276

p In practice the process of indirect measurement is clear: quantities connected with the measured one through a certain mathematically expressed dependence (relation) are measured directly, and the value of the measured quantity is determined from the dependence. But what is the 277 fundamental basis of this form of measurement? That is to say, of the measurement which, as noted above, removes the limitation of direct measurement and makes possible progress of scientific cognition? In particular, what is the unit in indirect measurement? Analysis of these questions is the theme of this section.

p Let us note to start with that’metrological direct measurement is indirect measurement in its formal content. Indeed, metrologically accurate measurement of a quantity is a measurement that can be reduced to ideal standards and instruments and to ideal conditions; and such reduction implies the use of dependences that relate the measured quantity to certain other quantities. In order to get a true result in metrological weighing, for instance, it is necessary to introduce corrections for the loss of weight in air, to exclude the effect of inequalities in the arms and the errors in the weights, let alone observing the scales’ state of sensitivity and determination of the zero point from the oscillation of balance arm.

p As regards its actual content, however, metrological direct measurement is direct measurement, because it is not the external circumstances in which the result is obtained that is essential but the method, the form of obtaining it.

p Metrological direct measurement is thus ideal direct measurement. It is the starting point of accurate indirect measurement.

p Indirect measurement is not only such according to its formal content, but also to its actual content. The heart of the matter in ideal direct measurement is reduction to ideal standards, instruments, and conditions; mathematical dependences are only used to introduce ‘corrections’ into the results, while the measurement itself can be done in principle without using them. In indirect measurement, however, the corresponding reduction to ideal standards is only a preliminary condition for obtaining the result. The very idea of ‘corrections’ (in the sense of ideal direct measurement) is totally alien to indirect measurement, and determination of the measured quantity without resorting to dependences does not make sense iii principle in indirect measurement.  [277•* 

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p Let us pass to the problem of the unit in indirect measurements.

p The equations of physics express dependences (relations) between quantities characterising not only individual, concrete systems and processes but also classes of systems and motions. Although the second dependences are the most essential, we shall begin our analysis with the first since they represent the elementary form of the relations that physics is dealing with.

p Assume that it is possible to say, on the basis of the appropriate experiments, that at a pressure of one atmosphere and a temperature of 20°C, a cubic centimetre of mercury weighs 13.6 grammes, two cubic centimetres 27.2 g, three cubic centimetres 40.8 g. We obtain a dependence between the volume of mercury and its weight that is expressed by the equation 

P^IS.BF,,

p (1)

p where P_l is the result of measuring the weight of the mercury (in grammes), and V_t is the result of measuring the volume of mercury in cubic centimetres.

p If the weight and the volume of mercury were measured, respectively, in any other units differing from the gramme and cubic centimetre, it could be demonstrated that the structure of all the corresponding equations would not differ from that of equation (1). In the symbolic form we have

= kV,

p

(2)

p where P is the weight of the mercury, V is its volume expressed in units which are not quantitatively specified, and k is a proportionality factor that depends on the choice of the units of weight and volume.

p Equation (2) can also be written as follows:

= W(V],

p

(3)

p where [P] and [F] are the units of measurement of weight and volume, respectively. Since k is the result of dividing 279

p P by V, we can denote it by

p i-Pl^p]

p " V[V] •

Let us divide the numerical value of P by the numerical value of F and introduce a symbol [P/F]; then

p

(4)

p Equation (4) can be interpreted as follows: the proportionality factor k is a certain quantity whose numerical value
is f-^r-) , and the measurement unit is TT •

p In general (as is illustrated by our example), the value of the proportionality factor is always associated with the unit of measurement, which differs from the units of quantities that are measured directly in its mediated nature; it depends on the units of other quantities (which figure in the equation), and is characterised by a structure with respect to those units. The proportionality factor accordingly functions as the embodiment of the dependence (relation) between the quantities.

p In a single equation the proportionality factor is a derivative of the other (primary) quantities. The importance of the concepts of the proportionality factor, dependent and independent units, and a derivative comes out quite clearly when we pass from single equations to systems of equations and to a system of systems of equations, i.e. to a physical theory.

