions, an a u t o m a t i c reciprocity b e t w e e n q- and p-space results, h a v i n g t h e aspects : 1) a n y e q u a t i o n of the t h e o r y in q-space gives a n o t h e r t r u e equation in p-space on the substitution x ~ - + p~, or (2.19)

x --> p,., etc. ,

t --> E / c 2 ,

~o ---> c 2 m / ] i ,

and conversely; 2) a n y solution for the fields relative to a q-frame tensorially t r a n s f o r m e d u n d e r (2.18) gets new c o m p o n e n t s representing the s a m e solution relative to the associated p-frame, a n d conversely. Call the t r a n s f o r m a t i o n (2.18) Bory~ c o n j u g a t i o n and the pair of properties (1) and (2) B o r n r e c i p r o c i t y a f t e r the foremost exponent of these ideas in m o d e r n physics. We r e m a r k t h a t the points on the ~'ones (2.20)

[x]~- = o ,

[p]'- = o ,

play an exceptional r o l e - their conjugate points p~ ---- ~ (~ = 1, .... 5). etc., coalesce and are called the infinite point. This a u t o m a t i c Born reciprocity is absent in theories based on special r e l a t i v i t y because their group of motions, (2.12) a) and/or b), which leave u n c h a n g e d the physical dimension, cannot c a r r y one f r o m one space to the conjugate space. The 5-uple x ~ will be i n t e r p r e t e d as the labels necessary to specify completely a t e s t p a r t i c l e a t a n e v e n t . Call this p a i r of things a p a r t i c l e s t a t e or s t a t e for short ((~ s t a t e ~) will also designate the 5-uple of n u m b e r s x ~ or (r, t, o~), etc., labelling the s t a t e where no confusion can arise (11)). The t e s t particle figures here only in its i n s t a n t a n e o u s a s p e c t ; one should not t h i n k of it as h a v i n g a n y (( i d e n t i t y ~) or (( continuous existence ~) in time. E.g., it is meaningless to

(11) Cf. the corresponding usage of ((event,, in special relativity.

CO N F O R M A L

GEOMETRY

AND ELEMENTARY

PARTICLES

~35

a s k for the position of * the t e s t particle ,~ of the state (r, t, a~) at some l a t e r time. The m o t i o n of t e s t particles (free and u n d e r forces) is described b y p a t h s of particle states which are integrals of the motion equations (w 3). Hence a t e s t particle in this t h e o r y has Jive degrees of freedom as c o m p a r e 4 to the v.sual four. In t h e m o m e n t u m pietm'e these are p , E, m: m o m e n t u m , energy, a n d rest mass, which are thus a s s u m e d in general independent (is). I n t h e position picture r and t label its position relative to a Lorentz f r a m e and w, the massgauge, or h(o/c, the associated C o m p t o n w a v e length of this mass-gauge, is some m e a s u r e of its finite size. We get a t e n t a t i v e idea of the model of the finite p~rticle implied b y the t h e o r y as follows. The s t a t e (r, t, c,J) corresponds to the sphere (hyperboloid as a real locus) of center r, t and radius hw/c (2.21)

( ~ - - r) 2 - C~(r-- t) ~ ~ e (heo/e) 2

in the space-time of coordinates (~, ~); indeed, this is its p r i m a r y g e o m e t r i c m e a n i n g in this theory. This hyperboloid is the locus of all events influenceable (v ~ t) b y our finite t e s t particle at the e v e n t (r, t) b y signals p r o p a g a t i n g with light v e l o c i t y - with the proviso t h a t relative to the point ~, it b e h a v e s like a disc of radius h~o/c centered at r perpendicular to ~ - - r and capable of radiating signals only f r o m its rim. Call t~w/c the size of the particle. Note t h a t b y (2.18) u n d e r B o r n conjugation w ~ 0 -+ m ~ 0 and ~ sa 0 --> ra r 0. I n other words, t e s t particles of finite and zero size h a v e finite and zero mass resp. in the conjugate m o m e n t u m picture, and conversely. For a null t e s t particle (zero size, hence zero mass) the hyperboloid (2.21) degenerates into the light cone of the associated event. We shall m a k e t h e h y p o t h e s i s t h a t the m e a n i n g o~ these h y p e r b o l o i d s is t h a t a t e s t particle has an intrinsic non-localization in space-time v a r y i n g directly with its size. Then it m i g h t be fruitful to identify the particle with this whole hyperboloid of e v e n t s - e v i d e n t l y this depends critically on the observational significance of (~particle ,. Similarly, in the m o m e n t u m - e n e r g y varm.bles (g, V) the hyperboloid

(2.21')

(~ -- p)~ --

c

~-(~ - -

E?

=

~(mc) ~

will m e a n an intrinsic non-localization of the test particle's m o m e n t u m a n d energy which varies directly with its mass. A (( Heisenberg conjugacy ~ (~a) of 5 - m o m e n t u m a n d 5-position for particle states follows directly f r o m t h e (~2) For the free particle, the expected dependence occurs as a consequence of the motion equations (3"2). (la) W. HE~SENB~RC.: Die Physikalischel~ Prinzipien der Quante,theorie (Leipzig), 1944), 4 Aufl., Kap. 2.

836

R.L. i~Gm~HA~t

definition (2.18) of Born conjugation. For the product [ p x ] ~ p . r - - E t - - ~h~om satisfies (2.22)

[px]

~- --

h .

