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Quantum mechanics as a broken symmetry
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Nuclear Physics B7 (1968) 574-582. North-Holland Publ. Comp., A m s t e r d a m
QUANTUM
MECHANICS
AS
A BROKEN
SYMMETRY
H. A. K A S T R U P
Universitiit Mi~nchen, Sektion Physik, lYr~nchen, Germany Received 7 August 1968 Abstract: The algebra of quantum mechanics is derived by breaking the exact Liouville s y m m e t r y of free n o n - r e l a t i v i s t i c stationary systems. The Liouville group consists in this case of the Euclidean group, the dilatations and the t h r e e - p a r a m e t e r special Liouville group. The two latter groups induce global and local geom e t r i c a l gauge t r a n s f o r m a t i o n s . The dilatations imply continuous energy spectra, and t h er efo re they have to be broken if we want to allow for the frequent d i s c r e t e energy s p ect ra in atomic physics, etc. Since physical states are represented by rays r a t h e r than v e c t o r s , it is sufficient to contract the dilatations into a phase transformation. These physical conditions on the breaking of the dilatations for interacting s y s t e m s imply that the g e n e r a t o r s of the special Liouville group go over into multiples of the usual position operators.
1. ~ T R O D U C T I O N Q u a n t u m m e c h a n i c s i s c e r t a i n l y one of t he m o s t b e a u t i f u l t h e o r i e s e v e r c o n c e i v e d by p h y s i c i s t s . I t s r i c h p h y s i c a l and c l e a r m a t h e m a t i c a l s t r u c t u r e h a s c a u s e d m a n y i n t e r e s t i n g i n v e s t i g a t i o n s f o r t h e p u r p o s e of d e r i v i n g o r explaining this theory from different or more basic viewpoints. The present p a p e r i s a n o t h e r a t t e m p t to s h e d s o m e l i g h t on t h e o r i g i n of q u a n t u m m e chanics. A f t e r a c c e p t i n g w h a t i s p r o b a b l y t h e m o s t b a s i c p r o p o s i t i o n of q u a n t u m m e c h a n i c s , t h a t p h y s i c a l s t a t e s a r e to be d e s c r i b e d by v e c t o r s o r r a t h e r r a y s in a H i l b e r t s p a c e , we e n c o u n t e r two d i f f e r e n t s i t u a t i o n s a s f a r a s t h e o b s e r v a b l e s - c h a r a c t e r i z e d by s e l f - a d j o i n t o p e r a t o r s - a r e c o n c e r n e d : t h e r e a r e t h o s e l i k e m o m e n t u m , a n g u l a r m o m e n t u m , and e n e r g y , w h i c h c o r r e s p o n d to t h e g e n e r a t o r s of s o m e s p a c e o r t i m e s y m m e t r y ; and t h e r e a r e o t h e r s , l i k e t h e p o s i t i o n o p e r a t o r s Qj, w h i c h do not s e e m to c o r r e s p o n d to any s p a c e s y m m e t r y w h i c h w o u l d a l l o w u s to d e r i v e t h e b a s i c c a nonical commutation relations *
Pj Vk - VkPj = i-1 5jk
(1)
in a s i m p l e p h y s i c a l way. We s h a l l show, h o w e v e r , t h a t t h e r e l a t i o n s (1) c a n b e d e r i v e d by s o m e s i m p l e p h y s i c a l a r g u m e n t f r o m a b a s i c g e o m e t r i c a l s y m m e t r y of t h e f r e e n o n - r e l a t i v i s t i c p a r t i c l e . T h e s y m m e t r y in q u e s t i o n i s g i v e n by t h e L i o u v i l l e g r o u p d i s c u s s e d in d e t a i l in a p r e c e d i n g p a p e r [1]. If we c o n s i d e r , f o r * We use the units Z - 1 and c - 1.
QUANTUM MECHANICS
575
r e a s o n s of s i m p l i c i t y , o n l y t h e s t a t i o n a r y c a s e (t = 0), t h e L i o u v i l l e g r o u p c o n s i s t s of t h e u s u a l E u c l i d e a n g r o u p ( r o t a t i o n s a n d s p a c e t r a n s l a t i o n s ) , t h e dilatations Dl(a) , and the special Liouville transformations SC3(c). The t w o l a t t e r g r o u p s i n d u c e t h e f o l l o w i n g t r a n s f o r m a t i o n s in x - s p a c e : D I ( ~ ) : x -~ e ~ x , a r e a l , SC3(c): x ~ RT3(c)Rx ,
Rx = x/x2
,
(2)
T3(c)x=x+
c.
(3)
T h e g r o u p s (2) and (3) a r e to b e i n t e r p r e t e d a s g l o b a l a n d l o c a l g e o m e t r i c a l g a u g e t r a n s f o r m a t i o n s w h i c h m a p a g i v e n l e n g t h ds = (dxidxi)~ onto a n o t h e r one w h i c h d i f f e r s f r o m t h e f i r s t one b y a f a c t o r e a o r 1 / ~ ( l ) , ~ ( x ) = 1 +2 v . x + v 2 x 2, r e s p e c t i v e l y . T h e p h y s i c a l m e a n i n g of t h e t r a n s f o r m a t i o n s (2) a n d (3) h a s b e e n d i s c u s s e d in d e t a i l e l s e w h e r e [ 1 - 3 ] , and we s h a l l not r e p e a t it h e r e . Important for our purpose are the commutation relations between the g e n e r a t o r s P J , j = 1 , 2 , 3, of t h e t r a n s l a t i o n s T 3 ( a ) a n d t h e g e n e r a t o r s KJ of t h e s p e c i a l L i o u v i l l e t r a n s f o r m a t i o n s S C 3 ( c ) :
p k g j - K j p k = 2i(6kjD - Mkj) ,
j , k = 1, 2, 3 .
(4)
H e r e t h e o p e r a t o r s D and MkJ g e n e r a t e t h e d i l a t a t i o n s and t h e r o t a t i o n s r e s p e c t i v e l y . Eq. (4) s a y s t h a t i n v a r i a n c e u n d e r t r a n s l a t i o n s and s p e c i a l L i o u v i l l e t r a n s f o r m a t i o n s i m p l i e s i n v a r i a n c e u n d e r r o t a t i o n s and d i l a t a tions ! Now, i n v a r i a n c e u n d e r f i n i t e c o n t i n u o u s d i l a t a t i o n s m e a n s t h a t a l l o b s e r v a b l e s w h i c h do not c o m m u t e w i t h D h a v e a c o n t i n u o u s s p e c t r u m . In p a r t i c u l a r , t h e m o m e n t a P J a r e t r a n s f o r m e d into e - a P J u n d e r f i n i t e d i l a t a t i o n s . S i n c e t h e m a s s m i s an i n v a r i a n t p a r a m e t e r u n d e r a l l n o n - r e l a t i v i s t i c s y m m e t r y o p e r a t i o n s , t h e k i n e t i c t e r m H o = (1~2re)P2 in any H a m i l t o n i a n i m p l i e s t h a t t h e d i l a t a t i o n s c a n o n l y b e a s y m m e t r y g r o u p of a g i v e n s y s t e m if t h e t o t a l e n e r g y o p e r a t o r H = Ho + V h a s a c o n t i n u o u s s p e c t r u m . T h i s f o l l o w s f r o m t h e f a c t t h a t t h e p o t e n t i a l V h a s to t r a n s f o r m in t h e s a m e w a y a s t h e k i n e t i c t e r m w h e n we a r e d e a l i n g w i t h a s y m m e t r y g r o u p . On t h e o t h e r hand: one of t h e m o s t c h a r a c t e r i s t i c f e a t u r e s of a t o m i c p h y s i c s i s t h e e x i s t e n c e of d i s c r e t e e n e r g y s p e c t r a . W e t h e r e f o r e h a v e to b r e a k t h e d i l a t a t i o n s y m m e t r y of t h e f r e e p a r t i c l e s f o r s u c h i n t e r a c t i n g s y s t e m s . B e c a u s e of eq. (4) t h i s c a n n o t b e done w i t h o u t b r e a k i n g t h e s p e c i a l L i o u v i l l e s y m m e t r y , too. In o r d e r to h a v e t h e d i l a t a t i o n o p e r a t o r D c o m m u t e w i t h t h e m o m e n t a , we h a v e to c o n t r a c t it a t l e a s t into a m u l t i p l e of t h e u n i t y o p e r a t o r . S i m u l t a n e o u s l y we w a n t to c o n t r a c t t h e f i n i t e d i l a t a t i o n s exp(i~D) into a p h a s e t r a n s f o r m a t i o n , b e c a u s e a n o v e r a l l c h a n g e of t h e p h a s e s of a l l s t a t e v e c t o r s d o e s not m a t t e r p h y s i c a l l y . T h e s e p h y s i c a l c o n d i t i o n s on t h e c o n t r a c t i o n s of t h e d i l a t a t i o n s i m p l y t h a t t h e g e n e r a t o r s KJ go o v e r into m u l t i p l e s of t h e u s u a l p o s i t i o n o p e r a t o r s QJ a n d eq. (4) g o e s o v e r into eq. (1)! T h u s , we a r r i v e at t h e f r a m e w o r k of q u a n t u m m e c h a n i c s b y s t a r t i n g f r o m a g e o m e t r i c a l s y m m e t r y g r o u p of t h e f r e e p a r t i c l e . T h e d e t a i l s of t h i s p r o c e d u r e a r e d i s c u s s e d in s e c t . 4. In s e c t s . 2 a n d 3 we d e a l w i t h u n i t a r y r e p r e s e n t a t i o n s of t h e 1 0 - p a r a m e t e r L i o u v i l l e g r o u p of t h e t h r e e - d i m e n s i o n a l Euclidean space. The describe a free stationary non-relativistic system.
