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1 Contrary to double quotation marks ("), single quotation marks (') are not used here in a derogatory sense, but rather to indicate the unity of an expression often consisting of a long string of words.
2 This definition of `faithful measurement' needs to be subjected to the qualification dealt with in this remark.
3 One might wish to restrict the arbitrariness of the labeling for reasons of correspondence, suggesting the labels am to obey certain relations. For instance, energy and momentum conservation in individual collision processes implies linear additivity of the corresponding quantities, which could be implemented by choosing the labels accordingly.
4 Usually the `strong form of the correspondence principle' is not presented separately, but introduced as a silent presupposition of the complementarity principle for standard observables. However, for didactic and semantic reasons I prefer to present the `correspondence between a quantum mechanical observable and an experimental measurement procedure' as a methodological principle valid independent of the notion of `incompatibility'. Moreover such an independence is required if the correspondence principle is applied also to generalized observables, for which `incompatibility' does not prevent their commeasurability.
5 In order to avoid normalization problems of improper eigenvectors I shall restrict myself mainly to observables having discrete spectra, like spin or polarization observables.
6 Note that with Bohr the classical description was the physically interpretable one, quantum mechanics being necessary to yield the restrictions that should be imposed in order to comply with quantization phenomena.
7 As a physicist I think not to be obliged to draw a sharp distinction between `ontic' and `ontological', nor between `epistemic' and `epistemological'. Since in physics anything referring to the `science of something' also refers to that `something' I feel justified to restrict myself to the expressions `ontological' and `epistemological', unless the alternative expressions are necessary for historical reasons (compare footnote 12).
8 With respect to quantum mechanical observables Bohr's attitude is less clear. He avoided the term `quantum mechanical observable', instead using the terms `physical quantity' and `variable', the quantum mechanical formalism being thought to be ``a purely symbolic scheme permitting only predictions, on lines with the correspondence principle, as to results obtainable under conditions specified by means of classical concepts.'' Although this might seem to point into the direction of an `instrumentalist interpretation of quantum mechanical observables' as well, Bohr's emphasis on a `classical account of the results of quantum measurement' can be understood only if to a measurement result `classical intersubjective reality' is attributed.
9 Here `observer' should be read as `observer including his measuring instruments' (compare).
10 I prefer the term `subquantum theories' because `hiddenness of variables' has largely lost its relevance after logical positivist/empiricist preoccupation with `observability' has become obsolete.
11 Usage of the term `anti-realism' in the sense defined here is rather confusing, because it is bound to be interpreted in the sense of the `realism versus idealism' dichotomy, denying also the phenomena an observer-independent existence. Nevertheless, I will stick to the terminology because it is the one employed in most of the physical literature, when referring to a belief in the existence of a `world described by hidden variables' as `realism'.
12 For historical reasons I here deviate from the general rule agreed upon in footnote 7.
13 Note that by a certain abuse of language `local causality' is often referred to as `locality'.
14 It is fair to note here that the possibility of a quantum mechanical description of quantum mechanical measurement processes was soon contemplated by Heisenberg (compare). However, by emphasizing the classical features of the macroscopic side of the measuring process its microscopic side tended to remain underexposed or even ignored. Originally, the Heisenberg cut was meant to be the dividing line between `observer' and `microscopic object', the measuring instrument even not being explicitly taken into account. Von Neumann's projection postulate was the result of an attempt to demonstrate the possibility of a measurement scheme consistent with the possibility of moving the Heisenberg cut between input and output of the measuring instrument. During a long time `necessity', or even `practical realizability' of such a scheme was seldom considered.
15 Another reason to choose the term empiricist interpretation rather than `operationalist interpretation' is that the qualification `operationalist' has been claimed by the `operational approach' advocated by Busch, Grabowski, Lahti, and Mittelstaedt, who during a long time have tried to implement the `generalized formalism of POVMs' in a realist way opposed to this `empiricist' one. `Operationalism' as endorsed in the `empiricist interpretation' is closely related to the operationalism advocated by Kraus.
16 Here `free evolution' may encompass e.g. `interaction with an electromagnetic field' if the latter is treated non-dynamically.
