PHYSICS LETTER
Physics LettersA 158 (1991) 1—8
North-Holland
A
Signal-locality, uncertainty, and the subquantum H-theorem. II
Antony Valentini
International Schoolfor Advanced Studies, Strada Costiera 11, 34014 Trieste, Italy
Received 10 September 1990; revised manuscript rerceived 20 June 1991; accepted for publication 21 June 1991
Communicated by J.P. Vigier
In the pilot-wave formulation, signal-locality and the uncertainty principle are shown to be valid only for the equilibrium
2 (which arises from the subquantum H-theorem proved earlier). The H-theorem then explains the emergdistribution P= I !P1
ence of effective locality and uncertainty from a deeper nonlocal and deterministictheory. In order to explain the present uneasy
“peaceful coexistence” (or “conspiracy”) between relativity and quantum theory, we suggest that a subquant.um analogue of
Boltzmann’s heat death has actually happened in the real universe.
1. Signal-locality and the uncertainty principle as consequences of p~I 12
The 1952 pilot-wave formulation of quantum theory is nonlocal and deterministic. In the preceding paper
[1] we have shown that the (coarse-grained) probability density P will tendtowards I !PI~for a complicated
system with a large number of variables. This “subquantum H-theorem” requires (of course) an assumption
about the initial conditions, which is similar to that of the classical H-theorem. It was also shown that, on extracting a single variable from the system, one obtains p= I iu~2~
Here we show that signal-locality (i.e. the absence of practical instantaneous signalling) and the uncertainty
principle are valid if and only ifp= I wI2 Thus, effective locality and uncertainty are merely properties of the
special state p= I wI~of subquantum equilibrium (or subquantum “heat death”). And the latter arises for purely
statistical (“thermodynamic”) reasons.
Signal-locality. We shall consider, for theoretical purposes, that p is known, even whenp,t I I~How p could
actually be known in practice will not be considered here, where we are occupied only with the general theoretical point that a known p would allow instantaneous signalling ifand only if~ I wI 2~ One might consider
that p is “known” by an imaginary “subquantumdemon”. Such a viewpoint is useful for the present theoretical
purpose, though its practical content is as yet unknown.
Consider, then, two non-interacting spatially-separated systems A and B, which are quantum-mechanically
“entangled”. We mean, of course, that the total Hamiltonian is at all times a sum of two independent commuting Hamiltonians, while the total wavefunction may not be factorised as WAWB.
For definiteness, we consider two (one-dimensional) “boxes” A and B, separated by a large distance, each
box containing a single particle with coordinate XA and XB respectively. For all I ~0, the total Hamiltonian is
taken to be
H=HA+HB,
where HA and H,, are both independent of time. At I ~ 0, each box-plus-particle has a ground state I E
0>
an excited2+fl2)~’2(aIEoEi>+flIE
state I E1>. The total state-vector at 1=0 is taken to be
Iwo>=(a
For simplicity we take a, fi,
1Eo>)
<XAIEI> and <X,,IE,> (i=0, 1) to be real.
0375-9601/911$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
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Volume 158, number 1,2
PHYSICS LETTERS A
19 August 1991
Having specified the initial Hamiltonian and wavefunction, it remains only to specify the initial probability
density p (XA, XB, 0). We first consider two special cases: (a) P(XA, XB, 0) = I W(XA, X,,, 0)12 and (b) P(XA,
0) (=f dXBP(XA, XB, 0), where now p(XA, XB, O)~IW(XA,XB, 0)12) is sharply peaked near some value
XA =x
0, while p( XB, 0) is arbitrary.
The question of interest is the following: If, at 1>0, the Hamiltonian HB of system B is suddenly altered to
H’,, ,& HB (for example, by suddenly moving the walls of box B), will the probability density P(XA, I) for the
distant particle A be affected? For the above cases (a) and (b), the answers turn out to be no and yes respectively. The first case then shows that Po = I Wo 12 is sufficient to prevent instantaneous signalling (as is already well known from standard quantum mechanics). The second case, together with a more general proof
below, shows that Po= I Wo 2 is necessary
as well as
2 guarantees
p=sufficient.
