Karakostas_Decoherence in Unorthodox Formulations of Quantum Mechanics

8/14/2019 Karakostas_Decoherence in Unorthodox Formulations of Quantum Mechanics

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V A S S I L I O S K A R A K O S T A S A N D M I C H A E L D I C K S O N

D E C O H E R E N C E I N U N O R T H O D O X F O R M U L A T I O N SO F Q U A N T U M M E C H A N I C S

A B S T R A C T . T h e c o n c e p t u al s tr u c tu r e o f o r t h o d o x q u a n t u m m e c h a ni c s h a s n o t p r o v i d eda f u l l y s a t i s f a c t o r y a n d c o h e r e n t d e s c r i p t i o n o f n a t u r a l p h e n o m e n a . W i t h p a r t i c u l a ra t t e n t i o n to t h e m e a s u r e m e n t p r o b l e m , w e r e v i e w a n d i n v e s ti g a t e t w o u n o r t h o d o x f o r m u -l a ti o n s. F i r s t , t h e r e i s th e m o d e l a d v a n c e d b y G R W P , a s t o c h a s ti c m o d i f i c a ti o n o f t h es t a n d a r d S c h r 6 d i n g e r d y n a m i c s a d m i t t i n g s t a t e v e c t o r r e d u c t i o n a s a r e a l p h y s i c a l p r o c e s s .S e c o n d , t h e r e i s t h e o n t o l o g i c a l i n t e r p r e t a t i o n o f B o h m , a c a u s a l r e f o r m u l a t i o n o f t h eu s u a l t h e o r y a d m i t t i n g n o c o l l a p s e o f t h e s t a t e v e c t o r . W i t h i n t h e s e t w o s e e m i n g l y q u it ed i f f e re n t a p p r o a c h e s , w e d i s cu s s i n a c o m p a r a t i v e m a n n e r , s e v e r a l p o i n ts : T h e m e a n i n go f t h e s t a t e v e c t o r , t h e s t a t u s o f q u a n t u m p r o b a b i l it y , t h e l e g i ti m a c y o f a t t r ib u t i n g m a c r oo b j e c t i v e p r o p e r t i e s t o p h y s i c a l s y s t e m s , a n d t h e p o s s i b i li t y o f r e t r i e v i n g t h e c l a ss i ca ll im i t . F i n a l l y , w e c o n s i d e r a s p e c t s o f n o n - l o c a l i t y a n d r e l e v a n t d i f fi c u lt ie s w i t h f o r m u l a t i n ga r e l a t i v is t i c g e n e r a l i z a t i o n o f t h e t w o a p p r o a c h e s .

1 . T H E P R O B L E M O F Q U A N T U M M E A S U R E M E N TE v o l u t i o n i n o r t h o d o x q u a n t u m m e c h a n i c s i s s t ra n g e l y d u al is ti c. O n t h eo n e h a n d , t h e s t a t e - v e c t o r o f a c l o s e d s y s t e m e v o l v e s i n a c o n t i n u o u s ,d e t e r m i n is t ic a n d r ev e r s ib l e m a n n e r a c c o r d i n g t o t h e t i m e d e p e n d e n tS c h r 6 d i n g e r e q u a t i o n

( 1) ~ o - - > ~ , = eiH'~o, ( ~ = 1 ) .O n t h e o t h e r h a n d , d i s c o n t i n u i t y , s t o c h a s t i c i ty a n d ir r e v er s ib i li ty a r ei n t r o d u c e d i n m e a s u r e m e n t s i t u a t i o n s t h r o u g h t h e p r o j e c t i o n p o s t u l a t e ,w h i c h l e a d s f r o m l in e a r s u p e r p o s i t i o n s in H i lb e r t - s p a c e v e c t o r s t o c l as s i-c a l s ta t i s t i ca l mix tu res o f s t a tes

( 2 ) 2 CnC*ml't~n)(at~mt~ 2 lCn[21a'Ifn)('f~nl.nm r~

T h i s s e c o n d t y p e o f e v o l u t i o n i s i n c o m p a t i b l e w i t h t h e S c h r 0 d i n g e re q u a t i o n a n d c o n s e q u e n t l y q u a n t u m t h e o r y i s u n a b l e t o c o n s i s t e n t l yp r o v i d e a d y n a m i c a l d e sc r ip t i o n o f t h e m e a s u r e m e n t p r o c e s s . I n p a rt i-c u l a r , i t i s u n c l e a r w h e r e t h e tr a n s i t i o n b e t w e e n ( 1 ) a n d ( 2 ) o c c u r s .T h i s u n s a t i s f a c t o r y c h a r a c t e r o f t h e s t a n d a r d f o r m u l a t i o n h a s b e e np r o n o u n c e d a s 'u n p r o f e s s i o n a l ' b y B e l l (1 9 8 6 , p . 5 4 ) . " W h e n I l o o k a tSynthese 102: 61-97 , 1995. 1995 Klu wer Academ ic Publishers. Printed in the N etherlands.


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62 V A S S I L I O S K A R A K O S T A S A N D M I C H A E L D I C K S O N

q u a n t u m m e c h a n i c s I s e e t h a t i t i s a d i r t y t h e o r y . T h e f o r m u l a t i o n s . . .t h a t y o u f i n d i n t h e b o o k s i n v o l v e d i v i d i n g t h e w o r l d i n t o a n o b s e r v e ra n d a n o b s e r v e d , a n d y o u a r e n o t t o l d w h e r e t h a t d i v i s i o n c o m e s . . .W h a t y o u le a r n is t h a t . . , t h e a m big uity " i n v o l v e s d e c i m a l p l a c e s r e m o t ef r o m h u m a n a b i l i t i e s t o t e s t " . E v i d e n t l y t h e p r o j e c t i o n p o s t u l a t e , a n di n d e e d a n y p o s t u l a t e t h a t a p p l i e s o n l y t o t h e a ct o f m e a s u r e m e n t , l o o k sl i k e a n a d h o c p h e n o m e n o l o g i c a l r u l e w h o s e s o l e p u r p o s e i s t o a v o i dt h e p r e d i c t i o n o f l in e a r m a c r o s c o p i c s u p e r p o s i t i o n s a n d t h u s t o s a v eq u a n t u m t h e o r y f r o m f la g r a n t c o n t r a d i c ti o n w i th e v e r y d a y e x p e r i e n c e .

H o w e v e r , i t w o u l d b e m i sl ea d i n g t o s u p p o s e t h a t t h e i n c o m p a t ib i li tyb e t w e e n p r o j e c t i o n p o s t u l a t e a n d S c h r O d i n g e r d y n a m i c s i s a n o p e np r o b l e m , a d m i s s i b l e t o f u r t h e r d e v e l o p m e n t s w i t h i n s t a n d a r d q u a n t u mt h e o r y . I n f a c t , t h i s ' m e a s u r e m e n t p r o b l e m ' h a s b e e n ' s o l v e d ' i n t h esense t h a t i t ha s bee n sh ow n tha t no s o lu t i on ex i s ts ( cf . F in e , 1970). 1T h e r e a r e m a i n l y t w o a l t er n a t iv e s t o t h e s t a n d a r d f o r m u l a t io n : t o in t r o -d u c e a n o n l i n e a r m o d i f i c a ti o n o f t h e S c h r 6 d i n g e r e q u a t i o n t h a t w o u l di n c o r p o r a t e a m e c h a n i s m f o r s t a t e r e d u c t i o n i n i t s d y n a m i c a l d e v e l o p -m e n t , o r to a d v a n c e a n i n t e r p r e ta t io n o f q u a n t u m m e c h a n i c s in w h ic h' s t a t e v e c t o r c o l l a p s e ' w o u l d p l a y e s s e n t i a l l y n o r o l e .

F r o m t h e l a r g e v a r i e t y o f ' a l t e r n a t i v e s c h e m e s ' i n t h e l i t e r a t u r e w el i m i t o u r s e l v e s t o t h e c o n s i d e r a t i o n o f(i)(ii)

G h i r a r d i , R i m i n i , W e b e r a n d P e a r l e ' s d y n a m i c a l r e d u c t i o nm o d e l , a n dB o h m ' s c a u sa l , o r o n t o lo g i c a l , i n t e r p r e t a t i o n o f q u a n t u mm e c h a n i c s .

T h e p a r t ic u l a r s e l e c ti o n o f t h e t w o a b o v e a l t e r n a t iv e s i s m o t i v a t e dpa r t l y b y Be l l ' s (1990 , p . 31) r e fe r enc e t o ( i ) and ( ii ) a s be ing t h e on lyt w o ' p r e c i s e p i c tu r e s ' o f n a t u r e a n d p a r t ly b y o u r b e l i e f t h a t a c o m p a r a -t iv e s t u d y o f t h e s e t w o s e e m i n g l y q u i t e d i f f e r e n t a p p r o a c h e s c a n s h e dlig h t o n t h e f o u n d a t i o n a l p r o b l e m s o f q u a n t u m m e c h a n i cs a n d h e lp t op r o v i d e a b e t te r u n d e r s t a n d i n g o f t h e a p p r o a c h e s t h e m s e lv e s .

2 . R E D U C T IO N A S A R E A L P H Y S IC A L P R O C E S SR e c e n t l y G h i r ar d i, R i m in i, W e b e r , a n d P e a r le ( G R W P ) h a v e p r o p o s e da s to c h a s t i c m o d i f ic a t io n o f s t a n d a r d q u a n t u m m e c h a n i c s i n w h i c h s t a t e-v e c t o r r e d u c t i o n i s d e s c r i b e d a s an o b j e c t i v e p h y s i c a l p r o c e s s . T h e t e r m' o b j e c t i v e ' r e f e r s t o t h e a u t h o r s ' r e f u s a l t o a t t r i b u t e a p e c u l i a r r o l e t o


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U N O RT H O D O X F O RM U L A T I O N S O F Q U A N T U M M E CH A N I CS 63

the obs erve r or to accept the ' shif ty spl it ' betw een q uantu m and classicalbehaviour in micro-macro in te rac t ions . Ins tead , the GRWP f rameworkaims at constructing a single fundamental dynamics governing all phe-nomena by unifying the two contradictory types of evolut ion we dis-cussed abov e into a s ingle propa gat ion law. Such a monist ic descr iptionoffers also the possibili ty of ascribing individual reality to th e w avefun c-t ion as corresponding to the s ta te of a s ingle system in nature.

As one might expect , the GRWP unif ied dynamics restr ic ts the abso-lute val idi ty of Schr6dinger 's equat ion. The la t ter is regarded as thel imit ing case of a more general equat ion of motion which leaves theevolut ion of systems in the quantum domain pract ical ly unal tered, butwhich suppresses the em barrassing su perposi t ions of different ly locatedstates a t the macroscopic level . The el iminat ion of coherence occursdynamically at the individual level of description according to an intrin-sically stochastic and irreversible dynamics which induces spatial spon-taneous local isat ions on the wavefunct ion.

The idea of spontaneous loca l isa tion has been formula ted in two mainapproaches. The ear l ies t and most intui t ive approach (DSL) introducesf ini te s tochast ic changes for the s ta tevector in the fo rm of discont inuousjumps occuring around appropriate posi t ions (Ghirardi , Pdmini , andWeber, 1986). For this reason, this kind of stochasticity has also beencal led a 'h i t t ing ' process . In the more ref ined cont inuous approach(CSL) the sudden jumps are replaced by a cont inuous s tochast ic evol-ut ion - a sor t of Brow nian m otion for the s ta tevector - in Hilb er t space(Pearle , 1989; Ghirardi , Pear le , a nd Rimini , 1990). The CSL mo del ismore genera l and powerfu l . However , we f i r s t cons ider the DSL typeof dynamics because of i ts more immediate physical content .2.1. Discontinuous Spontaneous Localisation (DSL)The basic assumption underlying the DSL model is that any par t ic le ,whether isolated or par t of a physical system, is subjected at randomlydistributed times, to approximate spatial localisations (hitt ings) whichact against the expansion of the system's wavefunct ion due to theSchrOdinger evolut ion. As a consequence of a local isat ion around apoint x in physical space (we con sider a on e-dimentiona l case for s impli-city), an initial w ave -fun ctio n I'-I ) rep rese ntin g a particle in posit ionbasis , col lapses instantaneously into a new wavefunct ion l~ , ) accordingt o ,


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I % > =a 1/4 1

w h e r e q is t h e p o s i t i o n o p e r a t o r o f t h e p a r t i c l e u n d e r g o i n g t h e l o c al is -a t io n p r o c e s s a n d a i s a p a r a m e t e r o f le n g t h d i m e n s i o n s w h o s e m e a n i n gis tha t l /V~a repre sen ts the accuracy (o r w id th ) o f the loca l i s a t iona r o u n d t h e ' h i t t i n g c e n t r e ' x . T h e c h a r a c t e r i s t i c l o c a l i s a t i o n l e n g t hl / ~ a a is a s s u m e d t o t a k e t h e v a l u e 1 0 . 5 c m .

