Fortschritte der Physik / Progress of Physics: Band 29, Heft 1 [Reprint 2022 ed.] 9783112655849

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

29. BAND 1981

A K A D E M I E

- V E R L A G

-

B E R L I N

Inhalt des 29. Bandes Heft 1 KASCHLTJHN, F . , Interaction and Measuring Process in Quantum Theory . . . G I E L E R A K , R., On the Grand Canonical Gibbs Ensembles in Euclidean Field Theory

1

. . . .

19

....

35

Heft 2 Ebert, D., and M. K.

VOLKOV,

Meson Interactions within Nonlinear Chiral Theories

Heft 3 SCHLICKEISER,

R., Gamma Ray Astronomy and the Origin of Cosmic Ray

95

Heft 4 G., Physical Principles, Geometrical Aspects, and Locality Properties of Gauge Field Theories 135

MACE,

Heft 5 F R A D K I N , E. S . , and D . M . GITMAN, Quantum Electrodynamics in Curved Space-Time 187 K L E I N E R T , H . , and K . M A K I , Lattice Textures in Cholesteric Liquid Crystals 219 K O B A Y A S H I , M . , Erratum (Polarization Phenomena in Elastic Electron-Proton Scattering) . 2 6 0 BUCHBINDER, L . I . ,

Heft 6 and D . Nucleon-Nucleon-Pion Coupling

GALI whose interaction is of finite range. Their asymptotically free states are described by wave groups with sufficiently sharp momenta or energies, respectively, i.e. by the complete or'thonormal systems %n and %pm with the (mean) momenta pn and pm, respectively. They obey the free equations of motion, in the non-relativistic case the corresponding Schrodinger equations. The initial state where the two particles are widely separated in space and their interaction can be ignored is assumed to be a pure product state XnWm involving correlations. If we restrict ourselves first to elastic 2-particle scattering then according to quantum theory this product state changes under the interaction into a superposition of such product states representing a correlation state (Fig. 1) Xnfm

27 an'.m';n.mXn'Vm'n'.ra'

o-* x„

(1)

W •IB

*

9 /

Fig. 1. Scattering process ( ! )

) Such a procedure is also not well defined with respect to the number or density of particles, respectively, actually necessary for a reduction of the wave function. ') We should keep in mind that quantum theory has to be completed by means of a statistical law for the single case at latest at the measuring process (if the wave function is related to the single system). 6

1*

F. Kaschlxthn

4

The case of particle production or annihilation, respectively,- will be treated later on8). In (1) space and time coordinates are not explicitly denoted. Because of the wide separation between the particles the final state is a free state again, i.e. the scattering amplitudes a»'.»»';)»,!» a r e time-independent. They fulfil the conservation laws for energy and momentum in a sufficient approximation, thus energy conservation En.

+ Em.

=

E

m

(2)

iV

+ Vm

= P n + Pm

(3)

n

+ E

and momentum conservation

hold (we consider states with sufficiently sharp momenta or energies, respectively). Because of these conservation laws different %n- are correlated with different ipm> in (1), i.e. the right-hand side of (1) represents a pure correlation state for the orthonormal systems chosen. Let us perform an interference experiment of the following type. After the scattering process at a sufficiently large distance from the interaction region we consider the parts 2

I C> . t^xL m

Xn" 1 Fig. 2. Interference experiment for process (1)

and a„", m " ; „, m Xn"fm" • Using appropriate external fields*which change only the direction but not the absolute values of the momenta, we superpose %n> with Xn" at 1 and y>m- with y>m" at 2 (Fig. 2). The corresponding interference term is of the form an',m';n,mXn'V>m'

a

n",m";n,man'.m',n,vtfm"iPm'X«"Xn'

"t" c - c -

(4)

where we did not note explicitly the change in the direction of the momenta. It is easy to verify that because of energy conservation the time dependence of the wave functions cancels in (4)9). From (4) it is seen that interference exists only if the corresponding parts are superposed at 1 and 2 simultaneously. The experimental test of the interference term (4) can be performed by coincidence measurements of the positions of the particles X and y> (where, of course, coincidence becomes irrelevant if the particle densities are so low that practically only one scattering process is involved in the measurements, respectively). Experiments of this type representing interference'measurements in the 6-dimensional configuration space of the two particles % and y> are directly connected with the problem 8

) The consideration of identical particles % and y> where we have to symmetrizise correspondingly proceeds similarly. i *) This holds exactly in the limit of plane waves.