576
H.A. KASTRUP
A s h o r t h i s t o r i c a l r e m a r k : the close connection between the L i e - a l g e b r a of the Liouville group of the Minkowski space and the a l g e b r a of quantum m e c h a n i c s was pointed out in ref. [4]. L a t e r I l e a r n t about group c o n t r a c tions [5] and an e a r l i e r m a t h e m a t i c a l p a p e r [6] by Segal in which he cont r a c t s the Lie a l g e b r a of the group SO(2, 4) into the a l g e b r a of r e l a t i v i s t i c quantum m e c h a n i c s . The physical a s p e c t s of a slightly different c o n t r a c tion of the s a m e group along the lines d i s c u s s e d above a r e investigated in ref. [7]. Since a r e l a t i v i s t i c canonical quantum m e c h a n i c s with position o p e r a t o r s analogous to the n o n - r e l a t i v i s t i c one s e e m s to meet with some difficulties [8], the r e l a t i v i s t i c c a s e may not be so i n t e r e s t i n g after all. However, the fact that the f r e e n o n - r e l a t i v i s t i c motion is Liouville invariant, opens the way to a new derivation of n o n - r e l a t i v i s t i c quantum m e c h a n i c s ! A final i n t r o d u c t o r y r e m a r k : the e m p h a s i s in this p a p e r will be on the physical a s p e c t s , not on m a t h e m a t i c a l r i g o u r . F o r instance, we shall not d i s c u s s the domains of definition of the differential o p e r a t o r s which o c c u r or those of their self-adjoint extensions.
2. UNITARY REPRESENTATIONS OF THE LIOUVILLE GROUP Our f i r s t b a s i c a s s u m p t i o n is that the s t a t e s of a s t a t i o n a r y f r e e nonr e l a t i v i s t i c s y s t e m f o r m a Hilbert space and that the full 1 0 - p a r a m e t e r Liouville group induces unitary r e p r e s e n t a t i o n s in this Hilbert space. We f u r t h e r a s s u m e the spin of the s y s t e m to be z e r o and c o n s i d e r r e p r e s e n t a tions in the space of functions g,(x) and t h e i r F o u r i e r t r a n s f o r m s ~(p) = (2~)-~ f d3x ~ ( x ) e - i P "x .
(5)
The s c a l a r product (@1, @2) of two functions ~hI and @2 is given by
(@l, ~2) = f d3x @l(X)@2(x) * = f d3p q){(p)~02(P) •
(6)
Since the full 1 0 - p a r a m e t e r Liouville group can be g e n e r a t e d [2] by the t r a n s l a t i o n s T3(a) and the d i s c r e t e length i n v e r s i o n R, one can c o n s t r u c t a u n i t a r y r e p r e s e n t a t i o n by defining the action of t h e s e subgroups on the functions ~(x). If we denote the r e p r e s e n t i n g unitary o p e r a t o r s by U(a) and U(R), we have
(7)
[u(a)~](x) = ~ ( x - a ) and [U(R)+](x) : r - 3 + ( R x ) ,
r = Ix l .
(8)
Defining the H e r m i t e a n g e n e r a t o r s A of the infinitesimal t r a n s f o r m a t i o n s g e n e r a l l y by U = exp(iA), we .~et the following explicit e x p r e s s i o n s for the g e n e r a t o r s P / , K / , D and MJ k in x - and p - s p a c e r e s p e c t i v e l y :
PJ = i-1 ~J '
~J = oxJ '
(9a)
577
Q U A N T U M MECHANICS
P j = p J,
(95)
(10a)
I
~k
-
apk '
(lOb)
(11a)
i-l(~ + xJ ~j) ,
(11b)
D = i({ +pJaj) , ~ j k = i-1 (xJ ~k - x k ~j) ,
02a)
MJk = i - l ( p J ~ k - p k a j )
(125)
.
The finite dilatations, for instance, have the f o r m 3
[eia/9~] (x) = e'~ a~h(e-ax) . We notice that the t e r m ~ i n / ) and D c o m e s f r o m the t r a n s f o r m a t i o n of the function ( ' i n t r i n s i c ' part) and the r e s t c o m e s f r o m the t r a n s f o r m a t i o n of the a r g u m e n t s x or p ( ' o r b i t a l ' part). The s a m e holds for the f i r s t t e r m and the r e s t in K} and KJ. The above g e n e r a t o r s f o r m the following Lie algebra:
[ p j, pl] : 0 ,
(13a)
[KJ, K t] = 0 ,
(135)
[P J, K l] = 2i(SjlD - MJ l) ,
(13c)
[D, P ~
= iP k ,
[D, KJ] : - i K J , [D, MJ k] = 0 ,
(13d) (13e) (13f)
[PJ, M kl] : i(6Jl P k - 6Jk P l) ,
(13g)
[KJ , M kl] = i(6jl K k - 5jk K l) .
(13h)
We notice that the adjunction of the o p e r a t o r Ho = ( 1 / 2 m ) P 2 does not lead to a Lie algebra: we have [D, Ho] = i2Ho, but [K),Ho] = 2 P ) / m +4//0 ~j. The fact that the o p e r a t o r Ho does not belong to the Lie a l g e b r a of the Liouville group does not have to disturb us, b e c a u s e we a r e dealing with a free. s y s tem for which the e n e r g y can always be e x p r e s s e d by the m o m e n t a P ) alone. We shall see in sect. 4 how the situation changes when i n t e r a c t i o n s c o m e in.
578
H.A. KASTRUP
3. E I G E N F U N C T I O N S T h e e i g e n f u n c t i o n s of the o p e r a t o r s PJ and MJ k in x - s p a c e a r e , of c o u r s e , w e l l - k n o w n . T h e y a r e the n o n - n o r m a l i z a b l e p l a n e w a v e s ep(x) = (2~)--~ e x p ( i p . x ) and the s p h e r i c a l h a r m o n i c s y/re. B e c a u s e of the r e l a t i o n KJ = R P J R (see eq. (3)) the n o n - n o r m a l i z a b l e e i g e n f u n c t i o n s of the o p e r a t o r s KJ a r e o b t a i n e d by a p p l y i n g th. e o p e r a t o r R to the e i g e n f u n c t i o n s of PJ. In the c a s e of the o p e r a t o r s KJ this p r e s c r i p tion g i v e s the e i g e n f u n c t i o n s
fh(x) = (2~)-~ r -3 e i h . x / x 2
,
(14)
w h e r e h is the e i g e n v e c t o r . We have
(15)
(fh 1, fh 2) = 6 ( h 1 - h2) . If we d e n o t e the e i g e n f u n c t i o n s o f / 9 by gs(X), we get
-i(~ + xJ ~j)gs(x) = s gs(x) , w h e r e s is a r e a l n u m b e r .