p The modern physical theories are usually logically closed systems of principles and basic concepts in accordance with the axiomatic method of their construction.  [279•*  From this position the concepts of basic and derivative quantities are legitimate since the first are defined (indirectly) in terms of the theory’s system of principles (in classical mechanics, for instance, in terms of Newton’s axioms of motion; in thermodynamics in terms of its two principles), while the second are derived when the axioms are employed in concrete 280 situations from the sphere of phenomena that is covered (or should be covered) by the given theory (and its system of axioms).

p From this the concepts of basic and derivative units emerge, as units of measurement of the corresponding basic and derivative quantities, respectively, that characterise a certain sphere of phenomena, and also the concept of a system of units that includes the basic units (as the basis of the system) and the derivative units. The metric system was a system of units for measuring geometrical quantities; but the first developed expression of a system of units for measuring physical quantities was the Gaussian and Weberian system of absolute units mentioned above.

p How do things stand with basic quantities and, correspondingly, with| the basic units, in non-classical theories? The approach to solving the problems arising is ultimately determined by the fact that (1) classical mechanics is the limiting case of relativistic mechanics (when c-*- oo, where c is the velocity of light) and the limiting case of quantum mechanics (when h -*• 0, where h is Planck’s constant); and (2) nonclassical theories cannot avoid using classical concepts in measurement, which are relativised in appropriate fashion.  [280•* 

p Thus, the sphere of both the theory of relativity and quantum mechanics includes classical basic quantities, but as approximate quantities (with respect to classical ones), with an accuracy determined by c and h as fundamental quantities of the theory of relativity and quantum mechanics, respectively. As for the basic units of measurement in these theories, we shall consider them below in connection with analysis of so-called dimensionless quantities.

p But let us return to basic and derivative units.

p By making use of axioms and the more complex dependences consistently obtained from them (which determine the equations) we can link the basic and derivate units by similar dependences, the proportionality factors being taken as unity. Dimensional theory is concerned with problems of this kind.^^20^^ Its fundamental concepts include dimensionality, which shows how the derivative unit is linked with the basic ones. In classical mechanics (with its basic quantities of length Z, mass m and time t and the basic units [L], [M], 281 [T] corresponding to these quantities), for example, the dimensional formula for all units of derivative quantities has the form of an exponential monomial L^l M^m T^^1^^.

p This form of dimensional formula is determined by the following condition, as dimensional theory demonstrates: the ratio of two numerical values of any derivative quantity is invariant in relation to the choice of dimensions for the basic units. The use of dimensional analysis is based on this invariance principle: the validity of physical equations can be checked by the dimensional formula and, in the appropriate conditions, the law governing one physical phenomenon or another determined. In this respect this invariance principle has the same heuristic value for establishing the laws of phenomena as other, deeper principles of invariance.

p A dimensional formula can serve as the definition of a derivative quantity in a logically closed classical theory. This method of defining a quantity or, on a broader scale, of defining a quantity by specifying the method of measuring it, is common in classical physics. Bridgman unjustifiably turned it into a certain philosophical principle (the operational method of definition) that supposedly embraced the whole of physics, but as Born correctly noted, this method of definition is limited to classical physics.^^21^^

p Now let us consider how many basic units there should be in a system of units and what their nature should be. The answer, it would seem, follows from the exposition above: the number and nature of the basic units are determined by the number and the nature of the basic quantities, i.e. by the system of units that forms the foundation of a theory. There are many systems, however, differing both in the number and nature of their basic units.^^22^^ In general, there is a common point of view in the literature that the number of basic units is arbitrary and can be increased or decreased.²³ On the other hand, there is also the view that it would be more useful to discard systems of units (it is supported, for instance, by the physicist Robert Pohl).

