- Email: [email protected]
Quantum cybernetics and its test in “late choice” experiments
Volume 118, number 8
PHYSICS LETTERSA
l 0 November 1986
Q U A N T U M CYBERNETICS AND ITS TEST I N "LATE C H O I C E " E X P E R I M E N T S Gerhard GR~)SSING Atominstitut der Osterreichischen Universitdten, Schiittelstrasse I15, A-1020 Vienna, Austria
Received 9 September 1986; accepted for publication 17 September 1986
A relativistically invariant wave equation for the propagation of wave fronts S = const (S being the action function) is derived on the basis of a cybernetic model of quantum systems involving "hidden variables". This equation can be considered both as an expression of Huygens' principle and as a general continuity equation providing a close link between classical and quantum mechanics. Although the theory reproduces ordinary quantum mechanics, there are particular situations providing experimental predictions differing from those existing theories. Such predictions are made for so-called "late choice" experiments, which are modified versions of the familiar "delayed choice" experiments.
1. Introduction As is well known, classical mechanics can be considered with Goldstein [1] as the "geometrical--optical approximation of wave mechanics", in the sense that the Hamiltoia-Jacobi equations reveal classical mechanics as the geometrical-optical limiting case of a wave movement: light rays being orthogonal to wave fronts correspond to particle trajectories which are orthogonal to surfaces with constant action functions S, where S ( q , p, t) = lee(q, p ) - E t ,
(l)
q, p, t denoting location, momentum, and time coordinates respectively, Wthe time-independent "characteristic function", and E the energy (fig. 1 ).
Our principal aim in this paper is to derive in the context of quantum theory a relativistically invariant wave equation describing this propagation of wave fronts S=const, which would thus provide a close link between classical and quantum theory by using the same language, and which would simultaneously be in agreement with relativity theory. To obtain such an equation, additional physical assumptions on the nature o r s will be necessary. Our proposal will be to take into account a modified version of de Broglie's "double solution" model in combination with his assumptions on a "hidden thermodynamics" [ 2,3 ], introduced by the author as "quantum cybernetics" [ 4 ].
2. Relativistic quantum cybernetics W=b
W=b+Edt %
To begin with, consider the two wavefunctions constituting de Broglie's "double solution":
-
'
I--
/\
] l (0) =a
~=R
S (dt)=b S (0) = b
S (dt) = a
Fig. 1. Surfaces of constant action function S corresponding to wave fronts propagatingin configuration space. 0375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
e -iS/~
and
w = a e -is/~ ,
(2)
with R, a, and S real, where ~ represents the ordinary solution and w represents a "singularity solution" of the wave equation having the same phase S as ~ , but with an amplitude involving a generally mobile singularity, i.e. a particle with an energy Eo = hog o = moc 2. Writing down the Klein-Gordon equation for w, 381
Volume 118, number 8
(D+moc2/h)w=O,
PHYSICS LETTERS A
I'-l=(1/c2)O2/Ot2-V 2,
(3)
The latter decomposes, as is well known, into two equations: a Hamilton-Jacobi-Bohm type equation [2,5] $2 = E 2 +c2(V S)2 + Q2 ,
(4a)
which can also be written as OuSOuS= mEc 2 + Q2/c2 = M E c 2 ,
(4a')
attributing a variable rest mass Mo to the particle with the generally variable "quantum potential" QZ=cEh2Da/a, and the equation for the conservation of the probability current
Ou[a2 OuS] =OuJU = 0 .
(4b)
j u is the four-probability current
(5)
J~' = w*wOu S ,
corresponding to a generalized four-momentum OuS. Rewriting (4b) as
2yavs-h~/c2 a
DS=0
(6)
we see that eq. (6) has three possible solutions with respect to the function S: (a) kS=0;
VS=S=0,
DS=0,
(b)~S=0;
VS~0,
S~0,
DS~0,
(c) 5S~0;
VS~0,
S~0,
FqS~0.