I f x ~ and p" are considered the 5-component quantities necessary to handle the size and mass covariantly, this says t h a t the (~5-size ,) and (~5-mass ~) are related in a bilinear equation of u n c e r t a i n t y relation form. The action constant h; is a measure of the intrinsic correlation between (( 5-size )> and (( 5-mass ~), i.e., between the localization in q- and p-space of the associated test particle. A conceptual difference with former theories should be noted. Here the q-coordinates of a particle state d e t e r m i n e the associated p-coordinates via (2.18), and conversely. The p~ are just a n o t h e r kind of coordinate, of different physical dimension from the xL The relation of the p's to the velocity for a moving test particle comes out of the motion equations (w 3). Force fields are defined b y means of their action on test particles. The field strengths at events unoccupied by particles not only do not come into question, but, strictly speaking, it is meaningless to ask for these values. This f u n d a m e n t a l feature of the field concept is here automatically t a k e n into account. For in this theory the fields will be defined on particle states, so that it will be impossible to give the value of a field at an event without specifying at the same time a test particle there on which it is to act. Whence the value of the field becomes the actual force on that test particle. The fact t h a t the fields are functions of ~o or m more complicated (1~) in general t h a n a simple proportionality implies also an a b a n d o n m e n t of the classical concept of the ((field strength ~ where it suffi('ed to know the force on a test particle of unit source strength (charge, mass, etc.) to specify it for test particles of :tll strengths. This p e r m i t t e d the suppression of the particle source strengths from the domain of definition of the fields. The question of the building of the M 6 b i u s ] t a m e s , i.e., b y what operations with physical nleters (~eneric term for measuring device) we arrive at the numbers (r, t, o~) or ( p , E , m) labelling a state is a deep one, meriting separate t r e a t m e n t , and cannot be properly gone into here. I f we call the gauge-meter 9t scale for convenience, a M o b i u s observer for position space will be a Lorentz observer provided with a scale at each event to measure the sizes of test particles there. The exact constitution of a scale, which obviously depends on the observational significance of the hyperboloids (2.21), is not clear at the moment. We emphasize that only the (.oincidence of states has absolute significance for the class of M6bius observers, not the coincidence of events or test

(it) Although in certain familiar cases (e.g., field of a resting gravitating particle) it reduces to the simple proportionality (of. sequel).

CONFORMAL GEOMETRY AND ELEMENTARY

PARTICLES

837

particles sizes separately (15). The null test particles are special in t h a t their zero size or zero mass remain zero on change of observer. I t is probabie, however, t h a t light and the electron can be used in the definition ab initio of the M6bius frames b y postulating t h a t t h e y propagate in t h e m according to

(2.22)

a)

dr 2

e2dt ~ : _ - - e2dT ~ ~ O,

b)

- - e "~d~ 2 - - e c~ d o 2 --- 0 ,

doz

0 [T ~: proper time]

resp. in position space and

a)

dk ~ - c - : d v

b)

--c-2dv~ - - e

(2.23)

2 . . . . . e- 2 d r o

O,

dm

0

c 2dm: h~" -- 0

[ ro - - proper frequency]

resp. in wave n u m b e r space. With the aid of a length-meter (rod) and light one would first lay out the Lorentz frames as in special relativity (equations a); then the scales would be superimposed using the electron (equations b). L a t e r results (w 3 and the sequel) for free p h o t o n and electron motion are compatible with these assumptions. Here c and ]i play the role of c o n v e n t i o n a l constants - arbitrary but fixed numbers which serve to define and calibrate clocks and scales in terms of a given rod b y r e q u i r i n g t h a t light and electron propagation obey (2.22) a) and b) resp. for position space; correspondingly for wave n u m b e r space. This description for constructing (properly set and calibrated) clocks and scales in terms of a given rod b y using light and electrons can be sadd to constitute the (~rigidity ~) of the ~I6bius observer. The above interpretation of the nngle metric then leads to the following enunciation. The quantities dv (proper time) and dr are separately noninvariant for the class of all M6bius observers. ] n the combination (2.15) however formed with the aid of the dimensional constants h and c, t h e y build the dimensionless invariant d02 of ~ sphere space M~. Correspondingly for dvo and dm in the conjugate space. Cf. obviously the analogous situation for length and time and the invariant dT ~ of the R4 of special relativity. 2"4. P h y s i c a l interpretation o/ the g r o u p o] m o t i o n s . - A physical interpretation of the m a t h e m a t i c a l transformations (2.12) m e a n s : given a M6bius observer M representing one system x ~, an operationally meaningful prescription for recognizing, or building, the unique other M5bius observer M representing the transformed system J~. We recall the basic fact t h a t because of the ((rigidity )) of the M6bius observer, it is not necessary to specify the behavior of all the elements making up the observer under the transformation. E.g., in (15) Cf. the analogous non-~bsol~teness of simultaneity.

~3~

R.L.

INGRAtIAM

(2.12) a) it is sufficient to give the (constant) velocity vector of one space point fixed in M relative to M. Since a MSbius observer is in particular a Lorentz observer, the proper Lorentz group a) and the space-time translations b) have their well k n o w n interpretations here. Their inclusion in the f u n d a m e n t a l group makes a n y t h e o r y built on sphere space both isotropic and homogeneous in space-time. Under both groups ~o is invariant, which means t h a t test particle sizes are not renormalized. B y (2.15) the cross-section X4 of the space composed of all particle states with the same size ?i~o/c behaves u n d e r these two groups like the special relativistic. R4 with the invariant length element ]io)dO/c. The motions c

are interpreted as uniform renormalizations and scales of a MSbius observer. B y rigidity the rods dilate in the ratio ~-I. Physical interpretation of d): Let M and (t--~ t ~ 0) coincident, and N and N resp. the Then d) is interpreted as a t r a n s l a t o r y relative the origin of X, performs the motion (2.24)

x -? (~/2)(x 2 - c2t 2) ~- 0 ,

(dilations) of the rods, clocks, it is sufficient to prescribe t h a t M be Mobms observers initially space frames belonging to t h e m . motion of N and N such t h a t O~

y :

z = 0

in 5 T. This is a uniform acceleration a --~ ~c ~ of 0 along the x axis of N such t h a t initially 0 coincides with O, 2g's origin, and is at rest with respect to it (1~). Relativistic uniform acceleration is of course m e a n t here ((( Hyperbelbewegung >>), defined shortly as ordinary uniform acceleration at every m o m e n t in an instantaneous rest frame. Then this physical interpretation of the mathematical transformation is a law; it makes experimentally verifiable predictions about the deformation of the material structures defining the MSbius observer on uniform acceleration. These deformations will be treated in detail below. The statement, (~The velocity of light is constant, equal to c, for all MSbius observers >), is another way of stating one aspect of this same law. Similarly for a s t a t e m e n t a b o u t electron propagation and the constant h, cf. (2.22) b). Note t h a t this group of accelerative motions renormalizes the sizes hco/v of test particles non-uniformly depending both on the sizes and on the associated events. Another feature of the accelerative motions: test particles of different size at the same event for an observer correspond to different events as seen b y an accelerated observer.

