Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*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 Δ*E* Δ*t* ≳ 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* = *mc*2, both Δ*E* and Δ*t* 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* = *i*ℏ, and although the energy of a system

is likewise represented as a Hermitian operator, the

Hamiltonian, the time variable cannot be represented

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

*R*̅ = 〈

*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).

Dictionary of the History of Ideas | ||