(7)
We know that the Hamilton-Jacobi equations follow from a minimum action principle 8S=0. However, eq. (6) is compatible with ~ S ~ 0 (viz. eq. (7c)) so that one could imagine the existence of a more encompassing equation in agreement with the continuity equation that contains 8 S = 0 as special case. To obtain such a "generalized continuity equation" requires, however, that one has additional information on the action function S. This will be provided by a cybernetic model for quantum systems as proposed by the author in ref. [ 4]. The motivation to introduce cybernetic concepts into quantum physics comes from the following observation by Maturana and Varela [ 6,7 ]: "If one says that there exists a machine M in which there is a feedback loop through the environment, so that the 382
10 November 1986
effects of its output affect its input, one is in fact talking about a larger machine M' which includes the environment and the feedback loop in its defining organization". This has led Maturana and Varela to propose the notion of an autonomous system as a dynamical s~stem maintaining its identity as defined by its organization, thus representing a self-referential unity in the specific domain of its realization. Our assumption is that the mathematical behavior of the action function S indicates the "organization" of a quantum system. In close analogy to Varela [ 7 ], we consider the latter to be a feedback system with a given reference signal that compensates disturbances only relative to the reference point (i.e. a basic frequency), and not in any way reflects the texture of the disturbance. Its behavior, then, is the process l>y which such a unit controls its "perceptual data" through adjusting the reference signal. Physically, this would correspond to a "particle" as harmonic oscillator and an accompanying "wave" w oscillating in phase and creating a feedback loop through the environment in the sense that an alteration of the wave's characteristic frequency alters the particle's frequency, and vice versa [4]. Our task is now to describe the behavior of the action function S in the framework as outlined above. To do so, we make use of de Broglie's thermodynamical description of a quantum system [ 3 ] which in turn draws on earlier ideas by Brillouin [ 8 ]. With Boltzmann's formula for the relation between a variation in action S and in dissipated heat Qo (equivalent to an energy Moc 2) of a periodic system characterized by its frequency ~,= l/T, 8S= -T~Qo = -z~Moc 2 ,
(8)
one gets v S S = T6Se ,
(9)
where Se denotes the entropy of the system and T the temperature of the surrounding heat bath. Considering a particle as containing some "internal heat" Qo (or variable rest mass Mo, respectively) that is in constant thermal contact with the heat bath of the vacuum of temperature T such that moc 2 = h l , = k T ,
(10)
where h and k are respectively Planck's and Boltzmann's constants, one obtains [ 3 ]
Volume 118, number 8
PHYSICS LETTERSA
S e / k = S/h .
( 11 )
Then, with (12)
8S/h= -~Qo/moc 2
systems, their organizational characterization, the action function S, obeys a wave equation itself, namely with its own four-gradient as source. Since we have also with (17) M 2 c 2 = 4h 2 Ou a 0 i, a/a 2,
one gets 8Se = - k S Q o / m o c 2 .
Se/k=ln P,
we see that to characterize the organization of a quantum system we only need to know the relative variation of its characteristic amplitude a:
(14)
8S=0,
P denoting the probability identified via the continuity equation (4b) as P = a 2, one obtaines both the familiar Boltzmann formula P=Po exp(-~Qo/kT),
(15)
and, with eq. (11 ), our new relation S / h = In a 2 .
(16)
Further, eq. (11 ) already indicates the equivalence of the minimal action principle and the second law of thermodynamics, which in fact has been shown by de Broglie [ 3 ] independently of the considerations made here. Consequently, on the natural trajectory of the particle the entropy Se is maximal, i.e. natural trajectories are more probable than others. One can therefore say that under the basic assumption of quantum cybernetics, the "resonance condition" (10) implying the balance of "internal" and "external" energies, the wave can be seen as mediator between the particle and the "hidden thermostat", i.e. the vacuum. This means that it is not only a "guiding wave" (as in de Broglie's model) determining the trajectory of the particle, but also that the wave's frequency can be looked at as altered by a sudden frequency change of the particle's oscillation. This, then, constitutes the particle's oscillation as "reference point", with the wave providing the feedback loop through the environment. We can now insert eq. (16) into (4b) to obtain: r-lP= Fla2 = 0 ,
(17) M2 c 2 .
( 19 )
(13)
Since
[ 3 S = - h - l OuSOu S = - h - l
10 November 1986
(18)
Eq. (18) is the wave equation for the wave fronts S=const we have been looking for, and it is easily identified as Huygens' principle in configuration space: because of the wave-like nature of quantum
or
2h6a/a.
(20)
We call eq. (20) a general action principle testing whether or not there exists a new source term 6a ~ 0 along the integration area ~ of the action function S = f ~ L d x producing new waves with wavefronts S = const. Moreover, eq. (17) shows that standing waves of the form a 2 = cos 09t cos k x = ½[ c o s ( o J t - kx) + cos(tot + kx)] are simple solutions of D P = n a 2= 0, implying that the particle oscillates in phase with its environment as long as 8 S = 0 , where the standing waves function as the mediators. This also gives an intuitive understanding of the ansatz hv = k T . One can also say that 6 S = 0 implies a situation of stability (conservation of organization) that can be represented by the existence of stable standing waves. Thus, on the one hand, the characteristic oscillation of each autonomous quantum system is defined as the resulting oscillation of a relevant environment, and, on the other hand, the central oscillation itself contributes to the overall frequency of the harmonically oscillating systems "particle plus wave". It is therefore wrong in this model to talk about a "wave coming into a apparatus with its particle" or "leaving the apparatus". Rather, the waves must be at any time understood more like "carrier waves" representing the whole system (particle plus apparatus) as long as this wholeness is not destroyed by an act of measurement (implying 8 S # 0). We therefore speak of "organizational coherence" as long as 8 S = 0 and of its destruction during the measurement process, 8 S ¢ 0. As we have already shown in ref. [ 4 ] that the geodesic equation of general relativity can be derived from eq. (18) such that the latter constitutes a c o m 383
Volume 118, number 8
PHYSICS LETTERSA
mon basis for both quantum theory and relativity theory, our main task will now be to obtain consequences of our model that can be tested in experiments.
