epistemological and mathematical foundations of quantum ... · epistemological and mathematical...

Foundations o f Physics, Vol. 7, Nos. 3/4, 1977

Epistemological and Mathematical Foundations of Quantum Mechanics

aerzy RayskP

Received July 12, 1974

The concepts o f measurement and measurable quantity are discussed. A probabilistie interpretation independent o f the arrow o f time is recommended and a defin#ion o f quantizabte physical systems is given. The space o f states o f information about the physical system is Schwarz space rather than Hilbert space.

1. THE CONCEPT OF MEASUREMENT IN QUANTUM MECHANICS

The concept of measurement in quantum mechanics has not become clearer over the years; on the contrary, much misunderstanding has accumulated around this concept in consequence of many unfortunate analyses by several authors in recent decades.

The most frequently encountered view is that - - in agreement with the probabilistic interpretation of the wave function--quantum mechanics is a theory of a manifestly statistical character and, therefore, only measurements performed on ensembles of systems (particles) and statistical assertions are physically meaningful. Moreover, several authors sharply discriminate between the concepts of initial measurement (preparation of an initial state) and final measurement and, consequently, formulate a probabilistic interpretation that is quite asymmetric with respect to time reversal and is valid only in one direction: from the known past toward the unknown future.

We emphatically disagree with the above-mentioned views and intend to point out the weak points, or even gross mistakes, in analyses leading to the above-mentioned conclusions.

1 Institute of Physics, Jagellonian University, Krakow, Poland.

151

© 1977 Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission of the publisher.

8 Z 5 / 7 / 3 / 4 - I

152 Rayski

Let us take, as a typical example, the concept of measurement introduced by Schwinger. m In the first chapter of his book he writes, "In the most elementary type of measurement an ensemble of independent similar systems is sorted by the apparatus into subensembles, distinguished by definite values of the physical quantity being measured. Let M(a') symbolize the selective measurement that accepts systems possessing the value a' of property A and rejects all others." From this excerpt the reader infers (i) that the measurement is an action performed on an ensemble, and (ii) it yields a presetected value, say a', as the result. Which one of the values a', a",.., the quantity A is able to assume resulting from such a "measurement" depends upon the type of apparatus being used.

No doubt, devices of the above type exist, and are called "filters." The point is, however, that--besides filters--other, more subtle devices are also possible, to be called "analyzers," An analyzer rejects nothing, accepts any one of the values a', a",.., the quantity A is able to assume, but, at the same time, registers which particular one of the cases a', a",.., occurred when a system passed through the analyzer.

It is the use of analyzers, not of filters, that constitutes a genuine measurement, because it answers the question of which one of a set of possible values

a quantity A possesses, and not which value we want it to possess. Using filters to yield a preselected value of an observable necessitates

the existence of a supply of many independent similar systems, because several systems may be rejected before an acceptable one will be encountered. On the other hand, a determination of a particular value of a physical quantity by means of an analyzer may be always achieved in the case of a single object, not necessarily an ensemble. Also, measurements of complete sets of observables (described by commuting operators and serving to determine the wave function) may be performed upon a single physical system, and not necessarily upon an ensemble. A "measurement" on an ensemble means in fact an ensemble of measurements, not a single act of measurement.

It is not difficult to see where the mistake comes from that so many authors, involved in the problematics of measurements in the microworld, make in identifying the determination of a wave function (or of the state vector) with a "measurement on an ensemble." They discuss the following problem: "Assume that a preparation of an initial state has occurred but the knowledge of the resulting state vector is lacking to an observer who would like to find out subsequently the state of the system." To this end he can measure an arbitrarily chosen (but complete) set of observables. As a result he will obtain, of course, a certain set of eigenvalues of the measured quantities but he is by no means entitled to say that the state was just the eigenstate of the actually measured observables to the actually obtained values. In order to find out which was the state prior to his encroachment

Epistemological and Mathematical Foundations of Quantum Mechanics 153

upon the system with his measuring device, he has to repeat his measurement several times, each time upon another member of an ensemble of similarly prepared systems; and then, from the statistics of his results, he can estimate all components of the state vector in the basis of the actually measured observables, i.e., determine the state.