Interaction and Measuring Process

5

to determine the (relative) phases of the scattering amplitudes an\m';KtOT10). At present these phases are experimentally unknown for correlation processes of the type (1). The scattering experiments done till now yield only information about the modulus of the amplitudes an',m'-,n.m • I11 other words, the superposition principle of quantum theory has not been tested hitherto for such processes. Let us add the following remark. If we observe only particle % at 1, i.e. proceed inclusively with respect to particle ip, we have to integrate over the coordinate of the latter particle. Because of the orthogonality of the states under discussion the integrated interference term (4) vanishes (we recall that different %n- are coupled to different ipm-). Thus such inclusive measurements representing ordinary interference experiments on particle X are useless for a test of the interference term (4), arid consequently for a determination of the (relative) phases of the amplitudes . In the following the assumption will be made that for the correlation process (5) the scattering theory has to be altered, and this in such a way that a statistical splitting takes place between the undisturbed transition and the disturbed ones. The consequences following from this assumption for the description of the interaction between a system and a macroscopic body, in particular a measuring apparatus, will be discussed in the next section. In the manner indicated we postulate instead of (1) a statistical law of the form an',

m'\n,

m'/.n'fm'

XnVmin.mXnVm

•

It states a spontaneous choice between the undisturbed transition and the disturbed ones, i.e. the alternative consists in the two possibilities that after the process the particles % and y> either remained in the initial state (then no interaction occurred at all) or they moved to the disturbed part (where the states changed mutually). Within the ensemble the transition from a pure state to a statistical mixture takes place. Of course, we have to normalize the final states of (6) anew by means of over-all factors. Each of the two transitions represents a definite reduction of the wave function. However, the transition probabilities for the individual final states Xn'Wm' a r e given as usual by the absolute squares of the (unmodified) scattering amplitudes a„\m--n,m (consideration within the ensemble). They are determined by the unmodified scattering theory in agreement with the fact that the statistical modification concerns only phase relations between orthogonal states. The transition probability for the two processes in (6) are given by the absolute squares of their normalization factors. It follows from (6) that • 10) The interference term (4) does not contain arbitrary phases (according to quantum theory). E.g. if we take into account only events at 2 with the same position of particle y>, an observation at 1 at neighboring directions should lead to interference of equal inclination as usual (provided, of course, we superpose at 2).

F . KASCHLUHN

the undisturbed transition and the disturbed ones do not interfere with each other and that there exist no correlations between them. The initial state ixnWm) decouples thus in the final state from the disturbed transitions ( X n ' f m ' with.«' =|= n, m! 4= TO). Now let us study the case that before scattering the particle % was decomposed in space into two coherent parts and /„(2> each with momentum pn (by means of external fields appropriately chosen). If sufficiently widely separated only one part, e.g. will interact with particle f (Fig. 3). Then according to quantum theory the transition is given by the following expression (Z„(1) +

XN™) W M ^ E W.M-V.MTIHM' + n',m'

XN^VM •

V)

The wave groups Xn' represent a complete orthogonal system considered (at first) in space region 1. They obey the free equation of motion and are normalized together with %„ Xn(1)))- The ipm- are defined as ^ii'

Fig. 3. Scattering process (7)

usual. The transition amplitudes a^m'-.n.m obey the conservation laws for energy and momentum (compare with (2) and (3)). We discuss the following interference experiment. At a sufficient distance from the scattering region 1 we superpose the part xnwith /M (»' =(= n) using appropriate external fields which change only the direction of the momenta. Then we get an interference term of the form +

C.C.