3. "Late choice" experiments Comparing analytical mechanics with geometrical optics one can show [ 8 ] an identity of the principle of least action 8 S = 0 and Fermat's principle of a light ray normal to surfaces S= const following the "shortest path" along its way from initial to final point, provided the velocity u of the wave front is given by
u = E / V S > c.
(21)
Let us now try to find a corresponding relation between wave and particle properties of a quantum system in the framework of quantum cybernetics. From eq. (17) we see that the four-probability current is proportional to the four-gradient of the probability P itself:
DP=O*-*OuJU=Ou[POuS ] =Ou[hOuP] = 0 .
(22)
There are, however, two ways to write out explicitly our "generalized" continuity equation" (22) corresponding to the introduction of two different velocities, v=dx/dt and u = Ox/Ot, that can also be defined in the following way:
v= - - c 2 ~ ,
u=c 2dS/dx -~-~,
uv=c 2 .
(23)
One can easily verify eq. (23) by inserting S = E t - p x with E=~_
°c2 v2/c2
and
~m°v P = x / " -- v2/c2
Under conservation of energy S = const, eq. (22) then provides both the "ordinary" continuity equation for particles with velocities v:.
OP/Ot + V ( Pv) =O ,
(24)
and a continuity equation for the waves with velocities u' = -u=c2/v:
OPlOt+ V (Pu') = 0 .
(25)
Note that the velocities u in (23) and (21) are iden384
l0 November 1986
tical. Moreover, the difference between v and u' involving total derivatives and partial derivatives, respectively, is also one of sign. One can therefore identify the waves S = const with shock waves whose velocity is inversely proportional to the particle's velocity. This is a result analogous to the classical case [ 1,8 ], except that now we can give a direct physical meaning to those waves: the waves S = const can now be considered as real shock waves due to a compression of the vacuum where an infinite velocity u' (i.e. v = 0 ) implies that the whole system (particle plus wave) oscillates harmonically at any point in spacetime. The possibility of velocities u' greater than the speed of light in the vacuum may look contradictory to relativity theory. However, a closer look shows that here it is the vacuum itself whose compressions are propagating, and not a particle travelling in the vacuum! Moreover, the accurate way to provide statements in agreement with relativity theory is to write the corresponding equations in a covariant form. This is, however, what we did in eq. (25) involving the velocity u'. We therefore conclude that, in general, velocities larger than the vacuum speed of light are not a priori forbidden by relativity theory. More importantly, for particular situations they even provide the basis for experimental predictions differing from those of existing theories, although - in all other c a s e s - there is full agreement between our model and both relativity and quantum theory. To obtain experimental predictions different from those of ordinary quantum theory, we consider the following interferometry experiments. Suppose we want to perform a modified "delayed choice experiment" [9] in the sense that one inserts or removes the first beamsplitting mirror (figs. 2 and 3) shortly after the main fraction of the wave packet has entered the interferometer. We call this kind of experiment a "late choice experiment" since a choice on mirror 1 is made after the bulk of the wave packet has entered the interferometer, as opposed to making a "delayed choice" on mirror 2 shortly before the wave packet leaves the interferometer device. Consider first the situation as shown in fig. 2. Quantum theory predicts no major effect of the removal of mirror l (fig. 2b) after the bulk of the wave packet has already passed it, since it is only the
Volume 118, number 8
PHYSICS LETTERS A
S
S mirror I removed
A
B
A
B
a
D1
D2
D1
D2
Fig. 2. (a) Schematized interferometer with source S, two halfsilvered mirrors 1 and 2, two totally reflecting mirrors A and B, and two detectors D~ and D2. (b) Schematized interferometer with beam-splitting mirror 1 removed. Contrary to orthodox quantum theory, quantum cybernetics predicts destruction of interference at detectors D, and D2 for the case that mirror 1 is removed shortly after the bulk of the wave packet has passed it.
