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On the covariant description of wavefunction collapse
Volume 108A, number 8
PHYSICS LETTERS
22 April 1985
ON T H E COVARIANT D E S C R I P T I O N OF WAVEFUNCTION COLLAPSE D. DIEKS Fysisch Laboratorium, Rijksuniversiteit Utrecht, Postbus 80.000, 3508 TA Utrecht, The Netherlands Received 20 February 1985; accepted for publication 28 February 1985
The instantaneous change of the wavefunction known as the "collapse" of the wavefunction leads to ambiguities in relativistic quantum mechanics. It is here shown that a relativistically satisfactory generalization follows from the treatment of the measurement as a physical interaction.
1. Introduction. According to the usual account of measurement in quantum mechanics an ideal measurement can be described by means of the instantaneous collapse of the wavefunction (or, more generally, of the density matrix). As this procedure is not relativistically covariant, it leads to ambiguities in relativistic quantum theory [1-3]. For instance, if by some local measurement the position of a particle is measured, the wavefunction collapses to become a mixture at all points which are simultaneous with the measurement event. However, application of this prescription by different Lorentz observers results in different assignments of the equal time hyperplane along which the collapse takes place. It depends therefore on which Lorentz frame is taken as the frame of reference whether a pure state or a mixture is attributed to one and the same space-time point. Clearly, these candidates for the state description cannot be Lorentz transforms of one another. The lack of covariance of the usual prescription has induced various authors [2,3] to propose relativistically covariant generalizations of the collapse postulate. It is the purpose of this note to show that the consistent treatment of the measurement as a quantum mechanical interaction unambiguously leads to the requ~ed generalization; in fact, we shall corroborate the solution proposed by Aharonov and Albert [3]. 2. Measurement and collapse. We take as the basis of our account the treatment of the measurement as 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
a physical interaction. Let Iq~)= ~,kCk [~k ) be the statevector of the system under consideration before the interaction; I~k) is an eigenvector of the operator corresponding to the physical quantity which is about to be measured. Similarly, I~0) is the statevector of the measuring device before the interaction. An ideal measurement interaction effects the following change of state: IxI') I~0) ~ ~k Ck I~kk)I~k).
(1)
Here the I~k) are the apparatus states which become correlated, through the interaction, with the object states [¢Jk). The criterion for a "good" measurement interaction, which discriminates between the various values of the physical quantity which is measured, is that the Iek) are (practically) orthogonal: (~kl~k') = 8kk' •
(2)
It is important to note that by virtue of relation (2) the expectation value of any operator pertaining to the object system alone, calculated in the final state of (1), can be written as
(0) = ~ [Ckl2 (~klOl~k). k
(3)
That is, the interaction with the measurement device "decouples" the various Iffk) so that there is no effect of interference between them. (We have assumed in the foregoing that there is only one "pointer basis" {lek)}, see refs. [4,5] .) 379
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We now propose the following rule to give a physical interpretation to the above formalism: ira physical
system, in interaction with its surroundings, is described by a statevector of the form (1), while condition [2) is fulfilled, this means that for all observations on the system alone it can be regarded as being in one of the eigenstates 14k), with probability [Ck[2. As can easily be seen, eq. (3) ensures the consistency of the above interpretative rule. In our opinion, the rule is largely sufficient for the physical interpretation of the quantum mechanical formalism. It gives the same probabilities as orthodox measurement theory; but the complete wavefunction is retained. Consequently, no collapse of the wavefunction occurs on the level of the theoretical description. On the level of observations, however, only one of the 14k) manifests itself, so that there is a collapse here as long as the total wavefunction can be written in the form of the right-hand side of (1). Yet, the possibility that by a reverse interaction the various [4k) recombine and that a total wavefunction with the form of the left-hand side of (1)reemerges is not excluded. In that case the collapse on the level of the observations would be undone. The theoretical possibility of such an event must be admitted if one is not prepared to sacrifice the universal validity of quantum theory; Wigner [6] has given a simple example of how such a reversal of the measurement interaction could come about. The above treatment of the measurement process, with the interpretation of the formalism as stated, makes it possible to answer the question o f whether or not a collapse has taken place on the basis of the theory itself. It is not necessary to introduce an additional "projection postulate". One only needs to consider the physical interaction between object and measurement device and see where and when the total wavefunction assumes the form o f the right&and side of (1). In the sequel we shall do this for a typical case, with special attention for the covariance properties of the collapse. However, it should be noted that the following analysis does not strictly depend on the proposed interpretative rule. For also if one does employ the "projection postulate", the requirement of consistency of the projection postulate on the one hand and Schr6dinger-type o f evolution on the other (in the manner o f Von Neumann) leads to the demand that conditions (1) and (2) be satisfied. 380