17 Note that Heisenberg clearly expressed allegiance to these (Aristotelian) ideas only much later in his 1955/56 Gifford lectures. Around 1925 Heisenberg was probably still thinking in terms of the `ontology of classical physics'. For this reason I keep referring to Jordan's assertion rather than to a `Heisenberg-Jordan one'.
18 As a matter of fact, Schrödinger was fooled by happening to consider a particle in a harmonic oscillator potential, which yields an exceptional case having solutions without dispersion.
19 For instance, the neutron interference experiments discussed in Publ. 27.
20 This interpretation is closely related to Popper's propensity interpretation, although not necessarily implying the latter interpretation's `dependence on the experimental arrangement' (since there does not exist any theorem, generalizing the Kochen-Specker theorem0, forbidding to attribute `probabilities of measurement results, to be obtained if the measurement is actually carried out' to a microscopic object as objective (non-contextual) properties). Perhaps we should look upon Popper's propensity interpretation (as suggested by him in chapter I.3 of Quantum Theory and the Schism in Physics) as a correction of the subjectivist tendencies to be observed within the Copenhagen interpretation, rather than as an interpretation of probability fundamentally opposed to the Copenhagen instrumentalist one.
21 When we disregard colour, all men are equal (rather than more unequal).
22 Note that in general the transition |ψ> → |ψm> is not a projection since in general it is not idempotent.
23 It is not improbable that Einstein has been inspired by the Compton-Simon experiment (compare) in which measurements are performed on two correlated objects (viz. an electron and a photon) after these objects have been interacting.
24 Remember that I restrict myself to non-degenerate spectra, thus ignoring the possibility to generalize this expression to Lüders projection.
25 A non-existence proof by Wigner (Am. J. Phys. 31, 6 (1963)) requires the existence of an additive constant of the motion of the system object+measuring instrument.
26 As a `preparation principle' the notion of `von Neumann projection' can be generalized to `measurements of the second kind' (cf. Publ. 47), be it that, due to lack of idempotency, it then is a reduction or collapse rather than a projection.
27 Note that the reasoning is independent of the distinction between Bohr's epistemological view and von Neumann's more ontological approach.
28 It should be noted that this preference can be reached independent of `Schrödinger's cat paradox'. Actually the `ensemble interpretation' does not yield the final solution to that paradox. It is able to do so only through the empiricist interpretation (of which it is an essential property). The latter interpretation is immune to the `cat paradox' because the state |ψcat> simply does not have a place in it. Note that the cat is nothing but a measuring instrument monitoring a `microscopic event' (i.c. the decay of a nucleus) different from the `event that is directly observed' (i.e. `the cat being either dead or alive' are the pointer positions taken into account in the `empiricist interpretation').
29 Although I consider von Neumann's interpretation as a variation of the Copenhagen one, I have no quarrel with anyone wishing to consider it a separate interpretation. It all depends on one's definition of the `Copenhagen interpretation' (see also footnote 65).
30 The `empiricist interpretation' has much in common with the `structuralist approach of theories' as developed by Sneed, Stegmüller, Balzer and Moulines. In particular, the existence of a `domain of (intended) applications' is a useful assumption exploited by this approach.
31 Remember my restriction to a realist interpretation of the state vector. Note that with Bohr's instrumentalist view the problem would not arise, since Bohr attributed only an epistemological significance to the quantum mechanical state vector, allowing him to treat a state vector and a density operator to represent equivalent information on an individual object.
32 In order to express the peculiarity of this phenomenon a comparison with `dancing Wu Li masters' has been made.
33 Von Neumann's idea of `homogeneity' can be seen as a translation of the Copenhagen idea of completeness in the restricted sense into `ensemble' language.
34 A possible measurement of this kind could be a Stern-Gerlach measurement in which the lateral position X of the atom is taken as the pointer observable. In such an experiment the corresponding momentum observable Px could be measured at a longitudinal position where the two beams have been duly separated, but the atom has not yet reached the particle detector measuring X, i.e. while the measurement process is still in the pre-measurement phase.
35 In this section I do not distinguish between two different versions of the `realist interpretation', viz. the `individual-particle version' and the `ensemble version', which will be considered here. Moreover, it should be realized that the term `realist interpretation' refers to a broad class of interpretations encompassing such different ones as Schrödinger's wave interpretation and Heisenberg's realist individual-particle interpretation endorsing ontological probability.