S
S mirror 1 ...-'"'" i n s e r t e d
AI
B
A
B
a
D1
D2
DI
D2
Fig. 3. (a) Schematized interferometer without the first beamsplitting mirror I. (b) Schematized interferometer with beamsplitting mirror 1. Contrary to orthodox quantum theory, quantum cybernetics predicts interference to occur at D2 even for the case that mirror 1 is inserted only shortly after the bulk of the wave packet has passed the area of not yet inserted mirror 1.
10 November 1986
m i n o r rest o f the wave packet that constitutes the noninterfering fraction o f counts at detectors D1 a n d D2. Q u a n t u m cybernetics, however, provides a different answer: since organizational coherence has been destroyed through the removal o f m i r r o r 1, there m u s t be a new " s o u r c e " o f waves S = const due to eq. (18). Since the particle and the surrounding wave are oscillating in phase with each other, a change in the e n v i r o n m e n t m u s t affect the particle as well. This "change in the e n v i r o n m e n t " , however, is such that the new experimental situation is o f the kind as shown in fig. 2b, i.e. no interferability. The question now is whether this change at the location o f m i r r o r 1 can still affect the particle that has already passed it. The answer is "yes", since we know from eq. (23), or (25) respectively, that the waves S = const propagate with velocities u' = -c2/v, where v is the velocity o f the object creating the disturbance, i.e. m i r r o r 1. Since one can neglect the time span until m i r r o r 1 is inserted (i.e. until v = 0 ) , one can say that the shock waves S = c o n s t practically i m m e d i a t e l y prevent interference to occur for the case shown in fig. 2b. The opposite can be observed in situations as shown in fig. 3. There, one starts with a non-interfering situation a n d waits with the insertion o f m i r r o r 1 until the m a j o r part o f the wave packet has entered the interferometer. Again, this insertion produces a different organizational situation in the sense that the source term o f eq. (18) changes such as to create an e n v i r o n m e n t that establishes interference. Since one can consider the shock waves S = c o n s t as propagating with u' >>c, q u a n t u m cybernetics predicts interference for the case shown in fig. 3b. Summarizing, q u a n t u m cybernetics reflects the high sensitivity o f the q u a n t u m system "particle plus wave" towards organizational changes in the surroundings o f the particle which are m e d i a t e d through the wave, wave front S = c o n s t representing a symbolical description o f a " p r o p a g a t i n g d i s t u r b a n c e " in the vacuum. Such a high sensitivity is well known in o r d i n a r y q u a n t u m theory, manifesting itself in the Heisenberg uncertainty relations, for example. Pertaining to a "realist" or " o b j e c t i v e " description o f q u a n t u m systems, it is h a r d to imagine that e.g. in a double slit experiment, the b e h a v i o u r o f a particle whose path goes through one slit is affected by what goes on at the other slit only as long as it has not actually passed the slit. Considering the delicate 385
Volume 118, number 8
PHYSICS LETTERS A
wholeness o f the system "particle plus a p p a r a t u s " a n d its sensitivity to a n y changes w i t h i n it, the experimental predictions of quantum cybernetics only reflect properly this high sensitivity i n t e r m s o f the creation or d e s t r u c t i o n o f o r g a n i z a t i o n a l coherences.
Acknowledgement T h i s work was s u p p o r t e d i n part b y the B u n d e s m i n i s t e r i u m f'tir W i s s e n s c h a f t u n d F o r s c h u n g o n the contract n u m b e r Z1.19.153/3/26/85.
References [1] H. Goldstein, Classical mechanics (Addison-Wesley, Reading, 1959).
386
10 November 1986
[ 2 ] L. de Broglie, Non-linear wave mechanics (Elsevier, Amsterdam, 1960). [3] L. de Broglie, La thermodynamique de la particule isol6e (Gauthier-Villars, Paris, 1964). [ 4 ] G. Gr6ssing, How does a quantum system perceive its environment?, talk given at Int. Conf. on Microphysical reality and quantum formalism (Urbino, Italy, Oct. 1985), proceedings to appear, eds. G. Tarozzi and A. van der Merwe ( Reidel, Dordrecht). [5] D. Bohm, Phys. Rev. 85 (1952) 166. [ 6 ] H. Maturana and F. Varela, Autopoiesis and cognition (Reidel, Dordrecht, 1980). [7] F. Varela, Principles of biological autonomy (North-Holland, Amsterdam, 1979). [ 8] L. Brillouin, Tensors in mechanics and elasticity (Academic Press, New York, 1964). [9 ] J.A. Wheeler, The past and the "delayed-choice" double-slit experiment, in: Mathematical foundations of quantum theory, ed. A.R. Marlow (Academic Press, New York, 1978).
