[time 748] About L_0: reply to Hitoshi

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sat, 11 Sep 1999 16:42:26 +0300 (EET DST)

Below answer to your question about L_0.

On Sat, 11 Sep 1999, Hitoshi Kitada wrote:

> [HK]
> Dear Matti,
> Thank you for your explanation and responses. I will confine this time
> following points:
> Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
> Subject: [time 743] Holy trinity summarizes the mathematical structure
> quantum TGD
> [HK]
> Dear Matti, Lance, Stephen et al.,
> To me three kinds of time of Matti look like a taxonomy of phenomena.
> Matti, do you have any relations between these three times?
> [MP] These three times emerge automatically from the basic
> structure of theory. They are by no means assumed in ad hoc manner.
> Holy trinity basically follows from 'holy trinity' of
> a) General Coordinate Invariance,
> b) existence of Poincare invariant and GCI S-matrix and
> c) existence of classical physics, not as approximation, but
> exact part of quantum theory. The existence of classical
> physics actually follows from the definition of configuration
> geometry in GCI invariant manner so that
> *it is basically Poincare invariance and GCI which lies behind holy
> trinity!*
> At least for me it was shock to realize how enormously forceful
> principle GCI is when combined with spacetime as 4-surface idea.
> Here are some details.
> a) *General Coordinate Invariance* makes impossible to define
> state as t=constant snapshot. States are configuration space
> spinor fields in entire configuration space consisting of 3-surfaces
> in M^4_+x xCP_2 with all possible values of lightcone proper time.
> Thus the phrase "quantum history".
> Quantum histories would be the description of manyworld enthusiast for
> entire universe: I think we know that this approach does not lead to any
> understanding of consciousness and splitting of worlds is at least to me
> intolerably ill defined concept.
> b) Quantum jumps between quantum histories occur.
> This sequence of quantum jumps is subjective time development and
> subjective
> time is measures as quantum jumps occurred after say wakeup of some
> Quantum jumps *CANNOT* occur just as
> Psi_i-->Psi_f-->...
> This would be totally inconsistent with which we know about quantum
> field theories which summarizes the predictions of the theory. Rather,
> each quantum jump involves unitary 'time' development acting
> in the space of quantum histories.
> Psi_i-->UPsi_i -->Psi_f,
> Uexp (iL_0(tf-ti)),
> tf--> infty, tf-->-infty.
> [HK]
> Here what is your L_0? It seems a Hamiltonian of some system, but what
> are you referring to by L_0? If it is the system, say L, corresponding
> state Psi_i (and I assume it is the same system as for Psi_f), the jump
> UPsi_i --> Psi_f
> must occur by a perturbation from the outside of the system L.

[MP] This requires some explaining.

 1. Basic equations state Super Virasoro invariance

a) Super conformal transformations are basic gauge symmetries of TGD
and string models besides GCI at spacetime level and at the level
of configuration space.

b) Super Virasoro algebra is the complex algebra of infinitesimal
conformal transformations generated by generators

L_n= z^(n+1)d/dz

Thus L_0 refers to infinitesimal generator
zd/dz. Depending on whether its coefficient is real of purely imaginary
it acts as scaling or rotation in plane in defining representation
(In TGD realization L_0 acts as radial scaling of light cone radial
coordinate r=t). Also super Virasoro generators G_n are included
and these anticommute to Virasoro generators.

c) Super Virasoro invariance is analogous to GCI and gauge invariances
in that Virasoro generators either annilate state or create a zero
norm state. L_n(tot), n>=0 annihilates state and L_n, n>0 creates
zero norm state.