36 ``Damit ist nicht gemeint, dass das Korrespondenzprinzip etwa eine Brücke zwischen der Quantentheorie und der klassischen Theorie sei, denn eine Versöhnung zwischen beiden Theorien ist überhaupt nicht denkbar; das Korrespondenzprinzip besagt nur, dass in der Quantentheorie eine Korrespondenz zwischen intra-atomistischer Bewegung und ausgesandter Strahlung vorhanden ist, die eine weitgehende Analogie mit der in der klassischen Elektronentheorie auftretenden Korrespondenz zwischen der Bewegung elektrischer Teilchen und der von diesen ausgehender Strahlung aufweist.'' (H.A. Kramers, Naturwiss. 11, 550-9 (1923), p. 551).
From this quotation it can be inferred that what is meant by `correspondence' may be even less well-defined than the `correspondence of descriptions of a phenomenon by different theories' (viz. the correspondence between `intra-atomic motion' and the `radiation that is produced by it'), our `correspondence' being referred to as an `analogy'.
37 In general I will not draw a terminological distinction between a standard observable and its mathematical representation by the Hermitian operator A, referring to the observable as `observable A'. Fortunately, this sloppy neglect of the difference between ontology and epistemology is less consequential than the analogous identification of `state' and `wave function' often is.
38 In my Publ. 52 and Publ. 53 I have not referred to this form of `correspondence' because of a restriction to `quantum mechanics proper'.
39 The term `operational' is used here in the `empiricist' sense referred to in footnote 15, rather than in the sense of Bridgman's operationalism0 (which fits into the logical positivist tradition, each observable being defined by the measuring operations used).
40 Actually, von Neumann's `quantum mechanical approach of quantum measurement' had not the intention to provide an unbiased account; it just had as a goal to prove the consistency of a `description of measurement in which the measuring instrument is ignored as an intermediary between microscopic object and observer' with a `description in which the measuring instrument is explicitly dealt with'. His projection postulate achieved this goal.
41 For instance, in the `empiricist interpretation' a `nonideal measurement of observable A' is considered as a `nonideal version of a measurement of that same observable A' rather than as a `measurement of an independent observable B ≠ A' (contrary to Bridgman's proposal to consider `measurements corresponding to distinct measurement arrangements' as measurements of `different quantities'). Another reason not to equate Bridgman's operationalism with the `empiricist interpretation' is that, with Bohr, he considers a measurement event "unanalyzable".
42 Note, however, that this does not imply `endorsement of scientific realism': by choosing the characterization `empiricist' it is expressed that in my view logical positivism/empiricism has not become obsolete because it was not sufficiently objectivistic-realist (as is the allegation of scientific realism), but because, notwithstanding its professed empiricism, it still had too much of a `realist flavour', introduced by a too classical view on `measurement' (compare).
43 The neutrality of the notion of `correspondence' with respect to the `direction of the mapping between theory and reality' may be useful to reconcile the different ways the notion of `interpretation' is used by philosophers and physicists.
44 It is important to note here that the notion of `state' Bohr had in mind was the `classical notion (q,p)' rather than the `quantum mechanical state |ψ'. Failure to recognize this is an important source of confusion.
45 ``A new background for the attitude towards such problems was, however, created by the discovery of the quantum of action in the first year of our century, which revealed a feature of individuality in atomic processes going far beyond the ancient doctrine of the limited divisibility of matter.'' (N. Bohr, in Essays 1958-1962 on Human Understanding, INTERSCIENCE PUBLISHERS, New York, London, 1963, p. 24). Also: ``[... the] essence [of the formulation of the quantum theory] may be expressed in the so-called quantum postulate, which attributes to any atomic process an essential discontinuity or rather individuality, completely foreign to classical theories and symbolized by Planck's quantum of action.'' (N. Bohr, The Quantum postulate and the recent development of atomic theory, Nature (Supplement) 121, 580-590 (1928), p. 580.)
46 See e.g. the revealing `Note added in proof' in Heisenberg's 1927 paper on `The physical content of quantum kinematics and mechanics', to the effect that according to Bohr's new insights the characteristic difference between classical and quantum mechanics is not exclusively a result of `discontinuity' but that particle-wave duality/complementarity should be taken into account.