P u t t i n g gs(x) = Us(r ) Y ~ ,
r T ; us(r)
= (is
- ~)us(r)
(16) we get
.
T h e s o l u t i o n s of this equation we a r e i n t e r e s t e d in a r e •
3
Us(r) = A rzs-~ . With A = (2~)-½ we have ( g l , g 2 ) = 6 r n l m 2 5/1/2 5 ( S l - S 2 ) • An e x p a n s i o n of a given f u n c t i o n in t e r m s of the e i g e n f u n c t i o n s Us(r) is the well-known Mellin transform. T h e e i g e n f u n c t i o n s of KJ in m o m e n t u m s p a c e a r e given by the m a t r i x elements
( p l h ) = (ep, Reh) = (ep,fh) = (27r)-3 f d3x ~.~ e - i P " x e i h " x / x 2
(17)
It is o b v i o u s f r o m the s e c o n d p a r t of t h e s e e q u a t i o n s that the m a t r i x e l e m e n t s (17) a l s o give the r e p r e s e n t a t i o n R i p , h ) of the length i n v e r s i o n R in momentum space ! T h e i n t e g r a l in eq. (17) is s i m i l a r to one d i s c u s s e d in ref. [9]. T h e r e we had _2h(2~)-3 f -d3x ~ e - i p . x e i h . x fix 2 = (27r)-1 Jo[(2ph- 2p.h)½] ,
h = I h) ,
(18)
QUANTUM
where gives
MECHANICS
579
Jo(y) i s t h e B e s s e l f u n c t i o n of o r d e r z e r o . C o m p a r i s o n w i t h eq. (17) R(p, h) = ( i 4 ~ h ) - 1 (Ap)~1 Jo[(2ph - 2 p . h ) ~ ]1 .
(19)
Now t h e B e s s e l f u n c t i o n in eq. (19) i s an e i g e n f u n c t i o n [9] of t h e o p e r a t o r
-pAp w i t h t h e e i g e n v a l u e h. T h u s we g e t 1
1
1
R(p, h) = -~ (Dh)-~Jo[(2Ph - 2 p . b ) ~ ] .
(20)
T h e s e f u n c t i o n s a r e t h e e i g e n f u n c t i o n s of t h e o p e r a t o r s KJ w i t h e i g e n v a l u e hJ. In a d d i t i o n , t h e y a r e t h e m a t r i x e l e m e n t s of t h e u n i t i i r e a n d H e r m i t e a n t r a n s f o r m a t i o n R in m o m e n t u m s p a c e . We n o t i c e t h a t t h e y a r e t h e k e r n e l of a generalized Hankel transformation. T h e e i g e n f u n c t i o n s of t h e o p e r a t o r D in m o m e n t u m s p a c e h a v e t h e s a m e f o r m a s t h o s e o f / 5 in x - s p a c e , t h e o n l y d i f f e r e n c e b e i n g t h a t t h e i h a s to b e replaced by -i.
4. QUANTUM
MECHANICS
AS A BROKEN
LIOUVILLE
SYMMETRY
S t a r t i n g f r o m o u r e x p e r i e n c e w i t h c l a s s i c a l m e c h a n i c s we a s s u m e t h e H a m i l t o n o p e r a t o r f o r t h e f r e e p a r t i c l e to be H o = ( 1 / 2 r n ) P 2 . It i s s u p p o s e d to g i v e t h e k i n e t i c e n e r g y of t h e f r e e p a r t i c l e . If we t u r n on an i n t e r a c t i o n now, we e x p e c t a n d a s s u m e t h e t o t a l e n e r g y o p e r a t o r H to b e c o m e t h e s u m of H o a n d s o m e p o t e n t i a l e n e r g y V. In t h e s a m e w a y a s Ho t h e t e r m V w i l l , w e a s s u m e , b e l o n g to t h e e n v e l o p i n g a l g e b r a of t h e L i e a l g e b r a of t h e L i o u v i l l e g r o u p . S i n c e t h i s a l g e b r a c a n b e g e n e r a t e d b y t h e o p e r a t o r s PJ and KJ, it w i l l b e s u f f i c i e n t to c o n s i d e r H a s a f u n c t i o n of t h e s e o p e r a t o r s . If we a s s u m e t h e p o t e n t i a l to b e v e l o c i t y i n d e p e n d e n t and to b e i n v a r i a n t u n d e r r o t a t i o n s , we h a v e V = V(K 2) a n d t h e r e f o r e H = 2~/)2
+ V(K 2) ,
(21)
T h e e x p r e s s i o n (21) h a s a c r u c i a l p h y s i c a l f l a w , h o w e v e r : if it i s to r e p r e s e n t t h e t o t a l e n e r g y , we e x p e c t it to h a v e d e f i n i t e t r a n s f o r m a t i o n p r o p e r t i e s u n d e r d i l a t a t i o n s , if t h i s s y m m e t r y . e x i s t . F r o m eq. (13c) we k n o w t h e d i l a t a t i o n s to e x i s t , if t h e g e n e r a t o r s KY e x i s t . S i n c e we know, a l s o , how H o t r a n s f o r m s u n d e r d i l a t a t i o n s (Ho ~ e - 2 a H o ) , t h e p o t e n t i a l V s h o u l d t r a n s f o r m in t h e s a m e w a y . T h i s i m p l i e s V ~ ( K 2 ) - 1 . Nature is much richer than this very special potential would allow for. T h e r e f o r e , we h a v e to b r e a k t h e s p e c i a l L i o u v i l l e s y m m e t r y f o r i n t e r a c t i n g s y s t e m s . T h i s c a n d e d o n e in t h e f o l l o w i n g p h y s i c a l l y s t r a i g h t f o r w a r d way: We h a v e n o t i c e a l r e a d y t h a t t h e o p e r a t o r s D a n d K J c o n s i s t of two t e r m s , t h e f i r s t one b e i n g c h a r a c t e r i s t i c f o r t h e t y p e of f u n c t i o n s we u s e f o r t h e d e s c r i p t i o n of s t a t e s , a n d t h e s e c o n d one b e i n g c h a r a c t e r i s t i c f o r t h e t r a n s f o r m a t i o n of t h e c o o r d i n a t e s x o r p . F o r D t h e f i r s t t e r m i s a m u l t i p l e of t h e i d e n t i t y o p e r a t o r , i . e . it c o m m u t e s with any l i n e a r o p e r a t o r in
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o u r H i l b e r t s p a c e , w h e r e a s t h e s e c o n d one d o e s not, in p a r t i c u l a r it d o e s not c o m m u t e w i t h H o o r H. S i n c e a d i s c r e t e s p e c t r u m of H i s not c o m p a t i b l e w i t h c o n t i n u o u s f i n i t e d i l a t a t i o n s , we h a v e to d r o p t h i s s e c o n d t e r m in D in o r d e r to m a k e r o o m f o r d i s c r e t e e n e r g y s p e c t r a . T h i s p r o c e d u r e i s in a c c o r d a n c e w i t h o u r i n t u i t i v e p i c t u r e : In o r d e r to a l l o w f o r d i s c r e t e w a v e l e n g t h s , f o r i n s t a n c e , we h a v e to d i s c a r d t h e d i l a t a t i o n s a s a g e o m e t r i c a l g a u g e t r a n s f o r m a t i o n , but it i s not n e c e s s a r y to d i s c a r d i t s a c t i o n s on t h e f u n c t i o n s of t h e H i l b e r t s p a c e c o m p l e t e l y , b e c a u s e an o v e r a l l c h a n g e of t h e p h a s e of t h e f u n c t i o n s in t h i s s p a c e d o e s not c h a n g e t h e c o n t e n t of i n f o r m a t i o n t h e y a r e a b l e to p r o v i d e . C l e a r l y , t h e i d e n t i t y o p e r a t o r , of w h i c h t h e f i r s t p a r t - t h e ' i n t r i n s i c ' one - of D i s a m u l t i p l e , i s t h e g e n e r a t o r of p h a s e t r a n s f o r m a t i o n s - ~ e i a ~ . W e s h a l l d i s c u s s t h e m a t h e m a t i c a l p r o c e d u r e of t h e ' c o n t r a c t i o n ' of D into a m u l t i p l e of t h e i d e n t i t y o p e r a t o r l a t e r . T h e c r