defence of a measurement theory

IL NUOVO CI~IENTO VOL. 1 B, N. 1 11 Gennaio 1971

Defence of a Measurement Theory.

t I . K~ ies

Department o] Mathematical Physics, University o] Adelaide - Adelaide

(ricevuto il 30 Giugno 1970; manoseritto revisionato ricevuto il 3 Luglio 1970)

S u m m a r y . - - In this paper the author theory of measurement is defended against criticisms, by an examination of the role of the density operator in quantum mechanics. The SchrSdinger cat paradox, and the Einstein, Podolski, Rosen praadox are discussed.

I n t r o d u c t i o n .

This paper provides an extension to, and a defence of, the phi losophy behind the theory of measurement presented b y the author in four previous papers (1.4),

in the light of m a n y helpful criticisms received in p r iva te correspondence (5). One of the ma in criticisms has concerned the au thor ' s in terpre ta t ion of the den- s i ty operator. The reason for this seems to be a lack of clari ty in notat ion, because, cont ra ry to accusations, the au thor ' s in terpre ta t ion does not (at least initially) s t ray f rom the tradit ional.

The t radi t ional in terpre ta t ion will be t aken as saying the following. Each sys tem S at t ime t has a densi ty operator W(~), which is a positive-definite Hermi t i an operator of uni t t race in the Hi lber t space Ws containing the s ta te

(1) ]:L P. ](RIPS: Suppl. -Vuovo Cimento, 6, 1127 (1968). (2) It. P. KmPs: 2~uovo Cimento, 68B, 278 (1969). (a) H. P. KRIes: Nuovo Cimento, 61B, 11 (1969). (4) It. P. KRIPS: Philosophy o/ Science, 36, 145 (1969). (5) The author would particularly like to acknowledge the comments of Prof. M. BUNGle, Prof. C. A. HOOKER, Dr. P. J. VAN H~V, RDEN, Dr. A. JASS]~LETTE and Prof. W. WEmLICm

23

2 4 H. KI%IPS

kets of S. W~) can be diagonalized so tha t W(]) ---- ~PiJW} (~ [ , where {1~>} t

is an orthonormal (o.n.) set of state kets for S, p~ ~ 0, and ~p~----1. I t is

then taken as a postulate of quantum mechanics tha t if W(] ~ = ~Pi[q~}(~[ i

then there is a probabi l i ty p~ of S at t having (or being in) a state ket (or state) ]~i}. I f p~---- O,, then S at t is said to be in a pure state ]~i}. I f p ~@ 6 , , then the state of S at t is said to be mixed (or impure).

The importan~ point to note about mixtures (at the risk of stressing the obvious) is t ha t they even occur in classical theory ; e.g. if a penny is tossed then when it hits the floor there is a probabi l i ty �89 of it having a state in which heads face up, and probabi l i ty �89 of i t having a state in which tails face up. YVhat does not occur in classical theory are superpositions (in the Hilbert-space sense) of s ta tes- - th is point becomes impor tan t in the resolution of the first of the paradoxes to be considered.

1. - The two main problems confronting measurement theory today are neat ly demonstra ted by the two famous paradoxes ( 9 - - t h e SchrSdinger cat paradox, and the Einstein, Podolski, l~osen (E.P.I%) paradox (6).

For ease of discussion of these paradoxes and their resolution, we will assume the following idealized measurement process. Consider a system S in a pure state I~}----~c~]~), where {[~i)} is a complete or thonormal (c.o.n.) set of

eigenkets of a measured variable A (assumed nondegenerate). The set of values of A is given by the set of eigenvalucs ai of A~ defined by A[~}----a~[~}; and A is said to have the value ai for S at t if and only if S at t is in the state [~}. Le t M be the measuring apparatus for A, which interacts with S over the t ime interval It, t']. M at t is supposed to be in a pure state [~}, so tha t S - ? M at t is in the pure state I~} x IV}- The Hamil tonian H for S ~ - M during [t, t'] is supposed to commute with A, so tha t S ~ - M at t' is in the pure state ~ c , / ~ ) X l ~ , where i

M being in the state [~fi> is supposed to correspond to M registering the value ai on some sort of scale.

