UNCERTAINTY RELATIONS AS A CONSEQUENCE OF THE LORENTZ TRANSFORMATIONS. V. N. Matveev and O. V. Matvejev

Transcription

1 UNCERTAINTY RELATIONS AS A CONSEQUENCE OF THE LORENTZ TRANSFORMATIONS V. N. Matveev and O. V. Matvejev Joint-Stok Company Sinerta Savanoriu pr., 159, Vilnius, LT-315, Lithuania matwad@mail.ru Abstrat A marosopi objet onsisting of a rod equipped with a pair of synhronized loks is examined. General physial relations are diretly derived from Lorentz transformations for the ase of the rod's onedimensional motion (along the X axis) the unertainty relation of the objet's x oordinate and the projetion of its impulse along the X axis, p x, and the unertainty relation of the objet's observation time, t, and its energy, E. The relations take the form: p x x H and E t H. The H value in the relation has ation dimensions and is dependent upon the preision of the rod's loks and its mass. It is shown that if the marosopi objet in and of itself performs the funtion of an ideal physial lok, the relations derived in the limiting ase then take the form of p x x h and E t h, where h is the Plank onstant. Introdution The unertainty relation of a miropartile oordinate and the projetion of its impulse along the oordinate axis, as well as the unertainty relation of miropartile time and energy, falls into the ranks of the most important quantum relations that express the known unertainty priniple in a mathematial form. Quantum relations are not observed in the maroosm in the sense that they are vanishingly small with respet to marobodies and do not show themselves in pratie. Aording to our data, no general physial relations have been reported in extant literature that extend to the maroosm and that limit the simultaneous speifiation of the impulse and the oordinate of physial bodies. In light of existing notions onerning the properties of marosopi bodies, the absene of suh general physial unertainty relations seems natural, sine unertainty relations have not been onneted to measuring instrument errors, while it is possible to talk about marobody oordinate, impulse, and energy inauraies, as they seem just as obvious today as measurement errors. The objetive of the work at hand was to demonstrate the existene of general physial unertainty relations that extend to marobodies. This objetive was ahieved over the ourse of solving a problem that onsisted of determining the veloity and the veloity-related physial values of objets based on the degree of desynhronization of moving loks. The relations p x x H and E t H were diretly derived from Lorentz transformations. The first relation onnets the unertainty of the projetion of the impulse, p x, of the body under onsideration, p x, to the unertainty of the x oordinate, x, while the seond relation onnets the unertainty of the energy, E, of the objet, E, to the observation time, t, unertainty, t. The H value in the relations has an ation dimension and is dependent upon the preision of the rod's loks and its mass. Taking into aount the fat that in Russia the reommendations formulated by metrologists for replaing the terms "error" with the term "unertainty" are not ompulsory in nature, and bearing in mind that these reommendations are still in the disussional stage [1-], we used both terms in the work at hand. We will take error to mean the inauray of measurement results that is due to purely metrologial auses. Inreasing an instrument's measurement auray an redue the error. The absolute error of the distane between two points an serve as an example of an error. We will all the inauray of measurement results that annot be eliminated by means of inreasing measurement instrument auray, and whih may be due to terminologial, 1

2 oneptual, or linguisti auses, the unertainty of physial values. The unertainty of the distane between two spheres that are loated lose to one another an serve as an example of this unertainty. This distane remains unertain with an auray of up to the dimensions of the spheres, even in the presene of ideal measurement auray, while the thing that remains unlear is what is meant by the sought distane the distane between the enters of mass of the spheres, the distane between their geometri enters, the distane between the losest points of the spheres, or something else. Construing unertainty in this manner is mentioned in extant literature, albeit in general terms. In this vein, for example, taking into aount the fat that it is only possible to speify the loation of a spatially extended body by determining the position of a single solitary point that belongs to it with some degree of unertainty, the unertainty of the position of a sphere determined by the position of its enter, whih equals the radius of this sphere, is written about in referene [3]. The unertainty of speifying the moment in time of a short-term proess that does not our instantaneously, but rather oupies a ertain finite, let's say, very short time interval, an serve as another example of an unertainty of this type. This unertainty an be regarded as equal to half the duration of the proess, if the moment in time of its transit is alled the moment into whih the middle of the proess falls. These oordinate and time unertainties, x and t, are preisely the ones that play a part in the unertainty relations we derived. 