L_n(tot)Psi=0, n>=0
L_n(tot)Psi= zero norm state for n<0

 2. Configurations space spinor fields provide representation for Super

a) In TGD configuration space spinor fields provide representation
for Super Virasor algebra. Just like ordinary spinor fields
have spin degrees of freedom in which rotation group acts, configuration
space have configuration space spinors have spin degrees of freedom:
gamma matrices and their complex conjugates create/destroy
fermions/antifermions and sigma matrices change either fermion
quantum numbers or create fermion pairs. Rotation creates
superposition of fermion-antifermion pairs among other things!
 Thus rotation is rather abstract concept and rather far from the
picture of Euclides or even Einstein about how rotation acts
physically! Every super Virasoro generator has ordinary
geometric part which acts on the argument of configuration
space spinor field plus performs spin rotation.

b) The form of L_0(tot) decomposes to the following one

L_0(tot)= p^2 -L_0

where p^2 is the counterpart of mass squared operator. M^4 d'Alembertian.
and L_0 is what almost I have called call Virasoro generator in above and
in the exponential exp(i*L_0(tf-ti)).

  3. Mass squared operator is defined by Diff^4 invariant momentum

a) Mass squared operator is

p^2= p_kp^k,

where p_k must form representation for Poincare translations. The
problem is to understand how these translations act in the space
of configuration space spinor fields. The problem is far
from trivial since ordinary Poincare translations do not commute with
Diff^4. One must construct what might be called Diff^4 invariant
Poincare. Intuitively Diff^4 invariant Poincare corresponds to Poincare
transformations at the limit when lightcone proper time a--> infty
so that symmetry breaking effect of lightcone boundary is not felt
anymore. I add short appendix about Diff^4 invariant Poincare.
What is essential is that Diff^4 invariant Poincare acts
on the *reduced space of configuration space spinor fields*
on delta M^4_+xCP_2. Just like Hamiltonian acts on t=constant
3-space in case of standard QM.

   3. Introduce lightcone coordinates for the space
of group parameters of Diff^4 invariant translations

a) The next step is to write p^2 as

p_+p_- -p_T^2

using lightcone coordinates

(x_+,x_-,x_T), x_T= (x,y).

x_+= x0+x3 and x_-= x0-x3 in terms of Minkowski coordinates.
Note that these parameters are *purely* group theoretical
parameters associated with the group
of Diff^4 invariant translations: they have nothing to do with lightcone

b) What is important is that these coordinates are unique since
each quantum jump involves localization in zero modes and leads to a
set of 3-surfaces for which directions of classical 4-momentum and
spacelike classical spin vectors are fixed so that these directions imply
the decomposition (u,v,x_T) uniquely. This decomposition is
also crucial for the definition of reals to p-adics mapping
as *phase preserving canonical identification*.

c) Next step is analogous to performed in string models.

h1) I ASSUME that the coordinate x_- is totally nondynamica and
that physical states are eigenstates of x_-:

p_- Psi= P Psi

Similar assumption is made in string models and I believe
that this condition follows as internal consistency condition
from the requirement that translations act properly in
the reduced configuration space.

d) Furthermore, I identify p_+ corresponds as
purely group theoretical translation generator

p_+ = id/dx_+

acting on group parameter x_+==t.

e) One could also require that states are eigenstates
of p_T^2

p_T^2 Psi = P_T^2 Psi

These conditions make sense since p_k(a) are assumed to
commute at the limit a--> infty.

  4....and you get 'Schroedinger equation'

a) This implies that the equation of motion

(p_+p_- -p_T^2 -L_0)Psi =0

reduces to

i(d/dt)*Psi = [p_T^2/P -L_0/P]Psi==L_0* Psi

This is Schrodinger equation with right hand side
containing the 'Hamiltonian' which I somewhat called somewhat
sloppily L_0 for pedagogical reasons.

b) Free Schrodinger equation at single particle level
gives mass squared conditions and universal mass spectrum.
Interaction terms are however present: they result from
the spinor connection of configuration space.

c) L_0 decomposes into free part which
is sum over single particle L_0:s
say, and interaction part. Free part gives universal mass squared
spectrum: m^2= n m_0^2 + m_0^2.

d) What is remarkable is that the Schrodinger equation derives


This means that one can avoid the problems of infinite volume limit
resulting in case of ordinary Scrodinger equation resulting from infinite
energy eigenvalues of free Hamiltonian. This suggests that in principle
there is no need to restrict the considerations to finite subsystem
in order to have well defined theory. Of course restriction
to finite system is in practical calculations what one must do.