PHYSICS LETTERSA
l 0 November 1986
Q U A N T U M CYBERNETICS AND ITS TEST I N "LATE C H O I C E " E X P E R I M E N T S Gerhard GR~)SSING Atominstitut der Osterreichischen Universitdten, Schiittelstrasse I15, A-1020 Vienna, Austria
Received 9 September 1986; accepted for publication 17 September 1986
A relativistically invariant wave equation for the propagation of wave fronts S = const (S being the action function) is derived on the basis of a cybernetic model of quantum systems involving "hidden variables". This equation can be considered both as an expression of Huygens' principle and as a general continuity equation providing a close link between classical and quantum mechanics. Although the theory reproduces ordinary quantum mechanics, there are particular situations providing experimental predictions differing from those existing theories. Such predictions are made for so-called "late choice" experiments, which are modified versions of the familiar "delayed choice" experiments.
1. Introduction As is well known, classical mechanics can be considered with Goldstein [1] as the "geometrical--optical approximation of wave mechanics", in the sense that the Hamiltoia-Jacobi equations reveal classical mechanics as the geometrical-optical limiting case of a wave movement: light rays being orthogonal to wave fronts correspond to particle trajectories which are orthogonal to surfaces with constant action functions S, where S ( q , p, t) = lee(q, p ) - E t ,
(l)
q, p, t denoting location, momentum, and time coordinates respectively, Wthe time-independent "characteristic function", and E the energy (fig. 1 ).
Our principal aim in this paper is to derive in the context of quantum theory a relativistically invariant wave equation describing this propagation of wave fronts S=const, which would thus provide a close link between classical and quantum theory by using the same language, and which would simultaneously be in agreement with relativity theory. To obtain such an equation, additional physical assumptions on the nature o r s will be necessary. Our proposal will be to take into account a modified version of de Broglie's "double solution" model in combination with his assumptions on a "hidden thermodynamics" [ 2,3 ], introduced by the author as "quantum cybernetics" [ 4 ].
2. Relativistic quantum cybernetics W=b
W=b+Edt %
To begin with, consider the two wavefunctions constituting de Broglie's "double solution":
-
'
I--
/\
] l (0) =a
~=R
S (dt)=b S (0) = b
S (dt) = a
Fig. 1. Surfaces of constant action function S corresponding to wave fronts propagatingin configuration space. 0375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
e -iS/~
and
w = a e -is/~ ,
(2)
with R, a, and S real, where ~ represents the ordinary solution and w represents a "singularity solution" of the wave equation having the same phase S as ~ , but with an amplitude involving a generally mobile singularity, i.e. a particle with an energy Eo = hog o = moc 2. Writing down the Klein-Gordon equation for w, 381
Volume 118, number 8
(D+moc2/h)w=O,
PHYSICS LETTERS A
I'-l=(1/c2)O2/Ot2-V 2,
(3)
The latter decomposes, as is well known, into two equations: a Hamilton-Jacobi-Bohm type equation [2,5] $2 = E 2 +c2(V S)2 + Q2 ,
(4a)
which can also be written as OuSOuS= mEc 2 + Q2/c2 = M E c 2 ,
(4a')
attributing a variable rest mass Mo to the particle with the generally variable "quantum potential" QZ=cEh2Da/a, and the equation for the conservation of the probability current
Ou[a2 OuS] =OuJU = 0 .
(4b)
j u is the four-probability current
(5)
J~' = w*wOu S ,
corresponding to a generalized four-momentum OuS. Rewriting (4b) as
2yavs-h~/c2 a
DS=0
(6)
we see that eq. (6) has three possible solutions with respect to the function S: (a) kS=0;
VS=S=0,
DS=0,
(b)~S=0;
VS~0,
S~0,
DS~0,
(c) 5S~0;
VS~0,
S~0,
FqS~0.