(8)

showing that interference exists only if the states ipm and ipm> are superposed simultaneously. As in case (4) an inclusive measurement which leaves the particle y> unobserved is useless for a test of the interference term (8). The process under discussion allows a more precise determination of the position of particle x by performing a measurement on particle tp. If we find e.g. by means of a momentum measurement particle y> in the state y>m> different from the undisturbed state y>m, we determine at the same time that particle x stays in space region 1. We have to reduce the expression on the right-hand side of (7) correspondingly, i.e. the final state is then given by Xn^Wm' (UP to a normalization factor). That f m - is related only to xW is a consequence of the conservation laws for energy and momentum. Of course, we may also perform a position measurement on particle y>. If we find particle y> in a space region not reached by the undisturbed state y>m (at least at that time), we state at the same time that particle x stays in sp&ce region 1. We have then again to reduce correspondingly.

7

Interaction and Measuring Process

From this we conclude the following. A determination that particle % is either in region 1 or 2 assumes necessarily that the interference term (8) vanishes since then the states of the particle ip must not superpose. This statement is valid in ordinary space (particle ip at a position where ipm vanishes) as well as in momentum space (pm- =(= pm). The vanishing of the interference term is of fundamental importance for an interpretation of quantum theory which regards the description of a particle by means of a wave function as a complete one. Otherwise a more precise determination of a physical quantity (in our case the position of particle %) would be possible than it can be expressed by a wave function.11) In accordance with the considerations leading to expression (6) we modify now the quantum theoretic result (7). We state that the scattering actually occuring is not described by relation (7) but proceeds in such a way that the final state splits into those transitions mutually orthogonal to the initial states of the two particles and the remain-

Fig. 4. Scattering process (10)

ing ones. The mutual change in states is considered again to be essential for the statistical law. I t is typical for processes where an exchange of physical quantities like energy and momentum takes place. Thus we replace (7) by the statistical law

(Xn{1)

+

Xn^)

y>m~

»'^»»ro'+r» {a«,m\n.mXnm

9) +

Xn™)

Vm-

The final states have to be normalized anew (compare with the discussion following (6)). The interference term (8) is absent for n' 4= n. The transition (9) above represents the reduction of the wave function of particle % to the space region l 1 2 ). If the part £„(2> vanishes the process (9) leads back to the previous one (6). Now we study the case that not only the part of particle % interacts with a particle y> but also the part Xn(2) with a further particle

would interfere with Xn 2) further on. 12 ) Moreover, the development in space region 2 where Xn 2i disappears during the process is completely unobservable as we should expect from an acausal process of such kind.

P. Kaschluhn

8 the transition is given by13) (Xn{1)

+

Xn(2)) Vm and

mn'-,nXn'V>

) In case of identical particles we have to symmetrizise correspondingly. ) At the individual processes not the states but only their amplitudes change. 15 ) Only energy is conserved in this case. 13

14

(12)

Interaction and Measuring Process

9

The intermediate case between the statistical law (6) and relation (12) comes into play when the particles to be scattered have no sufficiently sharp momenta or energies, respectively, or one particle is relatively heavy with respect to the other. Then the conservation laws for energy and momentum or only for momentum, respectively, are no longer satisfied. For the intermediate case the statistical law may be assumed in the form ri,m'=¥(.«,tri) y, ¿j ri ,m'=(n,m)

Xn VnC

„. a

n'.m';»,mXn'Vm''

Here a statistical splitting takes place in such a manner that the transitions in (13) above connected only with a mutual change in states do not involve those states %n', ipmwhich are coupled to the initial ones %pm, %„, respectively (noted as n', m' =(= (n, m)), whereas (13) below sums up all the remaining transitions («', m! = (n, m)), i.e. the undisturbed one m