u c i a l p o i n t i s t h a t one c a n n o t b r e a k t h e d i l a t a t i o n s w i t h o u t b r e a k ing t h e s p e c i a l L o u i v i l l e g r o u p , if one w a n t s to k e e p t h e t r a n s l a t i o n s ! T h i s f o l l o w s c l e a r l y f r o m t h e c o m m u t a t i o n r e l a t i o n s (13c). It t u r n s out t h a t we h a v e to d r o p t h e s e c o n d ( o r b i t a l ) p a r t of t h e o p e r a t o r s K J, too, if t h e o r b i t a l p a r t of D s h o u l d not a p p e a r on t h e r i g h t h a n d s i d e of eq. (13c). T h e r e m a i n ing i n t r i n s i c f i r s t t e r m of KJ i s a m u l t i p l e of t h e u s u a l p o s i t i o n o p e r a t o r in quantum mechanics ! In t h i s w a y t h e b a s i c o b s e r v a b l e s PJ a n d QJ of q u a n t u m m e c h a n i c s a p p e a r a s l i m i t s of t h e b a s i c g e n e r a t o r s PJ and K J of t h e L i o u v i l l e g r o u p , t h e s y m m e t r y g r o u p of t h e f r e e s t a t i o n a r y s y s t e m ! We s e e t h a t t h e p h y s i c a l c o n t e n t f o r t h e p r e s c r i p t i o n of how t h i s l i m i t h a s to b e , i s an i m m e d i a t e c o n s e q u e n c e of t h e q u a n t u m m e c h a n i c a l s u p e r p o s i t i o n p r i n c i p l e , w h i c h a l l o w s f o r an o v e r a l l p h a s e t r a n s f o r m a t i o n of t h e s t a t e v e c t o r s . T h i s p o s s i b i l i t y of a p h a s e t r a n s f o r m a t i o n p e r m i t s u s to k e e p a m u l t i p l e of t h e ' i n t r i n s i c ' p a r t of t h e d i l a t a t i o n s , and t h i s in t u r n i m p l i e s t h a t we c a n k e e p t h e ' i n t r i n s i c ' p a r t of K3! Mathematically, we can describe the procedure just discussed by the m e t h o d of g r o u p c o n t r a c t i o n s [5, 6]: l e t ¢ b e a r e a l p a r a m e t e r w i t h v a l u e s b e t w e e n 0 and 1 : 0 ~< ¢ ~< 1. We d e f i n e t h e f o l l o w i n g o p e r a t o r s : (22a)
PJe = PJ , M jk = i - l ( p j a k _ p k Oj),
(22b)
D E = i(~ +¢pJaj) ,
(22c)
K ¢j = 3 a j + ¢ ( 2 p k a k a j - p J ak ak) .
(22d)
F o r E = 1 we h a v e t h e p r e v i o u s g e n e r a t o r s of t h e L i o u v i U e g r o u p a n d f o r e = 0 we h a v e
QJ = ½i K J =0 I=~iD
=0 "
(23a) (23b)
QUANTUM MECHANICS The operators
581
(22a) - (22d) o b e y t h e f o l l o w i n g c o m m u t a t i o n r e l a t i o n s :
[PJ, K l] = 2i(6jl D~ - cMJ l) ,
(24a)
[D~,PJ] = i• P J ,
(245)
IDa,
= -i•
(24c)
[De, M J ~ : 0 ,
(24d)
[KJ, K l] = O,
(24e)
[KJ, M kZ].
6jk
(24f)
T h e c o m m u t a t i o n r e l a t i o n s of t h e E u c l i d e a n g r o u p a r e t h e s a m e a s b e f o r e . We s e e t h a t t h e L i e a l g e b r a (24a) - (24f) g o e s o v e r into t h e a l g e b r a of q u a n t u m m e c h a n i c s f o r ~ = 0. T h e f u n d a m e n t a l r e l a t i o n (1) i s t h e l i m i t of eq. (24a). We wish to point out that the operators DE= 0 and K~= 0 are purely imaginary. In order to obtain the observables we are interested in, we merely have to divide by i, as has been done in the eqs. (23a) and (23b). The situation is more involved when we consider the finite transformations U(G ~) = exp(iaDe) and U(c, •) = exp(icJK~)..In o.rder to have these operators unitary for • = 0, we consider a = a(E) and c9 = c)(~) as complex-valued function of • which are real for • = I and imaginary for • = 0. The operator U(a, E=0) is a phase transformation.
5. DISCUSSION In sect. 4 we saw how the algebra of quantum mechanics can be considered to be the limit of the Lie algebra of the Liouville group of the free system. We mainly used the fact that physical states are represented by rays rather than individual vectors. The following comments on this procedure seem to be appropriate: (i) In deriving the position operator QJ we had to break the dilatation symmetry by using arguments applicable only in the presence of interactions. However, why should there be no position operator for the free particle, when the Liouville symmetry, is exact ? Of course, nothing can prevent us from using the operator QY, obtained by breaking the dilatation symmetry, as a position operator for the free particle, without destroying its Louiville symmetry. (ii) It has been mentioned before that any free physical system with a dispersion law E = Ap ~ is Louiville invariant [I]. This is generally no longer true for the interactions between those elementary excitations. As a special example of how to deal with this situation we have discussed the non-relativistic particles with E =(I/2m)P 2. The breaking of the Louiville symmetry leads in this case to the framework of quantum mechanics. However, it is by no means clear whether the same breaking procedure
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i s a p p r o p r i a t e f o r o t h e r p h y s i c a l s y s t e m s . C o n s i d e r , f o r i n s t a n c e , the r e l a t i v i s t i c r e l a t i o n E = p, w h i c h i s c o m p a t i b l e with L i o u v i l l e i n v a r i a n c e , too [2]. If we a s s u m e t h e s e ' b a r e ' e l e m e n t a r y e x c i t a t i o n s to a c q u i r e m a s s e s - E = (p2 + m2)½ _ by m e a n s of i n t e r a c t i o n s w h i c h b r e a k the L i o u v i l l e s y m m e t r y , we h a v e to l e a r n how to do t h i s . A s a t i s f a c t o r y a n s w e r to t h i s p r o b l e m i s not y e t k n o w n , b u t we s e e t h a t the a r t we m u s t l e a r n s e e m s to c o n s i s t of k n o w i n g how to b r e a k t h e L i o u v i l l e s y m m e t r y in e a c h s p e c i a l c a s e u n d e r c o n s i d e r a t i o n ; o r e v e n b e t t e r , to l e a r n m o r e a b o u t the d e e p e r p h y s i c a l r e a s o n s f o r the w i d e - s p r e a d b r e a k i n g of t h e L i o u v i l l e g r o u p by n a t u r e .
RE F E R E N C E S [1] H.A.Kastrup, Nuel. Phys. B7 (1968) 545. [2] H.A.Kastrup, Phys. Rev. 150 (1966) 1183. [3] H.A. Kastrup, Proc. VII Int. Univ. weeks for nuclear physics at Schladming (Austria), ed. Paul Urban, to be published. [4] H.A. Kastrup, Ann. der Physik 9 (1962) 388. [5] E.In~Snii and E . P . W i g n e r , Proc. Nat. Acad. Sciences (USA) 39 (1953) 510; 40 (1954) 119. [6] I . E . Segal, Duke Math. J. 18 (1951) 221. [7] H . A . K a s t r u p , Phys. Rev. 143 (1966) 1021. [8] D . G . C u r i e , T . F . J o r d a n a n d E . C . G . S u d a r s h a n , Revs. Mod. Phys. 35 (1963) 350. [9] H. A. Kastrup, Phys. Rev. 142 (1966) 1060.