E q u a t i o n ( 3) sh o w s t h a t t h e G R W d i s c o n ti n u o u s a p p r o a c h c o n si st si n m u l t i p l y in g t h e i n i ti al w a v e f u n c t i o n b y a G a u s s i a n ( j u m p ) f u n c t i o nw h i c h i s d i s t r i b u t e d a r o u n d x w i t h s p r e a d 1 t ~ . I n t u r n t h i s m u l t i p l i -c a t i o n c o r r e s p o n d s t o t h e e f f e c t o f an a p p r o x i m a t e l o c a li s a ti o n : i f t h ep a r t i c le is i n it ia l ly a l r e a d y l o c a t e d w i th i n a n in t e r v a l ~ l / V a a r o u n d x ,i t s w a v e f u n c t i o n i s w e l l l o c a l i s e d a n d t h e r e f o r e r e m a i n s p r a c t i c a l l y u n -a l t e r e d b y t h e h i tt i n g p r o c e s s ; i f, o n t h e c o n t r a r y , t h e i n i ti a l w a v e f u n c -t ion sp reads ove r a d i s tance la rge r than the cha rac te r i s t i c loca l i s a t ionl e n g t h , i t i s c o m p r e s s e d b y t h e p r o c e s s .

T h e p r o b a b i l i t y d e n s i t y f o r s u c h a h i t t i n g t o o c c u r a t a n y p a r t i c u l a rp o i n t x is a s s u m e d t o b e( 4 ) e ( x ) = I I , : I , A

T h u s l o c a l i s a t i o n p r o c e s s e s a r e m o s t l i k e l y t o a p p e a r a t t h o s e p l a c e sw h e r e t h e i n it ia l w a v e f u n c t io n is l ar g es t. N o t e t h a t t h e G R W p r o b a b i l i t yp r e s c r i p t i o n i s a n a l o g o u s t o B o r n ' s p r o b a b i l i t y r u l e f o r t h e o u t c o m e so f a m e a s u r e m e n t , i n th a t r e l a t i o n ( 4 ) e n s u r e s t h a t h i t t in g s o c c u r w i t hh i g h e r p r o b a b i li t y a t t h o s e r eg i o n s w h e r e i n s t a n d a r d q u a n t u m m e c h a n -ic s the re i s a h ighe r p rob ab i l i t y o f f ind ing the p a r t i c le .F i n a l ly , i t is a l s o a s s u m e d t h a t h i tt in g s o c c u r a t r a n d o m t i m e s , a c c o r d -i n g t o a P o i s s o n d i s t r ib u t i o n , w i t h m e a n f r e q u e n c y A = 1 0 -1 6 s e c - 1 a n dt h a t i n t h e t i m e i n t e r v a l b e t w e e n t w o s u c c e s si v e h i tt in g s t h e s t a t e v e c t o re v o l v e s f o l l o w i n g t h e S c h r 6 d i n g e r e q u a t i o n .

I n t h e D S L a p p r o a c h j u s t d e s c r i b e d , h o w t h e r e d u c t i o n m e c h a n i s mw o r k s i s q u i t e t r a n s p a r e n t . A s e x p l a i n e d , i t i s i n c o r p o r a t e d i n t h e b a s i cp r i n c i p l e o f t h e D S L d y n a m i c s . T h e a p p l i c a t io n o f p r o c e s s ( 3 ) w i t h t h ep r o b a b i l i t y r u l e ( 4) q u i c k l y r e d u c e s a li n e a r s u p e r p o s i t i o n o f m a c r o -s e o p i c a ll y d i s t in g u i s h a b l e s t a t e s s e p a r a t e d b y a d i s t a n c e m u c h l a r g e rt h a n l / ~ a a t o o n e o f i ts c o m p o n e n t s . T h e p r o b a b i l i t y t h a t a p a r t i c u l a r


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U N O R T H O D O X F O R M U L A T I O N S O F Q U A N T U M M E C H A N I C S 6 5

c o m p o n e n t e m e r g e s i s p r o p o r t i o n a l t o it s w e i g h t i n t h e in i ti a l s u p e r p o s i -t i o n . T h i s g u a r a n t e e s t h a t t h e p r o b a b i l i s t ic p r e d i c t i o n s o f t h e o r t h o d o xm e a s u r e m e n t t h e o r y a r e r e p r o d u c e d . M o r e p r e c is e ly , a s w e s h al l s e e ,t h e y a r e r e p r o d u c e d u p t o s m a l l b u t i r r e d u c i b l e a n o m a l i e s a k i n t o t h ea n o m a l o u s s p o n t a n e o u s l o c a l i s a t i o n s .

2 .2 . C o n t i n u o u s S p o n t a n e o u s L o c a l is a t io n ( C S L )T h e C S L m o d e l , b a s e d o n t h e c o n s i d e r a t io n o f c o n t i n u o u s d i f fu s io np r o c e s s e s i n H i l b e r t s p a c e , a v o i d s t h e p h y s i c a l l y u n s a t i s f a c t o r y i n s t a n -t a n e o u s c h a n g e s o f t h e s t a t e v e c t o r , f o r m u l a t e s t h e s t o c h a s ti c p a r t o ft h e e v o l u t i o n p r i n c i p l e t h r o u g h a s i n g l e u n i f i e d e q u a t i o n a n d p r e s e r v e st h e s y m m e t r y p r o p e r t i e s o f t h e s t a t e v e c t o r i n t h e c a s e o f s y s t e m s w i t hi d e n t i c a l p a r t i c l e s . I n t h i s f o r m u l a t i o n , t h e u s u a l H a m i l t o n i a n d y n a m i c si s s u b j e c t e d t o a n o n - H e r m i t i a n r a n d o m l y f l u c t u a t i n g p o t e n t i a l w h i c hd e p e n d s u p o n a s e t o f w h i t e n o i s e f u n c ti o n s { w ( x , t)} , that is to say, t inyG a u s s i a n r a n d o m p r o c e s s e s , c o u p l e d t o a s e t o f s e lf - ad j o in t c o m m u t i n go p e r a t o r s { A (x )} w h o s e j o i n t e i g e n v e c t o r s ( a s e x p l a i n e d s h o r t l y b e l o w )r e p r e s e n t t h e d e n s i t y o f p a r ti cl e s a r o u n d x .

T h e s t a t e v e c t o r o b e y s a l i n e a r s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n o f t h eI t 6 t y p e

(5 ) d l ~ w ( t ) ) = - i H l ~ w ( t ) ) d t +[ ~ d x A ( x ) w ( x , t ) - /2 ~ , d x A 2 ( x ) j ,% (O ) tw i t h t h e r a n d o m f u n c t i o n s w ( x , t ) s a t i s f y i n g t h e f o l l o w i n g e x p e c t a t i o nv a l u e s

(6 ) ( w ( x , t)) = 0, ( w ( x , t ) w ( x ' , t ' ) ) = y 6 ( x - x ' ) 6 ( t - t ') .T h u s , w h i l e w ( x , t ) is p e r m i t t e d t o f l u c t u a t e p o s i t i v e l y as w e l l a s n e g a -t ive ly wi th eq ua l l ike l ihood , i ts f luc tua t ions t end to be s ta t i s t ic a l ly co r re -l a t e d t h r o u g h t h e p a r a m e t e r 7 w h i c h c o n t r o ls t h e s t r en g t h o f t h e s t o ch -a s t i c p r o c e s s . I t i s i n t e r e s t i n g t o n o t e t h a t u n d e r a n a p p r o p r i a t e c h o i c eof 7 and in the in f in i te f r eq ue ncy l im i t (A ~ m, a --> 0 ) the d i scon t in uou sp r o c e s s ( 3) c a n b e s h o w n t o t r a n s f o r m i ts e lf i n t o t h e c o n t i n u o u s p r o c e s s( 5 ) ( N i c r o s in i a n d R i m i n i , 1 9 9 0 ) . F r o m t h i s p o i n t o f v i e w , E q u a t i o n ( 5 )a c q u i r e s a n i m m e d i a t e p h y s i c a l c o n t e n t a s d e s c r i b i n g a s u c c e s s i o n o f' t i n y s p o n t a n e o u s l o c a l i s a t i o n s ' w h i c h c o n t i n u o u s l y s t r i v e t o r e d u c e t h e


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66 V A S S I L I O S K A R A K O S T A S A N D M I C H A E L D I C K S O Ns t a t e v e c to r i n to o n e o f th e c o m m o n e i g e n s ta t e s o f th e s e t o f c o m m u t i n gope ra to r s {A(x)} .I n e f f e c t , t h e o p e r a t o r s A ( x ) , f o r d i f fe r e n t v a l u e s o f t h e p a r a m e t e rx , d e t e r m i n e t h e e i g e n m a n i f o l d s o n w h i c h r e d u c t i o n t a k e s p l a c e . T h e i rs p e ci fi c f o r m , t h e s o - c a l l ed ' p r e f e r r e d b a s i s ' , is a s s u m e d t o b e d e f i n e db y

(7) Z ( x ) = ( ~ ) 3 / 2 f d q n ~ ( q ) n ( q ) e - m " ( q - x ) 2 ,w h e r e n * ( q ) an d n ( q ) a r e t h e c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s f o r ap a r t i c l e a t p o i n t q i n p h y s i c a l s p a c e . T h e n t h e e i g e n v a l u e s o f A ( x ) ca nb e t a k e n a s r e p r e s e n ti n g t h e a v e r a g e n u m b e r o f p a r ti c le s c o n t a i n e dw i t h in a s p h e r e c e n t e r e d a t x w i th v o l u m e o f t h e o r d e r o f a - 3 / 2 a n dr a d i u s a - m . P a r t i c l e n u m b e r ( o r m a c r o s c o p i c d e n s i t i e s ) i s t h u s t h ep r o p o s e d c a r r i e r f o r r e d u c t i o n i n C S L . T h e c o n s e q u e n c e i s t h a t as t a t e v e c t o r i n a s u p e r p o s i t i o n o f s t a te s d e s c r i b in g m a c r o s c o p i c c o l le c t io nof pa r t i c l e s i n d i f fe ren t p l aces wil l r ed uc e t o a s t a t e wi th de f in i t e num be r

o f p a r -t i c l e s . The pa r t i cu l a r e igens t a t e t ' . I t w ( t ) ) = j a i ) i n t o w h i c h t h e CS L d y -nam ics d r ive s t he s t a t ev ec tor i s de t e rm ined by a spec i fi c r ea l i s a ti onw i ( t ) o f t h e B r o w n i a n p r o c e s s .

P r o v i d e d t h a t d i f f e r e n t s a m p l e f u n c t i o n s w i ( t ) g e n e r a t e d i f f e r e n tn o r m s f o r l ~ w ( t ) ) , G R W P i n o r d e r t o o b t a i n c o n s i s t e n c y w i t h t h ep r e d i c t i o n s o f s t a n d a r d q u a n t u m m e c h a n i c s , r e s o r t t o a n a s s u m p t i o np a r a l l e l t o t h e o n e m a d e w i t h i n D S L f o r t h e p r o b a b i l i t y d e n s i t y o ft h e h i t t i n g p o s i t i o n s . T h e y a s s u m e t h a t t h e p r o b a b i l i t y f o r a s p e c i f i cr e a l i s a t i o n o f t h e s t o c h a s t i c p r o c e s s w i , o r e q u i v a l e n t l y o f t h e s t a t eve c to r Iq~w(t )) , is not the ' raw ' on e Praw[W,] asso c ia te d w i th the w hi ten o i s e d i s tr i b u t i o n o f E q u a t i o n ( 6 ) , b u t is g i v e n b y t h e ' c o o k e d ' p r o b a b i l -i ty P c [ w ~ ] ( a s P e a r l e ( 1 98 9 ) e x p la i n s) w h i c h d e p e n d s n o n l i n e a r ly o n t h es t a t e v e c t o r a t t i m e t a c c o r d i n g t o ,

(8 ) P c [ w J = e r .w [ W i ]l l 1 % ( 0 > 1 12 ,Th e pro bab i l i t y ru l e (8 ) mak es , fo r a g iven Pr,w[W~], s t a t evec tor s w i thl a r g e n o r m t o w e i g h m o r e , i . e . m o r e l i k e l y t o o c c u r , i n m e a s u r e m e n ts i t ua t i ons .

