Wave function of particle and the coordinates distribution in relativistic quantum theory
aa r X i v : . [ qu a n t - ph ] J u l Wave Fun tion of Parti le and Coordinates Distribution inRelativisti Quantum TheoryV. F. KrotovInstitute of Control Problems, Russian A ademy of S ien es, Russia,117997, Mos ow, Profso'uzna'a 65The onditions for observation of the individual parti le oordinates, required by logi of theSpe ial Relativity and (cid:28)ltering the quantum (cid:28)eld e(cid:27)e ts, are des ribed. A general relation betweenthe orresponding density of probability and the wave fun tion is found. It is a relativisti invariantdes ribing probability of the parti le emergen es in spa e(cid:21)time. This density is on retized forbosons, both s alar and ve tor (in luding photon), harged and neutral, also ele tron. The Heisen-berg's un ertainty relations have been approved in regards to relativisti parti le. As applied tothe quantum (cid:28)eld, this new onstru tion is transformed to new hara teristi of the parti lesdistribution in spa e-time, whi h omplete distribution throughout impulses. The operators ofthese distributions and the invariant relativisti des ription for free quantum (cid:28)elds have beenobtained. These new properties of the parti les and (cid:28)elds are proposed for experimental investigations.PACS 03+p; 03.65-w; 03.65 Ta1. Introdu tionModel of the Quantum Parti le (QP) ombines the deterministi dynami sof Wave Fun tion (WF) and the statisti al relation of WF with observedvalues: the dynami part of QP's des ription and the statisti al one. Whilethe former one is quite well formalized, the se ond one for relativisti QP(RQP) auses problems: no formal onstru ts with oordinates distributionproperties, and the restri tions on observation possibilities aused by e(cid:27)e tsof the Quantum Field (QF) are available. These problems have aused ageneral opinion that (cid:16)in onsistent relativisti quantum me hani s the parti le's oordinates at all annot serve as dynami variables, . . . and WF, as information arriers, annot be present in it(cid:17) [1℄, page 16. We have to add: relations ofun ertainty of Heisenberg are devoid of the theoreti al rationale as appliedto RQP, as there is no law of oordinate's distribution ne essary for that.We demonstrate that both problems mentioned above are solved when thestatisti al part of RQP have been in luded in logi of the Spe ial Relativity(SR) su(cid:30) iently fully. Despite the ommon opinion, there appears a theoreti alpossibility to preserve information properties of the WF in the extent omparable1o non-relativisti quantum me hani s (QM), and there are observation pro edures,(cid:28)ltering e(cid:27)e ts of QF. This in lusion require orre tion of the full set of theobservables and of the observation pro edures. A orre tion of the requirementsof ovarian e follow from this. Formal onstru ts expressed via WF areobtained, that possess all ne essary properties of probabilisti distributionsof the observed values, in luding parti le oordinates. In simplest ases thisexpression is formally similar to probability density of the oordinates of thenon-relativisi parti le, but has di(cid:27)erent ontent. Other observation onditions,other transformational properties, other meaning: probability density of theevents in spa e-time, instead of the probability density of positions in spa e.Instead of sto hasti dan e of the parti le, des ribed by the (cid:29)ow of probability,we have probabilisti distribution of the parti le appearan e in spa e-time, whi h an not be, generally, represented by the motion in spa e. On thisbasis relations of un ertainty obtain the justi(cid:28) ation as applied to RQP.As applied to the quantum (cid:28)eld, this new onstru t is transformed to new hara teristi of the parti les distribution in spa e-time, whi h ompletedistribution throughout impulses. The operators of these distributions andthe invariant relativisti des ription for free quantum (cid:28)elds have been obtained.2. Íåðåëÿòèâèñòñêàÿ ÷àñòèöàSin e the statisti al part is su(cid:30) iently formalized only for the non-relativisti QM, let us start here with de(cid:28)ning its ne essary details limitedto spinless QP. Let x = { x i } = ( x , x , x ) be the ve tor of parti le's spatial oordinates, the element of the related Eu lidian spa e E . Complex WF ψ ( x ) is de(cid:28)ned on the E , and is onsidered as the element of the Hilbert spa e H with the norm L ( E ) and respe tive produ t ( ψ , ψ ) = Z E ψ ( x ) ψ ∗ ( x ) d E ,where d E is elementary volume of E . As a rule, the WF is (cid:28)nite in a ube V ⊂ E : ψ ( x ) = 0 , x / ∈ V , and this integral is de(cid:28)ned.The evolution of theWF ψ ( t, x ) on the interval (0 , T ) des ribes the QP movement. It satis(cid:28)esS hrodinger equation, whereas k ψ ( t, x ) k is its dynami invariant, normalizedby the ondition k ψ ( t, x ) k = 1 .On observation pro edure. Full set of hara teristi s with values y == { y i } is being observed. They are either omponents of the ve tor x , orthe energy, impulse, and momentum. Let the operations of realization ofQP movement during the time T be de(cid:28)ned in identi al physi al onditions, orresponding to given WF evolution ψ ( t, x ) . At a given moment of time t asingle session of observation (session) is being performed over ea h realization.2alues y in ea h session are de(cid:28)ned, and are random would they repeat.Observation of traje tories y ( t ) is also admissible in ases when they areidenti al to the time series of results of sessions of the independent realizations.For ea h (cid:28)xed fun tion ψ ( t, x ) and ea h t the averages h y i [ ψ ( t, x )] = {h y i i ( ψ ) } are de(cid:28)ned, orresponding to in(cid:28)nite set of the sessions.On formal des ription. Given (cid:28)xed t the averages h y i [ ψ ( t, x )] = {h y i i ( ψ ) } are Hermit's Quadrati Forms (HQF) in H . If y = x , then the existen e ofthe probability density f ( t, x ) and its relation to WF f ( t, x ) = | ψ ( t, x ) | arepostulated. Correspondingly, the probability measure P ( Q/t ) = P [ Q, ψ ( t, x )] == Z E f ( t, x ) d E is de(cid:28)ned. If y = { y i } is an aggregate of energy, impulse, andmomentum, than the equality h y i = J ( ψ ) is postulated, where the right sideis the aggregate of related dynami invariants of the Wave Equation (WE),HQF J i ( ψ ) = ( ψ, L i ψ ) , and L i is a respe tive operator (operator of the i -th observable). In this ase the family of distributions P ( Q, ψ ) is expressedthrough omponents of spe tral fun tions of the operators L i in a knownmanner.There exist ex eptions from the rule ψ ∈ L ( E ) , when the norm is notde(cid:28)ned. In this ase the ondition P ( E , ψ ) = 1 does not hold. But relativeprobabilities P ( Q , ψ ) /P ( Q , ψ ) are de(cid:28)ned. Let us note that ψ ( x ) ∈ L ( V ) in any (cid:28)xed ube V ⊂ E , and WF is the element of Hilbert spa e H ( V ) with this norm. By (cid:28)xing here k ψ k = 1 , one an give P ( Q, ψ ) , Q ⊂ V , themeaning probability measure of the positions x , and f ( t, x ) (cid:22) its density,with the following amendment to the observation pro edure: only the sessionsregistering x ∈ V are onsidered.3. Relativisti Parti le. General Model Des riptionLet E be real pseudo-Eu lidian spa e with oordinate ve tor x == { x α } = ( x , x ) , x ∈ E ; x = ct , t is time, c is speed of light; metri tensor e = { e αβ } is diagonal: e = − , e ii = 1 , i > . On a (cid:28)nite bar V = (0 , cT ) × V ⊂ E the WF ψ ( x ) is de(cid:28)ned, that maps V to an Eu lidianspa e U with elements u and produ t u u . It satis(cid:28)es the respe tive waveequation (WE). The WF is onsidered as an element of the Hil-bert spa e H ( V ) with the norm k ψ ( x ) k = Z V u dE , u = ψ ( x ) , dE = dx d E (let usdenote norm of the spa e H as k ψ ( t, x ) k ), and respe tive produ t ( ψ , ψ ) .For ea h WF the identi al physi al onditions are de(cid:28)ned, in whi h the RQPis observed. 