22 April 1985
3. Measurements in relativistic quantum theory. We shall consider the case in which the number of particles can be considered constant, so that there is no need to work with the full field theory. In particular, let us focus our attention on a one-particle theory which operates with a wavefunction
[qo = f c(x, t) Ix) d 3 x , where Ix) is a localized state. Let us suppose that a device designed to measure the position of the particle is located at the space-point X and is switched on at t = 0. Along the lines of the measurement scheme explained above we have for the result of the interaction between particle and measurement device: I~)[~ 0) -+ f c(x, t)lx)[~ 0) d3x + cO(, t)lX)lqbl). x ~:x (4) The right-hand side of (4) applies for times t > 0; 1~1 ) is the excited state of the counter. It has been assumed in (4) that the counter has 100% efficiency and interacts with the particle only locally. The final statevector in (4) has been written as a function over an equal time hyperplane. It is well known [7,8] that the formalism can be generalized to apply to an arbitrary space-like hypersurface (by means of the so-called many-time formalism of quantum mechanics). If such a space-like hypersurface is specified by coordinates (x, tx) we get as a generalization of the final state in (4):
f c(x, tx)lX) [qb0,X, t X) d3x x 4~X
+ c(X, tx) t~ Iq'i, x, tx>,
(5)
where (X, t x ) is the point of intersection of the worldline of the counter with the specified hypersurface, and where i = 1 if (X, tx) lies on the part of the worldline where the counter has been activated and i = 0 if (X, t x ) lies elsewhere. If we now apply the interpretative rule of section 2 to the state (5), we see that a "collapse" (in the sense discussed above) of the particle state into either 141) - const f x . x C ( X , tx) Ix) d3x or 142) - IX) has taken place along every space-like hypersurface which intersects the worldline of the counter in a point where it has been activated (see fig. 1). This is a relativistically covariant description of the "wavefunction collapse",
Volume 108A, number 8
PHYSICS LETTERS
22 April 1985
counter is activated
r '
--... x
counter not activated
Fig. 1. Interaction of an initially non-localized particle with a localized counter. Along the hypersurface ~ a "collapse" has taken place, along E' there is no "collapse" o f the particle wavefunction.
as the space-like hypersurfaces have a geometrical meaning which is independent of the choice of a particular Lorentz frame. It should be noted that the question whether or not a collapse has taken place can onlybe answered with respect to a space-like hypersurface. Two hypersurfaces, one intersecting the wofldline of the counter where it has been activated and the other in a point where the counter is not in operation, will in general have a space-time region in common. With respect to this region alone the question of whether a collapse has taken place clearly does not admit an unambiguous answer: depending on whether the region is considered as belonging to the one or the other of the two hypersurfaces the answer is yes or no. The crux is that the concept of "wavefunction collapse" pertains to the loss of coherence between parts of the wavefunction at different places; it has no local meaning. This becomes clear if we calculate, in the state (5), expectation values of operators belonging to the particle system alone. For local operators B C which operate in a spatial region C that does not include the point X we find (Bc) = f c
fc c* (X,tx)c(x', tx)(xlBbc')d3x
d3x ' . (6)
This result is independent of whether or not a collapse is assumed to have happened. However, if we consider operators BC, that operate in regions that do include X it makes a difference whether or not a collapse has taken place. If there has been no collapse the expectation value contains a term
2 Ref c*(c, tx)c(X, tx) (xlBIX)d3x,
(7)
C' whereas this term is absent if there has been a collapse of the wavefunction (that measurement procedure for which (xlBIX) 4= 0 actually exist follows from the work of Aharonov and Albert [3]). Therefore, the question of wavefunction collapse has an experimental meaning only for those measurement procedures which extend over a space-like region which includes the position of the counter; depending on whether the counter has been activated or not collapse has or has not taken place. The preceding analysis can easily be generalized to more complicated cases. For instance, if more than one counter is present, interaction with these counters splits the total wavefunction into more components than the two components of (5); in fact, if n co,:nters are present at positions X1, X2, ..., X n the 381
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total wavefunction splits into (n + 1) terms. If the hypersurface intersects all the counter worldlines in points where the counters have been switched on, all these terms are mutually orthogonal, which means that along this hypersurface a collapse of the particle state in one of the states [~1 ) - c°nst"
Podolsky-Rosen type. Two spin ~ particles in an S = 0 state travel apart; the position parts of their wavefunctions are ~1 and xP2 respectively. Let us assume that these functions are zero except in a very small volume, so that it is possible to represent their paths in a space-time diagram (see fig. 2). Let a measuring device, which is able to measure the spin of an incoming particle, be located at a fixed position X. This device possesses a neutral quantum state IA0) and two states IA+) and [A_) which correspond to measurement outcomes + and - for the spin in a given direction. The initial state of the total system on a space-like hypersurface (x, tx) is
c(X'tx) Ix) d3x'
of
x4:X1, X2, ..., X n I~b2) ~ [XI> . . . . .