p In order to get an understanding of all this, let us first consider some examples. As we have already mentioned, there are three basic quantities in classical mechanics in accordance with Newton’s axioms of motion: length /, mass /re, and time t, with the units [L], [M], and [T], respectively. When Newton’s axioms are extended (or generalised) to 282 include (weak) gravitational fields, and certain observation data are taken into account,  [282•*  we obtain Newton’s gravitational theory, with the law of universal gravitation in which a dimensional constant 7 appears that does not come into the equations of mechanics (it is called the universal gravitational constant, and its value is determined experimentally).  [282•** 

p Thus, in Newton’s theory of gravitation, the basic quantities known from classical mechanics are supplemented by the universal gravitational constant.

p The gravitational constant 7 makes it possible to ‘rid’ the LMT system of units of the basic unit [M\. To this end, we take [7] as unity, in other words, use the equation [7] = = [L^^3^^M ^^1^^T-*]. Hence we obtain [M] = [L^ST *], i.e. the formula of dimensionality of the mass unit in the unit system LT.

p This unit system is natural in the problems in which the gravitational force is taken into account. Besides, in this case it is not that the basic unit [M\ is ‘removed’ from the unit system LMT, but this system is transformed into the system of units 7 LT in which [7] is considered as a dimensionless unit.

p Another example. When the axioms of motion of classical mechanics are extended to include electromagnetic phenomena (taking into account such data as the results of the Michelson-Morley experiment, etc.), we get the equations of the special theory of relativity, containing the universal constant c—the velocity of light in vacua. It makes it possible to get rid of the basic unit of time. For that, the constant c is regarded as a dimensionless unit, and after a certain argument the conclusion is reached that the time during which light travels a unit of length in vacua should be taken as the unit of time. In several branches of physics and astronomy dealing with phenomena in which the velocity of light iTi vacua is essential this unit of time (with the dimension [L]) is more natural than the second, the definition of which is based on the Earth’s rotation around its axis. In this example it can also be shown that it is not so 283 much that one basic dimensional unit is ‘removed’ from the system of units as that another dimensionless unit is ’ substituted’ for it.

p As a last example in our argument let us consider the International System of Units (denoted as SI), adopted in 1960 by the International Committee of Weights and Measures. There are six basic units forming its foundation: length (metre), mass (kilogram), time (second), electric current (ampere), thermodynamic temperature (Kelvin), luminous intensity (candela). By establishing uniformity in the units of measurement, it covers all spheres of pure and applied science, linking measurements of mechanical, electrical, thermal, and other quantities, and taking their specificity into account. From the angle of their practical application it is very convenient. On the other hand, for certain areas of measurement, and for theoretical analysis if one has in mind their special features, other systems of units prove to be convenient (as was briefly mentioned above).

p Thus, by generalising the material cited and other material like it we reach the following conclusions: (1) the number and nature of basic (dimensional) units is adequate, generally speaking, to the number and nature of the basic quantities, but in certain theoretical and practical conditions such correspondence is not necessary; systems of units are possible only on the basis of the laws appertaining to certain spheres of phenomena, and the connection between them reflects on the logical plane (or should reflect) the connection between the spheres of natural phenomena belonging to them; (2) the basic units of a system may include both dimensional and dimensionless units; the number of dimensional and dimensionless basic units equals the number of basic quantities in a given theory (excluding universal constants); (3) the different systems of units (in the sense of the number and nature of the basic units) correspond to different classes of dimensional and dimensionless quantities,  [283•*  with the possibility of quantities’ changing their dimensions or non-dimensionality in passing from one system to another; (4) there is no predominant system of ( 284 dimensional) units; all possible systems of (dimensional) units are equivalent in measurement but that does not mean that choice of the system for actual use is independent of the conditions of the mensuration.

p In the context of these conclusions let us consider the so-called natural system of units. The father of the idea of this system, Max Planck, was interested, above all, in getting rid of the special features of those concrete bodies and phenomena that form the basis of the modern system of measurement units and measures, i.e. in eliminating the arbitrary and random elements that are to some extent inevitable in the modern system of measurement. Planck continued the work of the authors of the metric system in the conditions of twentieth century science. His natural units are based on four universal constants: the velocity of light in vacua, the gravitational constant, the quantum of action, and Boltzmann’s constant; these should, in his opinion, be preserved for all times and all extraterrestrial and extrahuman cultures, as long as the laws of nature determining these universal constants remained invariable.