(7)
We know that the Hamilton-Jacobi equations follow from a minimum action principle 8S=0. However, eq. (6) is compatible with ~ S ~ 0 (viz. eq. (7c)) so that one could imagine the existence of a more encompassing equation in agreement with the continuity equation that contains 8 S = 0 as special case. To obtain such a "generalized continuity equation" requires, however, that one has additional information on the action function S. This will be provided by a cybernetic model for quantum systems as proposed by the author in ref. [ 4]. The motivation to introduce cybernetic concepts into quantum physics comes from the following observation by Maturana and Varela [ 6,7 ]: "If one says that there exists a machine M in which there is a feedback loop through the environment, so that the 382
10 November 1986
effects of its output affect its input, one is in fact talking about a larger machine M' which includes the environment and the feedback loop in its defining organization". This has led Maturana and Varela to propose the notion of an autonomous system as a dynamical s~stem maintaining its identity as defined by its organization, thus representing a self-referential unity in the specific domain of its realization. Our assumption is that the mathematical behavior of the action function S indicates the "organization" of a quantum system. In close analogy to Varela [ 7 ], we consider the latter to be a feedback system with a given reference signal that compensates disturbances only relative to the reference point (i.e. a basic frequency), and not in any way reflects the texture of the disturbance. Its behavior, then, is the process l>y which such a unit controls its "perceptual data" through adjusting the reference signal. Physically, this would correspond to a "particle" as harmonic oscillator and an accompanying "wave" w oscillating in phase and creating a feedback loop through the environment in the sense that an alteration of the wave's characteristic frequency alters the particle's frequency, and vice versa [4]. Our task is now to describe the behavior of the action function S in the framework as outlined above. To do so, we make use of de Broglie's thermodynamical description of a quantum system [ 3 ] which in turn draws on earlier ideas by Brillouin [ 8 ]. With Boltzmann's formula for the relation between a variation in action S and in dissipated heat Qo (equivalent to an energy Moc 2) of a periodic system characterized by its frequency ~,= l/T, 8S= -T~Qo = -z~Moc 2 ,
(8)
one gets v S S = T6Se ,
(9)
where Se denotes the entropy of the system and T the temperature of the surrounding heat bath. Considering a particle as containing some "internal heat" Qo (or variable rest mass Mo, respectively) that is in constant thermal contact with the heat bath of the vacuum of temperature T such that moc 2 = h l , = k T ,
(10)
where h and k are respectively Planck's and Boltzmann's constants, one obtains [ 3 ]
Volume 118, number 8
PHYSICS LETTERSA
S e / k = S/h .
( 11 )
Then, with (12)
8S/h= -~Qo/moc 2
systems, their organizational characterization, the action function S, obeys a wave equation itself, namely with its own four-gradient as source. Since we have also with (17) M 2 c 2 = 4h 2 Ou a 0 i, a/a 2,
one gets 8Se = - k S Q o / m o c 2 .
Se/k=ln P,
we see that to characterize the organization of a quantum system we only need to know the relative variation of its characteristic amplitude a:
(14)
8S=0,
P denoting the probability identified via the continuity equation (4b) as P = a 2, one obtaines both the familiar Boltzmann formula P=Po exp(-~Qo/kT),
(15)
and, with eq. (11 ), our new relation S / h = In a 2 .
(16)
Further, eq. (11 ) already indicates the equivalence of the minimal action principle and the second law of thermodynamics, which in fact has been shown by de Broglie [ 3 ] independently of the considerations made here. Consequently, on the natural trajectory of the particle the entropy Se is maximal, i.e. natural trajectories are more probable than others. One can therefore say that under the basic assumption of quantum cybernetics, the "resonance condition" (10) implying the balance of "internal" and "external" energies, the wave can be seen as mediator between the particle and the "hidden thermostat", i.e. the vacuum. This means that it is not only a "guiding wave" (as in de Broglie's model) determining the trajectory of the particle, but also that the wave's frequency can be looked at as altered by a sudden frequency change of the particle's oscillation. This, then, constitutes the particle's oscillation as "reference point", with the wave providing the feedback loop through the environment. We can now insert eq. (16) into (4b) to obtain: r-lP= Fla2 = 0 ,
(17) M2 c 2 .
( 19 )
(13)
Since
[ 3 S = - h - l OuSOu S = - h - l
10 November 1986
(18)
Eq. (18) is the wave equation for the wave fronts S=const we have been looking for, and it is easily identified as Huygens' principle in configuration space: because of the wave-like nature of quantum
or
2h6a/a.
(20)
We call eq. (20) a general action principle testing whether or not there exists a new source term 6a ~ 0 along the integration area ~ of the action function S = f ~ L d x producing new waves with wavefronts S = const. Moreover, eq. (17) shows that standing waves of the form a 2 = cos 09t cos k x = ½[ c o s ( o J t - kx) + cos(tot + kx)] are simple solutions of D P = n a 2= 0, implying that the particle oscillates in phase with its environment as long as 8 S = 0 , where the standing waves function as the mediators. This also gives an intuitive understanding of the ansatz hv = k T . One can also say that 6 S = 0 implies a situation of stability (conservation of organization) that can be represented by the existence of stable standing waves. Thus, on the one hand, the characteristic oscillation of each autonomous quantum system is defined as the resulting oscillation of a relevant environment, and, on the other hand, the central oscillation itself contributes to the overall frequency of the harmonically oscillating systems "particle plus wave". It is therefore wrong in this model to talk about a "wave coming into a apparatus with its particle" or "leaving the apparatus". Rather, the waves must be at any time understood more like "carrier waves" representing the whole system (particle plus apparatus) as long as this wholeness is not destroyed by an act of measurement (implying 8 S # 0). We therefore speak of "organizational coherence" as long as 8 S = 0 and of its destruction during the measurement process, 8 S ¢ 0. As we have already shown in ref. [ 4 ] that the geodesic equation of general relativity can be derived from eq. (18) such that the latter constitutes a c o m 383
Volume 118, number 8
PHYSICS LETTERSA
mon basis for both quantum theory and relativity theory, our main task will now be to obtain consequences of our model that can be tested in experiments.