T o e x p l a i n h o w t h e CS L s t o c h a s t i c e v o l u t i o n i n d u c e s , w i t h t h er e q u i r e d q u a n t u m m e c h a n i c a l p r o b a b i l i t i e s , a n i n d i v i d u a l r e d u c t i o n o f


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t h e s t a t e v e c t o r in t o o n e o f th e c o m m o n e i g e n m a n i f o ld s o f t h e o p e r a t o r s{ A ( x ) } , w e c o n s i d e r a s i m p l e q u a n t i t a t i v e e x a m p l e i n w h i c h o n l y o n eo p e r a t o r A a p p e a r s i n E q u a t i o n ( 5 ), i .e ., 2

( 9 ) d I ' t r w ( t ) } - i H l ' I t w ( t ) } d t + [ A w ( t ) - 1 / 2 7 A 2 ] i g ~ w ( t ) } d t .I f w e d i s r e g ar d t h e H a m i l t o n i a n p a r t , s i n c e w e w a n t t o s tu d y t h e p u r er e d u c t i o n p r o p e r t i e s o f t h e C S L d y n a m i c s , t h e s o l u t io n o f E q u a t i o n ( 9)is

(10) I~B(t) ) = e x p [ A B ( t ) - v A 2 t ] l , I ' ( 0 ) ) ,w h e r e B ( t ) i s a B r o w n i a n m o t i o n f u n c t i o n r e l a t e d t o t h e w h i t e n o i s eb y

f o1 1 ) B ( t ) = w ( t ) d t ,w i t h ' r a w " p r o b a b i l i t y d e n s i t y

(12) P ~ , , ~ [ B ( t ) ] = 1__e_B2/avt,N

N b e i n g a n o r m a l i s a t i o n f a c t o r .L e t u s f u r th e r a s s u m e t h a t t h e o p e r a t o r A is s p a n n e d b y o n l y t w o

e i g e n m a n i f o l d s M z , M r w i t h e i g e n v a l u e s l , r c o r r e s p o n d i n g t o t w om a c r o s c o p i c a l l y d i f f e r e n t o u t c o m e s , s o t h a t t h e i n i t i a l s t a t e v e c t o r o ft h e s y s te m u n d e r m e a s u r e m e n t h a s th e f o r m ,

(13) lxI r (O)) I . [ ~ z ) + R l a ' ~ r ) .T h e n , t h e t i m e e v o l u t i o n o f t h is t w o w a v e p a c k e t s y s t e m c a n b e w r i t te n ,a c c o r d i n g t o E q u a t i o n ( 1 0 ) , a s

(14) I ' trB(t)) = L e x p [ I B ( t ) - y l Z t] t* ~ ) + R e x p [ r B ( t ) - yr2t] l~r) ,w h e r e t h e s u m o f t h e t w o s t a te v e c t o r s a t t im e 0 b e c o m e s , a t t i m e t , as t a t e v e c t o r w h i c h i s t h e s u m o f t h e t w o e v o l v e d s t a te v e c t o rs .

T h e s o l u t io n o f E q u a t i o n ( 14 ) fo r ] ~ B ( t) ) d e p e n d s u p o n a p a rt i cu l a rr e a li s at io n o f th e B r o w n i a n m o t i o n B ( t ) t h a t d r i v e s t h e s t a t e v e c t o r . I nt h e a b s e n c e , h o w e v e r , o f a n y k n o w l e d g e o f t h e a c t u a l B ( t ) , w e r e s o r tt o t h e c o o k i n g p r e s c r i p t i o n ( 8 ) w h i c h a s s i g n s t h e f o l l o w i n g p r o b a b i l i t yd i s t r i b u t i o n t o t h e B r o w n i a n p r o c e s s a t t i m e t :


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68 V A S S I L I O S K A R A K O S T AS A N D M I C H A E L D I C K S O N

(15) 1 f [B(t) 2lyt] 2Pc[B(t)] = ~ tILl 2 ex pN U " 2~t[B(t) - 2r N2 1+ I e I 2 exp 2- - -i.z g

Accord ing to Equa t ion (15) the mos t prob able va lues of B( t ) ' s occurwhen e i ther or both exponents in (15) a re smal l , i . e . , when the fol lowingconditions are satisf ied,

(16) IB(t) - 213't1 < (2 yt ) v2, IB(t ) - 2r3,tl < (2 yt ) ~/2.Clear ly, re la t ions (16) ensure tha t , a t t imes la rge compared with y- l ,the Brownian process B(t) f luc tua te s wi th ove rwhe lming probabi l i tya round an in te rval o f wid th (2yt) v2 cente red on e i the r the va lue 2lytor the va lue 2ryt. W he n , t he r e f o r e , B(t) takes a va lue e i ther near to2lyt or nea r to 2ryt the corresponding s ta te vec tor I~B(t) ) wi l l be dr ivenei ther wi thin the e igenm anifold Mz or wi thin Mr with p robabi l i ty ILl zor [RI2, respectively.

This pic ture ma kes ap pare nt tha t the de ter min at ion of a specif icmeas urem ent -o utco me in the G R W P theory is ach ieved through a pa rt i -cular rea l isa t ion of the Brown ian process . In the above example , forins tance , B(t) gene ra te s a sor t o f Brownian mot ion for the s ta tevec tor ,so tha t , [~B(t) ) undergoes a di f fus ion process with two poss ible 'dr i f ts 'corresp ondin g to two incom pat ible m acroscop ic s i tua tions ( ' right ' and' le ft ' pointer readings) . I t i s the choice of the dr i f t, ran dom ly mad ewith probabi l i ty weights reproducing the quantum mechanica l s ta t is t ics ,tha t dr ives the s ta tevec tor into a par t icular 'pointer reading ' e igens ta te ,and the fur the r s tochas t ic i ty becomes redundant .The re la t ionship be tween the s ta tevec tor lq~> and the stochastic pro-cess B(t) can be be t te r r ea li sed by cons ide r ing B(t) as a 'pilot s tochasticfield' which g uides the evo luti on of l ',I ) in such a w ay that the statevec to r be com es loca l ised arou nd those pos i t ions x wh ere Iq~l is la rge ,while avo iding region s w her e l 't rl is small . Th e gu iding field relationimpl ic i t ly in t roduced in the GRWP theory and the pr iv i leged ro le ex-pl ic i t ly ass igned to pos i t ion are centra l fea tures in Bohm's pi lot wavetheory, to be discussed next . In Sec t ion 4, we analyse the pros andcons of these two seemingly qui te d i f fe ren t a t t empts to r e formula tequantum mechanics .


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U N O RT H O D O X F O RM U L A T I O N S O F Q U A N T U M M E CH A N I CS 69

3. BO H M ' S CA U S A L I N T E RP RE T A T I O NIn 1952, B oh m in t rodu ced a re- interp reta t ion of nonrelat ivis t ic quantu mmechanics which, s ince then, has seen considerable development (cf .Bo hm and Hi ley, 1993; and re ferenc es therein) . This 'causal interpreta-t ion ' permits a descr ipt ion of qua ntum processes in which no reduct ionof the wavefunct ion occurs . Rather than being a process requir ing i t sown pos tu la te , measurement in the Bohm theory i s descr ibed by thegeneral dynamics . The theory 's central feature is that par t ic les fol lowcont inuous t ra jec tor ies (whether observed or no t ) , de te rmined by afield that is always associated with the evolu tion o f a part icle. This 'O-field', O(x, t) = Re is(''/~ , sat isf ies the s tandard Schr tdinger equat ion(just as the electr om agn etic f ield satisfies Maxw ell 's equa tions) an dther efo re i t se lf evolves d eterminist ically .3.1. Fundamentals of Boh m TheoryTo revea l the impor tan t fea tures of the theo ry , we cons ider the physi -cally interest ing case of a two-part icle wavefunction, where for simpli-c i ty we assume tha t the par t ic les have equ al mass:

(17) i / iO0=[O t 2m~-~2V~z + V 2) + V ]~ .No w su bsti tute the 0-field into (17) an d let P = R 2 = I~ 2. (We suppressthe arguments of R and S for notat ional s impl ic i ty . ) St ra ightforwardcalculation yields two eq uat ions corresp onding to th e real a nd comp lexpar ts of the Schr tdinger equat ion respect ively:

__ ~ (V~+ V~)R 018) OS + (V1S)z + (V2S)2 + VOt 2m 2m 2m R(19) O--~P+ vI " (P v1 S) + v2" (p vz S) =

O ne can interpret Eq uat io n (18) as a c lassical Ha m il ton -Jac ob i equa-t ion,OS OS ) aS- - , . . . , ; t + - - = 0 ,(20) H q l, , q6; Oql Oq6 Ot


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70 V A S S I L I O S K A R A K O S T A S A N D M I C H A E L D I C K S O Nw h e r e q i ( i = 1 . . . . 6 ) a r e th e g e n e r a l iz e d c o o r d i n a t e s . I n m a k i n g th i si n t e r p r e t a t i o n , o n e d e f i n e s a ' q u a n t u m p o t e n t i a l '

(21) U = _/~z (V~ + V~)R2 m Rs o t h a t ( 1 8 ) m a y b e w r i t t e n

O S ( r i g ) 2 ( V 2 S ) 2(2 2) - - + + + V + U = 0.O t 2 m 2 r nE q u a t i o n ( 2 2 ) c a n b e u s e d t o j u s f i t y3 t he p re sc r ip t i on (cen t ra l i n

B o h m i a n m e c h a n i c s ) t h a t t h e v e l o c i t y o f a p a r t ic l e is g i v en b y V S / m =v ( x ) . U n d e r t h i s i n t e r p r e t a t i o n , E q u a t i o n ( 1 9 ) b e c o m e s

O P( 2 3 ) - - + V l " P v + V 2 " P v = 0 .OtE q u a t i o n ( 23 ) is i n th e f o r m o f a c o n s e rv a t io n e q u a t i o n , a n d B o h maccord ing ly i n t e rpre t s P a s a p robab i l i t y dens i t y ( i n conf igura t i ons p a c e ) . Be c a u s e ( 2 3 ) g u a r a n t e e s t h a t t h e f l o w o f p r o b a b i l i t y i s c o n -se rved , one can t h ink o f P a s g iv ing t he i n i t i a l d i s t r i bu t i on o f pa r t i c l e si n a n e n s e m b l e w h o s e m o t i o n i s t h e r e a f t e r g o v e r n e d b y E q u a t i o n ( 22 )v i a t h e p r e s c r i p t i o n V S / m = v(x) .

T h e q u a n t u m e q u a t i o n o f m o t i o n c a n th e r e f o r e b e s e e n as d et e rm i n -i n g a n e n s e m b l e o f p o s s i b l e t r a j e c to r i e s f o r p a r t ic l e s w i t h i n it ia l m o m e n -t u m P = V S , m o v i n g u n d e r t h e i n f lu e n c e o f t h e t o t a l p o t e n t i a l V + U .T h e ' g u i d i n g ' o r ' p i lo t ' 0 - f ie ld d e t e r m i n e s ( t h r o u g h R) t h e p r o b a b i l it yd e n s i ty P a n d t h e q u a n t u m p o t e n t ia l U , a n d ( t h r o u g h S ) t h e m o m e n t aof t he pa r t i c l e s .