3bservation pro edure reprodu es it for non-relativisti QP with the followingdi(cid:27)eren es. Session is not aused by moment of time t . Thus, time is ex ludedfrom the observation pro ess as a parameter, but is present as a part ofargument x , and, possible, of the observed value y . Session lasts duringthe time T . Other onditions will hold good. The main ondition amongthem is preservation of uniqueness of y value at session of measurement innew onditions. We will dis uss this ondition below in appli ation to the on rete observable values. A measure P ( Q, ψ ) is de(cid:28)ned for ea h WF, andrespe tively, the average value h y i [ ψ ( x )] .Formal des ription reprodu es the same for non-relativisti QP with thefollowing di(cid:27)eren es. HQF's P ( Q, ψ ) , J i ( ψ ) = ( ψ, L i ψ ) , and respe tively, theoperators of the observables are de(cid:28)ned in H , but not in H . Chara teristi s Y , and thus, their average J ( ψ ) , possess relativisti transformational properties.Therefore, P ( Q, ψ ) is relativisti invariant. The latter statement follows fromequality J ( ψ ) = h y i ( ψ ) = Z ydP ( ψ ) , sin e J ( ψ ) and y have the same tensordimension. The dimension of probability density g ( y, ψ ) is de(cid:28)ned by theequality dP = gdY = inv , where dY is elementary volume of the spa e ofobservable values. Detailed elaboration of model with referen e to on reteobservable values and types of parti les follows below.4. Coordinates ObservationOn observation pro edure. The full set of the observed is ve tor y = x of spatial-time oordinates of a parti le; the result of ea h session is the(cid:28)xation of the event that a parti le has appeared in the point x of the spa e-time E . A ontent of the session: let the measuring devi e be an ele troni mi ros ope; a bun h of ele trons probes the spa e, and at some point in timethere happens a ollision with a parti le-obje t; su h ollision, generally,distorts the observation onditions, orresponding to the given WF (theleast ase is the obje t-photon, that disappears as a result of the ollision);be ause of this, only this (cid:28)rst ollision has to be taken into onsideration;the position of the parti le is identi(cid:28)ed with the point on the s reen, (cid:22) atra e of the single s attered ele tron. The measurement should also re ordthe moment of ollision for omplete (cid:28)xation of the event. Only the sessions,that demonstrate events x ∈ V has to be taken into onsideration. The auses of the distortion of this ondition an be the following: the sessiondemonstrate an event x / ∈ V , the ollision is not the only, e(cid:27)e ts of the QFthere are. Admissibility of the latter is not evident, be ause QP and QF aredi(cid:27)erent systems. But it is valid (see below, item 6.4).4n the formal des ription. The fun tion g ( x ) = ψ ( x ) , k ψ ( x ) k = 1 (1)posses all ne essary properties of the probability density of the events x ∈ V .This density is a relativisti invariant (elementary volume dE is invariant). Insimplest ases this expression is formally similar to probability density of the oordinates of the non-relativisti parti le, but has di(cid:27)erent ontent. Otherobservation onditions, other transformational properties, other meaning ofthe fun tion ψ ( x ) : probability density of the events in spa e-time, insteadof the probability density of positions in spa e. The hange of meaning orresponds to the logi of SR and predi ts a new property of the RQP.Instead of sto hasti dan e of the parti le, des ribed by the (cid:29)ow of probability,we have probabilisti distribution of the parti le appearan e in spa e-time ,whi h an not be, generally, represented by the motion in spa e.Knowing g ( x ) , one an determine density g ( x ) of probability of positions x , g ( x ) of time x , density of spatial oordinates at a (cid:28)xed moment g ( x /x ) : g ( x ) = Z (0 , ∞ ) g ( x ) dx , g ( x ) = Z E g ( x ) d E , g ( x /x ) = g ( x ) /g ( x ) . A ordingly to it, we an proof distribution (1) without observing timebut observing g ( x ) .4.1. S alar Boson. Spa e U is one-dimensional, real or omplex, density: g ( x, ψ ) = | ψ ( x ) | , k ψ ( x ) k = 1 , in a limited beam V .4.2. Ve tor Boson. WF of su h parti les is the ve tor u ( x ) , mapping E into E , or into its omplex analog E ∗ . We have: u = u − ( u ) . Eu lideanspa e U is separated from E by ondition: u ≥ . For the latter, it isne essary and su(cid:30) iently that u ′ = 0 in a (cid:28)xed frame of referen e x ′ . Thus,spa e U is de(cid:28)ned a urate to transformation u ′ → u , i.e. to ve tor v ofspeed of the system x relatively to x ′ . For massive boson x ′ is the system of oordinates related to the parti le. Therefore, ve tor v is (cid:28)xed, and spa e U isEu lidean se tion of the pseudo-Eu lidean spa e E . Density: g ( x, ψ ) = u ( x ) .Massless ve tor boson, photon, has not the system of oordinates relatedto the parti le, but the la k of the system x ′ , u ′ = 0 is not follow fromthis. Moreover, if this system is available, then it is not the only, ontrary tomassive parti le. Really, let's onsider a (cid:29)at wave pa kage su h that its waveve tors are parallel to a ve tor k . In luding Lorentz ondition in the set of(cid:28)eld equations provides: u ′ = u ′ = 0 . Assuming v ↑↑ k , similar to massiveboson, it is easy to make sure that density is de(cid:28)ned: u = u = u ′ = u ′ = , and spa e U is a plane with basis u , u , independently from | v | . I. e., alibration u = u = 0 is invariant in subgroup v ↑↑ k of the Lorentz group.It seems natural the following Postulate of photon: the system x ′ , u ′ = 0 isavailable. It determines the density g ( x ) = u ( x ) and onsistently minimizesdistin tion of bosons properties: the system of oordinates related to theparti le in this ase is absent, but its property u ′ = 0 is kept in any system,moving in parallel to wave ve tors. But it is obtained with destru tion of theprin iple of gradient invarian e of ele trodynami s. The latter is on(cid:28)rmed byits experien e. But all of it deals with values and distributions of tensions,energy, impulses, the moments and does not on ern distributions of thephotons oordinates. Only an experiment an determine alternative hoi e:either this prin iple is unappli able here and (1) is valid, or (1) is not valid4.3. Ele tron. U is Eu lidean spa e with elements, u = { u α } , α == 1 , , , , have transformational properties of the spinor, and ( u ) is time omponent of the ve tor. WF u ( x ) is usually onsidered as a traje tory inHilbert spa e H ( V ) with norm L ( V ) : k u ( t, x ) k = Z u ( t, x ) d E ; u = X α | u α | . It satis(cid:28)es the Dira 's WE, and k u ( t, x ) k is its dynami invariant. Theseproperties give the bases for equation: u ( x ) = g ( x /x ) , k u ( t, x ) k = 1 , [2℄.Let us introdu e a new WF ψ ( x ) , su h that g ( x ) = g ( x ) u ( x ) = ψ ( x ) , k ψ ( x ) k = 1 . In virtue of WE and typi al boundary onditions it oin ideswith u ( x ) a urate to normalization. We have: g ( x ) = const = 1 /cT , ψ == ( cT ) − / u , k ψ ( x ) k = 1 . While T → ∞ , V → E the limit k ψ ( x ) k isde(cid:28)ned here, if similar integral over E is de(cid:28)ned. Densities g ( x ) , g ( x /x ) oin ides a urate to normalization. They an be onstru ted both observing x , with parameter x , and observing dire tly x .5. Observation of energy-impulseLet us begin onsideration with real s alar boson, WF of whi h is de(cid:28)nedin a (cid:28)nite beam V ⊂ E . Observable value y is a ve tor of 4-impulse p == ( p , p ) ∈ E . Its average is a dynami invariant, quadrati form in H .Their eigen WF form a family: ψ k = a k exp( ip k x/ ¯ h ) with parameters a k > , p k ∈ E , is known dis rete row, p k = − ( mc ) , values a k are de(cid:28)ned bynormalization (cid:16)parti le in unit volume(cid:17), and distribution is des ribed interms of average quantities of parti les n k with given 4-impulse p k as aprototype of QF. More pre isely, n k is an average