Wha t the Scln'6dinger paradox points out (4) is tha t S - ~ M at t r is in a

superposition of states of the form J~0~) x Ivan. The system M however is macroscopic, and therefore must objectively be in one of the lydia--not in a super-

(s) 2k. EINS~r]~IN, ]3. I~ODOLSKI and N. ROSE~: Phys. Rev., 47, 477 (1935).

DEF~NCE OF A MEASUREMENT THEORY 25

position of various [y~}. This follows from the classical concept of the state of a system, which only allows mixtures of states (see Introduct ion) .

The answer to this paradox lies in introducing the concept of a density operator into the picture. The density operator for S + M at t' is

(1) W'~+M Eei ~' ]~ 9i> <~9,, I X IV,> <~,'1 t ,f '

The densi ty operator for M only at t' can then be obtained b y using the rule for gett ing the density operator of ~ part ial system from tha t of the whole system. This rule was suggested originally by vo~ NEEMA~ (7), but its appli- cation does not commit one to the yon Neumann theory of measurement. Indeed yon :Neumann's theory of measurement is to be strenuously avoided, because of its appeal to humun consciousness us a deus ex mavhina--see later. Von ~eumann ' s densi ty operator rule simply states tha t W(,~,), the density operator for M at t', is given by T r sW 8+~ where (~ TrS~) denotes the operation of taking the trace in 9~ s. Therefore

(2) W(,~,, ]~ ~+@"

and hence M at t' is in a mixed state in which the various [~o,} are not super- posed bu t simply mixed. Thus the Schrddinger paradox is resolved, provided one accepts the consequences of the yon Neumann rule tha t a system can be in a pure state whereas one of its subsystems is in a mixed state.

The yon ~ e u m a n n densi ty-operator rule appealed to above has nothing to do with the yon Neumann measurement rule, which stipulates tha t the intervent ion of human consciousness at t' causes a discontinuous change in the state of S § at t' so tha t

For the reason stated previously, this la t ter rule of vo~r N~V~A~r is to be avoided at all costs.

The above simple resolution of the SchrSdinger paradox is purchased at a price however. I t is easily seen tha t

t

(7) j . vo~ NxU~A~: Mathematical Eoundations of Quantum Mechanics (Princeton, 1955).

26 ~. KRIPS

and hence, jus t as M at t ' is object ively in one of the ]~vi), we have S a t t ' object ively in one of the I~i). Fo r the measurement process to have been a

<( good one )) however (in some intui t ive sense of <( good ~)), we have the ext ra requi rement t ha t M registers the value ai (qua being in the s ta te I~0~)) a t t ' if and only if S is in I ~ ) at t. I t is not obvious f rom (1), (2), or (4) t ha t any such correlation holds; and hence the SchrSdinger paradox, suppressed a t one point , crops up in another form.

One obvious answer to this new pa radox is to introduce yon Neumann ' s

measurement rule, because if (3) holds then the requisite correlation holds. For the reason a l ready given however, this seems a bad way out. Ins tead, the solution to this pa radox seems to be in admi t t ing t ha t the requisite correlation does not hold exact ly, bu t does hold in an approximate sense.

This does, of course, mean t h a t good measurements are not always possible

in quan tum mechanics; bu t since approx ima te ly good measurements are always possible, this does not seem too grave a consequence. (By (~ an approx ima te ly good measurement )~ I mean one for which the requisi te correlation holds ap- proximately . ) An a t t e m p t a t a solution along these lines is made in (2), where it is shown tha t , under certain restrictions C, on H and the {]~>}, if one con-

siders the two cases where W~ +~ respect ively takes the forms given in (1) and (3), and if one t ime-averages w ~+~ --w~ over a var iable t ime interval about t ' in bo th these cases, then the respective t ime-averaged operators converge (in the strong norm) to the same l imit in bo th cases. A justification for the use of t ime- averaging can be found in (s), where it is shown t h a t errors in locating t ime co- ordinates of events are irreducible.