p In Planck’s single natural system of units the randomness that is inevitable when physical realities are taken as standards of measurement is, reduced almost to nothing. The role of standards in this system is fulfilled by the universal laws of nature. These laws become ideal standards in the fullest sense of the term, while the concept of dimensionality itself disappears, which is the great significance in principle of Planck’s natural system for the problematics of mensuration. But the system itself is inconvenient to use either for molecular or atomic phenomena (let alone macroscopic ones), because of the smallness of the units for length, time and mass.

p The main drawback of Planck’s system in principle is that the introduction of such a unified system of natural units opens no perspectives for physical theories: in quantum mechanics, for instance, the construction of units on the basis of gravitational constant, the velocity of light and Boltzmann’s constant is unnatural since they play no important role in the phenomena studied by it.

p In this case the natural unit systems for individual physical theories that have appeared since Planck’s system are more promising. The point is that the laws of nature are not invariant with respect to the change of scale of certain 285 domains of physical phenomena, and the universal constants now known are on the boundaries of these domains; for example, the laws of classical mechanics are applicable in a domain in which velocities are small compared with the velocity of light in vacua, and effects are large in comparison with Planck’s constant. This corresponds to the fact that, in (non-relativistic) quantum mechanics, classical electrodynamics, and quantum electrodynamics, say, there are their own natural systems of units. For quantum mechanics, for instance, the basic units are Planck’s constant h, electron charge e (or rather its square), and electron mass m₀; the length scale here, in particular, is Bohr’s atomic radius

p r_B = - =-, while the velocity scale is the atomic unit of

p 77106^

p velocity -^7-; (h= -^—} . Use of these systems has its
theoretical and practical advantages, which have been discussed in the physical literature.^^24^^

p Is the perspective of ’dimensionless physics’ broader in any sense? It is still difficult to answer. As yet relativistic quantum mechanics (in which h and c play an important role) does not exist as a logically closed theory. There are no logical bridges between Einstein’s gravitational theory and quantum theory. Synthetic theories of this kind are knocking on the doors of contemporary physics. Their creation would possibly mean the discovery of universal constants as yet unknown, and appearance of new basic concepts and principles that might include qualitatively new notions about the most profound properties of space and time.

p Questions like this are on the boundary of modern physical knowledge, and only assumptions of one kind or another are possible at present. At the same time there is no doubt that the explanation of ’universal non-dimensionals’ embodying the most fundamental foundations of modern’ physics is not to be found in its known theories but at a deeper level.

p To conclude this section, let us return to the questions of measurement that are met in present-day physical literature.

p It was noted above that the concept of measurement implies acceptance of the idea that measurement does not alter the properties of the object measured, that there are sufficiently constant bodies (solids) and sufficiently constant processes to serve as the appropriate standards. All these 286 features are inconceivable if one ignores classical physics and (a) its understanding of moving matter as an aggregate of moving particles, and space and time as receptacles of changing bodies of various degrees of complexity, and (b) its methods of cognising nature. Classical physics did not so much find the basis for and explain the properties of the hardness of bodies, and the stability of atoms, and interpret the properties of motion, space, and time in a definite way, as simply accepted them as experimental data.

p Non-classical physical theories have already found the basis for such an understanding and explanation of the existence of properties and relations postulated by classical theories. Quantum theory, for instance, on the basis of knowledge of the properties of electromagnetic interactions has demonstrated the stability of the atom as an electromagnetic system, i.e. the stability of atoms has been explained by laws of nature. Atomic dimensions and energies have been determined and explained on this basis, and the quantummechanical description of atomic properties led in turn to understanding of a host of characteristics of matter, and of the constants on which so much empirical data had been collected in classical physics.^^25^^

p All this was expressed by G. Chew in his comments on Wick’s paper The Extension of Measurement’^^1^^’^^6^^ presented at the jubilee meeting devoted to the 400th anniversary of the birth of Galileo, in the following way: ’if there were no electromagnetic interactions but only short-range interactions, it seems unlikely that matter would assume the required sort of configuration.’^^27^^ According to Chew, if the interaction between the measuring instrument and the object were nuclear rather than electromagnetic, it would not be at all obvious that the measurement would retain its meaning, because ’the measurement notion depends on the possibility that when one system is looking at another system it does not completely change its nature’.^^28^^

p Many other physicists spoke in the discussion on Wick’s paper. Without going into its details, let us refer to the comments of Wheeler and Feynman.

