Aharonov NEGATIVE KINETIC ENERGY BETWEEN PAST AND FUTURE STATE VECTORS

Aharonov's POV seems a bit startling to some used to the local Hamiltonian formulation. On the other hand, it is quite natural in the nonlocal Lagrangian formulation from Dirac to Feynman's path-dependent quantum amplitudes that naturally require both past preparation (pre-selection) and future detection (post-selection). The novel aspect is the intermediate detection, but that is already implicit in the Feynman path integrals of the exponential of the classical path-dependent action with an appropriate "measure" to do the integration - "mathematical nonsense" if we use the standards of Waldyr Rodrigues, Jr of UNICAMP. Theoretical physics is built on the shaky foundation of mathematical nonsense justified only by the miracle that it works - makes spectacularly accurate predictions that agree with experiment.

Note "quantum time machine" below.

"An analysis of errors in measurement yields new insight into classically
forbidden quantum processes. In addition to “physical” values, a realistic
measurement can yield “unphysical” values; we show that in sequences of measurements,
the “unphysical” values can form a consistent pattern. An experiment to
isolate a particle in a classically forbidden region obtains a negative value for its
kinetic energy. It is the weak value of kinetic energy between past and future state
vectors. ...

Barrier penetration, such as tunnelling into a potential wall, is a classically
forbidden quantum process. Quantum particles can be found in regions where
a classical particle could never go: it would have negative kinetic energy. But the
eigenvalues of kinetic energy cannot be negative. How, then, can a quantum particle
“tunnel”? The apparent paradox is resolved by noting that the wave function of a
tunnelling particle only partly overlaps the forbidden region, while a particle found
within the forbidden region may have taken enough energy from the measuring
probe to offset any energy deficit. Nevertheless, actual measurements of kinetic
energy can yield negative values. Here, we present a model experiment in which
we measure the kinetic energy of a bound particle to any desired precision. We
then attempt to localize the particle within the classically forbidden region. The
attempt rarely succeeds, but whenever it does, we find that the kinetic energy
measurements gave an “unphysical” negative result; moreover, these results cluster
around the appropriate value, the difference between the total and the potential
energy. This consistency, which seems to come from nowhere – a background of
errors – suggests strongly that the notion of a quantum observable is richer than
generally realized. Previous papers making this suggestion analyze a measurement
of spin1 and a quantum time machine2 as well as negative kinetic energy.3−4 ...

Then we measure the position of each particle and select only those cases where
the particle is found within some region “far enough” from the well – with “far
enough” depending on how precisely the kinetic energy was measured. In almost
all such cases, we find that the measured kinetic energy values are negative and
cluster around the particular negative value appropriate to particles in the classically
forbidden region. ...

any measured value is possible, although large errors are exponentially suppressed.
There is no mystery in such errors; they are expected, given the uncertainty associated
with the measuring device. Measurements can even yield negative values.
The negative values may be unphysical, but they are part of a distribution representing the measurement of
a physical quantity. They should not be thrown out, since they give information
about the distribution and contribute to the best estimate of the peak value. Since
these errors originate in the measuring device, and not in the system under study,
it seems that they cannot depend on any property of the system. However, closer
analysis of these errors reveals a pattern which clearly reflects properties of the
system under study. The pattern emerges only after selection of a particular final
state of the system. ...

We thus obtain a correlation between position measurements and prior kinetic
energy measurements: nearly all particles found far outside the potential well
yielded negative values of kinetic energy. On the other hand, we could consider all
particles that produced negative values of kinetic energy, and ask about their final
position. We would find nearly all these particles inside the well. The correlation
works one way only. Prior kinetic energy measurements on particles found far from
the well cluster around a negative value, but position measurements on particles
yielding negative values of kinetic energy cluster around zero. How do we interpret
this one-way correlation? ...

Our example suggests that particles in a classically forbidden region have negative kinetic energy.
The conventional interpretation of quantum mechanics has no place for negative kinetic energy.
However, the conventional interpretation involves an assumption about how measurements are made.
The conventional interpretation considers measurements on ensembles of systems
prepared in an initial state, without any conditions on the final state of the systems.
Such an ensemble, defined by initial conditions only, may be termed a pre-selected
ensemble. By contrast, we consider measurements made on pre- and post-selected
ensembles, defined by both initial and final conditions. The experiment of the previous
section is an example of a measurement on a pre- and post-selected ensemble.
It is natural to introduce pre- and post-selected ensembles in quantum theory: in
the quantum world, unlike the classical world, complete specification of the initial
state does not determine the final state.

Also, the measurements we consider are not ideal. Real measurements are subject
to error. At the same time, the disturbance they make is bounded. These
two aspects of non-ideal measurements go together. Suppose our measuring device
interacts very weakly with the systems in the ensemble. We pay a price in
precision. On the other hand, the measurements hardly disturb the ensemble, and
therefore they characterize the ensemble during the whole intermediate time. Even
non-commuting operators can be measured at the same time if the measurements
are imprecise. When such measurements are made on pre- and post-selected ensembles,
they yield surprising results. An operator yields weak values that need
not be eigenvalues, or even classically allowed.1,6 The negative kinetic energy of the
previous section is an example of a weak value. Another is a measurable value of
100 for a spin component of a spin-1/2 particle.

Let us briefly review how weak values arise ... “game of errors” displays a remarkable consistency,
and this consistency allows negative kinetic energies to enter physics in a natural
way. The concept of a weak value of a quantum operator gives precise meaning to
the statement that the kinetic energy of a particle in a classically forbidden region
is negative: namely, the weak value of the kinetic energy is negative."


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