1. Single-time and single-oordinate data. Determining the veloity of a rod based on the readings of the synhronized loks appurtenant to it We will visualize a thin rod of proper length, L, at two points, a and b, on whih synhronously running loks, A and B, are installed at a distane of d from one another. Let's say that loks A and B, like the loks appurtenant to the K referene system, where the rod is at rest, show this system's time; i.e., the readings of loks A and B are always in agreement with the readings of the K system's loks. The length, d, of setion ab, whih is loated between points a and b, may be equal to or less than the rod length, L; i.e., the ondition L d holds true in the general ase. If L > d, then the rod will look something like this: A B Here, the A and B haraters onditionally designate lok A and lok B, while the broken line shows the body of the rod. If the distane, d, is equal to the rod length, L i.e., if L = d, loks A and B are then found at the ends of the rod. We will all the arrangement that onsists of this rod and the two running loks, A and B, situated on it rod, R; i.e., loks A and B will be regarded as integral parts of rod R, and we will treat the readings of loks A and B as harateristi attributes of rod R. Let's say that rod R, whih is positioned parallel to the Х axis of inertial system K, is at rest relative to this system and moves at a onstant veloity, V', along the Х' axis of another referene system, K', remaining parallel to the Х' axis and its diretion of movement (the Х and Х' axes of the K and K' systems slide along one another over the ourse of their relative motion). We will all this rod motion longitudinal motion (with respet to its orientation in spae), and it alone will be referred to in the future. Pursuant to inverse Lorentz transformations, at a moment in time of t' of the K' system, the readings, A,t' and B,t', of loks A and B, whih are in agreement with the readings of the K system's loks, are determined by the relations

3 t' + A, t ' t' + B, t ' A, t ' = and B, t ' =, 1 ( ) 1 ( ) where A,t' and B, t' the oordinates of loks A and B of rod R within system K' at a moment in tie of t', while the speed of light in a vauum. As follows from the relations presented above, the differene in the readings of loks A and B of rod R at a moment in time of t' for system K' equals ( B,t' A,t' ) B,t' A,t' =. (1) 1 ( ) Introduing the following notation for the purpose of saving writing spae U' =, () 1 ( ) then from formula (1), with allowane for notation (), we obtain: ( U = B,t' A,t' '. (3) B,t' A.t' ) Thus, having the data on the oordinates and readings of loks A and B at a moment in time of t', formula (3) an be used to find the value of U', and this value an in turn be used to find the veloity, V', of rod R in system K'. We will all the data that haraterize the objet's spatially distributed elements, but that relate to one and the same moment in time, t', single-time data. In addition to the possibility of determining the rod's veloity using single-time data, the possibility also exists of determining the veloity, V', of rod R in system K' using singleoordinate data. We will all the data suessively reorded at different moments in time, but at one and the same point (with one and the same oordinate), whih haraterize the elements of a spatially extended objet at the times that they are loated at this (or near this) point, singleoordinate data. Data of this type inlude, for example, the A, and B, readings of loks A and B of rod R that are suessively reorded at moments in time of t' A, and t' B, at a point in system K' with a oordinate of, past whih it is moving within this referene system. Aording to inverse Lorentz transformations, the single-oordinate A, and B, readings of loks A and B are onneted to the moments in time, t' A, and t' B,, during whih loks A and B, at a point with a oordinate of, show the relations t' A, + t' B, + A, = and B, =. 1 ( ) 1 ( ) As follows from the relations presented above, the differene in the readings of loks A and B of rod R at a point with a K' system oordinate of equals t' B, t' A, =. (4) B, A, 1 ( ) 3