number of the measurement6essions with result p k . In respe tive spa e l of oe(cid:30) ients C = { C k } ofde omposition ψ = X k C k ψ k : n k = | C k | . And under an additional ondition k C k = 1 it is the un onditional distribution of probability of a individualparti le o urren es in spa e of the impulses, the relativisti invariant. Thetraditional distributions oin ide with the latter a urate to normalization,but they are attributed with the sense of onditional distribution at themoment of time t . This sense ontradi ts their relativisti invarian e, andmoreover, generates the known ontradi tion, [1℄: it requires instant (cid:28)xingof an impulse at measurements, whereas restri tions of a ura y of RQPobservation require a long (cid:28)xing. In ase of omplex WF the harge is addedto energy and impulse, and in multi omponent ase spin is added. Let usemphasize, that presen e of negative frequen ies in de omposition of WFand, a ordingly, observation of a (individual) parti le with di(cid:27)erent valuesof harge in (cid:28)xed pair of sessions, is admissible, but not pair o urren e(su h results are not taken into a ount). It does not ontradi t the laws of onservation whi h should be arried out only on the average. But it an belimited by external for QM laws, us law of the ele tri harge onservation.6. Quantum Field . The o upation of spa e-timeNew properties of RQP are transformed as applied to the QF in the formof hara teristi s of the parti les distributions into the spa e-time. A suitablefoundation for this: the Dira 's and Jordan's on eption of QF as a systemof identi al parti les, [2℄. Let's onsider a system of the N identi al parti leswith ommon wave fun tions ψ ( x ) ⊂ H ( V ) . Let y = { y k } be set of the QPobservables and h y i = { Y k } is this of the system. Let { ψ i } be the eigen basisof some physi al quantity; n = { n i } be the set of its o upation numbers, n i = 0 , , , . . . , N ; Φ( n ) be the symmetrized (respe tively, antisymmetrized)WF of the system expressed in terms of n .On observation pro edure. The operations of the system realization inidenti al physi al onditions orresponding to given WF have been de(cid:28)ned.A single session of observation is being performed over ea h realizationduring the time T . In ea h session appears, in general, not simultaneously, N parti les. In doing so, for every parti le is (cid:28)xed value y. Aggregate hara teristi sof the system are expressed dire tly through these values. For ea h WF theiraverages h Y i (Φ) orresponding to in(cid:28)nite set of sessions are de(cid:28)ned. Thetime is ex luded from the pro ess of observation as a parameter.Formal des ription. WF Φ( n ) maps a set of values n onto omplex Eu lideanspa e Υ with the elements γ and produ t γ γ and is onsidered as an element7f a Hilbert spa e with the produ t (Φ , Φ ) , normalized: k Φ( n ) k = 1 .The averages h Y i (Φ) are the HQF: h Y i (Φ) = (Φ , ΛΦ) , Λ (cid:22) orrespondingoperators. In parti ular, P ( Q, Φ) = (Φ( n ) , f ( n )Φ( n )) , where f ( n ) = 1 , n ∈∈ Q , f ( n ) = 0 , n / ∈ Q , is probability of the event n ∈ Q . Se ond quantizationreprodu es nonrelativisti analog, in luding, in addition to Φ( n ) , operatorsof the disappearan e and the birth of parti le a i , a + i and they attributed toa point x , (cid:22) wave operators (WO): Ψ( x ) = X i ψ i ( x ) a i , Ψ + ( x ) = X i ψ ∗ i ( x ) a + i ; (2)with the follow di(cid:27)eren es: WO as fun tions of x are de(cid:28)ned in H , but notin H , and time is in luded in them symmetri ally with spatial oordinates;formalism must be relativisti ally invariant; if { ψ i } is a olle tion of theplane waves, then this basis is (cid:16)doubled(cid:17) in virtue of the appearing stateswith negative frequen ies and, a ordingly, (cid:22) additional feature of QP, the harge, and the related omponents in (2) gain an uni(cid:28)ed view: ψ i ( x ) a i == ψ + i ( x ) b + i , ψ + i ( x ) a + i = ψ i ( x ) b i , where b i , b + i , are operators of the appearan eand the birth of these parti les. Also the synthesis te hnique of the operators Λ = { Λ k } of the system hara teristi s Y = { Y k } is reprodu ed, in ludinga rule: we re ord the average for an individual parti le, and produ e therepla ement: h y i ( ψ ∗ , ψ ) = ( ψ, Lψ ) = Z ψ ∗ ( x ) Lψ ( x ) dE ; ψ → Ψ( x ) ,ψ ∗ → Ψ + ( x ) , Λ = h y i (Ψ + ( x ) , Ψ( x )) . (3)In this produ t WO are onsidered as the elements of H . Respe tively: h Y i (Φ) = (Φ , ΛΦ) . Let { ψ i } , be the eigen basis of the observed, and P i be probabilities of the orresponding values y i when an individual RQP isobserved (or orresponding average numbers of measurements). Then withoutusing WO and (3) and making repla ement P i → n i , we may write down Λ = X i n i y i ; h Y i (Φ) = (cid:18) Φ , X i y i n i Φ (cid:19) = X i y i h n i i ; h n i i (cid:22) the average o upation numbers.Operators hara terizing the oordinates distribution of parti les arela king in RQM, as for the individual RQP, and (1), (3) (cid:28)ll this gap.Operator Λ( Q ) of the parti les amount in a domain Q ⊂ E is the follow: Λ( Q ) = (cid:0) Ψ + ( x ) , f ( x )Ψ( x ) (cid:1) = Z Q Ψ + ( x )Ψ( x ) dE, (4)8here f ( x ) = 1 , x ∈ Q , f ( x ) = 0 , x / ∈ Q . Operator Λ( Q ) of the parti lesamount in a domain Q ⊂ E oin ides with operator Λ( Q ) , Q = Q × (0 , T ) ⊂ E . The average o upation number of the domain Q : h N i ( Q, Φ) =(Φ , Λ( Q )Φ) .Let S ( ψ ∗ , ψ ) = ( ψ, L S ψ ) be a tion fun tional of QP, and S (Φ ∗ , Φ) == (Φ , Λ s Φ) (cid:22) of the system. Varying the latter with respe t to Φ ∗ ( n ) , weget the following WE: Λ s Φ( n ) = 0 , Λ s = (cid:0) Ψ + ( x ) , L s Ψ( x ) (cid:1) . (5)Let now the number N be not (cid:28)xed but varies from session to session.This is onsistent with the model of QF in the orpus ular on ept with ana ura y of the non-observed hara teristi s of the va uum state. WF shouldnow be symmetrized also in N , [2℄. And the rule (2), and respe tively the on rete representations of operators Λ , in luding Λ( Q ) , remain valid as wellas WE.6.1. Relativisti invarian e of the (cid:28)eld des ription. The numberof parti les in a given state is a result of observation, whi h does not dependon the hoi e of oordinates. A ordingly, a set of o upation numbers isrelativisti invariant, as well as operations on them a i , a + i . Therefore, WOhave relativisti transformation properties of the parti le WF ψ , and operators Λ have properties of their analogues L . Furthermore, Λ = (Ψ + ( x ) , L Ψ( x )) = X ij l ij a + i a j , l ij = ( ψ i , Lψ j ); h y i (Φ) = (Φ , ΛΦ) = X ij l ij (Φ , a + i a j Φ) . But (Φ , Φ) is invariant, as a orresponding value of the probability measure,as well as operators a + i a j . Thus, (Φ , a + i a j Φ) are invariants too, and theform (Φ , ΛΦ) possesses the relativisti transformation properties of the form ( ψ, Lψ ) . This des ription is fully invariant, unlike de omposition of (cid:28)eldinto system of the os illators, whi h is invariant in general, but ontainsnoninvariant fragments. Also this de omposition is not suitable to des riptionof the oordinates distributions.6.2. Representation of the Quantum Field Chara teristi s inTerms of O upation Numbers of the Impulse States. Let { ψ i ( x ) } be the eigen basis of impulse, of spin and of harge. With regard to energy,impulse, spin, harge (3) provides the textbook operators. Operator Λ( Q ) ofthe parti le amount in the domain Q ⊂ E was given with (4), where basis9 ψ i ( x ) } in WO should be renormalized: k ψ i ( x ) k = 1 instead of (cid:16)parti le inthe unit volume(cid:17). Let write (4) for the on rete parti les.S alar neutral boson. a i = b i , Ψ + ( x ) = Ψ( x ) ; Λ( Q ) = (1 / R Q Ψ ( x ) dE .Photon. In framework of model of item 4.2: Ψ + ( x ) = Ψ( x ) == (Ψ ( x ) , Ψ ( x )) ; Λ( Q ) = (1 / Z Q Ψ ( x ) dE . Here the WO Ψ ( x ) , Ψ ( x ) orrespond to the omponents of basis u , u .The s alar harged boson. Ψ( x ) = X i a i ψ i ( x ) + b + i ψ ∗ i ( x ) , Ψ + ( x ) == X i a + i ψ ∗ i ( x ) + b i ψ i ( x ) ; Λ( Q ) = Z Q Ψ + ( x )Ψ( x ) dE .6.3. Representation of the Quantum Field Chara teristi s inTerms of O upation Numbers of the Spa e (cid:22) time Cells. Letintrodu e into onsideration eigen basis of oordinates. For uni(cid:28) ation withdis rete basis of the impulse we make it at a prelimit level of Riemann integralsums. We split the beam V into olle tion of beams v ( ξ ) having volumes w ( ξ ) , every of them marked with belonging to one value x = ξ . De(cid:28)ne a set offun tions ψ ( x, ξ ) with ξ parameter: ψ ( x, ξ ) = w − ( ξ ) , x ∈ v ( ξ ) ; ψ ( x, ξ ) = 0 , x / ∈ v ( ξ ) . Approximate ψ ( x ) with step-fun tion ψ ′ ( x ) = ψ ( ξ ) , x ∈ v ( ξ ) .Fun tions ψ ( x, ξ ) , ψ ′ ( x ) are the elements of (cid:28)nite-dimensional subspa e H ′ ⊂ H ( V ) with orthogonal basis ψ ( x, ξ ) with a ura y of approximation: ( ψ ′ ( x ) , ψ ′ ( x )) = X ξ ψ ′∗ ( ξ ) ψ ′ ( ξ ) w ( ξ ); k ψ ′ ( x ) k = 1 , k ψ ( x, ξ ) k = w ( ξ ) − ; ψ ′ ( x ) = X ξ ψ ( ξ ) ψ ( x, ξ ) w ( ξ ) → ψ ( x ) = Z ψ ( ξ ) δ ( x − ξ ) dξ,w ( ξ ) → .P ( Q, ψ ′ ( x )) = Z Q ′ ψ ′ ( x ) dE = X ξ P ( ξ ) , P ( ξ ) = ψ ( ξ ) w ( ξ ) . (6)Here Q ′ is the minimal set of beams v ( ξ ) overing Q . A ordingly, ompletethe se ond quantization apparatus to the eigen basis of oordinates ψ ( x, ξ ) : n = { n ( ξ ) } (cid:22) the set of o upation numbers of the beams v ( ξ ) , n ( ξ ) == 0 , , , . . . , N . Making the repla ement P ( ξ ) → n ( ξ ) , we obtain the operator10 ′ ( Q ) of the parti les amount in the domain Q ′ ⊂ E : Λ( Q ′ ) = Z Q ′ Ψ ′ + ( x )Ψ ′ ( x ) dE = X ξ n ( ξ ) , ξ : v ( ξ ) ∈ Q ′ ; n ( ξ ) are a ting as eigen values of operator Λ ′ ( Q ) . The average amount ofparti les in Q ′ : h N i ( Q ′ , Φ) = Φ( n ) , X ξ n ( ξ )Φ( n ) = X ξ h n ( ξ ) i , ξ : v ( ξ ) ∈ Q ′ ; where h n ( ξ ) i is average parti les amount in v ( ξ ) .6.4. An Individual Parti le as a Quantum Field subsystem.Let onsider an individual parti le with WF ψ ( x ) in terms of the se ondquantization as subsystem N = 1 of the system (cid:16)QF(cid:17). We de(cid:28)ne it in thefollowing way: we observe a QF, and only the sessions of QF, that bringout N = 1 . We have: n = (0 , , . . . , , , . . . ) , n i = 0 , . The term (cid:16)identi alparti les(cid:17) loses its meaning, and WF Φ( n ) does not ontain the permutationoperator. We (cid:28)nd an average amount of parti les in domain Q ⊂ E : h N i ( Q, Φ) = (Φ , Λ ′ ( Q )Φ) , using the (cid:28)nite dimensional approximation givenabove. { ψ ( x, ξ ) } (cid:22) is the eigen basis of the observed. To every n orrespondsthe event x ( n ) ∈ E . Identify the point x ( n ) with the mark ξ of the beam v ( ξ ) . We have: WF Φ( n ) = ψ ( x ( n )) map the set of all n into U . In agreementwith (6) should be put (Φ , Φ ) = X n Φ ( n )Φ ( n ) w ( ξ = x ( n )) . We have: k Φ( n ) k = 1; h N i ( Q, Φ) = (Φ , Λ ′ ( Q )Φ) = X n Φ ( n ) = X ξ ψ ( ξ ) w ( ξ ) = P ( Q ′ , ψ ) → Z Q ψ ( ξ ) dξ, w ( ξ ) → n : x ( n ) ∈ Q, ξ : v ( ξ ) ∈ Q ′ . Thus, this fun tion Φ( n ) meets the de(cid:28)nition of WF, and observing su hsubsystem of QF gives the probability density (1) of the individual parti le.7. About Relations of Un ertainties and ObservationA ura y EstimatesIn non-relativisti QM the Heisenberg relations of un ertainties are thee(cid:27)e ts of the statisti al postulates des ribed above. Within the framework11f traditional model of relativisti QP they no longer have this theoreti albasis due to the absen e of the required law of the oordinates distribution.Nevertheless, they are used in the same way, stri tly speaking, now as anindependent postulate. In the model onsidered these relations obtain substantiation.Therewith, while in non-relativisti QM the relation oordinate-impulse andtime-energy are dedu ed in di(cid:27)erent ways and have di(cid:27)erent sense, [3℄ ñòð. 185 (cid:21)188, but herewith, they possess full formal and semanti symmetry. Theoreti albottom threshold of un ertainty of oordinates: △ x > △ x min = ¯ hc/ε ( ε (cid:22)energy), aused by these relations and absen e of the QF e(cid:27)e ts (for photonthis is an order of its wave length), loses validity, be ause under the observationin framework of our pro edure QF e(cid:27)e ts are admissible. This theoreti albottom threshold is repla ed with the minimal pra ti ally a eptable valueof probability of the individual parti le (cid:28)xing under the observation of QF.It is required to give a spe ial interpretation of the un ertainties relationtime-speed-impulse v △ t △ p > ¯ h , and a derived from it relation of impulseobservation a ura y with duration of an observation session △ t △ p > ¯ h/c △ t .First of all, does the on ept (cid:16)QP speed(cid:17) make sense in the given model, andif yes, whi h one?8. Material for the experimentIt provided by the new properties of RQP and QF being predi ated here:the probability density formulas for various types of RQP oordinates andformulas for distributions of amounts of parti les into spa e-time for QF.The simplest variant to he k the latter: to lo ate WF Φ at eigenbasis of 4-impulse, to al ulate the orresponding theoreti al h N i ( Q, Φ) and omparingit with dire tly measured h N i ( Q, Φ) . A signi(cid:28) ant new feature of the modelobservation: non onditionality of the measure from the moment of time t . Fora time T the repeated rea tions of parti les with instrument of observationare possible, and respe tively (cid:22) nonuniqueness value y when measurement.The latter should be eliminated in any way. Unlike textbook views, anindividual RQP observation under pro edures des ribed are not burdenedwith problem of preventing e(cid:27)e ts of QF. The parti ular interest representthe photon oordinates observation. From one side, it is ideally provided theuniqueness of the ollision with the parti le-devi e as far as it disappearsunder the rea tion. From the other side the validity of (1) for it is stipulatedwith the additional postulate, whi h is alternate to ele trodynami prin ipleof gradient invarian e. And an experiment will determine the alternative:either the given prin iple is invalid here, and the (1) is valid, or our postulate12f photon is invalid, and it's oordinate distribution is not determined.9. Con lusionsWhen the statisti al part of RQP has been in luded in logi of theSR su(cid:30) iently fully, there appears, despite ommon opinion, a theoreti alpossibility to preserve informative proper-ties of the WF in the extent omparableto non-relativisti QM. There appears a theoreti al op-portunity to observespatial-time oordinates of the RQP, and the presentation of the probabilitydensity g ( x ) = ψ ( x ) satisfying all ne essary requirements gets de(cid:28)ned.Here WF ψ ( x ) maps spa e-time E into Eu lidean spa e U , hara teristi forea h type of parti les: boson, real and omplex, s alar and ve tor, in ludingphoton, and ele tron. In appli ation to photon this formula of the density is onditioned by an additional assumption, that is alternative to the ele tro-dynami prin iple of gradient invarian e. Density g ( x ) in a simplest aseis formally alike the density of the probability of non-relativisti parti le's oordinates, but has other ontent: other observation onditions, other properties,other interpretation of density. Observation sessions are not aused by moment t , i.e. time is ex luded from the observation pro ess as a parameter, but ispresent in the argument x and the observable value y = x . Density now isrelativisti invariant. Instead of sto hasti dan e of the parti le, des ribedby the probability (cid:29)ow with spatial density f ( t, x ) = | ψ ( t, x ) | , we haveprobabilisti distribution of parti le's appearan e in spa e-time with density g ( x ) , not resulted, in general, to motion in the spa e. In a simplest ase theidea of su h distribution of oordinates was onsidered in [4℄, and [5℄ is a shortversion of this paper. The interpretation of distributions of energy-impulse ofRQP is also lari(cid:28)ed here. Relations of un ertainty of Heisenberg are the yieldof the postulates of non-relativisti QM. In the frame-work of traditionalmodel of RQP they no longer have this theoreti al rationale, as there is nolaw of oordinates' distribution, ne essary for that. These relations obtainthe justi(cid:28) ation in the model onsidered. An observation pro edure, (cid:28)lteringe(cid:27)e ts of QF, have been proposed and justi(cid:28)ed. Respe tive restri tions ofthe RQP observation pre ision, aused by these e(cid:27)e ts, are abandoned. Asapplied to the QF, this new onstru ts are transformed to new hara teristi sof the parti les distribution in spa e-time, whi h omplete distribution throughoutimpulses. The operators of these distributions and the invariant relativisti des ription for free QF have been obtained.Abbreviations: Quantum Parti le (cid:22) QP; Wave Fun tion (cid:22) WF; quantumme hani s (cid:22) QM; relativisti QP (cid:22) RQP; Spe ial Relativity (cid:22) SR; Hermitian13uadrati Form (cid:22) HQF; Wave Equation (cid:22) WE; Relativisti QM (cid:22) RQM.