Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 8 Jun 1999 08:49:31 +0300 (EET DST)
On Mon, 7 Jun 1999, Stephen P. King wrote:
> Dear Matti,
>
> I am reminded of an on-going conversation that my off-line friend Paul
> Hanna and I have been having concerning how to generalize the definition
> of velocity (and motions of "particles" in general) into the language of
> probabilities. My ideas here are very rough so I beg for your patience
> and critique. In [time 392] you said:
>
> [MP]
> "In quantum field theory situation is different since it is not possible
> to interpret time evolution as evolution in any kind of configuration
> space (the required assignment of the space of quantum states to single
> point of 3-space does not make sense). Problems are also caused by the
> fact that probability density is not scalar quantity anymore but time
> component of a 4-vector."
>
> [SPK]
> Could we not see this as a problem and consider its implications? If
> the probability density is the time component of a 4-vector, what would
> the spatial components be?
In wave mechanics probability current is Galilean four vector with
time compoment behaving like scalar. The spatial compoments of
probability current form 3-vector equal to
j= Psi^* i nabla Psi/m.
You use standard rules and just substutute v= p/m=
i*Nabla/M inside Psi^* and Psi. You can derive continuity equation
partial_t rho/partial_t + Nabla.j =0
expressing conservation of probability from Schrodinger equation.
In Lorentz invariant complex scalar field theory one can
derive conserved current associated with phase invariance of
action (Phi is multiplied by a constant phase factor). This
current is however not positive definite and has interpretation
as charge density rather than probability density.
The Hamiltonian evolution equation for the quantum states
of QFT is of form idPsi/dt = H Psi and in case of free field
theory is simply the energy operator for many particle system
H= SUM_k E_k n_k ,
where E_k is the energy of the state with momentum k and n_k =
a^dagger_k a_k is the occupation number of state with momentum k.
It is possible to associate any continuity equation
to this equation at spacetime level unlike to the corresponding equation
in case of single particle wave mechanics. The reason is simply
that many particle states are not localized to single point.
[I would put emphasize on *'single particle'*: in quantum TGD
configuration space spinor fields describes modes of classical spinor
field describing states of single classical 'fermion' (in very abstract
sense): there is *no* second quantization at the level of configuration
space: universe is single gigantic classical 'fermion'. This feature
together with special features of lightcone geometry makes it possible
to associate well defined measure of information to configuration space
spinor field.]
>
> and:
>
> [MP]
> "b) One can also worry about General Coordinate Invariance.
> In case of a nonrelativistic Schroedinger equation the information
> is Galilei invariant. In case of QFT Lorentz invariance
> is lost since probability density behaves like a component
> of a four-vector."
>
> [SPK]
> Can we construct a Lorentz group for the 4-vector of a single particle
> (as an observer), such that it would have an *illusion* of a preferred
> inertial frame? Thus a *space-time* is identified with the range of
> possible states (wrong word!) of the group and such for any particle. We
> modify this construction to account for the space-times of different
> types of particles, e.g. photons, neutrinos, electrons, etc.
> I effectively reversing the usual way that physics is done, instead of
> assuming an a priori space-time and then figuring out the group
> theoretic behaviors of objects "in it", I am saying that we consider the
> group theoretic properties of a particle as defining, a postiori, the
> particular space-time that it would have. Does this make any sense?
One could define preferred Lorentz fram as the rest frame of
particle. The direction of angular momentum fixes also the 'z-axis'
of the Minkowski coordinates so that frame is defined only apart
from rotations around z-axis.
Exactly this was what I found when pondering about the problem of
preferred frame while trying to understand how canonical identification
mapping real imbedding space to its p-adic counterpart could be
made GCI and PI. In my case, the classical four momentum and angular
momentum associated by absolute minimization of Kahler action
to the 4-surface X^4(Y^3) to surface Y^3 on lightcone boundary
made the choice of the Minkowski coordinates unique apart
from roatation around z-axis. Phase preserving canonical identification
map allowed to achieve PI and GCI.
>
> and further:
>
> [MP]
> "One is accustomed to speak about communication as information flow.
> Therefore one could wonder whether it is possible to define the concept
> of information current somehow in quantum TGD framework.
> U_a, a-->infty is indeed defined as a time evolution operator associated
> with Virasoro generator L_0 playing the role of Hamiltonian.
>
> Hence it should be possible to formally associate with
> the time evolution U_a a conserved probability current having time
> component I^a plus spatial components in the degrees of
> freedom characterized by the coordinates of the reduced configuration
> space. This assignment would be completely analogous to that performed
> for
> the ordinary Schroedinger equation and the Lorentz invariance of
> the lightcone proper time coordinate a would make this assignment
> possible."
>
> [SPK]
> Could we think of this U_a as identified with individual LSs, such that
> it would represent the "perceived" space-time and act as the external
> counterpart to the internal unitary propagator group that defines the
> LSs internal clock. This notion would make explicit the subject-object
> duality existing between each LS and its set of observables.