(1G) j . tIAANTJES: Proc. Ned. Akad. Wet., 43, 1288 (1940); E. L. HILL: Phys. Rev., 72, 143 (1947).

CON1;'ORMAL G E O M E T R Y

AND

ELEMENTARY

PARTICLES

839

The physical i n t e r p r e t a t i o n of t h e group g) is not k n o w n at present, t h o u g h it m u s t r e p r e s e n t some sort of u n i f o r m relative acceleration of clocks. Again t h e size of t e s t particles is non-uniformly renormalized, etc.. We k n o w t h a t t h e physical i n t e r p r e t a t i o n of B o r n c o n j u g a t i o n (2.18) is t h e change f r o m position observer to m o m e n t u m observer (q-observer t o p-observer for short) or vice-versa, hence its precise prescription m u s t be a n idealization of the change f r o m q-measuring a p p a r a t u s t o p - m e a s u r i n g a p p a r a t u s as it is a c t u a l l y p e r f o r m e d in the l a b o r a t o r y . This difficult job will not be a t t e m p t e d here. Two general r e m a r k s will h a v e to suffice. F i r s t : well k n o w n a r g u m e n t s (1,) on t h e n a t u r e of o u r m e a s u r i n g a p p a r a t u s show t h a t there is an ineradicable conjugate i n d e t e r m i n a c y in our readings of p a r t i c l e position a n d m o m e n t u m , due to the disturbance of one on m e a s u r i n g the other. This t h e n m e a n s t h a t in a n y consistent t h e o r y of particles which talks a b o u t their positions and m o m e n t a , this conjugate i n d e t e r m i n a c y m u s t be derivable m a t h e m a t i c a l l y f r o m the axioms of the theory, if the t h e o r y claims to describe particles in the domain where these effects are not negligible. E.g., in q u a n t u m mechanics, whose m a t h e m a t i c a l f r a m e w o r k is an operator ring on H i l b e r t space, the w a y in which the i n d e t e r m i n a c y relations are derived f r o m t h e axioms is well known. I n the case of this theory, whose m a t h e m a t i c a l f r a m e work differs f r o m t h a t of q u a n t u m mechanics, these s a m e physical a r g u m e n t s for t h e necessity of h a v i n g a conjugate p-q i n d e t e r m i n a c y in t h e t h e o r y t h e n support some such conjugate non-localizability i n t e r p r e t a t i o n as we h a v e m a d e of eq. (2.22). Second: Born conjugation takes all the states with [ x ] ~ > 0 into unphysical images (pS ~_ m c ~ 0), which m u s t be i n t e r p r e t e d as t h e p-unobservability, or unobservability of the m o m e n t a , of these states. We will say t h a t the p - f r a m e is p o l a r i z e d with respect to the cone [x]~-~ 0 which m a r k s the b o u n d a r y b e t w e e n the p-observable (Ix] 2 ~ 0) a n d p - u n o b s e r v a b l e ([x] 2 > 0) states. This p-unobservability can be t e n t a t i v e l y explained as follows. The associated p - f r a m e cannot be regarded as i n s t a n t a n e o u s l y established throughout position space, r a t h e r it m u s t be initiated at a definite t i m e and place and built out f r o m there with a finite velocity of extension. (E.g., if it consists of devices for sending" and receiving signals, the velocity of extension cannot exceed the speed of the signals.) This explanation accords with general q u a n t u m mechanical views on the limitations of m e a s u r i n g a p p a r a t u s , and is s u p p o r t e d b y the analysis of free particle m o t i o n u n d e r B o r n conjugation (3"2) (is) L e t C s t a n d for Born conjugation. T h e n the motion T in the w a v e n u m b e r

(17) W. HEISENBERG: Op. Cir., Chap. II and III. (is) Then the new q-frames defined by (2.12)d), e) and f), which take the cones A = 0 , etc., D = 0 into the infinite state (in position space) must evidently be regarded as polarized in the same sense with respect to these cones.

840

R.L.

INGRAItAM

picture can be physically i n t e r p r e t e d as the m o t i o n C T C in t h e position picture~ which l a t t e r has been discussed above. Due to t h e perfect reciprocity b e t w e e n the two pictures, however, T should a d m i t a d i r e c t i n t e r p r e t a t i o n withouv reference to the position picture (~). The d e f o r m a t i o n of ~ smM1 r o d fixed iu M at the s t a t e ( r / t , o)) as seen b y M , related to M b y the accelerative motion (2.12)d), is o b t a i n e d b y c o m p u t i n g ~Y~/3x (parallel orientation) a n d ~ / ~ y (transverse orientation, along y) and s u b s t i t u t i n g for the x ~ their values in t e r m s of the ~ . We get ~

(2.25)

~=

/~- (~/2)(~ ~

+

~-

c~- t~/

~ c~-]'

The similarly defined clock-period and mass-gauge (or test particle size) deformations at the state (F, t , ~ ) are, resp.