Ñïèñîê ëèòåðàòóðû[1℄ V.B. Berestetsky, E.M. Lifshits, L.P. Pitaevsky. Relativistska'aKvantova'a Teori'a (L.D. Landau, E.M. Lifshits. Theoreti heska'aPhysi a, T.IV ), Chast' 1, Moskva, Nauka, 1968, pp. 13 (cid:21) 17 (in Russian).[2℄ P.A.M. Dirak. Prini ples of Quantum Me hani s. Oxford, ClarendonPress, 1958.[3℄ L.D. Landau, E.M. Lifshits. Theoreti heska'a Physi a, T. III, Kvantova'aMe hanika. Phys-MathGis, Mos ow, 1963, 702 p. (in Russian).[4℄ V.F. Krotov. // Quadrati Parametri al Families of the Measures andBasi s of Quantum Me- hani s. International Journal of ComputingAnti ipatory Systems, 2006, v. 17, p. 77 (cid:21) 92.[5℄ V.F. Krotov. O statisti heskih svoistvah volnovoj fun tsii svobodnojrelativists oj hastitsy. // Trudy Instituta Problem Upravleni'a RAN,v. XXVIII, Moskva, 2008, p. 5 (cid:21) 13 (in Russian).14 r X i v : . [ qu a n t - ph ] J u l Âîëíîâàÿ óíêöèÿ ÷àñòèöû è ðàñïðåäåëåíèÿ êîîðäèíàò âðåëÿòèâèñòñêîé êâàíòîâîé òåîðèèÂ.Ô. ÊðîòîâÈíñòèòóò ïðîáëåì óïðàâëåíèÿ îññèéñêîé àêàäåìèè íàóê,Ïðîñîþçíàÿ, 65, 117997, Ìîñêâà, îññèÿ(Äàòà: Àïðåëü 24, 2008)Îïèñûâàþòñÿ óñëîâèÿ íàáëþäåíèÿ êîîðäèíàò åäèíè÷íîé ÷àñòèöû, äèêòóåìûå ëîãèêîéñïåöèàëüíîé òåîðèè îòíîñèòåëüíîñòè è èëüòðóþùèå ýåêòû êâàíòîâîãî ïîëÿ. Íàõîäèòñÿîáùàÿ ñâÿçü ñîîòâåòñòâóþùåé ïëîòíîñòè âåðîÿòíîñòè ñ âîëíîâîé óíêöèåé. Ýòî (cid:22) ðåëÿòè-âèñòñêèé èíâàðèàíò, îïèñûâàþùèé âåðîÿòíîñòü ïîÿâëåíèé ÷àñòèöû â ïðîñòðàíñòâå-âðåìåíè.Îíà êîíêðåòèçèðóåòñÿ äëÿ áîçîíà, íåéòðàëüíîãî è çàðÿæåííîãî, ñêàëÿðíîãî è âåêòîðíîãî,âêëþ÷àÿ îòîí, ýëåêòðîíà. Îòíîøåíèÿ íåîïðåäåëåííîñòåé åéçåíáåðãà ïîëó÷àþò îáîñíîâà-íèå ïðèìåíèòåëüíî ê ðåëÿòèâèñòñêîé ÷àñòèöå. Ïðèìåíèòåëüíî ê êâàíòîâîìó ïîëþ ýòè êîí-ñòðóêöèè òðàíñîðìèðóþòñÿ â íîâûå õàðàêòåðèñòèêè ðàñïðåäåëåíèÿ ÷àñòèö â ïðîñòðàíñòâå-âðåìåíè, äîïîëíÿþùèå òàêîâûå äëÿ èìïóëüñíûõ ñîñòîÿíèé. Ïîëó÷åíû îïåðàòîðû ýòèõ ðàñ-ïðåäåëåíèé äëÿ ñâîáîäíûõ ïîëåé è ðåëÿòèâèñòñêè-èíâàðèàíòíîå îïèñàíèå ïîñëåäíèõ. Ýòèíîâûå ñâîéñòâà ÷àñòèö è ïîëåé ïðåäëàãàþòñÿ äëÿ ýêñïåðèìåíòàëüíîãî èññëåäîâàíèÿ.PACS 03+p; 03.65-w; 03.65 Ta1. Ââåäåíèå ìîäåëè êâàíòîâîé ñèñòåìû, è â ÷àñòíîñòè, ÷àñòèöû (Ê×), ñèíòå-çèðîâàíû äåòåðìèíèðîâàííàÿ äèíàìèêà âîëíîâîé óíêöèè (ÂÔ) è ñòà-òèñòè÷åñêàÿ ñâÿçü ïîñëåäíåé ñ íàáëþäàåìûìè âåëè÷èíàìè: äèíàìè÷å-ñêàÿ è ñòàòèñòè÷åñêàÿ ÷àñòè îïèñàíèÿ Ê×. Åñëè ïåðâàÿ âïîëíå îð-ìàëèçîâàíà, òî ïðîáëåìû âòîðîé äëÿ ðåëÿòèâèñòñêîé Ê× (Ê×) ïðè-âåëè ê îáùåìó ìíåíèþ, ÷òî (cid:16)â ïîñëåäîâàòåëüíîé ðåëÿòèâèñòñêîé êâàí-òîâîé ìåõàíèêå êîîðäèíàòû ÷àñòèö âîîáùå íå ìîãóò èãóðèðîâàòü âêà÷åñòâå äèíàìè÷åñêèõ ïåðåìåííûõ, . . . à ÂÔ êàê íîñèòåëè èíîðìàöèèíå ìîãóò èãóðèðîâàòü â åå àïïàðàòå(cid:17) [1℄, ñòð. 16. Ýòè ïðîáëåìû: îòñóò-ñòâèå îðìàëüíûõ êîíñòðóêöèé, îïèñûâàþùèõ ðàñïðåäåëåíèÿ êîîðäè-íàò, è îãðàíè÷åííîñòü ïðèìåíåíèÿ ìîäåëè ââèäó ýåêòîâ êâàíòîâîãîïîëÿ (ÊÏ). Äîáàâèì: óòðà÷èâàþò òåîðåòè÷åñêîå îñíîâàíèå îòíîøåíèÿíåîïðåäåëåííîñòåé åéçåíáåðãà, ïîñêîëüêó íåîáõîäèìûé äëÿ èõ âûâîäàçàêîí ðàñïðåäåëåíèÿ êîîðäèíàò îòñóòñòâóåò. Çäåñü ïîêàçûâàåòñÿ, ÷òî,1îïðåêè ñëîæèâøåìóñÿ ìíåíèþ, ïðè äîñòàòî÷íî ïîëíîì âêëþ÷åíèè ñòà-òèñòè÷åñêîé ÷àñòè Ê× â ëîãèêó ñïåöèàëüíîé òåîðèè îòíîñèòåëüíîñòè(ÑÒÎ), îáå ýòè ïðîáëåìû íàõîäÿò ðåøåíèå, ñîõðàíÿþòñÿ èíîðìàòèâ-íûå ñâîéñòâà ÂÔ â îáúåìå, ñðàâíèìîì ñ íåðåëÿòèâèñòñêîé êâàíòîâîéìåõàíèêîé (ÊÌ) è èìåþòñÿ ïðîöåäóðû íàáëþäåíèÿ ÷àñòèöû, èëüòðó-þùèå ýåêòû êâàíòîâîãî ïîëÿ (ÊÏ). Ïîìèìî îáû÷íûõ óñëîâèé, ýòîâêëþ÷åíèå òðåáóåò êîððåêöèè ïîëíîãî íàáîðà íàáëþäàåìûõ è ïðîöåäóðíàáëþäåíèÿ. Èç ïîñëåäíåãî ñëåäóåò è êîððåêöèÿ òðåáîâàíèé êîâàðèàíò-íîñòè. Ïîëó÷åíû âûðàæåííûå ÷åðåç ÂÔ êîíñòðóêöèè, îáëàäàþùèå âñå-ìè íåîáõîäèìûìè ñâîéñòâàìè âåðîÿòíîñòíûõ ðàñïðåäåëåíèé íàáëþäàå-ìûõ, âêëþ÷àÿ êîîðäèíàòû ÷àñòèöû. Ïëîòíîñòü âåðîÿòíîñòè ïîñëåäíèõâ ïðîñòåéøèõ ñëó÷àÿõ îðìàëüíî ïîäîáíà åå íåðåëÿòèâèñòñêîìó àíàëî-ãó, íî èìååò èíîå ñîäåðæàíèå: èíûå óñëîâèÿ íàáëþäåíèÿ, èíûå ñâîéñòâà,èíîé ñìûñë. Ñåàíñû íàáëþäåíèÿ íå îáóñëîâëåíû ìîìåíòîì âðåìåíè, ò. å.âðåìÿ èñêëþ÷àåòñÿ èç ïðîöåññà íàáëþäåíèÿ â êà÷åñòâå ïàðàìåòðà, íîïðèñóòñòâóåò â ñîñòàâå àðãóìåíòà ÂÔ è íàáëþäàåìûõ; ñîîòâåòñòâåííî,ïëîòíîñòü òåïåðü (cid:22) ðåëÿòèâèñòñêèé èíâàðèàíò; âìåñòî ñòîõàñòè÷åñêî-ãî òàíöà ÷àñòèöû, îïèñûâàåìîãî ïîòîêîì âåðîÿòíîñòè, èìååì âåðîÿò-íîñòíîå ðàñïðåäåëåíèå ïîÿâëåíèé ÷àñòèöû â ïðîñòðàíñòâå-âðåìåíè, íåñâîäèìîå, âîîáùå, ê äâèæåíèþ â ïðîñòðàíñòâå. Íà ýòîé áàçå ïîäòâåð-æäàþòñÿ äëÿ Ê× îòíîøåíèÿ íåîïðåäåëåííîñòåé. Ïðèìåíèòåëüíî ê ÊÏýòè íîâûå êîíñòðóêöèè òðàíñîðìèðóþòñÿ â íîâûå õàðàêòåðèñòèêè ðàñ-ïðåäåëåíèÿ ÷àñòèö â ïðîñòðàíñòâå-âðåìåíè, äîïîëíÿþùèå òàêîâûå äëÿèìïóëüñíûõ ñîñòîÿíèé.2. Íåðåëÿòèâèñòñêàÿ ÷àñòèöàÏîñêîëüêó ñòàòèñòè÷åñêàÿ ÷àñòü äîñòàòî÷íî îðìàëèçîâàíà òîëüêîäëÿ íåðåëÿòèâèñòñêîé ÊÌ, òî íà÷íåì ñ èêñàöèè íåîáõîäèìûõ çäåñü ååäåòàëåé, îãðàíè÷èâøèñü áåññïèíîâîé Ê×. Ïóñòü x = { x i } = ( x , x , x ) (cid:22)âåêòîð ïðîñòðàíñòâåííûõ êîîðäèíàò ÷àñòèöû, ýëåìåíò ñîîòâåòñòâóþ-ùåãî ýâêëèäîâà ïðîñòðàíñòâà E . Êîìïëåêñíàÿ ÂÔ ψ ( x ) îïðåäåëåíà íà E è ðàññìàòðèâàåòñÿ êàê ýëåìåíò ãèëüáåðòîâà ïðîñòðàíñòâà H ñ íîð-ìîé L ( E ) è ñîîòâåòñòâóþùèì ïðîèçâåäåíèåì ( ψ , ψ ) = R E ψ ( x ) ψ ∗ ( x ) d E , d E (cid:22) ýëåìåíòàðíûé îáúåì E , ∗ (cid:22) çíà÷îê êîìïëåêñíîãî ñîïðÿæåíèÿ. Êàêïðàâèëî, ÂÔ èíèòíà â íåêîòîðîì êóáå V ⊂ E : ψ ( x ) = 0 , x / ∈ V , è ýòîòíåñîáñòâåííûé èíòåãðàë îïðåäåëåí ýëåìåíòàðíî. Ýâîëþöèÿ ÂÔ ψ ( t, x ) íà èíòåðâàëå (0 , T ) îïèñûâàåò äâèæåíèå Ê×. Îíà óäîâëåòâîðÿåò óðàâ-íåíèþ Øðåäèíãåðà, ïðè÷åì k ψ ( t, x ) k (cid:22) åãî äèíàìè÷åñêèé èíâàðèàíò,2îðìèðîâàííûé óñëîâèåì k ψ ( t, x ) k = 1 .Ê ïðîöåäóðå íàáëþäåíèÿ. Ïîëíàÿ ñîâîêóïíîñòü íàáëþäàåìûõ y == { y i } (cid:22) ëèáî âåêòîð x , ëèáî èìïóëüñ. Ïóñòü îïðåäåëåíû îïåðàöèè ðå-àëèçàöèè äâèæåíèÿ Ê× â òå÷åíèå âðåìåíè T â òîæäåñòâåííûõ èçè÷å-ñêèõ óñëîâèÿõ, îòâå÷àþùèõ äàííîé ýâîëþöèè ÂÔ ψ ( t, x ) . Íàä êàæäîéðåàëèçàöèåé â çàäàííûé ìîìåíò âðåìåíè t ïðîâîäèòñÿ åäèíñòâåííûé ñå-àíñ íàáëþäåíèÿ (çàìåð). Çíà÷åíèÿ y â êàæäîì ñåàíñå îïðåäåëåíû, à ïðèèõ ïîâòîðåíèè (cid:22) ñëó÷àéíû. Äîïóñòèìî è íàáëþäåíèå òðàåêòîðèé y ( t ) âñëó÷àÿõ, êîãäà îíè îòîæäåñòâèìû ñ âðåìåííûì ðÿäîì ðåçóëüòàòîâ çà-ìåðîâ íåçàâèñèìûõ ðåàëèçàöèé. Äëÿ êàæäîé èêñèðîâàííîé óíêöèè ψ ( t, x ) è êàæäîãî t îïðåäåëåíû ñðåäíèå h y i [ ψ ( t, x )] = {h y i i ( ψ ) } , ñîîòâåò-ñòâóþùèå áåñêîíå÷íîìó íàáîðó çàìåðîâ.Ê îðìàëüíîìó îïèñàíèþ. Ñðåäíèå ñóòü ýðìèòîâû êâàäðàòè÷íûåîðìû (ÝÊÔ) â H : h y i i ( ψ ) = ( ψ, L i ψ ) , L i (cid:22) ñîîòâåòñòâóþùèé îïåðàòîð(îïåðàòîð i -é íàáëþäàåìîé). Åñëè y = x , òî ïîñòóëèðóåòñÿ íàëè÷èå ïëîò-íîñòè âåðîÿòíîñòè f ( t, x ) è ñëåäóþùàÿ åå ñâÿçü ñ ÂÔ: f ( t, x ) = | ψ ( t, x ) | ,è ñîîòâåòñòâåííî, (cid:22) ìåðà âåðîÿòíîñòè: P ( Q/t ) = P [ Q, ψ ( t, x )] == R | ψ ( t, x ) | d E , Q ∈ E . Åñëè y (cid:22) èìïóëüñ, òî ïîñòóëèðóåòñÿ ðàâåí-ñòâî h y i [ ψ ] = J ( ψ ) , ãäå ïðàâàÿ ÷àñòü (cid:22) ñîîòâåòñòâóþùèé äèíàìè÷åñêèéèíâàðèàíò âîëíîâîãî óðàâíåíèÿ (ÂÓ), ÝÊÔ.  ýòîì ñëó÷àå ìåðà âåðî-ÿòíîñòè P ( Q, ψ ) íà ïîäìíîæåñòâàõ Q ïðîñòðàíñòâà íàáëþäàåìûõ âû-ðàæàåòñÿ èçâåñòíûì îáðàçîì ÷åðåç êîìïîíåíòû ñïåêòðàëüíûõ óíêöèéîïåðàòîðîâ L i .