Honoring Epimenides of Crete (
±Δx): From Quantum
Paradoxes, through Weak Measurements, to the Nature of
Time
Eliahu Cohen
1a
, Avshalom C. Elitzur
2
1
School of Physics and Astronomy, Tel Aviv University, Tel-Aviv 69978, Israel
2
IYAR, The Israeli Institute for Advanced Research, Rehovot, Israel
Abstract.
Quantum temporal peculiarities, involving ordinary and weak measurements,
are explored. We introduce the foundations of weak measurement and outline some novel
theoretical and experimental predictions derived from it. We then show how weak values,
which explicitly depend on both forward and backward evolving state-vectors, can serve
as important tools for gaining new insights into the nature of time.
1 Introduction
Thankful for this invitation to present our work in the stunning island of Crete, we take the
liberty of indulging in some historical reflections as an introduction.
Crete was the birthplace of Epimenides (7
th
/6
th
Century BC), author of the famous paradox
based on the self-contradicting statement "all Cretans are liars." The Quantum Liar Paradox
[1] described below is a physical manifestation of that ancient millstone, suggesting that
Nature herself is capable of creating self-contradictions.
Second, not far from this place is the village Milatos, which in ancient times gave its name to
the more famous town Miletus, birthplace of the first scientist known to history. Thales
(624–546 BC) was the discoverer of electricity and magnetism, and it so happened that
another work of ours presented in this conference [2,3] deals with the very nature of the
electric and magnetic fields.
And of course there were many other giants in the neighboring islands and shores during that
golden age. Archimedes (287–212 BC), long before the advent of calculus, recognized the
importance of infinitesimals. The "slope of a line segment as short as a point" sounds just as
absurd as Epimenides' "this statement is false," yet it eventually turned out to be one of
mathematics' most powerful tools. Archimedes was inspired by the earlier legendary dispute
between Parmenides (5th century BC) and Heraclitus (535–475 BC), about the nature of
time. For Parmenides, change was an illusion of the senses, reality being eternal and
a
eliahuco@post.tau.ac.il
DOI: 10.1051
/
C
Owned by the authors, published by EDP Sciences, 2014
,
/
0 002
(2014)
2 01
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immutable. Heraclitus, in contrast, held change to be reality's main attribute. Parmenides'
disciple, Zeno (490–430 BC), has derived one of the most famous paradoxes concerning the
alleged atomicity of time, a challenge that provoked not only Archimedes' infinitesimals but
also, in 20
th
Century, the quantum-mechanical realization [4].
All these thinkers belonged to the small genius nation of which Crete was part, whose
scholars have first raised the great questions of science and philosophy with utmost clarity
and acuity. Our research aspires to resolve quantum paradoxes as well as problems about the
nature of space-time, guided by the intuition that these two realms are closely related.
Presenting this work in this ancient cradle of science is therefore another source of awe and
inspiration.
2 Quantum Temporal Paradoxes
The most paradoxical effects displayed by quantum measurements involve spatial and
temporal anomalies, e.g., respectively, the EPR [5] and the delayed-choice [6] experiments.
Because time is the most elusive and unique dimension of space-time, we shall focus on
some novel quantum effects that strongly strain common-sense intuitions about time.
2.1 Motivation
Several years ago, during a class on quantum mechanics, one of us (AE) was presented with
a question from an inquisitive student. Why, she asked, should one think that Schrodinger's
cat was superposed before the box’s opening? This possibility can be ruled out by allowing a
certain time interval, say, three days, pass between the potentially lethal event and the
opening. Then, if the cat is found to be dead, it would be also decomposed, whereas if it is
alive it would be also lean and starved. In both cases, it should be clear that it has never been
superposed!
It takes some reflection in order to realize why this clever reasoning does not rule out that
superposition has prevailed within the box all along: Quantum observation determines not
only the cat’s state at that moment, but also its entire history since the lethal event’s
(non)occurrence!
This example indicates that some kind of retrocausality is inherent in nearly all quantum-
mechanical paradoxes, thereby giving additional credence to models that take this peculiarity
as their cornerstone. The most daring attempt of this kind is Aharonov’s two state-vector
formalism [7,8], which furthermore derives many surprising predictions, some of which are
being verified nowadays. Cramer’s [9] transactional interpretation also invokes this kind of
peculiar causality, yielding an interpretation which is elegant and parsimonious. Novel
elaborations of this model, attempting to accommodate it to recently-discovered quantum
phenomena, merit further interest [10-15].
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