L e t u s s u m m a r iz e t h e m a i n p o i n t s o f t h e t h e o r y a s p r e s e n t e d h e r e .B ( i )B ( i i )B ( i i i )

T h e p a r t i c l e s g o v e r n e d b y t h e w a v e f u n c t i o n a r e r e a l e n t i t i e sw i t h a w e l l - d e f i n e d , c o n t i n u o u s l y v a r y i n g , p o s i t i o n .A p a r t i c le ' s m o m e n t u m is r e s t r i c te d t o 7 S , w h i c h it se l f isg i v e n b y t h e ( d e t e r m i n i s ti c ) e q u a t i o n o f m o t i o n ( 2 2 ).P i s a p r o b a b i l i t y d e n s i t y f o r a n e n s e m b l e o f p a r t ic l e s m o v i n ga c c o r d i n g t o ( 2 2 ) . T h e c o n d i t io n t h a t p a r t ic l e s b e d i s t r i b u t e daccord in g t o P wi l l ho ld a t a ll t im es , i f i t ho ld s i n i ti a ll y , d uet o t h e c o n s e r v a t i o n r e l a t i o n ( 2 3 ) .


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3.2. Consequences of the TheoryThe c rude s l ogan fo r t he Bohm in t e l~ re t a t i on i s : ' I t ' s c l a s s i ca l a l l t hew a y d o w n . ' Bo t h c l a s s i c a l a n d q u a n t u m o b j e c t s a r e w e l l - l o c a l i z e d i np o s i t i o n s p a c e .

B u t t h a t s l o g a n is n o t t o s a y t h a t t h e B o h m i n t e r p r e t a t i o n r e v e r t se n t i r e ly t o a cl as s ic a l w o r l d v i e w . I n d e e d , t h r e e f e a t u r e s o f t h e q u a n t u mpoten t i a l , U , a re pa r t i cu l a r l y no t ab l e . F i r s t , a s shown by (21) , i t ss t r eng th i s un a f fec t ed b y mul t i p l i ca t ion o f ~0 by a cons t an t . In o the rw o r d s , i t i s n o t t h e s t r e n g t h o f U , b u t i t s form w h i c h is i m p o r t a n t . F o rt h e s a m e r e a s o n , U d o e s n o t d i s s i p a t e w i t h d i s t a n c e . T h e s e f a c t s l e a dt o th e i n t e r p r e t a t i o n o f t h e q u a n t u m p o t e n t i a l a s a 'g u i d i n g p o t e n t i a l'r a t h e r t h a n a s a ' f o r c e f ie l d '. O n e m a y i m a g i n e t h e q u a n t u m p o t e n t i a lt o be a se r i e s o f channe l s o f a l l owed t r a j ec to r i e s .

S e c o n d , u n l ik e a cl as s ic a l f ie ld , U h a s n o s o u r c e . I n s t e a d , w h a t e v e ri n f o r m a t i o n o n e f e e d s i n t o t h e u s u a l S c h r 6 d i n g e r e q u a t i o n , t h a t in f o r -m a t i o n i s w h a t d e t e r m i n e s U . I n g e n e r a l, s u c h in f o r m a t i o n c o n t a i n s th ee x p e r i m e n t a l a r r a n g e m e n t , a s i n t h e d o u b l e - s l it e x p e r i m e n t ( d i s c u s s e din 3.3) .

T h i r d , t h e q u a n t u m p o t e n t i a l i s n o n l o c a i i n t w o w a y s : F ir s t, t h e s t a teo f a p i e c e o f a p p a r a t u s c a n i n f l u e n c e t h e m o t i o n o f a d i s t a n t p a r t i c l eb y a l t er in g t h e f o r m o f U - t h e d o u b l e - s l it e x p e r i m e n t i s a n e x a m p l e .S e c o n d , t h e q u a n t u m p o t e n t i a l l i v e s i n c o n f i g u r a t i o n s p a c e , n o t i no r d i n a r y t h r e e - d i m e n s i o n a l p o s i t i o n s p a c e . ( N o t e t h a t U is a f u n c t i o n o ft h e p o s i t i o n o f e a c h p a r t i c le . ) T h e r e f o r e , t o n g - d i s ta n c e , e v e n s p a c e l i k e ,c o r r e la t io n s a m o n g p a r t ic l es m a y b e m a d e m a n i f e s t w h e n t h e c on f ig u r -a t i o n s p a c e t r a j e c t o r y i s t r a n s l a t e d i n t o a p o s i t i o n s p a c e t r a j e c t o r y f o reach pa r t i c l e .

F o r t h e s e t h r e e r e a s o n s , t h e q u a n t u m p o t e n t i a l i s h i g h ly n o n - cl as s ic a l.Bu t o n e w a y in w h i c h t h e B o h m t h e o r y i s ' cl as s ic a l' i s t h e w a y itr e c o v e r s t h e s t at is t ic a l n a t u r e o f q u a n t u m m e c h a n i c s . W e h a v e a l r e a d yn o t e d t h a t P = I ~ t 2 is ta k e n t o g i v e th e d i s t r ib u t i o n o f p a r ti c le s , s o t h a tt h e s t a ti st ic a l p r e d i c ti o n s o f t h e B o h m t h e o r y a r e i d e n ti c a l t o th e s ta n -d a r d q u a n t u m p r e d i c t io n s . 4 B u t b e c a u s e p a r t ic l e s h a v e w e l l - d e f in e dp o s i t i o n s a n d e v o l v e d e t e r m i n is i ti c a ll y , c a n t h e B o h m t h e o r y , u n l i k es t a n d a r d q u a n t u m m e c h a n i c s , m a k e n o n - p r o b a b i l i s t i c p r e d i c t i o n s f o r as i n g l e r u n o f a n e x p e r i m e n t ?

T h e a n s w e r i s ' n o ' . T w o f a c t s a b o u t t h e q u a n t u m p o t e n t i a l c o n s p i r eto p rec lude such pred i c t i ons . F i r s t , a c a re fu l ana lys i s o f U revea l s t ha t


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U N O R T H O D O X F O R M U L A T I O N S O F Q U A N T U M M E C H A N I C S 73

a t i o n s i n t h e f l u i d l e a d r a p i d l y t o P = ] 0 ] 2 . [0 12 i s t h e r e f o r e a n e q u i l i b -r i u m c o n d i t i o n i n t h is c a se , a n d is m a i n t a i n e d b y t h e c o n s e r v a t i o nE q u a t i o n ( 23 ) . M o r e r e c e n t l y , V a l e n t in i ( 1 9 9 1a , 1 9 9 1 b ) h a s d e r i v e d as u b q u a n t u m H - t h e o r e m f o r s y s te m s w it h a l a r g e n u m b e r o f d e g r e e s o ff r e e d o m , a n d s h o w e d t h a t m a x i m i z a t i o n o f e n t r o p y e n ta i l s th e p r o b a b i l -i t y d i s t r i b u t i o n P = t 01 2 . I n p a r t i c u l a r , i t i s s h o w n h e r e t h a t s i n g l ep a r t i c le s e x t r a c t e d f r o m ( o r is o l a t e d w i t h in ) s u c h a s y s t e m ( w h i c h o n ec o u l d t a k e t o b e t h e u n i v e r s e ) w i lt h a v e t h e d i s t r i b u t i o n P = t~12, w h e r e~ (x ) i s t h e w a v e f u n c t i o n o f t h e p a rt i c le . T h e s e a r g u m e n t s - a n d o t h e r st o o , e . g . , D t i r r e t a l ., 1 9 9 2 a , 1 9 9 2 b - d e m o n s t r a t e t h a t in B o h m ' s p i lo t -w a v e t h e o r y , t h e r e a r e i n d e p e n d e n t r e a s o n s f o r t ak i n g P = t 0t 2.B e f o r e c o n s id e r in g m e a s u r e m e n t , w e n o t e t h at t h e r e a r e tw o d e -v e l o p e d a c c o u n t s o f sp i n in th e B o h m t h e o r y . I n o n e ( c f. B e l l, 1 9 6 6 ;B o h m a n d H i l e y , 1 9 9 3 , c h . 1 0 ) , s p i n is n o t a p r o p e r t y o f a p a r t i c le , b u tis r a t h e r e n c o d e d i n t o t h e q u a n t u m p o t e n t i a l , s o t h a t p a rt ic l e s b e h a v ea s i f t h e y h a d s p i n . ( E . g . , a p a r t i c l e w i t h ' s p i n ' w i ll f o l l o w th e c o r r e c tt r a je c t o r ie s in a S t e r n - G e r l a c h e x p e r i m e n t . ) I n t h e o t h e r (c f . D e w d n e ye t a l . , 1 9 8 6 , 1 98 7 ; H o l l a n d , 1 9 9 3 ) , p a r t i c l e s d o h a v e s p i n , a n d t h i s s p i nc a n b e c o n s i d e r e d q u i t e l i t e r al l y to b e a n a c t u a l r o t a t i o n o f t h e p a r t i c le .T h i s l a tt e r v i e w h a s c o m e u n d e r s o m e c r i ti c i sm , b u t i t r e m a i n s a n o p e nq u e s t i o n a s t o w h i c h is t h e m o r e f ru i tf u l li n e to t a k e . H e r e w e a d o p tt h e f i r s t .3 . 3 . M e a s u r e m e n t in th e B o h m T h e o r yN o w , t o il lu s tr a t e h o w m e a s u r e m e n t w o r k s in t h e B o h m t h e o r y , l e t u sc o n s i d e r b r ie f ly a s t a n d a r d d o u b l e - s li t e x p e r i m e n t . A c c o r d i n g t o th eB o h m t h e o r y , t h e p a r t i c l e i n f a c t p a s s e s t h r o u g h j u s t a s in g l e sl it . B u tt h e 0 - f ie l d , a n d h e n c e t h e q u a n t u m p o t e n t i a l , p a s s e s t h r o u g h b o t ha n d t h e o u t g o i n g w a v e s i n t e rf e r e . T h e i n t e r fe r e n c e p a t t e r n s e s t ab l is hc h a n n e l s o f a l l o w a b l e t r a je c t o r ie s , a n d a l t h o u g h t h e p a r t ic l e 's m o t i o nt o w a r d s t h e s li t is c o m p l e t e l y d e t e r m i n i s t i c - w a s a lw a y s in s i d e o n e o ft h e c h a n n e l s - i g n o r a n c e o f i ts i n it ia l p o s i t i o n p r e c l u d e s a n y d e f i n i tep r e d i c t i o n f o r t h e o u t c o m e . I t c o u l d h a v e b e e n i n a n y o f t h e ( v e r yc l o s e l y s p a c e d ) c h a n n e l s a t t h e s li ts . C e r t a i n p r o p o r t i o n s o f t h e s e c h a n -n e l s l e a d t o v a r i o u s p o s i t i o n s a t t h e s c r e e n , a n d o v e r s e v e r a l r u n s o ft h e e x p e r i m e n t , a p a t t e r n d e v e l o p s o n t h e s c re e n , r e f l e ct in g n o t t h es e l f - in t e r f e r e n c e o f w a v e - l i k e p a r ti c l e s, b u t t h e p r o p o r t i o n s o f c h a n n e l sl e a d in g t o v a r i o u s s p o t s o n t h e s c r e e n . O f c o u r s e , i f o n l y o n e s l i t


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7 4 V A S S I L I O S K A R A K O S T A S A N D M I C H A E L D I C K S O N

were open, the quantum potential would be quite different, and theexperimental results would reflect the resulting lack of interference.Furthermore, there is no 'collapse' of the wavefunction once theparticle hits the screen. The only sense in which one can say that a'collapse' occurs is an epistemological one - prior to the particle'shitting the screen, we had no idea where it was (except that it was inone of the channels), and after it hits the screen, we do know where itis. Thus our merely probabilistic knowledge of the location 'collapses'into definite information about the location. We may call this event an'effective collapse', keeping in mind that is not a physical event.

Moreover, the quantum potential itself retains all of the wavepackets(channels) - the channels where the particle is not do not disappearafter measurement. Bohm and Hiley (1984) show that after measure-ment these 'empty' channels will not affect the particle (via interfer-ence). The Bohm theory therefore avoids the need for an arbitraryapplication of the projection postulate; a measurement has nothing todo with the participation of human observers, but is instead an ordinaryphysical interaction.

The example above illustrates that the essential observational fea-tures of any measurement are contained entirely in the quantum poten-tial. In the double-slit experiment , the particles do not themselves haveany property which causes the interference patterns to arise - thosepatterns are due entirely to the quantum potential (given by the wavefunction). In fact, the only property really and intrinsically possessedby a particle is its position.Likewise, particles in a Stern-Gerlach spin measurement do notactually have a spin (under the version adopted here). Instead, thequantum potential allocates the proper number of channels to eachdetector (half to the spin-up detector, and half to the spin-down detec-tor), and the particles merely follow these channels. Correlations in theEP R- Bo hm experiment are guaranteed by the quantum potential - theonly allowed trajectories in this case are those where one particle movesas though it has spin-up and the other spin-down in the same spin basis.