Èìåþòñÿ èñêëþ÷åíèÿ èç ïðàâèëà ψ ∈ L ( E ) , êîãäà íîðìà íå îïðåäå-ëåíà è íå óäîâëåòâîðÿåòñÿ óñëîâèå P ( E , ψ ) = 1 . Íî îïðåäåëåíû îòíîñè-òåëüíûå âåðîÿòíîñòè P ( Q , ψ ) /P ( Q , ψ ) . Îòìåòèì òàêæå, ÷òî â ëþáîìèêñèðîâàííîì êóáå V ⊂ E ÂÔ åñòü ýëåìåíò ãèëüáåðòîâà ïðîñòðàíñòâà H ( V ) ñ íîðìîé L ( V ) . Ôèêñèðóÿ çäåñü k ψ k = 1 , ìîæíî ñîõðàíèòü çà P ( Q, ψ ) , Q ⊂ V , ñìûñë ìåðû âåðîÿòíîñòè íàáëþäåíèÿ ïîçèöèé x , à çà f ( t, x ) (cid:22) åå ïëîòíîñòè, ïðè ñëåäóþùåé ïîïðàâêå ê ïðîöåäóðå íàáëþäå-íèÿ: ó÷èòûâàþòñÿ òîëüêî ñåàíñû, ïîêàçûâàþùèå x ∈ V .3. åëÿòèâèñòñêàÿ ÷àñòèöà. Îáùåå îïèñàíèå ìîäå-ëè Ïóñòü E (cid:22) äåéñòâèòåëüíîå ïñåâäîýâêëèäîâî ïðîñòðàíñòâî ñ êîîð-äèíàòíûì âåêòîðîì x = { x α } = ( x , x ) , x ∈ E ; x = ct , t (cid:22) âðåìÿ, c (cid:22) ñêîðîñòü ñâåòà; ìåòðè÷åñêèé òåíçîð e = { e αβ } (cid:22) äèàãîíàëüíûé: e = − , e ii = 1 , i > . Íà êîíå÷íîì áðóñå V = (0 , cT ) × V ⊂ E ψ ( x ) , îòîáðàæàþùàÿ V â ýâêëèäîâî ïðîñòðàíñòâî U , m -ìåðíîå, äåéñòâèòåëüíîå ëèáî êîìïëåêñíîå, ýëåìåíòàìè u è ïðîèç-âåäåíèåì u u = m X ( u ∗ ) i u i . Îíà óäîâëåòâîðÿåò ñîîòâåòñòâóþùåìó ÂÓè ðàññìàòðèâàåòñÿ êàê ýëåìåíò ãèëüáåðòîâà ïðîñòðàíñòâà H ( V ) ñ íîð-ìîé k ψ ( x ) k = Z V u dE , u = ψ ( x ) , dE = dx d E (íîðìó ïðîñòðàíñòâà H îáîçíà÷àåì k ψ ( t, x ) k ) è ñîîòâåòñòâóþùèì ïðîèçâåäåíèåì ( ψ , ψ ) . Äëÿêàæäîé ÂÔ îïðåäåëåíû òîæäåñòâåííûå èçè÷åñêèå óñëîâèÿ, â êîòîðûõíàáëþäàåòñÿ Ê×.Ïðîöåäóðà íàáëþäåíèÿ âîñïðîèçâîäèò òàêîâóþ äëÿ íåðåëÿòèâèñòñêîéÊ× ñî ñëåäóþùèìè îòëè÷èÿìè. Çàìåð íå îáóñëîâëåí ìîìåíòîì âðåìå-íè t . Òî åñòü âðåìÿ èñêëþ÷àåòñÿ èç ïðîöåññà íàáëþäåíèÿ â êà÷åñòâåïàðàìåòðà, íî ïðèñóòñòâóåò â ñîñòàâå àðãóìåíòà x è, âîçìîæíî, íàáëþ-äàåìîé y . Èñêëþ÷åíèå îñîáîé ðîëè âðåìåíè â çàìåðàõ (cid:22) íåîáõîäèìûéýëåìåíò âíåäðåíèÿ ëîãèêè ÑÒÎ. Çàìåð ïðîäîëæàåòñÿ â òå÷åíèå âðåìåíè T . Îñòàëüíûå óñëîâèÿ îñòàþòñÿ â ñèëå. ëàâíîå èç íèõ: ñóùåñòâîâàíèåè åäèíñòâåííîñòü çíà÷åíèÿ y ïðè çàìåðå â íîâûõ óñëîâèÿõ. Ìû îáñóäèìåãî íèæå ïðèìåíèòåëüíî ê êîíêðåòíûì íàáëþäàåìûì. Ïðåäïîëàãàåòñÿ,÷òî îïðåäåëåíà ìåðà P ( Q, ψ ) äëÿ êàæäîé ÂÔ, è ñîîòâåòñòâåííî, ñðåäíåå h y i [ ψ ( x )] .Ôîðìàëüíîå îïèñàíèå âîñïðîèçâîäèò òàêîâîå äëÿ íåðåëÿòèâèñòñêîéÊ× ñî ñëåäóþùèìè îòëè÷èÿìè. ÝÊÔ h y i i ( ψ ) = ( ψ, L i ψ ) , è ñîîòâåòñòâåí-íî, îïåðàòîðû íàáëþäàåìûõ îïðåäåëåíû â H , à íå â H . Õàðàêòåðèñòèêè y , è ñîîòâåòñòâåííî, èõ ñðåäíèå, îáëàäàþò ðåëÿòèâèñòñêèìè òðàíñîð-ìàöèîííûìè ñâîéñòâàìè. Ïðè ýòîì, P ( Q, ψ ) (cid:22) ðåëÿòèâèñòñêèé èíâàðè-àíò. Ïîñëåäíåå ñëåäóåò èç ðàâåíñòâà h y i ( ψ ) = R ydP ( ψ ) , ïîñêîëüêó h y i ( ψ ) è y èìåþò îäèíàêîâóþ òåíçîðíóþ ðàçìåðíîñòü. àçìåðíîñòü ïëîòíîñòèâåðîÿòíîñòè g ( y, ψ ) îïðåäåëÿåòñÿ ðàâåíñòâîì dP = gdY = inv , ãäå dY (cid:22)ýëåìåíòàðíûé îáúåì ïðîñòðàíñòâà íàáëþäàåìûõ. Äàëåå ñëåäóåò äåòà-ëèçàöèÿ ìîäåëè ïðèìåíèòåëüíî ê êîíêðåòíûì íàáëþäàåìûì è òèïàì÷àñòèö.4. Íàáëþäåíèå êîîðäèíàòÊ ïðîöåäóðå íàáëþäåíèÿ. Ïîëíûé íàáîð íàáëþäàåìûõ ðàñøèðÿåò-ñÿ äî: y = x ; ðåçóëüòàò êàæäîãî çàìåðà (cid:22) èêñàöèÿ ñîáûòèÿ: ÷àñòè-öà ïîÿâèëàñü â òî÷êå x ïðîñòðàíñòâà-âðåìåíè E . Ñîäåðæàíèå çàìåðà:ïóñòü ïðèáîð (cid:22) ýëåêòðîííûé ìèêðîñêîï; ïó÷îê ýëåêòðîíîâ (cid:16)ïðîùóïû-4àåò(cid:17) ïðîñòðàíñòâî, è â íåêîòîðûé ìîìåíò ïðîèñõîäèò ñòîëêíîâåíèå ñ÷àñòèöåé-îáúåêòîì; îíî, âîîáùå, íàðóøàåò óñëîâèÿ íàáëþäåíèÿ, îòâå-÷àþùèå äàííîé ÂÔ (êðàéíèé ñëó÷àé (cid:22) îáúåêò-îòîí, èñ÷åçàþùèé ïðèñòîëêíîâåíèè), ïîýòîìó äîëæíî ó÷èòûâàòüñÿ òîëüêî îäíî ñòîëêíîâåíèå;ïîçèöèÿ ÷àñòèöû èêñèðóåòñÿ òî÷êîé íà ýêðàíå, (cid:22) ñëåäîì åäèíñòâåííî-ãî ðàññåÿííîãî ýëåêòðîíà. Äëÿ ïîëíîé èêñàöèè ñîáûòèÿ, íåîáõîäèìàòàêæå èêñàöèÿ ìîìåíòà ñòîëêíîâåíèÿ. Ó÷èòûâàþòñÿ òîëüêî ñåàíñû,ïîêàçûâàþùèå ñîáûòèå x ∈ V , ïðèòîì åäèíñòâåííîå. Ñåàíñû, íàðóøà-þùèå ýòî óñëîâèå, èñêëþ÷àþòñÿ èç ðàññìîòðåíèÿ. Ïðè÷èíû íàðóøå-íèé: ïîêàçàíèå x / ∈ V , íååäèíñòâåííîñòü ñòîëêíîâåíèé, ýåêòû ÊÏ.Ñïðàâåäëèâîñòü äîïóùåíèÿ ïîñëåäíèõ è èõ èëüòðàöèè òàêèì ñïîñî-áîì íåî÷åâèäíà, ïîñêîëüêó Ê× è ÊÏ ñóòü ðàçíûå ñèñòåìû. Íî ýòî òàê,ñì. íèæå, ï. 6.4.  ýòîì ñìûñëå äîïóñêàåòñÿ íàáëþäåíèå ÊÏ.Ê îðìàëüíîìó îïèñàíèþ. Âûðàæåíèå g ( x ) = ψ ( x ) , k ψ ( x ) k = 1 (1)îáëàäàåò âñåìè íåîáõîäèìûìè ñâîéñòâàìè ïëîòíîñòè âåðîÿòíîñòè ñî-áûòèé x ∈ V è ïðåäëàãàåòñÿ äëÿ ýêñïåðèìåíòàëüíîé ïðîâåðêè â ýòîìêà÷åñòâå. Îíî îðìàëüíî ïîäîáíî åãî íåðåëÿòèâèñòñêîìó âàðèàíòó, íîèìååò èíîå ñîäåðæàíèå. Èíûå óñëîâèÿ íàáëþäåíèÿ, èíûå òðàíñîðìàöè-îííûå ñâîéñòâà (ðåëÿòèâèñòñêèé èíâàðèàíò), èíîé ñìûñë: âåðîÿòíîñòüñîáûòèé â ïðîñòðàíñòâå-âðåìåíè âìåñòî ïîçèöèé â ïðîñòðàíñòâå. Èçìå-íåíèå ñìûñëà ñîîòâåòñòâóåò ëîãèêå ÑÒÎ è ïðåäñêàçûâàåò íîâîå ñâîéñòâîÊ×. Âìåñòî ñòîõàñòè÷åñêîãî òàíöà, îïèñûâàåìîãî ïîòîêîì âåðîÿòíîñòè,èìååì âåðîÿòíîñòíîå ðàñïðåäåëåíèå ñëó÷àéíûõ ïîÿâëåíèé ÷àñòèöû âïðîñòðàíñòâå-âðåìåíè, íå ñâîäèìîå, âîîáùå, ê äâèæåíèþ â ïðîñòðàí-ñòâå.Çíàÿ g ( x ) , ìîæíî íàéòè ïëîòíîñòü g ( x ) âåðîÿòíîñòè ïîçèöèé x , g ( x ) (cid:22) âðåìåíè x , ïðîñòðàíñòâåííûõ êîîðäèíàò â èêñèðîâàííûé ìî-ìåíò êàê óñëîâíóþ g ( x /x ) : g ( x ) = Z (0 , ∞ ) g ( x ) dx , g ( x ) = Z E g ( x ) d E , g ( x /x ) = g ( x ) /g ( x ) . Ýòî çíà÷èò, ÷òî ïðåäñêàçûâàåìûå ðàñïðåäåëåíèÿ ìîæíî ïðîâåðÿòü, íåíàáëþäàÿ âðåìåííóþ êîìïîíåíòó ñîáûòèé.Çàìå÷àíèå. Ôèíèòíîñòü ÂÔ ψ ( x ) íà V , âîîáùå, íå ñîâìåñòíà ñ ÂÓ,âî âñÿêîì ñëó÷àå, äëÿ áîçîíîâ. Ýòî çíà÷èò, ÷òî â çàìåðàõ âîçìîæíûïîêàçàíèÿ x / ∈ V , òàêæå êàê ïîÿâëåíèå ïàð è äðóãèõ ýåêòîâ ÊÏ. Ò. å.5àêîí (1) îïèñûâàåò óñëîâíîå ðàñïðåäåëåíèå âåðîÿòíîñòè íà èçáðàííîììíîæåñòâå çàìåðîâ, ãäå îòñóòñòâóþò òàêèå ðåçóëüòàòû.4.1. Ñêàëÿðíûé áîçîí. Ïðîñòðàíñòâî U îäíîìåðíîå, äåéñòâèòåëü-íîå ëèáî êîìïëåêñíîå, ïëîòíîñòü: g ( x, ψ ) = | ψ ( x ) | .4.2. Âåêòîðíûé áîçîí. ÂÔ åñòü âåêòîð u ( x ) , îòîáðàæàþùèé E â E ,ëèáî â åãî êîìïëåêñíûé àíàëîã E ∗ . Èìååì: u = u − ( u ) . Ýâêëèäîâîïðîñòðàíñòâî U âûäåëÿåòñÿ óñëîâèåì: u ≥ . Äëÿ ïîñëåäíåãî íåîáõîäè-ìî è äîñòàòî÷íî: u ′ = 0 â êàêîé(cid:21)ëèáî èêñèðîâàííîé ñèñòåìå îòñ÷åòà x ′ . Ïðîñòðàíñòâî U òåì ñàìûì îêàçûâàåòñÿ îïðåäåëåíî ñ òî÷íîñòüþ äîïðåîáðàçîâàíèÿ u ′ → u , ò. å. (cid:22) äî âåêòîðà ñêîðîñòè v ñèñòåìû x îòíîñè-òåëüíî x ′ . Äëÿ ìàññèâíîãî áîçîíà òàêàÿ ñèñòåìà x ′ ñóùåñòâóåò. Ýòî (cid:22) åãîñèñòåìà ïîêîÿ. Òåì ñàìûì âåêòîð v èêñèðîâàí è ïðîñòðàíñòâî U åñòü3 ìåðíîå ýâêëèäîâî ñå÷åíèå ïñåâäîýâêëèäîâîãî ïðîñòðàíñòâà E . Ïëîò-íîñòü: g ( x, ψ ) = u ( x ) .Áåçìàññîâûé áîçîí, îòîí, íå èìååò ñèñòåìû ïîêîÿ, íî îòñþäà íåñëåäóåò îòñóòñò-âèå ñèñòåìû x ′ , u ′ = 0 . Áîëåå òîãî, â îòëè÷èå îò ìàñ-ñèâíîãî áîçîíà, åñëè òàêàÿ ñèñòåìà x ′ ñóùåñòâóåò, òî îíà íå åäèíñòâåí-íà. Äåéñòâèòåëüíî, âûäåëèì ñëó÷àé, êîãäà ïîëå åñòü ïëîñêèé âîëíî-âîé ïàêåò ñ âîëíîâûìè âåêòîðàìè êîìïîíåíò, ïàðàëëåëüíûìè âåêòîðó k . Âêëþ÷èâ óñëîâèå Ëîðåíöà â ñîñòàâ óðàâíåíèé ïîëÿ, è ïðåäïîëàãàÿ,÷òî âåêòîð v ïàðàëëåëåí k , ïîëó÷èì: u ′ = 0 . Ëåãêî óáåäèòüñÿ, ÷òî u = u = u ′ = u ′ = 0 , ò. å. ïðîñòðàíñòâî U åñòü ïîïåðå÷íàÿ ïëîñêîñòüñ áàçèñîì u , u íåçàâèñèìî îò | v | . Òî åñòü, êàëèáðîâêà u = u = 0 èíâàðèàíòíà íà ïîäãðóïïå ãðóïïû Ëîðåíöà v ↑↑ k . Íà îñíîâàíèè ñêà-çàííîãî ïðåäñòàâëÿåòñÿ åñòåñòâåííûì äîîïðåäåëèòü îòîí ïîñòóëàòîì:ñóùåñòâóåò ñèñòåìà x ′ , u ′ = 0 . Îí èñêëþ÷àåò íåîïðåäåëåííîñòü âûðà-æåíèÿ g ( x ) = u ( x ) â ñèëó åãî ãðàäèåíòíîé íåèíâàðèàíòíîñòè è îáåñïå-÷èâàåò åãî ïîëîæèòåëüíîñòü. Äîîïðåäåëÿåòñÿ íà îòîí (1) è íåïðîòè-âîðå÷èâî ìèíèìèçèðóåòñÿ ðàçëè÷èå ñâîéñòâ áîçîíîâ: ñèñòåìà ïîêîÿ äëÿîòîíà îòñóòñòâóåò, íî åå ñâîéñòâî u ′ = 0 ñîõðàíÿåò öåëîå ñåìåéñòâîñèñòåì. Íî äîñòèãàåòñÿ ýòî öåíîé îòêàçà îò ïðèíöèïà ãðàäèåíòíîé èí-âàðèàíòíîñòè ýëåêòðîäèíàìèêè. Ïîñëåäíèé ïîäòâåðæäåí åå îïûòîì (çàåäèíñòâåííûì èçâåñòíûì èñêëþ÷åíèåì). Íî âåñü îí ñâÿçàí ñî çíà÷åíè-ÿìè è ðàñïðåäåëåíèÿìè íàïðÿæåííîñòåé, ýíåðãèè, èìïóëüñîâ, ìîìåíòîâè íå êàñàåòñÿ ðàñïðåäåëåíèÿ êîîðäèíàò îòîíîâ. Òîëüêî ýêñïåðèìåíòìîæåò îïðåäåëèòü àëüòåðíàòèâíûé âûáîð: ëèáî äàííûé ïðèíöèï çäåñüíåïðèìåíèì, è ñïðàâåäëèâî g ( x ) = u ( x ) , ëèáî ðàñïðåäåëåíèå êîîðäèíàòîòîíà íå îïðåäåëåíî.4.3. Ýëåêòðîí. Ïðîñòðàíñòâî U u = { u α } , α = 1 , , , , òðàíñîðìàöèîííûå ñâîéñòâà ýëåìåíòîâ (cid:22)ñïèíîðû, à u (cid:22) âðåìåííàÿ êîìïîíåíòà âåêòîðà. ÂÔ u ( x ) = u ( x , x ) ðàññìàòðèâàåòñÿ îáû÷íî êàê òðàåêòîðèÿ â ãèëüáåðòîâîì ïðîñòðàíñòâå H ( V ) . Îíà óäîâëåòâîðÿåò ÂÓ Äèðàêà, ïðè÷åì k u ( x , x ) k (cid:22) åãî äèíàìè-÷åñêèé èíâàðèàíò. Ýòè ñâîéñòâà äàþò îñíîâàíèÿ äëÿ ðàâåíñòâà: u ( x ) == g ( x /x ) , k u ( t, x ) k = 1 , [2℄. Ââåäåì íîâóþ ÂÔ ψ ( x ) , òàêóþ ÷òî g ( x ) == g ( x ) u ( x ) = ψ ( x ) , k ψ ( x ) k = 1 .  