Nuclear Physics B7 (1968) 574-582. North-Holland Publ. Comp., A m s t e r d a m
QUANTUM
MECHANICS
AS
A BROKEN
SYMMETRY
H. A. K A S T R U P
Universitiit Mi~nchen, Sektion Physik, lYr~nchen, Germany Received 7 August 1968 Abstract: The algebra of quantum mechanics is derived by breaking the exact Liouville s y m m e t r y of free n o n - r e l a t i v i s t i c stationary systems. The Liouville group consists in this case of the Euclidean group, the dilatations and the t h r e e - p a r a m e t e r special Liouville group. The two latter groups induce global and local geom e t r i c a l gauge t r a n s f o r m a t i o n s . The dilatations imply continuous energy spectra, and t h er efo re they have to be broken if we want to allow for the frequent d i s c r e t e energy s p ect ra in atomic physics, etc. Since physical states are represented by rays r a t h e r than v e c t o r s , it is sufficient to contract the dilatations into a phase transformation. These physical conditions on the breaking of the dilatations for interacting s y s t e m s imply that the g e n e r a t o r s of the special Liouville group go over into multiples of the usual position operators.
1. ~ T R O D U C T I O N Q u a n t u m m e c h a n i c s i s c e r t a i n l y one of t he m o s t b e a u t i f u l t h e o r i e s e v e r c o n c e i v e d by p h y s i c i s t s . I t s r i c h p h y s i c a l and c l e a r m a t h e m a t i c a l s t r u c t u r e h a s c a u s e d m a n y i n t e r e s t i n g i n v e s t i g a t i o n s f o r t h e p u r p o s e of d e r i v i n g o r explaining this theory from different or more basic viewpoints. The present p a p e r i s a n o t h e r a t t e m p t to s h e d s o m e l i g h t on t h e o r i g i n of q u a n t u m m e chanics. A f t e r a c c e p t i n g w h a t i s p r o b a b l y t h e m o s t b a s i c p r o p o s i t i o n of q u a n t u m m e c h a n i c s , t h a t p h y s i c a l s t a t e s a r e to be d e s c r i b e d by v e c t o r s o r r a t h e r r a y s in a H i l b e r t s p a c e , we e n c o u n t e r two d i f f e r e n t s i t u a t i o n s a s f a r a s t h e o b s e r v a b l e s - c h a r a c t e r i z e d by s e l f - a d j o i n t o p e r a t o r s - a r e c o n c e r n e d : t h e r e a r e t h o s e l i k e m o m e n t u m , a n g u l a r m o m e n t u m , and e n e r g y , w h i c h c o r r e s p o n d to t h e g e n e r a t o r s of s o m e s p a c e o r t i m e s y m m e t r y ; and t h e r e a r e o t h e r s , l i k e t h e p o s i t i o n o p e r a t o r s Qj, w h i c h do not s e e m to c o r r e s p o n d to any s p a c e s y m m e t r y w h i c h w o u l d a l l o w u s to d e r i v e t h e b a s i c c a nonical commutation relations *
Pj Vk - VkPj = i-1 5jk
(1)
in a s i m p l e p h y s i c a l way. We s h a l l show, h o w e v e r , t h a t t h e r e l a t i o n s (1) c a n b e d e r i v e d by s o m e s i m p l e p h y s i c a l a r g u m e n t f r o m a b a s i c g e o m e t r i c a l s y m m e t r y of t h e f r e e n o n - r e l a t i v i s t i c p a r t i c l e . T h e s y m m e t r y in q u e s t i o n i s g i v e n by t h e L i o u v i l l e g r o u p d i s c u s s e d in d e t a i l in a p r e c e d i n g p a p e r [1]. If we c o n s i d e r , f o r * We use the units Z - 1 and c - 1.
QUANTUM MECHANICS
575
r e a s o n s of s i m p l i c i t y , o n l y t h e s t a t i o n a r y c a s e (t = 0), t h e L i o u v i l l e g r o u p c o n s i s t s of t h e u s u a l E u c l i d e a n g r o u p ( r o t a t i o n s a n d s p a c e t r a n s l a t i o n s ) , t h e dilatations Dl(a) , and the special Liouville transformations SC3(c). The t w o l a t t e r g r o u p s i n d u c e t h e f o l l o w i n g t r a n s f o r m a t i o n s in x - s p a c e : D I ( ~ ) : x -~ e ~ x , a r e a l , SC3(c): x ~ RT3(c)Rx ,
Rx = x/x2
,
(2)
T3(c)x=x+
c.
(3)
T h e g r o u p s (2) and (3) a r e to b e i n t e r p r e t e d a s g l o b a l a n d l o c a l g e o m e t r i c a l g a u g e t r a n s f o r m a t i o n s w h i c h m a p a g i v e n l e n g t h ds = (dxidxi)~ onto a n o t h e r one w h i c h d i f f e r s f r o m t h e f i r s t one b y a f a c t o r e a o r 1 / ~ ( l ) , ~ ( x ) = 1 +2 v . x + v 2 x 2, r e s p e c t i v e l y . T h e p h y s i c a l m e a n i n g of t h e t r a n s f o r m a t i o n s (2) a n d (3) h a s b e e n d i s c u s s e d in d e t a i l e l s e w h e r e [ 1 - 3 ] , and we s h a l l not r e p e a t it h e r e . Important for our purpose are the commutation relations between the g e n e r a t o r s P J , j = 1 , 2 , 3, of t h e t r a n s l a t i o n s T 3 ( a ) a n d t h e g e n e r a t o r s KJ of t h e s p e c i a l L i o u v i l l e t r a n s f o r m a t i o n s S C 3 ( c ) :
p k g j - K j p k = 2i(6kjD - Mkj) ,
j , k = 1, 2, 3 .
(4)
H e r e t h e o p e r a t o r s D and MkJ g e n e r a t e t h e d i l a t a t i o n s and t h e r o t a t i o n s r e s p e c t i v e l y . Eq. (4) s a y s t h a t i n v a r i a n c e u n d e r t r a n s l a t i o n s and s p e c i a l L i o u v i l l e t r a n s f o r m a t i o n s i m p l i e s i n v a r i a n c e u n d e r r o t a t i o n s and d i l a t a tions ! Now, i n v a r i a n c e u n d e r f i n i t e c o n t i n u o u s d i l a t a t i o n s m e a n s t h a t a l l o b s e r v a b l e s w h i c h do not c o m m u t e w i t h D h a v e a c o n t i n u o u s s p e c t r u m . In p a r t i c u l a r , t h e m o m e n t a P J a r e t r a n s f o r m e d into e - a P J u n d e r f i n i t e d i l a t a t i o n s . S i n c e t h e m a s s m i s an i n v a r i a n t p a r a m e t e r u n d e r a l l n o n - r e l a t i v i s t i c s y m m e t r y o p e r a t i o n s , t h e k i n e t i c t e r m H o = (1~2re)P2 in any H a m i l t o n i a n i m p l i e s t h a t t h e d i l a t a t i o n s c a n o n l y b e a s y m m e t r y g r o u p of a g i v e n s y s t e m if t h e t o t a l e n e r g y o p e r a t o r H = Ho + V h a s a c o n t i n u o u s s p e c t r u m . T h i s f o l l o w s f r o m t h e f a c t t h a t t h e p o t e n t i a l V h a s to t r a n s f o r m in t h e s a m e w a y a s t h e k i n e t i c t e r m w h e n we a r e d e a l i n g w i t h a s y m m e t r y g r o u p . On t h e o t h e r hand: one of t h e m o s t c h a r a c t e r i s t i c f e a t u r e s of a t o m i c p h y s i c s i s t h e e x i s t e n c e of d i s c r e t e e n e r g y s p e c t r a . W e t h e r e f o r e h a v e to b r e a k t h e d i l a t a t i o n s y m m e t r y of t h e f r e e p a r t i c l e s f o r s u c h i n t e r a c t i n g s y s t e m s . B e c a u s e of eq. (4) t h i s c a n n o t b e done w i t h o u t b r e a k i n g t h e s p e c i a l L i o u v i l l e s y m m e t r y , too. In o r d e r to h a v e t h e d i l a t a t i o n o p e r a t o r D c o m m u t e w i t h t h e m o m e n t a , we h a v e to c o n t r a c t it a t l e a s t into a m u l t i p l e of t h e u n i t y o p e r a t o r . S i m u l t a n e o u s l y we w a n t to c o n t r a c t t h e f i n i t e d i l a t a t i o n s exp(i~D) into a p h a s e t r a n s f o r m a t i o n , b e c a u s e a n o v e r a l l c h a n g e of t h e p h a s e s of a l l s t a t e v e c t o r s d o e s not m a t t e r p h y s i c a l l y . T h e s e p h y s i c a l c o n d i t i o n s on t h e c o n t r a c t i o n s of t h e d i l a t a t i o n s i m p l y t h a t t h e g e n e r a t o r s KJ go o v e r into m u l t i p l e s of t h e u s u a l p o s i t i o n o p e r a t o r s QJ a n d eq. (4) g o e s o v e r into eq. (1)! T h u s , we a r r i v e at t h e f r a m e w o r k of q u a n t u m m e c h a n i c s b y s t a r t i n g f r o m a g e o m e t r i c a l s y m m e t r y g r o u p of t h e f r e e p a r t i c l e . T h e d e t a i l s of t h i s p r o c e d u r e a r e d i s c u s s e d in s e c t . 4. In s e c t s . 2 a n d 3 we d e a l w i t h u n i t a r y r e p r e s e n t a t i o n s of t h e 1 0 - p a r a m e t e r L i o u v i l l e g r o u p of t h e t h r e e - d i m e n s i o n a l Euclidean space. The describe a free stationary non-relativistic system.