We notice that the above problem consisted in determining a wave function (or a vector in Hilbert space) valid up to, but not tater than, the instant of its determination (except if by pure chance this state vector happened to be an eigenvector of the measured quantities). Thus, the above procedure was meant to give us some information about the past of the system. This discussion contributed to the view that the final measurement is something essentially different from the initial measurement, the latter using filters serving to prepare a preselected state for the futme. Moreover, from the above analysis it seems to follow that measurement in quantum mechanics is and must be a statistical affair.

However, both above-mentioned conclusions are completely false. The reason will become obvious on realizing that the state vector (or the wave function) is not an observable and, consequently, one does not measure it. The above procedure of finding out a previously prepared state vector must not be called a measurement if we want to avoid confusion. In fact, it is a statistical treatment of the results of many individual measurements, performed on separate systems (forming an ensemble), and it can lead to nothing but grave misunderstanding if one calls such a statistical treatment "measurement of the state of the system." The "state" or the "wave function" is not an observable but is a piece of information. One cannot "measure an information" but one can get some information by measuring something else. The concept and the name "measurement" should be reserved exclusively for the single acts of measurements of observables, performed on separate physical systems. More exactly, we should call "measurement" a result obtained by means of an analyzer but not of a filter, the latter accepting and/or allowing systems to emerge which are distinguished by some preselected values of the observables in question. The result of a genuine measurement should be completely independent of the will of the observer instead of yielding a preselected value of the observable being measured. If the latter alternative meant "measurement," then quantum mechanics would confirm at most itself (its internal consistency) without saying much about physical systems and their dynamics.

With the proper understanding of the word "measurement," as an action of an analyzer that yields some values of the (complete set of) observables, causing at the same time as little disturbance of the system as possible, we do not need to discriminate between initial and final measurements. Fach measurement of a complete set of observables yields some information about

154 Rayski

the dynamical situation of the system in the past as well as preparing a definite situation for the future.

With regard to the future the initial (complete) measurement determines the wave function whose probabilistic interpretation is well known: I t provides us with probability amplitudes for the results of(possible future measurements. But it plays also the role of a "future measurement" with respect to another wave function determined still earlier in the past.

Although the connection of the results of a measurement with the future as well as with the past is of a probabilistic nature, so that a full and conclusive confirmation of the totatility of probabilistic predictions can be obtained only by taking a statistics on ensembles, "~ this connection itself is in fact a property of each single system considered at different time instants) Contrary to the widespread opinion tha t - - in contradistinction to the classical theory of point part icles--quantum mechanics is a theory of a statistical character, whose predictions are meaningful only for ensembles, we claim that the epistemological situation is just the opposite: Quantum mechanics constitutes the first example of a theory dealing successfully with single microobjects, whereas classical mechanics is a statistical theory because it deals only with expectation values, i.e., with averages over quantum phenomena.

2. PROBABILISTIC INTERPRETATION SYMMETRIC UNDER TIME REVERSAL

By introducing as measuring apparatus analyzers causing a minimum of disturbance into the process of measurement (e.g., by avoiding the use of filters which absorb the "unwanted" particles), we avoid the introduction of irreversible processes into the act of measurement. Consequently, there is no difference in principle between the initial and the final measurements. i f this is so, we do not see any reason whatsoever why the probabilistic interpretation should "work" only in one direction of the time flow, i.e., allow us to predict the future probabilistically. In quantum mechanics there is no intrinsic reason for an asymmetry of interpretation.