47 ``...there is ... no question of a mechanical disturbance of the system under investigation...'' but ``there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system.'' (N. Bohr, Phys. Rev. 48(1935), p. 700).
48 From the theory of quantum tomography0 it is known that for that purpose an infinite number of standard observables has to be measured. Observables Q and P are not complementary in the sense defined here, even though they are maximally incompatible.
49 ``Speaking, as it is often done of disturbing a phenomenon by observation, or even of creating physical attributes to objects by measuring processes is liable to be confusing, since all such sentences imply a departure from conventions of basic language which even though it can be practical for the sake of brevity, can never be unambiguous.'' (N. Bohr, in: New Theories in Physics, International Institute of Intellectual Co-operation, Paris, 1939, p. 24).
50 The `logical-positivist/empiricist aversion of metaphysics' should be distinguished from an `anti-realism negating the existence of a reality behind the phenomena'. To 'deem hidden variables not apt to be discussed scientifically' need not imply to `exclude their possible existence'. It is just as unscientific to identify these two views as to deny the `possibility of internal vibrations of a billiard ball' on the basis of the existence of the `classical theory of rigid bodies'.
51 A trivial solution is given by the product of the probability distributions of the separately measured observables. However, this solution is not thought to have physical relevance because it does not reflect any correlation obtaining between the observables.
52 Reference to `uncertainty' may be somewhat misleading because of its suggestion of `uncertainty about the sharp value allegedly possessed by an observable'. Yet, I will stick to the usual terminology, even though perhaps `indeterminacy principle' would be a better term.
53 I here ignore theories like that of Ghirardi, Rimini, and Weber0, attempting to explain strong von Neumann projection rather than the weak projection considered here. In an ensemble interpretation there is no reason to consider the strong form.
54 ``There exist, therefore, mutually exclusive though complementary [emphasis added, WMdM] experiments which only as a whole embrace everything which can be experienced with regard to an object. This idea of complementarity [emphasis in the original, WMdM]...'' (Max Born, The statistical interpretation of quantum mechanics, Nobel Lecture, December 11, 1954, p. 266). From this quotation it can be seen that this feature has even been considered the primary constituent of `complementarity'. In view of footnote 48 emphasis may have shifted toward the `uncertainty principle'.
55 Since von Neumann's reference to `psychophysical parallelism' does not necessarily imply that he attributed an active role to the human observer in realizing `strong projection', we cannot attribute to him an equally harmful role as it is possible to attribute to Wigner. Nevertheless, by ignoring in his "proof" of the arbitrariness of the position of the Heisenberg cut the fundamental difference between the `microscopic interaction of object and measuring instrument' and the `macroscopic interaction of measuring instrument and human observer' von Neumann may yet have contributed to a certain mystification of `quantum measurement'.
56 For instance, Dirac: ``The new theory, which connects the wave function with probabilities for one photon, gets over the difficulty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs.''
57 L.E. Ballentine, The statistical interpretation of quantum mechanics, Rev. Mod. Phys. 42, 358-381 (1970).
58 Note that it is not always clear whether this is thought to be a consequence of `complementarity', or whether its origin is already contained in the quantum postulate, position and momentum not having simultaneously sharp values because they do not separately have them (compare). As a consequence of the classical reasonings required by Bohr's strong correspondence principle it often appears that an object is thought to initially possess well-defined values of position and momentum, which values allegedly cannot be simultaneously measured as a consequence of the complementarity principle.
59 Note that, in view of failure of the `possessed values principle', `simultaneous sharp values' can in general not be attributed to `quantum mechanical measurement results'. However, this need not be relevant if quantum mechanics would not be complete in the wider sense (compare).
60 A more general derivation is indicated here.
61 Here `objectivity' should be equated with non-contextuality. Note that in the following a distinction will be drawn between `(objective) hidden variables λ' and `(contextual) elements of physical reality determining quantum mechanical measurement results'.
62 It actually is sufficient to assume the existence of `quadrivariate relative frequencies', the requirement of the existence of the limit N →∝ of relative frequencies not being necessary if realistic measurements are involved (compare).
63 For this reason de Broglie's idea that this wave should be described by a solution of the Schrödinger equation (even if thought to be a `singular one', different from the `regular solution representing the wave function') does not seem to have any physical basis.