Note: Althought the derivation of Shrodinger contains some
questionable assumptions these assumptions are also made
in string model formalism and are consistent with
commutativity of p_k.

> Then you need
> to consider a larger system L' that includes some part of the outside of
> Then the Schroedinger equation should be different than the one that
> the time evolution

 Uexp (iL_0(tf-ti)).

> In your description of the jump, you thus need to speak about another
> L' than L, and the states Psi_i and Psi_f are neither the eigenstates of
> larger system L' nor describe the system L'. In this larger system L',
> the time evolution would follow the Schroedinger propagator of L'. The
> is absorbed into the Shroedinger propagator of the larger sysmtem L' and
> looked like a jump of the system L was merely the ignorance about the
> system L'.

[MP] I would claim that since the basic equation results from the
gauge invariance condition L_0(tot)Psi=0 and since gauge invariance
of the state cannot depend by no means on the size of the system, the
condition makes sense for entire universe and hence in
principle there is no need to take U as model for dynamics of
finite sized system.

Of course, there are many delicacies and
I believe that p-adic numbers, in particular, p-adic numbers
associated with infinite primes are necessary for the proper
quantum description of S-matric for infinite-sized systems.
One could even claim that p-adic numbers is what makes possible
proper description of infinite-sized systems without introducing
of particle densities (which fails if universe is fractal).

Appendix: Diff^4 invariant translations

Diff^4 invariant translations define the momentum generators
p_k appearing in the Virasoro conditions and following
is the definition of them.

a) GCI implies that configuration space spinor field has same
value for all Diff^4 related 3-surfaces at X^4(Y^3) defined
as absolute minimum of Kaehler action for Y^3 on 7-dimensional
delta M^4_+xCP_2, 'lightcone boundary', or big bang for short.

b) GCI in absence of classical nondeterminism of Kahler
action would imply that configuration space spinor field is completely
determined by its values on lightcone boundary: that is the space
of 3-surface Y^3 on M^4_+xCP_2. Classical nondeterminism implies
that absolute minimum spacetime surface X^4(Y^3) is not unique
but this brings in only summation over nondegenerary so that
one conclude that the representation of Diff^4 invariant
algebra acts on reduced space of 3-surfaces 'on lightcone boundary'.

Note: The reduction to lightcone boundary
is is 'almost counterpart' for the loss of time
in the quantization of GRT. Cognitive spacetime sheets
however save psychological time. Also the fact that
p_k (a) corresponds to a--> generators makes it impossible to
reduce theory totally to 'lightcone boundary'.

Note: in string models so called Maldacena conjecture states
that string models reduce to the boundary of space M in which
strings move.

c) Consider now the definition of Diff^4 invariant Poincare
transformations. Take 3-surface X^3(a) defined by the intersection
of a= constant hyperboloid of future lightcone and X^4(Y^3).
Perform infinitesimal Poincare transformation in standard manner
for X^3(a) and define the Diff^4 invariant Poincare translate
of X^4(Y^3) as absolute minimum going through infinitesimally
 translated X^3(a). This spacetime surface is not translate
of X^4(Y^3) but is deformed and intersects lightcone boudary
at (Y^3)', which is also deformed.

c1) For finite values
of lightcone propertime p_k(a) need not satisfy commutation
relations of Poincare and do not need even close to
any algebra.

c2) Intuitively it however seems clear that at the limit
a--> infty the effect of lightcone boundary is not felt and
one can hope that resulting Diff^4 invariant Poincare translations
have standard commutation relations.

c3) It is however possible in principle that the algebra is deformed:
Kehagias has show that Poincare algebra allows nontrivial deformation
which is Lorentz invariant with respect to Lorentz group
leaving the dip of lightcone invariant and for which
new energy is some function of old energy.


Matti Pitkanen

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