Let us stress the following circumstance: Even if the Hamiltonian of interaction were not invariant under time reversal (as seems to be the case for the kaon interaction), probabilistic interpretation in both directions along the time axis should be possible, because the (complete) measurement at a given time instant, say t = 0, yields an intitial value for the solution

As a matter of fact, one would need an infinite number of systems in order to get a full confirmation of the statistical predictions, which is impossible! To see this, let us think of cases where the probabilities assume the value 0 or 1. Such predictions are already verifiable by experiments performed once, on a single system.


of the Schr~Sdinger equation not only for t > 0 but equally well for t < 0. A probabilistic interpretation symmetric with respect to time reversal was discussed previously, ~,~ so that we do not need to deat with it here in great detail. We shall recall only the main results of our previous m~alysis.

With each time interval closed by two complete measurements at tl and t2 there may be connected not a single but two wave functions, one serving for probabilistic predictions, the other for probabilistic retrodictions. These two wave functions are, in general, quite different from one another. The mere fact that with one and the same system in one and the same time interval two observers may associate two different wave functions shows how wrong it is to say that the wave function (or the vector in ~ space) represents the state of the system, or that the system is in the state ~. I f we stubbornly refuse to abandon the word "state," then we may use it, but in a different sense: The wave function represents the state of our knowIedge about the system, 4 achieved in a complete measurement performed at a single instant. Thus, the wave function is not an immanent characteristic of the system, but is a piece of information about the system. This information may be used for probabilistic predictions as well as for retrodictions about the evolution of the system. If the measurement were not performed on the system, the information would be lacking and we could not say that the system is in a "state ~//' which remains unknown but nonetheless exists.

From the above it follows that the concept of wave function contains a less naive sense of objectivity than it does according to the traditional view (suggested by the habit of saying that the wave function "determines the state of the system"). On the other hand, the new probabilistic interpretation, symmetric under time reversal, allows us to attach a more objective sense to the values of the parameters (observables) characterizing the system. To this end one has, above all, to define the meaning of the statement that "a quantity A possesses a value a' objectively."

Definition. t f the result of a measurement at the instant t may be made certain in another way, beyond this measurement (e.g., by some other measurement, earlier or later, performed on the same object, or even by a simultaneous measurement but performed on a different object, situated far away so that this measurement cannot disturb the object in question), then we say that the respective observable possesses this value objectively.

This definition of objectivity is not metaphysical, because it refers to measurements, too. Nevertheless, it goes beyond the frequently encountered view that the sentence, "A quantity A possesses a value a' at the instant t,"

Contrary to the opinion of Ballentine, ~ such an interpretation is not "subjective,'" because it is by no means knowledge about a human mind, but knowledge (or information) about the physical system!

156 Rayski

means nothing more than the sentence, "A measurement of A at the instant t yielded the result a'," but is in fact synonymous with it. The point is that science is not merely an amorphous set of direct results of measurements (observations), but establishes relations between them. A particular and highly characteristic type of interconnection between some pairs (sets) of different measurements may just serve to define a more subtle and more general sense of "objective existence" of a value of a physical quantity than the isolated, direct perception of the quantity.

According to the above definition, a particle can possess simultaneously and objectively some values of its coordinates and momenta with an exactitude surpassing the uncertainty relations. Indeed, we can measure at tl the particle's position and at t2 its momentum and then extrapolate the result from tz backward in time to the instant tl -- e. In this sense a particle possesses at the instant tl ÷ E both a well-defined position and momentum. This result does not contradict Heisenberg's uncertainty relations because the "surplus" of information concerns the time interval already closed by two measurements. On the other hand, it explains satisfactorily the "paradox" of Einstein and Podolsky (see Ref. 3).

The most general conclusion of the interpretation symmetric with respect to time reversal is that, in spite of, or rather just because of, the fact that the concept of state is not a fully and naively objective term of the quantum mechanical description (i.e., that the wave function does not characterize immanently the system, but rather our knowledge about the system), quantum mechanics yields a satisfactory description of physical reality which may be called an "objective description."