I~bn+l>~- [Xn>,
has occurred. With other orientations of the hypersurface other types o f collapse take place. If the surface intersects the worldlines of the counters at X 1 ..... X k in points where these counters have been switched on, and the other worldlines in points where there are unactivated counters, the collapse is into one of the following states: I~bl> = const,
q'l(Xl,
tx 1) 'I'2(x 2, tx 2) {l+> I-> - I-> I+>)[Ao, X, tx> (8)
After a local interaction with the particle whose path traverses the space-time region where the measuring device is situated the total state becomes [on a spacelike hypersurface (x', tx)]:
f c(x, tx) Ix> d3x, x # X , X2,... , X k
1~2) = IX1) . . . . .
22 April 1985
~ l ( X ' l , t X'l) ~2(X2, tx'2)
Iffk+l> = IXk>,
× (l+) IA+, X, t X) I-> - I-) [A_, X, txl+>).
and so forth. As a last example to make the idea clear consider a spin measurement in an experiment of the Einstein-
(9)
This means that on this hypersurface there is no longer coherence between the states I+) and I-) describing the spins of the particles (considered as observables
t
/
/ / /
/
/
/ / /
measurement vent
\ \\
~
cle
L
~
/ /
/
/
ticle
2
Fig. 2. Two localized spin-l/2 particles, one of which interacts with a localized counter. Along Z a "collapse" has taken place. 382
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22 April 1985
belonging to the particles alone). Therefore, a collapse has taken place, along this hypersurface, into either state I+) or I-) for each particle. This result applies to every space-like hypersurface which intersects the worldline of the measuring device in a point which is later than the measurement event (see fig. 2).
surement-like interaction with the surroundings). It is found that the collapse takes place with respect to a well-defined set of space-like hypersurfaces. This is sufficient to ensure that no difficulties arise in the relativistically covariant description of wavefunction collapse.
4. Conclusion. The above examples show that the question whether or not a collapse has taken place can be answered in a relativisticaUy invariant manner if the measurement process itself is treated as a quantum mechanical interaction. The criterion for the occurrence of collapse is the vanishing of interference between the various states which are the candidates for the "collapsed state". The destruction of interference can be brought about by the coupling of the object system with a measuring device (or by a mea-
References [1] I. Bloeh, Phys. Rev. 156 (1967) 1377. [2] K.E. Hellwig and K. Kraus, Phys. Rev. D1 (1970) 566. [3] Y. Aharonov and D.Z. Albert, Phys. Rev. D29 (1984) 228, and references therein. [4] W.H. Zurek, Phys. Rev. D24 (1981) 1516. [5] W.H. Zurek, Phys. Rev. D26 (1982) 1862. [6] E.P. Wigner, Am. J. Phys. 38 (1970) 1005. [7] S. Tomonaga, Prog. Theor. Phys. 1 (1946) 27. [8] F. Bloch, Phys. Z. Sow. Un. 5 (1934) 301.