(2.25)'

-ei-s + (~/2)e,~,, ~t

~~o) - /x + - ~ ( ~ / 2 ) - e 2

Unlike the case of uniform relative velocity, the deformations depend on position in t h e space. I n the range ( ~ ) ~ << a~, ( f l z l , ..., 5) the p r o p o r t i o n a l deformations ~ x - - 1 , (2.26)

etc., are all the same and equal to - - a~/e 2 .

T h e y are thus proportionM b o t h to t h e acceleration a n d t h e abscissa. A case for which these effects m i g h t be observable is the recession of the galaxies. The empirical law of red-shifts (s0) for the galaxies Gi ( i ~ 1 , 2,...) at distances r~ f r o m our g a l a x y G is

where k--~5.7.10 - ~ c m -1. Now if t h e galaxies Gi are a s s u m e d to be fixed on t h e x-axes of M6bius f r a m e s M<,, initially coincident with f r a m e s M(i) in each of which our g a l a x y G is fixed at the space origin, all in the same u n i f o r m relative acceleration (magnitude A > 0) along the ~-axes of t h e corresponding

(~9) We mention in particular that test particle masses are invariant under translations and Lorentz transformations but are non-uniformly renormalized by the accelerative motions. (20) E. HUBBL]~: The Observational Approach to Cosmology (Oxford, 1937),p. 25.

CONFORI~IAL G E O M E T R Y A N D E L E M E N T A R Y

PARTICLES

841

J

M~,), t h e n b y (2.26) t h e w a v e length X of an a t o m i c spectral line e m i t t e d f r o m G, will suffer t h e shift

t o w a r d the red (5~/A > 0) as seen b y G. This agrees with the empLrical law, a n d gives t h e small acceleration A_~_ 5.7.10 -5 cm s -2. F r o m t h e exact expressions (2.25), (2.25)' it is seen t h a t for a given accele r a t i o n a the distance d f r o m the origin for which the p r o p o r t i o n a l d e f o r m a t i o n is of the order of u n i t y is defined b y a-d/c~---1. I t is interesting t h a t this f o r m u l a also defines the beginning of the w a v e zone of an accelerated particle in classical Maxwell theory, n a m e l y the distance d f r o m a charged particle moving with acceleration a at which the r a d i a t i v e begins to d o m i n a t e t h e C o u l o m b field.

3. -

Free Particles.

3"1. T h e geodesics. - The geodesics are the p a t h s of m i n i m u m (or s t a t i o n a r y ) angle in the sphere space Ms. A simple intuitive idea of t h e m m a y be o b t a i n e d b y picturing our space as the surface of a ball as in footnote (9); t h e y are t h e n the (( great circles )~. I n n a t u r a l coordinates t h e y come out to be (Appendix I I ) x TM = n ~ n - ~ ( a + 7) -~ + 5 " ,

(3.1)

(null geodesic) x~ = n-~(a + 7) -~

and

,(3.2)

x" :

umu-2 tg (0 + 7) + om ,

x~ :

~-1 see (0 + 7)

(non-null geodesic)

where y and 5" are constants a n d x ~ > 0 requires 0 < a § 7 < cx~, - - ~ / 2 < < 0 + y < z~/2. The condition t h a t x b be rest imposes ~he condition on t h e ~ Sgn (~)' = e . 3"2. T h e free particle p a t h s . - L e t t i n g t i m e be the running variable, a n d i n t r o d u c i n g new constants to facilitate the physical i n t e r p r e t a t i o n , we get the 53 - I I N u o v o

Ci-rnen$o.

842

R.L.

INGRAHAM

p a t h s in position space x=v~(t--to)~-xo, (3.3)

(null geodesic)

c[1

-

-

etc..

V2/C2[ 89

to)

--

(to< t < c~)

and

x: (3.4)

(non-null geodesic)

a) =

v~(t--to) Jr Xo ,

etc.

I 1 - - v2/c2189

- - t~ -~ \ 1 - - v2/c ~] J

(--c~<

t< ~)

where we h a v e set ~4 : 401, ~ 1 = v,/C~o ~, etc., t,2 = r~ ~-v~ § v~, ~1 =X0, et('.~ 5 ~ = cto, and the sign condition on (u)2 becomes Sgn ( 1 - - v 2 / c 2) = s. N o t e t h a t the p a t h (3.3) is i n d e p e n d e n t of 40, corresponding to its invariance u n d e r the dilation group (2.12)c). The geodesics i n t e r p r e t e d as p a t h s in w a v e n u m b e r or m o m e n t u m space are given at once b y m a k i n g t h e substitution of Born reciprocity x '~ -+ k ~ or x ~ - ~ p ~ in (3.1), (3.2). E q u i v a l e n t l y , for m o m e n t u m space say, we m a k e the substitution (2.19) in (3.3), (3.4), r e n a m e the constants 40, v, ro, a n d to resl). ti/lo, u, Po, and Eo/C ~. We get

I P~--Po~ = u~ (null geodesic)

( E - - Eo)

c~

I m=tl--u2/c=[ 89

, etc.,

(E--E~ C2

(3.5) ( E - - Eo) (non-null geodesic.) ! P~--P~ = u ~ -

I o:

c2 , etc., E - - Eo 2

/

where the constants u a.re subjected to the sign condition on the v. more familiar in the f o r m

mux px--po~=-iI__u2/c21 89 , etc.,

(3.5)' P x - - Pox--

( E - - Eo) C

ix, etc.,

E--Eo= ( i = u/u),

(3.5) is

mc2 il_u.~/c2! 89

(~ # c)

m ----- O

(u = c )

in which m r a t h e r t h a n E is t h e running variable for u ~ ~.