This feature of the theory constitutes an ontological departure fromstandard quantum mechanics, where one assumes that (any mathema-tically conceivable) observables, as given by self-adjoint operators, cor-respond to real physical magnitudes possessed after measurement bythe measured system. Nonetheless, it appears to be an experimentalfact about observation - and a principle of the Bohm theory - that in


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t h e e n d a l l m e a s u r e m e n t s a r e r e d u c i b l e t o p o s i t i o n m e a s u r e m e n t s .T h e r e f o r e , i n sp i t e o f u n d e r l y i n g o n t o l o g i c a l d i f fe r e n c e s , t h e r e i s n oo b s e r v a t i o n a l d i f fe r e n c e . 4 I n d e e d , t h e f a c t t h a t t h e q u a n t u m p o t e n t i a lis c a l c u l a t e d i n e x a c t l y t h e s a m e w a y a s t h e u s u a l s o l u t i o n o f t h eS c h r / S d i n g e r e q u a t i o n ( i n p o s i t i o n r e p r e s e n t a t i o n ) p r o v i d e s t h e o b s e r -v a t io n a l a g r e e m e n t b e t w e e n t h e s ta n d a r d q u a n t u m m e c h a n i cs a n d t h eB o h m t h e o r y . ( N o t e , h o w e v e r , t h a t in t h e s to c h a s ti c v e r s i o n m e n t i o n e da b o v e , s u b q u a n t u m f l u c t u a t i o n s f o r c e i n p r i n c i p l e s l i g h t d i f f e r e n c e sf r o m t h e p r e d ic t i o n s o f s t a n d a r d q u a n t u m m e c h a n i c s .)

L e t u s n o w s u m m a r iz e t h e m a i n f e a t u re s o f m e a s u r e m e n t in th eB o h m t h e o r y :

B - M ( i ) M e a s u r e m e n t i s n o t a s p e c ia t p r o c e s s w h i c h d if fe r s ( d u e t oh u m a n o b s e r v a t i o n ) f r o m o t h e r p h y s i c a l i n t e r a c t i o n s .

B - M ( i i) T h e r e i s n o p h y s i c a l 'c o l l a p s e ' o f t h e w a v e f u n c t i o n a t a n yt i m e , b u t o n l y a n ' e f fe c t i v e c o l l a p se ' a s d e s c r i b e d a b o v e .

B - M ( i ii ) O b s e r v e d p r o p e r t i e s a r e t h e r e s u l t o f a p a r t i c l e ' s f o l lo w i n gt h e c h a n n e ls a l l o w e d b y th e q u a n t u m p o t e n t ia l . T h e o n l yp r o p e r t y p o s s e s s e d b y a p a r t i c l e i s i t s p o s i t i o n , a l l o t h e r sb e i n g b y - p r o d u c t s o f i t s tr a j e c t o r y .B - M ( i v ) A l l k n o w n p r e d ic t io n s f o r m e a s u r e m e n t - o u t c o m e s a r e t h es a m e a s f o r s t a n d a r d q u a n t u m m e c h a n i c s .

4 . C O M P A R A T I V E E V A L U A T I O N O F T H E T W O A P P R O A C H E SW h a t s t r i k e s o n e i m m e d i a t e l y i s t h a t b o t h a p p r o a c h e s a g r e e a b o u tB - M ( i ) , a b o v e . W e b e l i e v e t h a t i t i s i n p a r t t h i s a g r e e m e n t w h i c h B e l l( 1 9 9 0 ) h a d i n m i n d w h e n h e c a l l e d t h e s e t w o t h e o r i e s ' p r e c i s e p i c t u r e s ' .W e a r e in a p o s it io n n o w t o i n v es ti g at e t h e G R W P a n d B o h m a p -p r o a c h e s t o q u a n t u m m e c h a n i c s w i t h a v i e w t o w a r d s s u g g e s ti n g w h a ti s, a n d w h a t i s n o t , e s s e n t i a l to a p r e c i s e p i c t u r e o f q u a n t u m r e a l i ty .

4.1. Evolution and OntologyO n e f e a t u r e w h i c h c l e a r l y d i s t i n g u i s h e s t h e t w o t h e o r i e s i s t h a t t h eB o h m i n t e r p r e t a ti o n l e a v e s u n c h a n g e d t h e s t a n d a r d e q u a t i o n o f m o -t io n , w h e r e a s C S L d e p e n d s o n m o d i f y in g it. T h i s d i f f e re n c e h a s a p p a r -e n t l y o b v i o u s c o n s e q u e n c e s f o r t h e e v o l u t i o n o f p h y si c a l sy s t em s :B o h m i a n e v o l u t i o n i s d e t e r m i n i s t i c ; C S L e v o l u t i o n i s i n d e t e r m i n i s t i c .


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76 V A S S I L I O S K A R A K O S T A S A N D M I C H A E L D I C K S O NH o w e v e r , t h e d i f f e r e n c e b e t w e e n C S L a n d B o h m o n t h i s p o i n t i s n o ts o fu n d a m e n t a l a s m i g h t f ir st a p p e a r . W e h a v e a l r e a d y m e n t i o n e d t h ep o s s i b i l i t y o f a v a r i a n t B o h m t h e o r y t h a t p o s t u l a t e s a s u b q u a n t u ms t o ch a s ti c it y in t h e e v o l u t i o n o f q u a n t u m s y s te m s . M o r e o v e r , t h e r e i sa n e l e m e n t o f s to c h a s t i c i ty in t h e o r i g i n a l ( d e t e r m i n i s t ic ) v e r s i o n , d u et o t h e u n c o n t r o l l a b l e f l u c t u a t i o n s i n t h e q u a n t u m p o t e n t i a l . ( S e e 3 . 2 . )W e m a y s p e c u l a te a ls o t h a t e v e n t h e q u a n t u m p o t e n t i a l m i g h t b e m a d et o c o n t a i n a n a d d i t i o n a l e l e m e n t o f s to c h a s t i ci t y b y i n t ro d u c i n g a f o r c et e r m t o t h e S c h r r d i n g e r e q u a t i o n t h a t i s l a r g e o n l y f o r p r o c e s s e s i n v o l v -i n g m i n u t e d i s t a n c e s o f t h e o r d e r o f 1 0 - 13 c m , t h e s o - c a ll e d ' s u b - a t o m i cl e n g t h ' . H e n c e , a l t h o u g h a t a f i r s t g l a n c e t h e B o h m i n t e r p r e t a t i o na p p e a r s t o e m b r a c e d e t e r m i n i s m b y e m b r a c i n g t h e S c h r r d i n g e r e q u a -t i o n , t h e r e i s n o n e e d f o r i t t o d o s o . A s w e d i s c u s s e d i n 3 . 2 , w h a t i se s s e n ti a l t o t h e n o t i o n o f a p i l o t - w a v e i s n o t d e t e r m i n i s m , b u t c a u s a li t y .

O n t h e o t h e r h a n d , C S L d o e s m o d i f y t h e u s u a l Sc hr/S din ge r d y n a m i c st h r o u g h t h e a d d i t i o n o f n e w t e r m s . T h e s t a t e v e c t o r in t e r a c ts w i t h n e wp h y s i c a l v a r i a b l e s d e s c r i b e d b y w h i t e n o i s e s t o c h a s t i c p r o c e s s e s . A sm e n t i o n e d , i t i s t h e s e d i f f u s i o n p r o c e s s e s w h i c h p r o b a b i l i s t i c a l l y g u i d et h e s t a t e v e c t o r i n t o a p a r t i c u l a r ' p o i n t e r r e a d i n g ' s t a t e . P r o b a b i l i s t i cb e h a v i o u r i n th e G R W P p i c t u r e is t h e r e f o r e n o t a s s o c i a t e d e x c lu s iv e l yw i t h t h e a c t o f m e a s u r e m e n t . I t i s i n c o r p o r a t e d i n t h e e v o l u t i o n p r in -c i pl e o f t h e t h e o r y i n t h e f o r m o f r a n d o m s p o n t a n e o u s l o c a l is a ti o n s( j u m p s ) . H o w e v e r , n o e x p l a n a t i o n i s g i v e n a s t o h o w r a n d o m n e s s e m -e r g e s a t t h e q u a n t u m l e v e l o f d e s c r i p t i o n a n d c o n s e q u e n t l y t h e w a v e -f u n c t i o n i s a f f e c t e d b y t h e s t o c h a s t i c b e h a v i o u r o f s o m e t h i n g , s t i l l t ob e d e t e r m i n e d . I t s e e m s t h a t t h e G R W P s t o c h a s ti c d y n a m i c s re q u i r e sr e c o u r s e t o s o m e e x t e r n a l p h y s i c a l i n p u t i n o r d e r t o a c c o u n t f o r t h ed e f i n i te n e s s o f t h e w o r l d a t t h e m a c r o s c o p i c le v e l . S u c h a n i n p u t m i g h tb e p r o v i d e d b y c o n n e c t i n g t h e r e d u c t i o n m e c h a n i s m t o q u a n t u m g r av -i t y , a s h a s b e e n s u g g e s t e d i n t h e l i t e r a t u r e ( c f . D i o s i , 1 9 9 2 ) , a l t h o u g hn o v e l p r o b l e m s i n r e l a ti o n t o e n e r g y n o n c o n s e r v a t i o n m a y a r is e. 5 I n a n yc a s e , w h a t s e e m s c l e a r i s t h a t a l t h o u g h C S L i s a t p r e s e n t a n i n h e r e n t l ys t o c h a s t i c m o d e l , t h i s s t o c h a s t i c i t y m i g h t w e l l b e d u e t o p r e s e n t l y u n -k n o w n o r u n c o n t r o l l a b l e o r u n p r e d i c t a b l e e f f e c t s . I n t h a t c a s e , t h es t o c h a s ti c i ty i n C S L w o u l d b e g i n t o l o o k v e r y m u c h l i k e t h e s to c h a s t i c it yi n B o h m .B u t e v e n i f t h e t w o t h e o r i e s c o u l d a p p e a r m u c h t h e s a m e a s r eg a r d st h e s o u r c e o f r a n d o m n e s s , t h e i r d i ff e r e n c e as r e g a r d s o n t o l o g y l o o k si r re c o n c il a b le . W e m e n t i o n e d i n t h e i n t r o d u c ti o n t h a t t h e t w o m a i na p p r o a c h e s t o s ol v in g t h e m e a s u r e m e n t p r o b l e m a r e to m o d i f y t h e