The me thod of approach suggested looks then to the na ture of the measuring appara tus to provide a resolution of the paradox. One advan tage of this method is the very fact t h a t it does make certain restrictions C on M; because the very general i ty of the systems employed as measur ing appara tus so far, is an

embar ras smen t ra ther t han an advantage. I t is an embar rassment because,

in the absence of physical restrictions on M, the theory of measurement is not falsifiable (in the Popper ian sense) and hence could be t aken as little more than a word game. The au thor is indebted to :BUNGE for point ing this out ; however, in reply, i t carl be said tha t even without providing the restrictions C on M, there is a point to measurement theory, viz. the point of showing t h a t a consistent (albeit not a highly falsifiable) measurement theory can be set up within the f ramework of q u a n t u m mechanics. Surely failure to give satisfaction on this point would weigh heavi ly against the completeness of quan tum mechanics; and therefore this aspect of measurement theory is not a t r ivial one.

An a l ternat ive answer to the above pa radox which seems to be in favour ('),

(s) G. I~. ALLCOCK: Ann. o/ .Phys., 53, 253 (1969). (9) j . M. JAUCH, E. P. W~G~]~R and M. M. YANASE: -N'UOVO Cimento, 48 B, 144 (1967).

DEFENCE OF A MEASUREMENT THEORY 27

consists of pointing out t ha t the difference between (1) and (3) could not be discovered except by measuring the submacroscopic structure of M (assuming the commuta t iv i ty of all macroscopic variables). Therefore, the argument runs, since such measurements are never in pract ice carried out, the difference between (1) and (3) can be ignored. This way out seems less satisfactory than the one suggested above however, because one would like to th ink tha t the correlation (albeit an approximate one) between M registering at at t' and S being in [%> at t', is objective in the sense tha t the correlation exists independent ly of any fu ture measurements.

In conclusion therefore, i t would seem tha t the SchrSdinger paradox (and the related paradox introduced above), can be adequate ly dealt with in the framework of the tradi t ional quantum mechanics, wi thout resorting to the

positivist solution suggested in the preceding paragraph.

2. - The point around which the E .P .R. paradox 1,evolves (4) is t ha t a case may arise for which

[ [

(5) w,% = ~p , l%~ <v,I = ~ p , lv,> <v~[,

where (Iv:>} (as well as (IV,>}) is an o.n. set of kets in Z~, and (p~} (as well as {p~}) is a set of positive scalars which sum to unity. In fact such a case does arise if and only if at least two of the Pi are equal (i.e. the spectrum of W~) is degenerate), since if p ~ p~ -~ p~, then

pilv,> <vii + p~]%> <wl = p(lv,> <v,] + Iv,> <v,I) = p(iv~> <v~l ~- [%~ <v',[), !

for any IV> and IV'c> which are orthogonal kets in the space spanned b y [Vi> and IV>. F rom (5) we get the strange picture tha t there are two coexisting bu t distinct (( modes >> for S at t'. In one of these modes, there is a probabi l i ty p~ t ha t S at t' is in ]W> for any i, and in the other mode there is a probabil i ty p~ tha t S at t' is in lye> for any i. The answer to this question cannot be <~ both ,>, if one adopts the convent ional restr ict ion tha t there is a unique s tate-ket for S at t'. The answer <( both )> does seem the most sat isfactory however (4), and we will therefore drop the conventional restriction. Then later on we will a t t empt to explain away the coexistence of the two modes as being some- thing quite natural . Thus the E .P .R. paradox will have been faced b y simply accepting its conclusion, and showing it to be nonparadoxical if one reframes the convent ional restrictions on state-kets (lo).

(10) The E.P.R. paradox was given an extra bite in its original formulation by a proof that the I~'~> can be position eigenstates and the [Soi> can be momentum eigenstates. In the author's opinion--spelled out in (4)--there is no additional paradox resulting from this proof. That Einstein did not place too much importance on this extra point is indicated by his letter to Popper--see The Logic o] Scienti[ic Discovery, by K. R. PoPPE~ (Hutchinson, 1968).

