Geen diatitel

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Utrecht University
the
Determinististic
QUANTUM
Gerard t Hooft
Tilburg, Keynote Address
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QM vs determinism opposing religions?
Newtons laws not fully deterministic
The heuristic theory local determinism,
info-loss and quantization (discretization)
Quantum statistics QM as a tool.
Quantum states The deterministic Hamiltonian
The significance of the info-loss
assumption equivalence classes
Energy and Poincaré cycles
Free Will, Bells inequalities the
observability of non-commuting operators
Symmetries, local gauge-invariance
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Determinism
Omar Khayyam (1048-1131)
in his robaiyat
And the first Morning of creation wrote
/ What the Last Dawn of Reckoning shall
read.
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Starting points
Our present models of Nature are quantum
mechanical. Does that prove that Nature
itself is quantum mechanical?
We assume a ToE that literally determines
all events in the universe determinism
Theory of Everything
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Newtons laws not fully deterministic
CHAOS
t0 x 1.23456789012345
prototype example of a chaotic mapping
t1 x 2.41638507294163
t2 x 4.62810325476981
How would a fully deterministic theory look?
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t0 x 1.23456789012345
t1 x 1.02345678901234
t2 x 1.00234567890123
There is information loss.
Information loss can also explain
QUANTIZATION
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Information is not conserved
This is a necessary assumption
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Two (weakly) coupled degrees of freedom
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One might imagine that there are equations
of Nature that can only be solved in a
statistical sense. Quantum Mechanics appears
to be a magnificent mathematical scheme to
do such calculations.
Example of such a system the ISING MODEL
L. Onsager, B. Kaufman 1949
In short QM appears to be the solution
of a mathematical problem. As
if We know the solution, but what EXACTLY
was the problem ?
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The use of Hilbert Space Techniques as
technicaldevices for the treatment of the
statistics of chaos ...
TOP DOWN BOTTOM UP
Beable
Diagonalize
Changeable
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Quantum States
With this Hamiltonian, the quantum system
is identical to the classical system.
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If there is info-loss, this formalism will
not change much, provided that we introduce
EQUIVALENCE CLASSES
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Consider a periodic system
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¹/3T
2
½T
1
T
E 0
a harmonic oscillator !!
? 1
? 2
? 3
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The equivalence classes have to be very
large
these info - equivalence classes are
very reminiscent of local gauge equivalence
classes. It could be that thats what
gauge equivalence classes are
Two states could be gauge-equivalent if
the information distinguishing them gets lost.
This might also be true for the
coordinate transformations
Emergent general relativity
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Other continuous Symmetries such as
rotation, translation, Lorentz, local gauge
inv., coordinate reparametrization invariance,
may emerge together with QM ... They may
be exact locally, but not a property of
the underlying ToE, and not be a
property of the boundary conditions of the
universe
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momentum space
Rotation symmetry
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Renormalization Group how does one
derive large distance correlation features
knowing the small distance behavior?
K. Wilson
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momentum space
Unsolved problems
Flatness problem, Hierarchy problem
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A simple model
generating the following quantum theory for
an N dimensional vector space of states
2 (continuous) degrees of freedom, f and
?
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In this model, the energy ? is a beable.
stable fixed points
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Bell inequalities
And what about the
?
John S. Bell
electron
vacuum
Measuring device
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Quite generally, contradictions between QM
and determinism arise when it is assumed
that an observer
may choose between non-commuting
operators, to measure whatever (s)he wishes
to measure,
without affecting the wave functions,
in particular their phases.
But the wave functions are man-made
utensils that are not ontological, just as
probability distributions.
A classical measuring device cannot be
rotated without affecting the wave functions
of the objects measured.
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The most questionable element in the usual
discussions concerning Bells inequalities, is
the assumption of
FREE WILL
Propose to replace it with
Unconstrained Initial State
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Free Will
Any observer can freely choose which
feature of a system he/she wishes to
measure or observe.
Is that so, in a deterministic theory ?
In a deterministic theory, one cannot
change the present without also changing the
past.
Changing the past might well affect the
correlation functions of the physical degrees
of freedom in the present the phases of
the wave functions, may well be modified
by the observers change of mind.
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Do we have a FREE WILL , that does not
even affect the phases?
Using this concept, physicists prove that
deterministic theories for QM are
impossible.
The existence of this free will seems to
be indisputable.
Citations
R. Tumulka we have to abandon one of
Conways four incompatible premises. It seems
to me that any theory violating the freedom
assumption invokes a conspiracy and should be
regarded as unsatisfactory ... We should
require a physical theory to be
non-conspirational, which means here that it can
cope with arbitrary choices of the experimenters,
as if they had free will (no matter whether or
not there exists genuine" free will). A
theory seems unsatisfactory if somehow the
initial conditions of the universe are so
contrived that EPR pairs always know in advance
which magnetic fields the experimenters will
choose.
Conway, Kochen free will is just that the
experimenter can freely choose to make any one
of a small number of observations ... this
failure of QM to predict is a merit rather
than a defect, since these results involve free
decisions that the universe has not yet made.
Bassi, Ghirardi Needless to say, the the
free-will assumption must be true, thus B is
free to measure along any triple of directions.
...
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General conclusions

• At the Planck scale, Quantum Mechanics is
  not wrong, but its interpretation may have
  to be revised, not only for philosophical
  reasons, but also to enable us to
  construct more concise theories, recovering
  e.g. locality (which appears to have been
  lost in string theory).
• The random numbers, inherent in the
  usual statistical interpretation of the wave
  functions, may well find their origins at
  the Planck scale, so that, there, we have
  an ontological (deterministic) mechanics
• For this to work, this deterministic system
  must feature information loss at a vast
  scale
• Holography any isolated system, with
  fixed boundary, if left by itself for long
  enough time, will go into a limit cycle,
  with a very short period.
• Energy is defined to be the inverse of
  that period E h?

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THE END
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In search for a
Lock-in mechanism
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Lock-in mechanism
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The vacuum state must be a Chaotic
solution Just as in Conways Game of
Life ...
stationary at large distance scales
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Free will is limited by laws of physics
The ultimate religion ( Moslem ?? ) The
will of God is absolute ...
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Conway's Game of Life
1. Any live cell with fewer than two neighbours
dies, as if by loneliness.
2. Any live cell with more than three neighbours
dies, as if by overcrowding.
3. Any live cell with two or three neighbours
lives, unchanged, to the next generation.
4. Any dead cell with exactly three neighbours
comes to life.
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This allows us to introduce quantum
symmetries
Example of a quantum symmetry A 11
dimensional space-time lattice with only
even sites x t even.
? t
Law of nature
x ?
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? t
Classically, this has a symmetry
x ?
an EMERGENT QUANTUM SYMMETRY
But in quantum language, we have
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What about rotations and translations?
One easy way to use quantum operators
to enhance classical symmetries
The displacement operator
Eigenstates
Fractional displacement operator
This is an extension of translation symmetry
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Consider two non - interacting periodic
systems
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The allowed states have kets with
and bras with
Now,
and
So we also have
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The combined system is expected again to
behave as a periodic unit, so, its energy
spectrum must be some combination of
series of integers
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In the energy eigenstates,
the equivalence classes coincide with the
points of constant phase of the wave
function.
Limit cycles
The phase of the wave function tells us
where in the limit cycle we will be.