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d y n a m i c s o r t o m o d i f y t h e i n t e r p r e ta t io n o f s ta n d a r d q u a n t u m m e c h a n -i c s . T h e B o h m i n t e r p r e t a t i o n p r e s e r v e s t h e s t a n d a r d d y n a m i c s a n dt h e r e f o r e m o d i f ie s t h e i n t e r p r e t a t i o n , a s s u m i n g th a t t h e s t a t e v e c t o rm u s t b e s u p p l e m e n t e d b y p r e c i s e p o s i t i o n v a r i a b l e s . O n t h e o t h e rh a n d , t h e G R W P m o d e l r e ta i n s th e o r d i n a r y in t e r p re t a ti o n o f th ew a v e f u n c t i o n a s p r o v i d i n g t h e m o s t a c c u r a t e d e s c r i p t i o n o f p h y s i c a lp h e n o m e n a , a n d t h e r e f o r e m o d i fi es th e S c h r 6 d i n g e r d y n a m i cs .O n e w a y not t o con s t r ue t h i s d i f f e r en ce is w i t h r e f e r en ce to ' h i dd env a r ia b l e s' . T h e B o h m t h e o r y is n o t a ' h i d d e n - v a r i a b le s t h e o r y ' i n t h eus ua l s ens e . I n gene r a l , a pos t u l a t ed ' h i dden va r i ab l e ' i s a h i t he r t ou n d i s c o v e r e d p h y s ic a l q u a n t i t y , b u t in B o h m , t h e r e a r e n o u n d i s c o v e r e dquan t i t i e s - pos i t i on is t he on l y phys i ca ll y f un da m en t a l va r i ab l e , an d i tis c le a r ly n o t ' h i d d e n ' , e x c e p t p e r h a p s f r o m " t h e e y e s o f t h o s e w h oi n v e n t e d q u a n t u m t h e o r y " ( B e l li n f a n te , 1 97 3, p . 8 ). F a r f r o m i n t ro d u c -in g t h e m , t h e B o h m t h e o r y eliminates m os t phys i ca l quan t i ti e s , r educ i ngth em to func t ions o f pos i t ion . (See the d i scuss ions in 3 .3 an d 4 .3 . )T h e e s s en t i a l ne w f ea t u r e , o f cou r s e , i s t ha t t he p os i t ion va r i ab l e s a r es u b j e c t t o th e a c t i o n o f t h e q u a n t u m p o t e n t i a l ( b e c a u s e , m a t h e m a -ti ca ll y, t h e y a r e a r g u m e n t s i n t h e w a v e f u n c ti o n ). T h e G R W P m o d e l ,o n t h e o t h e r h a n d , d o e s c o n t a i n i n i t s p o s t u l a t e d e q u a t i o n o f m o t i o nw h a t m i g h t b e c a ll e d ' h i d d e n v a r i a b le s ', n a m e l y , t h e r a n d o m p o t e n t ia l .H o w e v e r , h e r e t o o t h e t e r m i s n o t a p p r o p r i a t e , f o r h i d d e n v a r i a b l e sa r e g e n e r a l l y t h o u g h t t o b e p h y s ic a l q u a n t it ie s t o w h i c h k n o w n q u a n t i-t ie s a r e re d u c i b l e , a n d s u c h is n o t t h e c a s e in t h e G R W P m o d e l . I n d e e d ,t h e G R W P m o d e l d i f fe rs f r o m t h e B o h m t h e o r y in t h a t i t r e ta i n s all o ft h e p h y s i c a l q u a n t i t i e s o f s t a n d a r d q u a n t u m m e c h a n i c s a n d t h e w a v e -l i ke na t u r e o f quan t um r ea l i t y , i nc l ud i ng s upe r pos i t i ons . I t wou l d bemi s l ead i ng , howeve r , t o s ay t ha t t he r e a r e no s upe r pos i t i ons i n t heB o h m t h e o r y . W h i l e i t i s t r u e t h a t B o h m , u n l i k e G R W P , d o e s n o tc o u n t e n a n c e m a t e r i a l o b j ec t s in s u p e r p o s it io n s ta t e s , n o n e t h e l e s s l i n e a rc o h e r e n c e is b u il t in t o t h e q u a n t u m p o t e n t i a l, d u e t o i ts o b e y i n g th eS c h r 6 d i n g e r e q u a t i o n . B o h m p l a c e s a l l q u a n t u m b e h a v i o u r i n t o t h eq u a n t u m p o t e n t i a l, a n d l e a v e s t h e m a t e r i a l w o r l d a l o n e ( i .e . , c la ss ic a l) .W e s ha l l s ee i n t he nex t s ec t i on t ha t t he r e i s a si gn if ican t adv an t ag e t ob e g a i n e d b y d o i n g s o .

4.2. Recapturing Quantum Dynamics and the Classical LimitB e c a u s e o f t h e w a y i n w h i c h it u s e s t h e S c h r 6 d i n g e r e q u a t i o n s u p pl e-m e n t e d b y t h e c h a r a c t e ri s ti c B o r n p r o b a b i li ty r u l e a s f u rn i s h in g t h e


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78 VASSI LI OS KARAKOSTAS AND M I CHAEL DI CKSONd i s tr i b u t io n o f p a r t ic l e s in c o n f i g u r a ti o n s p a c e , t h e B o h m t h e o r y i s t h u sf a r e x p e r i m e n t a l l y i n d i s t i n g u i s h a b l e f r o m s t a n d a r d q u a n t u m m e c h a n i c s .T h i n g s a r e n o t s o c l e ar fo r C S L . T h e G R W P m o d i f i e d d y n a m i c s ca p -t u r e s t h e e m p i r i c a l c o n t e n t o f s ta n d a r d q u a n t u m m e c h a n i c s , i n al l c a s e si n w h i c h t h e l a t t e r h a s s o f a r b e e n c o n f i r m e d e x p e r i m e n t a l l y , t h r o u g ha c a r e f u l c h o i c e o f t h e n u m e r i c a l v a l u e s o f t h e t w o n e w p a r a m e t e r s aand i t (o r y - A ( a / 4 ~ ) - 3 /2 ) a p p e a r i n g i n t h e m o d e l . I t i s u n d e r s t o o d ,h o w e v e r , t h a t t h e v a l u e s o f t h e s e q u a n t i ti e s - q u a n t i t ie s w h i c h d e s e r v et o b e c a l le d c o n s ta n t s o f n a t u r e i f t h e G R W P d y n a m i c s is t a k e n a sd e s c r i b i n g f u n d a m e n t a l p h y s i c a l p r o c e s s e s - a r e n o t t o b e d e t e r m i n e dm e r e l y b y r e q u i r in g c o n s i s t e n c y w i t h k n o w n d a t a , b u t i n s t e a d s h o u l db e d e t e r m i n e d e x p e r i m e n t a l l y w i th a h i g h d e g r e e o f a c c u r a c y . 6 N a t -u r a l ly , o n e w o u l d e x p e c t t h a t a s y s t e m a t i c a n d p h y s i c a ll y m o t i v a t e dd e v e l o p m e n t o f s to c h a s ti c q u a n t u m d y n a m i c s s h o u ld b e r e l a te d t o s ta n -d a r d H a m i l t o n i a n d y n a m i c s t h r o u g h s o m e l i m i t i n g p r i n c i p l e .

W e e m p h a s i s e i n t h is r e s p e c t t h a t t h e li m it y ~ 0 i n w h i c h t h e m o d i -f ie d d y n a m i c s g o e s i n t o t h e u s u a l S c h r 6 d i n g e r e v o l u t i o n 7 w o u l d n o t d o .F i r s t , t h i s l imi t i ng p rocess i s comple t e ly c i rcu l a r b r i ng ing one back t ot h e p o i n t o f d e p a r t u r e , t h a t is t h e s t an d a r d q u a n t u m d y n a m i c s ; o n ep o s i t s s t o c h a s ti c it y in t h e w a v e f u n c t i o n t o a c h i e v e s t a t e v e c t o r r e d u c t i o na n d t h e n r e m o v e s i t u n j u s t i f i a b l y t o r e c o v e r t h e ' H a m i l t o n i a n l i m i t ' .The van i sh ing o f y (o r i t ) i s i n no way suf f i c i en t t o gua ran t ee t hep h y s i c a l r e l e v a n c e o f t h e l i m i ti n g t h e o r y . I f o n e is to s t u d y t h e l o g i c o ft h e p o s s i b l e l i m i t s o f a t h e o r y , o n e m u s t s t a r t f r o m t h i s t h e o r y i t s e l f ,a s i s e x p r e s s e d w i t h i n it s a u t o n o m o u s s y s t e m o f c o n c e p t s . I n o t h e rw o r d s , t h e G R W P t h e o r y n e e d s a n a r g u m e n t f o r w h y t h e u n iv e rs alc h a r a c t e r o f t h e s t o c h a s t i c e v o l u t i o n f a d e s o u t f o r s y s t e m s t y p i c a l l yg o v e r n e d b y H a m i l t o n i a n d y n a m i c s . S e c o n d , t h e li m it o f a c o n s t a n t toze ro i s neve r r ea l i s ed : ne i t he r y nor /~ a re ac tua l l y ze ro . Such a l imi t i ngp r o c e s s i s, a t b e s t , p h y s i c a ll y o b t a i n e d w h e n t h e r a t i o s o f c e r ta i n p h y s i -c a l q u a n t i t i e s a r e s m a l l ( o r l a r g e ) i n c o m p a r i s o n t o t h e c o n s t a n t w h i c hr e l a t e s t h e t w o q u a n t i t i e s . H o w e v e r , t h e y ( o r i t ) p a r a m e t e r d o e s n o to b v i o u s l y e n j o y a n y s u c h i n te r r e la t io n a l s ta t u s w it h in t h e G R W P c o n -t e x t . I t h a s b e e n a d d e d f r o m o u t s i d e t h e t h e o r y t o i m p o r t t h e r i g h tk i n d o f s to c h a s t ic i ty i n t o t h e e v o l u t i o n o f th e w a v e f u n c t i o n f o r a c h i e v in gs t at e v e c t o r r e d u c t io n w i t h o u t a p p r e c i a b l y d i st u rb i n g t h e q u a n t u m -mechanica l s t a t i s t i c s . As t h ings s t and , i t appea rs t ha t t he de t e rmin i s t i cH a m i l t o n i a n d y n a m i c s a n d t h e s t o c h a s t i c l o c a l i s a t i o n d y n a m i c s , r a t h e rt h a n b e i n g r e l a t e d b y s o m e g u i d i n g p h y s ic a l p r i n c ip l e , c o - e x i st b y d e c r e e


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i n th e e q u a t i o n o f m o t i o n . A g a i n , a p o s si b l e c o n n e c t i o n o f t h e i n t ro -d u c e d s t o c h a s t ic i t y t o t h e r e s t o f p h y s i c s ( e . g . , g r a v i t y , c o s m i c b a c k -g r o u n d r a d i a ti o n , e l e c t r o m a g n e t i s m ) m i g h t p a v e t h e w a y t o w a r d s am o r e p h y s ic a l ly s a ti sf y in g v e r s io n o f t h e G R W P t h e o r y . I n t h a t c a s e ,i t m a y b e p o s s i b l e t o r e t r i e v e t h e o r d i n a r y q u a n t u m d y n a m i c s f r o mC S L i n s o m e p h y s i c a l l i m i t .

B u t e v e n t h e n , C S L w o u l d i n h e ri t t h e s t a n d a r d p r o b l e m o f a tt a in i n gt h e c l a ss i ca l l im i t o f q u a n t u m m e c h a n i c s . W e s h a ll n o t e n t e r i n t o ad e t a i le d d i s c u ss io n o f t h a t p r o b l e m h e r e , b u t o n l y m a k e r e f e r e n c e t os o m e r e c e n t ( t h o u g h c o n t r o v e r si a l ) a r g u m e n t s i n c h a o s t h e o r y a c c o r d i n gt o w h i c h q u a n t u m m e c h a n i c s c a n n o t b e s h o w n c o n s i s t e n t l y t o e x h i b i tc l a ss ic a l s to c h a s t i c b e h a v i o u r , a n d n e i t h e r c a n i t s c l as s ic a l l im i t ( F o r d ,1 98 9). T h e g e n e r a l v a l i d i t y o f t h e c o r r e s p o n d e n c e p r i n c i p l e is h e r e a ts t a k e . I t a p p a r e n t l y b r e a k s d o w n in t h e c a s e o f n o n i n t e g r a b l e c l a s s ic a ls y s t e m s w h i c h s h o w c h a o t i c m o t i o n ( L a n e t a l. , 1 9 9 1) . N o t e , t h a t t h el a t t e r a r i s e s o u t o f t h e c o m p l e x i t y o f p h y s i c a l s y s te m s e x h i b i t in g as u f fi c ie n t d e g r e e o f in s t a b i li t y ( e . g . , m i x i n g , K - f lo w ) a n d i s n o t n o r m a l l yr e g a r d e d a s b e i n g o f t h e t y p e o f t h e l o c a li si n g s to c h a s ti c it y o f C S L .N e v e r t h e l e s s , o n e m a y a r g u e t h a t C S L h a s c l a s s i c a l c h a o t i c f e a t u r e sb u i l t i n d u e t o t h e i n c l u s i o n o f t h e w h i t e n o i s e r a n d o m p r o c e s s e s int h e e v o l u t i o n e q u a t i o n . H e n c e , a m a c r o s c o p i c o b j e c t in C S L , w i th i tsr e p e a t e d c o l l a p se s a s m e n t i o n e d a b o v e , m i g h t v e r y w e ll b e s u b j e c t t oc h a o t i c b e h a v i o u r , s in c e a r e p e a t e d l y l o c al is e d w a v e p a c k e t w il l f e e l as t o c h a s t i c f o r c e th a t r e s e m b l e s m o r e t h e k i n d o f f o r c e g iv i ng ri s e t oc la ss ic a l c h a o ti c b e h a v i o u r r a t h e r t h a n t h e s p a c ia U y a v e r a g e d f o r c e o fq u a n t u m b e h a v i o u r . T o t h e b e s t o f o u r k n o w l e d g e , h o w e v e r , t hi s q u e s -t io n r e m a i n s u n s e t t le d , a n d i n o u r o p i n i o n m o r e w o r k a t th e f o u n d a t i o n so f cl a ss ic a l a n d q u a n t u m c h a o s n e e d s t o b e d o n e i f f u r t h e r i n si g h t is t ob e g a i n e d i n s u c h i s su e s , sO n t h e o t h e r h a n d , i t is a n a d v a n t a g e o f B o h m ' s c a u s a l i n t e r p r e t a ti o n( o v e r b o t h t h e m o d e l o f G R W P a n d c o n v e n ti o n a l q u a n t u m t h e o r y ) th a ti t p r o v i d e s a s i m p l e a n d i n te l li g ib l e c r i t e r io n f o r a c o h e s i v e a p p r o a c h t oc l a s s i c a l b e h a v i o u r . ( W e a r e s p e a k i n g h e r e a b o u t a n o v e r a l l a p p r o a c ht o t h e c l a s s ic a l l im i t , a n d n o t a b o u t t h e s p e c i a l is s u e o f t h e s u p r e s s i o no f m a c r o s c o p i c s u p e r p o s i ti o n s . ) I n t h e B o h m t h e o r y , c la ss ic a l b e h a v i o u ra r is e s w h e n e v e r t h e q u a n t u m p o t e n t i a l t e r m i n E q u a t i o n ( 22 ) is n e gl ig i-b l e w it h r e s p e c t t o t h e o t h e r q u a n t i ti e s i n t h e e q u a t i o n o f m o t i o n . T h ev a l id i ty o f s u c h a n e g l e c t m a y b e a s c e r t a i n e d t h r o u g h a s t a n d a r d W K Ba p p r o x i m a t i o n , w h i c h i s p h y s i c a l l y p e r m i s s i b l e w h e n t h e w i d t h o f t h e


