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Maximal entropy random walk and euclidean path integrals

Duda Jarek

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While thinking about random walk on a graph, standard approach is that every possible edge is equally probable - kind of maximizing local entropy.

There is new approach (MERW) - which maximizes global entropy (of paths) - for each two vertexes, each path of given length between them is equally probable.

For regular graph they give the same, but generally they are different - in MERW we get some localizations, not met in the standard random walk:



This approach can be generalized to random walk with some potential - something like discretized euclidean path integrals.

Now taking infinitesimal limit, we get that

p(x) = psi^2(x)

where psi is normalized eigenfunction corresponding to the ground state (E_0) of corresponding Hamiltonian H=-1/2 laplacian + V.

This equation is known - can be got instantaneously from Feynman-Kac equation.


But we get also analytic formula for the propagator:

K(x,y,t)=(<x|e^{-2tH}|y>/e^{-2E_0}) * psi(y)/psi(x)

Usually we variate paths around the classical one getting some approximation - I didn't met with not approximated equations this type (?)

In the second section is the derivation:


Bravely we could say that thanks of analytic continuation, we could use imaginary time, and we get solution to standard path integrals ?


Have You heard about this last equation?

Is physics local - particles decide locally, or global - they see the space of all trajectories and choose with some probability... ?


Ok - it was to be rhetoric question. Physicist should (?) answer, that the key is the interference - microscopically is local, than interfere with itself, environment ... and for example it looks like a photon would go around negative refractive index material...


I wanted to emphasize, that this question has to be deeply understand ... especially while trying to discretize physics, for example: which random walk corresponds to the physics better?

It looks like that to behave like in MERW, the particle would have to 'see' all possible trajectories ... but maybe it could be the result of macroscopic time step?

Remember that an edge of such graph corresponds to infinitely many paths ...


To translate this question into lattice field theories, we should also think how does discrete laplacian really should look like...?

Edited by Duda Jarek
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To argument that MERW corresponds to the physics better, let's see that it's scale-free. GRW chooses some time scale - corresponding to one jump.


Observe that all equations for GRW works not only for 0/1 matrices, but also for all symmetric ones with nonnegative terms.

k_i=\sum_j M_ij (with diagonals)

We could use it for example on M^2 to construct GRW for time scale twice larger. But there would be no straight correspondence between these two GRWs.

MERW for M^2 would is just the square of MERW for M - like in physics, no time scale is emphasised.

P^t_ij= (M^t_ij / lambda^t) psi(j)/psi(i)

Edited by Duda Jarek
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I've just corrected in (0710.3861) the derivation of the equation for propagator - probability density of finding a particle in position y after time t, which started in position x:


K(x,y,t)=<x | e^-{tH} |y> / e^{-tE_0} * psi(y)/psi(x)


where psi is the ground state of H with energy E_0.

At first look it's a bit similar to the Feynman-Kac equation:


K_FK (x,y,t)= < x |e^-{tH} | y >


The difference is that in FK the particle decays - the potential says the decay rate. After infinity time it will completely vanish.

In the first model the particle doesn't decay (\int K(x,y,t)dy=1), but approach to stationary distribution:


The potential defines that the probability of going through given path is proportional to

e^{-integral of potential over this path).


The question if the physics is local or global seems to be deeper than I thought.

Statistical physics would say that this distribution of paths should really looks like that ... but to achieve this distribution, the particle should see all paths - behave globally.

Statistical physics would also say that the probability that a particle would be in a given place, should behave like p(x) ~ e^{-V(x)}.

We would get this distribution for GRW like model - with choosing behavior locally...


Fulfilling statistical mechanics globally (p(x)=psi^2(x)) would create some localizations, not met in models assuming fulfilling statistical mechanics locally (p(x) ~ e^{-V(x)}).


Have You met with such localizations?

Is physics local or global?

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  • 2 weeks later...

Quantum physics says that really atoms should approach their ground state (p(x)~psi^2(x)), as in my model. Single atom does it by emitting energy in portions of light. But from the point of view of statistical physics, if there were many of them their average probability distribution should behave locally more or less like in my propagator.