4 Introduing the notation Γ' ( = 1 1 ), then from formula (4), we obtain Γ' = t' B, A, B, t' A,, (5) from whih the veloity, V', of rod R an be learned as neessary. The determination of veloity using single-oordinate data is interesting in that it does not require distane measurements, but rather is based on the measurement of the B, A, and t' B, t' A, time intervals.. Relation of rod veloity and oordinate unertainties alulated using single-time data In talking about the determination of the veloity of rod R using the differene in the readings of loks A and B, we taitly proeeded on the basis of the fat that loks A and B of rod R run ideally; i.e., the readings of loks A and B are in absolutely preise agreement with the ideally aurate readings of the K system loks. We speulate that a ertain maximum absolute error exists in the time readings of eah of loks A and B, whih are appurtenant to rod R. We will designate the absolute error of these loks as. Here, adhering to the generally aepted assumption, we will presume that all the loks appurtenant to any inertial referene system, inluding the K system, run with ideal auray. The existene of an absolute error,, in the readings of eah of loks A and B means that, at a given moment in time, t, within the K system, the readings of eah of these loks of rod R an differ from the readings of the K system loks loated lose to them and running with ideal auray by a value that does not exeed. In this regard, the remarks made above onerning the agreement of the reading of the K system loks with those of loks A and B must be interpreted with allowane for the finite auray of the latter. If the rate of the K' system loks, the measured B, t' A,t' value, and the speed of light value are as aurate as desired, the error in the U' value alulated using formula (3) will then be solely due to the existene of an absolute error, ( B,t' A,t' ), in the differene of the B,t' A,t' readings of loks A and B. In this instane, the U' error, with allowane for formula (3), is expressed by the equality U ( B,t' A,t' ' = (6) B,t' A,t' ) In instanes when the maximum absolute error of the differene in the readings of loks A and B onsists of the errors of eah of these loks i.e., when ( B,t' A,t' ) =, it follows from equality (6) that: ( ) U ' =. (7) B,t' A,t' We note that the error is not dependent upon referene system seletion, sine it onsists of the maximum possible differene between the readings of eah of loks A and B and the readings of the К system loks loated lose to them, where rod R is at rest. It is lear that this differene is not dependent upon the referene system within whih it is piked up. We will now imagine that, in addition to the ondition of the single-time nature of the readings of loks A and B within system K', the requirement of the single-oordinate nature of the speifiation of the loation where rod R is situated at a moment in time of t' is satisfied. Let's say the essene of this requirement onsists of using a single solitary R oordinate to 4

5 speify the loation of the projetion of rod R on the X' axis. This requirement an be satisfied if the rod length is ignored and it is regarded as a point. But if, due to the neessity of taking the property of the rod's spatial extent into aount, it is impossible to ignore its length, the requirement of the single-oordinate nature of the speifiation of the position of rod R an only be satisfied in part. For example, the oordinate of any point appurtenant to rod R an be speified as its R oordinate and a referene to its unertainty an aompany the speifiation of this oordinate. In partiular, the oordinate of the geometri enter of rod R, or the oordinate of its enter of mass, an serve as the oordinate of this point. In suh ases, the distane from the point with a oordinate of R to the point of the rod's projetion on the X' axis farthest away from it an be regarded as the unertainty, R, of the R oordinate. When the position of rod R is speified in this manner, the R oordinate indiates the loation of one of a set of points of the projetion of rod R that lies on the X' axis. If we give this point preferene for one reason or another, the R unertainty will then determine a range of point oordinates that, in the presene of other onsiderations, ould also be regarded at the point oordinates of rod R. For example, if the oordinate, ab, of the geometri enter (the midpoint) of the ab setion of rod R parallel to the X' axis is seleted as this setion's oordinate, the unertainty, ab, of the ab oordinate of the rod's ab setion an by definition be regarded as a value equal to half the length of the setion of the moving (within the K' system) rod. Sine the length, d' = d 1 ( ), of the rod's moving ab setion within the K' system equals B,t' A,t', the unertainty, ab, of the ab oordinate of setion ab of rod R then equals ½( B,t' A,t' ). Thus, formula (7) an be presented in the form U ' =. (8) ab Within the framework we have provided when introduing the determination