I WI2 for all 1>0. For 1>0 take
First (a): The condition Po I Wo1
H=HA+H’B,
where H’,, is independent of time for 1>0 and H’,, ~ HB (so that the Hamiltonian of B changes suddenly across
1=0). We then have
IW(t)>=exp[—(i/h)(HA+H’B)t]IWO>
which gives
Iw(t)> =(a2+fl2) _U2{a exp[—(i/h)E
0t] lEo> exp[—(ijh)H~z]IE,>
(1)
+/Jexp[—(i/h)E,t]lE,>exp[—(i/h)H~t}IEo>}
(noting of course that H’,, IE,> ~E,lE1>). The fact that exp[ (i/1~)H’,,t]is unitary then implies that
2=(a2+fl2)~(a2I<XAIEo>12+fl2I <XAIEI >12),
dXB I <XA~’~BIW(t)>
I
showing that P(XA, 1) is completely independent of the value ofH~.Thus, for this case, the probability density
of particle A is independent of any operations on the distant system B, and no instantaneous signalling is possible. (It should be noted, however, that ‘A itself is affected nonlocally by H’,, (see below). It is only the probability distribution of XA, for an ensemble of systems, which is not so affected.)
Now case (b): At 1=0, p(XA, 0) is sharply peaked near XA=XO. We wish to show that P(XA, 1) depends
strongly on H~.Note first that if H’,, = HB, then
—
J
W(XA,XB,t)=W(XA,XB,O)expE—(i/1l)(EO+EI)t]
which implies (taking Wo as real)
S(XA,XB,
t)=—(E
0+E,)t,
so that XA = X,, = 0. Clearly, the continuity equation for p=p ( XA, X,,, 1),
ÔP~8PXA~ÔPXBØ
öt
ÔXA
8XB
then implies thatp(XA, XB, t)=p(XA, ‘B, 0) andp(XA, t)=p(XA, 0) for all 1. Thus, ifH’B=H,,, thenp(XA,
1) is static, and remains sharply peaked at x0 for all 1>0, so that a measurement of the particle position at 1>0
would yield a value near .,~. In contrast, if H’,, # H,,, we would have (from (1))
2+fl2) ~‘2(a<XAIEo>RI (X,,, 1) exp{(i/ft) [S, (XB, I) —E
W(XA, ‘B, 1) = (a
01]}
+fJ<XAIEJ>RO(X,,,t)exp{(l/h)[SO(XB,t)—EIt]}),
where we have defined (for i= 1, 2)
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Volume 158, number 1,2
PHYSICS LETTERS A
19 August 1991
<X,, exp[—(i/h)H’Bt]IEI>=RI(X,,,t) exp[(i/h)S1(X,,,t)]
with R,, 5, real functions. Writing
W(XA, XB, t) =R (XA, XB, I) exp [(i/h )S(XA, XB, I)]
we then have
tan (S/h)—
— a<XAIEO>R,
a<XaIEO>R,sin[(S,—Eot)/h]+fl<XAIE,>Rosin[(So—E,t)/h]
cos[(S, —Eot)/h]+fl<XAIE, >Rocos[(So—E,t)/h]
(2
The essential point is that, in general, the phase S(XA, ‘B, I) will depend on both ‘A and ‘B, in a way that
involves the functions R(X,,, I) and S,(XB, t) (where the latter depend on H’,,). Thus the now nonvanishing
value of ‘A = (1/rn) ÔS/ÔXA ~ 0 will depend on both XA and X,, and, more importantly, will depend on R and
5, and therefore on H’,,. The distribution p(XA, 1) will no longer be static, P(XA, I) ~p(XA, 0), and measurement
of ‘A will then yield a value which may differ significantly from x0, in a way which depends on the value of
H’B, i.e. on the sudden movement of the walls of the distant box B. Thus, if x0 (i.e. P(XA, 0)) were known,
instantaneous signalling would be possible. (For this simple example, the evolution of P(XA, t) will clearly
depend on the value of H’,,.)
