7. The Einstein-Bohr Controversy about Indeter-
minacy. Although having decidedly furthered the de-
velopment of the probabilistic interpretation of quan-
tum phenomena through his early contributions to the
photo-electric effect and through his statistical deriva-
tion of Planck's formula for black-body radiation,
Einstein never agreed to abandon the principles of
causality and continuity or, equivalently, to renounce
the need of a causal account in space and time, in favor
of a statistical theory; and he saw in the latter only
an incomplete description of physical reality which has
to be supplanted sooner or later by a fully deterministic
theory. To prove that the Bohr-Heisenberg theory of
quantum phenomena does not exhaust the possibilities
of accounting for observable phenomena, and is conse-
quently only an incomplete description, it would
suffice, argued Einstein correctly, to show that a close
analysis of fundamental measuring procedures leads to
results in contradiction to the uncertainty relations. It
was clear that disproving these relations means dis-
proving the whole theory of quantum mechanics.
Thus, during the Fifth Solvay Congress in Brussels
(October 24 to 29, 1927) Einstein challenged the cor-
rectness of the uncertainty relations by scrutinizing a
number of thought-experiments, but Bohr succeeded
in rebutting all attacks (Bohr, 1949). The most dramatic
phase of this controversy occurred at the Sixth Solvay
Congress (Brussels, October 20 to 25, 1930) where these
discussions were resumed when Einstein challenged the
energy-time uncertainty relation ΔE Δt ≳ ℏ with the
famous photon-box thought-experiment (Jammer
[1966], pp. 359-60). Considering a box with a shutter,
operated by a clockwork in the box so as to be opened
at a moment known with arbitrary accuracy, and re-
leasing thereby a single photon, Einstein claimed that
by weighing the box before and after the photon-
emission and resorting to the equivalence between
energy and mass, E = mc2, both ΔE and Δt can be
made as small as desired, in blatant violation of the
relation (2). Bohr, however (after a sleepless night!),
refuted Einstein's challenge with Einstein's own
weaponry; referring to the red-shift formula of general
relativity according to which the rate of a clock de-
pends on its position in a gravitational field Bohr
showed that, if this factor is correctly taken into ac-
count, Heisenberg's energy-time uncertainty relation
is fully obeyed. Einstein's photon-box, if used as a
means for accurately measuring the energy of the
photon, cannot be used for controlling accurately the
moment of its release. If closely examined, Bohr's
refutation of Einstein's argument was erroneous, but
so was Einstein's argument (Jammer, 1972). In any
case, Einstein was defeated but not convinced, as Bohr
himself admitted. In fact, in a paper written five years
later in collaboration with B. Podolsky and N. Rosen,
Einstein showed that in the case of a two-particle
system whose two components separate after their
interaction, it is possible to predict with certainty
either the exact value of the position or of the momen-
tum of one of the components without interfering with
it at all, but merely performing the appropriate meas-
urement on its partner. Clearly, such a result would
violate the uncertainty relation (1) and condemn the
quantum-mechanical description as incomplete (Ein-
stein, 1935). Although the majority of quantum-
theoreticians are of the opinion that Bohr refuted this
challenge also (Bohr, 1935), there are some physicists
who consider the Einstein-Podolsky-Rosen argument
as a fatal blow to the Copenhagen interpretation.
Criticisms of a more technical nature were leveled
against the energy-time uncertainty relation (2). It was
early recognized that the rigorous derivation of the
position-momentum relation from the quantum-
mechanical formalism as a calculus of Hermitian oper-
ators in Hilbert space has no analogue for the energy-
time relation; for while the dynamical variables q and
p are representable in the formalism as Hermitian
(noncommutative) operators, satisfying the relation
qp - pq = iℏ, and although the energy of a system
is likewise represented as a Hermitian operator, the
Hamiltonian, the time variable cannot be represented
by such an operator (Pauli, 1933). In fact, it can be
shown that the position and momentum coordinates,
q and
p, and their linear combinations are the only
canonical conjugates for which uncertainty relations
in the Heisenberg sense can be derived from the oper-
ator formalism. This circumstance gave rise to the fact
that the exact meaning of the indeterminacy Δ
t in the
energy-time uncertainty relation was never unam-
biguously defined. Thus in recent discussions of this
uncertainty relation at least three different meanings
of Δ
t can be distinguished (duration of the opening time
of a slit; the uncertainty of this time-period; the dura-
tion of a concomitant measuring process c.f., Chyliński,
1965; Halpern, 1966; 1968). Such ambiguities led L. I.
Mandelstam and I. Tamm, in 1945, to interpret Δ
t
in this uncertainty relation as the time during which
the temporal mean value of the standard deviation of
an observable
R becomes equal to the change of its
standard deviation: Δ̅
R̅ = 〈
Rt + Δt〉 - 〈
Rt.
now, denotes the energy standard deviation of the
system under discussion during the R-measurement,
then the energy-time uncertainty relation acquires the
same logical status within the formalism of quantum
mechanics as that possessed by the position-momentum
relation.
A different approach to reach an unambiguous in-
terpretation of the energy-time uncertainty relation
had been proposed as early as 1931 by L. D. Landau
and R. Peierls on the basis of the quantum-mechanical
perturbation theory (Landau and Peierls, 1931; Landau
and Lifshitz [1958], pp. 150-53), and was subsequently
elaborated by N. S. Krylov and V. A. Fock (Krylov
and Fock, 1947). This approach was later severely
criticized by Y. Aharonov and D. Bohm (Aharonov and
Bohm, 1961) which led to an extended discussion on
this issue without reaching consensus (Fock, 1962;
Aharonov and Bohm, 1964; Fock, 1965). Recently at-
tempts have been made to extend the formalism of
quantum mechanics, as for instance by generalizing the
Hilbert space to a super-Hilbert space (Rosenbaum,
1969), so that it admits the definition of a quantum-
mechanical time-operator and puts the energy-time
uncertainty relation on the same footing as that of
position and momentum (Engelmann and Fick, 1959,
1964; Paul, 1962; Allcock, 1969).