Particle doesn't behave locally only - they don't use other particle's positions only to choose what to do in given moment, but use also their histories - stored in fields (electromagnetic, fermionic ...).

So it would suggest that the statistics should be made among paths ending in given moment, but it would give stationary probability distribution p(x)~psi(x).

To get the square, paths should go into the future also.


Intuitively - I'm starting to think that particles should be imagined as one-dimensional paths in four-dimensional timespace. The are not ending in given moment and being slowly created further, but they are already somehow entanglement in the future - it just comes from the statistics...

So the time passing we feel is only going through time dimension of some four-dimensional construct with strong boundary conditions (bigbang)... ?

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I'm sorry - I didn't realized it's supported. Here are the main equations again:

Assuming Bolzman distribution among paths: that the probability of a path is proportional to

exp(- integral of potential over the path)

gives propagator


where [math]\hat{H}=-\frac{1}{2}\Delta+V[/math], [math]E_0[/math] is the ground (smallest) energy, [math]\psi[/math] is corresponding eigenfunction (which should be real positive).

Propagator fulfills [math]\int K(x,y,t)dy=1,\quad \int K(x,y,t)K(y,z,s)dy=K(x,z,t+s) [/math]

and have stationary probability distribution [math]\rho(x)=\psi^2(x)[/math]

[math]\int \rho(x)K(x,y,t)dx = \rho(y) [/math]




To summarize, we can interpret physics:

- locally (in time) - in given moment particle chooses behavior according to situation in this moment - standard approach, or

- globally (in time) - interaction is between trajectories of particles in four dimensional spacetime.


In the local interpretation the timespace is being slowly created while time passes, in the global we go along time dimension of more or less created timespace.


In local interpretation particles in fact uses the whole history (stored in fields) to choose behavior.

If according to it, we would assume that the probability distribution among paths ending in given moment is given by exp(- integral of potential over the path) we would get that the probability distribution of finding the particle is [math]\rho(x)\cong\psi(x)[/math].

To get the square, paths cannot finish in this moment, but have to go to the future - their entanglement in both past and future have to influence the behavior in given moment.

The other argument for that both past and future is important to choose behavior, is that we rather believe in CPT conservation, which switches past and future.


Observe also that in this global interpretation, two-slits experiment is kind of intuitive - the particle is generally smeared trajectory, but can split for some finite time and have a tendency to join again (collapse).

For example because in split form it has higher energy.

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There is nice picture from another forum, that time-space looks like four-dimensional jello - both tensions from past and future influence present.

Observe that this picture intuitively corresponds to general relativity also.


I believe a discussion has just started there, feel welcome:


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  • 3 weeks later...

First of all it occured that the way of thinking that we are moving along the time dimension of some already created timespace is known and called eternalism/block universe. It's main arguments are based on general relativity, but also that there is a problem with CPT conservation and wave function collapse...

The fact that bolzmanian distribution among paths gives statistical behavior similar to known from QM, suggest even more - that QM is just the result of such structure of the timespace.

...that wave function collapse is for example reversed split of the particle (to go through two-slits).


This simple statistical physics among trajectories gives similar behavior of QM, but still qualitatively different - particles leaves an excited state exponentially instead of making it in quick jumps, producing a photon for energy conservation.

I think this difference is because we assumed that in given time, the particle is in a given point, but in fact it's rather its density spread around this point. If instead of considering a single trajectory for a particle, we take some density of trajectories with some force which wants to hold them together, the particle's density instead of slowly leaking, should wait for a moment to quickly jump to a lower state as a whole.

This model should be equivalent to simpler - use a trajectory in the space of densities (instead of density of trajectories).

But I don't see how to explain the production of the photon - maybe it will occur as an artifact, maybe the energy conservation should be somehow artificially added ?


The question is what holds them together, to form exactly the whole particle - not more, not less? Kind of similar question is why charge/quantum numbers are integer multiplicities?