ñèëó ÂÓ îíà ñîâïàäàåò ñ u ( x ) òî÷íîñòüþ äî íîðìèðîâêè. Èìååì: g ( x ) = const = 1 /cT , ψ = ( cT ) − / u , k ψ ( x ) k = 1 . Ïðè T → ∞ , V → E ïðåäåë k ψ ( x ) k çäåñü îïðåäåëåí, åñëèîïðåäåëåí àíàëîãè÷íûé íåñîáñòâåííûé èíòåãðàë ïî E . Ïëîòíîñòè g ( x ) , g ( x /x ) ñîâïàäàþò ñ òî÷íîñòüþ äî íîðìèðîâêè è ìîãóò ñèíòåçèðîâàòüñÿêàê ïóòåì íàáëþäåíèÿ x ñ x â êà÷åñòâå ïàðàìåòðà, òàê è íåïîñðåäñòâåí-íî x .5. Íàáëþäåíèå ýíåðãèè-èìïóëüñàÏóñòü Ê× (cid:22) äåéñòâèòåëüíûé ñêàëÿðíûé áîçîí. Íàáëþäàåìàÿ (cid:22) âåê-òîð 4-èìïóëüñà p = ( p , p ) ∈ E . Åãî ñðåäíåå åñòü ñîîòâåòñòâóþùèé äèíà-ìè÷åñêèé èíâàðèàíò, ÝÊÔ â H . Ñîáñòâåííûå ÂÔ îáðàçóþò ñåìåéñòâî: ψ k = a k exp( ip k x/ ¯ h ) ñ ïàðàìåòðàìè a k > , p k ∈ E (cid:22) èçâåñòíûé äèñ-êðåòíûé ðÿä, p k = − ( mc ) , çíà÷åíèÿ a k îïðåäåëÿþòñÿ íîðìèðîâêîé (cid:16)÷à-ñòèöà â åäèíè÷íîì îáúåìå(cid:17), è ðàñïðåäåëåíèå îïèñûâàåòñÿ â òåðìèíàõñðåäíèõ êîëè÷åñòâ ÷àñòèö n k ñ äàííûì 4-èìïóëüñîì p k â êà÷åñòâå íà-ãëÿäíîãî ïîëóàáðèêàòà êâàíòîâîãî ïîëÿ. À òî÷íåå, n k (cid:22) ñðåäíåå êî-ëè÷åñòâî çàìåðîâ ñ ðåçóëüòàòîì p k .  ñîîòâåòñòâóþùåì ïðîñòðàíñòâå l êîýèöèåíòîâ C = { C k } ðàçëîæåíèÿ ψ = X k C k ψ k : n k = | C k | , à ïðèäîïîëíèòåëüíîì óñëîâèè k C k = 1 ýòî (cid:22) ðåëÿòèâèñòñêè-èíâàðèàíòíîåáåçóñëîâíîå ðàñïðåäåëåíèå âåðîÿòíîñòè ïîÿâëåíèé åäèíè÷íîé ÷àñòèöûâ ïðîñòðàíñòâå 4-èìïóëüñîâ. Õðåñòîìàòèéíûå ðàñïðåäåëåíèÿ ñîâïàäàþòñ ïîñëåäíèìè ñ òî÷íîñòüþ äî íîðìèðîâêè, íî èì ïðèïèñûâàåòñÿ ñìûñëóñëîâíîãî ðàñïðåäåëåíèÿ â ìîìåíò âðåìåíè t . Ýòîò ñìûñë ïðîòèâîðå÷èòèõ ðåëÿòèâèñòñêîé èíâàðèàíòíîñòè, è êðîìå òîãî, ïîðîæäàåò èçâåñòíîåïðîòèâîðå÷èå [1℄: îí òðåáóåò ìãíîâåííîé èêñàöèè èìïóëüñà ïðè çàìå-ðàõ, òîãäà êàê îãðàíè÷åíèÿ òî÷íîñòè íàáëþäåíèÿ Ê× òðåáóþò ïðîäîë-æèòåëüíîé èêñàöèè.  ñëó÷àå êîìïëåêñíîé ÂÔ ê ýíåðãèè è èìïóëüñóäîáàâëÿåòñÿ çàðÿä, à ìíîãîêîìïîíåíòíîé (cid:22) ñïèí. Ïîä÷åðêíåì, ÷òî äî-ïóñêàåòñÿ, âîîáùå, è íàëè÷èå îòðèöàòåëüíûõ ÷àñòîò â ðàçëîæåíèè ÂÔ è,ñîîòâåòñòâåííî, (cid:22) íàáëþäåíèå ÷àñòèöû (åäèíñòâåííîé) ñ ðàçíûìè çíà-÷åíèÿìè çàðÿäà â èêñèðîâàííîé ïàðå ñåàíñîâ, íî (cid:22) íå ïîÿâëåíèå ïàðû7òàêèå ïîêàçàíèÿ íå ó÷èòûâàþòñÿ). Ýòî íå ïðîòèâîðå÷èò êâàíòîâîìó çà-êîíó ñîõðàíåíèÿ çàðÿäà, âûïîëíÿåìîìó òîëüêî â ñðåäíåì, íî ìîæåò íåäîïóñêàòüñÿ âíåøíèìè äëÿ ÊÌ çàêîíàìè, òàêèìè, êàê âñåîáùèé çàêîíñîõðàíåíèÿ ýëåêòðè÷åñêîãî çàðÿäà.6. Êâàíòîâîå ïîëå. Çàïîëíåíèå ïðîñòðàíñòâà-âðå-ìåíèÍîâûå ñâîéñòâà Ê× äîîïðåäåëÿþòñÿ íà ÊÏ â âèäå õàðàêòåðèñòèêðàñïðåäåëåíèÿ ÷àñòèö â ïðîñòðàíñòâå-âðåìåíè. Àäåêâàòíàÿ áàçà äëÿ ýòî-ãî: êîíöåïöèÿ ÊÏ êàê ñîîòâåòñòâóþùåé ñèñòåìû òîæäåñòâåííûõ ÷àñòèöÄèðàêà è Èîðäàíà, [2℄. àññìîòðèì ñèñòåìó N òîæäåñòâåííûõ ÷àñòèö ñîäèíàêîâîé ÂÔ ψ ( x ) ⊂ H ( V ) . Ïóñòü y = { y k } (cid:22) íàáëþäàåìàÿ ñîâîêóï-íîñòü õàðàêòåðèñòèê îäíîé Ê× è Y = { Y k } (cid:22) ñèñòåìû. Ïóñòü { ψ i } (cid:22) ñîá-ñòâåííûé áàçèñ íåêîòîðîé èçè÷åñêîé âåëè÷èíû; n = { n i } (cid:22) íàáîð åãî÷èñåë çàïîëíåíèÿ, n i = 0 , , , . . . , N ; Φ( n ) (cid:22) ñèììåòðèçîâàííàÿ (ëèáî,ñîîòâåòñòâåííî, àëüòåðíèçèðîâàííàÿ) ÂÔ ñèñòåìû, âûðàæåííàÿ ÷åðåç n .Ê ïðîöåäóðå íàáëþäåíèÿ. Îïðåäåëåíû îïåðàöèè ðåàëèçàöèè ñèñòåìûâ òîæäåñòâåííûõ èçè÷åñêèõ óñëîâèÿõ, îòâå÷àþùèõ äàííîé ÂÔ. Íàäêàæäîé ðåàëèçàöèåé ïðîâîäèòñÿ åäèíñòâåííûé ñåàíñ íàáëþäåíèÿ (çà-ìåð), ïðîäîëæàþùèéñÿ â òå÷åíèå âðåìåíè T .  êàæäîì ñåàíñå ïîÿâëÿ-åòñÿ, âîîáùå, íå îäíîâðåìåííî, N ÷àñòèö. Ïðè ýòîì äëÿ êàæäîé ÷àñòèöûèêñèðóåòñÿ åäèíñòâåííîå çíà÷åíèå y . Àãðåãèðîâàííûå õàðàêòåðèñòèêè Y ñèñòåìû âûðàæàþòñÿ íåïîñðåäñòâåííî ÷åðåç ýòè çíà÷åíèÿ. Äëÿ êàæ-äîé ÂÔ îïðåäåëåíû èõ ñðåäíèå h Y i (Φ) , ñîîòâåòñòâóþùèå áåñêîíå÷íîìóíàáîðó çàìåðîâ. Âðåìÿ èñêëþ÷àåòñÿ èç ïðîöåññà íàáëþäåíèÿ â êà÷åñòâåïàðàìåòðà.Ôîðìàëüíîå îïèñàíèå. ÂÔ Φ( n ) îòîáðàæàåò ìíîæåñòâî çíà÷åíèé n â êîìïëåêñíîå ýâêëèäîâî ïðîñòðàíñòâî Υ ñ ýëåìåíòàìè γ è ïðîèçâåäå-íèåì γ γ è ðàññìàòðèâàåòñÿ êàê ýëåìåíò ãèëüáåðòîâà ïðîñòðàíñòâà ñïðîèçâåäåíèåì (Φ , Φ ) , íîðìèðîâàííûé: k Φ( n ) k = 1 . Ñðåäíèå h Y i (Φ) ñóòü ÝÊÔ: h Y i (Φ) = (Φ , ΛΦ) , Λ (cid:22) ñîîòâåòñòâóþùèå îïåðàòîðû.  ÷àñò-íîñòè, P ( Q, Φ) = (Φ( n ) , f ( n )Φ( n )) , ãäå f ( n ) = 1 , n ∈ Q , f ( n ) = 0 , n / ∈ Q ,åñòü âåðîÿòíîñòü ñîáûòèÿ n ∈ Q . Âòîðè÷íîå êâàíòîâàíèå âîñïðîèçâîäèòíåðåëÿòèâèñòñêèé àíàëîã, âêëþ÷àÿ, ïîìèìî Φ( n ) , îïåðàòîðû óíè÷òîæå-íèÿ è ðîæäåíèÿ ÷àñòèöû a i , a + i , è òàêîâûå, îòíåñåííûå ê òî÷êå x , (cid:22)âîëíîâûå îïåðàòîðû (ÂÎ): Ψ( x ) = X i ψ i ( x ) a i , Ψ + ( x ) = X i ψ ∗ i ( x ) a + i . (2)8òëè÷èÿ ñëåäóþùèå: ÂÎ êàê óíêöèè x îïðåäåëåíû â H , à íå â H , èâðåìÿ âõîäèò â íèõ ñèììåòðè÷íî ñ ïðîñòðàíñòâåííûìè êîîðäèíàòàìè;îðìàëèçì äîëæåí áûòü ðåëÿòèâèñòñêè èíâàðèàíòåí; åñëè ψ i (cid:22) ïëîñêèåâîëíû, òî ýòîò áàçèñ èçâåñòíûì îáðàçîì ðàñøèðåí, ñîîòâåòñòâåííî, ïî-ÿâëÿåòñÿ äîïîëíèòåëüíàÿ õàðàêòåðèñòèêà Ê×,œ(cid:22) çàðÿä, à ñîîòâåòñòâó-þùèå íîâûì ñîñòîÿíèÿì ñëàãàåìûå â (2) îáðåòàþò óíèèöèðîâàííûéâèä ψ i ( x ) b i , ψ ∗ i ( x ) b + i , ãäå b i , b + i (cid:22) îïåðàòîðû óíè÷òîæåíèÿ è ðîæäåíèÿ÷àñòèöû â ýòèõ ñîñòîÿíèÿõ. Âîñïðîèçâîäèòñÿ òàêæå òåõíèêà ñèíòåçà îïå-ðàòîðîâ Λ = { Λ k } õàðàêòåðèñòèê ñèñòåìû Y = { Y k } , âêëþ÷àÿ ïðàâèëî:çàïèñûâàåì ñðåäíåå äëÿ åäèíè÷íîé ÷àñòèöû, è ïðîèçâîäèì çàìåíó: h y i ( ψ ∗ , ψ ) = ( ψ, Lψ ) = R ψ ∗ ( x ) Lψ ( x ) dE ; ψ → Ψ( x ) ,ψ ∗ → Ψ + ( x ) , Λ = h y i (Ψ + ( x ) , Ψ( x )) . (3)Çäåñü â ïðîèçâåäåíèè ÂÎ ðàññìàòðèâàþòñÿ êàê ýëåìåíòû H . Ñîîòâåò-ñòâåííî: h Y i (Φ) = (Φ , ΛΦ) . Åñëè { ψ i } (cid:22) ñîáñòâåííûé áàçèñ íàáëþäàåìîé,òî Λ = X i n i y i ; h Y i (Φ) = (cid:18)X i y i n i Φ (cid:19) = X i y i h n i i ; y i (cid:22) ñîáñòâåííûå ÷èñ-ëà îïåðàòîðà L , à n i y i (cid:22) îïåðàòîðà Λ , h n i i (cid:22) ñðåäíèå ÷èñëà çàïîëíåíèÿ.Îïåðàòîðû, õàðàêòåðèçóþùèå ðàñïðåäåëåíèÿ êîîðäèíàò ÷àñòèö, îòñóò-ñòâóþò â ÊÌ, êàê è äëÿ åäèíè÷íîé Ê×, è (1), (3) âîñïîëíÿþò ýòîòïðîáåë. Îïåðàòîð Λ( Q ) êîëè÷åñòâà ÷àñòèö â îáëàñòè Q ⊂ E : Λ( Q ) = (cid:0) Ψ + ( x ) , f ( x )Ψ( x ) (cid:1) = Z Q Ψ + ( x )Ψ( x ) dE, (4)ãäå f ( x ) = 1 , x ∈ Q , f ( x ) = 0 , x / ∈ Q . Îïåðàòîð Λ( Q ) êîëè÷åñòâà ÷àñòèöâ îáëàñòè Q ⊂ E ñîâïàäàåò ñ îïåðàòîðîì Λ( Q ) , Q = Q × (0 , T ) ⊂ E .Ñðåäíåå ÷èñëî çàïîëíåíèÿ îáëàñòè Q : h N i ( Q, Φ) = (Φ , Λ( Q )Φ) .Ïóñòü S ( ψ ∗ , ψ ) = ( ψ, L S ψ ) (cid:22) óíêöèîíàë äåéñòâèÿ Ê×, à S (Φ ∗ , Φ) == (Φ , Λ s Φ) (cid:22) ñèñòåìû. Âàðüèðóÿ ïîñëåäíèé ïî Φ ∗ ( n ) , ïîëó÷èì ÂÓ: Λ s Φ( n ) = 0 , Λ s = (cid:0) Ψ + ( x ) , L s Ψ( x ) (cid:1) . (5)Ïóñòü òåïåðü ÷èñëî N íå èêñèðîâàíî, à ìåíÿåòñÿ îò çàìåðà ê çàìå-ðó. Ýòî ñîîòâåòñòâóåò ìîäåëè ÊÏ â ðàìêàõ êîðïóñêóëÿðíîé êîíöåïöèè ñòî÷íîñòüþ äî íåíàáëþäàåìûõ õàðàêòåðèñòèê âàêóóìíîãî ñîñòîÿíèÿ. ÂÔòåïåðü äîëæíà áûòü ñèììåòðèçîâàíà åùå è ïî N , [2℄. Ïðàâèëî æå (2),è ñîîòâåòñòâåííî, (cid:22) êîíêðåòíûå ïðåäñòàâëåíèÿ îïåðàòîðîâ Λ , âêëþ÷àÿ Λ( Q ) , îñòàþòñÿ ñïðàâåäëèâûìè, òàêæå êàê è ÂÓ.9.1. åëÿòèâèñòñêàÿ èíâàðèàíòíîñòü îïèñàíèÿ ïîëÿ. Êîëè÷å-ñòâî ÷àñòèö â äàííîì ñîñòîÿíèè åñòü ðåçóëüòàò íàáëþäåíèÿ, íå çàâè-ñÿùèé îò âûáîðà ñèñòåìû êîîðäèíàò. Ñîîòâåòñòâåííî, íàáîð ÷èñåë çà-ïîëíåíèÿ ðåëÿòèâèñòñêè èíâàðèàíòåí, òàê æå êàê è îïåðàöèè íàä íèì a i , a + i . Ïîýòîìó ÂÎ îáëàäàþò ðåëÿòèâèñòñêèìè òðàíñîðìàöèîííûìèñâîéñòâàìè ÂÔ ÷àñòèöû ψ , à îïåðàòîðû Λ (cid:22) ñâîéñòâàìè èõ àíàëîãîâ L .Äàëåå, Λ = (Ψ + ( x ) , L Ψ( x )) = X ij l ij a + i a j , l ij = ( ψ i , Lψ j ) ; h y ih Y i (Φ) == (Φ , ΛΦ) = X ij l ij (Φ , a + i a j Φ) . Íî (Φ , Φ) åñòü èíâàðèàíò êàê ñîîòâåò-ñòâóþùåå çíà÷åíèå ìåðû âåðîÿòíîñòè, òàêæå êàê è îïåðàòîðû a + i a j , ïî-ýòîìó (Φ , a + i a j Φ) (cid:22) òîæå èíâàðèàíòû, à îðìà (Φ , ΛΦ) îáëàäàåò ðåëÿ-òèâèñòñêèìè òðàíñîðìàöèîííûìè ñâîéñòâàìè îðìû ( ψ, Lψ ) . Äàííîåîïèñíèå (cid:16)íàñêâîçü(cid:17) ðåëÿòèâèñòñêè èíâàðèàíòíî, â îòëè÷èå îò äåêîìïîçè-öèè ïîëÿ íà îñöèëëÿòîðû, èíâàðèàíòíîé â öåëîì, íî ñîäåðæàùåé íåèí-âàðèàíòíûå çâåíüÿ. Íå ãîâîðÿ óæå î òîì, ÷òî ïîñëåäíÿÿ íèêàê íå ñîîò-íîñèòñÿ ñ ðàñïðåäåëåíèÿìè êîîðäèíàò.6.2. Ïðåäñòàâëåíèå õàðàêòåðèñòèê êâàíòîâîãî ïîëÿ â ÷èñ-ëàõ çàïîëíåíèÿ èìïóëüñíûõ ñîñòîÿíèé. Ïóñòü { ψ i ( x ) } (cid:22) ñîáñòâåí-íûé áàçèñ ýíåðãèè, èìïóëüñà, ñïèíà è çàðÿäà. Ïðèìåíèòåëüíî ê ýíåðãèè,èìïóëüñó, ñïèíó, çàðÿäó (3) äàåò õðåñòîìàòèéíûå îïåðàòîðû. Îïåðàòîð Λ( Q ) êîëè÷åñòâà ÷àñòèö â îáëàñòè Q ⊂ E çàäàí (4), ïðè÷åì áàçèñ { ψ i ( x ) } â ÂÎ äîëæåí áûòü ïåðåíîðìèðîâàí: k ψ i ( x ) k = 1 âìåñòî (cid:16)÷àñòèöû â åäè-íè÷íîì îáúåìå(cid:17). Çàïèøåì (4) äëÿ êîíêðåòíûõ ÷àñòèö.