In my case U_a acts on '\phi' using your terminology and is counterpart
of time evolution operator exp(iHt), with t replaced with Lorentz
invariant proper time of future lightcone. In case of LS time would
be most naturally the proper time of LS and hence uniquely determined
so that one would obtain Lorentz invariance. The information
current in question would however flow inside LS.
There is also problem about information flow between different
LS:s. How can one define information current between LS:s if
these systems correspond to 'different spacetimes'?
>
> [MP]
> "In p-adic context n= Log_pR is pseudo constant
> for finite values of the integer n and this would mean
> that information current would be conserved locally in p-adic sense.
> This would *not* imply the conservation of information even
> in the case that n is pseudo constant everywhere.
>
> [SPK]
> This "everywhere" is not absolute, but bounded relative to a given p?
>
Everywhere is in sector D_p of configuration spacee (quite huge
infinite-dimensional space). If n has only finite values it is
pseudo constant. There is NO unitary violation. Probability
current is NOT equal to information current! Log_p(R)!
The nonconservation of information is *what one wants* in TGD framework
at least! In p-adic context however p-adic pseudoconstancy can give
this nonconservation even without explicit breaking of
current conservation.
Pseudoconstancy is not necessarily needed
but it simplifies the situation enormously and somehow
'quantizes' the nonconservation of information. Information
nonconservation is perhaps localized on boundaries of regions
where n changes or something like this. In fact, Log_pR=n
could be interpreted as *quantization of information*.
It is good to recall the physical picture. The time evolution by U_a
generates a lot of potential information by dispersion to various sectors
so that in quantum jump net information gain defined as difference of
informations associated with U_a*initial quantum history and final quantum
history is achieved.
One can visualize quantum jumps as sequence of steps consisting of
information generating time development by U_a and quantum
jump transforming this potential information into conscious information.
> Thus we do not have a unitarity violation problem when changing
> framings[1], since each framing (defined as a p-adic space-time patch)
> would have its own convex set of information that is "conserved".
As I already noticed, there is no unitarity violation problem: probability
is conserved by Schrodinger equation. Information is not conserved but
this is highly desirable in TGD framework: the (potential) information
generated by U_a is eaten by quantum jump giving rise to localization in
sector D_p (this eats the information).
One can also visualize the time development as a Holy Spirit
which again and again disperses itself over the waters (all sectors D_p)
and then collapses into single D_p and learns something about that which
is.
Or one could also talk about ghost, is it Djinn?, coming from the
bottle D_p_i! There is free will associated with
quantum jump so that the ghost from the bottle satisfies
a wish and at the very moment of satisfying the wish goes back to a
possibly new bottle D_p_f!(;-)
> [1] This "framing" notion relates to how the composition of an LS is
> altered when we shift between perspectives, like the situation discussed
> by Hitoshi in Section 9 of http://www.kitada.com/bin/time_I.pdf and what
> I understand of your discussion of biological p-adic physics [time 377]
> etc.
I believe that there are no problems related with probability
conservation or with definition of information current if
one can model LS using single particle QM. The problems
are related with information flow between LS:s.
>
> [MP]
> If this indeed works, one could assign
> with a given time evolution U_a transfer of information in
> the reduced configuration space of 3-surfaces. The zero modes
> characterizing the nondeterminism of the K\"ahler action
> contain information about the moments for multifurcations in the
> time development of the spacetime surface and this
> gives hopes of approximate reduction of this
> information flow to an effective information flow
> occurring at the level of 'quantum average effective spacetime'."
>
> [SPK]
> I am thinking of the motions of "particles" (consistent with Hitoshi's
> center-of-mass definition of the exteriors of LSs) as being modelable,
> in a fundamental sense, by a "probability density current" quantity.
I think that 'Schrodinger equation' is more fundamental since it
gives automatically rise to conserved probability density.
If frame is rest frame then one can also define information density
without any problems with covariance.
> It is "probabilistic" since its possible directions are not a priori
> definite but are defined by the observational act, which is an
> interaction subject to finite constraints between pairs of LSs. It is a
> "density" because it is a quantity that is not a priori single valued
> nor a priori localized to a single point, and a "current" because there
> is a continuous "flow" to the individual components due to the existence
> of a potential, which I think exists due to the non-absoluteness of the
> optimization. We can see that if the potential vanishes, so does the
> possibility of a change in the velocity and the information is identical
> everywhere in the U^T (this is another way of saying that U^T is a bound
> state and is at absolute equilibrium with all proper subsets of itself).
> Perhaps these notion is nonsensical! :) I am just trying to work out
> how it is that we observe an illusion of continuous motions and can
> communicate about these to each other when we can prove that they are
> not real, e.g. they are an illusion! :)
> see:
> http://lobelia.physics.wisc.edu/duncan/phys531/wavemechanics/node2.html
>
> Onwards to the Unknown,
>
> Stephen
>
Best,
Matti
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