of unertainty, the ab value is absolutely an unertainty and not an error, sine it is dependent upon the length of the rod's ab setion and annot be redued by means of inreasing measurement auray. Beause L d in the general ase, the unertainty of the oordinate of rod R within the arbitrary positioning of loks A and B thereon ( = ½L') an then generally both equal and exeed the ab value. And sine the relation ab holds true, then in the general ase of the arbitrary positioning of the loks on the rod, formula (8) takes the form: U '. (9) Relation (9) pertains to the general ase of the arbitrary positioning of the loks on the rod and makes the transition to the equality U' = in the speial and most favorable ase (as far as the determination of the U' value using single-time data) of lok positioning on the ends of the rod ( = ab ). The produt of the error, U', of the U' value and the unertainty,, of the oordinate of rod R during an instantaneous observation will only be dependent upon the error in the readings of loks A and B of rod R. Therefore, relation (9) remains unhanged in any inertial referene system and an be written for an arbitrary system in the form: U x x (1) If the mass of rod R with loks A and B is known without a doubt and equals M R (here and further on, the onepts of Lorentz-invariant mass [4] will be used), relation (1) an be transformed into the relation M R U x x M R by multiplying its left-hand and right-hand members times M R, whene, taking into aount the fat that M R U x = p x, we obtain: 5

6 p x M. (11) x R Introduing the notation H = M, (1) R then from formula (11), we obtain p x x H. (13) The error of loks A and B of rod R is one of the internal parameters of rod R and is not a part of the parameters of the measuring devies that are external relative to rod R. Therefore, the U x and p x errors in relations (1) and (13) an tentatively be referred to as external (relative to rod R) unertainties. The internal properties of rod R itself (a salient feature of loks A and B appurtenant to it) determine the external unertainties U x and p x, and in the presene of a speifi x unertainty, they annot be eliminated by means of inreasing the auray of the measuring instruments loated beyond rod R. 3. Relation of rod energy and time unertainties alulated using single-oordinate data The error, Γ', of the Γ' value an be derived from formula (5). Only the B, A, value error due to the error in the readings of loks A and B determines the Γ' error of the Γ' value, sine the t' B, t' A, time interval at point, aording to our initial assumption, an be measured by the K' system loks with an ideal auray. Therefore, from formula (5), we obtain: Γ' ( t' B, A, =, B, t' A, ) or with allowane for the fat that ( B, A, ) =, Γ' = t' B, t' A,, whene it follows that: Γ'(t' t' ) =. (14) B, A, We will now assume that, in addition to the ondition of the single-oordinate nature of the observation lok readings, the requirement of the single-time nature of the speifiation of the observation time of the ab setion of rod R, moving past point, is imposed. Let's say the requirement onsists of using a single solitary moment in time, t' ab,, to speify the setion ab observation time at point. Sine the observation at a point with a oordinate of is arried out over a time interval of t' B, t' A,, it is then only possible to approximately speify the observation time by indiating, for example, the moment in time, t' ab,, of the middle of the time interval, t' B, t' A,, that goes toward observation. Here, the unertainty, t' ab,, of the moment in the observation time that equals half the t' B, t' A, observation time an be speified; i.e., it an be assumed that t' ab, = ½(t' B, t' A, ). Then formula (14) an be presented in the form: 6

7 Γ' t' ab, =. (15) For the arbitrary positioning of the loks on the rod, the t' observation time for the entire rod, R, with a length of L proves to be greater than the observation time, t' ab,, for the ab rod setion; i.e., the ondition t' t' ab, is satisfied, as a result of whih it follows from formula (15) that Γ' t'. (16) During an aurate observation, the produt of the unertainty, t', of a moment in time, t', of the observation of rod R and the Γ' error of the Γ' value will only be dependent upon the error of the readings of loks A and B of rod R. Therefore, relation (16) remains unhanged within any inertial referene system and an be written for an arbitrary system in the form: Γ t x. (17) Multiplying the left-hand and right-hand members of relation (17) times M R, bearing in mind that ΓM R = E, and using notation (1), we obtain E t H. (18) x If the error is regarded as a rod parameter that is not a part of the measuring equipment parameters, the Γ and E values an then be referred to as unertainties. 