3. "Late choice" experiments Comparing analytical mechanics with geometrical optics one can show [ 8 ] an identity of the principle of least action 8 S = 0 and Fermat's principle of a light ray normal to surfaces S= const following the "shortest path" along its way from initial to final point, provided the velocity u of the wave front is given by
u = E / V S > c.
(21)
Let us now try to find a corresponding relation between wave and particle properties of a quantum system in the framework of quantum cybernetics. From eq. (17) we see that the four-probability current is proportional to the four-gradient of the probability P itself:
DP=O*-*OuJU=Ou[POuS ] =Ou[hOuP] = 0 .
(22)
There are, however, two ways to write out explicitly our "generalized" continuity equation" (22) corresponding to the introduction of two different velocities, v=dx/dt and u = Ox/Ot, that can also be defined in the following way:
v= - - c 2 ~ ,
u=c 2dS/dx -~-~,
uv=c 2 .
(23)
One can easily verify eq. (23) by inserting S = E t - p x with E=~_
°c2 v2/c2
and
~m°v P = x / " -- v2/c2
Under conservation of energy S = const, eq. (22) then provides both the "ordinary" continuity equation for particles with velocities v:.
OP/Ot + V ( Pv) =O ,
(24)
and a continuity equation for the waves with velocities u' = -u=c2/v:
OPlOt+ V (Pu') = 0 .
(25)
Note that the velocities u in (23) and (21) are iden384
l0 November 1986
tical. Moreover, the difference between v and u' involving total derivatives and partial derivatives, respectively, is also one of sign. One can therefore identify the waves S = const with shock waves whose velocity is inversely proportional to the particle's velocity. This is a result analogous to the classical case [ 1,8 ], except that now we can give a direct physical meaning to those waves: the waves S = const can now be considered as real shock waves due to a compression of the vacuum where an infinite velocity u' (i.e. v = 0 ) implies that the whole system (particle plus wave) oscillates harmonically at any point in spacetime. The possibility of velocities u' greater than the speed of light in the vacuum may look contradictory to relativity theory. However, a closer look shows that here it is the vacuum itself whose compressions are propagating, and not a particle travelling in the vacuum! Moreover, the accurate way to provide statements in agreement with relativity theory is to write the corresponding equations in a covariant form. This is, however, what we did in eq. (25) involving the velocity u'. We therefore conclude that, in general, velocities larger than the vacuum speed of light are not a priori forbidden by relativity theory. More importantly, for particular situations they even provide the basis for experimental predictions differing from those of existing theories, although - in all other c a s e s - there is full agreement between our model and both relativity and quantum theory. To obtain experimental predictions different from those of ordinary quantum theory, we consider the following interferometry experiments. Suppose we want to perform a modified "delayed choice experiment" [9] in the sense that one inserts or removes the first beamsplitting mirror (figs. 2 and 3) shortly after the main fraction of the wave packet has entered the interferometer. We call this kind of experiment a "late choice experiment" since a choice on mirror 1 is made after the bulk of the wave packet has entered the interferometer, as opposed to making a "delayed choice" on mirror 2 shortly before the wave packet leaves the interferometer device. Consider first the situation as shown in fig. 2. Quantum theory predicts no major effect of the removal of mirror l (fig. 2b) after the bulk of the wave packet has already passed it, since it is only the
Volume 118, number 8
PHYSICS LETTERS A
S
S mirror I removed
A
B
A
B
a
D1
D2
D1
D2
Fig. 2. (a) Schematized interferometer with source S, two halfsilvered mirrors 1 and 2, two totally reflecting mirrors A and B, and two detectors D~ and D2. (b) Schematized interferometer with beam-splitting mirror 1 removed. Contrary to orthodox quantum theory, quantum cybernetics predicts destruction of interference at detectors D, and D2 for the case that mirror 1 is removed shortly after the bulk of the wave packet has passed it.