More generally, consider an arbitrary p(XA, X,,, 0) ~ I W(XA, X,,, 0)1 2~ Integrating the continuity equation
for P(XA,1B, t) over all ‘B implies
äp(XA,t)
~
~
Ut
~
U.,tBP~ItA,AB,t)AA,
UILA
where in general XA depends on ‘B. We have seen that, if H’,, =HB, then an arbitrary P(XA, 0) will be static,
~(‘A, I) = 0 for all 1. To show that Po ~ I Wo 12 generally leads to instantaneous signalling, we must show that it
leads to the nonvanishing of the right-hand-side of (3) for H’,, ~ HB.
To do this, we consider the case where 1 is infinitesimally small, I= e. We have
xA(E)~ 1 aS(XA,X,,,C)
0XA
or equivalently
h
1
ötan(S/h)
axA
(4)
‘
where tan(S/h) is given by (2). It is convenient to take a=/J= 1.
To calculate tan(S/h), we need to know R.(XB, 1) and S(XB, t). By definition, and for t=e—.0, we have
R,(X,,,e)exp[(i/h)S,(X,,,e)]=<X,,IEI>—(i/h)c<XBIH’,,IEI>
It is convenient to take both <X,,I E> and <X,, I H’,, I E1> as real. We may then simply take
RI(XB, )=R1(X,,, 0)= <‘BE1>
(5)
and
S ,
C<XBIHBIEI>
,
B,~
<X,,IE1>
We wish to calculate ~(XA, ~) to first
order
NowFurther,
from (5)
and‘A(6),
(S/h) is
of order
e, and
may
2 (S/h)
] in‘ine. (4).
since
(e) tan
is already
oforder
e, we
maywe
replace
therefore
neglect
the
factor
[1
+
tan
P(XA, XB, e) by just p(X~,X,,, 0) in (3). We thus have, to first order in ,
—
3
Volume 158, number 1,2
h
a
rn
UAA
PHYSICS LETTERS A
(f dXBp(XA,X,,,O) ätan[S(e)/h]
~
VitA
\.l
19 August 1991
( )
.
Putting (5) and (6) into tan (S/h), a straightforward calculation then shows that
a
(
b
P(XA~)=—~—~--\a(XA)JdXB (XB)
p(XA,XB,0)—Iw(XA,XB,0)12
IW(XA,XB,O)I2
8
‘
(
where
a(XA)=(<XAlE,>)2~~
1_
<XAIEO>
and
b(XB)=<E, IXB><XBIH’BIEo>—<EI IH’,,IXB><XBIEO>+(E, .—E0)<E, IXB><XBIEO>
We note the following: (i) If H~= HB, than b( ‘B) = 0 and ~ (‘A, C) vanishes, as expected. (ii) In general, for
H’,, ~ H,,, and for po ~ IWo I 2, we2)have
e) ~ 0, so thatbut
instantaneous
may be sent
fromthen,
system
B
/ I Wo~(XA,
12 is nonvanishing,
happens tosignals
be independent
of X,,,
since
to
system
A.
(iii)
If
(Po
I
Wo
I
5 dX,, b (X,,)= 0, we have ~ (Xx, e) = 0, and no instantaneous signal may be sent from B to A. Clearly, however,
instantaneous signalling would be possible from A to B. Thus, any Po ~ IWo 12 leads to instantaneous signalling.