383
PHYSICS LETTERS
22 April 1985
ON T H E COVARIANT D E S C R I P T I O N OF WAVEFUNCTION COLLAPSE D. DIEKS Fysisch Laboratorium, Rijksuniversiteit Utrecht, Postbus 80.000, 3508 TA Utrecht, The Netherlands Received 20 February 1985; accepted for publication 28 February 1985
The instantaneous change of the wavefunction known as the "collapse" of the wavefunction leads to ambiguities in relativistic quantum mechanics. It is here shown that a relativistically satisfactory generalization follows from the treatment of the measurement as a physical interaction.
1. Introduction. According to the usual account of measurement in quantum mechanics an ideal measurement can be described by means of the instantaneous collapse of the wavefunction (or, more generally, of the density matrix). As this procedure is not relativistically covariant, it leads to ambiguities in relativistic quantum theory [1-3]. For instance, if by some local measurement the position of a particle is measured, the wavefunction collapses to become a mixture at all points which are simultaneous with the measurement event. However, application of this prescription by different Lorentz observers results in different assignments of the equal time hyperplane along which the collapse takes place. It depends therefore on which Lorentz frame is taken as the frame of reference whether a pure state or a mixture is attributed to one and the same space-time point. Clearly, these candidates for the state description cannot be Lorentz transforms of one another. The lack of covariance of the usual prescription has induced various authors [2,3] to propose relativistically covariant generalizations of the collapse postulate. It is the purpose of this note to show that the consistent treatment of the measurement as a quantum mechanical interaction unambiguously leads to the requ~ed generalization; in fact, we shall corroborate the solution proposed by Aharonov and Albert [3]. 2. Measurement and collapse. We take as the basis of our account the treatment of the measurement as 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
a physical interaction. Let Iq~)= ~,kCk [~k ) be the statevector of the system under consideration before the interaction; I~k) is an eigenvector of the operator corresponding to the physical quantity which is about to be measured. Similarly, I~0) is the statevector of the measuring device before the interaction. An ideal measurement interaction effects the following change of state: IxI') I~0) ~ ~k Ck I~kk)I~k).
(1)
Here the I~k) are the apparatus states which become correlated, through the interaction, with the object states [¢Jk). The criterion for a "good" measurement interaction, which discriminates between the various values of the physical quantity which is measured, is that the Iek) are (practically) orthogonal: (~kl~k') = 8kk' •
(2)
It is important to note that by virtue of relation (2) the expectation value of any operator pertaining to the object system alone, calculated in the final state of (1), can be written as
(0) = ~ [Ckl2 (~klOl~k). k
(3)
That is, the interaction with the measurement device "decouples" the various Iffk) so that there is no effect of interference between them. (We have assumed in the foregoing that there is only one "pointer basis" {lek)}, see refs. [4,5] .) 379
Volume 108A, number 8
PHYSICS LETTERS
We now propose the following rule to give a physical interpretation to the above formalism: ira physical
system, in interaction with its surroundings, is described by a statevector of the form (1), while condition [2) is fulfilled, this means that for all observations on the system alone it can be regarded as being in one of the eigenstates 14k), with probability [Ck[2. As can easily be seen, eq. (3) ensures the consistency of the above interpretative rule. In our opinion, the rule is largely sufficient for the physical interpretation of the quantum mechanical formalism. It gives the same probabilities as orthodox measurement theory; but the complete wavefunction is retained. Consequently, no collapse of the wavefunction occurs on the level of the theoretical description. On the level of observations, however, only one of the 14k) manifests itself, so that there is a collapse here as long as the total wavefunction can be written in the form of the right-hand side of (1). Yet, the possibility that by a reverse interaction the various [4k) recombine and that a total wavefunction with the form of the left-hand side of (1)reemerges is not excluded. In that case the collapse on the level of the observations would be undone. The theoretical possibility of such an event must be admitted if one is not prepared to sacrifice the universal validity of quantum theory; Wigner [6] has given a simple example of how such a reversal of the measurement interaction could come about. The above treatment of the measurement process, with the interpretation of the formalism as stated, makes it possible to answer the question o f whether or not a collapse has taken place on the basis of the theory itself. It is not necessary to introduce an additional "projection postulate". One only needs to consider the physical interaction between object and measurement device and see where and when the total wavefunction assumes the form o f the right&and side of (1). In the sequel we shall do this for a typical case, with special attention for the covariance properties of the collapse. However, it should be noted that the following analysis does not strictly depend on the proposed interpretative rule. For also if one does employ the "projection postulate", the requirement of consistency of the projection postulate on the one hand and Schr6dinger-type o f evolution on the other (in the manner o f Von Neumann) leads to the demand that conditions (1) and (2) be satisfied. 380