Ñêàëÿðíûé íåéòðàëüíûé áîçîí. a i = b i , Ψ + ( x ) = Ψ( x ) ; Λ( Q ) == (1 / R Q Ψ ( x ) dE .Ôîòîí.  ðàìêàõ ìîäåëè ï. 3.3: Ψ + ( x ) = Ψ( x ) = (Ψ ( x ) , Ψ ( x )) ; Λ( Q ) = (1 / R Q Ψ ( x ) dE . Çäåñü ÂÎ Ψ ( x ) , Ψ ( x ) ñîîòâåòñòâóþò êîìïî-íåíòàì áàçèñà u , u .Ñêàëÿðíûé çàðÿæåííûé áîçîí. Ψ( x ) = X i a i ψ i ( x )+ b + i ψ ∗ i ( x ) , Ψ + ( x ) == X i a + i ψ ∗ i ( x ) + b i ψ i ( x ) ; Λ( Q ) = R Q Ψ + ( x )Ψ( x ) dE .6.3. Ïðåäñòàâëåíèå õàðàêòåðèñòèê êâàíòîâîãî ïîëÿ â ÷èñëàõçàïîëíåíèÿ ÿ÷ååê ïðîñòðàíñòâà-âðåìåíè. Ââåäåì â ðàññìîòðåíèåñîáñòâåííûé áàçèñ êîîðäèíàò, äëÿ óíèèêàöèè ñ äèñêðåòíûì áàçèñîìèìïóëüñà (cid:22) íà äîïðåäåëüíîì óðîâíå èíòåãðàëüíûõ ñóìì. àçîáúåì áðóñ V ⊂ E íà áðóñû v ( ξ ) ñ îáúåìîì w ( ξ ) , êàæäûé ïîìå÷åí çíà÷åíèåì x = ξ ,ïðèíàäëåæàùèì åìó. Îïðåäåëèì ñåìåéñòâî óíêöèé ψ ( x, ξ ) ñ ïàðàìåò-ðîì ξ : ψ ( x, ξ ) = w − ( ξ ) , x ∈ v ( ξ ) ; ψ ( x, ξ ) = 0 , x / ∈ v ( ξ ) . Àïïðîêñèìèðóåì10 ( x ) ñòóïåí÷àòîé óíêöèåé ψ ′ ( x ) = ψ ( ξ ) , x ∈ v ( ξ ) . Ôóíêöèè ψ ( x, ξ ) , ψ ′ ( x ) ñóòü ýëåìåíòû ïîäïðîñòðàíñòâà H ′ ⊂ H ( V ) êîíå÷íîé ðàçìåðíîñòèñ îðòîãîíàëüíûì áàçèñîì ψ ( x, ξ ) è, ñ òî÷íîñòüþ äî àïïðîêñèìàöèè: ( ψ ′ ( x ) , ψ ′ ( x )) = X ξ ψ ′∗ ( ξ ) ψ ′ ( ξ ) w ( ξ ); k ψ ′ ( x ) k = 1 , k ψ ( x, ξ ) k = w ( ξ ) − ; ψ ′ ( x ) = X ξ ψ ( ξ ) ψ ( x, ξ ) w ( ξ ) → ψ ( x ) = R ψ ( ξ ) δ ( x − ξ ) dξ,w ( ξ ) → .P ( Q, ψ ′ ( x )) = Z Q ′ ψ ′ ( x ) dE = X ξ P ( ξ ) , P ( ξ ) = ψ ( ξ ) w ( ξ ) . (6)Çäåñü Q ′ (cid:22) ìèíèìàëüíûé íàáîð áðóñîâ v ( ξ ) , ïîêðûâàþùèé Q . Ñîîòâåò-ñòâåííî, äîîïðåäåëèì àïïàðàò âòîðè÷íîãî êâàíòîâàíèÿ íà ñîáñòâåííûéáàçèñ êîîðäèíàò ψ ( x, ξ ) : n = { n ( ξ ) } (cid:22) ìíîæåñòâî ÷èñåë çàïîëíåíèÿ áðó-ñîâ v ( ξ ) , n ( ξ ) = 0 , , , . . . , N . Ïðîèçâîäÿ çàìåíó: P ( ξ ) → n ( ξ ) , ïîëó÷èìîïåðàòîð êîëè÷åñòâà ÷àñòèö â îáëàñòè Q ′ ⊂ E : Λ( Q ′ ) = X ξ n ( ξ ) , ξ : v ( ξ ) ∈ Q ′ ; n ( ξ ) âûïîëíÿþò çäåñü ðîëü ñîáñòâåííûõ ÷èñåë îïåðàòîðà Λ( Q ′ ) . Ñðåäíååêîëè÷åñòâî ÷àñòèö â Q ′ : h N i ( Q ′ , Φ) = (cid:18) Φ( n ) , X ξ n ( ξ )Φ( n ) (cid:19) = X ξ h n ( ξ ) i , ξ : v ( ξ ) ∈ Q ′ ; h n ( ξ ) i (cid:22) ñðåäíåå êîëè÷åñòâî ÷àñòèö â v ( ξ ) .6.4. Åäèíè÷íàÿ ÷àñòèöà êàê ïîäñèñòåìà êâàíòîâîãî ïîëÿ. àñ-ñìîòðèì åäèíè÷íóþ ÷àñòèöó ñ ÂÔ ψ ( x ) â òåðìèíàõ âòîðè÷íîãî êâàíòî-âàíèÿ êàê ïîäñèñòåìó N = 1 ñèñòåìû (cid:16)ÊÏ(cid:17), îïðåäåëÿåìóþ ñëåäóþùèìîáðàçîì: ó÷èòûâàþòñÿ òîëüêî çàìåðû ÊÏ, ïîêàçûâàþùèå N = 1 . Èìååì: n = (0 , , . . . , , , . . . ) , n i = 0 , . Òåðìèí (cid:16)òîæäåñòâåííûå ÷àñòèöû(cid:17) òåðÿ-åò ñìûñë, è ÂÔ Φ( n ) íå ñîäåðæèò îïåðàòîðà ïåðåñòàíîâêè. Íàéäåì ñðåä-íåå êîëè÷åñòâî ÷àñòèö â îáëàñòè Q ⊂ E : h N i ( Q, Φ) = (Φ , Λ ′ ( Q )Φ) , ïîëü-çóÿñü êîíå÷íîìåðíîé àïïðîêñèìàöèåé, ïðèâåäåííîé âûøå. { ψ ( x, ξ ) } (cid:22)ñîáñòâåííûé áàçèñ íàáëþäàåìîé. Êàæäîìó n îòâå÷àåò ñîáûòèå x ( n ) ∈ E .Îòîæäåñòâèì òî÷êó x ( n ) ñ ïîìåòêîé ξ âêëþ÷àþùåãî åå áðóñà v ( ξ ) . Èìå-åì: ÂÔ Φ( n ) = ψ ( x ( n )) îòîáðàæàåò ìíîæåñòâî âñåõ n â U .  ñîãëàñèèñ (6) ñëåäóåò ïîëîæèòü (Φ , Φ ) = X n Φ ( n )Φ ( n ) w ( ξ = x ( n )) . Èìååì: k Φ( n ) k = 1 ; h N i ( Q, Φ) = (Φ , Λ ′ ( Q )Φ) = X n Φ ( n ) = X ξ ψ ( ξ ) w ( ξ ) = P ( Q ′ , ψ ) → Z Q ψ ( ξ ) dξ , w ( ξ ) → ; n : x ( n ) ∈ Q , ξ : v ( ξ ) ∈ Q ′ . Òàêèìîáðàçîì, äàííàÿ óíêöèÿ Φ( n ) îòâå÷àåò îïðåäåëåíèþ ÂÔ, à íàáëþäå-íèå òàêîé ïîäñèñòåìû ÊÏ äàåò òîò æå ðåçóëüòàò, ÷òî è îïèñàííîå âûøåíåïîñðåäñòâåííîå íàáëþäåíèå åäèíè÷íîé ÷àñòèöû.7. Îá îòíîøåíèÿõ íåîïðåäåëåííîñòåé è îöåíêàõòî÷íîñòè íàáëþäåíèÿ íåðåëÿòèâèñòñêîé ÊÌ îòíîøåíèÿ íåîïðåäåëåííîñòåé åéçåíáåðãàñóòü ñëåäñòâèå ñòàòèñòè÷åñêèõ ïîñòóëàòîâ, ïðèâåäåííûõ âûøå.  ðàìêàõòðàäèöèîííîé ìîäåëè ðåëÿòèâèñòñêîé Ê× îíè óæå íå èìåþò ýòîãî òåîðå-òè÷åñêîãî îñíîâàíèÿ, ïîñêîëüêó íåîáõîäèìûé äëÿ ýòîãî çàêîí ðàñïðåäå-ëåíèÿ êîîðäèíàò îòñóòñòâóåò. Òåì íå ìåíåå, îíè èñïîëüçóþòñÿ â òîì æåâèäå, ñòðîãî ãîâîðÿ, óæå â êà÷åñòâå ñàìîñòîÿòåëüíîãî ïîñòóëàòà. Çäåñüýòè îòíîøåíèÿ ïîëó÷àþò îáîñíîâàíèå. Ïðè ýòîì, åñëè â íåðåëÿòèâèñò-ñêîé ÊÌ îòíîøåíèÿ êîîðäèíàòû-èìïóëüñ è âðåìÿ-ýíåðãèÿ âûâîäÿòñÿðàçíûì ñïîñîáîì è èìåþò ðàçíûé ñìûñë [3℄ ñòð. 185 (cid:21) 188, òî çäåñü îíèîáëàäàþò ïîëíîé îðìàëüíîé è ñìûñëîâîé ñèììåòðèåé. Òåîðåòè÷åñêèéíèæíèé ïîðîã íåîïðåäåëåííîñòè êîîðäèíàò Ê×: △ x > △ x min = ¯ hc/ε ( ε (cid:22) ýíåðãèÿ), îáóñëîâëåííûé ýòèìè îòíîøåíèÿìè è íåäîïóùåíèåì ý-åêòîâ ÊÏ (äëÿ îòîíà, (cid:22) ïîðÿäêà äëèíû åãî âîëíû), òåðÿåò ñèëó,ïîñêîëüêó ýåêòû ÊÏ ïðè íàáëþäåíèè â ðàìêàõ îïèñàííîé çäåñü ïðî-öåäóðû äîïóñòèìû. Îí çàìåíÿåòñÿ ìèíèìàëüíûì ïðàêòè÷åñêè ïðèåì-ëåìûì çíà÷åíèåì âåðîÿòíîñòè èêñàöèè åäèíñòâåííîé ÷àñòèöû ïðè íà-áëþäåíèè ÊÏ.Òðåáóåò ñïåöèàëüíîãî îñìûñëåíèÿ îòíîøåíèå íåîïðåäåëåííîñòåé âðå-ìÿ (cid:22) ñêîðîñòü (cid:22) èìïóëüñ v △ t △ p > ¯ h è âûòåêàþùàÿ èç íåãî ñâÿçü òî÷-íîñòè íàáëþäåíèÿ èìïóëüñà ñ ïðîäîëæèòåëüíîñòüþ ñåàíñà íàáëþäåíèÿ △ p > ¯ h/c △ t . Ïðåæäå âñåãî, èìååò ëè ñìûñë ïîíÿòèå (cid:16)ñêîðîñòü Ê×(cid:17) âäàííîé ìîäåëè, è åñëè äà, òî êàêîé?8. Ìàòåðèàë äëÿ ýêñïåðèìåíòàÅãî äàþò íîâûå ñâîéñòâà Ê× è ÊÏ, ïðåäñêàçûâàåìûå çäåñü: îð-ìóëû äëÿ ïëîòíîñòè âåðîÿòíîñòè êîîðäèíàò ðàçëè÷íûõ òèïîâ Ê× èðàñïðåäåëåíèé êîëè÷åñòâ ÷àñòèö â ïðîñòðàíñòâå-âðåìåíè äëÿ ÊÏ. Ïðî-ñòåéøèé âàðèàíò ïðîâåðêè ïîñëåäíåãî: èêñàöèÿ ÂÔ íà ñîáñòâåííîì12àçèñå 4-èìïóëüñà, ïîäñ÷åò ñîîòâåòñòâóþùåãî òåîðåòè÷åñêîãî h N i ( Q, Φ) è ñðàâíåíèå åãî ñ íåïîñðåäñòâåííî çàìåðåííûì h N i ( Q, Φ) . Ñóùåñòâåí-íàÿ íîâàÿ îñîáåííîñòü ìîäåëè íàáëþäåíèÿ: íåîáóñëîâëåííîñòü çàìåðàìîìåíòîì âðåìåíè t .  òå÷åíèå âðåìåíè T âîçìîæíû ïîâòîðíûå ðåàê-öèè ÷àñòèö ñ ïðèáîðîì, è ñîîòâåòñòâåííî, (cid:22) íååäèíñòâåííîñòü çíà÷åíèÿ y ïðè çàìåðå. Ïîñëåäíÿÿ äîëæíà áûòü òàê èëè èíà÷å óñòðàíåíà.  îòëè-÷èå îò õðåñòîìàòèéíîãî ìíåíèÿ, íàáëþäåíèå åäèíè÷íîé Ê× ñîãëàñíîîïèñàííûì ïðîöåäóðàì íå îáðåìåíåíî ïðîáëåìîé íåäîïóùåíèÿ ýåê-òîâ ÊÏ. Îñîáûé èíòåðåñ ïðåäñòàâëÿåò íàáëþäåíèå êîîðäèíàò îòîíà. Ñîäíîé ñòîðîíû, èäåàëüíî îáåñïå÷èâàåòñÿ åäèíñòâåííîñòü ñòîëêíîâåíèÿñ ÷àñòèöåé-ïðèáîðîì, ïîñêîëüêó îí èñ÷åçàåò ïðè ðåàêöèè. Ñ äðóãîé, (cid:22)ïðèìåíèìîñòü ê íåìó (1) îáóñëîâëåíà äîïîëíèòåëüíûì ïîñòóëàòîì, àëü-òåðíàòèâíûì ïðèíöèïó ãðàäèåíòíîé èíâàðèàíòíîñòè ýëåêòðîäèíàìèêè.È ýêñïåðèìåíò îïðåäåëèò àëüòåðíàòèâó: ëèáî äàííûé ïðèíöèï çäåñüíåïðèìåíèì, è ñïðàâåäëèâî (1), ëèáî íåñïðàâåäëèâ íàø ïîñòóëàò î-òîíà, è åãî ðàñïðåäåëåíèå êîîðäèíàò íå îïðåäåëåíî.9. ÂûâîäûÏîêàçàíî, ÷òî ïðè äîñòàòî÷íî ïîëíîì âêëþ÷åíèè ñòàòèñòè÷åñêîé ÷à-ñòè Ê× â ëîãèêó ÑÒÎ èíîðìàòèâíûå ñâîéñòâà åå ÂÔ îáðåòàþò îáúåì,ñðàâíèìûé ñ íåðåëÿòèâèñòñêîé ÊÌ. Ïîÿâëÿåòñÿ, âîïðåêè õðåñòîìàòèé-íîìó ìíåíèþ, òåîðåòè÷åñêàÿ âîçìîæíîñòü íàáëþäåíèÿ êîîðäèíàò Ê×,è îïðåäåëåíî ïðåäñòàâëåíèå èõ ïëîòíîñòè âåðîÿòíîñòè g ( x ) = ψ ( x ) ,îòâå÷àþùåå âñåì íåîáõîäèìûì òðåáîâàíèÿì. Çäåñü ÂÔ ψ ( x ) îòîáðàæà-åò ïðîñòðàíñòâî-âðåìÿ E â ýâêëèäîâî ïðîñòðàíñòâî U , õàðàêòåðíîå äëÿêàæäîãî èç ðàññìîòðåííûõ òèïîâ ÷àñòèö: áîçîíà, äåéñòâèòåëüíîãî è êîì-ïëåêñíîãî, ñêàëÿðíîãî è âåêòîðíîãî, âêëþ÷àÿ îòîí, ýëåêòðîíà. Ïëîò-íîñòü g ( x ) â ïðîñòåéøèõ ñëó÷àÿõ îðìàëüíî ïîäîáíà ïëîòíîñòè âåðî-ÿòíîñòè êîîðäèíàò íåðåëÿòèâèñòñêîé ÷àñòèöû, íî èìååò èíîå ñîäåðæà-íèå: èíûå óñëîâèÿ íàáëþäåíèÿ, èíûå ñâîéñòâà, èíîé ñìûñë. Ñåàíñû íà-áëþäåíèÿ íå îáóñëîâëåíû ìîìåíòîì âðåìåíè t , ò. å. âðåìÿ èñêëþ÷àåòñÿèç ïðîöåññà íàáëþäåíèÿ â êà÷åñòâå ïàðàìåòðà, íî ïðèñóñòâóåò â ñîñòà-âå àðãóìåíòà ÂÔ è íàáëþäàåìîé y = x ; ñîîòâåòñòâåííî, ïëîòíîñòü òå-ïåðü (cid:22) ðåëÿòèâèñòñêèé èíâàðèàíò; âìåñòî ñòîõàñòè÷åñêîãî òàíöà ÷àñòè-öû, îïèñûâàåìîãî ïîòîêîì âåðîÿòíîñòè ñ ïðîñòðàíñòâåííîé ïëîòíîñòüþ f ( t, x ) = | ψ ( t, x ) | , èìååì âåðîÿòíîñòíîå ðàñïðåäåëåíèå ïîÿâëåíèé ÷à-ñòèöû â ïðîñòðàíñòâå-âðåìåíè ñ ïëîòíîñòüþ g ( x ))