CONFORMAL G E O M E T R Y

AND E L E M E N T A R Y

843

PARTICLES

W e a r e t h u s l e d t o i n t e r p r e t t h e g e o d e s i c s as p a t h s of f r e e p a r t i c l e s in u n i f o r m m o t i o n w i t h c o n s t a n t v e l o c i t y v o r u (2~). r u n d h~o/c, a r e t h e p o s i t i o n a n d size a t t i m e t of t h e f r e e p a r t i c l e for a q - o b s e r v e r ; a l t e r n a t i v e l y p a n d m a r e i t s m o m e n t u m a n d r e s t m a s s a t e n e r g y E for a p - o b s e r v e r (22). F o r m u l ~ (3.5)' is ~ p a r t i c u l a r l y s t r o n g t h e o r e t i c a l s u p p o r t of t h e p h y s i c a l i n t e r p r e t a t i o n of w 2. A n d i t h a s p e r m i t t e d us t o n o r m a l i z e a w a y a p u r e n u m b e r f a c t o r in t h e fifth c o m p o n e n t of t h e m e t r i c ( 2 . ] 7 ) ' u n d e t e r m i n e d on d i m e n s i o n a l g r o u n d s a,lone. T h e m o m e n t a of t h e free p a r t i c l e s (3.3), (3.4) a r e c a l c u l a t e d b y B o r n c o n j u g a t i n g t h e s e s o l u t i o n s . T h e d e t a i l s a r e g i v e n in A p p . I I I . H e r e we will t r e a t o n l y t h e case of t h e n u l l geodesics. W e g e t t r a j e c t o r i e s (3.5)' w i t h t h e c o n s t a n t s (~3) (3.7)

Po--

~

~

C t o -- r 0

(~

22

--

2~ u =

c $0 - - ro

1

.2~2--..2

C ~0-

ro /

v

c2502 _ _ r02

ro ,

where

(3.s)

to--

.ro.

S i n c e t h e t r a n s f o r m a t i o n g i v e s m (or E) a s a f u n c t i o n of t, w e c a n also g i v e these their time-dependent form:

(3.9)

m(v+rol(t--to))

m:(t)

E --

(o :/: c).

with ~2t2

inch

]1-- v21e~lt

~2\--I

2C(to),~, + c ( t - - t o ) ]

and (V ~ - ~)

with t

22

2 --1

(21) The sign condition on t h e v and u seems to exclude ~ = + l , as giving space-like motion to free particles. Moreover, e : + 1 make t h e spheres (2.21), (2.21)' representing a test particle related space-like to t h e i r centers. (22) Incidentally, the problem of the <(i d e n t i t y )) of a particle t h r o u g h t i m e disappears. We simply agree to call a 1-dimensional schar of particle states in which the position a n d size coordinates depend on the time coordinate a~ in (3.5) or (3.6) <(a >) free partiele e n d u r i n g in time. (~a) u ~ c~---,v = e; cf, App. I I I .

844

R.

L. I N G R A H A M

A s y m p t o t i c a l l y in t i m e p a n d E go over into the expressions (3.5)' for c o n s t a n t s Po--~ Eo ~ 0, u ~ v in t e r m s of the (asymptotically} c o n s t a n t rest mass m~ and energy E ~ resp. given b y

(3.10)

m=c

Ii--v~Ic~189 -

=

(to),o~]

p , E, and m are thus (asymptotically) c o n s t a n t in t i m e for the free particle, moreover, t h e p - v e l o c i t y u is just the q-velocity v. The following i n t e r p r e t a t i o n of the t e m p o r a l b e h a v i o r (3.9) and (3.9)' of t h e free particle m o m e n t a is a plausibility a r g u m e n t for the t e n t a t i v e explanation of the existence of p - u n o b s e r v a b l e states as due to their situation outside t h e domain of the gradually established p - f r a m e (cf. 2"4). The key, we repeat, is to i n t e r p r e t u n p h y s i c a l values as the p - u n o b s e r v a b i l i t y of t h e corresponding state. Now for the particles (3.3) [x]2 X 0 a s y m p t o t i c a l l y if and only if (to)~t ~ 0. A particle with (tO)ro~~ 0 would thus be a s y m p t o t i c a l l y p-unobservable, f r o m which we conclude t h a t it is c o m p l e t e l y outside the domain of t h e p - f r a m e and the t e m p o r a l b e h a v i o r (3.9), (3.9)' without significance (24). A particle with (tO)ret> 0 (whose p a t h states are thus e v e n t u a l l y p-observable) m a y be called a p - o b s e r v a b l e particle, and its h i s t o r y analyzed t h u s : b y (3.9), (3.9)' the i n t e r v a l (to, c~) breaks into tile two p a r t s (to, te~), (tr co), where tr = to + (r~o- c~to)/2c~(to)r~(. E (or m ) < 0 in the first interval, m e a n i n g t h a t the p - f r a m e has not y e t (( arrived at ~>the free particle. At the critical t i m e t----tr the m o m e n t of (~arrival)), the m o m e n t a are infinite. I n the second i n t e r v a l E (or m) > 0 and decreases m o n o t o n e l y to the positive lower b o u n d (3.10) at t ~ c~. I n this t r a n s i e n t p h a s e t h e m o m e n t a are observable a n d settling down to c o n s t a n t values as t h e p e r t u r b a t i o n due to the e s t a b l i s h m e n t of the p - f r a m e a t t e n u a t e s . A certain s p e c t r u m of a s y m p t o t i c masses or frequencies depending on (t0)~ is o b t a i n e d (2~). These t r a n s i e n t phases are exp e c t e d to be v e r y short u n d e r usual o b s e r v a t i o n a l conditions. A fuller account of this will be published elsewhere. B y (3.3) the size (v:/:c) increases u n i f o r m l y to infinity, hence these free particles would dislocalize or (~disperse ~) n a t u r a l l y in t i m e with the nonlocalization m e a n i n g of size. On the o t h e r hand, if it is p-observable, the m a s s decreases m o n o t o n e l y in t i m e to a lower b o u n d m~ > 0. T h a t is, its localization in p-space increases up to a limit m e a s u r e d b y m ~ . F o r a free p a r t icle, then, the passage of t i m e is an agent tending to dislocalize its position a n d localize its m o m e n t u m . (24) The tacit assumption here is that a particle can in time enter the domain of a p-frame, but once in, can never leave it. (25) In the case of the non-null geodesics, this spectrum will depend also on ~o. i.e., on the initial sizes.