p In Wheeler’s opinion, when measurements are made there is the fact that the observer is spatially separated from the object looked at, ’but in the case of a closed universe, we have no platform on which to stand to look at the universe— there is no place on the outside to poke it from’,^^29^^ i.e. we 287 have no observation point^’rom which the universe could be considered as an external object. He, therefore, put a number of questions; he asked whether or not it meant that ’the whole character of physics is different when we look at the universe as a whole than when we look at a part of it...’.^^30^^

p Feynman raised objections to Chew’s remarks: ’I do not see any sense whatever in discussing how measurements might look if there were no electromagnetic interactions. On the other hand, it is perfectly clear that when we learn more about strong interactions we may find that our concepts of physics, of philosophical bases of physics, just as many times before, have been changed, may be changed in such a way that ideas of measurement are in fact altered.’^^31^^

p It would seem that Feynman was much closer to the truth than Chew. His argument about measurement followed the line of development of scientific cognition: from absence of knowledge to its existence, and from shallow knowledge to deeper knowledge. All physical knowledge about nature and its laws is found, in the final analysis, from experimental material and the readings of measuring instruments, which (data) are described in the language of classical concepts. This was shown with extreme clarity by Bohr when he studied measurement in quantum mechanics. A physical theory, however, that reflects the phenomena of nature and their regular connections is not a set of instrument readings or a set of some sort of formal equations, but, as already established, a combination of these parts in some higher synthesis, which, as a matter of fact, is the only one that can be called a physical theory. Every fundamental (closed) theory, in particular, has its own rules linking its formalism and instrument readings: quantum mechanics enunciated this most convincingly for the first time in history. It is necessary, of course, to find not only the formalism (without which there is no theory in physics) but also the rules of its connection with the experimental data (without which there can be no physical theory), and such a successful search is the constructing of a physical theory.

p Problems of this kind are resolved by the development of physics as a science; the problematics of measurement are a part of them. In this respect modern physics provides excellent material.

288

p The same considerations apply to the problems of measurement posed by Wheeler. The progress of modern astrophysics, which studies stars, galaxies, and the Universe in general in their change and development, and not from a static point of view, as was the case before the 1940s, can be taken to provide concrete answers to these questions. We have in mind the new element in the understanding of measurement, which is inseparable from solution of problems and study of phenomena that have been still unexplained in the conditions of the twentieth century revolution in astronomy (the discovery of quasars, pulsars, etc.), rather than general formulations of ‘answers’.

p Modern astrophysics is closely related to the modern theory of elementary particles, and not just because it has been recognised that nuclear reactions are the source of stellar energy. These two lead ing spheres of modern physical science are also connected methodologically. The problem of the formation of a quantum of electric charge, of the spectrum of masses and charges of the elementary particles now known, and why they, and not other particles exist, questions that even now lack theoretical interpretation, fall within the competence of both disciplines. And does the idea that the existence of each elementary particle cannot be independent of the existence of each and all of the other elementary particles, which figures in the modern theory of strong interactions, have nothing to do with Wheeler’s questions?

p All these issues relate to the problem of measurement in various modern physical theories. Our job, however, was not to analyse it (in any sense thoroughly), or not even to analyse the formulation of the problem but to stress the fact that the problem of measurement becomes meaningful only when it is considered as an inalienable part of theory (in accordance with the concreteness of the theory and its coverage of reality). As a theory arises and is built, acquiring a certain shape, the problem of measurement becomes deeper and more meaningful in it.

This has been concretely established by quantum mechanics in its own sphere, and in this respect it can serve as a sort of prototype for other fundamental theories of nonclassical physics, including those being developed. The next two sections of this chapter will be devoted to the problem of measurement in quantum mechanics.