An idea of my own....

[_heretic]

Shouldn't the universal wavefunction - as described by the Wheeler-DeWitt equation - be subject to the quantum mechanical version of the Poincaré recurrence theorem?

 

"Quantum Recurrence" http://prola.aps.org.../v107/i2/p337_1

"Note on the quantum recurrence theorem" http://pra.aps.org/a.../v18/i5/p2379_1

A paper discussing the Wheeler-DeWitt equation and quantum gravity http://www.physics.d...o%20Rovelli.pdf

 

Thoughts?

[_heretic]

...?

 

Is there an answer to this :< Does the universal wavefunction obey the quantum mechanical Poincaré recurrence theorem?

[Ronald Hyde]

I'm thinking that you shouldn't be thinking so much in terms of a wavefunction as a propagator, and maybe trying to find the rules for writing and solving the propagator equation. And no, I don't think that the Universe will exactly repeat itself in Time, and the propagator will be purely a function of time, so that space and everything in it will be a function of time.

[_heretic]

I'm thinking that you shouldn't be thinking so much in terms of a wavefunction as a propagator, and maybe trying to find the rules for writing and solving the propagator equation. And no, I don't think that the Universe will exactly repeat itself in Time, and the propagator will be purely a function of time, so that space and everything in it will be a function of time.

 

Thanks. I'll give that some thought.

[MigL]

Poincare's recurrence says that any statistical aggregate of particles will, given enough time, return to its original state. It was a rebuff of the simplistic Boltzmann statistical interpretation of entropy ( which says that an ideal gas will become more and more disorganized and that is a measure of entropy ). As such it makes use of 'determinism', ie present follows past and future follows present or past and future can be interchanged . A wave function is also 'deterministic' in the sense that past states are encoded into it as are future states, and it is also time symmetric. The difference is that on observation, measurement or anything which causes the wave function collapse into a specific state, this symmetry and 'determinism' is lost.

 

What lies in the realm of philosophy is that ,the Wheeler-deWitt equation being a wave function of the universe, who or what is making the observation or measurement which collapses it ??

Edited September 1, 2012 by MigL

[_heretic]

Poincare's recurrence says that any statistical aggregate of particles will, given enough time, return to its original state. It was a rebuff of the simplistic Boltzmann statistical interpretation of entropy ( which says that an ideal gas will become more and more disorganized and that is a measure of entropy ). As such it makes use of 'determinism', ie present follows past and future follows present or past and future can be interchanged . A wave function is also 'deterministic' in the sense that past states are encoded into it as are future states, and it is also time symmetric. The difference is that on observation, measurement or anything which causes the wave function collapse into a specific state, this symmetry and 'determinism' is lost.

 

What lies in the realm of philosophy is that ,the Wheeler-deWitt equation being a wave function of the universe, who or what is making the observation or measurement which collapses it ??

 

Perhaps it doesn't collapse, a.la. Many Worlds

[ajb]

The trouble here is that the Wheeler-DeWitt equation is not the same as the Schrödinger equation. In particular the Hamiltonian acts as a 1st class constraint and does not determine the evolution of the system. I am not sure then if one can apply the quantum recurrence theorem, at least not directly.

[_heretic]

The trouble here is that the Wheeler-DeWitt equation is not the same as the Schrödinger equation. In particular the Hamiltonian acts as a 1st class constraint and does not determine the evolution of the system. I am not sure then if one can apply the quantum recurrence theorem, at least not directly.

 

Since the Wheeler-DeWitt equation supposes a universal wavefunction, then does that wavefunction not evolve according to the Schrödinger equation and therefore...the quantum poincare recurrence theorem....?

[ajb]

Since the Wheeler-DeWitt equation supposes a universal wavefunction, then does that wavefunction not evolve according to the Schrödinger equation and therefore...the quantum poincare recurrence theorem....?

 

No, the Hamiltonian is really a constraint, which is a large part of the difficulty with canonical approaches to quantum gravity. Also, there is no clear notion of a time parameter here for the wavefunction to evolve with respect to.

[_heretic]

No, the Hamiltonian is really a constraint, which is a large part of the difficulty with canonical approaches to quantum gravity. Also, there is no clear notion of a time parameter here for the wavefunction to evolve with respect to.

 

Isn't there a time-independent version of the schrodinger equation? Perhaps that would apply...

[ajb]

Isn't there a time-independent version of the schrodinger equation? Perhaps that would apply...

 

The time-independent version is obtained from the time-dependent via a formal solution [math]\phi(x,t) = \psi(x) e^{-i H t}[/math], assuming that the Hamiltonian does not depend on time.

 

Again the Hamiltonian is just a constraint in quantum general relativity, it does not really describe the time evolution. All quantum theories of gravity will have this problem, time should play exactly the same role as space; they are all just coordinates on a 4-d manifold in the classical theory.

 

One way round this, classically at least is to use multisymplectic geometry, which comes from the de Donder-Weyl formalism. As far as I know, quantising in this set-up is difficult, but some progress has been made [1]. There is some suggestion that this formalism maybe the way to understand the quantisation of 2-branes in M-theory.

 

References

 

[1] Christian Saemann, Richard J. Szabo. Quantization of 2-Plectic Manifolds. arXiv:1106.1890v1 [hep-th]

[_heretic]

The time-independent version is obtained from the time-dependent via a formal solution [math]\phi(x,t) = \psi(x) e^{-i H t}[/math], assuming that the Hamiltonian does not depend on time.

 

Again the Hamiltonian is just a constraint in quantum general relativity, it does not really describe the time evolution. All quantum theories of gravity will have this problem, time should play exactly the same role as space; they are all just coordinates on a 4-d manifold in the classical theory.

 

One way round this, classically at least is to use multisymplectic geometry, which comes from the de Donder-Weyl formalism. As far as I know, quantising in this set-up is difficult, but some progress has been made [1]. There is some suggestion that this formalism maybe the way to understand the quantisation of 2-branes in M-theory.

 

References

 

[1] Christian Saemann, Richard J. Szabo. Quantization of 2-Plectic Manifolds. arXiv:1106.1890v1 [hep-th]

 

Thanks. I guess the issue is really an unknown at this point then (?) It will be interesting to watch all of these hypotheses develop

[ajb]

Thanks. I guess the issue is really an unknown at this point then (?) It will be interesting to watch all of these hypotheses develop

 

It is an interesting idea, after some finite time the wavefunction of the Universe will return to its initial state. Classically we envisage a "big crunch", though this looks impossible given what we know about the expansion of the Universe.

 

However, even the statement "after some finite time" in the context of a quantum theory of gravity and the Wheeler-DeWitt equation is ambiguous at best.