S
S mirror 1 ...-'"'" i n s e r t e d
AI
B
A
B
a
D1
D2
DI
D2
Fig. 3. (a) Schematized interferometer without the first beamsplitting mirror I. (b) Schematized interferometer with beamsplitting mirror 1. Contrary to orthodox quantum theory, quantum cybernetics predicts interference to occur at D2 even for the case that mirror 1 is inserted only shortly after the bulk of the wave packet has passed the area of not yet inserted mirror 1.
10 November 1986
m i n o r rest o f the wave packet that constitutes the noninterfering fraction o f counts at detectors D1 a n d D2. Q u a n t u m cybernetics, however, provides a different answer: since organizational coherence has been destroyed through the removal o f m i r r o r 1, there m u s t be a new " s o u r c e " o f waves S = const due to eq. (18). Since the particle and the surrounding wave are oscillating in phase with each other, a change in the e n v i r o n m e n t m u s t affect the particle as well. This "change in the e n v i r o n m e n t " , however, is such that the new experimental situation is o f the kind as shown in fig. 2b, i.e. no interferability. The question now is whether this change at the location o f m i r r o r 1 can still affect the particle that has already passed it. The answer is "yes", since we know from eq. (23), or (25) respectively, that the waves S = const propagate with velocities u' = -c2/v, where v is the velocity o f the object creating the disturbance, i.e. m i r r o r 1. Since one can neglect the time span until m i r r o r 1 is inserted (i.e. until v = 0 ) , one can say that the shock waves S = c o n s t practically i m m e d i a t e l y prevent interference to occur for the case shown in fig. 2b. The opposite can be observed in situations as shown in fig. 3. There, one starts with a non-interfering situation a n d waits with the insertion o f m i r r o r 1 until the m a j o r part o f the wave packet has entered the interferometer. Again, this insertion produces a different organizational situation in the sense that the source term o f eq. (18) changes such as to create an e n v i r o n m e n t that establishes interference. Since one can consider the shock waves S = c o n s t as propagating with u' >>c, q u a n t u m cybernetics predicts interference for the case shown in fig. 3b. Summarizing, q u a n t u m cybernetics reflects the high sensitivity o f the q u a n t u m system "particle plus wave" towards organizational changes in the surroundings o f the particle which are m e d i a t e d through the wave, wave front S = c o n s t representing a symbolical description o f a " p r o p a g a t i n g d i s t u r b a n c e " in the vacuum. Such a high sensitivity is well known in o r d i n a r y q u a n t u m theory, manifesting itself in the Heisenberg uncertainty relations, for example. Pertaining to a "realist" or " o b j e c t i v e " description o f q u a n t u m systems, it is h a r d to imagine that e.g. in a double slit experiment, the b e h a v i o u r o f a particle whose path goes through one slit is affected by what goes on at the other slit only as long as it has not actually passed the slit. Considering the delicate 385
Volume 118, number 8
PHYSICS LETTERS A
wholeness o f the system "particle plus a p p a r a t u s " a n d its sensitivity to a n y changes w i t h i n it, the experimental predictions of quantum cybernetics only reflect properly this high sensitivity i n t e r m s o f the creation or d e s t r u c t i o n o f o r g a n i z a t i o n a l coherences.
Acknowledgement T h i s work was s u p p o r t e d i n part b y the B u n d e s m i n i s t e r i u m f'tir W i s s e n s c h a f t u n d F o r s c h u n g o n the contract n u m b e r Z1.19.153/3/26/85.
References [1] H. Goldstein, Classical mechanics (Addison-Wesley, Reading, 1959).
386
10 November 1986
[ 2 ] L. de Broglie, Non-linear wave mechanics (Elsevier, Amsterdam, 1960). [3] L. de Broglie, La thermodynamique de la particule isol6e (Gauthier-Villars, Paris, 1964). [ 4 ] G. Gr6ssing, How does a quantum system perceive its environment?, talk given at Int. Conf. on Microphysical reality and quantum formalism (Urbino, Italy, Oct. 1985), proceedings to appear, eds. G. Tarozzi and A. van der Merwe ( Reidel, Dordrecht). [5] D. Bohm, Phys. Rev. 85 (1952) 166. [ 6 ] H. Maturana and F. Varela, Autopoiesis and cognition (Reidel, Dordrecht, 1980). [7] F. Varela, Principles of biological autonomy (North-Holland, Amsterdam, 1979). [ 8] L. Brillouin, Tensors in mechanics and elasticity (Academic Press, New York, 1964). [9 ] J.A. Wheeler, The past and the "delayed-choice" double-slit experiment, in: Mathematical foundations of quantum theory, ed. A.R. Marlow (Academic Press, New York, 1978).