We conclude that, if the initial distribution Po is considered as known, then instantaneous signalling is possible if and only if Po ~ IWo I 2~
If Wo is not entangled (e.g. IWo> = I E
0E,>), then of course no instantaneous signalling is possible, whatever
the relation between Po and IWo I 2, This follows immediately from (2): putting, for example, fl= 0 leads to
—
tan(S/h)=tan[SI(XB, t)—E0t]/h,
8S/ÔXA will vanish, so that P(XA, I) will be static, and therefore
which
is independent
independent
of H~. of XA. Thus XA = (1/rn )
The above conclusion with regard to instantaneous signalling also applies for momentum “measurements”.
To discuss this, it is first of all crucial to note the distinction, in the pilot-wave formulation, between “actual”
and “measured” values of momentum [2,3]. For the case of position, the following may be assumed: A socalled “measurement” of X yields a value which is equal to the actual value which existed prior to the “measurement”. For the case of momentum, however, the result of what is usually termed a “measurement” generally differs from the actual value rnX= asia, which existed prior to the “measurement”.
For example, for a single particle in the ground state of a box, S(X, 1) = —E
0t, and so mX= ÔS/ôx= 0. The
particle is at rest. If one then “measures” the momentum by, for example, opening the box, this physical action
alters the phase S of the wavefunction, and thereby alters (via as/ax,t~
0) the position of the particle in such
a way that, as t—~~x,
the particle is found to be at .Y= ±t( 2E0/rn)’ /2• The value (plus or minus) depends on
the initial position of the particle1 in
box. Measurement
of~ at large
then
a “measurement”
of
/2, the
unrelated
to the actual vanishing
valuet at
t = yields
0. As stressed
by Bohm [2]
momentum,
with
result
±
(2
mE0)
and by Bell [3], the value of a momentum “measurement” is an outcome of the whole “apparatus-plus-system”
set-up, and is not a pre-existing property of the system alone. The “measurement” must not be thought of as
yielding a result which is related in any simple way to the true value prior to “measurement” [3]. Clearly, as
stressed with great clarity by Bell, the word “measurement” is profoundly inappropriate in quantum theory,
and should perhaps be replaced by the word “experiment”. (The general confusion generated by mis-use of
the word “measurement”, in particular in leading to mistaken “impossibility proofs” regarding “hidden variables”, has been clearly shown by Bell [3].) For the present case, it might be more appropriate to regard the
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PHYSICS LETTERS A
19 August 1991
“measured” momentum value as being in a sense emitted by the whole processof opening the box. This emis-
sion has some associated probability distribution ~(PA).
Returning to the question of signalling for the momentum case, the question becomes: Is the emitted distribution P(PA) affected by the distant H’,,? For the case p= I WI2 Bohm [2] has shown that such “measured”
values have the usual distribution predicted by standard quantum mechanics (involving the Fourier transform
of the wavefunction). Thus, again, no instantaneous signalling would be possible for this case, as in standard
quantum mechanics. However, for our above example of p# I WI 2, the value of x
0 around which Po is peaked
may be chosen so that, on opening the box at t= 0, ‘A~+ ~ rather than + ~ (as t—+oo), yielding a “measured”
value PA = — (2mE0) I/2~ But then, HB-+H’B for the distant box will alter 2).
XATaway
the
hus, iffrom
Po ~ x0,
I WosoI 2,that
we opening
find instanbox would
no longer
necessarily
XA—’ ~ (i.e. PA
= — (2mEo)~’
taneous
signals
also with
regard toyield
momentum-related
experiments,
and a lack thereof when Po = I Wo 12, just as
for the case of position.
Uncertainty principle. It is straightforward to show that the uncertainty principle holds ifp= I WI 2, but is generally violated otherwise, noting again our viewpoint that p be regarded as theoretically known, independently
of W.