CONFORMAL G ~ O M E T R Y AND E L E M E N T A R Y PARTICL]~S

8~5

We now c o n s t a t e the r e m a r k a b l e f a c t t h a t our free particles m o v e w i t h uniform velocity with respect to all M6bius observers. These include relatively accelerated observers {this will always m e a n in the precise sense of t h e prescription of 2"4). F o r in (a) of App. I I , x ~ referred to a n y M6bius frame. I n Einstein r e l a t i v i t y , on the o t h e r hand, if one of the Lorentz observers of 2"4 is a s s u m e d an inertial system, t h e n the other observer, not m o v i n g with unif o r m velocity relative to the first, would be non-inertial, i.e., would feel a gravitational field. A free particle, b y definition m o v i n g uniformly for t h e first oberver, would p e r f o r m non-uniform m o t i o n for t h e second, accelerated b y this induced gravitational field. L e t us say a t e s t particle has the Galilean property u n d e r a group of changes of f r a m e if w h e n e v e r it m o v e s u n i f o r m l y in one frame, it m o v e s um_'formly in all f r a m e s connected with t h e first b y a t r a n s f o r m a t i o n of the group. T h e n we can characterize t h e situation shortly b y saying t h a t in this theory, free particles h a v e t h e Galilean p r o p e r t y u n d e r all u n i f o r m accelerations; in E i n s t e i n relativity, u n d e r only t h e subgroup of u n i f o r m velocities. T h e explanation of this a p p a r e n t l y p a r a d o x i c a l b e h a v i o r of our free p a r t icles on an accelerative motion lies in t h e b e h a v i o r of t h e fifth coordinate. A c o m p e n s a t o r y change of size t a k e s place, just sufficient to t r a n s f o r m u n i f o r m m o t i o n again into u n i f o r m m o t i o n (e.g., p u t {3.1) or {3.2) into (2.12) d)). Size, a n d this b e h a v i o r of it, is obviously not to be p r e d i c a t e d of macroscopic , p a r t icles ) ~ - complicated agglomerations of simpler e l e m e n t s - - t h e r e f o r e we restrict our i n t e r p r e t a t i o n of the geodesics to free elementary particles, and t h e null geodesics, to p h o t o n s a n d electrons for reasons which a p p e a r in the sequel. v ~ c defines t h e photon, which implies b y (3.3) and (3.5) t h a t it is a null particle {zero rest m a s s a n d size). This is of course a M6bius i n v a r i a n t definition. The a b o v e suggests a f u n d a m e n t a l e x p e r i m e n t of the Michelson-Morley" t y p e capable of choosing for physics b e t w e e n ordinary r e l a t i v i t y a n d sphere space g e o m e t r y . Experiment: Observe a free p h o t o n or electron f r o m two L o r e n t z f r a m e s relatively accelerated as prescribed in 2"4, one of which, say L, is an inertial system. Then this t h e o r y predicts a u n i f o r m m o t i o n relative to L also. Einstein relativity, predicts some non-uniform velocity in ~ (the order of m a g n i t u d e 6f deflection f r o m uniform m o t i o n being t h a t due to a g r a v i t a t i o n a l field of s t r e n g t h a ~ cce~ in t h e ~ direction) (2s). N o t e t h a t t h e o b s e r v a t i o n of u n i f o r m space-time m o t i o n in L would suffice to r e j e c t Einstein r e l a t i v i t y , quite a p a r t f r o m the b e h a v i o r of t h e size whose o b s e r v a t i o n depends on the (2e) A r conjugate ~ experiment in the momentum picture is immediately formulizab le.

846

R. L. I N G R A I I A M

as y e t p r o b l e m a t i c a l construction of the scales. The new uniform velocity predicted here is in general different f r o m the o l d - the formulas are worked out in App. I I I . The p a r t i c u l a r p a t h (3.3) with r o ~--to ~ 0 however is inv a r i a n t : ~ ~- v, ~o ~ {~ ~ O. Fine point: the prescription of the e x p e r i m e n t assumes t h a t we can recognize the force-free world (or force-free p a r t thereof); equivalently, t h a t we k n o w w h a t we m e a n b y (( a free particle ~ and (( an inertial s y s t e m ~. The difficulties of defining both of these in r e l a t i v i t y without circularity, necessitating the introduction of elements logically e x t r a n e o u s to the t h e o r y like the (( fixed stars ~), etc., are well known. W i t h o u t f u r t h e r s t u d y we cannot say whether the same logical trouble exists in this theory. Therefore, faute de mieux, (( free particle ~) and (( Lorentz f r a m e ~) for the purposes of the exp e r i m e n t are u n d e r s t o o d to m e a n w h a t e x p e r i m e n t a l physics is generally agreed to designate b y those names. One should not conclude f r o m the a b o v e - p r e d i c t e d behavior of free p a r t icles t h a t t h e r e is no (~principle of equivalence )) for sphere space theories. There exists such a principle here of course, stating t h a t a n y change of f r a m e (27) not one o~ the m o t i o n s (2.12) manifests itself as v,~rious induced physical forces, including g r a v i t a t i o n (cf. the sequel). I t differs f r o m Einstein r e l a t i v i t y however in having a wider group of motions, whence there exist changes of f r a m e inducing a gravitational field there which induce no force fields m this theory. I n concluding this section, a c a v e a t should be inserted. I f the hyperboloids (2.21), (2.21)' really m e a n non-localized t e s t particles, the observational significance of r and t, (, the position at a t i m e of the particle )) ill the language of point particles used above, becomes uncle.~r. E v i d e n t l y an e x p e r i m e n t designed to t e s t w h e t h e r free particles h a v e the Galilean p r o p e r t y under unif o r m acceleration i n the coordinates r, t has little chance of success if we do not k n o w how to m e a s u r e r a n d t. E v e r y t h i n g depends on the observational m e a n i n g of the hyperboloids. The photon, however, is perfectly localized (co--~ m--~ 0), hence it seems t h a t at least for this particle r and t m a y be given point particle meaning. I n which case the proposed e x p e r i m e n t goes through for light. 3"3. C o n c l u d i n g general r e m a r k s . - The b e h a v i o r (Galilean p r o p e r t y under uniform accelerations) predicted here in particular for free p h o t o n a n d electron~ though strange f r o m the viewpoint of macroscopic intuition, is really no odder t h a n the relativistic c o n s t a n c y of ]s velocity. I t suggests t h a t e l e m e n t a r y particles are even less like N e w t o n i a n billiard balls t h a n we imagine today.