28 ~. KRIPS

We will now examine the existence of modes other than those due to the degeneracy of W~,). To do this it is convenient to work with the secondary in te rpre ta t ion rule tha t if W~,) = ~,Pi P,~ where Pi is a project ion operator onto

i

a subspace V~ (not just a single ket 19~}, as previously) of ~ s , p , p j = 0 for i S j, and p~ > 0, ~p~ = 1, then there is a probabil i ty p, of S at t' having a

state ket somewhere in V~, for any i, provided tha t (p~P,} is maximal in a sense to be explained. The sense of (( maximal >> used here is given by taking (p,t~i} as maximal if the members of no subset of (p,P~} sums to pP, for some scalar p and some projection P. In the case tha t (P~} is orthogonal, the condition of maxmal i ty reduces to tha t of p~ ~e p~ for i r j. Note tha t this emended interpreta t ion enables one to let the p, be zero (not just positive, as before), and hence to let the (P,} in general be complete. This was not possible under the old in terpre ta t ion because having several zero p, would have in t roduced the troublesome multiple modes.

There is a more serious paradox which is a generalization of the E.P .R. paradox. Suppose a system S at t has been prepared in a s tate [9~} with probabil i ty p: and in a s tate [9~} with probabi l i ty P2. I f (9~192} = 0 ther~ the density operator for S at t is P119,}(9~1 +P~[92}(92[, because this is the only form which W(~ ) can take which is consistent with the way S has been prepared at t. W ha t if (9~[92} ~ 0 however? The in terpre ta t ion of the densi ty operator does not determine W(~) in this case, in contrast with the case tha t (9:19~} = ~). Yet quan tum mechanics must be able to say what W(~) is in the second case, or else be accused of incompleteness. I t cannot be argued tha t there is no need in practice to say what W~) is in the second case, because i t is easy to imagine practical circumstances in which this second case would arise, given the usual sort of f requency interpreta t ion of probabi l i ty espoused in quan tum mechanics. For example, the case would arise if two beams of particles prepared in pure stntes ]9:} and [9~} respectively, were blended together in the proportions p: :P2, where (9~[92} r 0. To cut a long s tory short, the only form suitable for W(~) in this second ease seems to be P:I9:} (9:[ +P219~} (92J- That this la t ter operator is indeed positive-definite Hermi t ian with unit trace (which is a minimal requirement if it is to be identified as a densi ty operator), even though (9~19~} ~ 0, can be seen as follows.

Le t p~>0, ~ p~ = 1~ and set W ~ ~P~]9~> <9~[, where there are no restric-

tions on the (19~)} beyond (9i19~ = 1. Obviously Tr W = 1, zn4 W is Her- mitian. Therefore an or thonormal set of eigenkets (In)} exists for W corresponding to set of eigenvalues %~, where ~ ~ - - 1 . Therefore, by the spec- t ral theorem (::),

n

(::) N. DUNFORD and J. SCHWAnTZ: Linear 01oerators (New York, 1958).

D E F E N C E OF A MEASUREMENT THEORY 2 ~

The orthonormal i ty of the {In}} then gives

i

and therefore W is positive-definite t t e rmi t i an with uni t trace. The above suggestion for the form of W~> in the case (~11~} r 0~ can

however be seen to have the consequence of reintroducing the problem of a system which has multiple ~ modes >>, as follows. Any nontr ivial density operator (i.e. one for which p~ r ~,,) can be diagonalized in an uncountably infinite number of w~ys into ~p~P~, where {P~} is ~ complete maximal set of projections onto subspaces of J ~ , p~>~O, and ~p~ = 1 (:~). Therefore the same density operator will be used to describe aa uneount~bly infinite number of distinct situations; and hence, if the density operator is to be taken seriously ~s ~< defi~:ing >> the state of a system (in the sense tha t two systems are physically identical if and only if they have the same density operator), then all the distinct situations described by a given density operator must be considered a~ corresponding to coexisting modes of the same system. In other words the already extended tradi t ional in terpreta t ion must be extended further, so tha t if W~-~ ~.p~P~, whether (P~} is an orthogonal set of projections or not, then there is probabil i ty p~ of S at t having a s tate-ket somewhere in V~ (where V~ is the subspace of %zz onto which P~ projects). The condition of maximal i ty on the {p~P~} must be fur ther supplemented with a condition of linear independence of the {P~} to guarantee uniqueness of the p~. (l~ote tha t the above problem does not arise classically--for example, if we define the s ta te of a penny at t ime t' by s~ying tha t there is probabi l i ty �89 of it coming up heads ~nd probabil i ty �89 of it coming up tails, then there is no question of some ~< in between >> states --<~ hails >> or ~< tends >>-coming up !)