Firstly, if p= I WI 2, it is known that [2] momentum-related experiments yield “measured values” with a distribution I ~~‘I2
where ~t,is the Fourier transform of W. The standard deviation L~pof these values then nec—
—
essarily obeys the usual statistical dispersion relation (“uncertainty principle”)
where i~xis the standard deviation of the distribution p= I WI2
To show that p I WI2 may violate the uncertainty relation, consider the simple case of a single particle in
a box, with ground-state wavefunction Wo(X), where the particle position has a probability distribution (for
an ensemble of similar systems) which is sharply peaked at some x=x
0. We again choose x0 to be such that,
on opening the box, the particle position X—~ ~ as t—’oo, yielding a “measurement” p= — (2mEo) 1/2, For
this simple case, the standard deviation of “measured” momentum values vanishes, while & is finite (and
small compared to the size ofthe box). This clearly violates the usual Heisenberg statistical dispersion relation.
—
2. Discussion
2 is an equilibrium state, arising from a “subquantumH-theorem”. We have
Weseen
havethat
shown
[1] that P=
I WJthe uncertainty principle are valid ifand only if P= I WI 2~ Signal-locality and
now
signal-locality
and
uncertainty therefore emerge merely as properties of equilibrium, from an underlying nonlocal and deterministic theory.
The subquantum H-theorem may be thought of as describing, for complicated systems, the scrambling of
holistic information present at the subquantum level. After sufficient scrambling, the coarse-grained P and I WI2
are no longer distinguishable, and such information may no longer be seen directly. Instead, its presence must
be deduced indirectly, via Bell-type theorems. The H-theorem thus shows how a fundamentally nonlocal theory
may nevertheless yield an effectively local equilibrium state. It also shows how a deterministic theory may lead
to the uncertainty principle, this being just a property of the equilibrium probability distribution. No under-
lying stochastic process seems necessary.
The universe presumably begins in a state P,t I WI 2, where instantaneous signalling would be possible. As
one approaches the state P= I WI2 of maximum subquantum entropy, the possibility of instantaneous signalling
fades away, and statistical uncertainty takes over. The equilibrium P= I WI2 may then be seen as a kind of
subquantum “heat death”, where the nonlocal connections of quantum theory may no longer be used for practical signalling. While nonlocality is present for individual events, it is not directly observable in the equilibrium distribution. This distribution leads to the uncertainty principle, so that both “systems” and “apparatus”
5
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PHYSICS LETTERS A
19 August 1991
are subject to a mutually-consistent uncertainty from which one cannot escape. This limits our knowledge of
particle trajectories, by an amount that prevents us from sending instantaneous signals. This situation is clearly
analogous to that of a “Maxwell demon” whose equipment happens to be in thermal equilibrium with the gas
which he is attempting to study. (For thermal fluctuations then render the equipment useless for the study of
molecular trajectories.) The above “subquantum heat death” is then somewhat analogous to the classicaithermodynamic “heat death” (in a heuristic sense of course), and accounts for the universal operation of the uncertainty principle at the present time.
The equilibrium distribution P= I WI2 has special features which are not fundamental to the underlying theory (as generally happens with equilibrium distributions). Two such features are signal-locality and uncertainty, as shown above. One also generally expects that a maximum-entropy (statistical) equilibrium state will
show an especially high degree of symmetry. And indeed, the symmetry of Lorentz covariance holds only for
I WI2 (in the context of the pilot-wave theory of fields [4]). We show elsewhere [5] that the same is true
for Einstein’s local principle of equivalence. “General covariance” is then also merely a symmetry of subquanturn equilibrium.
The above picture of emergent locality uses the 1952 pilot-wave formulation. However, the main idea, that
I W~represents a special equilibrium, actually follows rather compellingly from standardquantum mechanics.
For consider the uneasy “peaceful coexistence” between relativity and quantum theory [6]. It is increasingly
realised that there exists a peculiar “conspiracy” between these two theories [5].This conspiracy prevents one
from using quantum nonlocality for practical signalling, and is usually enforced by the uncertainty principle.
The uncertainty principle “noise” seems to act so as to “save” relativity from being violated by quantum theory ~‘. As Bell put it: “It is as if there is some kind of conspiracy, that something is going on behind the scenes
which is not allowed to appear on the scenes” [9] 52~ So far, no satisfactory explanation for this conspiracy
seems to have been put forward. Certainly, such a conspiracy in the presently-known standard laws ofphysics
can hardly be regarded as fundamental.
