Isaac Newton Institute, December 15, 2004

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Isaac Newton Institute, December 15, 2004
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Conventional Quantum Mechanics
Deterministic Quantum Mechanics
The rules for physical calculations are
identical
All choices of basis are equivalent
There is a preferred basis
Locality applies to commutators outside the
light cone
Locality can only be understood in this
basis
Gauge equivalence classes of states
Ontological equivalence classes
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The use of Hilbert Space Techniques as
technicaldevices for the treatment of the
statistics of chaos ...
Diagonalize
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(An atom in a magnetic field)
An operator that is diagonal in the
primordial basis, is a BEABLE .
In the original basis
Other operators such as H, or
are CHANGEABLES
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Deterministic evolution ofcontinuous degrees
of freedom
but, this H is not bounded from below !
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The harmonic oscillator
Theorem its Hilbert Space is that of
a particle moving along a circle
?
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Our assignment Find the true beables of our
world!
Beables can be identified for
An atom in a magnetic field
Second quantized MASSLESS, NON-INTERACTING
neutrinos
Free scalar bosons
Free Maxwell photons
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Beables for the first quantized neutrino
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But, single neutrinos have
empty
Diracs second quantization
full
But, how do we introduce mass? How do we
introduce interactions? How do the flat
membranes behave in curved space-time ?
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A key ingredient for an ontological theory
Information loss
Introduce equivalence classes
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Neutrinos arent sheets ...
They are equivalence classes
Note
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Two coupled degrees of freedom
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Does dissipation help to produce a lower
bound to the Hamiltonian ?
Consider first the harmonic oscillator
The deterministic case write
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Two independent QUANTUM harmonic oscillators!
We now impose a constraint (caused by
information loss?)
Important to note The Hamiltonian nearly
coincides with the Classical conserved
quantity
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This oscillator has two conserved quantities
Write
Alternatively, one may simply remove the
last part, and write
Or, more generally
Then, the operator D is no longer needed.
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Compare the Hamiltonian for a (static) black
hole.
We only see universe I.
Information to and from universe II is
lost. We may indeed impose the constraint
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Let H be the Hamiltonian and U be
an ontological energy function.
But, even in a harmonic oscillator, this
lock-in is difficult to realize in a model.
The classical quantization of energy
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This way, one can also get into grips
with the anharmonic oscillator.
Since H must obey
where T is the period of the (classical)
motion, we get that only special orbits
are allowed.
Here, information loss sets in. The special
orbits are the stable limit cycles!
If T is not independent of , then the
allowed values of H are not
equidistant, as in a genuine anharmonic
oscillator.
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The perturbed oscillator has
discretized stable orbits. This is what
causes quantization.
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A deterministic universe may show POINCARÉ
CYCLES
Equivalence classes form pure cycles
Gen. Relativity time is a gauge parameter !
Dim( ? ) different Poincaré cycles
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For black holes, the equivalence classes
are very large!
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The black hole as an information processing
machine
These states are also equivalence
classes. The ontological states are in the
bulk !!
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The cellular automaton
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Suppose ? a theory of ubiquitous
fluctuating variables ? not resembling
particles, or fields ...
Suppose ? that what we call particles
and fields are actually complicated
statistical features of said theory ...
One would expect ? statistical features
very much as in QM (although more probably
resembling Brownian motion etc.
? Attempts to explain the observations in
ontological terms would also fail, unless
wed hit upon exactly the right theory ...
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