How ~re we to reconcile ourselves to the existence of these various modes? One w~y is to simply shrug them off by s~ying tha t if we perform a measurement on S ~t t, then ~ll the modes of S at t give the same result, and hence their existence c~n be ignored. Al ternat ively one could say tha t there is really only one m o d e - - t h e so-called natura l mode - - fo r S at t, which corresponds to the case where the {P~} s~re the orthogoual eigenprojections of the W~). This part icular mode is the (< natural mode ~> because i t is the only mode for which the actual probabil i ty of S ~t t having a state l e t in V~ is the same ~s the measured probabi l i ty of S at t having a s t ~ e l e t in V,. (I t is this last p roper ty of the

(12) In (~) it is proved that if W is any one of the uneountably infinite number of density operators for which l] W-- W~II < ~, for ~ small, then W(~ can be diagonMized, in the required fashion, into a weighted sum of projections which are characteristic of the eigcnkets of W. These projections are linearly independent; this becomes of importance later.

3 0 H. KRIPS

<~ natural ~> mode, which presumably consitutes the reason why the t radi t ional interpretat ion of the density operator only concerns itself with the (~ na tura l ~ mode.) The former of these alternatives is not really sat isfactory however, because it works only by shutt ing its eyes to what is a b la tant inelegance (qua rendundancy) in quantum mechanics. That elegance is indeed a fac tor to which eyes should not be shut is evident from the fact tha t one of the impor tan t features which separates a scientific theory from an exper imenta l mnemonic, is elegance (qua simplicity). The la t ter of the above alternatives is not satisfactory either, because it has to face the awkward possibility tha t a system ma y be prepared in a nomtatural mode, and then has to undergo a discontinuous and instantaneous change of modes into the natural mode

(cf. the <~ reduct ion of the wave packet ~). There is however a third way of reconciling oneself to the above paradox,

but one which involves a radical change in the way of looking at quan tum mechanics. This new way of looking at quan tum mechanics will not differ empirically from the old, since it has been suggested only to solve an isolated conceptual problem. Nevertheless, the replacement of the old with the new way of looking at things, may have some empirical justification in so far as the new way may suggest new generalizations of (as opposed to replacements for) quantum mechanics in areas, like field theory, where only part ial success has so far been achieved. Even without this la t ter minimal enlpirical justification however, the new way is surely justified if it can solve a conceptual problem which the old way cannot. Fur thermore , the new way will be seen to pu t quantum mechanics in a common framework with the tradit ional classical theories, thus making it easier to compare quan tum mechanics with the t radi t ional classical theories. This in turn makes it easier to answer contentious questions like ~, Is quantum mechanics deterministic? ~ or <~ Is quantum mechanics objective? ~ (We will not pursue these questions here.) The new way is in t rodnceh as follows.