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8 0 V A S S I L I O S K A R A K O S T A S A N D M I C H A E L D I C K S O N

w a v e p a c k e t is m u c h g r e a t e r t h a n t h e w a v e le n g t h . U n d e r t h is a s s u m p -t ion ,

~2 (V ~ + V ~ ) R ( V ~ S ) ( V 2 S )( 24 ) ~ + - - ,2 m R 2 m 2 ma n d t h e a p p r o x i m a t i o n o f ( 1 8 ) b e c o m e s

(25) 0S + (V1S)z + (VzS)z + V = 0 ,Ot 2m 2m

w h i c h i s e x a c t ly a c la s si ca l H a m i l t o n - J a c o b i e q u a t i o n .N o t e t h a t t h e c r i t e r i o n i n v o l v ed in t h is ap p r o x i m a t i o n t o t h e c l a ss ica ll imi t d i spenses en t i r e ly w i th l e t t ing z~ ap pro ach 0 . P lan ck ' s c on s tan t isac t u a l l y a l w ay s co n s t an t , w h e r e a s t h e r e l a t iv e s i g n i fi c ance o f t h e q u an -t u m a n d c la ss ic a l p o t e n t i a l t e rm s i n t h e e q u a t i o n o f m o t i o n i s w h a t m a yc h a n g e w i t h c o n d i ti o n s . F o r e x a m p l e , it c a n b e e a s il y s h o w n ( B o h m e ta l . , 1 9 8 7 ) t h a t u n d e r sp ec i a l co n d i t i o n s o f l o w t em p e r a t u r e s , a s t h o sem e t i n t h e l a r g e- s c a le p h e n o m e n a o f s u p e r c o n d u c t iv i t y a n d s u p e r -f lu i d it y , t h e q u an t u m p o t en t i a l e f f ec t s d o m i n a t e o v e r t h o se o f it s cl as s i-ca l co u n t e r p a r t , m ak i n g i t c l e a r t h a t t h e c l a s s i ca l l i m i t i s n o t a l w ay sv a l i d a t t h e m ac r o sco p i c l ev e l , a s is ex p e r i m en t a l l y t h e ca se . T h e B o h mt h e o r y t h u s a p p e a r s t o p r o v i d e a s m o o t h t r a n s i ti o n b e t w e e n t h e c l as si ca la n d q u a n t u m m e c h a n i c a l e q u a t i o n o f m o t i o n , a n d m a k e s p h y si ca l s e n seo f th i s tr a n s it io n . T h e r e i s n o ' c u t ' b e t w e e n t h e s e t w o t y p e s o f e v o l u t io n ,a f e a t u r e n o t g e n e r a l l y a v a il a bl e in o t h e r i n t e r p r e ta t i o n s . ( F o r f u r t h e rd i scuss ion , see Dickson , 1994 . )

4.3. MeasurementI n b o t h C S L a n d B o h m , t h e r e is a p u r e l y p h y s ic a l m e c h a n i s m b y w h i c ho u t c o m e s f o r m e a s u r e m e n t s a r e s e l e c t e d . T h e r e i s n o n e e d i n e i t h e rt h e o r y f o r a n o b s e r v e r to m a k e t h e c h o i c e b e tw e e n S c h r 6 d in g e r d y n a m -ic s a n d t h e p r o j e c t i o n p o st u la t e . T h e r e s u lt is t h a t m e a s u r e m e n t in t h e s em o d e l s i s n o t a sp ec i a l p r o ces s r eq u i r i n g sp ec i a l co n s i d e r a t i o n , b u t i ss u b s u m a b l e u n d e r t h e s a m e e q u a t i o n o f m o t i o n u s e d f o r a l l p h y s i c a lp r o ces se s .I n B o h m , t h i s o b j ec t i v i t y i s a ch i ev ed b y e l i m i n a t i n g t h e p r o j ec t i o np o s t u l a t e a l t o g e th e r . W e h a v e a l r e a d y n o t e d t h a t i n t h e B o h m i n t e rp r e -t a t i o n t h e w a v e f u n c t i o n n e v e r co l l apses . 9


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I n G R W P , o b j e c t iv i t y is a ch i e v e d b y c o m b i n i n g S c h r 6 d i n g e r d y n a m -i cs a n d r e d u c t i o n d y n a m i c s i n t o a si n g le e q u a t i o n o f m o t i o n . T h e l a t t e rc o n t i n u o u s l y s t ri v e s t o c o l l a p s e t h e w a v e f u n c t i o n i n to o n e o f t h e a p p a r -a t u s e i g e n m a n i f o l d s . E v e n t u a l l y , t h e w a v e f u n c t i o n d o e s c o l l a p s e , o ra l m o s t s o . T h e e l u s i v e n e s s o f t h e a p p r o x i m a t e r e d u c t i o n p o i n t s to a ne m b a r r a s s in g s i tu a t io n w i th i n th e G R W P d y n a m i c a l a p p r o a c h : t h e C S Lr e d u c t i o n d y n a m i c s r e q u i r e s a m e a s u r e m e n t t o b e a r b i t r a r i l y l o n g i nd u r a t i o n . I n t h e t w o w a v e p a c k e t e x a m p l e o f 2 . 2 t h e i n it ia ll y su p e r -p o s e d s t a t e v e c t o r ( 1 3 ) is d r i v e n b y t h e CS L d y n a m i c s in t o a ' fi x e dp o i n t e r p o s i t i o n ' (M ~ o r M r ) o n l y i n t h e u n a t t a i n a b l e m a t h e m a t i c a l l i m itt ~ ~ ( K a r a k o s t a s , 1 9 9 4 ). F o r a n y f in i te t i m e t , n o o n e o f t h e p a c k e t se v o l v e d f r o m t h i s i n i t i a l s u p e r p o s i t i o n c a n b e i d e n t i f i e d w i t h a n e x a c te i g e n s t a t e o f t h e p o i n t e r ' s p o s i t io n o b s e r v a b l e . T h i s i s a d i r e c t c o n s e -q u e n c e o f t h e m o d e l ' s s to c h a s t i c ( w h i t e n o i s e ) a s s u m p t i o n s ( m u l ti p li c a-t io n o f th e s t a t e v e c t o r b y t in y G a u s s i a n r a n d o m p r o c e s s e s ) , a c c o r d i n gt o w h i c h f o r a l l v a l u e s o f t h e r a n d o m v a r i a b l e B(t) w h i c h h a v e a no v e r w h e l m i n g p r o b a b i l i ty t o o c c u r ( cf . E q u a t i o n ( 1 6 )) , t h e n o r m o f th ec o r r e s p o n d i n g n o r m a l i s e d s t a t e v e c t o r n e v e r g r o w s t o u n i t y f o r a l lt i m e s . F o r i n s t a n c e w h e n B(t) f l u c t u a t e s a r o u n d t h e v a l u e 2 r y t t h es q u a r e d a m p l i t u d e l ( ~ ' r [ ~ I ~ B ( t ) ) [ z a p p r o a c h e s t h e v a l u e 1 e x p o n e n t i a ll yw i t h t i m e , s o t h a t t h e a s s o c i a t e d s t a t e v e c t o r w i ll a l w a y s h a v e a n o n -z e r o a m p l i tu d e i n te r fe r in g w i t h t h a t o f th e l ~t ) c o m p o n e n t w h i c h d e c a y st o 0 f o r t ~ ~ H e n c e e a c h s t a t e v e c t o r i n t h e e n s e m b l e r e m a i n s a l w a y si n a s u p e r p o s i t i o n w h o s e e x i s t e n c e p r e v e n t s o n e f r o m a s s e r t i n g ( c o n -t r a r y t o m a c r o s c o p i c e x p e r i e n c e ) t h a t t h e p o i n t e r o c c u p i e s a d e f i n i t epos i t i on i n spa ce . 1

T h e i m p l i c at io n is t h a t i f o n e w i s h es t o a t t r i b u t e o b j e c t i v e p r o p e r t i e sto i nd iv idua l sys t ems i n f i n i t e t ime i n t e rva l s (norma l ly spec i f i ed by t her e a c ti o n t i m e o f t h e a p p a r a t u s a t h a n d ) , o n e h a s t o a c ce p t , a s G R W Pd o , t h a t t h e ' o u t c o m e r ' h a s b e e n o b j e c ti v e l y r e a li se d e v e n w h e n](XI'trlXI'tB(0)[2 i s no t exac t l y eq ua l t o 1 bu t s i gn if ican t ly c lose t o i t. 11

W e b e l i e v e h o w e v e r t h a t a s a t is f a ct o ry w a y t o r e s to r e t h e c o n c e p t o fi n d e p e n d e n t r e a li ty w i th i n t h e d y n a m i c a l r e d u c t i o n p r o g r a m m e w o u l db e , i n s t ea d o f a d o p t in g a n im p r e c i se c o r r e s p o n d e n c e b e t w e e n t h e o r et i-c al e l e m e n t s a n d r e s u l ts o f o b s e r v a t i o n s , t o c o n s i d e r t h e p o s s i b il it y o fa m o r e r e a l i s t i c ( t h a n w h i t e n o i s e ) s t o c h a s t i c s o u r c e w h i c h m a y l e a d t oc o m p l e t e s t a t e v e c t o r r e d u c t i o n i n c h a r a c t e r i s t i c m e a s u r e m e n t t i m e s .T h e r e a l i s a t i o n o f t h e f o r m e r p o s s i b i l i t y , h o w e v e r m a t h e m a t i c a l l y d i f -f ic u lt , is n o t i m p o s s i b l e a n d P e a r l e ( 1 9 9 3 ) s k e t c h e s a f o r m u l a t i o n o f CS L


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b y m e a n s o f w h i c h t h is a im m i g h t b e p u r s u e d . N o d e f in i te c o n c l u s io n h a sb e e n r e a c h e d h o w e v e r a n d f u r t h e r w o r k i s n e c e s s a r y if t hi s p r o j e c t ist o b e c o m p l e t e d .