By saying that quantum mechanics yields a satisfactory description of physical reality, we mean "satisfactory in its main features." It does not mean that every detail of the description is fully satisfactory or perfect and cannot undergo a revision in the future. We have to keep open for the possibilities of some future improvements of the quantum formalism and its interpretation, The next sections will be devoted to a brief survey of those features of the quantum mechanical description that give rise to doubts and may be subject to refinement.

3. ovg-SPACE OR g-SPACE ?

In quantum mechanics it is assumed that the physical states are repre- sented by square-integrable wave functions, or, equivalently, by vectors in a Hilbert space and vice versa: To each vector in ~ there corresponds a possible state of the system. The question arises as to whether these statements are beyond doubt.


Besides the fact that one should say "the state of our knowledge about the system," rather than "the state of the system" (as has been explained in the foregoing sections), some other doubts may be raised: Each vector representing possible information about the actual dynamical properties of the system must belong to ~ space, but not necessarily vice versa. Or, in the language of representations by wave functions: Each wave function, being a physically meaningful solution of the Schr6dinger equation, must be square-integrable, but it is not excluded that this is a necessary but not yet sufficient condition. First of all, there is a correspondence argument: Classical values of physical quantities (which obviously must be finite) should corre- spond to expectation values of observables in quantum mechanics. Thus, the latter should be finite, too, for any physically meaningful state. Therefore, the condition

o < (4~, ~) < o~ (1)

seems to be too weak. In particular, besides (1), it may be necessary to require the expectation values of the Hamiltonian and its square to be finite

0 <~ (~b, H~b) < oo, 0 ~ (tt~b, H~b) < oo (2)

Inasmuch as the Hamiltonian is not a bounded operator, the conditions (2) constitute restrictions upon the space of functions {~b}. The second condition (2) seems to be of interest also, for the following reason.

In our opinion, one of the most fundamental postulates of the theory is the postulate of repeatability of measurements. A measurement of one and the same physical quantity, repeated after a short time, must yield, with a probability very close to certainty, the same result as the foregoing measurement. This postulate guarantees that measurements are not childish games with gadgets equipped with pointers and scales, but are genuine physical experiments. Since possible outcomes of measurements are connected with wave functions satisfying the Schr6dinger equation

H4~ = i ~4,/~t (3)

it may be not sufficient to assume ~b to be a function of class C1 of the variable t, but to require that

to) = [ + ( q ) ! 4 , ( t ) - ¢(to)l ~ A2( t, (4) d

where/~(q) means the measure in the space of the variables q] goes smoothly to zero, like (t -- t0f for t --+ to, and taking advantage of (3), one finds the second condition (2). This condition means a requirement of stability of the solution of the Schr6dinger equation.

158 Rayski

Inasmuch as the usually considered Hamiltonians are either polynomials in p and q or differ from a polynomial by a bounded operator, it seems plausible to restrict (J to be of class C~ (except perhaps for a set of points of measure zero) and such that, for each finite N, there exists

(~, q ~ ) < oo (5)

and

(~0 eN~b/eq N) = (--l)N(eN~h/eq N, ~) "< O0 (6)

The space of functions satisfying these requirements is the Schwarz space ~. There exists a reach class of Hamiltonians (bounded, or increasing like

a polynomial) for which the one-parameter group of time evolution determined by the Schr~Sdinger equation maps ~ onto itself.

4. CONSTRUCTION OF QUANTUM MECHANICS IN SCHWARZ SPACE

In order to construct a model of a dynamical theory, classical or quantum, one has to possess a priori some knowledge about the physical system to be submitted to a dynamical description. It must be possible to identify the system, i.e., to say whether two systems U and U' are identical or differ from each other. Each physical system is describable by means of a certain number of parameters, variable over some sets of real values, either continuous or discrete, but finite, and possibly also a certain number of additional parameters with finite sets of values. Inasmuch as a finite number of parameters with finite sets of values may be replaced by a single parameter (a set enumerated by an index assuming a finite number of values), the physical system may be described by a certain number of parameters ~: which assume infinite sets of values and eventually by one additional parameter ~ assuming a finite number of values.