In classical theory the state of S at t is defined b y asserting a set of sen- tences of the form <, The variable X has the value X~ for S at t' ~, for each of a set of so-called state variables X, where there is a unique X~ for each X, S, t. Imagine now a classical theory for which there is the following ra ther peculiar (but not obviously nonclassical) measurement theory. Within the set C of s ta te variables there is a subset C', whose members we will call <~ in principle measurable variables~. To each X, X~C' , there corresponds an apparatus M~, which, when interacted with S at t, registers one of a set of values X~, with probabi l i ty p(X~). Suppose however, tha t it was not allowable to deduce, f rom the fact tha t M~ registers the value X~ say, tha t X has the value X1 for S at t. (This could be due to errors cropping up in the measurement, for example.) Instead, suppose tha t what one must do to find out the state of S at t, is to do calculations on a subset of the set of p(X~) for varying X and ~. (This might be possible because, given enough part ial information about each of the various

D E F E N C E OF A M E A S U R E M E N T T ~ E O R Y ] l

interactions of S at t with the Ms, one can piece together enough information to deduce the actual state of S at t.) Suppose fur thermore tha t the calculations indicate tha t there exists one of the X for which the interact ion of S at t with Ms was actually a measurement, in the sense tha t S at t actual ly had the value X~ for X if and only if M~ registered the value X~. (This would happen if the calculation indicated that , by coincidence, the experimental error for measuring one of the X happened to cancel out to zero, for the special case of S at t.) Tha t part icular X, for which interact ion of Mx with S at t turns out to actually be a measurement, we denote <( X(S, t)~>. I t is impor tan t to note tha t in the classical theory being imagined here, the part icular member of C' which turns out to be X(S, t) is not predictable without knowledge of the state of S at t; and tha t any one of the in principle measurable variables may turn out to be actually measurable for S at t.

Fur thermore , imagine tha t this already peculiar classical theory is even more peculiar in tha t there exists a set C", C"c C, of which the members are not in principle measurable (within the envisaged measurement theory), bu t for which i t can be inferred, f rom the state of S at t, tha t the variable Y has the value ] z for S at t, with probabi l i ty p(Y~), for all Y e C". (A similar si tuation to this exists in classical statistical mechanics, where there is no measurement theory for the molecular momenta or positions, and yet one assumes tha t these lat ter state variables do have definite values. Fur the rmore one infers tha t the probabi l i ty distributions over these values have a certain f o r m - - t h a t given by the (~ postulate ~ of equal a priori probabi l i t ies--on the basis of certain measurable macroscopic variables, like pressure and temperature , exhibiting equilibrium behaviour.) Lastly, imagine tha t there is a set C", C'~'c C, of which the members are those members of C which are neither in C' or C". The point in imagining this last set of variables, is t ha t it will now be supposed tha t the set C" (but not C') may be different for different S and t. In particular, for any Z, Z ~ C H', there is a possible S and t for which Z e C".

The point of asking" the reader to imagine the above theory is tha t i t will be seen to provide a model for a new way of looking at quan tum mechanics, in which the last of the above-mentioned paradoxes does not appear.

The suggestion is to adopt the above (~ classical ~) theory (without its paren- thetical comments) as a model for quan tum mechanics, by, firstly, identifying the class C with the class of complete sets of projection operators onto subspaces of 5~f s (with one fur ther qualification to be mentioned). X(S, t) is then identified as tha t (necessarily unique) set {P~} for which W~,)~--Zp, P,, pi>/O,

i

~p~ = 1, P~P,, = 0 for i r i', and {p,P~} is m~ximal. The class C' is taken

as the class of complete sets of orthogonal projections, and the M~, corresponding to (P~}, is taken as tha t apparatus which we tradi t ional ly call (( the measuring apparatus for the {P~} ~>. (More usually perhaps, one talks about the measuring apparatus for the variable A, which corresponds to the Hermi t ian

3 2 m K R ~ S

operator A for which ( ~ i } constitutes the set of eigenprojections, i.e. for which AP, = a~_P~, where a~ =/= a~, for i # i', for all i, i '.) To just i fy this identifica- t ion one does, of course, need to show just what are the calculations on the various p(X~), for varying X and g, which tell us the s ta te of S at t. They are as follows.