(27) Involving relative motion.

CONFORMAL G E O M E T R Y A N D E L E M E N T A R Y P A R T I C L E S

847

I t is however at variance with the predictions of Einstein relativity, and therefore a new Michelson-Morley t y p e experiment, this t i m e with uniform relative accelerations, would be v e r y welcome. A clear cut choice b e t w e e n this a n d the Lorentz g e o m e t r y could be made, provided only t h a t a p p a r a t u s could be devised to raise the predicted effects into the observable range. Needless to say, these m u s t be free p a r t i c l e s - e.g., no strong electromagnetic fields to accelerate measuring devices could be allowed. I n general one m i g h t expect t h a t the observation of all e l e m e n t a r y particles, free and u n d e r forces, under uniform relative acceleration of observers would yield interesting results. Both classical and qut~ntum e l e c t r o d y n a m i c s suffer f r o m infinities associated with point model particles; the f o r m e r with divergent electromagnetic masses, etc., the l a t t e r with divergent additions to the (~mechanical )~ mass due to the coupling of the m a t t e r field with the e l e c t r o m a g n e t i c v a c u u m , etc. ]n an electrodynamics couched in this, r a t h e r t h a n the Lorentz g e o m e t r y , changes could be expected. F o r one thing, this g e o m e t r y a u t o m a t i c a l l y requires finite particles. E v e n if these infinities subsisted, a re-appraisal of their significance would certainly be d e m a n d e d . F o r e l e m e n t a r y particle mass here is not an absolute. We found t h a t free particle mass is (after a t r a n s i e n t phase) c o n s t a n t in t i m e (2"2), but in the presence of forces it will v a r y along the path. Moreover, test particle masses are renormMized non-uniformly b y a n y accelerative change of observer. The masses of the particle states on the cone A = 0 in (2.12)d) are in fact infinite for the new observer. B u t this infinity has no coordinate-free significan(.e. These m a t t e r s will be t r e a t e d in the sequel.

APPENDIX

I

We m u s t e v a l u a t e

(i)

Gt,~(X~' -4- dX~')X ~

[e ~ Sgn (X) ~ = S g n ( X q- dX)~]. We have I(X q- d X ) I-1 = I(X)IL-1{1 + 2Xt, dX~' -4- dX,dX~}- 89 with X~, ~ S~,~X~, dX~, =_ S , , d X L S,~ as in (2.8). I f :r is of first order smallness a n d fl--~ 0(a2), t h e n (1 if- r162-,~ fl)-89 = 1 - - a/2 q- (~ a * -

fl/2)

848

R. L. INGRAHAM

good to second order terms. S u b s t i t u t i n g a - ~ 2XudX~' , f l - ~ d X u d X ~ ' in this, we get the factor ( 1 - - X u d X ~ ' q - ~ ( X ~ , d X ~ ' ) 2 - - 8 9 in (1). Multiplying out, first order t e r m s cancel, and we get 1 - - 89(Su, - - XuX~) dX~' d X 9

for the right member, from which (2.8) follows.

APPENDIX

II

The geodesics are the integral curves of (1)

d2x~'/d02 ~-

fi) j (dx~/dO)(dxr/dO) = 0

where for the null geodesics (dO ~ 0 along the curve), 0-+ a, a a n y running p a r a m e t e r . I n natural coordinates the Christoffel symbols are ,

55~z--l/x',

other

{fl~}=O.

P u t t i n g these in (1) we get d2x~/dO 2

2(d log xS/dO)(dx"/dO) --~ 0,

d2xS/dO 2 - (1/xS)(dxS/dO) 2 - (e/xS)(dx/dO)2 --= O, [ (dx/dO)2 ~ gm~(dxm/d0)(dxn/dO)],

0--~ a for t h e null geodesics.

(3)

This has t h e i m m e d i a t e first integral dxm/dO --~ nm(xS) 2

[n ~ constants].

F r o m (2.11) (dxS/dO)2 --_ _ (x5)2 + e(dx/d0) 2

and in the null P u t t i n g (dx/dO) 2 integrable. T h e n formed. We get

case 0 - + g and t h e --(xS) 2 t e r m on the right is absent. ~- (z)~(xS) 4, (z) 2 ~ g,,~u~z ~, from (3) into this, it is easily this solutions for x 5 is p u t into (3) and the integration perthe solutions (3.1).