4. A physial lok. Relation of the impulse and oordinate unertainties of a spatially extended body We will assume that eah of loks A and B disretely hanges its reading with frequeny, ν. The maximum absolute error in the readings of the moments in time of eah of the loks will equal 1/ν. We will also assume that this hange in the lok readings ours synhronously; i.e., not only are the lok readings at a given moment in time synhronized, but also the times of the hange in readings. In this instane, the maximum error of the differene in the readings of loks A and B will equal not the sum of the two errors, but rather a single error. Then relations (13) and (18), dedued from the ondition of the equality of the lok A and B differene error to the two errors, take the form: and p x x H / (19) E t H /. () x Let's imagine that eah of loks A and B onsists of two omponents, one of whih, being artifiial ("manmade"), performs the funtion of a display and provides disrete time readings, hanging them in a keeping with external signals, while the other one we will all it a physial lok generates these signals in a natural, easy manner and ontrols the hange in the display's readings. 7

8 For the sake of larity, we will visualize the seond part of the physial lok as a piee of radioative material with a long half-life (the material's radiant power an be regarded as onstant for a suffiiently long time frame). If the display reats to a speifi portion, ε, of the physial lok material's absorbed gamma-radiation energy by hanging its readings, then in the presene of a material radiant power than equals P, the frequeny, ν, of the hange in the display readings will equal P/ε. We will all the ε portion of the absorbed energy that leads to a hange in the display's reading the energy of the pereived (by the display) physial lok signals. We will assume that, regardless of the quantity of the physial lok material, the display absorbs all of the energy radiated, and that eah ε portion of the energy absorbed performs the funtion of a reading hange signal that is pereived by the display. The frequeny, ν, of the physial lok materials signals pereived by the display, and aordingly the frequeny of the hange in the physial lok's readings, will then be proportional to the quantity of the physial lok's material. This means that if the frequeny of the pereived signals equals ν in the presene of a physial lok material unit mass of m, it will then equal M ν /m in the presene of a mass of M ; i.e., ν = M ν, (1) m Sine the maximum absolute error of the readings of eah of the loks equals 1/ν, the following expression an then be derived from equality (1): m =. () M ν Taking the mass of the rod without the loks to be negligible as ompared to the mass of the physial lok material onentrated in loks A and B, or assuming that the rod R onsists entirely of the physial lok's material, the M value (sine the mass of the material in both loks equals M ) an be set to equal the mass, M R, of rod R with loks A and B. Taking this into aount, it follows from formulas (1) and () that m ν H = (3) The H value is dependent upon physial lok type and display sensitivity, and is not a physial onstant. The H beomes different when the display's sensitivity hanges, or when the physial lok's radioative material is replaed with a radioative material that has a different radiant power. In the ontext of the physial lok, relations (19) and () pertain to the ase of the arbitrary distribution of the physial lok's material along the rod and make the transition to equalities in the speial ase when the physial lok's material is onentrated at the ends of rod R. At first glane, the relations derived only hold true for the tehniques of instantaneous and point observations of an objet, and the unertainties going into these relations onsist exlusively of the unertainties inherent in these tehniques. In atuality, however, it is impossible to measure even the onstant veloity of this rod R, equipped with loks A and B, with absolute auray using onventional methods (based on the path traversed and the time), if the word ombination "rod R with loks A and B" is taken to mean a speifi objet, the omplement of harateristi traits of whih inludes the differene in the readings of loks A and B (or the events that haraterize this objet). In suh ases, the veloity and the values 8