64 This analogy might be substantiated by the quantum electrodynamical view of an elementary particle (e.g. an electron), not as a `point particle', but possessing an `extended sphere of vacuum fluctuations' which might play the role of a bow wave (here it should be remembered that quantum electrodynamics too should be interpreted as describing an ensemble; hence, the vacuum fluctuations of quantum electrodynamics account for statistical averages of `individual de Broglie waves').
Note that, strictly speaking, this picture to a certain extent justifies the often-heard assertion that `the electron passes through both slits', although probably the main part of the electron is passing through one slit only.
65 Note that `weak projection' has become manifest only by considering the pre-measurement phase in which measurement is dealt with in a `completely quantum mechanical way in general not addressed by the Copenhagen interpretation'. Von Neumann's `strong projection' was fitting into that interpretation because it seemed to be in a correspondence-like manner consistent with a classical account of measurement, `classical point particles' being replaced by `wave packets'. This illustrates von Neumann's problematic relation with the `Copenhagen interpretation' as well as the problematic relation of the `Copenhagen interpretation' with a `quantum mechanical description of measurement'.
`Weak projection' could be ignored by von Neumann (while endorsing `strong projection') because he did not have in mind a `full-blown quantum mechanical description of quantum measurement, but he just wanted to illustrate the arbitrariness of the position of the Heisenberg cut if `strong projection' is assumed (cf. section 3.2.2 of Publ._52).
66 ``The last remarks apply equally well to the special problem treated by Einstein, Podolsky and Rosen, ..., and which does not actually involve any greater intricacies than the simple examples discussed above.'' (N. Bohr, Phys. Rev. 48 (1935), p. 699).
67 Another confusing use of the contrasting terminology `objectivity versus subjectivity' will be discussed here.
68 I will in general not follow Bohr's terminology but refer to `observables' also in case of Bohr's `physical quantities' (compare footnote 8), moreover, for reasons mentioned there attributing to Bohr a `realist interpretation of quantum mechanical observables' (as is usual in classical physics) rather than an instrumentalist one.
69 I follow here the somewhat sloppy way `resolutions of the identity' and `POVMs' are usually indicated by the same symbol although these concepts are not identical, a POVM comprising all operators obtainable by addition of elements of the `resolution of the identity'.
70 Note that at the time of the EPR discussion the Kochen-Specker theorem did not yet exist. Actually, the reason why the EPR proposal was rejected by the physics community was different, viz. `abhorrence of metaphysics', which I consider equally dubious (compare), mathematical proofs (like that of the Bell inequality) being based on `presuppositions referred to as quasi-objectivity' which are related to `Einstein's assumption of objectivity'.
71 It is often heard that experimental violation of the Bell inequality would be experimental evidence of nonlocality. This statement is analyzed here and here. A simple reasoning why `violation of the Bell inequality by standard quantum mechanics' is not related to `nonlocality' is given here.
72 Note that for obtaining these conclusions it is not necessary to assume that the Heisenberg-Kennard-Robertson inequality is the mathematical implementation of the idea of `mutual disturbance in a joint measurement of incompatible observables'.
73 A. Fine, Hidden variables, joint probability, and the Bell inequalities, Phys. Rev. Lett., 48, 291 (1982); P. Rastall, The Bell inequalities, Found. of Phys. 13, 555 (1983).
74 Note that this contradicts the often-heard `"refutation" of counterfactual definiteness' to the effect that `` unperformed experiments have no results'' or ``values of unmeasured observables would not exist'' (although they do exist in a rather trivial way, compare).
75 Note that in the mathematical literature `stochastic matrices' are sometimes defined in such a way that the `nonideality matrix' is the `transpose of a stochastic matrix'.
76 Remember that I restrict myself here to the case of non-degenerate spectra. A more general expression, valid for non-maximal standard observables, can be found in the publications referred to.
77 For instance, R. Horodecki, P. Horodecki, and M. Horodecki, Violating Bell inequality by mixed spin-1/2 states: necessary and sufficient condition, Physics Letters A 200, 340-344 (1995).
78 For instance, Ryszard Horodecki, Pawel Horodecki, Micha Horodecki, and Karol Horodecki, Quantum entanglement, Reviews of Modern Physics 81, 865-942 (2009).