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H.A. KASTRUP
A s h o r t h i s t o r i c a l r e m a r k : the close connection between the L i e - a l g e b r a of the Liouville group of the Minkowski space and the a l g e b r a of quantum m e c h a n i c s was pointed out in ref. [4]. L a t e r I l e a r n t about group c o n t r a c tions [5] and an e a r l i e r m a t h e m a t i c a l p a p e r [6] by Segal in which he cont r a c t s the Lie a l g e b r a of the group SO(2, 4) into the a l g e b r a of r e l a t i v i s t i c quantum m e c h a n i c s . The physical a s p e c t s of a slightly different c o n t r a c tion of the s a m e group along the lines d i s c u s s e d above a r e investigated in ref. [7]. Since a r e l a t i v i s t i c canonical quantum m e c h a n i c s with position o p e r a t o r s analogous to the n o n - r e l a t i v i s t i c one s e e m s to meet with some difficulties [8], the r e l a t i v i s t i c c a s e may not be so i n t e r e s t i n g after all. However, the fact that the f r e e n o n - r e l a t i v i s t i c motion is Liouville invariant, opens the way to a new derivation of n o n - r e l a t i v i s t i c quantum m e c h a n i c s ! A final i n t r o d u c t o r y r e m a r k : the e m p h a s i s in this p a p e r will be on the physical a s p e c t s , not on m a t h e m a t i c a l r i g o u r . F o r instance, we shall not d i s c u s s the domains of definition of the differential o p e r a t o r s which o c c u r or those of their self-adjoint extensions.
2. UNITARY REPRESENTATIONS OF THE LIOUVILLE GROUP Our f i r s t b a s i c a s s u m p t i o n is that the s t a t e s of a s t a t i o n a r y f r e e nonr e l a t i v i s t i c s y s t e m f o r m a Hilbert space and that the full 1 0 - p a r a m e t e r Liouville group induces unitary r e p r e s e n t a t i o n s in this Hilbert space. We f u r t h e r a s s u m e the spin of the s y s t e m to be z e r o and c o n s i d e r r e p r e s e n t a tions in the space of functions g,(x) and t h e i r F o u r i e r t r a n s f o r m s ~(p) = (2~)-~ f d3x ~ ( x ) e - i P "x .
(5)
The s c a l a r product (@1, @2) of two functions ~hI and @2 is given by
(@l, ~2) = f d3x @l(X)@2(x) * = f d3p q){(p)~02(P) •
(6)
Since the full 1 0 - p a r a m e t e r Liouville group can be g e n e r a t e d [2] by the t r a n s l a t i o n s T3(a) and the d i s c r e t e length i n v e r s i o n R, one can c o n s t r u c t a u n i t a r y r e p r e s e n t a t i o n by defining the action of t h e s e subgroups on the functions ~(x). If we denote the r e p r e s e n t i n g unitary o p e r a t o r s by U(a) and U(R), we have
(7)
[u(a)~](x) = ~ ( x - a ) and [U(R)+](x) : r - 3 + ( R x ) ,
r = Ix l .
(8)
Defining the H e r m i t e a n g e n e r a t o r s A of the infinitesimal t r a n s f o r m a t i o n s g e n e r a l l y by U = exp(iA), we .~et the following explicit e x p r e s s i o n s for the g e n e r a t o r s P / , K / , D and MJ k in x - and p - s p a c e r e s p e c t i v e l y :
PJ = i-1 ~J '
~J = oxJ '
(9a)
577
Q U A N T U M MECHANICS
P j = p J,
(95)
(10a)
I
~k
-
apk '
(lOb)
(11a)
i-l(~ + xJ ~j) ,
(11b)
D = i({ +pJaj) , ~ j k = i-1 (xJ ~k - x k ~j) ,
02a)
MJk = i - l ( p J ~ k - p k a j )
(125)
.
The finite dilatations, for instance, have the f o r m 3
[eia/9~] (x) = e'~ a~h(e-ax) . We notice that the t e r m ~ i n / ) and D c o m e s f r o m the t r a n s f o r m a t i o n of the function ( ' i n t r i n s i c ' part) and the r e s t c o m e s f r o m the t r a n s f o r m a t i o n of the a r g u m e n t s x or p ( ' o r b i t a l ' part). The s a m e holds for the f i r s t t e r m and the r e s t in K} and KJ. The above g e n e r a t o r s f o r m the following Lie algebra:
[ p j, pl] : 0 ,
(13a)
[KJ, K t] = 0 ,
(135)
[P J, K l] = 2i(SjlD - MJ l) ,
(13c)
[D, P ~
= iP k ,
[D, KJ] : - i K J , [D, MJ k] = 0 ,
(13d) (13e) (13f)
[PJ, M kl] : i(6Jl P k - 6Jk P l) ,
(13g)
[KJ , M kl] = i(6jl K k - 5jk K l) .
(13h)
We notice that the adjunction of the o p e r a t o r Ho = ( 1 / 2 m ) P 2 does not lead to a Lie algebra: we have [D, Ho] = i2Ho, but [K),Ho] = 2 P ) / m +4//0 ~j. The fact that the o p e r a t o r Ho does not belong to the Lie a l g e b r a of the Liouville group does not have to disturb us, b e c a u s e we a r e dealing with a free. s y s tem for which the e n e r g y can always be e x p r e s s e d by the m o m e n t a P ) alone. We shall see in sect. 4 how the situation changes when i n t e r a c t i o n s c o m e in.
578
H.A. KASTRUP
3. E I G E N F U N C T I O N S T h e e i g e n f u n c t i o n s of the o p e r a t o r s PJ and MJ k in x - s p a c e a r e , of c o u r s e , w e l l - k n o w n . T h e y a r e the n o n - n o r m a l i z a b l e p l a n e w a v e s ep(x) = (2~)--~ e x p ( i p . x ) and the s p h e r i c a l h a r m o n i c s y/re. B e c a u s e of the r e l a t i o n KJ = R P J R (see eq. (3)) the n o n - n o r m a l i z a b l e e i g e n f u n c t i o n s of the o p e r a t o r s KJ a r e o b t a i n e d by a p p l y i n g th. e o p e r a t o r R to the e i g e n f u n c t i o n s of PJ. In the c a s e of the o p e r a t o r s KJ this p r e s c r i p tion g i v e s the e i g e n f u n c t i o n s
fh(x) = (2~)-~ r -3 e i h . x / x 2
,
(14)
w h e r e h is the e i g e n v e c t o r . We have
(15)
(fh 1, fh 2) = 6 ( h 1 - h2) . If we d e n o t e the e i g e n f u n c t i o n s o f / 9 by gs(X), we get
-i(~ + xJ ~j)gs(x) = s gs(x) , w h e r e s is a r e a l n u m b e r .