22 April 1985
3. Measurements in relativistic quantum theory. We shall consider the case in which the number of particles can be considered constant, so that there is no need to work with the full field theory. In particular, let us focus our attention on a one-particle theory which operates with a wavefunction
[qo = f c(x, t) Ix) d 3 x , where Ix) is a localized state. Let us suppose that a device designed to measure the position of the particle is located at the space-point X and is switched on at t = 0. Along the lines of the measurement scheme explained above we have for the result of the interaction between particle and measurement device: I~)[~ 0) -+ f c(x, t)lx)[~ 0) d3x + cO(, t)lX)lqbl). x ~:x (4) The right-hand side of (4) applies for times t > 0; 1~1 ) is the excited state of the counter. It has been assumed in (4) that the counter has 100% efficiency and interacts with the particle only locally. The final statevector in (4) has been written as a function over an equal time hyperplane. It is well known [7,8] that the formalism can be generalized to apply to an arbitrary space-like hypersurface (by means of the so-called many-time formalism of quantum mechanics). If such a space-like hypersurface is specified by coordinates (x, tx) we get as a generalization of the final state in (4):
f c(x, tx)lX) [qb0,X, t X) d3x x 4~X
+ c(X, tx) t~ Iq'i, x, tx>,
(5)
where (X, t x ) is the point of intersection of the worldline of the counter with the specified hypersurface, and where i = 1 if (X, tx) lies on the part of the worldline where the counter has been activated and i = 0 if (X, t x ) lies elsewhere. If we now apply the interpretative rule of section 2 to the state (5), we see that a "collapse" (in the sense discussed above) of the particle state into either 141) - const f x . x C ( X , tx) Ix) d3x or 142) - IX) has taken place along every space-like hypersurface which intersects the worldline of the counter in a point where it has been activated (see fig. 1). This is a relativistically covariant description of the "wavefunction collapse",
Volume 108A, number 8
PHYSICS LETTERS
22 April 1985
counter is activated
r '
--... x
counter not activated
Fig. 1. Interaction of an initially non-localized particle with a localized counter. Along the hypersurface ~ a "collapse" has taken place, along E' there is no "collapse" o f the particle wavefunction.
as the space-like hypersurfaces have a geometrical meaning which is independent of the choice of a particular Lorentz frame. It should be noted that the question whether or not a collapse has taken place can onlybe answered with respect to a space-like hypersurface. Two hypersurfaces, one intersecting the wofldline of the counter where it has been activated and the other in a point where the counter is not in operation, will in general have a space-time region in common. With respect to this region alone the question of whether a collapse has taken place clearly does not admit an unambiguous answer: depending on whether the region is considered as belonging to the one or the other of the two hypersurfaces the answer is yes or no. The crux is that the concept of "wavefunction collapse" pertains to the loss of coherence between parts of the wavefunction at different places; it has no local meaning. This becomes clear if we calculate, in the state (5), expectation values of operators belonging to the particle system alone. For local operators B C which operate in a spatial region C that does not include the point X we find (Bc) = f c
fc c* (X,tx)c(x', tx)(xlBbc')d3x
d3x ' . (6)
This result is independent of whether or not a collapse is assumed to have happened. However, if we consider operators BC, that operate in regions that do include X it makes a difference whether or not a collapse has taken place. If there has been no collapse the expectation value contains a term
2 Ref c*(c, tx)c(X, tx) (xlBIX)d3x,
(7)
C' whereas this term is absent if there has been a collapse of the wavefunction (that measurement procedure for which (xlBIX) 4= 0 actually exist follows from the work of Aharonov and Albert [3]). Therefore, the question of wavefunction collapse has an experimental meaning only for those measurement procedures which extend over a space-like region which includes the position of the counter; depending on whether the counter has been activated or not collapse has or has not taken place. The preceding analysis can easily be generalized to more complicated cases. For instance, if more than one counter is present, interaction with these counters splits the total wavefunction into more components than the two components of (5); in fact, if n co,:nters are present at positions X1, X2, ..., X n the 381
Volume 108A, number 8
PHYSICS LETTERS 1
total wavefunction splits into (n + 1) terms. If the hypersurface intersects all the counter worldlines in points where the counters have been switched on, all these terms are mutually orthogonal, which means that along this hypersurface a collapse of the particle state in one of the states [~1 ) - c°nst"
Podolsky-Rosen type. Two spin ~ particles in an S = 0 state travel apart; the position parts of their wavefunctions are ~1 and xP2 respectively. Let us assume that these functions are zero except in a very small volume, so that it is possible to represent their paths in a space-time diagram (see fig. 2). Let a measuring device, which is able to measure the spin of an incoming particle, be located at a fixed position X. This device possesses a neutral quantum state IA0) and two states IA+) and [A_) which correspond to measurement outcomes + and - for the spin in a given direction. The initial state of the total system on a space-like hypersurface (x, tx) is
c(X'tx) Ix) d3x'
of
x4:X1, X2, ..., X n I~b2) ~ [XI> . . . . .