79 A more direct proof makes use of the easily proven inequality
80 H. Maassen and J.B.M. Uffink, Phys. Rev. Lett. 60, 1103 (1988).
81 An obvious exception is, for instance, observation of a stick partly immersed in water, that is seen as if being broken.
82 `Non-contextuality' could be equated with `objectivity' if the latter notion is changed from `independence of the experimenter/observer' into `independence of the experimenter/observer including his preparing and measuring instruments'. However, in order to avoid confusion of `contextuality' with `subjectivity' it seems preferable to avoid reference to `objectivity' in case it should be understood in an ontological sense. Note, however, that for historical reasons I will maintain the term objectivistic-realist interpretation rather than change it into `non-contextualistic-realist interpretation'.
83 I think Popper (chapt. I.3) is right when he concludes that Born in his publications used the term `statistical' where nowadays `probabilistic' would be used. Although this is not evident from Born's scientific publications, it can be inferred from The Born-Einstein Letters, exhibiting the fundamental misunderstandings between Born and Einstein, the latter maintaining a statistical interpretation in the sense of reducible probability, whereas Born seems to have favoured irreducibility. The confusion was aggravated because both Born and Einstein used a subjectivistic terminology, be it that with Born `lack of knowledge as expressed by the quantum mechanical standard deviations' was assumed to be fundamental/irreducible because it had an ontological basis (compare).
84 As an early example I mention here Dirac's relativistic theory of stationary single electron states, inducing him to interpret `negative energy states' as states of holes in an `electron sea'. Later this ontological picture was substituted by the quite different picture of a positron (or "anti-electron"), explaining the phenomena in a more natural way (e.g. G. Farmelo, The strangest man, The hidden life of Paul Dirac, Mystic of the atom,, Basic books, 2009, chapt. 13, 15). On the other hand, similar phenomena in semiconductors are better explained by absence of an electron (a hole in a semiconductor band). Evidently, theories describing analogous phenomena may be compatible with different ontological pictures.
85 John Bell, Against `measurement', Physics World, 33-40 (August 1990).
86 A proof of a Gleason-like theorem, valid for generalized measurements (described by POVMs), has been given by Carlton M. Caves, Christopher A. Fuchs, Kiran K. Manne, and Joseph M. Renes, Found. of Phys. 34, 193-209 (2004).
87 See e.g. Stanford Encyclopedia of Philosophy: Holism and Nonseparability in Physics.
88 The notion of `entanglement' has been introduced by Schrödinger (Naturwissenschaften 23, 807-812, 823-828, 844-849 (1935), in German; English translation in Section I.11 of J.A. Wheeler and W.H. Zurek, eds., Quantum Theory and Measurement, Princeton University Press, New Jersey, 1983). Schrödinger used the term `Verschränkung', which is translated as `entanglement'.
89 In order not to complicate matters I shall ignore the historical issue that in defining the notion of `latitude of an observable (physical quantity)', Bohr initially did not have in mind the mathematical representation of an observable by a Hermitian operator introduced by Heisenberg, but considered such quantities as `classical quantities' (compare). I shall refer also to Bohr's `physical quantities' as `observables A' of the standard formalism. However, it is necessary to distinguish the `latitude δA' from the `standard deviation ΔA' of that formalism.
90 e.g. Renate Wahsner, Horst-Heino von Borzseszkowski, Die Wirklichkeit der Physik, Peter Lang, Frankfurt am Main, etc., 1992, section 3.2.
91 It is taken for granted here that the interpretational mapping is actually on equivalence classes of preparing and measuring procedures/instruments.
92 The notion of `interpretation' is used by philosophers and physicists in different ways, philosophers usually referring to a `mapping from reality/ontology into the theory' whereas physicists refer to the inverse mapping. In the literature of the philosophy of science the former mapping (i.e. from reality into theory) is usually referred to as `semantics' or `theory of meaning'.
93 In 2012 the Nobel Prize in physics was awarded to Haroche and Wineland "for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems". A number of the experiments performed by Haroche are analyzed in Publ._49 and sections 8.5.3 - 8.5.6 of Publ._52, and demonstrated to be joint nonideal measurements of incompatible standard observables in the sense defined here.
94 For the general case, see H. Martens and W. de Muynck, Found. of Phys. 20, 357-80 (1990).