P u t t i n g gs(x) = Us(r ) Y ~ ,
r T ; us(r)
= (is
- ~)us(r)
(16) we get
.
T h e s o l u t i o n s of this equation we a r e i n t e r e s t e d in a r e •
3
Us(r) = A rzs-~ . With A = (2~)-½ we have ( g l , g 2 ) = 6 r n l m 2 5/1/2 5 ( S l - S 2 ) • An e x p a n s i o n of a given f u n c t i o n in t e r m s of the e i g e n f u n c t i o n s Us(r) is the well-known Mellin transform. T h e e i g e n f u n c t i o n s of KJ in m o m e n t u m s p a c e a r e given by the m a t r i x elements
( p l h ) = (ep, Reh) = (ep,fh) = (27r)-3 f d3x ~.~ e - i P " x e i h " x / x 2
(17)
It is o b v i o u s f r o m the s e c o n d p a r t of t h e s e e q u a t i o n s that the m a t r i x e l e m e n t s (17) a l s o give the r e p r e s e n t a t i o n R i p , h ) of the length i n v e r s i o n R in momentum space ! T h e i n t e g r a l in eq. (17) is s i m i l a r to one d i s c u s s e d in ref. [9]. T h e r e we had _2h(2~)-3 f -d3x ~ e - i p . x e i h . x fix 2 = (27r)-1 Jo[(2ph- 2p.h)½] ,
h = I h) ,
(18)
QUANTUM
where gives
MECHANICS
579
Jo(y) i s t h e B e s s e l f u n c t i o n of o r d e r z e r o . C o m p a r i s o n w i t h eq. (17) R(p, h) = ( i 4 ~ h ) - 1 (Ap)~1 Jo[(2ph - 2 p . h ) ~ ]1 .
(19)
Now t h e B e s s e l f u n c t i o n in eq. (19) i s an e i g e n f u n c t i o n [9] of t h e o p e r a t o r
-pAp w i t h t h e e i g e n v a l u e h. T h u s we g e t 1
1
1
R(p, h) = -~ (Dh)-~Jo[(2Ph - 2 p . b ) ~ ] .
(20)
T h e s e f u n c t i o n s a r e t h e e i g e n f u n c t i o n s of t h e o p e r a t o r s KJ w i t h e i g e n v a l u e hJ. In a d d i t i o n , t h e y a r e t h e m a t r i x e l e m e n t s of t h e u n i t i i r e a n d H e r m i t e a n t r a n s f o r m a t i o n R in m o m e n t u m s p a c e . We n o t i c e t h a t t h e y a r e t h e k e r n e l of a generalized Hankel transformation. T h e e i g e n f u n c t i o n s of t h e o p e r a t o r D in m o m e n t u m s p a c e h a v e t h e s a m e f o r m a s t h o s e o f / 5 in x - s p a c e , t h e o n l y d i f f e r e n c e b e i n g t h a t t h e i h a s to b e replaced by -i.
4. QUANTUM
MECHANICS
AS A BROKEN
LIOUVILLE
SYMMETRY
S t a r t i n g f r o m o u r e x p e r i e n c e w i t h c l a s s i c a l m e c h a n i c s we a s s u m e t h e H a m i l t o n o p e r a t o r f o r t h e f r e e p a r t i c l e to be H o = ( 1 / 2 r n ) P 2 . It i s s u p p o s e d to g i v e t h e k i n e t i c e n e r g y of t h e f r e e p a r t i c l e . If we t u r n on an i n t e r a c t i o n now, we e x p e c t a n d a s s u m e t h e t o t a l e n e r g y o p e r a t o r H to b e c o m e t h e s u m of H o a n d s o m e p o t e n t i a l e n e r g y V. In t h e s a m e w a y a s Ho t h e t e r m V w i l l , w e a s s u m e , b e l o n g to t h e e n v e l o p i n g a l g e b r a of t h e L i e a l g e b r a of t h e L i o u v i l l e g r o u p . S i n c e t h i s a l g e b r a c a n b e g e n e r a t e d b y t h e o p e r a t o r s PJ and KJ, it w i l l b e s u f f i c i e n t to c o n s i d e r H a s a f u n c t i o n of t h e s e o p e r a t o r s . If we a s s u m e t h e p o t e n t i a l to b e v e l o c i t y i n d e p e n d e n t and to b e i n v a r i a n t u n d e r r o t a t i o n s , we h a v e V = V(K 2) a n d t h e r e f o r e H = 2~/)2
+ V(K 2) ,
(21)
T h e e x p r e s s i o n (21) h a s a c r u c i a l p h y s i c a l f l a w , h o w e v e r : if it i s to r e p r e s e n t t h e t o t a l e n e r g y , we e x p e c t it to h a v e d e f i n i t e t r a n s f o r m a t i o n p r o p e r t i e s u n d e r d i l a t a t i o n s , if t h i s s y m m e t r y . e x i s t . F r o m eq. (13c) we k n o w t h e d i l a t a t i o n s to e x i s t , if t h e g e n e r a t o r s KY e x i s t . S i n c e we know, a l s o , how H o t r a n s f o r m s u n d e r d i l a t a t i o n s (Ho ~ e - 2 a H o ) , t h e p o t e n t i a l V s h o u l d t r a n s f o r m in t h e s a m e w a y . T h i s i m p l i e s V ~ ( K 2 ) - 1 . Nature is much richer than this very special potential would allow for. T h e r e f o r e , we h a v e to b r e a k t h e s p e c i a l L i o u v i l l e s y m m e t r y f o r i n t e r a c t i n g s y s t e m s . T h i s c a n d e d o n e in t h e f o l l o w i n g p h y s i c a l l y s t r a i g h t f o r w a r d way: We h a v e n o t i c e a l r e a d y t h a t t h e o p e r a t o r s D a n d K J c o n s i s t of two t e r m s , t h e f i r s t one b e i n g c h a r a c t e r i s t i c f o r t h e t y p e of f u n c t i o n s we u s e f o r t h e d e s c r i p t i o n of s t a t e s , a n d t h e s e c o n d one b e i n g c h a r a c t e r i s t i c f o r t h e t r a n s f o r m a t i o n of t h e c o o r d i n a t e s x o r p . F o r D t h e f i r s t t e r m i s a m u l t i p l e of t h e i d e n t i t y o p e r a t o r , i . e . it c o m m u t e s with any l i n e a r o p e r a t o r in
580
H.A. KASTRUP
o u r H i l b e r t s p a c e , w h e r e a s t h e s e c o n d one d o e s not, in p a r t i c u l a r it d o e s not c o m m u t e w i t h H o o r H. S i n c e a d i s c r e t e s p e c t r u m of H i s not c o m p a t i b l e w i t h c o n t i n u o u s f i n i t e d i l a t a t i o n s , we h a v e to d r o p t h i s s e c o n d t e r m in D in o r d e r to m a k e r o o m f o r d i s c r e t e e n e r g y s p e c t r a . T h i s p r o c e d u r e i s in a c c o r d a n c e w i t h o u r i n t u i t i v e p i c t u r e : In o r d e r to a l l o w f o r d i s c r e t e w a v e l e n g t h s , f o r i n s t a n c e , we h a v e to d i s c a r d t h e d i l a t a t i o n s a s a g e o m e t r i c a l g a u g e t r a n s f o r m a t i o n , but it i s not n e c e s s a r y to d i s c a r d i t s a c t i o n s on t h e f u n c t i o n s of t h e H i l b e r t s p a c e c o m p l e t e l y , b e c a u s e an o v e r a l l c h a n g e of t h e p h a s e of t h e f u n c t i o n s in t h i s s p a c e d o e s not c h a n g e t h e c o n t e n t of i n f o r m a t i o n t h e y a r e a b l e to p r o v i d e . C l e a r l y , t h e i d e n t i t y o p e r a t o r , of w h i c h t h e f i r s t p a r t - t h e ' i n t r i n s i c ' one - of D i s a m u l t i p l e , i s t h e g e n e r a t o r of p h a s e t r a n s f o r m a t i o n s - ~ e i a ~ . W e s h a l l d i s c u s s t h e m a t h e m a t i c a l p r o c e d u r e of t h e ' c o n t r a c t i o n ' of D into a m u l t i p l e of t h e i d e n t i t y o p e r a t o r l a t e r . T h e c r