9 derived from it (the impulse and energy) prove to be tied not to an extraneous referene system, but rather to this objet's harateristis traits, for example, to time or event marks [5]. In order to grasp the essene of the latter omment, we will address the onepts of relativity and unertainty. It was shown in referene [5] that many of the questions arising in the quagmire of physial relativism are erased if attention is direted to the presene of unertainty in physial relativity. Rather than talking about the relativity of the veloity of a body and about the fat that the veloity of a body without speifying a referene system makes no physial sense, it is proposed in referene [5] that the term "unertain veloity" be used with respet to the irrelative veloity of a body. In this ase, the veloity of a body that is not tied to a speifi referene system should be referred to as unertain veloity. The unertainty of the irrelative veloity of a body equals the onstant. It an be said of the irrelative veloity of this body, for example, that it equals zero, while its unertainty equals. This speifiation of veloity does not differ qualitatively in any way from the onventional speifiation of a veloity value supplemented by the speifiation of its error. We draw attention to two methods that make it possible to avoid the unertainty of the veloity of a body and to give it ertainty. The first generally aepted method onsists of speifying the referene system within whih the veloity of a body is examined. After the referene system is speified, the veloity of this body beomes ertain. The seond method, despite its obviousness, is sarely mentioned at all in physis. This method onsists of individualizing an objet, whih in turn onsists of desribing it in greater detail, and under ertain onditions, it is apable of replaing the seletion of a referene system. In the example we examined, the objet with respet to whih relations (19) and () hold true is not a rod, R, equipped with loks A and B, but rather a rod, R, equipped with loks A and B, and possessing a predetermined differene, B A, in the readings, A and B, of loks A and B. If one and the same rod, R, equipped with loks A and B, is by definition regarded as a set of more onrete subobjets, eah of whih has an inherent differene, B A, in the readings of loks A and B, then these different objets will possess different veloities. The possibility of thus dividing an objet into more onrete subobjets eludes relativists, sine it puts an end to the objetive nature of physial relativism. Relativists are inapable of understanding something that was lear to Herales, who saw the differene between one and the same river and partiular onrete "subobjets" of this river that differed from one another. At the same time, relativists themselves note that one and the same objet differs in different referene systems due to the relativity of its single-time nature; i.e., it breaks down into subobjets of sorts. It is strange that they don't omprehend the fat that, having desribed a subobjet, its veloity an often be determined based on "external appearane" without speifying a referene system. For example, in the ase we examined, a subobjet suh as a rod, R, that has a speifi differene, B,t' A,t', in the readings, A,t' and B,t', of loks A and B, moves at a speifi longitudinal veloity, V x. When the loks run with ideal auray, the V x veloity value is funtionally dependent upon the B,t' A,t' differene value. A subobjet suh as a rod, R, with a differene, B,t' A,t', in the readings, A,t' and B,t', of loks A and B that equals zero is at rest. Its longitudinal veloity, V x, equals zero and annot differ from zero in any referene system, sine subobjets with a differene, B,t' A,t', in the readings of loks A and B that equals zero do not exist in any referene system where this rod moves longitudinally. This rod, R, with loks A and B, has different longitudinal veloities in different referene systems, but the subjet rod, R, with a differene, B,t' A,t', in the readings of loks A and B that equals zero, being a onrete subobjet, only has a longitudinal veloity that equals zero. This fat is entirely independent of the manner in whih the veloity is measured based on the lok readings or using the path and distane traversed. 9

10 In addition to this, if loks A and B provide disrete readings, the B,t' A,t' differene will also have a ertain disreteness. Here, the longitudinal veloity value, V x, will have an unertainty of V x. In this ase, a rod, R, with a given differene in the readings of loks A and B, being a subobjet, may be found in a ertain set of referene systems in a state of motion at various longitudinal veloities that differ from one another by a magnitude that does not exeed a ertain value, whih is the V x unertainty. This unertainty in auray equals the unertainty that ours during the single-time determination of the V x veloity of a given subobjet. If loks A and B do not run, but rather stand still, ontinuously indiating an idential time, then the V x unertainty of the longitudinal veloity of rod, R, with a differene of B,t' A,t' in the readings of loks A and B that equals zero, will equal the speed of light,. Suh a rod, R, with a differene of B,t' A,t' in the readings of loks A and B that equals zero, an be found in all referene systems