A minimal number of parameters (sets of real numbers each belonging to R1) necessary and sufficient for the description of a given physical system is called the number of degrees of freedom f of this system. I f the number of degrees of freedom is finite, we have to deal with a mechanical system; if it is infinite, then we leave the territory of mechanics and go over to field theories.

Classical mechanical systems differ from quantum systems in that in the former case the sets of values of all parameters ~: always form a continuum, whereas in the quantum case some of the parameters assume infinite but enumerable sets of discrete values. Therefore in classical mechanics it is


sufficient to prescribe the intervals of variability of the parameters, whereas in quantum mechanics one is confronted with a difficult task of pointing out all the values these parameters might assume.

In classical mechanics the generalized coordinates q~ and their canonically conjugate momenta PI~ appear on equal footing (within the framework of Hamiltonian theory). Therefore it seemed that in quantum theory it would be the same (the more so as, historically, the Hamiltonian formalism played an important role in the discovery and in the preliminary formulations of quantum mechanics). However, this does not need to be so, in view of the fact that in quantum mechanics one often encounters cases where a generalized coordinate assumes a continuum, while its canonically conjugate momentum assumes a discrete set of values. In these cases it is not a priori certain that their roles must (or even can) be completely equivalent (i.e., whether by assuming p as coordinate and - -q as momentum, Q = p, P = - q , one will obtain an equivalent theory). The construction of quantum mechanics presented in this section is characterized by the fact that, within its framework, the roles of generalized coordinates and momenta appear equivalent if and only if both assume continuous sets of values.

Definition. A quantizable mechanical system is called a system with f (or possibly f + 1) degrees of freedom if it may be described in terms o f f continuous parameters ql ..... q~ to be called generalized coordinates 5,6 and possibly by one more parameter c~ capable of assuming a finite set of values.

In all known cases, and, it seems, the only physically interesting cases, the parameters assuming infinite but discrete sets of values either appear pairwise with quantities that are canonically conjugate to them and assume continuous sets of values (e.g., the z component of angular momentum possesses a discrete set of values but its canonically conjugate quanti ty-- the angle q~--assumes a continuum of values) or they cannot play the role of generalized coordinates at all (e.g., the energy). Therefore, for a quantum- theoretic construction we may choosefcont inuous parameters.

Having selected f (or possibly f + 1) generalized coordinates so that ql,.-., ql assume a continuum of real values in well-defined domains of

Not every physical quantity whose set of values forms a continuum may play the role of generalized coordinate. E.g., the energy of a free particle assumes a continuum of values but is not a generalized coordinate, because the momentum canonically conjugate to it is not well defined.

6 The measure in the space of generalized coordinates is settled by the fact that they describe sets of point particles in Euclidean space, subjected eventually to some holonomic- scleronomic constraints.

160 Rayski

variability and, possibly, a parameter ~, we may consider a set of rapidly decreasing functions of class C~

¢(ql ,..., qf, ~) ~ ¢~(ql .... , qf) (7)

forming a Sehwarz space. Let us consider one-parameter families of such functions ~b(q, t)

representing one-parameter sets of vectors in ~. We assume an action integral

t2 h ~ HI W([2 ' /1) = ~T,f" 1 d[ 2a f d[z(q) ~* (~ ~ @- / ¢ (8)

to be given, where/~(q) is the measure in the space of q, and H is a certain operator constructed of the operators of multiplication by q~ and partial differentiation with respect to qk, but which is at the same time a matrix with the number of rows and of columns equal to the number of values assumed by c~; thus,

H = H~B(q~, ~/Oqi) (9)

This operator must be linear and symmetric in ~, i.e., such that

(H~, V) = (¢, HV) (lO)

if % ~ belong to ~. The action integral (8) is, by definition, a scalar with respect to the

coordinate transformations

Q5 -- QJ(q,~) (11)

which are reversible and yield a one-to-one correspondence between a set of continuous parameters qk and another set of continuous parameters Qj.