CONFORMAL

G]~OMETRY

ANn

ELEMENTARY

PARTICLES

84~

API:,E~ DI x I I I

I n c o m p u t i n g t h e change of c o n s t a n t s in (3.1) resulting f r o m a n y m o t i o n (2.12), t h e t y p i c a l i d e n t i t y to be satisfied is

(1)

J_'~(a + 7) -~ + B~' = A"(,~ -~ ;~)-~ + B ~' c(o + y)-~ + D

W e w a n t t h e new (burred) c o n s t a n t s in t e r m s of t h e old. The f~ct t h a t the quadratic t e r m s cancel out of the d e n o m i n a t o r on t h e right is the characteristic p r o p e r t y of t h e p a t h s (3.1) which m a k e s such an i d e n t i t y possible. Crossmultiplying, etc., and e q u a t i n g coefficients of a ~, a, .~,nd 1, we get D/3 ~ = B ~ , (2)

D/3~'(7 + ~) + C/~ ~ + D A ~' = B~'(~, + ~) -]- A ~',

DB~vy + DA~v + CB~79 + -A~C_= A~p + B~y~. These h a v e t h e solutions (3)

/ ~ = B~'/D,

and, if B ~ - - 0 ,

Ar = D'(A ~'- CB~

[D ~ 0]

C~e0 B~ = A~'/C,

(4)

~ = y -~ C/D

2 ~ = 0,

~ = anything

[ D = 0].

I n all o t h e r cases t h e t r a n s f o r m e d p u t h is i n d e t e r m i n a t e . F o r B o r n conjugation, our c o n s t a n t s are (cf. (3.1) a n d (2.18))

A,~ =

__ hx,~•

C

2(~t)~-~

=

A ~= [(~b)

~

__ ?/~-t,

~b,.1d

a m "a ,'1

B m = __/~,~,

,

D =

Bs=O,

(~)z,

w h e r e t h e s e are linked w i t h the (~physical )) c o n s t a n t s in p- and q-space v i a

(~olh)11--u'-Ic21L a = p o ,

~=(u/e)(~of~),

~'=~o/~,

~

x=(vle)(V2o),

x'=l/~o,

~=(1/2o) Ii--v~tc2189, ~ = r o ,

Eolc, ~'-=eto.

850

R. L. INGRAHAM

T h e n t h e a p p l i c a t i o n (3) g i v e s (3.7) as w e l l as

lo

fi=

-

2tov "r o - - (e2t] -~ r'~) 20 '

-

I

2ctoe(to),o~_ 1 , e t o - ro ]

(

1

*) __~ ~C ~0(__!.0)ret

- ~ -F \ I _ _ v2/c 2] c*t2o-- r x '

p

i,

1-

t-

T h e e x p l i c i t f o r m u l a e f o r t h e o t h e r i n t e r e s t i n g m o t i o n s , t h e a c c e l e r a t i v e ones, are now easily obtainable.

RIASSUNT0

(*)

Nel presente lavoro si esaminano le conseguenze m a t e m a t i c h e derivanti dal basare una teoria di campo (classiea o quantistica) sulla geometri.a conforme. Lo spazio in questione quello di t u t t e le sfere in Ra (spazio lineare quadridimensionale di segnatura ( + q- + --)) invariante fondamentale, l'angolo d'intersezione di due sfere. Nelle premesse matematiche (w l) si sviluppa un opportuno formalismo non omogeneo, che permette di trattare la sfera come un punto in uno spazio Riemanniano a 5 dimensioni di c u r v a t u r a unitaria costante il cui elemento di lunghezza ~ l'angolo infinitesimo i r a due sfere adiacenti. I1 gruppo conforme (trasformazioni di Lorentz proprie, traslazioni, dilatazioni uniformi, e inversioni nelle sfer6) ~ il gruppo di moti a 15 p a r a m e t r i (trasformazioni conservanti la metrica) di questo spazio. Nel w 2 le sfere si interpretano come particelle di prova finite (non localizzate). I campi fisici sono cosi definiti su particelle di prova anzich~ (per esempio, nello spazio di posizione) soltanto sugli eventi occupati dalle particelle di prova. Le sfere possono distinguersi o con la 5-posizione (posizione e dimensione nello spazio-tempo della particella di prova) in un sistema q, o col 5-momento (4-momento e massa a riposo della particella di prova) in un sistema p. Esiste un moto ehe trasforma lo spazio q nello spazio p e inversamente (in altre parole gli osservatori in q e in p sono equivalenti), in virtfi del quale, si dimostra, ogni teoria conforme presenta a u t o m a t i c a m e n t e una reciprocit5 di B o r n i r a lo spazio p e lo spazio q. La 5-posizione e il 5-mornento soddisfano una equazione del tipo della relazione di indeterminazione; cio~, le non localizzazioni negli spazi q e p stanno in relazione inversa con l'h di Planck che misura la correlazione intrinseca, q)utti gli altri moti si possono formare da sottogruppi che portano gli spazi q e p in se stessi. Questi si interpretano sistematicamente come cambiamenti del sistema di riferimento. Quelli che coinvolgono moti relativi rappresentano accelerazioni relative u n i ] o r m i di osservatori di Lorentz (gruppo di Lorentz ~ ~ accelerazioni zero). La massa e le dimensioni delle particelle di prova sono invarianti nel gruppo di Lorentz ma non uniformemente rinormalizzate

(*) Traduzione a cura della Redazione.

C(t:NFOR~IAL G E O M E T R Y

AND EL]~M]~NTARY

PARTICLES

~5~

dai moti accelerati. Nel w 3 si dimostra che le geodetiche dello spazio delle sfere (equazioni di moto delle particelle di prova (elementari))descrivono m o t i u n i f o r m i nel caso presente (in assenza di forze). Le particelle di prova tendono a delocalizzarsi nello spazio q al crescere del tempo e nello spazio p al ereseere dell'energia. La co~ stanza nel tempo dei 5-momenti, la loro interdipendenza d a t a dalla dinamica delia relativit~ speciale seguono da queste equazioni di moto. Si dimostra che le particelle elementari libere mantengono in t u t t i i moti, in particolare iln quelli accelerati, uno stato di velocits uniforme. Ci5 b in contraddizio~e con la relativits ordinaria e suggerisce un'esperienza a t t a in linen di principio a fare per la Fisica una scelta fra la geometria conforme e la geometria di Lorentz.