and possesses any veloity over a range of zero to the speed of light,. Conlusion The purpose of the work at hand onsisted not of finding areas of ommon interest between Lorentzian physis and quantum mehanis, but rather of demonstrating the existene of general physial unertainty relations in the physis of the maroosm. The general physial relations derived are externally reminisent of the known unertainty relations of quantum mehanis; however, the physial essene of the values that go into the relations and that ontain the relations themselves are different than those in quantum mehanis. In relation (13), the x unertainty is taken to mean the unertainty in speifying the position of a projetion with a length of x of a spatially extended objet using a single solitary oordinate, x, while in the Heisenberg relation, x is a probabilisti harateristi of the position of a miropartile desribed by the root-mean-square deviation from the mean value. The p x value in relation (13) is the p x impulse error, whih in the presene of a predetermined x unertainty, generally speaking, an be redued by using another type of physial lok, while p x in the Heisenberg relation is an unertainty that it is fundamentally impossible to redue in the presene of a predetermined x value. Different meanings are plaed on the H and h values. The H value in relations (13), (18), (19), and () is dependent upon display sensitivity and physial lok type. If the sensitivity of the display is hanged or the physial lok's radioative material is replaed with a radioative material that has a different mass radiation frequeny, the H value will be different. However, the fundamental Plank onstant, h, is unrelated to the physial properties of any speifi material. The fat that these relations prove to be onneted to the Heisenberg relation, not only externally, but also internally, seems espeially strange. First, there is the question of the minimum value, H min, of an H parameter with an ation dimension. Can one assume that this value may be as small as desired in the maroosm and may orrespond, for example, to the ondition H min << h? And seond, relations (19) and () are formally transformed into the relations p x x h/ and E t h / via the simple substitution of the energy, hν, of a photon with a frequeny of ν in plae of the unit energy, m, of the physial lok in formula (3); i.e., by taking the unit mass of a photon, the ν frequeny of whih numerially equals the ν frequeny of the signals of a hypothetial hange in the lok readings as the physial lok. It would be more orret to proeed on the basis of the onept of Lorentz-invariant mass and the equality of photon mass to zero, then a pair of photons oming from opposite diretions with equal energies of E = hν and with a resultant impulse, P, that equals zero, should be regarded as a physial lok of a unit mass of x 1

11 m. Sine the unit mass, m, of this lok equals ( P ) ) ( E =E/, while the ν summation frequeny of the eletromagneti osillations equals ν, then taking the ν frequeny of the unit mass physial lok to equal the ν summation frequeny of the photon pair, it follows from formula (3) that H = H min = h. Another approah is also possible. In onsidering formula (3), if it is assumed that a ertain H min value exists that is ommon to all types of physial loks, then m /ν = H min. The latter equality only ours when m = ½ H min ν. If the lowest energy of a hypotheti signal of a hange in lok readings is taken as m, then H min must be equal to the Plank onstant h, sine the ½hν value is the minimum possible energy of the zero-point osillations of an osillator with a frequeny of ν. The relations expressed by formula (3), if we move to them from relations (19) and (), do not reflet the statistial nature of the generation of the lok reading hange signals; thus, when using this approah, referene an only be made to the order of the parameter present in the right-hand member of the relations and not its preise value. Referenes 1. Tarbeyev Yu. V., Slayev V. A. and Chunovkina A. G. Problems With Using the International Guide to the Expression of Unertainty in Measurement in Russia. Izmeritelnaya Tekhnika [Measurement Tehniques], No. 1, 1997, pp European Soiety for Analytial Chemistry/International Cooperation on Traeability in Analytial Chemistry (EURACHEM/CITAC) Guide Entitled "A Quantitative Desription of Unertainty in Analytial Measurement" ( nd Edition, ) Translated From the English. D. I. Mendeleyev Russian National Metrology Researh Institute (RNMRI), St. Petersburg,. 3. Sukhanov A. D., Golubeva O. N. Letures on Quantum Physis. Vysshaya Shkola [Higher Shool] Publishing House, Mosow, 6, p Okun L. B. Advanes in the Physial Sienes (APS), No. 178, 8, pp Matveev V. N. Into the Third Millennium Without Physial Relativity. CheRo Publishing House, Mosow,. 11