The above conspiracy then suggests the following hypothesis: That the distribution P= I WI2 ofquantum mechanics is not fundamental, but merely represents a special equilibrium (which the universe happens to be in
at present). Now, independently of any particular theory (such as the pilot-wave), this hypothesis offers a rational understanding of the above mysterious conspiracy. For only in this special equilibrium does the uncertainty principle “noise” exactly mask the quantum nonlocality, so as to prevent instantaneous signalling ~.
This delicate balance between nonlocality and uncertainty is, as mentioned above, rather analogous to the classical state of thermodynamic heat death. In the classical heat death, no further macroscopic changes are possible. In the “subquantum heat death”, the underlying nonlocal connections may no longer be used for practical
signalling, and relativistic symmetry emerges ~.
The standard laws of physics, as presently known, therefore suggest that: A subquantum analogue of Boltzmann’s heat death has actually happened in the real universe. The subquantum H-theorem gives an explicit
model, based on the pilot-wave formulation, of how this “subquantum heat death” came about. However, the
essential idea, and the above “conspiracy” argument in its favour, are independent ofthe pilot-wave formulation.
We have shown [1] that the coarse-grained value of S= — k 5 dIP in (P/ I WI2) can never decrease. To see
As a recent example, the proposal by Herbert [7] to use “photon cloning” for instantaneous signalling does not work essentially
because of uncertainty principle “noise” in the photon-replication process (see ref. [8]).
~2 Bell’s remark refers to apparent Lorentz symmetry at the statistical level, despite anunderlying absolute rest-frame (inwhich quantum
nonlocality occurs purely across space). Detection of this frame is prevented by the uncertainty principle (i.e. only if P= I ~PI2).
~ To see that P= I ~I’I2
is related to locality, without using the pilot-wave picture, consider a Bell-type inequality derived by Roy and
Singh [10]. This was derived for hidden-variables theories, not necessarily local ones, which have the constraint of no practical
instantaneous signalling. Quantum mechanics precisely saturates the inequality, so that it only just avoids practical instantaneous
“
‘~
6
signals. This alone suggests that P= I ~PI2is a state where quantum nonlocality and uncertainty noise “just balance”.
This “historical” view of present physical laws has some affinity with broken symmetry in particle physics, except that the present
state P= I p112 (being a statistical equilibrium) has more symmetry than the underlying laws.
Volume 158, number 1,2
PHYSICS LETTERS A
19 August 1991
if the entropy-maximum (S=0) is actually reached for a particular complicated system requires an analysis
of the mixing of P and I WI2 by the velocity field X= (h/rn) Im V in W. At some time t=0, X may be chosen
independently of I WI2 (since X depends only on the phase of W). In particular, X
0 may vary rapidly in configuration space, compared to P0 and I W0 I 2~ For such W0, mixing will definitely begin. Just how far the mixing
will go, and its rate, will depend on the system (as in classical statistical
the scale
chaotic
2 onmechanics).
a very small Presumably,
coarsè-graining
41~,
past history ofaour
universe
willofguarantee
that Ptoactually
reaches
I WI
Nevertheless,
detailed
study
the approach
equilibrium
is worthwhile,
not only as a point of principle,
but also for its possible implications for cosmology: If at early times P~I WI2 on a significantly large coarsegraining scale, then the resulting nonlocality may be relevant for explaining the existence of the preferred cosmological rest-frame (defined by the microwave background), and could shed light on the “horizon problem”.
Bohm and Vigier [11], and more recently Bohm and Hiley [121, have also discussed P= I WI2 as an equilibrium. They assume the existence of subquantum “fluid fluctuations”, which provide perturbations leading
very rapidly to P—~I WI2 even for a system with just one degree of freedom. The present approach differs in
several respects:
(i) We derive P= I WI2 from the deterministic 1952 formulation (also done by Bohm for a special case [13]).