I~bn+l>~- [Xn>,
has occurred. With other orientations of the hypersurface other types o f collapse take place. If the surface intersects the worldlines of the counters at X 1 ..... X k in points where these counters have been switched on, and the other worldlines in points where there are unactivated counters, the collapse is into one of the following states: I~bl> = const,
q'l(Xl,
tx 1) 'I'2(x 2, tx 2) {l+> I-> - I-> I+>)[Ao, X, tx> (8)
After a local interaction with the particle whose path traverses the space-time region where the measuring device is situated the total state becomes [on a spacelike hypersurface (x', tx)]:
f c(x, tx) Ix> d3x, x # X , X2,... , X k
1~2) = IX1) . . . . .
22 April 1985
~ l ( X ' l , t X'l) ~2(X2, tx'2)
Iffk+l> = IXk>,
× (l+) IA+, X, t X) I-> - I-) [A_, X, txl+>).
and so forth. As a last example to make the idea clear consider a spin measurement in an experiment of the Einstein-
(9)
This means that on this hypersurface there is no longer coherence between the states I+) and I-) describing the spins of the particles (considered as observables
t
/
/ / /
/
/
/ / /
measurement vent
\ \\
~
cle
L
~
/ /
/
/
ticle
2
Fig. 2. Two localized spin-l/2 particles, one of which interacts with a localized counter. Along Z a "collapse" has taken place. 382
Volume 108A, number 8
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22 April 1985
belonging to the particles alone). Therefore, a collapse has taken place, along this hypersurface, into either state I+) or I-) for each particle. This result applies to every space-like hypersurface which intersects the worldline of the measuring device in a point which is later than the measurement event (see fig. 2).
surement-like interaction with the surroundings). It is found that the collapse takes place with respect to a well-defined set of space-like hypersurfaces. This is sufficient to ensure that no difficulties arise in the relativistically covariant description of wavefunction collapse.
4. Conclusion. The above examples show that the question whether or not a collapse has taken place can be answered in a relativisticaUy invariant manner if the measurement process itself is treated as a quantum mechanical interaction. The criterion for the occurrence of collapse is the vanishing of interference between the various states which are the candidates for the "collapsed state". The destruction of interference can be brought about by the coupling of the object system with a measuring device (or by a mea-
References [1] I. Bloeh, Phys. Rev. 156 (1967) 1377. [2] K.E. Hellwig and K. Kraus, Phys. Rev. D1 (1970) 566. [3] Y. Aharonov and D.Z. Albert, Phys. Rev. D29 (1984) 228, and references therein. [4] W.H. Zurek, Phys. Rev. D24 (1981) 1516. [5] W.H. Zurek, Phys. Rev. D26 (1982) 1862. [6] E.P. Wigner, Am. J. Phys. 38 (1970) 1005. [7] S. Tomonaga, Prog. Theor. Phys. 1 (1946) 27. [8] F. Bloch, Phys. Z. Sow. Un. 5 (1934) 301.
383