I'll briefly present my intuitions to deeper physics. The first one is that the answer to these question is that particles are some topological singularities of the field.

That explains spontaneous creations of a pair/annihilation, that such pair should has smaller energy when closer - create attractive force. The qualitative difference between weak and strong interaction could be due to the topological difference between SU(2) and SU(3).


So the particle would be some relatively stable state of the field (in which for example the spin has spatial direction). It would have some energy, which should correspond to the mass of the particle. The energy/singularities densities somehow creates spacetime curvature...?

Now if particles are not just a point, the field which they consist of fluctuates - still have some orders of freedom (some vibrations). I think that quantum entanglement is just the result of these orders of freedom - when particles interact, they synchronise fluctuations of their field. But these orders of freedom are very sensible - easy to decoherence...


ps. If someone is interested in the inequality for the dominant eigenvalue ([math]\lambda[/math]) of real symmetric matrix ([math]M[/math]) with nonnegative terms from the paper:

[math]\ln(\lambda)\geq\frac{\sum_i k_i \ln(k_i)}{\sum_i k_i}\qquad [/math] where [math] k_i = \sum_j M_{ij}[/math], I've started separate thread: http://www.scienceforums.net/forum/showthread.php?t=36717



Simplifying the picture


The field theory says that every point of the space-time has some value for example from U(1)*SU(2)*SU(3).

This field doesn't have something like zero value, so the vacuum must have some nontrivial state and intuitively it should be more or less constant in space.

But it can fluctuate around this vacuum state - these fluctuations should carry all interactions.


It allow also for some nontrivial spatially localized, relatively stable states - particles. They should be topological singularities (like left/right swirl). Another argument is that if not, they could continuously drop to the vacuum state, which is more smooth - has smaller energy - they wouldn't be stable.

Sometimes the fluctuations of the field exceed some critical value and spontaneously create a particle/antiparticle pair.


Observe that this value around which vacuum fluctuates, should have huge influence in choosing the stable states for particles. Maybe even this value is the reason for weak/strong interactions separation (for high energies this separation should weaken). It could also be the reason for matter/antimatter asymmetry...


The problem with this picture is that it looks like the singularities could have infinity energy (I'm not sure if it's necessary?)

If yes - the problem could be with Lagrangian being to simple?

The other question is if the field theory is really the lowest level? Maybe it's the result of some lower structure...?



I was thinking about how energy/singularities could create spacetime curvature...

Let's think what is time?

It's usually described in the language of reason-result chains.

They can happen in different speeds in different points of reference.

These reason-result chains are microscopically results of some (four-dimensional) wave propagations.

But remember that for example light speed depends on the material ... the wave propagation speed depends on microscopic structure of ... the field it is going through.

This field should be able to influence both time and spatial dimensions - slow it down from the light speed.


In this picture spacetime is not some curved 4D manifold embedded in some multidimensional something, but is just flat - local microscopic structure of the field specifically slows down some time/space waves of the field.

...and for example we cannot go out of black hole horizon, because the microscopic structure of the field (created by the gravity) won't allow any wave to propagate.

This picture also doesn't allow for hyperspace jumps/time travels...



Finally something the most controversial - the whole picture...

Let's imagine such particle, for example one of created in the big bang.

This stable state of the field in 4D is well localized in some three of dimensions, and is very long in the last one ... most of created in the big bang should choose these directions in similar way ... choosing some general (average...) direction for time (the long one) and space (localized ones)...

These trajectories entangles somehow in the spacetime ... sometimes for example because of the field created by some gravity they change their (time) direction a bit - observed as general relativity...

... their statistics created (purely bolzmanian - without magic tricks like Wick's rotation) quantum mechanical like behaviour ...



What do You think about this picture?

Is internal curvature really necessary, or maybe it's only an illusion (like light 'thinks' that geometry changes when it changes the material)?

Are rules for time dimension really special? ... is time imaginary?

Or maybe it is only a result of some solutions for these rules, specially due to boundary conditions (big bang)...?

Edited by Duda Jarek
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