We regard the P= I WI2 seen today as the result of large scale mixing, not unlike Boltzmann’s heat death based
on classical deterministic mechanics. This avoids any arbitrariness, and gives a definite picture of how P—~I WI2
occurs. It also enables the relaxation rate to be calculated in principle.
(ii) Our derivation applies to a complicated system with a large number of variables. Any extracted single
variable will have p= I WI2 as shown before [1]. And as shown by Bohm [2], once p = I WI2 is initially given,
all later measurements (even rapidly repeated) will agree with standard quantum theory. On the other hand,
a single-variable system with an initial p I WI2 (unobtainable at present) will not relax to p = 1W 12 But since
any single variable is extracted from a larger system, and will therefore have an initial p= I WI2 there seems
to be no need to introduce a violent relaxation mechanism for a single variable. Of course it might turn out,
on closer analysis, that some sort of background perturbations are in fact necessary to ensure thorough mixing
of P and I WI2 for a complicated system. And such an analysis may suggest the required form of such perturbations. However, recalling again the long and violent history of known physical systems, it seems unlikely that
the perturbations would need to be anywhere near, as violent as those suggested by Bohm and Hiley. Instead
one expects that, if at all necessary, very slight perturbations should be sufficient to allow the past violence to
do its work (recalling that a “speck of dust” in a blackbody cavity ensures that equilibrium is reached). On
this ground we suggest that corrections to the 1952 theory could be very small, so that the predicted path
X_—(/l/m) Im V In Wis at least a good approximation. Further, there seems to be no ground for supposing
that such corrections have a stochastic nature [141.
(iii) We consider it reasonable that uncertainty and locality emerge as equilibrium properties without any
need for modification of the 1952 theory.
(iv) We stress that the essential idea, that P= I WI2 is a special equilibrium, suggests itself independently of
the pilot-wave picture. For this idea gives a natural explanation for the uneasy “peaceful coexistence”, or “conspiracy”, between relativity and quantum theory, as discussed above. So the above considerations may have
a value which is not dependent on the pilot-wave theory.
Acknowledgement
I would like to thank Professor D.W. Sciama, Sebastiano Sonego, and Gabriela Ronkainen, for many en‘~
One might be sceptical about the big-bang cosmology. But it cannot be doubted that all presently-observed systems have undergone a
long and violent astrophysical history.
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Volume 158, number 1,2
PHYSICS LETTERS A
19 August 1991
joyabie and beneficial discussions. This work was supported by the Science and Engineering Research Council
of Great Britain.
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A. Valentini, Phys. Lett. A 156 (1991) 5.
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J.S. Bell, Speakable and unspeakable in quantum mechanics (Cambridge Univ. Press, Cambridge, 1987).
D. Bohm, B.J. Hiley and P.N. Kaloyerou, Phys. Rep. 144 (1987) 349.
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A. Shimony, in: Foundations of quantum mechanics in the light of new technology, ed. S. Kamefuchi (Physical Society of Japan,
Tokyo, 1984).
[7]N. Herbert, Found. Phys. 12(1982)1171.
[8] R.J. Glauber, in: New techniques and ideas in quantum measurementtheory, ed. D.M. Greenberger (NewYork Academy ofSciences,
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[9] J.S. Bell, in: The ghost in the atom, eds. P.C.W. Davies and J.R. Brown (Cambridge Univ. Press, Cambridge, 1986).
[10] S.M. Roy and V. Singh, Phys. Lett. A 139 (1989) 437.
[11] D. Bohm and J.P. Vigier, Phys. Rev. 96 (1954) 208.
[12] D. Bohm and B.J. Hiley, Phys. Rep. 172 (1989) 93.
[13]D.Bohm,Phys.Rev.89
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[14] A. Valentini, On the pilot-wave, path-integral, and stochastic formulations ofquantum theory, preprint SISSA 19/91/A (1991), to
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[6]
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