The quantizable physical system is given if, besides the knowledge of a certain set of parameters qT~, ~, the operator H is known. Limitation of our considerations to ~ space imposes some restrictions upon the choice of H. These restrictions are part of the definition of a "quantizable physical system."

Quantum dynamics follows from the postulate called Hamilton's principle. It requires the vanishing of the first variation of the action integral

~W = 0 (12)

under independent variations ~b and ~b* (or independent variations of the real and imaginary parts of ~b), which vanish at tl and t2 and at the boundaries of the domain of integration, with respect to the coordinates q together with a sufficient number of partial derivatives. As a consequence of the above


principle, the SchrSdinger equation and its complex conjugate are satisfied. From now on the mathematical problems of quantum mechanics

reduce to a discussion of partial differential equations of the type (3). The remaining problematics concern a physical interpretation of the wave function and of the linear operator in ~.

5. THE P R O B L E M OF E Q U I V A L E N T R E P R E S E N T A T I O N S

The above description of quantizable systems is certainly not unique. If, e.g., one performs a transformation (11) to another set of continuous parameters, then the action integral turns into an equivalent form and an equivalent SchrSdinger equation is obtained. It is possible to perform, more generally, canonical transformations

Q = Q(q,p), P = P(q ,p) (13)

whereby P becomes - - i ~/~Q, but one has to limit oneself to such transformations that guarantee that the new coordinates Q will possess continuous sets of values.

Each permissible set of continuous coordinates (and with ~ assuming a finite set of values) may be called "representative physical quantities." Representative quantities are observables whose spectrum is continuous (with the exception of ~). The ~ spaces corresponding to different but equivalent choices of the representative parameters are isomorphic and may be regarded as constituting one and the same space.

On the other hand, transformations leading to a replacement of continuous by discrete parameters would mean a transition to another model, viz., matrix mechanics. But one may doubt whether matrix mechanics can be fully equivalent to SchrSdinger's mechanics. The point is that the laws of matrix algebra do not allow for an automatic transition from a finite to an infinite number of rows and columns. It is not always true that matrices associated with products of operators are matrix products of matrices representing the factors. If for the operators one has A B = C, then the matrix C,,~ is not necessarily equal to the product ~ AmkB~,~ if the number of rows and columns of these matrices is infinite. One is always allowed to introduce the unit operator of the form

1 = f d q l q > ( q [ (14)

but one is not always allowed to introduce ~_,7~]A~}(Ak I and to interchange the order of summation over the index k with other summations and integrations which possibly remain to be performed.

162 Rayski

6. MEASURABLES AND THEIR OPERATORS

The most conspicuous difference between classical and quantum mechanics consists in the fact that in classical mechanics the dynamical parameters of the system (generalized coordinates and momenta) always assume a continuum of values, whereas in quantum mechanics only some of the parameters assume a continuum of values, while others assume discrete sets of values.

In the case of continua it is impossible to determine a single, uniquely fixed value in the act of measurement; it is only possible to find out whether the Yalue ~: (belonging to the set of real numbers) lies in a given, finite interval or not. Such a "measurement" requires the use of an apparatus, with a finite power of resolution, which replies "yes" or "no" to the question whether the respective values lies in the given interval. In order to distinguish between such in-principle inexact measurements and exact measurements of those quantities that assume discrete sets of values, we shall call them "observations" and "measurements," respectively: The procedure by means of which one finds an answer to the question whether the value of an observed quantity belongs to a given interval of width A~: is called an "observation." On the other hand, a "measurement" is a procedure enabling one to obtain a single real number to be called "the value of the measured quantity."

A physical quantity that may be observed wilt be called an "observable." On the other hand, a quantity that may be measured exactly (i.e., there exists a prescription for constructing an apparatus and performing a reproducible experiment whose result will be a well-defined single number belonging to an enumerable set of real numbers) will be called "measurable." Each measurable is observable but not vice versa. The representative physical quantities are observables, but not measurables, because their spectrum is continuous.