Le t the state of S at t be given by a density operator W(~), operating on an iV-dimensional (finite) Hi lber t space 5~f ~. Le t {1i}} be a e.o.n, set in 5~ s. Then, applying the tradi t ional Born interpretat ion, observat ion of the interactions of S at t with the various M~, enables us to read off the values of s for any }~} e 5~ z. (The Born interpreta t ion equates (~[ W(~)I~}, with p(~), which is the probabi l i ty tha t the variable A, corresponding to a nondegenerate t t e rmi t ian operator A, takes the value a for S at t, where AI~} = a l ~ } . I t is assumed tha t probabilities e~n be read off f rom observing frequencies over ensembles-- this is a t least in principle possible, i.e. it is possible given a count- able infinity of operations.) Bu t [~} = 5 % I i}, and Wh) : ~ ~o,,[i} (i '], where

l,i' the % can be calculated f rom (i]~}; and therefore we have ~ set of equations for the co,,:

F rom the (A)ii, , the W(~) can then be calculated. When IV becomes infinite this calculation remains in principle, if not in practice, possible.

The class C" is taken as the class of sets of projections {P~} which are not in C', bu t for which W,~)~- ~p i~ ) i , pi •O, ~ i = 1, aIld (PiPi} is maximal.

t This identification is justified by the interpreta t ion of this last equation, viz. tha t there is probabil i ty p~ of the state ket of S at t being in the subspace for which /)j is the project ion (see earlier). Finally, any residual members of C, not in c ' oi" c", are t aken as comprising the class C".

The one fur ther qualification to be made about the membership of C is t ha t {P~} E C only if the {P~} are l inearly independent. The reason for this (men- t ioned earlier) is tha t linear independence of the {P~} is equivalent to unique- hess of the p~ in the expression ~ Pi Pi, which in tu rn is necessary if the Pi

t are to be in terpre ted as bona fide probabilities.

Wi th the above suggestions (which are extensions of those in (2)) i t imme- diately follows tha t the last of the paradoxes mentioned above, disappears for the following reason. The possibility of decomposing W~) in various ways no longer entails tha t there are various (~ modes )~. Ins tead this possibility simply arises f rom the fact tha t there are several state variables, each of which takes a value for S at t. Thus there is no paradox, just as there is no paradox in the fact t ha t in classical theories the state of a system is determined b y giving the values taken by several (not just one) variables. A_u interesting side effect of this new way of looking ~t quantum mechanics, is t ha t it no longer

DEFENCE OF A MEASUREMENT THE0~Y SS

makes sense to say <( S a t t has a s ta te ]~p) ~, jus t as i t makes no sense in clas-

sical t h e o r y to say (( S a t t has a va lue 40 }>. I n s t e a d one m u s t say (( S a t t

has a s ta te ]9) out o] the set o] subspaees (V~} }> (just as one m u s t say classically

(~ S u t t has the va lue 40 ]or the momentum variable (say) in e.g.s, units (say) }>).

Thus the new w a y of looking a t q u a n t u m mechanics is an extens ion of the

school of t h o u g h t accord ing to which i t is t he dens i ty opera to r (not the s ta te

ket) which is of p r i m a r y significance.

The a u t h o r g ra te fu l ly acknowledges the assis tance of Prof . C. A. HuI~sT,

over the course of m a n y discussions.

@ R I A S S U N T O (*)

Si difende dalle eritiehe 1~ teori~ della misura esposta preeedentemente dall 'autore. Allo scope si esamina il ruolo dell'operatore densits nella meee~niea quantistiea. Si diseutono il paradosso del gatto di SehrSdinger e quello di Einstein, Podolski, Rosen.

(*) Traduzione a cura della Redazione.

3an~HTa TeopHH H3Mepetmth

P e s m M e (*). ~ B 3TOH CTaTbe IIpOBO~RTC~I 3am~aTa TeOpt4H H3MepeHI4~I aBTopa ~pOTrm

KpltTIIKII , IIocpe~CTBOM ~ c c n e ~ o B a n n ~ p o y m o n e p a T o p a IIYlOTHOCT/~ B EBaHTOBO~I Mexa-

nnre. O6cy~rnaroTcz napa~oKc IIIpe~nHrepa c romKoft a napa~oKc ~rttuTefma, YIo- ~O.rlbCKOrO, Po3eHa.

(*) 1-1epeaec)eno pet)atatuert.

3 - I I N u o v o Cimenfo B .

defence of a measurement theory

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