I n a n y c a s e , i t s h o u l d b e c l e a r t h a t t h e r e i s n e v e r n e e d t o m a k er e c o u r s e t o t h e p r o j e c t i o n p o s t u l a t e i n C S L a n d B o h m . B u t b e c a u s et h e p r o j e c t i o n p o s t u l a t e i s t a k e n o u t o f o u r h a n d s , w e n o l o n g e r h a v et h e p o w e r t o p r o je c t o n t o t h e e i ge n m a n i f ol d s o f w h a t e v e r o p e r a t o r w ec h o o s e . I n s t e a d , t h e t h e o r ie s t h e m s e l v e s m u s t c h o o s e f o r u s . T h e y f a c et h e d a u n t i n g t a s k o f c h o o s i n g e i g e n m a n i f o l d s w h i c h w i l l a l w a y s sa t is f yo u r e x p e r i m e n t a l i n t e r e s t s . B o t h B o h m a n d G R W P p r o p o s e t o a n s w e rt h e c h a l l e n g e , a n d c h o o s e a ' p r e f e r r e d b a s i s ' w h i c h r e p r e s e n t s t h e b a s i so n t o w h i c h re d u c t io n t a k e s pl a c e ( G R W P ) o r t h e f u n d a m e n t a l q u a n t i-t ie s o f p h y si c a l r e a l it y ( B o h m ) . O f c o u r s e , t h e c h o i c e m u s t b e m a d ec a r e fu l ly - a t h e o r y i n w h i c h e v e r y m e a s u r e m e n t is f u n d a m e n t a l l y a na n g u la r m o m e n t u m m e a s u r e m e n t w o u l d n o t b e m u c h o f a t h e o r y .

I n t h e B o h m i n t e r p r e ta t i o n , t h e s e le c t i o n o f t h e p r e f e r r e d b a s is isi m p l i c it i n t h e u s e o f p o s i t i o n a s t h e f u n d a m e n t a l p h y s i c a l q u a n t i t y .B e c a u s e o f i ts p r i m a r y p h y s ic a l r o l e w i th i n B o h m i a n m e c h a n i c s , a llm e a s u r e m e n t o u t c o m e s - i n d e e d , a ll q u a n ti ti e s d i s c u s s ed in p h y s ic s -a r e d e s c r i b e d i n t e r m s o f p o s i ti o n . T h i s s e l e c t i o n is a m a t t e r o f p r in c i p l ei n B o h m ; i t f o l lo w s i m m e d i a t e l y f ro m t h e i n t e r p r e ta t i o n o f S a n d P .F o r t h i s r e a s o n , i n B o h m t h e p r e f e r r e d b a s i s h a s a m u c h w i d e r s i g n i f i -c a n c e t h a n j u s t d e t e r m i n i n g t h e b a s i s f o r r e d u c t i o n ; i t c h a r a c t e f i s e s t h ese t o f a l l pos s ib le in t r in s ic s ta tes o f a p hy s ic a l s y s te m . F o r B o h m , w h e nt h e s p i n m e t e r p o i n t s u p , i t t e l l s y o u n o t h i n g e l s e b u t t h a t t h e p a r t i c l ew a s a t th e s p i n -u p d e t e c t o r r a t h e r t h a n t h e s p i n - d o w n o n e . I f y o u p r e f e rt o s a y t h a t t h e p a r ti c le ' h a s' s p in + ~ / 2 y o u m a y d o s o. B u t t o B o h m ,such t a lk i s a t bes t e l lip ti ca l.12 Th e p laus ib i l i ty o f th is do c t r ine r es t s ont h e c l a i m t h a t i n t h e e n d , a l l o b s e r v a t i o n s a r e r e d u c i b l e t o p o s i t i o no b s e r v a t i o n s .

F o r C S L , m a t t e r s a r e s l i g h t l y m o r e c o m p l e x . B e c a u s e r e d u c t i o n i nC S L i s a r e a l p h y s i c a l p r o c e s s , t h e c h o i c e o f t h e e i g e n m a n i f o l d s t ow h i c h r e d u c t i o n o c c u r s is c o n s tr a i n e d b y p h y s i ca l c o n s id e r a t io n s . F o re x a m p l e , i f o n e c h o s e m o m e n t u m a s t h e r e d u c t i o n b a si s, t h e n t h ei n t e r n a l m o t i o n o f s e m i - r i g i d b o d i e s w o u l d b e m a s s i v e l y d i s r u p t e d b yt h e r e d u c t i o n p r o c e s s , d u e t o t h e e n s u i n g w i d e p o s i t i o n s p r e a d . 13 T h i sa n d s i m i l a r c o n s i d e r a t i o n s p o i n t s t r o n g l y t o w a r d s t h e C S L c h o i c e , e x h i -b i t e d i n E q u a t i o n ( 7 ). T h e r e s u lt is t h a t C S L a n d B o h m s h a r e a ' p r e fe r -e n c e ' f o r p o si t io n , t h e d i f f e re n c e b e i n g t h a t i n B o h m , p o s i t io n i s t h e


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U N O R T H O D O X F O R M U L A T I O N S O F Q U A N T U M M E C HA N I C S 83

basis of all physical quantit ies, while in CSL, i t is preferred for reduc-t ion, but is not fundamental in the same way.It is clear that the selection of a preferred basis affords a certainecono my to both theories ; i t al lows them to d ispense wi th the nee d fortwo types of physical evolut ion. But surely there is a pr ice to pay?Indeed. As one might suspect , Lorentz invariance does not restcomfor tab ly wi th the idea tha t pos i t ion a lone (as opposed to pos i t ion-t ime) enjoys a special rote in the space- t ime manifold. We turn now toinvestigate further.

4.4. Nonlocali ty and Lorentz Invariance4.4.1. Nonlocali tyOne useful way to examine the nontocal i ty in these models is in termsof t he Pa rame te r Independence (P I) and Ou tcom e Independence (OI )condit ions.14 PI says roughly that measurement outcomes at one detec-to r ( i n a s t anda rd EPR-Bohm expe r imen t , a GHZ expe r imen t , o r aHardy non-local i ty experiment) are probabi l is t icat ly independent ofparameter se t t ings a t another . Mathemat ica l ly ( for the EPR-Bohmexper iment ) ,

(26) Pr(rA = -+ 1IDA, D m if) = Pr(rA = + 1IDA,where rA is the resu lt at d ete cto r A, D~ is the setting at de tec tor i , and( is the co mp lete s ta te of the par t ic le-pair. OI says that ou tcom es atone de tec tor a re independent of ou tcom es a t the o ther de tec tor ,

(27 ) Pr( rA = --+ llrB = -+. 1, DA, D B, () =Pr(rA = +-- ll r~ W- 1, DA, DB, ~).Orth odox quantum mechanics ( in which ~ is the quantum s ta tevec tor )

clearly violates OI , bu t no t PI. T he B oh m the ory (in which ~" is the 0-field plus the pre cise position of each particle) violates PI. Th is violationis easi ly seen in the case of the two-sl i t experiment , discussed above,where a s l i t ' s being open or c losed can affect the quantum potent ia l ,and hence a measurem ent ou tcom e. A s imi la r e f fec t occurs in the E P R -Bohm exper iment , where a change in the parameter -se t t ing a t onedetec tor can immedia te ly a l te r the quantum poten t ia l a t the oppos i tedetector , thus al ter ing the motion of a par t ic le guided by the 0-f ie ld(Bohm and Hiley, 1984; Dewdney et a l . , 1987) . One might suppose


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t h a t t h is v i o l a ti o n o f P I w o u l d a l lo w f o r s u p e r l u m i n a l c o m m u n i c a t i o n .I n d e e d , S h i m o n y ( 1 98 4 ) h a s s h o w n t h a t v i o l a ti o n o f P I d o e s i n g e n e r a lp r e v e n t t h e p r o o f o f t h e n o - s i g n a ll in g t h e o r e m , a5 N e v e r t h e l e s s , t h eq u a n t u m p o t e n t i a l c a n n o t b e u s e d f o r si g na ll in g , b e c a u s e o f t h e i m p o s s i -b i l i t y o f s p e c i f y i n g p r e c i s e l y t h e c o m p l e t e s t a t e ( , a s w e d i s c u s s e d i n3.2 .

A n o t h e r w a y t o p u t t h e v i o l a ti o n o f P I is in t e r m s o f c o n te x t u a li t y .I n g e n e r a l , ' c o n t e x t u a l i t y o f a p h y s ic a l q u a n t i t y ' r e f e r s t o t h e d e p e n -d e n c e o f th a t q u a n t i t y o n t h e ' c o n t e x t ', o r t h e s t a te o f t h e e n v i r o n m e n tw i t h i n w h i c h i t r e v e a l s i t s e l f . M o r e s p e c i f i c a l l y , w h e n m e a s u r i n g t h ep h y s i c a l q u a n t i t y i n q u e s t i o n , o n e s o m e t i m e s r e f e r s t o t h e ' c o n t e x to f m e a s u r e m e n t ' , w h i c h in c l u d es , f o r e x a m p l e , p a r a m e t e r -s e t ti n g s o nd i s ta n t m e a s u r i n g a p p a r a t a ( t h u s th e c o n n e c t i o n b e t w e e n c o n t e x t u a l it ya n d n o n l o c a l i t y ) . I t is u s e f u l t o d i s t in g u i s h t w o t y p e s o f c o n t e x t u a l i ty ,' d i s p o s i t io n a l c o n t e x t u a l i t y ' a n d ' c a t eg o r i c a l c o n t e x t u a l i t y ' . 16 T h e f o r -m e r r e f e r s t o t h e m u n d a n e f a c t t h a t c e r t a i n d i s p o s i t i o n a l p r o p e r t i e s a r en o t r e v e a l e d e x c e p t u n d e r t h e r i g h t c i r c u m s t a n c e s . ( T h e f r a g i l i t y o fg l as s - i ts d i s p o s i ti o n t o b r e a k w h e n s t r u c k v i o l e n t l y - i s r e v e a l e d o n l yi f s t ru c k v i o l e n tl y .) I n t h e B o h m t h e o r y , s p in m a y b e c o n s i d e r e d a s th edisposition o f a p a r t ic l e t o m o v e o n e w a y o r a n o t h e r i n a m a g n e t i c f ie l d( d e p e n d i n g o n t h e p a r t i c l e ' s i n i t i a l p o s i t i o n ) , b u t t h i s d i s p o s i t i o n i sr e v e a l e d o n l y if t h e m a g n e t i c f i e ld is p r e s e n t , i . e ., o n l y i f o n e m e a s u r e st h e s p i n o f t h e p a r t i cl e . D i s p o s i t i o n a l c o n t e x t u a l i s m i s n o t d i r e c tl yc o n n e c t e d w i t h n o n l o c a li ty ; i t is c o m m o n i n e v e r y d a y e x p e ri e n c e . O nt h e o t h e r h a n d , c a t e g o r i c a l c o n t e x t u a l i t y m e a n s t h a t c e r t a i n p o s s e s s e d( c a t e g o r ic a l ) p r o p e r t i e s o f a p a r t i c le d e p e n d o n t h e c o n t e x t o f m e a s u r e -m e n t . F o r B o h m , ' p o s s e s s e d p r o p e r t i e s ' m e a n s ' p o s i ti o n - b a s e d p r o p e r -t i e s , ' a n d p o s i t i o n i s i n d e e d c o n t e x t u a l i n t h e B o h m i n t e r p r e t a t i o n . F o re x a m p l e , i n t h e E P R - B o h m e x p e r i m e n t , a p a r t i c l e ' s f u t u r e t r a j e c t o r yi n o n e r e g i o n d e p e n d s o n w h e t h e r a m e a s u r e m e n t i s p e r f o r m e d - a n di f s o , w h a t k i n d o f a m e a s u r e m e n t is p e r f o r m e d - i n t h e o t h e r r e g i o n .A s w e s a w , t h is d e p e n d e n c e is d u e t o t h e n o n l o c a li t y o f t h e q u a n t u mp o t e n t i a l , w h i c h i n t r o d u c e s a c o n t e x t u a l ( g l o b a l ) i n t e r d e p e n d e n c yamong the in i t i a l pos i t ion o f a g iven pa r t i c le , the in i t i a l conf igura t ions p a c e o f t h e w a v e f u n c t i o n a n d t h e s y s t em ' s H a m i l t o n i a n . H e n c e c a te -g o r i c a l c o n t e x t u a l i t y i n t h e B o h m i n t e r p r e t a t i o n i s i n t i m a t e l y l i n k e dw i t h t h e v i o l a t

Karakostas_Decoherence in Unorthodox Formulations of Quantum Mechanics

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