u c i a l p o i n t i s t h a t one c a n n o t b r e a k t h e d i l a t a t i o n s w i t h o u t b r e a k ing t h e s p e c i a l L o u i v i l l e g r o u p , if one w a n t s to k e e p t h e t r a n s l a t i o n s ! T h i s f o l l o w s c l e a r l y f r o m t h e c o m m u t a t i o n r e l a t i o n s (13c). It t u r n s out t h a t we h a v e to d r o p t h e s e c o n d ( o r b i t a l ) p a r t of t h e o p e r a t o r s K J, too, if t h e o r b i t a l p a r t of D s h o u l d not a p p e a r on t h e r i g h t h a n d s i d e of eq. (13c). T h e r e m a i n ing i n t r i n s i c f i r s t t e r m of KJ i s a m u l t i p l e of t h e u s u a l p o s i t i o n o p e r a t o r in quantum mechanics ! In t h i s w a y t h e b a s i c o b s e r v a b l e s PJ a n d QJ of q u a n t u m m e c h a n i c s a p p e a r a s l i m i t s of t h e b a s i c g e n e r a t o r s PJ and K J of t h e L i o u v i l l e g r o u p , t h e s y m m e t r y g r o u p of t h e f r e e s t a t i o n a r y s y s t e m ! We s e e t h a t t h e p h y s i c a l c o n t e n t f o r t h e p r e s c r i p t i o n of how t h i s l i m i t h a s to b e , i s an i m m e d i a t e c o n s e q u e n c e of t h e q u a n t u m m e c h a n i c a l s u p e r p o s i t i o n p r i n c i p l e , w h i c h a l l o w s f o r an o v e r a l l p h a s e t r a n s f o r m a t i o n of t h e s t a t e v e c t o r s . T h i s p o s s i b i l i t y of a p h a s e t r a n s f o r m a t i o n p e r m i t s u s to k e e p a m u l t i p l e of t h e ' i n t r i n s i c ' p a r t of t h e d i l a t a t i o n s , and t h i s in t u r n i m p l i e s t h a t we c a n k e e p t h e ' i n t r i n s i c ' p a r t of K3! Mathematically, we can describe the procedure just discussed by the m e t h o d of g r o u p c o n t r a c t i o n s [5, 6]: l e t ¢ b e a r e a l p a r a m e t e r w i t h v a l u e s b e t w e e n 0 and 1 : 0 ~< ¢ ~< 1. We d e f i n e t h e f o l l o w i n g o p e r a t o r s : (22a)
PJe = PJ , M jk = i - l ( p j a k _ p k Oj),
(22b)
D E = i(~ +¢pJaj) ,
(22c)
K ¢j = 3 a j + ¢ ( 2 p k a k a j - p J ak ak) .
(22d)
F o r E = 1 we h a v e t h e p r e v i o u s g e n e r a t o r s of t h e L i o u v i U e g r o u p a n d f o r e = 0 we h a v e
QJ = ½i K J =0 I=~iD
=0 "
(23a) (23b)
QUANTUM MECHANICS The operators
581
(22a) - (22d) o b e y t h e f o l l o w i n g c o m m u t a t i o n r e l a t i o n s :
[PJ, K l] = 2i(6jl D~ - cMJ l) ,
(24a)
[D~,PJ] = i• P J ,
(245)
IDa,
= -i•
(24c)
[De, M J ~ : 0 ,
(24d)
[KJ, K l] = O,
(24e)
[KJ, M kZ].
6jk
(24f)
T h e c o m m u t a t i o n r e l a t i o n s of t h e E u c l i d e a n g r o u p a r e t h e s a m e a s b e f o r e . We s e e t h a t t h e L i e a l g e b r a (24a) - (24f) g o e s o v e r into t h e a l g e b r a of q u a n t u m m e c h a n i c s f o r ~ = 0. T h e f u n d a m e n t a l r e l a t i o n (1) i s t h e l i m i t of eq. (24a). We wish to point out that the operators DE= 0 and K~= 0 are purely imaginary. In order to obtain the observables we are interested in, we merely have to divide by i, as has been done in the eqs. (23a) and (23b). The situation is more involved when we consider the finite transformations U(G ~) = exp(iaDe) and U(c, •) = exp(icJK~)..In o.rder to have these operators unitary for • = 0, we consider a = a(E) and c9 = c)(~) as complex-valued function of • which are real for • = I and imaginary for • = 0. The operator U(a, E=0) is a phase transformation.
5. DISCUSSION In sect. 4 we saw how the algebra of quantum mechanics can be considered to be the limit of the Lie algebra of the Liouville group of the free system. We mainly used the fact that physical states are represented by rays rather than individual vectors. The following comments on this procedure seem to be appropriate: (i) In deriving the position operator QJ we had to break the dilatation symmetry by using arguments applicable only in the presence of interactions. However, why should there be no position operator for the free particle, when the Liouville symmetry, is exact ? Of course, nothing can prevent us from using the operator QY, obtained by breaking the dilatation symmetry, as a position operator for the free particle, without destroying its Louiville symmetry. (ii) It has been mentioned before that any free physical system with a dispersion law E = Ap ~ is Louiville invariant [I]. This is generally no longer true for the interactions between those elementary excitations. As a special example of how to deal with this situation we have discussed the non-relativistic particles with E =(I/2m)P 2. The breaking of the Louiville symmetry leads in this case to the framework of quantum mechanics. However, it is by no means clear whether the same breaking procedure
582
H.A. KASTRUP
i s a p p r o p r i a t e f o r o t h e r p h y s i c a l s y s t e m s . C o n s i d e r , f o r i n s t a n c e , the r e l a t i v i s t i c r e l a t i o n E = p, w h i c h i s c o m p a t i b l e with L i o u v i l l e i n v a r i a n c e , too [2]. If we a s s u m e t h e s e ' b a r e ' e l e m e n t a r y e x c i t a t i o n s to a c q u i r e m a s s e s - E = (p2 + m2)½ _ by m e a n s of i n t e r a c t i o n s w h i c h b r e a k the L i o u v i l l e s y m m e t r y , we h a v e to l e a r n how to do t h i s . A s a t i s f a c t o r y a n s w e r to t h i s p r o b l e m i s not y e t k n o w n , b u t we s e e t h a t the a r t we m u s t l e a r n s e e m s to c o n s i s t of k n o w i n g how to b r e a k t h e L i o u v i l l e s y m m e t r y in e a c h s p e c i a l c a s e u n d e r c o n s i d e r a t i o n ; o r e v e n b e t t e r , to l e a r n m o r e a b o u t the d e e p e r p h y s i c a l r e a s o n s f o r the w i d e - s p r e a d b r e a k i n g of t h e L i o u v i l l e g r o u p by n a t u r e .
RE F E R E N C E S [1] H.A.Kastrup, Nuel. Phys. B7 (1968) 545. [2] H.A.Kastrup, Phys. Rev. 150 (1966) 1183. [3] H.A. Kastrup, Proc. VII Int. Univ. weeks for nuclear physics at Schladming (Austria), ed. Paul Urban, to be published. [4] H.A. Kastrup, Ann. der Physik 9 (1962) 388. [5] E.In~Snii and E . P . W i g n e r , Proc. Nat. Acad. Sciences (USA) 39 (1953) 510; 40 (1954) 119. [6] I . E . Segal, Duke Math. J. 18 (1951) 221. [7] H . A . K a s t r u p , Phys. Rev. 143 (1966) 1021. [8] D . G . C u r i e , T . F . J o r d a n a n d E . C . G . S u d a r s h a n , Revs. Mod. Phys. 35 (1963) 350. [9] H. A. Kastrup, Phys. Rev. 142 (1966) 1060.