Let us consider a class of all linear and symmetric operators in ~. Belonging to this class are operators of multiplication by the representatives q, the operators of differentiation with respect to them (if multiplied by an imaginary unit), and several functions of these operators. Moreover, Hermitian matrices acting as linear operators in the subspace {e~} also full in this class. A general operator is a matrix A~s whose elements are functions of ql~ and PI~ = -- i ~/a~.

Among these operators we distinguish a class of operators whose spectrum is enumerable, discrete, and real. We adopt the axiom that to each measurable physical quantity there corresponds a linear and symmetric operator in ¢ whose eigenfunctions belong to ~ and constitute a complete set in ~, such that the set of its eigenvalues is identical with the set of values that may be obtained in a measurement of the respective measurable.


Taking advantage of the symmetricity (Hermiticity), it is easy to show that the eigenfunctions belonging to different eigenvalues are orthogonal. Introducing a complete set of measurables, it is possible to remove degeneracy and to construct an orthonormal set of common eigenfunctions forming a basis in ~.

We assume that a (correctly performed) measurement of a measurable may yield, as a result, nothing but one of the eigenvalues of the corresponding operator. Thus, a measurement of a complete set ofmeasurables is equivalent to a determination of a function ~b(q, a) belonging to ~ (determined up to a constant factor).

From the well-known theorem according to which complete sets of common eigenvectors of two noncommuting operators do not exist, there follows the physical conclusion that simultaneous measurements (performed on one and the same physical object) of two measurables described by two noncommuting operators are impossible. Indeed, as a result of such measurements, one would obtain a common eigenfunction, which does not exist in general. The above impossibility should be understood, in that it is impossible to construct a measuring device (to point out a measuring prescription) serving the purpose of both measurements simultaneously. On the other hand, observables with continuous spectra may be observed simultaneously, with an exactitude limited by Heisenberg's uncertainty relation.

7. C O N C L U D I N G R E M A R K S

In the controversy between the "or thodox" and "statistical" inter- pretations, we prefer the former, i.e., we assert that quantum mechanics does not provide only a description of certain statistical properties of an ensemble, but, in the first instance, applies to single systems of the microworld.

Moreover, we have stressed the necessity of a less naively realistic understanding of the wave function (state vector). A wave function, or a vector, does not represent the state of the system as such but the state of maximal information about the system, derivable from a complete measurement (performed at a single time instant). This is not yet subjectivism; by no means!

Such an understanding of the wave function (state vector) resolves immediately all the difficulties connected with the discontinuous "reduction of the state vector" by the process of measurement. The discussions often encountered in the literature of the process of measurement by an analysis of the dynamical interaction between the microobject and another physical

164 Rayski

object--the appara tus--are in principle faulty. The roles of the apparatus as a measuring device and as a physical object, which may be included in the system that is being measured, belong to different categories and thus cannot be mixed up! This would lead to similar logical inconsistencies as, e.g., regarding a set {@ as another one of its elements ~.

Besides its interpretation, the formalism of quantum mechanics also seems to allow for refinement. There is a big difference between those observables whose spectrum is finite and those whose spectrum is discrete but infinite as well as those whose spectrum is continuous. The latter cannot be measured exactly and therefore are not genuine measurable quantities, but they are indispensable for the construction of the mathematical formalism of quantum mechanics.

R E F E R E N C E S

1. J. Schwinger, Quantum Kinematics and Dynamics (Benjamin, New York, 1970). 2. P. C. Bergman et aL, Phys. Rev. 134, B1410 (1964). 3. J. Rayski, Found. Phys. 3, 89 (1973). 4. L. Eo Ballentine, Rev. Mod. Phys. 42, 371 (1970).

epistemological and mathematical foundations of quantum ... · epistemological and mathematical...

Documents