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THE CURSE OF LIGHT BEAM RIDER
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THE
CURSE OF LIGHT BEAM RIDER
Quantum
mechanics was the only Science which Einstein
hated. Though very successful as an analytical
Science,
it has a lot of vexing problems. The first
problem is the problem of quantum entanglement.
Trying
to make its foundations weak, Einstein challenged
Bohr with various problems put forward in Copenhagen
meetings. These meetings are of much significance;
they helped the theory to develop to far greater
domains, something Einstein had never dreamt
of; but, they also showed that the interpretation
of theory was much more difficult than it seemed
to be. Copenhagen meetings led to very exciting
and unbelievable results on quantum mechanics.
One of the issues was the problem of Quantum
entanglement. Consider a state which can be
either up or down (representing upward or downward
spin
of particle). Adjust the states such that the
superposition of the states is always a null
state. Even if both the particles are kept
zillions of parsecs apart, the knowledge of state
of one
particle is sufficient to determine the state
of the other particle. |
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Thus,
the states are entangled even when they are very
far apart. Bohr never rejected this conclusion. Einstein
used to call it ‘spooky action at a distance’. The
problem was finally cleared by John S Bell who derived
some inequalities by assuming that particles were
unentangled if kept far apart (called the principle
of locality). He showed that if quantum laws
were true, then these inequalities would be violated.
Now, entanglement is being used extensively for quantum
computing and teleportation.
So, one big trouble is nearly clear. But, there is another
problem, and even a greater one. No present theory is
powerful enough to solve this problem. The evolution
of a quantum system is defined by the equation:
iħ(δ|yñ/δt)
= E|yñ
Under this equation, a quantum system evolves in a very
deterministic way with probabilities for each state.
But as an observer does his job, the whole wave
function reduces to a single point. The observer,
then, knows the state of function; it is no longer a
probability. As the measurement is stopped, the
original evolution starts again (called unitary
evolution). This unpredictable collapse of the whole
wave function to a single state is called state
reduction.
To elaborate this more, assume that |y,xñ
determines the amplitude of the particle to be at x.
This might be a fraction. But, once the observation is
made, this amplitude takes infinite values. This may
seem rather strange, but some information on dirac-delta
function will make it clear.
Once the particle is observed at a point, the wave function
dependence on position becomes something like this: |
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To determine h, we use the probability law:
∫y*
y
dx = 1
But, the only region having
y
non-zero is the small region dx. So, the contribution to
the integral is due to this area only. So h2dx = 1.
This implies that h is infinite.
This situation was first studied by Dirac. Such wave
function (or any function) with this form of jump is
called a dirac-delta function. Thus f(x-a)
is a dirac-delta if
(x-a)
= ∞ (x=a)
= 0 (x≠a)
Such unpredictable activities of quantum mechanics are the
least understood phenomena (even less understood than the
unified law; later, we will see that the solution to this
problem may come directly from a fully consistent
quantum gravity theory) of the nature. Niels Bohr held the
view that it’s the information about the system that gets
modified, and not the behaviour of the system. Other
interpretations say that the involvement of conscious
activity must be taken into account; humans themselves
work on quantum rules. For the contribution of other
elements, things called density matrices are
introduced. Other interpretations predict the presence of
alternative worlds, in which the observer enters to create
a new future of his/her own.
There are some quantities called projector operators,
which are applicable to space with finite number of
quantum states (there can be infinite dimensions, but that
is not too healthy for the understanding of the operator’s
behaviour). An operator, when applied to |yñ,
yields the answer ‘yes’ (1) or ‘no’ (0); 1, if the state |yiñ
associated with Ei is observed; else, 0. There
is a probability to observe 1 or 0. So, it is called the
projection operator. It projects the state into its
respective ‘yes’ or ‘no’ condition. Some physicists think
that a more general idea is required, of which the unitary
evolution and state reduction are just approximations.
A big trouble comes when the state reduction is
studied in the gravitational field. Suppose there are two
matrices, then
d/dt
operator will take different metrics. Suppose the metrics
are g1 and g2. Let the states in
the metrics be |g1ñ
& |g2ñ.
Suppose that we want a linear superposition of these
states. Then, for the evolution of the superposition, we need
the
d/dt
operators and the relation between them (else, you will
have two time operators). But, relating the two operators
must be for any coordinates we choose for the metric
spaces. If there is a relation, then the choice of the
metric in one space will force us to use a particular
metric for other spaces, again Einstein’s principle about
the choice of frame. Considering this trouble further
yields a surprising result that the state |yñ
(combination of |g1ñ
and |g2ñ)
must break into one of the constituent states after a time
interval of ћ/E, where E is the energy difference between
the two states. If there is any arbitrarily chosen state,
then it will break into any state during this time. This
form of reduction will cause a lot of problems in the
stability of the universe. The state reduction problem
does not seem to be cleared still by any of the present
quantum gravity theories. Moreover, many theories do not
take it very seriously. We will end this topic with some
information on some present perspectives in the next
section. |
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